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A Calibration system for the visible region with application to the NO₂ and BO₂ molecules Steunenberg, Dinie M. 1989

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A CALIBRATION SYSTEM FOR THE V I S I B L E REGION WITH APPLICATION TO THE NOa AND B0 2 MOLECULES By DINIE M. STEUNENBERG B.Sc. (Hon. Chemis try ) 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 , 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY 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 February 1989 © DINIE STEUNENBERG In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department Date 1 W ) V K f t < \ . DE-6 (2/88) Abstract A method f o r c a l i b r a t i n g h i g h r e s o l u t i o n l a s e r s p e c t r a i n t h e v i s i b l e r e g i o n has been d e v e l o p e d and t e s t e d by t a k i n g u l t r a - h i g h r e s o l u t i o n molecu lar -beam s p e c t r a o f N0 2 and s u b -D o p p l e r i n t e r m o d u l a t e d f l u o r e s c e n c e s p e c t r a o f B0 2 . The c a l i b r a t i o n system i s based on an evacuated F a b r y - P e r o t e t a l o n whose c a v i t y l e n g t h i s s e r v o - l o c k e d t o a s t a b i l i z e d HeNe l a s e r ; t h e a b s o l u t e o r d e r number o f the f r i n g e a t t h e HeNe f r e q u e n c y i s known and the f r e e s p e c t r a l range can be d e t e r m i n e d w i t h h i g h a c c u r a c y . F o r a b s o l u t e f r e q u e n c y measurements, the o r d e r number o f a t r a n s m i s s i o n f r i n g e i s o b t a i n e d from a commerc ia l wavemeter (whose a c c u r a c y i s s u f f i c i e n t t o i d e n t i f y the f r i n g e ) ; the a b s o l u t e f requency i s t h e n t h e HeNe f requency m u l t i p l i e d by the r a t i o o f t h e •unknown' o r d e r number t o the ' l o c k p o i n t ' o r d e r number. Two K=0 subbands o f the 2 B 2 - 2Aj e l e c t r o n i c t r a n s i t i o n o f N0 2 a t 593.5 nm and 585.1 nm were chosen f o r a n a l y s i s . The h y p e r f i n e t r a n s i t i o n s a s s o c i a t e d w i t h r o t a t i o n a l l e v e l s up t o N"=10 were s t u d i e d ; a l t h o u g h t h e system i s h i g h l y p e r t u r b e d , i t was p o s s i b l e t o o b t a i n a s e r i e s o f m e a n i n g f u l c o n s t a n t s d e s c r i b i n g t h e e x c i t e d s t a t e . V a l u e s f o r the e l e c t r o n s p i n -r o t a t i o n , F e r m i c o n t a c t and ( I ,S ) d i p o l a r parameters were o b t a i n e d . Good agreement was found w i t h l i t e r a t u r e v a l u e s where t h e y e x i s t . U s i n g t h e c a l i b r a t i o n system t h e s m a l l f r e q u e n c y s p l i t t i n g s o f the h y p e r f i n e i n t e r v a l s were measured a c c u r a t e t o ± 1 MHz, as de termined by compar i son w i t h e a r l i e r microwave work. L a r g e r f r e q u e n c y i n t e r v a l s (between r o t a t i o n a l l e v e l s ) were found t o be c o n s i s t e n t t o b e t t e r t h a n 10 MHz, t h e r e b y d e m o n s t r a t i n g the a c c u r a c y o f the c a l i b r a t i o n system o v e r s m a l l (a few MHz) and somewhat l a r g e r (~480 GHz) f r e q u e n c y r a n g e s . The r o t a t i o n a l a n a l y s i s o f B0 2 c o v e r e d the r e g i o n n e a r 5400A which a l l o w e d t h e (0,0) and l\ bands o f the A 2 ^ and X l^lg t r a n s i t i o n s t o be a n a l y z e d . The ground s t a t e c o n s t a n t s o b t a i n e d agreed w e l l w i t h p r e v i o u s r e s u l t s w h i l e t h e upper s t a t e e x h i b i t e d two v e r y i n t e r e s t i n g f e a t u r e s : t h e r e i s a K -resonance a v o i d e d c r o s s i n g between the 010 2 A 5 / 2 and 2 E * v i b r o n i c l e v e l s , w h i l e t h e 2 A 3 / 2 l e v e l l i e s e n t i r e l y between t h e Fj and F 2 s p i n components o f the 22" v i b r o n i c l e v e l . These f e a t u r e s have p e r m i t t e d t h e r e l a t i v e e n e r g i e s o f a l l t h e v i b r o n i c l e v e l s o f t h e 010 v i b r a t i o n a l l e v e l s t o be d e t e r m i n e d a c c u r a t e l y i n b o t h e l e c t r o n i c s t a t e s . T h i s i s not n o r m a l l y p o s s i b l e i n a - ^ e l e c t r o n i c t r a n s i t i o n . Both the v2=0 and v 2 = l upper s t a t e s are randomly p e r t u r b e d by h i g h l y i n g ground s t a t e v i b r o n i c l e v e l s ; t h i s i s an unexpected e f f e c t f o r such c o m p a r a t i v e l y low e n e r g i e s , and would have gone u n n o t i c e d but f o r t h e ex treme ly h i g h p r e c i s i o n o f t h e d a t a . Once a g a i n t h e c a l i b r a t i o n system p r o v e d i t s power, w i t h r o t a t i o n a l c o m b i n a t i o n d i f f e r e n c e s i n bands w i t h a common upper s t a t e , s e p a r a t e d by more t h a n 200 cm"1, b e i n g r e p r o d u c e d , on average , t o ± 0 . 0 0 0 3 cm"1. i i i Table of Contents Abstract i i L i s t of Tables v i i L i s t of Figures ix Acknowledgement x i i i Chapter 1 A C a l i b r a t i o n System f o r the V i s i b l e Region 1 1.1 Background 1 1.2 Theory 3 1.2.1 Interferometers 3 1.2.2 The Calculation of the Free Spectral Range 5 1.3 Experimental Details 7 Chapter 2 Application of the C a l i b r a t i o n System to the 13 Hyperfine Structure of N02 2.1 Introduction 13 2.2 Experimental Details 14 2.2.1 The Molecular Beam Apparatus 14 2.2.2 Fluorescence C o l l e c t i o n Device 16 2.2.3 Laser-Induced Fluorescence 18 2.2.4 The Doppler Width 19 2.2.5 Lamb Dips 20 2.2.6 The Laser Beam Arrangement 23 2.2.7 V e l o c i t y Measurement 25 2.2.8 Rotational Temperature 27 iv 2.2.9 Calculation of Frequencies 31 2.3 Theory 32 2.3.1 Background 32 2.3.2 Molecular Hamiltonian 36 2.3.3 Matrix Elements 40 2.3.4 Selection Rules and Intensity Calculations 46 2.4 Results 49 C h a p t e r 3 A p p l i c a t i o n t o t h e R o t a t i o n a l S t r u c t u r e o f t h e 55 B 0 3 M o l e c u l e 3.1 Introduction 55 3.2 Experimental Details 57 3.2.1 B02 Production 57 3.2.2 Intermodulated Fluorescence 57 3.2.3 Calculation of Frequencies 59 3.3 Theory 3.3.1 Background 61 3.3.2 Hund's Coupling Cases 69 3.3.3 Rotational Levels and Selection Rules 74 3.3.4 Case (a) Hamiltonian and Matrix Elements 78 3.3.5 The Hamiltonian and Matrix Elements f o r 81 Hund's Case (b) 3.3.6 The Hamiltonian for A2!!,,, v 2=l, and i t s 85 Matrix Elements 3.4 Results 98 3.4.1 Least Squares F i t t i n g 98 3.4.2 Numerical Results and Discussion 100 3.4.3 Conclusions 114 v R e f e r e n c e s Appendix I Appendix I I Append ix I I I Appendix IV Appendix V T r a n s i t i o n f r e q u e n i c e s o f N0 2 Sample i n t e n s i t y c a l c u l a t i o n Observed t r a n s i t i o n s ( i n cm"1) f o r the A^-X^l ig system o f B0 2 C a l c u l a t e d ground s t a t e energy l e v e l s o f U B 0 2 i n wave numbers (cm"1) The upper s t a t e energy l e v e l s < B0 2 i n wave numbers (cm"1) v i L i s t o f T a b l e s 2 .1 Comparison o f microwave and o p t i c a l l y measured 50 h y p e r f i n e s p l i t t i n g s i n N0 2 . 2 .2 Comparison o f a microwave and o p t i c a l l y measured 50 r o t a t i o n a l i n t e r v a l i n N0 2 a l o n g w i t h c a l c u l a t e d v a l u e s . 2 . 3 a M o l e c u l a r c o n s t a n t s f o r upper s t a t e o f N0 2 , band 99. 52 2 .3b Upper s t a t e r o t a t i o n a l c o n s t a n t s f o r N 0 2 / band 99. 52 2 . 4 a M o l e c u l a r c o n s t a n t s f o r upper s t a t e o f N0 2 , band 115. 53 2 .4b Upper s t a t e r o t a t i o n a l c o n s t a n t s f o r N0 2 , band 115. 53 3 .1 The H a m i l t o n i a n m a t r i x f o r the 000C^) v i b r o n i c 82 s t a t e s o f B0 2 . The ± r e f e r t o the e and f l e v e l s r e s p e c t i v e l y . The term v a l u e , T 0 , r e p r e s e n t s the o r i g i n o f t h e v i b r o n i c s t a t e . 3.2 The H a m i l t o n i a n m a t r i x f o r the X i^ig 010 (2A) v i b r o n i c 82 s t a t e o f B0 2 . The ± r e f e r t o t h e e and f l e v e l s r e s p e c t i v e l y . The term v a l u e , T 0 , r e p r e s e n t s the o r i g i n o f t h e v i b r o n i c s t a t e . 3.3 The H a m i l t o n i a n m a t r i x f o r the X 2 n g 010 (2S) v i b r o n i c 85 s t a t e o f B0 2 . The term v a l u e , T 0 , r e p r e s e n t s the v i b r o n i c o r i g i n . 3.4 The H a m i l t o n i a n m a t r i x f o r t h e A 2 ^ 010 (2A) and 95 ( 2S) v i b r o n i c s t a t e s o f B0 2 . The ± r e f e r t o t h e e and f p a r i t y l e v e l s r e s p e c t i v e l y . v i i 3.5 M o l e c u l a r c o n s t a n t s f o r the X 2 !^ 000 l e v e l o f 102 U B 0 2 , i n cm"1. 3 .6 M o l e c u l a r c o n s t a n t s f o r the X 2 ^ 000 l e v e l o f 102 1 0 B O 2 / i n cm"1. 3 .7 M o l e c u l a r c o n s t a n t s f o r the A 2 ^ 000 l e v e l o f 104 n B 0 2 , i n cm"1. 3 .8 M o l e c u l a r c o n s t a n t s f o r the A 2 ! ^ 000 l e v e l o f 104 1 0 B O 2 , i n cm"1. 3 .9 M o l e c u l a r c o n s t a n t s f o r t h e X 2 !^ 010 l e v e l s o f 107 n B 0 2 , i n cm"1. 3 .10 M o l e c u l a r c o n s t a n t s f o r the A 2 ^ 010 l e v e l s o f 109 n B 0 2 / i n cm"1. 3 .11 Fundamental c o n s t a n t s form f i t t e d parameters i n 115 B0 2 , i n cm"1. v i i i L i s t o f F i g u r e s 1.1 A c o n f o c a l F a b r y - P e r o t i n t e r f e r o m e t e r w i t h an 4 i n c i d e n t beam p a r a l l e l t o the a x i s o f t h e i n t e r f e r o m e t e r . 1.2 A s c h e m a t i c d iagram o f the c a l i b r a t i o n sys tem. 8 2 .1 " A b l o c k d iagram o f the m o l e c u l a r beam a p p a r a t u s . 15 2.2 I l l u s t r a t i o n o f the n o z z l e and skimmer c o m b i n a t i o n . 14 2.3 The m o l e c u l a r beam c r o s s e d a t r i g h t a n g l e s by t h e 17 l a s e r beam w i t h f l u o r e s c e n c e c o l l e c t e d p e r p e n d i c u l a r l y t o b o t h . 2.4 Two l e v e l sys tem. 19 2 .5 The arrangement t o c o l l e c t Lamb d i p s i g n a l s . 20 2.6 A G a u s s i a n v e l o c i t y p o p u l a t i o n d i s t r i b u t i o n . 21 2.7 The G a u s s i a n v e l o c i t y p o p u l a t i o n d i s t r i b u t i o n 22 showing a) t h e f o r m a t i o n o f Bennet h o l e s as t h e l a s e r beam i n t e r a c t s w i t h m o l e c u l e s s y m m e t r i c a l l y D o p p l e r s h i f t e d , b) the convergence o f the two Bennet h o l e s as t h e l a s e r i s tuned i n f r e q u e n c y and c) t h e Bennet h o l e s s e p a r a t i n g as t h e l a s e r i s tuned p a s t t h e t r a n s i t i o n f r e q u e n c y . 2 .8 Ri(2) and R S 1 2 (2) t r a n s i t i o n s from a) s i n g l e 24 c r o s s i n g L I F (1 .5 GHz scan) and b) t h e c o r r e s p o n d i n g Lamb d i p s (500 MHz s c a n ) . i x 2.9 E x p e r i m e n t a l s e t up f o r v e l o c i t y measurement. The 26 c i r c l e i n d i c a t e s the a r e a where b o t h l a s e r beams c r o s s t h e m o l e c u l a r beam and f l u o r e s c e n c e i s o b s e r v e d . 2.10 The m o l e c u l a r f l u o r e s c e n c e s i g n a l used t o o b t a i n 28 t h e v e l o c i t y measurement. 2 .11 The graph o f l n [ ( I H / I Q J / A H ] v e r s u s F „ / k used t o 30 o b t a i n the r o t a t i o n a l t emperature o f t h e N0 2 m o l e c u l a r beam. 2.12 N0 2 w i t h i n e r t i a l ( a , b , c ) and p o i n t group ( x , y , z ) 33 axes shown. 2.13 Gjv c h a r a c t e r t a b l e . 33 2.14 Walsh d iagram a p p r o p r i a t e t o N0 2 [ a f t e r r e f e r e n c e 34 3 1 ] . 2 .15 E x i s t i n g r o t a t i o n a l l e v e l s i n the ground e l e c t r o n i c 37 s t a t e o f N0 2 . 2 .16 V e c t o r d iagram o f Hund's c o u p l i n g case (b^) . 41 2.17 Shown a r e the observed a l l o w e d R(4) t r a n s i t i o n s 47 ( s o l i d l i n e s ) and the f o r b i d d e n t r a n s i t i o n s ( d o t t e d l i n e s ) . 3 .1 B l o c k d iagram showing the i n t e r m o d u l a t e d 58 f l u o r e s c e n c e exper iment . 3.2 A sample 3 0 GHz B0 2 s c a n . The 150 MHz markers 60 a r e not shown due t o t h e i r h i g h d e n s i t y . The a s s i g n e d t r a n s i t i o n s have been l a b e l l e d w i t h the v i b r o n i c symmetry, a s u p e r s c r i p t i n d i c a t i n g t h e b o r o n i s o t o p e , and a b r a n c h l a b e l . The b r a n c h l a b e l x i s R o r P (AJ=+1 o r A J = - 1 ) w i t h the s p i n component as a s u b s c r i p t when r e q u i r e d and the a s s o c i a t e d J " o r N" v a l u e i n p a r e n t h e s e s . 3.3 The DL^ p o i n t group c h a r a c t e r t a b l e . 62 3.4 Rough c o n t o u r d iagrams o f n g and n u o r b i t a l s . 64 3 .5 The p o t e n t i a l energy c u r v e s s e p a r a t i n g as a l i n e a r 65 m o l e c u l e bends . 3 .6 The degenerate bend ing mot ion o f B0 2 . A 9 0 ° 67 o u t - o f - p h a s e a d d i t i o n causes t h i s t o l o o k l i k e v i b r a t i o n a l a n g u l a r momentum. 3 .7 The p a t t e r n o f v i b r o n i c energy l e v e l s formed when 68 t h e b e n d i n g v i b r a t i o n o f a l i n e a r symmetric t r i a t o m i c m o l e c u l e i n a ^ g e l e c t r o n i c s t a t e i s e x c i t e d [36 ] . 3.8 Energy l e v e l d iagram showing the t r a n s i t i o n s s t u d i e d . 70 3 .9 V e c t o r d iagram o f Hund's c o u p l i n g case (a) w i t h 71 a) G=0, b) G*0. 3.10 V e c t o r d iagram o f Hund's c o u p l i n g case ( b ) . 74 3 .11 V e c t o r d iagram o f Hund's c o u p l i n g case ( c ) . 75 3.12 R o t a t i o n a l l e v e l s which e x i s t i n t h e ground s t a t e 77 2 v i b r o n i c l e v e l s . 3.13 R o t a t i o n a l l e v e l s which e x i s t i n the ground s t a t e 79 n and A v i b r o n i c l e v e l s . The d o t t e d l i n e s r e p r e s e n t t h e m i s s i n g l e v e l s . 3.14 D e f i n i t i o n o f the v i b r o n i c e n e r g i e s T ^ and A T z 97 used i n T a b l e 3 . 4 . 3 .15 A A 2 F " ( J ' ) c o m b i n a t i o n d i f f e r e n c e . 101 x i 3.16 R e s i d u a l s from t h e upper s t a t e f i t o f t h e 2 n 105 r o v i b r o n i c l e v e l s p l o t t e d a g a i n s t J * f o r a) U B 0 2 and b) 1 0 B O 2 . C i r c l e d d a t a p o i n t s i n d i c a t e a s s o c i a t e d t r a n s i t i o n s have been e x c l u d e d from f i n a l f i t . The c r o s s e s r e p r e s e n t the r e s i d u a l s w h i l e t h e diamonds r e p r e s e n t the ^ r e s i d u a l s . 3.17 R e s i d u a l s from the upper s t a t e f i t o f the 2 A and 2 S 110 r o v i b r o n i c l e v e l s p l o t t e d a g a i n s t J 1 f o r a) 2 A 5 / 2 ( r e p r e s e n t e d by c r o s s e s ) and 2 S l o w e r ( r e p r e s e n t e d by diamonds) and b) 2 A 3 / 2 ( r e p r e s e n t e d by c r o s s e s ) and ^upper ( r e p r e s e n t e d by diamonds) . C i r c l e d d a t a p o i n t s i n d i c a t e the a s s o c i a t e d t r a n s i t i o n s have been e x c l u d e d from f i n a l f i t . 3.18 A graph o f t h e r o t a t i o n a l energy l e v e l s o f t h e A 2 n u 112 v 2 = l v i b r o n i c s t a t e s . The c r o s s e s r e p r e s e n t t h e 2 A l e v e l s w h i l e the c i r c l e s r e p r e s e n t the 2 S l e v e l s . The 22~ and 2 A 3 / 2 e n e r g i e s have been s c a l e d by a f a c t o r o f -0 .3115J(J+1) w h i l e the V and 2 A 5 / 2 have been s c a l e d by a f a c t o r o f - 0 . 3 1 J ( J + 1 ) . T h i s s c a l i n g i s done t o ensure t h a t the d e t a i l s o f the energy l e v e l s can be seen c l e a r l y . 3.19 The " r o t a t i o n l e s s " e n e r g i e s o f a l l the v i b r o n i c 113 l e v e l s s t u d i e d w i t h r e s p e c t t o a z e r o a t the X 2 ! ^ , 000(^3/2) J = 3 / 2 l e v e l . x i i Acknowledgement I w i sh t o thank D r . A . J . Merer f o r p r o v i d i n g me w i t h the o p p o r t u n i t y t o work i n the c h a l l e n g i n g f i e l d o f the h i g h r e s o l u t i o n e l e c t r o n i c s p e c t r o s c o p y group a t UBC. H i s h e l p and s u p p o r t was much a p p r e c i a t e d as w e l l as h i s c o n t i n u i n g encouragement throughout t h i s work. I a l s o want t o thank D r . A . Adam w i t h whom I worked c l o s e l y on a l l a s p e c t s o f t h i s p r o j e c t . F i n a l l y , thanks go t o the v a r i e t y o f g r a d u a t e s t u d e n t s , p o s t d o c t o r a l f e l l o w s and t e c h n i c i a n s I have met and worked w i t h o v e r the p a s t few y e a r s . x i i i 1 C h a p t e r 1 A C a l i b r a t i o n System f o r the V i s i b l e R e g i o n 1.1 Background One problem facing high resolution l a s e r spectroscopists i s to know, with the highest possible degree of accuracy, the wavelength of the l i g h t with which they are probing t h e i r system. Various methods are currently used. Several methods depend on the simultaneous recording of a secondary wavelength standard while others make use of the coherency of lase r l i g h t and interferometric techniques. Systems that use secondary standards often use iodine fluorescence or absorption l i n e s [1] because there i s an iodine a t l a s [2] avai l a b l e which contains over 22 000 l i n e s between 14800 cm"1 and 20000 cm"1 measured to an accuracy of ±0.002 cm"1. Other secondary standards being used are iron-neon hollow cathode lamps [3] and uranium hollow cathode lamps [4] whose spectra are recorded using optogalvanic spectroscopy. When secondary standards are used for lase r work, one simultaneously records the secondary standard, the system being studied and interferometric markers for int e r p o l a t i o n between the l i n e s of the secondary standard. A method of c a l i b r a t i o n which does not involve recording a secondary standard uses a 'lambda-1 or 'wave-1 meter. There are several designs currently i n use although they are a l l s i m i l a r i n p r i n c i p l e [5-7]. They are based on a t r a v e l l i n g 2 Michelson interferometer and a comparison of the unknown laser frequency with the known frequency of a s t a b i l i z e d laser, often a HeNe laser s t a b i l i z e d on an I 2 hyperfine l i n e [8]. In t h e i r previous sub-Doppler work, the high resolution e l e c t r o n i c spectroscopy group at UBC ca l i b r a t e d t h e i r data against iodine fluorescence spectra. However, as the group has recently developed a molecular beam system which produces spectra with much narrower linewidths, a new c a l i b r a t i o n system with greater accuracy was required. There were two possible ways of upgrading the old system. One was to buy an "Autoscan" wavemeter upgrade system for t h e i r e x i s t i n g r i n g l a s e r from Coherent Inc., and the other was to b u i l d a new c a l i b r a t i o n system within the lab which would perform a s i m i l a r function. The "Autoscan" system [9] i s s p e c i f i e d to be able to acquire data 'seamlessly* over a 10 THz frequency range with ±2 5 MHz r e p r o d u c i b i l i t y , but i t cannot scan slowly enough, with a s u f f i c i e n t l y large number of data a c q u i s i t i o n points, to record the very high resolution data that the molecular beam system was capable of producing. Spectral l i n e s with f u l l - w i d t h at half-maximum (FWHM) of 2 MHz could be di s t o r t e d or even missed at the slowest data a c q u i s i t i o n rate av a i l a b l e . For t h i s reason, i t was decided to develop a new c a l i b r a t i o n system. The system i s centred around a 750 MHz free spectral range etalon whose cavity length i s servo-locked to a multiple of the HeNe laser wavelength; more d e t a i l s w i l l be given i n 3 S e c t i o n 1 .3 . S p e c t r a were r e c o r d e d u s i n g a computer -c o n t r o l l e d l a s e r s c a n n i n g and d a t a a c q u i s i t i o n programme w r i t t e n a t UBC. As the computer v a r i e d t h e l a s e r f r e q u e n c y , t h r e e c h a n n e l s o f d a t a , the m o l e c u l a r s p e c t r a , the 750 MHz c a l i b r a t i o n m a r k e r s , and 150 MHz i n t e r p o l a t i o n m a r k e r s , were r e c o r d e d . 1.2 T h e o r y 1.2.1 I n t e r f e r o m e t e r s B e f o r e d e s c r i b i n g the c a l i b r a t i o n sys tem, a s h o r t summary o f the t h e o r y o f the c o n f o c a l F a b r y - P e r o t i n t e r f e r o m e t e r ( a l s o c a l l e d an e t a l o n ) w i l l be g i v e n . An i n t e r f e r o m e t e r o f t h i s t y p e c o n s i s t s o f two concave s p h e r i c a l m i r r o r s s e p a r a t e d by t h e i r f o c a l l e n g t h , d . The m i r r o r s a r e no t 100% r e f l e c t i n g , b u t a r e d e s i g n e d t o t r a n s m i t a few p e r c e n t o f any i n c i d e n t l i g h t . When a beam o f l i g h t i s s e n t t h r o u g h t h e i n t e r f e r o m e t e r (as i n F i g u r e 1 . 1 ) , t h e r e w i l l be maxima i n the t r a n s m i t t e d i n t e n s i t y ( c a l l e d f r i n g e s ) when t h e o p t i c a l p a t h d i f f e r e n c e t r a v e l l e d by p a r t s o f the beam t h a t undergo d i f f e r e n t numbers o f r e f l e c t i o n s i s an i n t e g r a l m u l t i p l e o f t h e wave length o f t h e l i g h t . I f t u n a b l e monochromatic l i g h t from a l a s e r passes t h r o u g h t h e i n t e r f e r o m e t e r , t h e s e p a r a t i o n o f t h e i n t e n s i t y maxima w i l l be 6v = c / 4 d F i g u r e 1.1: A c o n f o c a l F a b r y - P e r o t i n t e r f e r o m e t e r w i t h an i n c i d e n t beam p a r a l l e l t o t h e a x i s o f t h e i n t e r f e r o m e t e r . 5 where 4d i s the path difference between two rays, c i s the speed of l i g h t and Su i s c a l l e d the free spectral range (FSR) of the interferometer quoted i n frequency u n i t s . For an etalon to have a FSR of 750 MHz, the mirrors must be 10 cm apart. 1.2.2 The C a l c u l a t i o n o f the F r e e S p e c t r a l Range To measure the frequency of a dye laser accurately, i t i s necessary to know the FSR of the s t a b i l i z e d 750 MHz etalon with great p r e c i s i o n . This requires that the wavelength of the HeNe laser be known, along with the absolute order number of the fringe to which the system i s locked. In the present case, these were determined by recording the etalon fringes together with the I 2 fluorescence spectrum over as wide a range as p r a c t i c a l , which was about 3 000 cm"1. I t was not necessary to record the f u l l 3000 cm"1 as the FSR could be determined with enough accuracy from a 10 cm"1 portion of the spectrum to obtain the r e l a t i v e order number of the fringes i n another 10 cm"1 portion some 100 cm"1 away. The FSR could then be refined, i n order to l i n k a t h i r d 10 cm"1 portion some 400 cm"1 away. In a l l , eleven 10 cm"1 i n t e r v a l s were c o l l e c t e d over the regions 16480 cm"1 to 17700 cm"1 (using R6G laser dye) and 14920 cm"1 to 15900 cm"1 (using DCM laser dye) . This large range was required so that the order number of the fringe at the HeNe las e r frequency to which the system was locked could be determined to an accuracy of approximately ±0.1. A l i n e a r least-squares programme was used to f i t the wave numbers of 6 the I 2 l i n e s [2] a g a i n s t t h e i r p o s i t i o n s i n terms o f r e l a t i v e f r i n g e numbers. The e q u a t i o n g o v e r n i n g t h i s r e l a t i o n s h i p i s "N = "int + N*FSR (1.1) where i s the f requency o f the l o n g e s t wave length f r i n g e r e c o r d e d (near 14920 cm"1) and «/„ i s the f r e q u e n c y o f the f r i n g e w i t h r e l a t i v e o r d e r number N i n wave number u n i t s . From the f i t the FSR was de termined t o be 0.025038797 cm"1 w i t h ± 3 on t h e l a s t d i g i t . S i n c e the e x a c t HeNe l a s e r f r e q u e n c y must l i e i n t h e range 15798.00 ± 0.01 cm"1, and g i v e n t h a t f k t w a s d e t e r m i n e d t o be 14923.19815 cm"1, t h e f r e q u e n c y o f our p o l a r i z a t i o n - s t a b i l i z e d HeNe l a s e r on t h a t day was found t o be 15798.00364 cm"1, w i t h t h e system l o c k e d t o f r i n g e number 630941. T h e r e f o r e , g i v e n t h e f r e q u e n c y o f a f r i n g e measured by a commerc ia l wavemeter, one can work out i t s o r d e r number and t h e r e f o r e o b t a i n i t s f r e q u e n c y w i t h h i g h e r a c c u r a c y from e q u a t i o n ( 1 . 1 ) . I t s h o u l d be ment ioned t h a t the above p r o c e d u r e o n l y had t o be done once . A f t e r t h e FSR o f the e t a l o n was known, the system c o u l d be shut down as l o n g as the p h y s i c a l arrangement o f t h e e t a l o n was no t d i s t u r b e d . The o n l y d i f f i c u l t y w i t h r e l o c k i n g the system was t o ensure t h a t i t was l o c k e d t o t h e c o r r e c t f r i n g e , and n o t , f o r example, t o an a d j a c e n t one. W i t h t h e l o c k i n g system i n i t s p r e s e n t arrangement (see S e c t i o n 1.3) i t can be r e l o c k e d t o any o f n i n e d i f f e r e n t 7 fringes. The method used to check for the correct fringe was to compare some previously c o l l e c t e d data, or more s p e c i f i c a l l y , the distance between the 750 MHz markers and several N02 hyperfine t r a n s i t i o n s . I f the system i s not locked to the correct fringe, the cavity length (and therefore the FSR) w i l l be s l i g h t l y d i f f e r e n t ; the e f f e c t i s to s h i f t the markers with respect to the fixed N02 t r a n s i t i o n s . The s i z e of the s h i f t may be calculated by multiplying the FSR by the r a t i o of the N02 frequencies to the HeNe frequency and then subtracting the FSR. For N02 t r a n s i t i o n s at 16850 cm"1, t h i s corresponds to a 50 MHz s h i f t . For future work at wavelength regions d i f f e r e n t from the N02 regions studied (see Chapter 2), the system would have to be locked f i r s t to the correct fringe using N02 and then switched to the new region where data c o l l e c t e d on a d i f f e r e n t molecule could be used as a 'reference lock point' f o r that region. 1.3 E x p e r i m e n t a l D e t a i l s The c a l i b r a t i o n system which was developed i s shown i n Figure 1.2. The system i s designed to measure the frequency of an argon ion-pumped r i n g dye l a s e r (Coherent Inc. model 699-21). A 750 MHz etalon was chosen over other commercially a v a i l a b l e etalons because the accuracy of our e x i s t i n g Burleigh WA-20VIS wavemeter (1 i n 106) i s such that i t can d i s t i n g u i s h between two fringes 750 MHz apart. With a smaller FSR etalon, the order numbers would be greater than 106, so fen. generator lock-in Q 6 O-automatic level - B Z 3 -control -0 A HV amp. J BS6 st a b i l i z e d _ j# \ P o 1 1 A H*&Hff ) V I 0 »t PD2 J \ / BS5 1 T PDl to computer —&—V i r i s 2 > 4H pol2 HeNe laser to computer passive 150 ) ^ ramped 750 to oscilloscope ( ) n\2 BS3 85T/15R Ar* ion laser BS4 70R/30T BS2 50T/50R ^ B S l 85T/15R wavemeter I to experiment Figure 1.2: A schematic diagram of the calibration system. 09 9 t h a t t h e wavemeter would not have the a c c u r a c y t o measure them, w h i l e l a r g e r FSR e t a l o n s l e a d t o too few c a l i b r a t i o n m a r k e r s . A s m a l l f r a c t i o n o f the dye l a s e r o u t p u t i s p i c k e d o f f by a b e a m s p l i t t e r (BS1) w h i l e the remainder o f the o u t p u t i s s ent t o the e x p e r i m e n t . The f r a c t i o n p i c k e d o f f i s f u r t h e r s u b d i v i d e d a t two o t h e r b e a m s p l i t t e r s (BS2 and BS3) , which send s m a l l amounts o f d y e - l a s e r l i g h t t o a 'ramped' 750 MHz e t a l o n and a p a s s i v e 150 MHz e t a l o n . A 'ramped' e t a l o n has one o f i t s end m i r r o r s mounted on a p i e z o e l e c t r i c t r a n s l a t o r such t h a t a ramped v o l t a g e a p p l i e d t o t h e t r a n s l a t o r changes t h e c a v i t y l e n g t h r e p e a t e d l y . The t r a n s m i s s i o n o f t h i s e t a l o n , as d e t e c t e d by a p h o t o d i o d e , i s d i s p l a y e d on an o s c i l l o s c o p e which a l l o w s one t o see i f t h e dye l a s e r i s s c a n n i n g c o r r e c t l y on a s i n g l e l o n g i t u d i n a l mode. The output from t h e p a s s i v e 150 MHz e t a l o n p r o v i d e s i n t e r p o l a t i o n markers which form a n e c e s s a r y p a r t o f t h e c a l i b r a t i o n sys tem; they w i l l be d i s c u s s e d f u r t h e r i n C h a p t e r 2, S e c t i o n 2 . 2 . 8 and C h a p t e r 3, S e c t i o n 3 . 2 . 3 . A f u r t h e r p o r t i o n o f t h e l a s e r beam i s p i c k e d o f f by b e a m s p l i t t e r BS4, and s e n t t o t h e B u r l e i g h wavemeter t o o b t a i n a wave number r e a d i n g . The purpose o f the i r i s ( i r i s l ) l o c a t e d a p p r o x i m a t e l y 1 meter i n f r o n t o f the wavemeter, i s t o d i r e c t t h e l a s e r beam r e p r o d u c i b l y such t h a t t h e a n g l e o f e n t r y i s the same from day t o d a y . T h i s i s i m p o r t a n t as t h e wave number r e a d i n g o b t a i n e d i s h i g h l y dependent on t h i s a n g l e . E r r o r s o f up t o 0.04 cm"1 can o c c u r because of t h i s e f f e c t . The remainder of the la s e r beam i s sent to the central component of the c a l i b r a t i o n system, the pressure- and temperature-stabilized 750 MHz etalon. This s t a b i l i z a t i o n i s done by means of a vacuum housing which i s kept at a pressure below 10 microns and a temperature of approximately 48°C ( s t a b i l i t y ±0.004°C [10]) by a diffusion/mechanical pump combination and a c i r c u l a t i n g o i l bath respectively. The path of t h i s l a s e r beam i s also defined by i r i s e s ( i r i s 2 and i r i s 3 ) , as the shape of the transmission fringes obtained from the etalon depends d i r e c t l y on the input angle [11]. The lens situated i n front of the etalon matches the e l e c t r i c f i e l d d i s t r i b u t i o n of the laser with the f i e l d d i s t r i b u t i o n required for a TEMQO mode i n the etalon. This allows the maximum amount of l i g h t through to the detector [11]. The p o l a r i z e r (poll) ensures that the dye laser beam i s v e r t i c a l l y polarized. The reason f o r using orthogonal p o l a r i z a t i o n s for the two beams (the HeNe la s e r beam i s h o r i z o n t a l l y polarized) i s so that they can be separated by a prism p o l a r i z e r a f t e r passing through the etalon, and sent to d i f f e r e n t detectors. The f i n a l beamsplitter (BS5) combines the HeNe la s e r beam with the dye l a s e r beam such that they follow the same path within the etalon. The HeNe laser (Spectra Physics model 117A) i s a p o l a r i z a t i o n - s t a b i l i z e d helium-neon l a s e r where s t a b i l i z a t i o n i s achieved by a l t e r i n g the cavity length to maintain the r e l a t i v e i n t e n s i t i e s of the two orthogonally-polarized modes 11 which are allowed to operate i n the cavity; only one of these modes i s allowed to emerge from the las e r . The HeNe laser beam passes through a p o l a r i z e r (pol2) and a quarterwave plate (A/4) combination which prevents feedback, that i s , r e f l e c t e d l i g h t returning to the laser, which would upset i t s s t a b i l i z a t i o n . The f i n a l p o l a r i z e r (pol3) ensures that the HeNe la s e r beam i s purely h o r i z o n t a l l y polarized on entering the etalon. The two laser beams are monitored by two photodiodes (PD1 and PD2). The HeNe laser signal (from PD2) i s sent to a p o s i t i v e feedback loop consisting of a l o c k - i n amplifier (EG&G Princeton Applied Research model 5101), a function generator (Circuitmate FG2), a high voltage amplifier (Burleigh Spectrum Analyzer RC-46) and an automatic l e v e l control. This loop locks a transmission fringe of the etalon to the HeNe laser frequency, using the f i r s t derivative of the signal to monitor any d r i f t away from the peak p o s i t i o n . A correction voltage i s then applied to the p i e z o e l e c t r i c t r a n s l a t o r on which the end mirror i s mounted. I n i t i a l l y i t was found that the system often l o s t lock a f t e r several hours because of d r i f t i n g i n the HeNe la s e r frequency and c i r c u i t i n s t a b i l i t i e s , which caused the correction voltage to become too large. The automatic l e v e l control c i r c u i t was then added; t h i s adjusts an o f f s e t voltage to the p i e z o e l e c t r i c t r a n s l a t o r such that the correction voltage from the lo c k - i n amplifier remains very near zero. The system can now be locked for several weeks. 12 The dye laser signal (from PD1) i s acquired by the computer as a data channel of 750 MHz c a l i b r a t i o n markers. In summary, with the accurate determination of the FSR of the s t a b i l i z e d 750 MHz etalon, i t i s now possible to calculate the order number of a transmission fringe given the wave number reading obtained from the Burleigh wavemeter. This allows c a l c u l a t i o n of the precise wave numbers of the transmission fringes, thereby c a l i b r a t i n g the spectral data obtained. The accuracy of the system i s discussed i n Section 2.4 of Chapter 2. Chapter 2 Application of the C a l i b r a t i o n System to the Hyperfine Structure of N02 2 . l Introduction The spectrum of N02 was f i r s t seen by Brewster i n 1834 [12] and since then many s c i e n t i s t s have put i n years of work attempting to unravel i t s mysteries. N02 was chosen for the f i r s t t e s t of the new molecular beam system at UBC for several reasons: N02 absorbs throughout the v i s i b l e region, which meant that the most convenient lase r dye, rhodamine-6G, could be used, - molecular beams of N02 are simple to make; one only needs a 10% mixture of N02 i n argon at a backing pressure of less than one atmosphere, several papers on molecular beam spectra of N02 [8,13-15] were available, so that comparisons with the e a r l i e r data could e a s i l y be made, ro t a t i o n a l analyses had already been done [16-18] for various bands i n the rhodamine-6G region, the UBC system should have s u f f i c i e n t resolution to see the nitrogen magnetic hyperfine structure. The actual band systems chosen for study were two K>0 subbands of the 2B2-2A1 e l e c t r o n i c t r a n s i t i o n at 593.5 nm and 585.1 nm. 14 They were chosen as they were both noted to be strong t r a n s i t i o n s by Smalley et a l . [15]; following Smalley et a l . ' s numbering scheme these w i l l be c a l l e d bands 99 and 115 respec t i v e l y . 2.2 Experimental Details 2 . 2 . 1 The Molecular Beam Apparatus A block diagram of the molecular beam apparatus, as set up to produce N02 fluorescence spectra, i s shown i n Figure 2.1. A molecular beam of N02 was formed by expanding a mixture of 10% N02 i n argon through a nozzle and skimmer combination as shown i n Figure 2.2. The nozzle diameter used was 50 fim and the skimmer diameter was 500 pm. pressure pressure pressure 300 Torr ~10 -3 Torr <10 -6 Torr nozzle skimmer Figure 2.2 I l l u s t r a t i o n of the nozzle and skimmer combination. t o c a l i b r a t i o n Figure 2.1 A block diagram of the molecular beam apparatus. 16 Before passing through the nozzle, the gas mixture obeys the normal Maxwell-Boltzmann v e l o c i t y d i s t r i b u t i o n for molecules i n thermal equilibrium at room temperature. The expansion through the nozzle converts the random motions of thermal equilibrium into a directed mass flow, which reduces the width of the v e l o c i t y d i s t r i b u t i o n [15]. The skimmer picks out the central portion of t h i s expansive beam to produce a well collimated molecular beam, where the v e l o c i t y range and angular divergence are both small. Within t h i s beam c o l l i s i o n s are infrequent since the molecules are a l l moving forward at nearly the same speed. As a r e s u l t of the expansion the molecules are r o t a t i o n a l l y cold, as w i l l be discussed further i n Section 2.2.8. 2.2.2 F l u o r e s c e n c e C o l l e c t i o n D e v i c e Two d i f f e r e n t experiments were done on N02. One involved a single l a s e r crossing where the t o t a l fluorescence was c o l l e c t e d and the linewidth was determined by the angular divergence of the beam (see Section 2.2.3). The second experiment used Lamb dip methods (see Section 2.2.5) where the linewidth was reduced by sampling the middle of the molecular beam and only a f r a c t i o n of the. t o t a l fluorescence was c o l l e c t e d . In both cases, the molecular beam i s crossed at r i g h t angles by the laser beam as shown i n Figure 2.3. The apertures i n the input and output arms reduce the scattered room and la s e r l i g h t reaching the c o l l e c t i o n system. The Figure 2.3 The molecular beam crossed at r i g h t angles by the laser beam with fluorescence c o l l e c t e d perpendicularly to both. 18 fluorescence c o l l e c t i n g device, also shown i n Figure 2.3, was devised to increase the amount of molecular fluorescence c o l l e c t e d ; i t includes a s p a t i a l f i l t e r to eliminate scattered l i g h t . The l a s e r beam and the molecular beam cross i n the centre of the c o l l e c t i n g system where N02 molecules fluoresce i s o t r o p i c a l l y . The c o l l e c t i n g system uses a series of lenses (L1-L4) to focus l i g h t upward to the photomultiplier tube (PMT) (RCA tube type C31034-02). The current from the PMT i s fed to a l o c k - i n amplifier (EG&G Princeton Applied Research Model 128-A) which measures the signal by phase s e n s i t i v e detection. The major reasons why scattered l i g h t i s a problem i s that i t i s modulated, by the external chopper, at the same frequency as the molecular fluorescence, so that the l o c k - i n a m p l i f i e r cannot separate the two signals. I t i s also possible f o r scattered l i g h t to overload the PMT. A cut-off f i l t e r (F) (Corning coloured glass f i l t e r 2-58) was placed i n front of the PMT to reduce the background signal from the l a s e r l i g h t while allowing molecular fluorescence to the red of the c u t - o f f wavelength (37% transmission between 637-648 nm) to pass. 2 . 2 . 3 L a s e r - I n d u c e d F l u o r e s c e n c e A molecule can absorb radiat i o n of the appropriate frequency and be excited from i t s ground state into an upper state (Figure 2.4), from which i t decays back to lower states, emitting i t s extra energy as spontaneous emission or 19 F i g u r e 2.4 Two l e v e l sys tem. f l u o r e s c e n c e . The m o l e c u l e does not n e c e s s a r i l y decay back to i t s o r i g i n a l s t a t e but i n l a s e r e x c i t a t i o n s p e c t r o s c o p y a l l t h a t i s i m p o r t a n t i s the t o t a l f l u o r e s c e n c e which r e s u l t s from the e x c i t a t i o n . In t h i s e x p e r i m e n t , the MicroVAX computer i s programmed such t h a t i t scans the l a s e r i n f r e q u e n c y , and c o l l e c t s and s t o r e s the e x c i t a t i o n spec trum. L a s e r - i n d u c e d f l u o r e s c e n c e (LIF) s i g n a l s w i t h a l i n e w i d t h o f a p p r o x i m a t e l y 10 MHz were a c h i e v e d i n t h i s manner. T h i s l i n e w i d t h i s narrower t h a n t h a t which can be a c h i e v e d w i t h an a b s o r p t i o n c e l l , o r u n c o l l i m a t e d f low sys tem, because the s m a l l , d i v e r g e n c e o f the m o l e c u l a r beam reduces t h e D o p p l e r w i d t h . 2.2.4 The D o p p l e r Width M o l e c u l e s w i t h v e l o c i t y v w i l l absorb r a d i a t i o n a t a f r e q u e n c y w = u > 0 ( l ± v / c ) where the ± r e f e r t o m o l e c u l e s moving away from o r towards the l i g h t s o u r c e . The w0 i s t h e t r a n s i t i o n f r e q u e n c y as shown i n F i g u r e 2 . 4 , where i t i s assumed t h a t t h e m o l e c u l e i s no t moving w i t h r e s p e c t t o the l i g h t s o u r c e . I n a gas c e l l , a G a u s s i a n v e l o c i t y d i s t r i b u t i o n 20 i s p r e s e n t as shown i n F i g u r e 2 . 6 , which r e s u l t s i n a w i d e r range o f f r e q u e n c i e s t h a n the t r u e t r a n s i t i o n f r e q u e n c y b e i n g a b s o r b e d . T h i s g i v e s a widened s p e c t r a l l i n e which i s s a i d to be D o p p l e r broadened w i t h a c e r t a i n D o p p l e r w i d t h . The m o l e c u l a r beam r e d u c e s the D o p p l e r w i d t h by n a r r o w i n g the v e l o c i t y d i s t r i b u t i o n o f the m o l e c u l e s w i t h r e s p e c t t o the l a s e r beam. 2 . 2 . 5 Lamb D i p s Lamb d i p s can be used t o reduce f u r t h e r the l i n e w i d t h of the t r a n s i t i o n s b e i n g d e t e c t e d . When the l a s e r f r e q u e n c y i s not e q u a l t o the t r a n s i t i o n f r e q u e n c y , the l a s e r beam a t c r o s s i n g s A and B (see F i g u r e 2.5) i n t e r a c t s w i t h m o l e c u l e s l a s e r beam 10% NOa "in A r F i g u r e 2 . 5 The arrangement t o c o l l e c t Lamb d i p s i g n a l s , s y m m e t r i c a l l y D o p p l e r - s h i f t e d i n o p p o s i t e senses from t h e t r u e f r e q u e n c y , as shown i n F i g u r e 2 . 7 a . S i n c e t h e l a s e r beam i s a v e r y i n t e n s e l i g h t s o u r c e , i t d i s t o r t s t h e v e l o c i t y 21 Figure 2 .6 A Gaussian v e l o c i t y population d i s t r i b u t i o n . 22 — — v e l o c i t y B A Figure 2.7 The Gaussian v e l o c i t y population d i s t r i b u t i o n showing a) the formation of Bennet holes as the laser beam interacts with molecules symmetrically Doppler s h i f t e d , b) the convergence of the two Bennet holes as the laser i s tuned i n frequency and c) the Bennet holes separating as the lase r i s tuned past the t r a n s i t i o n frequency. 23 d i s t r i b u t i o n i n the lower state by depleting the population at the two v e l o c i t i e s ±v, leading to what are known as "Bennet holes" [19]. As the laser frequency i s scanned toward the centre frequency of the molecular l i n e , the two "Bennet holes" move towards each other, coincide at the true t r a n s i t i o n frequency (Figure 2.7b) and then separate again (Figure 2.7c). A phase s e n s i t i v e detection method i s used, with the la s e r beam being chopped p r i o r to crossing A while a l o c k - i n amplifier i s used to detect the modulated fluorescence at crossing B. Linewidths down to 2 MHz were achieved using t h i s method. A comparison of the Ri(2) and RS 1 2(2) l i n e s obtained by singl e crossing LIF and the corresponding Lamb dips i s shown i n Figure 2.8. 2 . 2 . 6 The L a s e r Beam Arrangement A Coherent Inc. model 699-21 rin g dye laser, pumped by an argon ion laser, was used as the source to excite the N02 molecules i n t h i s experiment. The output of the dye laser (see Figure 2.1) i s focussed by two lenses, LI and L2, which together produce a large region where the la s e r beam i s t i g h t l y focussed. This reduces the scattered l i g h t and allows the l a s e r beam to be focussed down to the same spot s i z e at both crossings A and B. The path of the dye la s e r beam i s con t r o l l e d by a series of mirrors. The f i r s t two mirrors (ml and m2) r a i s e the laser beam to the same height as the molecular beam, since the laser i s on a separate o p t i c a l table 24 Rl(2) + *S12(2) LIF 1.5 GHz scan Figure 2.8 Ri(2) and RS 1 2(2) t r a n s i t i o n s from a) single crossing LIF (1.5 GHz scan) and b) the corresponding Lamb dips (500 MHz scan). The tr a n s i t i o n s are l a b e l l e d F' - F". 25 i n another room, below the axis of the molecular beam. When the l a s e r beam i s directed to B, two mirrors (m3 and m.4) are needed to control i t s horizontal and v e r t i c a l placement through the molecular beam apparatus. In a single crossing LIF experiment the laser beam crosses at B where the high s e n s i t i v i t y PMT i s situated, although i t i s possible to rotate the top flange of the beam chamber to put the PMT at A and, with the use of a beamsplitter (BS), send the la s e r beam through A. Mirrors m5 and m6 have the same purpose as mirrors m3 and m4 when using crossing A. To record Lamb dip signals, the la s e r i s f i r s t sent through crossing B and then r e f l e c t e d o f f two mirrors (m7 and m8, Figure 2.6) to bring i t back through crossing A, upstream of crossing B. In t h i s case, the mechanical chopper (CI) i s moved from i t s p o s i t i o n i n front of crossing B to between mirrors m7 and m8. 2.2.7 V e l o c i t y Measurement The speed of the molecules i n the molecular beam could be estimated by using Doppler s h i f t s . A portion of the la s e r beam was picked o f f by the beamsplitter (BS) and sent along the axis of the molecular beam as shown i n Figure 2.9. The Doppler s h i f t (Av) of the fluorescence signal i s related to 26 l a s e r beam 1 Figure 2.9 Experimental set up for v e l o c i t y measurement. The c i r c l e indicates the area where both laser beams cross the molecular beam and fluorescence i s observed. 27 the v e l o c i t y of the molecules (v) by A i / c V = (2.1) v s i n * The value of e i s 90° for t h i s experiment. A t y p i c a l signal i s shown i n Figure 2.10. I t i s in t e r e s t i n g to note that the hyperfine structure, which i s e a s i l y resolved when the laser beam crosses the molecular beam perpendicularly, i s completely unseen i n the very much Doppler broadened l i n e obtained from the coaxial la s e r beam because of the spread of v e l o c i t i e s i n the molecular beam. A value of 440 ± 30 m/s was found for the molecular v e l o c i t y . This i s a reasonable number as i t i s somewhat less than the terminal v e l o c i t y of argon which i s 558 m/s [20]. 2.2.8 Rotational Temperature I t i s well known that cooling occurs i n supersonic beams formed by the free expansion of a gas through a nozzle [20]. By means of in t e n s i t y measurements, i t i s possible to estimate the r o t a t i o n a l temperature of the molecular beam i n t h i s experiment. The i n t e n s i t y of a given r o t a t i o n a l l i n e i s proportional to [21] T _ „ •» „ -F(KJ)/kT ,1 I K J = ci/Afugoe (2.2) Figure 2.10 The molecular fluorescence signal used to obtain the v e l o c i t y measurement. 29 where c = c o n s t a n t depending on the e l e c t r o n i c and v i b r a t i o n a l t r a n s i t i o n , v = t r a n s i t i o n f requency , A K J = H o n l - L o n d o n l i n e s t r e n g t h f a c t o r , gK J = s t a t i s t i c a l weight o f lower s t a t e , F(KJ) = term v a l u e , i . e . energy o f lower s t a t e , k = Boltzmann c o n s t a n t , T = r o t a t i o n a l t emperature o f beam. In t h i s N0 2 e x p e r i m e n t , K i s z ero and the g f a c t o r s a r e 1; a l s o the r o t a t i o n a l quantum number i s w r i t t e n as N r a t h e r t h a n J . A f t e r some rearrangement , e q u a t i o n 2.2 l e a d s t o l n [ ( I « / I 0 ) / A I ] = - F „ / k T (2.3) The i n t e n s i t i e s o f t h e s e p a r a t e h y p e r f i n e components have been summed t o g i v e t h e t o t a l i n t e n s i t y f o r the r o t a t i o n a l t r a n s -i t i o n , I N . The v a l u e s o f An have been c a l c u l a t e d from [21] A«(R l i n e s ) = (N+l) / (2N+l) (2.4) A«(P l i n e s ) = N/(2N+1) Averages o f the l o g a r i t h m i c term have been used i n the graph o f l n [ (Ig/IoJ/Ag] v e r s u s F H / k (shown i n F i g u r e 2 . 1 1 ) . The r e a s o n f o r t h e l a r g e s c a t t e r i n t h e s e v a l u e s , which i s r e f l e c t e d i n the e r r o r b a r s shown, i s p r o b a b l y f l u c t u a t i o n i n t h e l a s e r power. The s l o p e o f t h e graph l e a d s t o a r o t a t i o n a l t e m p e r a t u r e o f 20.2 ± 1.6 K f o r t h e NO2 beam used i n t h e s e e x p e r i m e n t s . 30 31 2.2.9 Calculation of Frequencies Each N02 spectrum was recorded as 4096 data points, or channels, covering 1.5 GHz. Three separate signals were recorded; the N02 fluorescence s i g n a l , the 750 MHz markers and the 150 MHz markers, as was shown i n Figure 2.8a. The 150 MHz markers have been d i g i t a l l y smoothed to remove noise generated by the j i t t e r of the laser, and the peaks were then f i t t e d to a Gaussian function to obtain the peak centres i n terms of channel numbers. The positions of the 750 MHz c a l i b r a t i o n markers were found i n terms of channel numbers using a graphics cursor routine on the MicroVAX computer. These positions were refined with a Gaussian function f i t t i n g program which obtained the best peak p o s i t i o n i n f r a c t i o n a l channel numbers. A f i t to the 750 MHz markers was required because, regardless of how well the lase r beam was aligned with the Fabry-Perot cavity, the fringes were often s l i g h t l y asymmetric. This asymmetry made i t d i f f i c u l t to judge the centre p o s i t i o n of the marker accurately by eye. The N02 peaks were measured to ±\ channel using the same graphics cursor routine and then converted to corresponding f r a c t i o n a l 150 MHz marker numbers (as were the positions of the 750 MHz markers). F i n a l l y , the N02 frequencies were calculated from the known frequencies of the 750 MHz markers. 32 2.3 Theory 2.3.1 Background N02 i s a stable gaseous triatomic free r a d i c a l (meaning that i t has an odd number of electrons). I t i s shown schematically i n Figure 2.12 with i t s i n e r t i a l (a,b,c) and point group (x,y,z) axes l a b e l l e d . I t i s c l a s s i f i e d as an asymmetric top because i t s three p r i n c i p a l moments of i n e r t i a , I a, I b and I c, are a l l d i f f e r e n t . N02 belongs to the point group, whose character table i s shown i n Figure 2.13, and which has four symmetry species, A 1 # A2, B l f and B2. The el e c t r o n i c states can be c l a s s i f i e d according to how the el e c t r o n i c wavefunction transforms under the symmetry operations of the point group. In the ground state of N02, the 17 valence electrons are i n the following o r b i t a l s [22] (3a 1) 2(2b 2) 2(4a 1) 2(3b 2) 2(lb 1) 2(5a 1) 2(la 2) 2(4b 2) 2(6a 1) 1 which leads to an e l e c t r o n i c state with symmetry A^ The spin of the unpaired electron (S=^) leads to a spin m u l t i p l i c i t y (2S+1) of 2, so that the ground state i s 2At. The Walsh diagram for XY2-type molecules (Figure 2.14), which shows how the energies of the o r b i t a l s change with bond angle, i s consistent with N02 being bent i n i t s ground state. The reason i s that the unpaired electron i n the ai=irn o r b i t a l has a much lower energy i n the bent configuration, and outweighs the e f f e c t s of the four electrons i n the na o r b i t a l which are more 33 z b X c Figure 2.12 N02 with i n e r t i a l (a,b,c) and point group (x,y,z) axes shown. C2v E C2 o"v(xz) o"v' (yz) A l 1 1 1 1 A2 1 1 -1 -1 B l 1 -1 1 -1 B2 1 -1 -1 1 Figure 2.13 character table. 34 Figure 2.14 Walsh diagram appropriate to N02 [after reference 31]. 35 stable i n the l i n e a r configuration. In the v i s i b l e region, there are three possible e l e c t r o n i c states which can be reached through absorption of radiat i o n : 1st excited state . . . (la 2) 2(4b 2) 1 (6ai) 2 2Ba 2nd excited state ... ( l a 2 ) 1 (4b 2) 2(6a 1) 2 2A2 3rd excited state . . . (la 2) 2(4b 2) 2(2b 1) 1 2 B i The e l e c t r o n i c t r a n s i t i o n studied i n t h i s work goes to the f i r s t excited e l e c t r o n i c state, 2 B 2 , and i s shown i n Figure 2.14 by a dotted l i n e . The zero nuclear spins of the two equivalent oxygen atoms i n N02 lead to alternate r o t a t i o n a l l e v e l s being absent. S p e c i f i c a l l y , the sign of the t o t a l wavefunction must remain unchanged under rotation by 180° around the symmetry (z) axis (z) (C2 ) as t h i s i s the 'equivalent r o t a t i o n ' to exchanging the two oxygen n u c l e i . This means that the product of e l e c t r o n i c , v i b r a t i o n , rotation and nuclear spin wavefunctions must transform as Ax or A2. Since the ground state v i b r o n i c wavefUnction i s A 1 ; the product of the rot a t i o n and nuclear spin wavefunctions must transform as Aj or A2. The nuclear spin wavefunctions are c l a s s i f i e d under a sub-group of which i s C2 E A l l B 1 - 1 and the numbers of A and B symmetry spin wavefunctions are 36 given by [23] n A s y * = (21+1) (1+1) (2.5) n ^ y a = (21+1)1 Since nBsym=0, only r o t a t i o n a l l e v e l s with Ai or A2 symmetry can e x i s t . This i s shown i n Figure 2.15 where the l e v e l s that do not e x i s t have been cross-hatched. The l e v e l s are l a b e l l e d according to standard asymmetric top notation, v , where N i s the r o t a t i o n a l angular momentum quantum number and Ka and \%. are i t s projections onto the i n e r t i a l a and c axes respectively. Since N02 i s a near prolate asymmetric top, the Ka and Kc notation w i l l be dropped i n favour of using K, the projection of N onto the a-(prolate) axis. In the upper 2 B 2 v i b r o n i c state, the same symmetry conditions apply, the r e s u l t being that only Bx and Rj r o t a t i o n a l l e v e l s e x i s t . 2.3.2 Molecular Hamiltonian To f i t spectroscopic data, i t i s necessary to obtain expressions for the energy l e v e l s of the system being studied. This i s done by solving the time-independent Schrodinger equation H* = E* (2.6) using the appropriate Hamiltonian operator (H) and basis functions (*) to solve for the energy l e v e l s (E). For the 37 3 a / / / / / / 3j2 -303 / / / / / / 3 U / / / / / / 3 1 3  '11 2 a / / / / / / 2 a / / / / / / •02" l o i • • / / / / / / O o o K=0 l i o / / / / / / In K = l K = 2 Figure 2.15 E x i s t i n g r o t a t i o n a l l e v e l s i n the ground el e c t r o n i c state of N02. 38 systems studied, the e f f e c t i v e Hamiltonian can be written Heff - Hrot + + Hgj. + Hhfs (2.7) where Hrot i s the ro t a t i o n a l Hamiltonian, H^ i s the ro t a t i o n a l c e n t r i f u g a l d i s t o r t i o n term, Hsr i s the electron spin-rotation i n t e r a c t i o n and Hhfs i s the hyper fi n e i n t e r a c t i o n . The r o t a t i o n a l Hamiltonian for an asymmetric top can be written i n the form [24] Hrot = A N 2 + B N X + C N Y (2.8) where N X, N Y and N Z are components of the r o t a t i o n a l angular momentum operator (N) referred to the molecule-fixed axis system. The ro t a t i o n a l constants, A, B and C, are defined i n terms of the p r i n c i p a l moments of i n e r t i a as h h h A = B = C = (cm-1) 8 7 T 2 C l A 8 7 T 2 C l B 8 7 T 2 C l c The quartic c e n t r i f u g a l d i s t o r t i o n terms have been shown by Watson [25] to be Hcdl = - A ^ - A ^ ^ - A K N ^ - S H ^ ^ (2.9) where the ce n t r i f u g a l d i s t o r t i o n constants are A„, A ^ , A K , 5H and 6K. Several higher order c e n t r i f u g a l d i s t o r t i o n terms (sextic and octic) were also included, based on t h e i r requirement i n previous N02 work [26], and these are 39 ^ /V fi A 4 A 9 A 9<Vl A f i A o Hcd2 = HKNZ + H K H N / N 2 + H N K N . V + HMN - LjN,8 ( 2 . 1 0 ) HK, H K H , H N K , H K(sextic) and Lxfoctic) are the d i s t o r t i o n constants. The ce n t r i f u g a l d i s t o r t i o n terms are required since the r o t a t i o n a l Hamiltonian i s based on the r i g i d rotor approximation which does not allow the bonds of a molecule to stretc h while i t rotates. The electron spin-rotation Hamiltonian may be written [24] A A A A A A A A A A A A A A A A A Hsr = -a0(NxSx+NySy+N zS2)-a(2N zSz-N xSx-NySy)-b(NxSx-N yS y) ( 2 . 1 1 ) where Van Vleck's spin-rotation parameters (a, a 0 and b) [27] are related to the diagonal elements of the spin-rotation tensor by 9o = ( - 1 / 3 ) ( exx+ £yy+£zz) a = ( - 1 / 6 ) ( 2 e z z - e x x - £ y y ) ( 2 . 1 2 ) b = ( exx- £yy) The f i n a l term i s the hyperfine Hamiltonian operator, which has three terms a r i s i n g from interactions of the 14N nuclear spin ( 1 = 1 ) , [28] xyz A 1 xyz A Hhfs = a c I « S + E TiiliSi + E xi i l i 2 ( 2 . 1 3 ) i 2 1 ( 2 1 - 1 ) i The f i r s t term i s the Fermi contact i n t e r a c t i o n , which involves the overlap of the nuclear spin wavefunction with that of the electron spin. The second term i s the electron 40 spin-nuclear spin dipolar i n t e r a c t i o n , which treats the two spins as in t e r a c t i n g bar magnets. The t h i r d term i s the nuclear quadrupole i n t e r a c t i o n which i s needed because the nucleus has a spin greater than \. This term i s the in t e r a c t i o n of the nuclear quadrupole moment with the e l e c t r i c f i e l d gradient i n the molecule. The constants involved i n the hyperfine i n t e r a c t i o n are a c, the Fermi contact i n t e r a c t i o n constant, T^, Tyy and T^, the magnetic dipole-dipole i n t e r a c t i o n constants i n the p r i n c i p a l axis system; and X z z / X y y and X x x , the nuclear quadrupole coupling constants, also i n the p r i n c i p a l axis system. 2 . 3 . 3 M a t r i x Elements For N02 the most appropriate set of basis functions i n which to calcu l a t e the matrix elements of the Hamiltonian (in other words, that which gives the smallest off-diagonal elements) i s c a l l e d Hund's case ( b w ) . This corresponds to the physical s i t u a t i o n where there i s no f i r s t - o r d e r s p i n - o r b i t coupling, and where the electron spin-rotation i n t e r a c t i o n i s larger than the hyperfine e f f e c t s . The vector coupling scheme i s written N + S = J J + I = F (2.14) for which a coupling diagram i s given i n Figure 2.16. The case (b w) basis functions are written | NKSJIFMF>, where K i s the projection of the ro t a t i o n a l angular momentum N along the 4 1 Figure 2.16 Vector diagram of Hund's coupling case (bBJ) . 42 near-symmetric top axis, J i s the vector sum of N and S, and F i s the t o t a l angular momentum including nuclear spin; the space-fixed projection quantum number MF i s not needed i n t h i s work since no external f i e l d s are involved. The quantum numbers I and F are only needed for the hyperfine structure. The matrix elements of the e f f e c t i v e Hamiltonian i n a case (bBJ) basis are given below. In t h i s basis the asymmetric top r o t a t i o n a l Hamiltonian (Equation 2.8) has matrix elements of the form AK=0,±2, AN=0. The ro t a t i o n a l and electron spin f i n e structure matrix elements have been taken from reference 26, while the hyperfine matrix elements have been taken from reference 29. In those cases where primes (') are used to indicate a change i n quantum number, the se l e c t i o n rules from the 6-j and 9-j symbols are shown following the matrix element. <NJSK | Hrot+Hcd+Hsr | NJSK> = \ (B+C) N (N+l) + [A-^ (B+C) ] K2 - h[J(J+1)-N(N+1)-3/4] x [ A ° A { N(N+1) N(N+1) ] - A K K 4 - A H K N ( N + 1 ) K 2 - A N N 2 ( N + 1 ) 2 + H K K 6 + H^K'N (N+1 ) + H N K K 2 N 2 (N+1 ) 2 + H„N 3 (N+1) 3 " L g K 8 3K2 , , K* (2.15) <NJSK±2 | Hrot+H^+Hsr | NJSK> = < * ( B - C ) -*b[J(J+l)-N(N+1) - 3/4]/[N(N+l) ]-6"„N(N+l) (2.16) - ^5 K[K 2+(K±2) 2] } [N(N+1)-K(K±1) ]* X [N(N+1)-(K±l)(K±2)] 43 < N - U S K | H R O T + H C D + H S R | N J S K > = (3a/2-^uI^) KC^-k 2] */N ( 2 . 1 7 ) < N - l f J S K ± 2 | f i W T + H C D + H B P | N J S K > = ±^b [ N ( N + l ) -K ( K ± l ) ] * ( 2 . 1 8 ) x [ (NTK-1) (N=FK-2) ]*/N < N K S J ' I F | H F E R A L | N K S J I F > = (-1) N + S* J' (-1) J + I + F + 1 [ ( 2 J • + 1 ) ( 2 J + l ) ]* x [ S ( S + 1 ) ( 2 S + 1 ) I ( I + 1 ) ( 2 1 + 1 ) ] * ( 2 . 1 9 ) x ( i J 1 F ! \S J ' N I I J i i j (J s l j with A J = 0 , ± 1 AF=0 <N'K'SJ'IF|Hdip|NKSJIF> = - 7 3 0 gg„/x,/i«(-l) J + I * F X [(2J'+1)(2J+1)(2N'+1)(2N+1)]* (2.20) x [1(1+1)(2I+1)S(S+1)(2S+1)]* f l J ' F7 ( N » N 2 ( J I 1 \ - J S S 1 J ( J 1 J 1 E (-1)N'"K' / N ' 2 N \ T Q 2 ( C ) q l-K' q Kl W i t h AJ=0,±1 AN=0,±1,±2 AK=0,±2 [ (1+1) (21+1) (21+3) ]*[ (2J'+1) (2J+1) (2N'+1) (2N+1) ]* (2.21) <N'K,SJ,IF|Hquad|NKSJIF> = (-1) J + I + F(-1) N' + s + J [I (21-1) ] x (1+1 (21+1) (21+3) ]*[(x / l J ' F\ J N 1 J» S \ \j I 2 J ( J N 2j x E (-1)B'"K' / N ' 2 N \ T q 2(VE) q \-K' q Kj W i t h AJ=0,±1 AN=0,±1,±2 AK=0,±2 In the above matrix elements, several parameters have not been defined. These are n, the leading c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n to f a [26], and the dipolar and quadrupole coupling 44 constants which are related to the previous constants by ggNMcMNT02(C) = *sTB ggNjLi6MNT±22(C) = (1/724) (T^-Tyy) (2.22) eQT02(VE) = ^xzz eQTj22(VE) = (1/724) ( xxx " Xyy) Since t h i s study of N02 was undertaken as a t e s t of the c a l i b r a t i o n and molecular beam systems, the amount of data c o l l e c t e d was not extensive. Therefore the data was f i t t e d i n the following manner. The r o t a t i o n a l and hyperfine constants determined by Perrin et a l . [30] from a wide c o l l e c t i o n of microwave and infrared t r a n s i t i o n s , were used with the above matrix elements to determine the ground state energy l e v e l s . These energy l e v e l s were then used with the t r a n s i t i o n frequencies l i s t e d i n Appendix I to obtain upper state energy l e v e l s . These energy l e v e l s were then f i t t e d using much s i m p l i f i e d matrix elements. These matrix elements contain a ro t a t i o n a l term, TH, a Fermi contact parameter, b,,, a dipolar parameter, T^, and an electron spin-rotation parameter, i = i i ( e x x + e y y ) = 2 ( a _ a o ) • T n e matrix elements are <NKJF|HtenB|NKJF> = T„ (2.23) <NKJF | Hferai | NKJF> = [F (F+l)-J (J+l)-2 ] (2.24) 4J(J+1) X [N(N+1)-J(J+1)-3/4] 45 <NK,J ,=J±l /F|H f e i m i|NKJF> = (2.25) b^7(F+J+2)(J+l-F)(F+J-l)(F+2-J)(N+J+3/2)(J+^-N)(N+J-^)(N+3/2-J) 4 j / 4 J 2 - l <NK, J=N+ ^  , F | Hdip | NK, J=N+ \ , F> = (2.26) -^T Z 2[N 2+2N+ll/4-F(F+l) ] [SK^-NfN+l) ] (N+l)(2N+1)2N+3) <NK, J=N- ^  , F | Hdip | NK, J=N- \ , F> = (2.27) ^T22[N2+7/4-F(F+l) ] [3K2-N(N+1) ] N(2N-1)(2N+1) <NK, J=N- ^  , F | Hdip | NK, J=N+ \ ,Y> = (2.28) -T22[3K2-N(N+1) ]7(N+F+5/2) (N-F+3/2) J(N+F-^) (F-N+3/2) 8N(N+1)(2N+1) <NK, J=N+^,F|Hsr|NK, J=N+^,F> = -%7N (2.29) <NK, J=N- Hsr | NK, J=N- h ,F> = hi (N+l) (2.30) The r o t a t i o n a l term values were then further f i t t e d to the expression T„ = T0 + BN(N+1) - DN2(N+1)2 (2.31) to determine T 0 (the band o r i g i n ) , B (the r o t a t i o n a l constant) and D (the c e n t r i f u g a l d i s t o r t i o n constant) for the upper state. 46 2 . 3 . 4 S e l e c t i o n R u l e s and I n t e n s i t y C a l c u l a t i o n s Because more l i n e s appear i n the spectra than are expected from the s e l e c t i o n rules for case (b w) coupling, i t was necessary to calculate the r e l a t i v e i n t e n s i t i e s of the hyperfine components i n order to assign the spectra. In case (b w) coupling the expected t r a n s i t i o n s follow the s e l e c t i o n rules [16] AN=0,±1, AJ=0,±1 (except AJ^O, AN^O for K=0); AN=AJ; AK=0,±1; AF=0,±1; AS=0 For t r a n s i t i o n s between K=0 states these s e l e c t i o n rules lead to R branches (AN=+1) and P branches (AN=-1). An example of the 'allowed' t r a n s i t i o n s i s shown i n Figure 2.17, where the s o l i d l i n e s indicate the R(4) t r a n s i t i o n s (4 being the ground state N value). The Fx and F 2 notation r e f e r s to the two electron spin components, J=N+^ and J=N-^, respectively, of a r o t a t i o n a l l e v e l N. The 'extra' l i n e s are spin-forbidden t r a n s i t i o n s between Ft and F 2 l e v e l s . They break the AN=AJ se l e c t i o n rules and are shown as dotted l i n e s i n Figure 2.17. The assignment of the r o t a t i o n a l quantum number N for band 99 was taken from previous work (references 17 and 18) while the assignment of band 115 followed c l o s e l y that of band 99; i n order to assign the J and F quantum numbers, the r e l a t i v e l i n e strengths of the hyperfine components had to be 47 Figure 2.17 Shown are the observed allowed R(4) t r a n s i t i o n s ( s o l i d lines) and the forbidden t r a n s i t i o n s (dotted l i n e s ) . 4 8 calculated, as follows. The matrix elements of the e l e c t r i c dipole moment were calculated using [29] <N'J'SKIF1 |/i|NJSKIF> a (-1)J"+F (-1) r + J + 3 / 2 (-1)"'~K x x [ (2F'+1) (2F+1) ]*[ (2J'+1) (2J+1) ]* [(2N'+1)(2N+1)] (2.32) x (j« F' l l ( N ' J ' h I / N' 1 N 1 [F J l j J N 1 I-K1 0 K These matrix elements are eith e r zero or very small f o r AN=£AJ t r a n s i t i o n s but, because the electron spin and hyperfine s p l i t t i n g s i n the ground state are comparable i n s i z e , the matrix elements of the hyperfine Hamiltonian of the type Aj=±l,AN=AF=0 cause a departure from 'pure' case (bSJ) coupling. The ground state wavefunctions become mixed such that the J=N-^,F 2 l e v e l takes on some of the character of the J=N+^,Fj l e v e l of the other f i n e structure component. The degree of mixing i n the ground state was calculated by se t t i n g up 2x2 matrices (H) i n the case (bBJ) basis f o r the p a i r s of i n t e r a c t i n g F l e v e l s , using the spin-rotation, Fermi contact and di p o l a r matrix elements (equations 2.23-2.30) and diagonalizing these to obtain eigenvalues (A) and eigenvectors (S) : H S = A where S = cosfl - s i n * sinfl cos» (2.33) These eigenvectors were then used to take appropriate l i n e a r combinations of the basis states (tf1*818) to form the 49 e i g e n f u n c t i o n s (* e i 8 e n) c o s « s ine = - s i n e costf ptosis T1basis _ 2 (2.34) The m a t r i x o f case (b^) t r a n s i t i o n moments g i v e n by e q u a t i o n 2.32 was t h e n t r a n s f o r m e d t o the b a s i s g i v e n by the e i g e n f u n c t i o n s i n t h e upper and lower s t a t e s , i . e . ^ = S upperi basi S c.-l M b lower (2.35) I t was assumed t h a t no h y p e r f i n e m i x i n g o c c u r s i n the upper s t a t e so t h a t S u p p e r i s a u n i t m a t r i x i n t h i s c a s e . The r e l a t i v e l i n e s t r e n g t h s a r e the squares o f the e lements o f j i f i n a l , and i t <•>»/ was found t h a t the AN/AJ t r a n s i t i o n s a c q u i r e c o n s i d e r a b l e i n t e n s i t y as a r e s u l t o f t h e h y p e r f i n e m i x i n g o f t h e e l e c t r o n s p i n components. A sample c a l c u l a t i o n i s shown i n Appendix I I . W i t h the a i d o f t h e s e i n t e n s i t y c a l c u l a t i o n s i t was p o s s i b l e t o a s s i g n the ' e x t r a * l i n e s a s s o c i a t e d w i t h t h e h y p e r f i n e - i n d u c e d t r a n s i t i o n s . 2 . 4 Results The main aim o f t h i s a n a l y s i s was t o check t h a t t h e c a l i b r a t i o n system worked and t o see t h e q u a l i t y o f s p e c t r o s c o p i c d a t a t h a t t h e m o l e c u l a r beam system c o u l d p r o d u c e . As ment ioned i n C h a p t e r 1, t h e ground s t a t e h y p e r f i n e s p l i t t i n g s measured from our d a t a agreed v e r y w e l l w i t h t h o s e o b t a i n e d from microwave d a t a . A compar i son i s shown i n T a b l e 2 . 1 . The r e s u l t s a r e r a t h e r i m p r e s s i v e , w i t h 50 Table 2.1 Comparison of microwave and o p t i c a l l y measured hyperfine s p l i t t i n g s i n N02. N U p p e r F J - N L o w e r F J T h i s W o r k ( M H z ) M i c r o w a v e D a t a ( M H z ) D i f f e r e n c e ( M H z ) 2 2*5 2k 2 2 ^ lk 3 1 . 9 6 3 2 . 6 9 a + 0 . 7 3 2 3k 2 2k 111.12 1 7 7 . 6 5 a - 0 . 0 7 2 lk 2 k 2 2k 3k 5 2 . 1 9 5 2 . 5 4 3 + 0 . 3 5 2 lk lk 2 lk 2k 4 0 . 3 2 4 0 . 3 4 3 + 0 . 0 2 2 lk k 2 lk lk 2 4 . 8 8 2 4 . 4 4 a - 0 . 4 4 0 k lk 0 k k 2 2 1 . 1 6 2 2 0 . 8 8 a - 0 . 2 8 6 6k 6 k 6 6k 5k 6 1 . 2 2 6 0 . 8 6 b - 0 . 3 6 6 6 k lk 6 6k 6k 1 2 7 . 8 9 1 2 8 . 9 8 b + 1 . 0 9 6 5k 6k 6 6k lk 1 8 3 . 6 5 1 8 2 . 8 3 b - 0 . 8 2 6 5k 5k 6 5k 6k 6 4 . 5 2 6 3 . 3 4 b - 1 . 1 8 a - R e f e r e n c e 3 2 . b - R e f e r e n c e 3 3 . Table 2.2 Comparison of a microwave and o p t i c a l l y measured r o t a t i o n a l i n t e r v a l i n N02 along with a calculated value. U p p e r L o w e r N F J - N F J T h i s W o r k ( M H z ) C a l c u l a t e d ( M H z ) M i c r o w a v e D a t a ( M H z ) D i f f e r e n c e ( m i c r o w a v e -o p t i c a l l y m e a s u r e d ) 2 lk k 0 k k 2 2k lk 0 k lk 7 6 2 2 1 . 1 1 ± 2 . 4 7 5 6 7 3 . 2 1 1 1 . 7 7 6 2 1 6 . 5 6 7 5 6 6 7 . 8 2 7 6 2 1 6 . 3 3 7 5 6 6 7 . 7 2 - 4 . 7 8 - 5 . 4 9 51 the average difference between the o p t i c a l and microwave data being j u s t greater than 0.5 MHz. A comparison of the s p l i t t i n g between two r o t a t i o n a l l e v e l s was also made and i s shown i n Table 2.2. The agreement i s not as close although the values obtained are within 6 MHz, which i s remarkably good. These data also served as a check of the ground state energy l e v e l c a l c u l a t i o n s , and one can see that there i s agreement to within a few tenths of a MHz between the microwave data and our calculated values. The constants obtained from the upper state f i t s for the two subbands are given i n Tables 2.3 and 2.4, along with the avail a b l e comparisons. The main conclusion that can be drawn from these r e s u l t s i s that although the upper state of N02 i s highly perturbed, there i s enough r e g u l a r i t y for meaningful parameters to be determined. Looking at the r e s u l t s of band 99, where other data were available, one sees a f a i r l y close agreement between parameters. Of the f i v e r o t a t i o n a l l e v e l s examined, one of them (N=5) i s quite badly perturbed, as shown by i t s Fermi contact and dipolar coupling constants. In band 115 the N=3 r o t a t i o n a l l e v e l i s perturbed; i n fact the N=3 Fj spin component i s doubled, with the two parts bracketing the F 2 component i n energy. The hyperfine structure has collapsed i n both cases such that the F = 2\, 3\ and Ah l e v e l s have no apparent s p l i t t i n g at our resolution (<2-3 MHz). The Fermi contact parameter i s the most highly affected at t h i s perturbation; the dipolar coupling constant shows Table 2.3a Molecular constants for upper state of N02/ band 99. N 1 T N (cm X) b (MHz) This Work Previous Results a b 1 16850.319250(12) 70.7(09) 66.5 3 16854.559929(10) 70.8(17) 5 16862.124221(10) 137.6(25) 7 16872.985741(10) 61.2(46) 57.0(52)° 9 16887.051546(10) 44.7(46) 37.9(38) N» T Z Z < M H Z ) i (MHz) This Work Previous Results This Work Previous Results 1 3 5 7 9 -9.9(12) -11.7(31) 158.0(47) -28.8(71) -60.0(91) -11.2 b 1452.07(48) 1411.09(18) 1427.97(11) 1443.65(8) 1451.30(6) .1409(420)*? 1469(180)° 1379(120) 1444.9° 1451.6 a - E r r o r l i m i t s are l a i n terms of the l a s t d i g i t quoted, b - Reference 34. c - Reference 35. d - Reference 17. Table 2.3b Upper state r o t a t i o n a l constants for N02, band 9 T 0(cm _ 1) B (cm X) D (cm - 1) This Work Previous Results 16849.4775(65) a 16849.48b,16849.8C * 0.42368(39) (0.427,0.421)°,0.4224 b 6.89(41)xl0~ 5 3.5xl0~ 5 b a - Error l i m i t s are l a i n terms of the l a s t d i g i t quoted, b - Reference 18. c - Reference 15. 53 Table 2.4a Molecular constants f o r upper state of N02/ band 115. N» T N (cm - 1) b^ (MHz) T 2 Z (MHZ) 7 (MHz) 1 3 3 5 7 9 17092.895578(12) a 17096.601492(11) 17097.295595(11) 17104.320836(10) 17115.056881(10) 17128.939538(10) 55.33(96) -51.4(18) -50.4(19) 63.0(27) 89.6(36) 50.4 (92) -13.4(13) -153.9(33) -151.9(35) 2.3(52) 60.3(70) -20.1(92) -613.15(49) 6640.64(18) -3763.52(18) 1643.47(11) 1654.74(8) 1307.93(7) a - Error l i m i t s are la i n terms of l a s t d i g i t quoted. Table 2.4b Upper state r o t a t i o n a l constants for N02, band 115. T Q (cm"1) B (cm"1) This Work Previous Results 17092.181(62) a 17092.3b 0.410(32) (0.434,0.394) b a b - Error l i m i t s are l a i n terms of l a s t d i g i t quoted. - Reference 15. 54 i r r e g u l a r i t i e s f or a l l the observed r o t a t i o n a l l e v e l s , and the electron spin-rotation parameter i s most obviously perturbed for the N=l and N=3 l e v e l s . The negative electron spin-r o t a t i o n parameter for N=l i s a r e s u l t of the F 2 l e v e l l y i n g above the Ft l e v e l s . In conclusion, the use of N02 as a t e s t molecule has shown that not only does the c a l i b r a t i o n system work to a high degree of accuracy, but also that the molecular beam apparatus i s capable of producing very high resolution e l e c t r o n i c spectra which allow one to see very f i n e d e t a i l s of the energy l e v e l s of the molecule. 55 C h a p t e r 3 A p p l i c a t i o n t o the R o t a t i o n a l S t r u c t u r e o f the B0 2 M o l e c u l e 3 . l I n t r o d u c t i o n Despite our success at making molecular beams of N02 i t has proved to be very d i f f i c u l t to make molecular beams of unstable t r a n s i t i o n metal-containing r a d i c a l s such as TiN and NbN, using a microwave discharge source, and under conditions that normally led to strong laser-induced fluorescence spectra. I t was possible to produce TiN molecules i n a discharge flame i n front of the skimmer, but the number density and l i f e t i m e of these molecules was such that very few got through the skimmer and into the detection region; therefore fluorescence signals were not obtained. The more stable molecule B02, which can be made i n a discharge through BC13 and 02, was chosen for further study of the source c h a r a c t e r i s t i c s and discharge conditions necessary to get molecules of t h i s type into a molecular beam. I t has indeed proven possible to obtain LIF spectra of B02 i n a beam, but before any high resolution work could be accomplished i t was found necessary to do a survey spectrum and r o t a t i o n a l analysis of the B02 v i s i b l e band systems by using a lower resolution technique. Not only did t h i s study determine the frequencies of the B0 2 t r a n s i t i o n s more accurately, but i t also pinpointed other areas of inte r e s t such as upper state 56 p e r t u r b a t i o n s . The a c q u i s i t i o n o f d a t a over a l a r g e wave number r e g i o n , some 300 cm"1, has s u p p l i e d a f u r t h e r demanding t e s t o f the c a l i b r a t i o n sys tem. The B0 2 m o l e c u l e , which i s an a l m o s t - s t a b l e l i n e a r t r i a t o m i c m o l e c u l e w i t h an u n p a i r e d e l e c t r o n , i s an e x c e p t i o n a l l y good " t e s t i n g ground" f o r t h e t h e o r y o f such sys tems . The o n l y r o t a t i o n a l a n a l y s e s have been t h e o r i g i n a l f l a s h p h o t o l y s i s work by Johns [36] i n 1961, i n f r a r e d d i o d e l a s e r s p e c t r a t a k e n by Kawaguchi e t a l . [37,38] i n the e a r l y 1980's and r e c e n t F o u r i e r t r a n s f o r m i n f r a r e d s p e c t r a r e c o r d e d by M a k i e t a l . [ 39 ] . Our r e s o l u t i o n i s about 100 t imes b e t t e r t h a n Johns was a b l e t o a c h i e v e w i t h g r a t i n g s p e c t r a ( e s t imated a c c u r a c y ± 0 . 0 2 cm"1 [ 3 6 ] ) , and a l t h o u g h not as s e n s i t i v e as the Kawaguchi e t a l . o r Maki e t a l . r e s u l t s , a much more complete a n a l y s i s has been done. I n t h i s work the (0,0) and 2,1 bands o f t h e A 2 ^ and X 2 ^ t r a n s i t i o n n e a r 5400 A have been measured. The (0,0) band had been p r e v i o u s l y a n a l y z e d by Johns [36] ; we have however c o m p l e t e l y r e s o l v e d a l l the band head t r a n s i t i o n s and c a r r i e d t h e ass ignments out t o h i g h e r J ( t o t a l a n g u l a r momentum quantum number) . The i n f r a r e d d i o d e l a s e r [37,38] and F o u r i e r t r a n s f o r m i n f r a r e d s p e c t r a [39] i n v o l v e d o n l y t h e ground s t a t e , X ^ g , where c o n s t a n t s f o r a l l the v 2 = l v i b r o n i c l e v e l s i n v o l v e d i n our t r a n s i t i o n s have been d e t e r m i n e d , a l t h o u g h a g a i n our d a t a ex tend t o h i g h e r J . Two o f t h e f o u r upper s t a t e v 2 = l l e v e l s were a n a l y z e d by Johns b u t t h e o t h e r two 57 l e v e l s have been a n a l y z e d f o r the f i r s t t ime i n t h i s work. 3.2 Experimental Details 3.2.1 B02 Production U n l i k e N0 2 , the B0 2 f r e e r a d i c a l s had t o be a c t i v e l y p r o d u c e d . T h i s was done by f l o w i n g s m a l l amounts o f BC1 3 and 0 2 , i n A r c a r r i e r gas , t h r o u g h a microwave d i s c h a r g e i n t o a cube - shaped meta l f l u o r e s c e n c e c e l l ( the ' c u b e ' ) , as shown i n F i g u r e 3 . 1 . The d i s c h a r g e e m i t t e d a c h a r a c t e r i s t i c p a l e green glow a s s o c i a t e d w i t h B0 2 chemi luminescence . D u r i n g t h e r u n n i n g o f the e x p e r i m e n t , the p r e s s u r e was kept n e a r 700 mTorr t o produce maximum l a s e r - i n d u c e d f l u o r e s c e n c e i n t e n s i t y . The f l u o r e s c e n c e spec trum was p r o d u c e d by c r o s s i n g t h e f low o f BOj m o l e c u l e s w i t h two c o u n t e r - p r o p a g a t i n g l a s e r beams from t h e same dye l a s e r system used f o r N0 2 , and was m o n i t o r e d u s i n g a PMT p l a c e d a t r i g h t a n g l e s t o b o t h t h e l a s e r beams and the BO2 f l o w . Data were c o l l e c t e d over t h e range 18215 cm"1 t o 18390 cm"1 and 18444 cm"1 t o 18500 cm"1 u s i n g t h e l a s e r dye rhodamine 560. 3.2.2 Intermodulated Fluorescence I n C h a p t e r 2, t h e t h e o r y o f l a s e r - i n d u c e d f l u o r e s c e n c e was d e s c r i b e d . The method o f c o l l e c t i n g s u b - D o p p l e r s p e c t r a o f BO2 was t h e t e c h n i q u e o f i n t e r m o d u l a t e d f l u o r e s c e n c e ( IMF) , f i r s t d e s c r i b e d by Sorem and Schawlow [40 ] . F i g u r e 3 .1 shows pump PMT M i c r o V A X c o m p u t e r l o c k - i n HeNe l a s e r t o c a l i b r a t i o n BS A r * i o n l a s e r r i n g l a s e r ~7m F i g u r e 3 . 1 B l o c k d i a g r a m s h o w i n g i n t e r m o d u l a t e d f l u o r e s c e n c e e x p e r i m e n t . 00 59 the flow of B02 molecules crossed by two l a s e r beams, one modulated at frequency f t, and the other, t r a v e l l i n g i n the opposite d i r e c t i o n through the sample, modulated at frequency f 2 . As i n Chapter 2, the two lase r beams in t e r a c t with molecules that have opposite Doppler s h i f t s with respect to the axis of the lase r beam. I f the fluorescence i n t e n s i t y modulated at either f t or f 2 i s recorded, a Doppler-broadened l i n e p r o f i l e i s obtained. However i f the fluorescence i n t e n s i t y modulated at frequency (f!+f2) i s recorded, the sign a l obtained i s a sub-Doppler IMF signal because the molecules i n t e r a c t simultaneously with the f i e l d s of both l a s e r beams. This signal i s a non-linear saturation e f f e c t , and there i s also a component modulated at f i ~ f 2 [19]; the advantage of using the sum frequency f t+f 2 i s that i t avoids the low frequency amplitude noise of the l a s e r . The IMF data c o l l e c t e d for B02 had linewidths of approximately 100 MHz, mainly due to pressure broadening, and therefore no hyperfine structure was resolved. 3.2.3 Calculation of Frequencies The B02 spectrum was scanned i n 3 0 GHz i n t e r v a l s , of which an example i s shown i n Figure 3.2. Interferometer markers spaced by 750 MHz and 150 MHz were recorded simultaneously with the B02 fluorescence s i g n a l . The 150 MHz markers were used s o l e l y to check for laser mode hops, which show up as i r r e g u l a r i t i e s i n the marker spacing. Because of the large 60 F i g u r e 3.2 A sample 30 GHz B0 2 s c a n . The 150 MHz markers a r e no t shown due t o t h e i r h i g h d e n s i t y . The a s s i g n e d t r a n s i t i o n s have been l a b e l l e d w i t h t h e v i b r o n i c symmetry , a s u p e r s c r i p t i n d i c a t i n g t h e b o r o n i s o t o p e , and a b r a n c h l a b e l . The b r a n c h l a b e l i s R o r P (AJ=+1 o r AJ=-1) w i t h the s p i n component as a s u b s c r i p t when r e q u i r e d and t h e a s s o c i a t e d J " o r N" v a l u e i n p a r e n t h e s e s . 61 number of 750 MHz markers, wave numbers from the Burleigh wavemeter were only recorded for a few markers near the beginning and end of each scan. The order numbers of these markers were then calculated using a rearranged version of equation 1.1, along with the wave number of one marker near the beginning of the scan. The data channel numbers of the 750 MHz markers were found by the computer, which also calculated t h e i r wave numbers. These wave numbers were then f i t t e d to a seventh degree polynomial i n order to determine an equation for the frequency i n terms of channel number. The channel numbers of the B02 t r a n s i t i o n s were found using the cursor method (see Section 2.2.9) and the polynomial was then used to convert these channel positions into wave numbers. 3.3 Theory 3.3.1 Background B02 i s a l i n e a r symmetric triatomic molecule belonging to the point group D^; the character table i s given i n Figure 3.3. I t has 21 electrons, of which the outer eleven occupy the following o r b i t a l s i n the X 2^ ground state . . . K ) 2 ( a u ) 2 ( 7 r u ) V g ) 3 . . . x 2 i y In the f i r s t excited state, A2!!,,, an electron has moved from ir„ to 7rg, giving D , ooh I 20* oo 2 C ^ OO a h c o C 2 0 0 0 V 2 S ™ co 2S2<° 00 i 2 + + 1 + 1 +1 +1 + 1 + 1 + 1 + 1 +1 9+ 2 u + 1 + 1 +1 -1 -1 + 1 -1 -1 -1 2 + 1 + 1 +1 +1 -1 -1 + 1 + 1 +1 g_ 2 u + 1 + 1 +1 -1 + 1 -1 -1 -1 -1 n g +2 2cos«3 2cos2p -2 0 0 -2cos^ -2cos2p +2 n u +2 2cos^> 2cos2p +2 0 0 +2cosp +2cos2y> -2 A g +2 2cos2p 2cos4io +2 0 0 +2cosp +2cos4*3 +2 A U +2 2cos2»? 2cos4p -2 0 0 -2cos2y? -2cos4p -2 Figure 3.3 The D,^ point group character table. 63 ... (o- B) 2(a u) 2(7r u ) 3 ( 7 r B ) 3 ...A2nu The 'u' and 'g' subscripts are abbreviations for 'ungerade' and 1gerade 1, which mean antisymmetric and symmetric with respect to inversion. Rough contour diagrams of the ira and irg o r b i t a l s are shown i n Figure 3.4. The four v i b r a t i o n a l degrees of freedom i n B02 are the symmetric stretch, v w the antisymmetric stretch, v 3, and the doubly degenerate bending motion, v 2; the stretching modes are not dealt with i n t h i s analysis. When a l i n e a r triatomic molecule bends, the v i b r a t i o n a l degeneracy i s l i f t e d as the D«,h symmetry i s broken. I f there i s no o r b i t a l degeneracy the two components of the bending v i b r a t i o n become the a-axis rotation and the non-degenerate bending motion of the bent molecule. However i f the e l e c t r o n i c state i s o r b i t a l l y degenerate the same must occur for both o r b i t a l components; the r e s u l t i s that there are two e l e c t r o n i c p o t e n t i a l curves, one for each o r b i t a l component, which touch at the l i n e a r configuration, but which are otherwise separate, as shown i n Figure 3.5. These two e l e c t r o n i c p o t e n t i a l curves represent two d i f f e r e n t e l e c t r o n i c states of the bent molecule, which l i e very close i n energy and inter a c t strongly through the a-axis rotation. This constitutes a major breakdown of the Born-Oppenheimer separation of electron and nuclear motion, and i t s r e s u l t s are loosely c a l l e d the Renner-Teller e f f e c t [41]. F i g u r e 3 . 4 R o u g h c o n t o u r d i a g r a m s o f n g a n d n u o r b i t a l s . 65 Figure 3.5 The pote n t i a l energy curves separating as a l i n e a r molecule bends. 66 F i g u r e 3.5 r e p r e s e n t s the s i t u a t i o n f o r the ground s t a t e o f B0 2 . The upper c u r v e has A 2 symmetry ( i n symmetry n o t a t i o n ) c o r r e s p o n d i n g t o the e l e c t r o n c o n f i g u r a t i o n ( a 1 ) 2 ( b 2 ) 2 ( b 1 ) 2 ( a 1 ) 2 ( a 2 ) ( b 2 ) 2 , and c o r r e l a t e s w i t h i n the LV p o i n t g r o u p . The lower c u r v e has ^ symmetry, c o r r e s p o n d i n g to ( a 1 ) 2 ( b 2 ) 2 ( b 1 ) 2 ( a 1 ) 2 ( a 2 ) 2 ( b 2 ) , and c o r r e l a t e s w i t h a Z u * . The d i s c u s s i o n o f the o r b i t a l a n g u l a r momentum e f f e c t s g i v e n here assumes t h a t the a m p l i t u d e o f the b e n d i n g mot ion i s s m a l l , so t h a t the s p l i t t i n g o f the two p o t e n t i a l c u r v e s shown i n F i g u r e 3 .5 can be c o n s i d e r e d as a p e r t u r b a t i o n o f the d o u b l y - d e g e n e r a t e harmonic o s c i l l a t o r model f o r t h e bend ing v i b r a t i o n . A s s o c i a t e d w i t h the bend ing mot ion o f a l i n e a r m o l e c u l e i s a v i b r a t i o n a l a n g u l a r momentum, which can be thought o f as a r i s i n g from the a d d i t i o n o f t h e two o r t h o g o n a l m o t i o n s , one i n the xz p l a n e and one i n the yz p l a n e , w i t h a 9 0 ° phase d i f f e r e n c e , as shown i n F i g u r e 3 . 6 . The v i b r a t i o n a l a n g u l a r momentum i s denoted by 6 and i t s p r o j e c t i o n onto the i n t e r n u c l e a r a x i s i s c a l l e d I. In a degenerate e l e c t r o n i c s t a t e t h i s v i b r a t i o n a l a n g u l a r momentum c o u p l e s w i t h the e l e c t r o n i c a n g u l a r momentum, L, as f i r s t d i s c u s s e d by Renner [ 4 1 ] . The p r o j e c t i o n o f L onto t h e i n t e r n u c l e a r a x i s i s A , and a new quantum number K=A+I i s i n t r o d u c e d t o l a b e l the v i b r o n i c ( t h a t i s , e l e c t r o n i c and v i b r a t i o n a l ) energy l e v e l s . I f one i n c l u d e s t h e e l e c t r o n s p i n , t h e t o t a l i n t e r n a l a n g u l a r momentum, A+Z+Z, i s c a l l e d P . The p a t t e r n o f v i b r o n i c energy l e v e l s f o r a ^ e l e c t r o n i c s t a t e i s shown i n F i g u r e 3 . 7 . The 67 Figure 3.6 The degenerate bending motion of B02. A 90° out-of-phase addition causes t h i s to look l i k e v i b r a t i o n a l angular momentum. 68 Figure 3.7 The pattern of vib r o n i c energy l e v e l s formed when the bending v i b r a t i o n of a l i n e a r symmetric triatomic molecule i n a 2nz e l e c t r o n i c state i s excited [36]. 69 notation used for l a b e l l i n g the vibronic l e v e l s i s V i V 2 v 3 with the v i b r o n i c symmetry ^Kp, and the inversion symmetry, g or u, i n parentheses. Therefore one has l e v e l s such as 000( 2n 3 / 2 g) or 010 (2EU~) . The ground and f i r s t excited e l e c t r o n i c states are inverted s p i n - o r b i t states such that the 2n 3 / 2 l e v e l l i e s below the ^ l e v e l , i . e . the higher P states l i e lower i n energy. The t r a n s i t i o n s which have been studied i n t h i s t h e s i s are shown i n Figure 3.8; t h e i r l i n e frequencies are l i s t e d i n Appendix I I I . The 22~ - 2S + t r a n s i t i o n i s forbidden i n pure case (b) coupling, but the spin-orbit i n t e r a c t i o n i n the A2nu state of B02 mixes the vibronic l e v e l s associated with the two p o t e n t i a l s i n Figure 3.5, with the r e s u l t that the two 010(2S) states are more accurately described as n=h states i n case (c) coupling; i n consequence the 010 (22~) - 010 (2S+) t r a n s i t i o n takes on some of the character of a h-h t r a n s i t i o n i n case (c) coupling, so that i t becomes weakly allowed. 3.3.2 Hund's Coupling Cases Five possible ways i n which the angular momenta of a molecule may be coupled have been described by Hund. These represent f i v e choices of basis functions f o r the c a l c u l a t i o n of molecular energy l e v e l s , and are now c a l l e d Hund's coupling cases (a) to (e) . The B02 spectra reported i n t h i s thesis give examples of cases (a) to (c). Vector diagrams for case (a) coupling are shown i n Figure 3.9. I f the v i b r a t i o n a l angular momentum i s equal to F i g u r e 3 . 8 E n e r g y l e v e l d i a g r a m s h o w i n g t h e t r a n s i t i o n s s t u d i e d . 71 Figure 3.9 Vector diagram of Hund's coupling case (a) with a) G = 0, b) 6 ^ 0 . 72 zero, as i n the 000(2n)-000(2n) band, Figure 3.9a applies. The electron o r b i t a l and spin angular momenta, L and S, are strongly coupled to the internuclear axis and then further coupled to the r o t a t i o n a l angular momentum, R, to form the t o t a l angular momentum, J . Thus case (a) can be regarded as J = L + S + R (3.1) The sum of the projections of L and S i s A+S=n, the projection of J onto the internuclear axis. In Figure 3.9b, the v i b r a t i o n a l angular momentum i s not zero ( i . e . when the bending v i b r a t i o n i s excited) and the o r b i t a l , v i b r a t i o n a l and electron spin angular momenta are a l l coupled to the internuclear axis. The projection quantum numbers K and P (described previously) are shown. As i n Figure 3.9a, these angular momenta couple with the rotation to give the t o t a l angular momentum, J , i n t h i s case written as J = L + G + S + R (3.2) Both equations 3.1 and 3.2 represent case (a) coupling, where the coupling of the angular momenta to the internuclear axis takes precedence over any couplings between the momenta themselves. The basis functions for case (a) can be written |nA;SS;GZ;JP> where n i s the e l e c t r o n i c state. In Hund's coupling case (b), the electron spin angular momentum i s only weakly coupled to the internuclear axis such that i t s projection can no longer be defined. A vector 73 diagram i s shown i n Figure 3.10 and the coupling may be written L + G + R = N N + S = J (3.3) The basis functions can be written |t?;NKSJ>. Case (b) applies to S e l e c t r o n i c states where there i s no o r b i t a l angular momentum to couple the spin to the axis, or i n the case of B02, when the coupling of L and G i s such that a S vi b r o n i c state a r i s e s . However, case (b) also can ar i s e at large J values i n case (a) states, as the electron spin uncouples from the internuclear axis with increasing rotation. Hund's case (c) coupling arises when the spi n - o r b i t coupling i s so large that the o r b i t a l , v i b r a t i o n a l and electron spin angular momenta couple together to form what w i l l be termed J a as shown i n Figure 3.11. This intermediate angular momentum then couples with the r o t a t i o n a l angular momentum to give the t o t a l angular momentum, J . This i s written as L + G + S = J a J a + R = J (3.4) One can no longer s t r i c t l y define the projections A, I or E; however the t o t a l angular momentum s t i l l has a projection P as in case (a) coupling. 74 Figure 3.10 Vector diagram of Hund's coupling case (b). 75 Figure 3.11 Vector diagram of Hund's coupling case (c). 76 3.3.3 R o t a t i o n a l L e v e l s and S e l e c t i o n R u l e s B02 i s s i m i l a r to N02/ having two equivalent oxygen atoms with zero nuclear spin which lead to the absence of alternate r o t a t i o n a l l e v e l s . For the 2 vibro n i c l e v e l s t h i s implies that only odd or even values of N e x i s t ; these are shown i n Figure 3.12 for the v 2=l l e v e l s of the ground state. The opposite occurs i n the upper state because the symmetry with respect to inversion i s g rather than u. The Fi/F 2 notation i s the same as that used for N02 while the e/f notation labels l e v e l s of even and odd p a r i t y following the convention introduced by Brown et a l . [42]. This notation i s useful because i t leads to straightforward s e l e c t i o n rules, and the Hamiltonian matrix factors into two submatrices which are i d e n t i f i a b l e with the e and f l e v e l s . The s e l e c t i o n rules for e l e c t r i c dipole t r a n s i t i o n s are [42] AJ=0, e-H-f (3.5) AJ=±1, e«e, f«f whereas perturbations follow AJ=0, e<+e, f ~ f (3.6) For n and A vibronic l e v e l s , i t i s found that alternate lambda (A)-doubling components are missing, such that only one l e v e l e x i s t s f o r each J value. A-doubling i s a further breakdown of the Born-Oppenheimer approximation; i t arises when rot a t i o n does not allow the o r b i t a l angular momentum to N 77 . i t / a F t < e ) F 3 ( f ) 7 / a B/2 F l F a ( e ) ( f ) S/2 1/2 F i F 2 (e) ( f ) -t>/2 F x ( f ) •7/2 F 2 (e) 6/2 Ft ( f ) 8/2 F 2 (e) 2v " 1/2 F t ( f ) Figure 3.12 Rotational l e v e l s which e x i s t i n the ground state S vib r o n i c l e v e l s . 78 be s t r i c t l y quantized along the internuclear axis, so that the +A and -A components of a degenerate state have s l i g h t l y d i f f e r e n t energies. Each J l e v e l therefore s p l i t s into two A-components, one of which i s missing because of the zero spins of the equivalent oxygen atoms. The ro t a t i o n a l l e v e l s which e x i s t f o r the ground state are shown i n Figure 3.13. The absence of alternate A-components leads to a "staggering" i n the appearance of some of the branches i n the B02 spectrum. 3.3.4 Case (a) H a m i l t o n i a n and M a t r i x E lements For the ground state A vibronic l e v e l s and a l l the n v i b r o n i c l e v e l s case (a) basis functions were used, with the e f f e c t i v e Hamiltonian operator being A A A A A A A Heff = Hgo + Hrot + HCD + HCJKJ) + HLD + HLDCD (3.7) In t h i s equation Hgo i s the spin-orbit i n t e r a c t i o n , Hrot i s the ro t a t i o n a l Hamiltonian, i s the ce n t r i f u g a l d i s t o r t i o n correction to the ro t a t i o n a l Hamiltonian, Hs^ i s the ce n t r i f u g a l d i s t o r t i o n correction to the spi n - o r b i t A , A i n t e r a c t i o n , HLD i s the A-doubling operator and HLDCD i s i t s c e n t r i f u g a l d i s t o r t i o n correction. The s p i n - o r b i t i n t e r a c t i o n , H^ = AL-S, can be taken as Ha = AL ZS 2 (3.8) where the contributions from the x and y components of L are dropped as they are o f f diagonal i n el e c t r o n i c state. The J J 79 11/2 (f) 9/2 (e) 7/2 — — — — — ( f ) 5/2 (•) 3/2 (f) 9/2 _ (e) 7/2 (f) 5/2 —(e) 3/2 ( f ) 1/2 (e) 13/2 ( e ) 1 1 / 2 "> 9/2 (e) 1 1 / 2 <f> 7 / 2 (f) 9/2 ( e ) *'* C> 3/2 (f) 7/2 _ _ _ _ _ ( f ) 5/2 ( e ) 2 2 *S /2 A3/2 F i g u r e 3.13 R o t a t i o n a l l e v e l s which e x i s t i n t h e ground s t a t e n and A v i b r o n i c l e v e l s . The dashed l i n e s r e p r e s e n t t h e m i s s i n g l e v e l s . 80 sp i n - o r b i t constant i s A. The r o t a t i o n a l term i s A A n A A A A o H W T = B R = B(J-L-G-S) (3.9) where B i s the ro t a t i o n a l constant. Afte r some algebra, the terms that are diagonal i n A and the v i b r a t i o n a l quantum numbers may be written A /N O A O A 9 A O A A A /\ Hrot = B(J 2-J Z 2+S 2-3 Z 2) - B ( J + S . + J . S + ) (3.10) The c e n t r i f u g a l d i s t o r t i o n Hamiltonian i s A A A A A A Ho = -D(J-L-G-S) (3.11) where the matrix representation of the angular momentum term i s taken as the square of the c o e f f i c i e n t matrix for B from equation 3.10. The c e n t r i f u g a l d i s t o r t i o n correction to A, described with a parameter AD, arises because the spin-or b i t coupling i s a function of the bond length; t h i s correction i s HSOCD = ^AD[ (J-L-G-S) 2,L ZS Z] + (3.12) The square brackets and plus sign indicate that the anticommutator, [a,b] + = ab+ba, must be taken i n order for the Hamiltonian matrix to remain Hermitian. The A-doubling i n a 2n state can be treated by assuming there i s an e f f e c t i v e operator 81 HLD = -h (p+2q) (J+S_+J_S+) + ^q(J?+J-2) (3.13) acting between the |A=+I> and |A=-l> components. The determinable constants i n case (a) are q and p+2q. The c e n t r i f u g a l d i s t o r t i o n correction to the A-doubling i s determined i n the same manner as the correction to the sp i n - o r b i t coupling, that i s by taking the anticommutator of the A-doubling operator with the r o t a t i o n a l operator, such that HLDCD = hqnt (J+2+J.2) , (J-L-G-S) 2] + (3.14) A A A A A A A A o - h (pD+2qD) [ (J+S.+J.S+) , (J-L-G-S) 2] + The same combination of constants i s determinable: qj) and pD+2qD. The Hamiltonian matrices for the 2n and 2A v i b r o n i c states are shown i n Tables 3.1 and 3.2 respectively. The basis functions, which are l i n e a r combinations of the signed quantum number basis functions |A 2 J P>, have been written i n shorthand notation, they are: |2n3/2 e/f> = (1/72)( |1 -k j k> + I"! k J -k>) e/f> = (1/72){ |1 r *2 j 3/2> + I"! l 2^ J -3/2> } e/f> = (1/72) { |1 -k j 3/2> + I"! k J -3/2> } e/f> = (1/72) { |1 k J 5/2> + I"! -k J -5/2> } (3.15) The e and f p a r i t y labels can be dropped from the 2A functions since no A-doubling i s resolved. 82 Table 3.1 The Hamiltonian matrix for the 000 (^J vib r o n i c states of B02. The ± re f e r to the e and f l e v e l s respectively. The term value, T0, represents the o r i g i n of the vibronic state i X a e/f> | X e/f> T 0 + * A + B[(J+ J () a-21 - D[((J+^)2-2r+(J+^) !-l] + hAD[(J+h)-2] T hqs>{J+h) [ (J+h) - 1 ] -B[ (J+^) 2-l]*+2D[ ( J + ! s ) 2 - l ] 3 / 2 ± a«q(J+^) [(J+ * s) 2-i]\„ ± hq»(J+\) ] ( J + ^ ) 2 - l ] 3 / 2 ± \ (pD+ 2 q „ ) (J+h) [ (J+h)-!] symmetric T 0 - U + B ( J + ^ ) 2 - D [ ( J + ! i ) V ( J + ^ ) 2 - l ] - %Aa(J+5|)2 T !5(P+ 2 q ) (J+'i) =F *qb(J+*) [(J+^r-1] T li(PD+2qD) ( J + * ) O T Table 3.2 The Hamiltonian matrix for the X2^, 010 (2A) vibronic state of B02. The ± re f e r to the e and f l e v e l s respectively. The term value, T0, represents the o r i g i n of the vibronic state. | 2 A 5 / 2 e/f> | 2 A 3 / 2 e / f > < 2 A 5 / 2 e / f | T 0+^A+B[ ( J + y 2 - 6 ] - D [ ( J + ^ ) J - l l ( J + ^ ) 2 + 3 2 ] + ijAflt (J+^) - l l ( J + 1 i ) 2 + 3 2 ] -B[ (J+^) 2-4]*+2D[ ( J + h ) 2 - 4 ] 3 / 2 symmetric T 0-hA+B[ (J+3) 2-2] - D(J+!0 2 [ (J+^) 2-3] - hhoiiJ+hy-2] - *AH(J+fc) a[(J+5jr-3] 83 3.3.5 The H a m i l t o n i a n and M a t r i x E lements f o r Hund's case (b) The v 2=l 2E vibronic l e v e l s of the ground state were analyzed using Hund's case (b) basis functions. The e f f e c t i v e Hamiltonian operator i s A A A A A Heff = Hrot + Hrj) + HgR + Hsuco (3.16) A A where Hrot i s the ro t a t i o n a l Hamiltonian, Ho, contains quartic A and s e x t i c c e n t r i f u g a l d i s t o r t i o n corrections, Hgj i s the spin-r o t a t i o n i n t e r a c t i o n and Hgo i s i t s c e n t r i f u g a l d i s t o r t i o n correction. The r o t a t i o n a l Hamiltonian takes the form Hrot = BN2 (3.17) with the ce n t r i f u g a l d i s t o r t i o n corrections being Ho, = -DN4 + HN6 (3.18) The spin-rotation constant, 7 , represents the int e r a c t i o n between the electron spin angular momentum and the ro t a t i o n a l angular momentum. The operator form of t h i s i n t e r a c t i o n i s HSR = 7N-S (3.19) The c e n t r i f u g a l d i s t o r t i o n constant for the spin-rotation i n t e r a c t i o n , T d, has the simple form A A A AO H S R C D = T D(N'S ) i r 84 since N and 8 commute. The Hamiltonian matrix i s shown i n Table 3.3. The basis functions written as | Ft> and | F2> are |Fj> = |22, N 0 S J = H+h> (3.21) |F2> = | 2E, N 0 S J = N-%> I t i s i n t e r e s t i n g to note that i n case (b), the 2x2 matrix generated i s completely diagonal whereas i n case (a), the matrix has off-diagonal elements which require that the matrix be diagonalized before energy l e v e l s can be obtained. 3.3.6 The H a m i l t o n i a n f o r A 2 ! ^ , v 2 = l , and i t s M a t r i x Elements The most in t e r e s t i n g r e s u l t i n t h i s thesis i s the complete analysis of the manifold of spin and vibr o n i c l e v e l s belonging to the bending fundamental of the A 2n u state. The A2!!,, state of B02 i s unique among the triatomic molecules that have been studied so fa r because i t s spin-orbit coupling i s va s t l y larger than the e l e c t r o s t a t i c (or Renner-Teller) s p l i t t i n g between the o r b i t a l components. The consequences of t h i s are highly unusual, and allow much more information to be extracted from the spectrum than i s normally possible. A very d e t a i l e d derivation of the Hamiltonian and i t s matrix elements w i l l therefore be given. O r b i t a l angular momentum e f f e c t s are best introduced using the language for a bent molecule. Omitting electron 85 Table 3.3 The Hamiltonian matrix for the X 2n G,010( 2Z) vibronic states of B 0 2 . The term value, T0, represents the vibro n i c o r i g i n . I Fi> | F 2 > T 0 + B N ( N + 1 ) - D N 2 ( N + 1 ) 2 + H N ^ N + l ) 3 + 3 5 T D N 2 ( N + 1 ) 0 0 T 0 + B N ( N + 1 ) - D N 2 ( N + 1 ) 2 + H N 3 ( N + 1 ) - ^ ( N + l ) „ - J S 7 D N ( N + 1 ) 2 86 spin f o r the moment, the t o t a l angular momentum, J , i s the sum of the r o t a t i o n a l angular momentum, R, and the electron o r b i t a l angular momentum, L, J = R + L (3.22) which gives R - J - L (3.23) The r o t a t i o n a l Hamiltonian i s then Hrot = AR22 + BRV2 + C % 2 (3.24) = (AJz2+B x^2+c5y2) - (2I&jLz+2B3jtx+2c3yi\) + (AL,,2 + BL^2 + CL\2) Because the a-axis rotation of a bent molecule turns into one component of the degenerate bending v i b r a t i o n i n the l i n e a r molecule, i t forms a spe c i a l case, which must be separated from the b- and c-axis rotations which are s t i l l present i n the l i n e a r l i m i t . The f u l l Hamiltonian should then be written / \ A A A H— (Hbending+Hg-axis rotation) "'"H^- and c-axis rotations (3.25) Now, as explained above, an el e c t r o n i c n state of a l i n e a r molecule can be described as two Born-Oppenheimer po t e n t i a l curves, representing two d i f f e r e n t e l e c t r o n i c states, which touch at the l i n e a r configuration (see Figure 3.5). Each of these two bent-molecule Born-Oppenheimer states i s an 'ordinary' non-degenerate e l e c t r o n i c state, with i t s own 87 bending Hamiltonian H = hP 2 + ^ AQ2 + . . . (3.26) In t h i s equation Q i s the bending coordinate ( e s s e n t i a l l y , 180° minus the bond angle at the central atom, m u l t i p l i e d by a mass f a c t o r ) , P i s i t s conjugate momentum and A i s a force constant. The complication caused by the o r b i t a l angular momentum i s that these two Born-Oppenheimer states are coupled by the cross-term A A A ^electronic-rotation = ~2hJzLz (3.27) from equation 3.24 which has matrix elements between them. This may seem unimportant at f i r s t sight, but the term becomes enormous near the l i n e a r l i m i t , since the r o t a t i o n a l constant A i s i n f i n i t e when the molecule i s exactly l i n e a r . The bending and a-axis r o t a t i o n a l motions of the two e l e c t r o n i c states that touch i n the l i n e a r l i m i t to become a n state must be represented by a 2x2 matrix, H = |*> |*"> (3.28) i P ^ A V + A f l t f + A 2 ) -2AKA -2AKA ^P2+^A"Q2+A(K2+A2) The basis functions | of equation 3.28 are products of three factors: 88 i) An e l e c t r o n i c factor for the o r b i t a l motion of the electrons, *** = HTTP (ei(* ± e ™ ) = (2)-\ (|A> ± |-A>) (3.29) where * i s the angle between the averaged p o s i t i o n of the unpaired electron and the plane of the bent molecule. The sums and differences are needed so that these factors transform c o r r e c t l y under the symmetry operations of the point group, and i n fact the product of t h e i r i r r e d u c i b l e representations i s the same as that of l^, which i s why the operator of equation 3.27 couples them. i i ) A harmonic o s c i l l a t o r v i b r a t i o n a l function, which can be written |v>, and i i i ) An a-axis r o t a t i o n a l factor |K> = ( 2 7 r ) ~ V K * (3.30) where x i s the angle between the plane of the bent molecule and an a r b i t r a r y reference plane. At t h i s point the Renner-Teller e f f e c t can be seen to a r i s e n a t u r a l l y as an a-axis r o t a t i o n a l coupling between two bent-molecule e l e c t r o n i c states that happen to become degenerate when the molecule i s l i n e a r . However, when electron spin i s considered the spin-orbit coupling i s o f f -diagonal i n the basis of equation 3.28, and adds a term Aspin"orbit<Lz>2 to the e l e c t r o n i c - r o t a t i o n a l coupling (see equation 3.8). I t then becomes preferable to transform the e l e c t r o n i c 89 factors 9 E L £ to a new basis | A = ±1> = ( 2 7 r ) " V i # (3.31) by taking t h e i r sums and differences. In the new basis, equation 3.28 becomes H = h^+k (A++A~)Q2+A(K+A)2 h (A*-A~)Q2 ^ (A+-A')Q2 hP2+h (A++A-)Q2+A(K-A)2 (3.32) The terms i n equation 3.32 are these. h^+h (A*+A~)Q2 i s the bending Hamiltonian for a non-degenerate e l e c t r o n i c state where the pot e n t i a l i s the mean of the two Born-Oppenheimer potentials forming the n state; ^(A+-A")Q2 i s the difference between the two Born-Oppenheimer potentials, and the terms A(K±A) 2 are the angle-dependent terms that convert the one-dimensional bending Hamiltonian into the two-dimensional harmonic o s c i l l a t o r Hamiltonian with the v i b r a t i o n a l angular momentum quantum number I given by I = K-A (3.33) The new basis functions can therefore be written as products of |A=±I> e l e c t r o n i c factors and eigenfunctions f o r the two-dimensional harmonic o s c i l l a t o r , |vZ>; the matrix 3.32 i s of course diagonal i n K=Z+A, because i t i s a transformed version of equation 3.28. Returning to the v 2=l l e v e l of the state of B02, there are two d i f f e r e n t |K| values, 0 and 2, which a r i s e when A=±l 90 i s added to Z=±l. Two separate matrices are therefore required. The diagonal elements are very simple, being just the eigenvalues of the two-dimensional harmonic o s c i l l a t o r Hamiltonian, Ev.i = (va+l)« (3.34) where w i s the v i b r a t i o n a l frequency i n cm"1 u n i t s . The o f f -diagonal elements are f a i r l y complicated because, i n order to connect the v i b r a t i o n a l basis functions | v ' , Z ' = K - A > and | v , Z = K+ A > , the coordinate operator has to be a v i b r a t i o n a l ladder operator, which i s also assumed to act between the e l e c t r o n i c functions |A=+1> and A=-l>. The d e t a i l s of the c a l c u l a t i o n w i l l be found i n reference [43], though for the purposes of t h i s work the only important matrix element i s < A = l ; v , Z = K - l | \ ( A + - A ~ ) Q 2 | A=-l;v ,Z=K+l> = \zw [ (v+l^-K 2]* (3.35) In equation 3.35 the experimental parameter e i s defined as e = ( A + - A ~ ) / ( A + + A ~ ) . (3.36) The elements off-diagonal i n v can be treated by perturbation theory [43], although f o r K=0 they sim p l i f y to give the exact energy expression E(v,K=0) = w(v+l)(l±e)* (3.37) Equation 3.37 shows that the K=0 energy l e v e l s correspond to the two Born-Oppenheimer potentials making up the n e l e c t r o n i c 9 1 state, so that for t h i s special case there i s no v i b r o n i c coupling (as can also be seen from equation 3 . 2 7 , since the eigenvalue of J 2 i s zero) . The part of equation 3 . 3 2 that i s diagonal i n v for K=0 i s given by equations 3 . 3 4 and 3 . 3 5 : |A=l;v=l,Z=-l> |A=-l;v=l,Z=l> H(K=0) = 2u> 2w ( 3 . 3 8 ) The matrices for K=±2,v=l collapse to l x i ' s , since there are no basis functions with v=l,|Z | = 3 . The v i b r o n i c l e v e l s |K|=2,v=l are therefore not affected by v i b r o n i c coupling except i n higher order, as a r e s u l t of the elements o f f -diagonal i n v, and l i e near the centre of the manifold of states i n t h i s v i b r a t i o n a l l e v e l . The next step i s to introduce the spin- o r b i t coupling. Another factor, | S 2 > , where Y>=±\, i s added to the basis set, and the e f f e c t s of the spin-orbit operator, equation 3 . 8 , are seen to be as follows. Equation 3 . 3 8 becomes two separate, but i d e n t i c a l , matrices A=±l;v=l , Z= Tl;Z=s±*s> I A = T 1 ; V = 1 , Z = ± 1 ; S = ± ^ > H(K=0, 2 ) = 2u/+^A ( 3 . 3 9 ) while the energies of the K=2 ( A) components are E(K= 2 , A) = 2u.±^A ( 3 . 4 0 ) 92 Already i t can be seen that i f A i s much larger than ew, as i n the A2nn state of B02, there w i l l be two groups of spin-vibronic l e v e l s , each consisting of a 2Z state and a component of the 2A, separated by the spin-orbit coupling (see Figure 3.7) F i n a l l y , the b- and c-axis rotations, from equation 3.25, are included. In the l i n e a r l i m i t the r o t a t i o n a l Hamiltonian i s written A A A A A . 9 Hrot = B(J-L-G-S) 2 as i n equation 3.9, and i t s matrix elements diagonal i n A, v and I come from the s i m p l i f i e d form, equation 3.10. As before the r-dependence of the ro t a t i o n a l and spin-or b i t parameters, B and A, produces ce n t r i f u g a l d i s t o r t i o n e f f e c t s , which have been described i n equations 3.11 and 3.12. The major difference i n the ro t a t i o n a l l e v e l structure of the bending fundamental compared to the zero-point l e v e l l i e s i n the A-type and Z-type doublings. The A-type doubling Hamiltonian, equation 3.13, i s s t i l l present but i t s e f f e c t s are usually masked because i t i s diagonal i n Z and therefore acts between the A component and the two S components. Similar considerations usually apply to the Z-type doubling, but the A2!^ state of B02 i s not a "usual" case. Z-type doubling has not been discussed before i n t h i s t h e s i s ; i t i s a v i b r a t i o n a l C o r i o l i s e f f e c t which i s caused by the x- and y-components of the v i b r a t i o n a l angular momentum, G. The expansion of the ro t a t i o n a l Hamiltonian, equation 3.9, 93 produces many terms which are omitted from equation 3.10 because they are off-diagonal i n A, v and Z . Among these are -B(J +G_+J_G\) and B(G\§.+G_S +) . The matrix elements of G\ are of the type AV 2=AV 3=±1,AZ=±1, so that i n second order they connect the two degenerate basis l e v e l s for the bending fundamental, |v=l,Z=l> and |v=l,Z=-l>, through the antisymmetric stretching v i b r a t i o n , v 3. In the bending fundamental of a lS molecule such as C02 the r e s u l t i s a s p l i t t i n g of the r o t a t i o n a l l e v e l s , according to the formula Aw = q ^ f j + l ) (3.41) where q v i s the Z-type doubling constant. ( S t r i c t l y , of course, what i s observed i n COo, i s a staggering, since h a l f the l e v e l s are missing because of the zero nuclear spins of the oxygen atoms.) The Z-type doubling can be considered as a r i s i n g from a transformed Hamiltonian, H , whose matrix elements [44] are <v2,Z |H(2)|v2,Z±2> =?iqvy(v2TZ) (v2±Z+2) VJ(J+l)-P(P±1) (3.42) x yj(j+i)-(p±i)(P±2) This equation i s derived by the application of second-order degenerate perturbation theory to the matrix elements of -B(J*G_+J_G +) ; the derivation also gives an expression f o r q v i n terms of the v i b r a t i o n a l frequencies and C o r i o l i s constants, 94 ->t> 2 T 2 2 2B J 2 k w 2 q v = [1 + 4 2 — — ] (3.43) w2 k=l ,3 wk -w2 Another matrix element, s i m i l a r to equation 3.42, ari s e s i n mult i p l e t e l e c t r o n i c states [44] from the cross-term between A A A A A A A A -B(J+G_+J_G+) and B(G+S.+G.S+) : <v 2,Z;Z±l|H ( 2 ) |v2,Z±2;i:> =-*sqvy(V2=FZ) (v2±Z+2) (3.44) x yj(j+i)-(p±i)(p±2) ys(s+i)-s(s±i) Both these elements are present i n ^  states though, as explained above, they are not normally needed. The f u l l Hamiltonian matrices required f o r the v 2=l l e v e l s of the A ^ state of B02 are given i n Table 3.4. There are eight basis functions f o r each J i n the general case, but the 8x8 matrix that r e s u l t s can be fact o r i z e d into two 4x4 matrices by transforming to the p a r i t y basis |JP±> = 2"*[|A v I S;JP> ± | —A v -I -2;J,-P>] (3.45) The two 4x4 matrices correspond to the e and f p a r i t y l e v e l s , and only one of them i s needed f o r any p a r t i c u l a r J value, because of the zero spins of the oxygen n u c l e i . Written out i n d e t a i l the eight basis functions are 95 Table 3.4 The Hamiltonian matrix for the A 2^ 010 2A and 2S vibronic states. The ± re f e r to the e and f par i t y l e v e l s respectively. ±> 1^ 3/2 ±> | 2 S l o « r ±> | 2 S u p p e r ± > ( B 4 + % A D [ ( J + i s ) 2 - 6 ] - ( P L + ^ A H ) r (J+ii) 4 -11(J+^) +32] + i s A(l - 3 / 4 e " i ) [ ( J + ^ ) 2 - 4 ] * { B t - 2 D J (J+1-!) - 4 ] } » i q » [ ( J + ^ ) a - i ] * [(J+»0 -•*] ± ^ q [ ( J + ^ ) 2 r i ] * [(J+*) -4 ] * (B^-'iADtCJ+is)2-^] - ( D a - J i A a ) (J+H) [ (J+*) -3 ] + T C E + A T s - 1 i A(l - 3/4e ) [ (J + ! s ) 2 - l ] * ( ± ! s q ( J+4 ) - q v ) [ ( J + S 5) 2-l ] * [ ± ^ ( P + 2 q ) + ! s q v ( J + i s ) ] symin* 2 t r i c B » ( J + * ) a - D j . [ ( J + * ) * + (J+!s) ] +TQJ + ! j A(l-%€ ) ± ! S 7 r . ( J + ^ ) + a 7 a : . - [ * ( B B . + B f r ) ] (J+h) + (DE.+Dj;_) (J+*) 3 ±eu3 B I - ( J + ! 5 ) 2 - D £ . [ ( J + i S ) 4 +(J+%) ] +TeB;-!sA(l-v«e:l) a = !s (J+'s) {J-h)* f o r e l e v e l s b = -\(J+h) ( J + 3 / 2 ) 2 f o r e l e v e l s = -*(J+%) (J+3 /2 ) f o r f l e v e l s = h(J+h) (J-h) 2 f o r f l e v e l s 9 6 | 2A 5 / 2,±> = 2~*{|1 1 1 h;J 5 / 2 > ± | - 1 1 - 1 -h ; J - 5 / 2 > } | 2 A 3 / 2 / ± > = 2"*{|1 1 1 -\;3 3 / 2 > ± | - 1 1 - 1 k;J - 3 / 2 > } ( 3 . 4 6 ) I Slower,±> = 2"*{ 1 1 l - l h;J h> ± | - i l l - % ; J -^>} | 22 u pper/±> = 2"*{|i l - l -h;J -\> ± | - l l l h;J h>) The n o t a t i o n 2 S u p p e r and 2 E i o w e r has been used i n s t e a d o f the more c o n v e n t i o n a l 2Xf and 2S* because the two s t a t e s a r e mixed so s t r o n g l y by s p i n - o r b i t c o u p l i n g t h a t t h e group t h e o r y l a b e l s a r e i n a p p r o p r i a t e ; the 2 S s t a t e s behave as n=h s t a t e s i n st case (c) c o u p l i n g . An i n t e r e s t i n g r e s u l t o f t h i s work i s t h a t i t was n e c e s s a r y t o add a s p i n - r o t a t i o n i n t e r a c t i o n term t o the d i a g o n a l e lements f o r the ^ s t a t e s . T h i s must r e p r e s e n t the i n t r i n s i c 7N«S i n t e r a c t i o n o f t h e e l e c t r o n i c s t a t e w h i c h , because o f t h e completeness o f the d a t a , can be s e p a r a t e d from t h e v e r y l a r g e apparent s p i n - r o t a t i o n i n t e r a c t i o n i n t h e 2 S v i b r o n i c s t a t e s t h a t i s caused by t h e p r e s e n c e o f t h e e l e c t r o s t a t i c s p l i t t i n g i n the o f f - d i a g o n a l e lement between them. N o r m a l l y i t i s no t p o s s i b l e t o de termine t h e parameters AD and i s e p a r a t e l y i n a ^ s t a t e by l e a s t s q u a r e s because they a r e a lmos t t o t a l l y c o r r e l a t e d [45] ; the A 2 ^ s t a t e o f B0 2 p r o v i d e s one o f t h e r a r e examples o f where t h i s c o r r e l a t i o n i s b r o k e n . The v i b r o n i c e n e r g i e s TQ^ and A T S A i n T a b l e 3.4 a r e d e f i n e d i n t h e energy l e v e l d iagram o f F i g u r e 3.14; i t has been n e c e s s a r y t o r e f e r a l l v i b r o n i c e n e r g i e s t o t h e X 2 ! ^ 0 1 0 Figure 3.14 D e f i n i t i o n of the vibronic energies TQS and used i n Table 3.4 98 s t a t e as z e r o , so t h a t the measured t r a n s i t i o n f r e q u e n c i e s c o u l d be used i n the l e a s t squares f i t t i n g . I t wou ld , n a t u r a l l y , have been p r e f e r a b l e t o use a s i n g l e B r o t a t i o n a l c o n s t a n t f o r a l l f o u r s p i n - v i b r o n i c components o f t h e A 2 ^ 010 l e v e l , but w i t h the p r e c i s i o n o f the d a t a , t h i s has no t been p o s s i b l e because o f s m a l l h i g h e r - o r d e r e f f e c t s . 3 . 4 Results 3 .4.1 Least Squares F i t t i n g The g o a l o f a s p e c t r o s c o p i c a n a l y s i s i s t o reduce a l a r g e body o f d a t a t o a s m a l l e r number o f parameters t h a t appear i n the H a m i l t o n i a n m a t r i x . T h i s i s an i t e r a t i v e p r o c e s s which b e g i n s w i t h e s t i m a t e s o f t h e c o n s t a n t s which a r e g r a d u a l l y improved upon . T h i s t y p e o f prob lem i s s o l v e d by s e p a r a t i n g t h e H a m i l t o n i a n i n t o two p a r t s : the m o l e c u l a r parameters and t h e i r ' s k e l e t o n ' m a t r i c e s which c o n t a i n t h e i r a n g u l a r momentum c o e f f i c i e n t s , H = X • B (3.47) ss ~ ss I n t h i s e q u a t i o n X i s t h e v e c t o r o f parameters and B i s the a s s o c i a t e d ' s k e l e t o n ' m a t r i x . The energy l e v e l s o f t h e system a r e o b t a i n e d by d i a g o n a l i z a t i o n o f H a c c o r d i n g t o ss U T H U = E (3.48) SS SS SS SS where E c o n t a i n s t h e e i g e n v a l u e s and U i s a m a t r i x o f e i g e n v e c t o r s . The c a l c u l a t i o n s a r e done by a computer 99 programme which gives U, E and the residuals A E (observed minus calculated energies for each data point). These r e s i -duals are used to obtain corrections to the parameters, A X , by making use of the Hellmann-Feynman theorem. This states [46] SEi/SX = /¥ i *(6"H7cSX )* i dr ( 3 . 4 9 ) or, f o r a single parameter, X D , cSEi/cSXn = [ U T ( 6 H / 5 X , ) U ] 1 1 = D i B ( 3 . 5 0 ) The derivatives matrix, D , gives the dependence of the energy l e v e l s on changes i n the parameters, according to A E = D • AX ( 3 . 5 1 ) The l e a s t squares corrections to the estimates of the parameters, A X , as a function of D and the residuals A E , are r-j r^j AX = ( D T D ) _ 1 • D T • A E ( 3 . 5 2 ) where the extra matrix algebra i s needed because D i s a rectangular matrix which cannot be inverted d i r e c t l y . These corrections, A X , can be added to the appropriate parameters to be used i n the next i t e r a t i o n . The process i s continued u n t i l the residuals are reduced to the l e v e l of the experimental p r e c i s i o n . How well a set of energies or t r a n s i t i o n s f i t s a model i s measured by the standard deviation, a, f o r unweighted data defined by 100 a = (AE)T(AE) n-m (3.53) where n i s the number of independent measurements and m i s the number of parameters being determined. The f i t t i n g of the data was done i n two steps. F i r s t l y the ground state combination differences (A2F") were f i t t e d to obtain ground state constants. A combination difference A 2F"(J) i s the difference i n energy between an R(J-l) t r a n s i t i o n and a P(J+1) t r a n s i t i o n which gives the energy separation of the l e v e l s J - l and J+l i n the lower state; t h i s i s shown i n Figure 3.15. With the ground state constants determined, the ground state energies can be calculated and the l i n e frequencies used to determine the upper state constants. 3.4.2 Numerical Results and Discussion The r o t a t i o n a l constants for the X 2^ 000 l e v e l , as determined from f i t s to the combination differences, are given i n Table 3.5 f o r the boron-11 isotope and Table 3.6 for the boron-10 isotope. Constants determined by previous workers are also given for comparison. I t i s d i f f i c u l t to make d i r e c t comparisons as the matrix elements used i n previous works [37-39] d i f f e r from ours. Our diagonal B parameter corresponds to t h e i r B+^q, which means that the values agree to within experimental error. The form of the off-diagonal elements Figure 3.15 A A 2F"(J') combination difference. 102 Table 3.5 Molecular constants for the X 2^ 000 l e v e l of UB0 2, i n cm"1. Constant This Work Maki et a l . T391 A [-148.58]3 [-148.58]3 B 0.3293697(18)b 0.3293926(73) D 1.3207(53)x(10)~ 7 1.3117(130)xl0~ 7 AD -3.12(22)xl0~ 5 -1.322(166)xlO- 5 q -5.15(65)X10~ 5 -4.29(135)xl0~ 5 p+2q 5.76(17)xl0~ 3 5.670(235)xl0~ 3 a - Fixed to value given i n Reference [36]. b - Error l i m i t s are 3a i n units of the l a s t figure quoted. Table 3.6 Molecular constants for the X 2^ 000 l e v e l of 10BO2, i n cm"1. Constant This Work Maki et a l . T391 A [-148.58]3 [-148.58]3 B 0.3293521(17)b 0.3293815(15) D 1.3148(56)X10~ 7 1.297(29)xlO" 7 AD -3.65(15)xl0~ 5 -1.401(263)xl0~ 5 q -5.12(64)X10" 5 -5.46(254)X10~ 5 p+2q 5.84(14)X10~ 3 5.70(38)xlO" 3 a - Fixed at value given i n Reference [36]. b - Error l i m i t s are 3a i n units of the l a s t figure quoted. 103 involving B i s s l i g h t l y d i f f e r e n t ; t h i s confuses the correspondance of the AJ/AD terms (AD=2AJ) i n such a way that ^AD should be somewhat more negative than t h e i r Aj, which i s as we f i n d i t . Given our larger range of J values and extremely low standard deviations of 0.00030 cm"1 and 0.00026 cm"1 f o r UB0 2 and 10BO2 respectively, our r o t a t i o n a l constants for the 000 i 2!!) l e v e l s are a d e f i n i t e improvement over past f i t s . Our data do not however allow the spin-orbit parameter to be refined and thus Johns' value of -148.58 cm"1 remains the best available. The upper state f i t s f or A 2^, 000 (hi) were not as straight forward and the standard deviations could not be reduced as far as those of the ground state. The f i n a l nB0 2 f i t has a = 0.00154 cm"1; that f o r 10BO2 i s s l i g h t l y better, at 0.00141 cm"1. The f i n a l e f f e c t i v e constants are given i n Tables 3.7 and 3.8 for UB0 2 and 10B02 respectively, along with a v a i l a b l e comparisons. The residuals obtained are shown plott e d against J i n Figure 3.16. Several data points are widely scattered and were not included i n the f i t ; these are not i n c o r r e c t l y assigned l i n e s but r e f l e c t random perturbations i n the state caused by higher v i b r a t i o n a l l e v e l s of the ground state. The density of ground state l e v e l s was calculated, using Haarhoff's formula [47], to be 0.86 |K|-states per wavenumber at 18,300 cm"1, which i s e a s i l y high enough to cause the number of random perturbations shown i n Figure 3.16. Although the ce n t r i f u g a l d i s t o r t i o n corrections to the A-doubling parameters are not well 104 Table 3.7 Molecular constants for the A X 000 l e v e l of nB0 2, i n cm"1. Constant This Work Johns [36] T o 18291.59659(56)a 18291.5g A -101.28126(94) -101.3 Q B 0.31073912(95) 0.3106 D 1.2375(28)xl0 _ V 1.2xl0~ 7 AD -3.589(65)xl0"5 q -3.06(58)xl0 p+2q 1.1826(76)xl0~ 2 1.1(19)xl0~ 9 P D + 2 C 3 D -7.1(35)xl0~8 a - Error l i m i t s are 3a i n units of the l a s t figure quoted. Table 3.8 Molecular constants for the A X 000 l e v e l of 10BO2, in cm"1. Constant This Work T o 18313.23753(53) a A -101.35670(89) B 0.3107332(10) D 1.2345(37)xio" 7 AD -3.951(73)xl0~ 5 q -3.04(19)xl0 p+2q 1.1798(36)xl0" 2 a - Error l i m i t s are 3a i n units of the l a s t figure quoted. 105 a) ® a 0.0010 « > •o *> a 3 U -H « > u V » O « a : 0.0005 0.0000 •0.000* -0.0010 © 9.5 19.5 29.5 39.5 © 49.5 59.5 S -o.oio 59.5 F i g u r e 3.16 R e s i d u a l s from the upper s t a t e f i t o f t h e 1^ r o v i b r o n i c l e v e l s p l o t t e d a g a i n s t J ' f o r a) B0 2 and b ) 1 0 BO2. C i r c l e d d a t a p o i n t s i n d i c a t e a s s o c i a t e d t r a n s i t i o n s which nave been e x c l u d e d from the f i n a l f i t . T h e c r o s s e s r e p r e s e n t t h e TI 3 / 2 r e s i d u a l s w h i l e t h e diamonds r e p r e s e n t the *nl/2 r e s i d u a l s . 106 d e t e r m i n e d i n these f i t s , they a r e d e f i n i t e l y r e q u i r e d (as seen by t h e f a c t they lower the s t a n d a r d d e v i a t i o n ) . The c o n s t a n t s f o r the X 2 ^ , 010 2 A and 22 s t a t e s a r e g i v e n i n T a b l e 3 . 9 . On ly the boron-11 i s o t o p e was a n a l y z e d as the s p e c t r a r e c o r d e d d i d not a l l o w f o r a complete boron-10 a n a l y s i s . The d a t a s e t c o n s i s t e d o f the r o t a t i o n a l c o m b i n a t i o n d i f f e r e n c e s from the f o u r s p i n - v i b r o n i c s t a t e s , t o g e t h e r w i t h the s e p a r a t i o n s o f c o r r e s p o n d i n g J l e v e l s i n the 2 S U + , 2SU" and 2 A 5 / 2 u s t a t e s . These s e p a r a t i o n s can be o b t a i n e d because the weakly a l l o w e d A - X 010-010 2 Z g ~ - band has the same upper s t a t e as the s t r o n g 22g~ - 2SU" band and because l o c a l r o t a t i o n a l p e r t u r b a t i o n s between the A ^ 010 2 E g + and 2 A 5 / 2 i g upper s t a t e s cause v a r i o u s l i n e d o u b l i n g s which appear i n b o t h the 2 2 g + - 2SU* and 2 A 5 / 2 i g - 2 A 6 / 2 | 0 subbands; the s e p a r a t i o n s o f t h e s e groups o f d o u b l e d l i n e s a r e the s e p a r a t i o n s o f t h e l e v e l s o f t h e 2 S U + and 2 A 6 / 2 u l ower s t a t e s . However t h i s d a t a s e t does not a l l o w t h e r e l a t i v e p o s i t i o n o f t h e 2 A 3 / 2 n l e v e l t o be f i x e d . The p o s i t i o n o f the 2 A 3 / 2 „ l e v e l was l a t e r d e t e r m i n e d by an i t e r a t i v e p r o c e d u r e ; t h i s i n v o l v e d f i x i n g the lower s t a t e s p i n - o r b i t c o n s t a n t and i n t r o d u c i n g a p a r a m e t e r , i n the upper s t a t e f i t which a l l o w e d t h e e n e r g i e s o f t h e X 2 ^ , 010 2 a 3 / 2 . U r o t a t i o n a l l e v e l s t o f l o a t w i t h r e s p e c t t o t h e o t h e r ground s t a t e v 2 = l v i b r o n i c l e v e l s . The r e a s o n why t h i s has t o b e done f r o m t h e u p p e r s t a t e f i t i s t h a t a l t h o u g h t h e 2 A 3 / 2 i g u p p e r l e v e l i n t e r a c t s w i t h t h e nearby 2 X g " u p p e r l e v e l t h e o n l y e f f e c t o f t h e p e r t u r b a t i o n i s t o produce anomalous K - t y p e 107 Table 3.9 Molecular constants for the X*ns 010 l e v e l s of nB0 2, i n cm"1. Constant This Work Kawaauchi et a l . r 3 7 - 3 8 ! V B 0.3306791(34) a 0.330700(25) D 1.420(30)xl0~ 7 1.53(13)xl0~ 7 H -9.3(80)X10~ 1 3 7 0.15225(14) 0.15222(115) 2.160(64)X10~ 6 d -5.34(84)xl0" 7 d 2 " T0 228.1937(36) B 0.3303520(39) 0.33037696(69) D 1.244(48)X10~ 7 1.326(80) H -1.17(19)xio" 1 2 7 0.15367(15) 0.15348(17) 1.922(67)X10~ 6 d -5.92(77)X10" 7 d T Q 111.3599(25) A t-144.6721] b [-146.5]° B 0.3306500(21) 0.330662(28) D 1.3599(80)xio" 7 1.39(24)xlO - 7 AD -9.77(42)X10~ 5 - 3 . 5 ( l l ) x l 0 ~5 AH 1.32(16)xio" 8 a - Error l i m i t s are 3a i n units of the l a s t figure quoted, b - O r i g i n a l l y allowed to vary; then fi x e d at t h i s value, c - Fixed to value given i n Reference [36]. d - There i s a factor of -4 difference i n matrix element along with d i f f e r e n t N-dependence. 1 0 8 d o u b l i n g s , and no e x t r a l i n e s can be a s s i g n e d ; n e v e r t h e l e s s t h e K - t y p e d o u b l i n g s c a r r y enough i n f o r m a t i o n f o r the s e p a r a t i o n o f t h e 2 2 u ~ and 2 A 3 / 2 i U s t a t e s t o be d e t e r m i n e d . ( T h i s i s d e s c r i b e d i n more d e t a i l be low. ) I t t u r n s out t o be p o s s i b l e t o l o c a t e a l l t h e A 2 ! ^ , 0 1 0 s t a t e s a c c u r a t e l y w i t h r e s p e c t t o t h e X 2 ^ , 0 0 0 ( ^ ^ J s t a t e , s i n c e Kawaguchi e t a l . [ 3 8 ] have de termined the o r i g i n o f the xhigl 0 1 0 2 2 " - 0 0 0 ^ .^g band , near 6 3 0 cm"1. W i t h the J=3/2 l e v e l o f the ^3/2 s t a t e (the lowest r e a l energy l e v e l o f B 0 2 ) t a k e n as z e r o , t h e r e l a t i v e r o v i b r o n i c e n e r g i e s o f t h e ground s t a t e c o u l d be c a l c u l a t e d ; these a r e l i s t e d i n Appendix I I I . The ^ l e v e l i s no t as a c c u r a t e l y f i x e d as t h e o t h e r l e v e l s , because i t s energy i s t h e s p i n - o r b i t c o n s t a n t f o r t h e X 2 ^ , 0 0 0 l e v e l wh ich has been t a k e n a t Johns ' v a l u e as we have no i n f o r m a t i o n w i t h which t o r e f i n e i t . The c o n s t a n t s f o r t h e A 2 ^ , 0 1 0 2 A and 2 S upper s t a t e l e v e l s a r e g i v e n i n T a b l e 3 . 1 0 . V e r y few s u i t a b l e comparisons e x i s t as o n l y Johns has s t u d i e d t h e upper S s t a t e s ; a t h i s r e s o l u t i o n the A v i b r o n i c t r a n s i t i o n s were t o o o v e r l a p p e d t o p e r m i t r o t a t i o n a l a n a l y s i s . I t i s seen t h a t t h e i n t r i n s i c s p i n - r o t a t i o n parameter i s v e r y d i f f e r e n t from t h e a p p a r e n t s p i n - r o t a t i o n parameter de termined by J o h n s . The r e s i d u a l s o b t a i n e d a r e shown i n F i g u r e 3 . 1 7 ; a g a i n random p e r t u r b a t i o n s by t h e ground s t a t e have had t o be e x c l u d e d i n o r d e r f o r t h e s t a n d a r d d e v i a t i o n t o be comparable t o t h e e x p e r i m e n t a l p r e c i s i o n (a = 0 . 0 0 0 4 4 cm"1). The A 2!^, 0 1 0 l e v e l s a r e v e r y 109 Table 3.10 Molecular constants for the A 2n u 010 l e v e l s of UB0 2 / i n cm"1. Constant This Work Previous Work T OS 18434.97225(18) a 18435. l Q h AT SA 2.51275(22) T 03/2 0.14208(36) A -100.85127(25) 6 W2 -13.89603(83) - 1 3 . l b Bs- 0.3116669(10) 0.3124 Q° Bs+ 0.31160984(77) 0.3108^^ BA 0.31157124(58) Ds- 1.2423(82)xlO —i D s + 1 . 2 7 6 4(41)xl0 _*7 D A 1.2487(28)X10 -4 -4 f % 4.7459(52)xlO -2 d 4 . 7 1 4 X 1 0 p+2q [1.1826x10 ] -5 d q [-3.06X10 ] 7S+ -2.107(31)xl0~ 3 0 . 4 7 4 e 7S- -1.443(29)X10~ 3 0.444e 7DS+ 1.091(23)xl0 7DE- 9.50(32)xl0 — R AD 1.77(12)xl0 Q -6.81(59)xl0 a - Error l i m i t s are 3a i n units of the l a s t figure quoted, b - Reference [36]. c - Reference [36] absolute value ±0.001. d - Fixed at value from 000 upper state as not enough information to obtain, e - Apparent spin-rotation parameters, from Reference [36]. f - Calculated from equation 3.43. 110 0.0030 •a 01 •o 01 V) XI o 0.0020 0.0010 0.0000 -0.0010 3 •o -0.0020 49.5 b) 0.020 0.010 id > •o 01 a 3 O a 0.000 13 01 i i -0.010 3 -0.020 ® © © © © ® ® © © © ® © © 9.5 19.5 29.5 ® 39.5 49.5 Figure 3.17 Residuals from the upper state f i t of the A and 2S rovibronic l e v e l s plotted against J 1 for a) A5/2 (represented by crosses) and 2S l o w e r (represented by diamonds) and b) 2A 3 / 2 (represented by crosses) and 22 U pper (represented by diamonds) . C i r c l e d data points indicate associated t r a n s i t i o n s have been excluded from f i n a l f i t . I l l i n t e r e s t i n g i n t h a t t h i s i s the f i r s t r e c o r d e d case o f an a v o i d e d c r o s s i n g w i t h doub led l i n e s seen i n a K - t y p e r e s o n a n c e . T h i s a l l o w e d a c c u r a t e p lacement o f the upper s t a t e 22 and 2 A 5 / 2 v i b r o n i c l e v e l s r e l a t i v e t o each o t h e r . The 2 A 3 / 2 r o t a t i o n a l l e v e l s were a c t u a l l y found t o l i e between t h e F x and F 2 s p i n components o f the 22f s t a t e , a h i g h l y unusua l s i t u a t i o n . No d o u b l e d l i n e s c o u l d be i d e n t i f i e d s i n c e the c r o s s i n g o c c u r s n e a r J=h, t h a t i s b e f o r e the f i r s t r o t a t i o n a l l e v e l o f the 2 A 3 / 2 s t a t e a p p e a r s ; however the anomalous K - t y p e d o u b l i n g t h a t i s i n d u c e d a t t h e h i g h e r J v a l u e s (see F i g u r e 3.18) makes i t p o s s i b l e t o c a l c u l a t e the r e l a t i v e p o s i t i o n s o f the 2 £~ and 2 A 3 / 2 l e v e l s w i t h r e a s o n a b l e p r e c i s i o n . Once i t had been de termined t h a t t h e s t a t e s were v i r t u a l l y d e g e n e r a t e , t h e f i t t i n g program was a b l e t o p l a c e them w i t h h i g h a c c u r a c y . Appendix V l i s t s t h e r o v i b r o n i c energy l e v e l s o f a l l t h e upper v i b r o n i c s t a t e s s t u d i e d , w i t h the X 2 ] ^ , 000 ( ^ ^ g ) J = 3 / 2 l e v e l t a k e n as z e r o e n e r g y . I t i s p o s s i b l e t o m a n i p u l a t e t h e parameters used i n the f i t t i n g t o d e t e r m i n e t h e more fundamental c o n s t a n t s . S p e c i f i c a l l y , u s i n g t h e s e p a r a t i o n s o f t h e ' r o t a t i o n l e s s ' l e v e l s , (shown i n F i g u r e 3 . 1 9 ) , which a r e o b t a i n e d by p r o j e c t i n g back t o t h e *J=-\ % energy l e v e l , i t i s p o s s i b l e t o o b t a i n v a l u e s o f w2, e and g K . One uses e x p r e s s i o n s f o r t h e v i b r o n i c energy l e v e l s o f a ^ s t a t e which a r e [43] 112 18897.00 18895.00 188 93.00 18691.00-E (cm"') 18792 .00-18790.00 18788.00 18786.00-9.5 i3/2 A 5 / 2 2 V • 2v • 19.5 29.5 39.5 49.5 J Figure 3.18 A graph of the r o t a t i o n a l energy l e v e l s of the A X v 2=l vibronic states. The crosses represent the 2A le v e l s while the c i r c l e s represent the 2S le v e l s . The 2 Z f and 2 A 3 / 2 energies have been scaled by a factor of -0.3115J(J+l) while the 22* and 2 A 5 / 2 have been scaled by a factor of -0.31J(J+1). This sc a l i n g i s done to ensure that the d e t a i l s of the energy l e v e l s can be seen c l e a r l y . 113 18892.2984 5 18892.2866 6 18790.2531 5 18787.7084 7 18415.87! 18313.9679 9 E(cm ) 633.1480 8 588 .2067 9 442.0725 0 404.9550 8 147.92 3 -upper,g 5/2, g loiter ,g '3/2,u V , = 1 A 2!!, V 2 = 0 v 2 = 1 x 2 n V2 = 0 -1.31524 l 3 / 2 . g F i g u r e 3 .19 The " r o t a t i o n l e s s " e n e r g i e s o f a l l t h e v i b r o n i c l e v e l s s t u d i e d w i t h r e s p e c t t o a z e r o a t the X T i g , 0 0 0 ( ^ 3 / 3 ) J = 3 / a l e v e l . 114 E (2) = 2w2-he2w2-5/64e4w2-e2A2/ (16w2) ±\ [A 2 (1-he 2 ) 2+4e2u>22 ( l+e 2 /32) 2 ]* E (A) = 2w 2 -3 /4e 2 w 2 -2 i /64e 4 a; 2 -3e 2 A 2 / (16w 2 )±^A( l -3 /4e 2 )+2g K E(n) = w 2 - ^ e 2 w 2 - 7 / 6 4 e 4 w 2 - e 2 A 2 / ( 1 6 w 2 ) ± ^ A ( l - ^ e 2 ) + g K These a r e c o r r e c t t o f o u r t h o r d e r and can be used t o s o l v e f o r t h e above c o n s t a n t s . The gK parameter was i n t r o d u c e d by Brown [48] t o a l l o w f o r d i s c r e p a n c i e s which had been e x p e r i m e n t a l l y no ted b u t no t t h e o r e t i c a l l y a l l o w e d f o r . I t a r i s e s from m i x i n g o f a n e l e c t r o n i c s t a t e w i t h A and 2 e l e c t r o n i c s t a t e s . The v a l u e s o b t a i n e d a r e g i v e n i n T a b l e 3 .11 a l o n g w i t h v a l u e s f o r ew2 and a 2 ( t h i s f i n a l term b e i n g t h e v i b r a t i o n a l c o r r e c t i o n t o B ) . 3 . 4 . 3 Conclusion A v e r y complete r o t a t i o n a l a n a l y s i s f o r t h e 000 and 010 l e v e l s o f the X 2 ^ and A2!!,, e l e c t r o n i c s t a t e s o f B0 2 has been c a r r i e d o u t . I t has produced a c c u r a t e r o t a t i o n a l c o n s t a n t s f o r a l l t h e v i b r o n i c l e v e l s i n v o l v e d ; the s t a n d a r d d e v i a t i o n s f o r t h e ground s t a t e f i t s a r e l e s s t h a n 0.0003 cm"1 w h i l e those f o r t h e e f f e c t i v e upper s t a t e f i t s a r e l e s s t h a n 0.0015 cm"1. The f i r s t example o f an a v o i d e d c r o s s i n g i n a K - t y p e resonance has been s een . T h i s i n d i c a t e s t h a t s i t u a t i o n s such as t h i s must e x i s t i n o t h e r m o l e c u l e s when t h e s p i n - o r b i t parameter i s much l a r g e r t h a n t h e R e n n e r - T e l l e r p a r a m e t e r . A c c u r a t e wave numbers f o r t h e m o l e c u l a r t r a n s i t i o n s were o b t a i n e d and a r e p r e s e n t l y b e i n g used i n m o l e c u l a r beam work a t UBC aimed a t 115 Table 3.11 Fundamental constants from f i t t e d parameters i n B02, i n cm"1. Constant A 2 n u state X 2n State This Work Previous Results This Work Previous Results e w2 *K a2 -0.0291-. 14 477.29 3 1 [-13.896 03] 2' 8 304 0.00087? 484 a -13.I a -0.1939 1 448.18 1 9 -86.907Q 46 0.00119^^ 449.9(6) b -86.9(3) b 0.0011(2) b a b - Reference [36]. - Reference [49]. 116 measuring the hyperfine structure. The c a l i b r a t i o n system has again proven i t s e l f to be a convenient and accurate method of c a l i b r a t i n g spectral data. For example, the assignment of the 010-000 22" - ^Z' t r a n s i t i o n s was done using A 2F 1 combination differences obtained from the 22~ - 2S + t r a n s i t i o n s . The energy separation of the 2S + and 22~ lower states i s over 200 cm"1, but yet the combination differences were never worse than 0.0008 cm"1 with an average value being 0.0003 cm"1. Thus the c a l i b r a t i o n system has shown i t s e l f to be not only useful for accurate measurements of hyperfine s p l i t t i n g s but also convenient for extended coverage of e l e c t r o n i c t r a n s i t i o n s . 117 References 1. J.O. Schroder, B. Z e l l e r and W.E. Ernst, J . Mol. Spec. 127, 255 (1988). 2. S. Gerstenkorn and P. Luc, Atlas du Spectre d 1Absorption  de l a Molecule d'lode. CNRS Ed., Paris, France, 1978. 3. W.H. Hocking, M.C.L. Gerry and A.J. Merer, Can. J . Phys. 57, 54 (1979). 4. S. Gerstenkorn, P. Luc and A. Perrin, J . Mol. Spec. 64, 56 (1977). 5. J.L. H a l l and S.A. Lee, Appl. Phys. Lett. 29, 367 (1976). 6. F.V. Kowalski, R.E. Teets, W. Demtroder and A.L. Schawlow, J . Opt. Soc. Am. 68, 1611 (1978). 7. Javan, J . Opt. Soc. Am. 67, 1413 (1977). 8. H.J. Foth, H.J. Vedder and W. Demtroder, J . Mol. Spec. 88. 109 (1981). 9. Coherent Inc. Autoscan brochure and private communications with Coherent Inc. 10. Grant Instrument Buyers Guide, Barrington W6-KA bath system. 11. M. Hercher, Appl. Opt. 7, 951 (1968). 12. D. Brewster, Trans. Roy. Soc. (Edinburgh) 12., 519 (1834). 13. H.J. Foth and H.J. Vedder, J . Mol. Spec. 102. 148 (1983). 14. F. B y l i c k i and H.G. Weber, Chem. Phys. Lett. 79, 517 (1981). 15. R.E. Smalley, L. Wharton and D.H. Levy, J . Chem. Phys. 63, 4977 (1975). 16. D.K. Hsu, D.L. Monts and R.N. Zare, Spectral Atlas of  Nitrogen Dioxide 5530 to 6480A. Academic Press, 1978. 17. T. Tanaka, R.W. F i e l d and D.O. Harris, J . Mol. Spec. 56, 188 (1975). 18. C.G. Stevens and R.N. Zare, J . Mol. Spec. 56, 167 (1975). 19. W. Demtroder, Laser Spectroscopy. Springer-Verlag, New York, 1981. 118 20. R.B. Bernstein, Chemical Dynamics Via Molecular Beam and  Laser Techniques. Oxford University Press, 1982. 21. G. Herzberg, Molecular Spectra and Structure II Infrared  and Raman Spectra of Polyatomic Molecules. D. Van Nostrand Company Inc., 1966. 22. G. Herzberg, Molecular Spectra and Structure III  E l e c t r o n i c Spectra and E l e c t r o n i c Structure of Polyatomic  Molecules. D. Van Nostrand Company Inc., 1966. 23. CH. Townes and A.L. Schawlow, Microwave Spectroscopy. McGraw-Hill, New York, 1965. 24. I . e . Bowater, J.M. Brown and A. Carrington, Proc. R. Soc. Lond. A 333, 265 (1973). 25. J.K.G. Watson i n V i b r a t i o n a l Spectra and Structure (J.R. Durig, Ed.), Vol. 6, E l s e v i e r S c i . Pub. Co., New York, 1977. 26. K.-E.J. H a l l i n and A.J. Merer, Can. J . Phys. 54/ 1157 (1976). 27. J.H. Van Vleck, Rev. Mod. Phys. 23, 213 (1951). 28. Landolt-Bornstein, Numerical Data and Functional  Relationships i n Science and Technology, Vol. 6. Molecular Constants, edited by K.H. Hellwege and A.M. Hellwege, Springer-Verlag, New York, 1974. E. Hirota, High Resolution Spectroscopy of Transient  Molecules. Springer-Verlag, New York, 1985. A. Perrin, J.-M. Flaud and C. Camy-Peyret, Molec. Phys. 63, 791 (1988). J . I . S t e i n f e l d , Molecules and Radiation: An Introduction  to Modern Molecular Spectroscopy. The MIT Press, Massachusetts, 1978. W.C. Bowman and F.C. DeLucia, J . Chem. Phys. 77, 92 (1982). P.A. Baron, P.D. Godfrey and D.O. Harris, J . Chem. Phys. 60. 3723 (1974). G. Persch, H.J. Vedder and W. Demtroder, J . Mol. Spec. 123. 356 (1975). T. Tanaka and D.O. Harris, J . Mol. Spec. 59, 413 (1976). J.W.C. Johns, Can. J . Phys. 39, 1738 (1961). 29. 30. 31. 32 . 33. 34. 35. 36. 119 37. K. Kawaguchi, E. Hirota and C. Yamada, Mol. Phys. 44, 509 (1981). 38. K. Kawaguchi and E. Hirota, J . Mol. Spec. 116, 450 (1986). 39. A.G. Maki, J.B. Burkholder, A. Sinha and C.J. Howard, J . Mol. Spec. 130. 238 (1988). 40. M.S. Sorem and A.L. Schawlow, Opt. Commun. 5, 148 (1972). 41. R. Renner, Z. Physik 92, 172 (1934). 42. J.M. Brown, J.T. Hougen, K.-P. Huber, J.W.C. Johns, I. Kopp, H. Lefebvre-Brion, A.J. Merer, D.A. Ramsay, J . Rostas and R.N. Zare, J . Mol. Spec. 55, 500 (1975). 43. J.M. Brown and F. J^irgensen, Adv. Chem. Phys. 52, 117 (1983). 44. A.J. Merer and J.M. A l l e g r e t t i , Can. J . Phys. 49, 2859 (1971). 45. J.M. Brown and J.K.G. Watson, J . Mol. Spec. 65, 65 (1977). 46. H. Lefebvre-Brion and R.W. F i e l d , Perturbations i n the  Spectra of Diatomic Molecules. Academic Press Inc., Fl o r i d a , 1986. 47. P.C. Haarhoff, Mol. Phys. 7, 101 (1963). 48. J.M. Brown, J . Mol. Spec. 68, 412 (1977). 49. K.G. Weyer, R.A. Beaudet, R. Straubinger and H. Walther, Chem. Phys. 47, 171 (1980). APPENDIX I. Tra n s i t i o n frequencies for N02. APPENDIX IA. Band 99. 120 N' J ' F' ! 1.5 0.5 1 1.5 1.5 1 1.5 2.5 1 1.5 0.5 1 1.5 1.5 1 0.5 1.5 1 0.5 0.5 1 0.5 1.5 1 0.5 0.5 1 1.5 2.5 1 1.5 1.5 1 1.5 1.5 1 1.5 0.5 1 1.5 2.5 1 1.5 2.5 1 1.5 2.5 1 1.5 1.5 1 1.5 1.5 1 1.5 1.5 1 1.5 0.5 1 1.5 0.5 1 0.5 1.5 1 0.5 1.5 1 0.5 0.5 1 0.5 1.5 1 0.5 0.5 1 0.5 1.5 1 0.5 1.5 1 0.5 0.5 3 3.5 3.5 3 3.5 2.5 3 3.5 2.5 3 3.5 4.5 3 3.5 3.5 3 3.5 3.5 3 3.5 2.5 3 3.5 2.5 3 2.5 1.5 3 2.5 2.5 3 2.5 1.5 3 2.5 1.5 3 2.5 3.5 3 2.5 2.5 3 2.5 3.5 3 2.5 2.5 3 2.5 3.5 3 2.5 2.5 3 2.5 1.5 3 3.5 3.5 3 3.5 4.5 3 3.5 2.5 N" J " F" 0 0.5 1.5 0 0.5 1.5 0 0.5 1.5 0 0.5 0.5 0 0.5 0.5 0 0.5 1.5 0 0.5 1.5 0 0.5 0.5 0 0.5 0.5 2 2.5 2.5 2 2.5 1.5 2 2.5 2.5 2 2.5 1.5 2 2.5 3.5 2 1.5 2.5 2 1.5 1.5 2 1.5 2.5 2 1.5 1.5 2 1.5 0.5 2 1.5 1.5 2 1.5 0.5 2 1.5 0.5 2 1.5 1.5 2 1.5 0.5 2 1.5 2.5 2 1.5 1.5 2 1.5 2.5 2 1.5 1.5 2 1.5 1.5 2 2.5 2.5 2 2.5 1.5 2 2.5 2.5 2 2.5 3.5 2 2.5 3.5 2 1.5 2.5 2 1.5 2.5 2 1.5 1.5 2 1.5 2.5 2 1.5 2.5 2 1.5 1.5 2 1.5 0.5 2 1.5 2.5 2 1.5 1.5 2 2.5 3.5 2 2.5 3.5 2 2.5 2.5 2 2.5 1.5 2 2.5 1.5 4 4.5 3.5 4 4.5 4.5 4 4.5 3.5 Frequency 8 (cm"1) 16850.290581 16850.291784 16850.293821 16850.297983 16850.299192 16850.364864 16850.366369 16850.372205 16850.373726 16847.768572 16847.767550 16847.766448* 16847.766448* 16847.762744 16847.761017 16847.759663 16847.758949 16847.757600 16847.756774 16847.756387 16847.755536 16847.829730 16847.830590 16847.831216 16847.831919* 16847.831919* 16847.839579 16847.840648 16847.842206 16851.961138* 16851.961138* 16851.960212 16851.957115 16851.955518 16851.953805 16851.952588 16851.951221 16852.120254 16852.119324 16852.118903 16852.118001* 16852.118001* 16852.118001* 16852.119719 16852.121047 16852.125634 16852.128062 16852.129000 16846.055270* 16846.055270* 16846.054044 APPENDIX I, continued, Band 99. N« J ' F' 3 3.5 3.5 3 3.5 4.5 3 3.5 4.5 3 3.5 3.5 3 3.5 3.5 3 3.5 3.5 3 3.5 2.5 3 3.5 2.5 3 2.5 3.5 3 2.5 3.5 3 2.5 2.5 3 2.5 1.5 3 2.5 2.5 3 2.5 3.5 3 2.5 3.5 3 2.5 3.5 3 2.5 2.5 5 5.5 5.5 5 5.5 6.5 5 5.5 4.5 5 5.5 5.5 5 5.5 4.5 5 5.5 4.5 5 5.5 4.5 5 5.5 5.5 5 4.5 3.5 5 4.5 4.5 5 4.5 3.5 5 4.5 5.5 5 4.5 4.5 5 4.5 5.5 5 4.5 5.5 5 4.5 4.5 5 4.5 3.5 5 5.5 6.5 5 5.5 5.5 5 5.5 4.5 5 5.5 6.5 5 5.5 5.5 5 5.5 4.5 5 5.5 4.5 5 5.5 5.5 5 5.5 5.5 5 5.5 6.5 5 4.5 4.5 5 4.5 3.5 5 4.5 5.5 5 4.5 4.5 5 4.5 5.5 5 4.5 5.5 5 4.5 4.5 7 7.5 7.5 N" J " Y» 4 4.5 4.5 4 4.5 5.5 4 3.5 4.5 4 3.5 4.5 4 3.5 3.5 4 3.5 2.5 4 3.5 3.5 4 3.5 2.5 4 3.5 2.5 4 3.5 3.5 4 3.5 2.5 4 3.5 2.5 4 3.5 3.5 4 3.5 4.5 4 4.5 4.5 4 4.5 3.5 4 4.5 3.5 4 4.5 5.5 4 4.5 5.5 4 4.5 4.5 4 4.5 4.5 4 4.5 3.5 4 3.5 3.5 4 3.5 4.5 4 3.5 4.5 4 3.5 2.5 4 3.5 3.5 4 3.5 3.5 4 3.5 4.5 4 3.5 4.5 4 4.5 5.5 4 4.5 4.5 4 4.5 4.5 4 4.5 3.5 6 6.5 7.5 6 6.5 6.5 6 6.5 5.5 6 6.5 6.5 6 6.5 5.5 6 5.5 4.5 6 5.5 5.5 6 5.5 5.5 6 5.5 6.5 6 5.5 6.5 6 5.5 4.5 6 5.5 4.5 6 5.5 5.5 6 5.5 5.5 6 5.5 6.5 6 6.5 6.5 6 6.5 5.5 6 6.5 7.5 Frequency3 (cm"1) 16846.053623 16846.050290 16846.046604 16846.044980 16846.043208 16846.041926* 16846.041926* 16846.040721 16846.206032 16846.207429* 16846.207429* 16846.208381 16846.208675 16846.209176 16846.217734 16846.219395 16846.220751 16853.564708 16853.566227 16853.568370 16853.569685 16853.570087 16853.557867 16853.559705 16853.560995 16853.820776 16853.821662 16853.822068 16853.822969 16853.823536 16853.826677 16853.831583 16853.833819 16853.834199 16844.285431 16844.288166 16844.288891 16844.289620 16844.290165 16844.272646 16844.274191 16844.275541 16844.277711 16844.279236 16844.536390 16844.536767 16844.537331 16844.537885 16844.539440 16844.549986 16844.552604 16855.096152 APPENDIX I, continued, Band 9 9 . N' J 1 F' 7 7.5 8.5 7 7.5 6.5 7 7.5 7.5 7 7.5 6.5 7 7.5 6.5 7 7.5 6.5 7 7.5 7.5 7 6.5 5.5 7 6.5 6.5 7 6.5 7.5 7 6.5 5.5 7 6.5 6.5 7 6.5 7.5 7 6.5 7.5 7 6.5 6.5 7 7.5 8.5 7 7.5 7.5 7 7.5 6.5 7 7.5 8.5 7 7.5 7.5 7 7.5 6.5 7 7.5 6.5 7 7.5 7.5 7 7.5 7.5 7 7.5 8.5 7 6.5 7.5 7 6.5 6.5 7 6.5 7.5 7 6.5 5.5 7 6.5 6.5 7 6.5 7.5 7 6.5 7.5 7 6.5 6.5 9 9.5 9.5 9 9.5 10.5 9 9.5 8.5 9 9.5 9.5 9 9.5 8.5 9 9.5 8.5 9 9.5 9.5 9 8.5 7.5 9 8.5 8.5 9 8.5 9.5 9 8.5 7.5 9 8.5 8.5 9 8.5 9.5 9 8.5 8.5 9 9.5 10.5 9 9.5 9.5 9 9.5 8.5 9 9.5 8.5 9 9.5 8.5 N" J " F" 6 6.5 7.5 6 6.5 6.5 6 6.5 6.5 6 6.5 5.5 6 5.5 5.5 6 5.5 6.5 6 5.5 6.5 6 5.5 4.5 6 5.5 5.5 6 5.5 6.5 6 5.5 5.5 6 5.5 6.5 6 6.5 7.5 6 6.5 6.5 6 6.5 5.5 8 8.5 9.5 8 8.5 8.5 8 8.5 7.5 8 8.5 8.5 8 8.5 7.5 8 7.5 6.5 8 7.5 7.5 8 7.5 7.5 8 7.5 8.5 8 7.5 8.5 8 7.5 6.5 8 7.5 6.5 8 7.5 7.5 8 7.5 6.5 8 7.5 7.5 8 7.5 8.5 8 8.5 8.5 8 8.5 7.5 8 8.5 9.5 8 8.5 9.5 8 8.5 8.5 8 8.5 8.5 8 8.5 7.5 8 7.5 7.5 8 7.5 8.5 8 7.5 6.5 8 7.5 7.5 8 7.5 8.5 8 7.5 7.5 8 7.5 8.5 8 8.5 8.5 8 8.5 7.5 10 10.5 11.5 10 10.5 10.5 10 10.5 9.5 10 9.5 8.5 10 9.5 9.5 Frequency" (cm"1) 16855.097567 16855.099277 16855.100507 16855.101363 16855.086628 16855.088785 16855.090002 16855.449037 16855.449417 16855.450211 16855.450544 16855.451518 16855.456355 16855.460639 16855.463976 16842.445048 16842.447661 16842.448624 16842.448963 16842.449852 16842.429197 16842.431366 16842.432180 16842.434871 16842.436243 16842.791263 16842.792457 16842.792857 16842.793577 16842.794147 16842.795157 16842.807617 16842.811167 16856.460598 16856.461847 16856.463375 16856.464481 16856.465551 16856.448318 16856.451788 16856.909071 16856.909858 16856.910566 16856.910848 16856.911915 16856.923159 16856.926659 16840.441541 16840.443924 16840.445117 16840.422909 16840.424677 APPENDIX I, continued, Band 99. 123 N' J ' F' - N" J " 9 9.5 9.5 10 9.5 9 8.5 8.5 10 9.5 9 8.5 9.5 10 9.5 9 8.5 7.5 10 9.5 9 8.5 8.5 10 9.5 9 8.5 9.5 10 9.5 9 8.5 9.5 10 10.5 9 8.5 8.5 10 10.5 APPENDIX IB. Band 115. N 1 J 1 F 1 - N" J " 1 1.5 0.5 0 0.5 1 1.5 1.5 0 0.5 1 1.5 2.5 0 0.5 1 1.5 0.5 0 0.5 1 1.5 1.5 0 0.5 1 0.5 1.5 0 0.5 1 0.5 0.5 0 0.5 1 0.5 1.5 0 0.5 1 0.5 0.5 0 0.5 1 1.5 2.5 2 2.5 1 1.5 1.5 2 2.5 1 1.5 1.5 2 2.5 1 1.5 0.5 2 2.5 1 1.5 2.5 2 2.5 1 1.5 2.5 2 1.5 1 1.5 2.5 2 1.5 1 1.5 1.5 2 1.5 1 1.5 1.5 2 1.5 1 1.5 1.5 2 1.5 1 1.5 0.5 2 1.5 1 0.5 1.5 2 1.5 1 0.5 1.5 2 1.5 1 0.5 0.5 2 1.5 1 0.5 1.5 2 1.5 1 0.5 0.5 2 1.5 1 0.5 1.5 2 1.5 1 0.5 1.5 2 1.5 1 0.5 0.5 2 1.5 3 3.5 3.5 2 2.5 3 3.5 2.5 2 2.5 3 3.5 2.5 2 2.5 3 3.5 4.5 2 2.5 3 3.5 3.5 2 2.5 3 3.5 2.5 2 2.5 3 3.5 3.5 2 1.5 3 3.5 2.5 2 1.5 3 3.5 2.5 2 1.5 3 2.5 2.5 2 1.5 F" Frequency* (cm-1) 0 .5 16840 .428374 8 .5 16840 .884345 9 .5 16840 .884858 8 .5 16840 .885525 9 .5 16840 .886197 0 .5 16840 .887292 0 .5 16840 .902904 9 .5 16840 .906503 p i i Frequency 8 (cm"1) 1 .5 17092 .901736 1 .5 17092 .902771 1 .5 17092 .904302 0 .5 17092 .909095 0 .5 17092 .910156 1 .5 17092 .872271 1 .5 17092 .873630 0 .5 17092 .879338 0 .5 17092 .880801 2 .5 17090 .379283 1 .5 17090 .378819 2 .5 17090 .377781* 1 .5 17090 .377781* 3 .5 17090 .373284 2 .5 17090 .371524 1 .5 17090 .370068* 2 .5 17090 .370068* 1 .5 17090 .368638 0 .5 17090 .367838 0 .5 17090 .366754 0 .5 17090 .337255 1 .5 17090 .338065 0 .5 17090 .338668 2 .5 17090 .339415* 1 .5 17090 .339415* 2 .5 17090 .347144 1 .5 17090 .348156 1 .5 17090 .349605 2 .5 17094 .956324 1 .5 17094 .957344 2 .5 17094 .956324 3 .5 17094 .950441* 3 .5 17094 .950441* 3 .5 17094 .950441* 2 .5 17094 .948691* 2 .5 17094 .948691* 1 .5 17094 .947344 2 .5 17094 .509674 APPENDIX I, continued, Band 115. 124 N 1 J» F' 3 2.5 1.5 3 2.5 1.5 3 2.5 3.5 3 2.5 2.5 3 2.5 3.5 3 2.5 2.5 3 2.5 3.5 3 2.5 2.5 3 2.5 1.5 3 3.5 2.5 3 3.5 2.5 3 3.5 3.5 3 3.5 2.5 3 3.5 3.5 3- 3.5 4.5 3 3.5 2.5 3 3.5 3.5 3 3.5 2.5 3 3.5 3.5 3 3.5 4.5 3 3.5 2.5 3 3.5 3.5 3 3.5 4.5 3 3.5 4.5 3 3.5 4.5 3 3.5 3.5 3 3.5 3.5 3 3.5 3.5 3 3.5 2.5 3 3.5 2.5 3 3.5 4.5 3 2.5 3.5 3 2.5 3.5 3 2.5 2.5 3 2.5 1.5 3 2.5 2.5 3 2.5 3.5 3 2.5 3.5 3 2.5 3.5 3 2.5 2.5 3 3.5 3.5 3 3.5 2.5 3 3.5 2.5 3 3.5 3.5 3 3.5 4.5 3 3.5 3.5 3 3.5 4.5 3 3.5 4.5 3 3.5 4.5 3 3.5 3.5 3 3.5 2.5 3 3.5 3.5 3 3.5 4.5 N" J " 2 1.5 1.5 2 1.5 0.5 2 1.5 2.5 2 1.5 1.5 2 2.5 3.5 2 2.5 3.5 2 2.5 2.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 2.5 2 2.5 2.5 2 2.5 3.5 2 2.5 3.5 2 2.5 3.5 2 1.5 2.5 2 1.5 2.5 2 1.5 1.5 4 4.5 3.5 4 4.5 4.5 4 4.5 3.5 4 4.5 4.5 4 4.5 5.5 4 4.5 3.5 4 3.5 4.5 4 3.5 4.5 4 3.5 3.5 4 3.5 2.5 4 3.5 3.5 4 3.5 2.5 4 3.5 3.5 4 3.5 2.5 4 3.5 3.5 4 3.5 2.5 4 3.5 2.5 4 3.5 3.5 4 3.5 4.5 4 4.5 4.5 4 4.5 3.5 4 4.5 3.5 4 3.5 2.5 4 3.5 2.5 4 . 3.5 3.5 4 3.5 3.5 4 3.5 3.5 4 3.5 4.5 4 3.5 4.5 4 4.5 5.5 4 4.5 4.5 4 4.5 4.5 4 4.5 3.5 4 4.5 3.5 4 4.5 3.5 Frequency8 (cm"1) 17094.509181 17094.508435* 17094.508435* 17094.508435* 17094.510252 17094.511462 17094.516222 17094.518513 17094.519293 17093.742844 17093.741733* 17093.741733* 17093.735835* 17093.735835* 17093.735835* 17093.734084* 17093.734084* 17093.732542 17087.050346* 17087.048691* 17087.050436* 17087.048691* 17087.043767 17087.050346* 17087.040007* 17087.040007* 17087.038166* 17087.036873* 17087.038166* 17087 .036873* 17087.038166* 17088.596636 17088.597830* 17088.597830* 17088.598647 17088.599075 17088.599647 17088.608326 17088.610010 17088.611143 17087.822135* 17087.822135* 17087.823466* 17087.823466* 17087.825333* 17087.825333* 17087.825333* 17087.829030 17087.833992* 17087.833992* 17087.835462* 17087.835462* 17087.835462* APPENDIX I, continued, Band 115. 125 N' J " F 1 5 5.5 5.5 5 5.5 6.5 5 5.5 4.5 5 5.5 5.5 5 5.5 4.5 5 5.5 4.5 5 5.5 4.5 5 5.5 5.5 5 4.5 3.5 5 4.5 4.5 5 4.5 3.5 5 4.5 5.5 5 4.5 4.5 5 4.5 5.5 5 4.5 5.5 5 4.5 4.5 5 5.5 6.5 5 5.5 5.5 5 5.5 4.5 5 5.5 6.5 5 5.5 5.5 5 5.5 4.5 5 5.5 4.5 5 5.5 5.5 5 5.5 5.5 5 5.5 6.5 5 4.5 4.5 5 4.5 3.5 5 4.5 5.5 5 4.5 4.5 5 4.5 5.5 5 4.5 5.5 5 4.5 4.5 7 7.5 7.5 7 7.5 8.5 7 7.5 6.5 7 7.5 7.5 7 7.5 6.5 7 7.5 6.5 7 7.5 6.5 7 7.5 7.5 7 6.5 5.5 7 6.5 6.5 7 6.5 7.5 7 6.5 5.5 7 6.5 6.5 7 6.5 7.5 7 6.5 7.5 7 6.5 6.5 7 6.5 6.5 7 6.5 5.5 7 7.5 8.5 N" J " •p» 4 4 .5 5.5 4 4.5 5.5 4 4.5 4.5 4 4 .5 4.5 4 4.5 3.5 4 3.5 3.5 4 3.5 4.5 4 3.5 4.5 4 3.5 2.5 4 3.5 3.5 4 3.5 3.5 4 3.5 4.5 4 3.5 4.5 4 4.5 5.5 4 4.5 4 .5 4 4.5 3.5 6 6.5 7.5 6 6.5 6.5 6 6.5 5.5 6 6.5 5.5 6 6.5 5.5 6 5.5 4.5 6 5.5 5.5 6 5.5 5.5 6 • 5.5 6.5 6 5.5 6.5 6 5.5 4.5 6 5.5 4.5 6 5.5 5.5 6 5.5 5.5 6 5.5 6.5 6 6.5 6.5 6 6.5 5.5 6 6.5 7.5 6 6.5 7.5 6 6.5 6.5 6 6.5 6.5 6 6.5 5.5 6 5.5 5.5 6 5.5 6.5 6 5.5 6.5 6 5.5 4.5 6 5.5 5.5 6 5.5 6.5 6 5.5 5.5 6 5.5 6.5 6 6.5 7.5 6 6.5 6.5 6 6.5 5.5 6 6.5 6.5 6 6.5 5.5 8 8.5 9.5 Frequency 8 (cm"1) 17095.743526 17095.744680 17095.747358 17095.748523 17095.749025 17095.736877 17095.738722 17095.739810 17096.039388 17096.039759 17096.040521* 17096.040521* 17096.041562 17096.044234 17096.049291 17096.052105 17086.463691 17086.466756 17086.467744 17086.470068 17086.468803 17086.451781 17086.453215 17086.454249 17086.456379 17086.457618 17086.754726 17086.755596 17086.755230 17086.756224 17086.757397 17086.767665 17086.770768 17097.142429 17097.143604 17097.145602 17097.146690 17097.147620 17097.133259 17097.135343 17097.136390 17097.547993 17097.548648 17097.549914 17097.549451 17097.550816 17097.556083 17097.560365 17097.563306 17097.561333 17097.564104 17084.491739 APPENDIX I, continued, Band 115. N« J ' F ' 7 7.5 7.5 7 7.5 6.5 7 7.5 7.5 7 7.5 6.5 7 7.5 6.5 7 7.5 7.5 7 7.5 7.5 7 7.5 8.5 7 6.5 6.5 7 6.5 5.5 7 6.5 6.5 7 6.5 7.5 7 6.5 7.5 7 6.5 6.5 9 9.5 9.5 9 9.5 10.5 9 9.5 8.5 9 9.5 9.5 9 9.5 8.5 9 9.5 8.5 9 9.5 9.5 9 8.5 7.5 9 8.5 8.5 9 8.5 9.5 9 8.5 9.5 9 8.5 8.5 9 8.5 9.5 9 8.5 8.5 9 9.5 10.5 9 9.5 9.5 9 9.5 8.5 9 9.5 8.5 9 9.5 10.5 9 8.5 7.5 9 8.5 8.5 9 8.5 9.5 9 8.5 9.5 9 8.5 8.5 N " J " F " 8 8.5 8.5 8 8.5 7.5 8 8.5 7.5 8 7.5 6.5 8 7.5 7.5 8 7.5 7.5 8 7.5 8.5 8 7.5 8.5 8 7.5 6.5 8 7.5 6.5 8 7.5 7.5 8 7.5 8.5 8 8.5 8.5 8 8.5 7.5 8 8.5 9.5 8 8.5 9.5 8 8.5 8.5 8 8.5 8.5 8 8.5 7.5 8 7.5 7.5 8 7.5 8.5 8 7.5 6.5 8 7.5 7.5 8 7.5 8.5 8 8.5 9.5 8 7.5 8.5 8 8.5 8.5 8 8.5 7.5 10 10.5 11.5 10 10.5 10.5 10 10.5 9.5 10 9.5 8.5 10 9.5 9.5 10 9.5 8.5 10 9.5 9.5 10 9.5 10.5 10 10.5 10.5 10 10.5 9.5 Frequency 8 (cm"1) 17084.494334 17084.495446 17084.496523 17084.476514 17084.478232 17084.479309 17084.481624 17084.482906 17084.891638 17084.892321 17084.893252 17084.894722 17084.907528 17084.910643 17098.369950 17098.370967 17098.372980 17098.373816 17098.375158 17098.358057 17098.361232 17098.772589 17098.773367 17098.774619 17098.783349 17098.775649 17098.787359 17098.790635 17082.350949 17082.353529 17082.354948 17082.338188 17082.336936 17082.749614 17082.750509 17082.751870 17082.767376 17082.770668 a) The l a s t d i g i t i s not s t r i c t l y a s i g n i f i c a n t figure however they were included i n the f i t t i n g process. * Indicates a blended l i n e . 127 APPENDIX I I . Sample i n t e n s i t y calcuation. A sample c a l c u l a t i o n for the P(6) t r a n s i t i o n s i s shown followed by a table containing the measured and predicted l i n e i n t e n s i t i e s . P(6) t r a n s i t i o n s have N"=6 with J " = 13/2 (FJ J " = 11 /2 (F2) F" = 11/2 13/2 15/2 F" = B/2 11/2 13/2 There w i l l be mixing between the two F"=n/2 l e v e l s and the two F"=i3/2 l e v e l s . For the 13/2 l e v e l , the 2x2 matrix, i n units of MHz, i s H = Diagonalization leads to |F2> 85.044 103.399 |Fi> 103. 399~ -143.407 A = 124.893 0 0 -183.256 S = 0.933 0.360 -0.360 0.933 Two example matrix elements (in the notation |NJF> ) are <5 9/2 11/21 n I 6 11 /2 i 3 / 2 > = -2 . 523 <5 9/2 11/2 I /i I 6 13/2 1 3 /2> = 0.0 allowed t r a n s i t i o n forbidden t r a n s i t i o n The r o t a t i o n a l eigenfunctions are l i n e a r combinations of the basis functions ^ b a s l s = |g 1 1 / 2 1 3 / 2 > « a D , , u _ J g 1 3 / 2 i 3 / 2 > basis 1 2 8 and are obtained by the transformation eigen *1 eigen which gives v*1*™ = 0 . 9 3 3 - 0 . 3 6 0 0 . 3 6 0 0 . 9 3 3 basis basis eigen _ _ 0 . 9 3 3 * 1 B A S I S + 0 . 3 6 0 * 2 B " I B 0 . 3 6 0 * 1 B A S I S + 0 . 9 3 3 * 2 B A S I S The i n t e n s i t i e s of the t r a n s i t i o n s are then e i g e n | fj, | *°isenz 1 2 basis I I * _ _ _ _ basis^ , „ - basis i .. i - - , ~ . T . basis^ i 2 = |<0.933* 1 D A S X S|A*| 0 . 9 3 3 * 1 M 8 1 S > + <0 . 3 6 0 * 2 | M | 0 . 3 6 0 * 2 > I = | 0 . 9 3 3 ( - 2 . 5 2 3 ) + 0| 2 = 5 . 5 4 l n t 2 a |<*2eisen|M|*2ei8en>|2 = |<-0.3 6 0 4 * 1 B A S I S | M | - 0 . 3 6 0 * 1 B A S I S > + <0 . 9 3 3 * A B " L B | M | 0 . 9 3 3* 2 B A 8 L 8> | 2 = | - 0 . 3 6 0 ( - 2 . 5 2 3 ) + 0| 2 = 0 . 8 2 The P ( 6 ) FJFJ allowed and P ( 6 ) F ^ forbidden i n t e n s i t i e s are tabulated below and have been scaled such that the strongest i n t e n s i t y i s 1. J ' F i J " F " a Measured Predicted 1 1 / 2 9/2 1 1 / 2 9/2 0 . 0 6 1 0 . 0 0 9 1 1 / 2 9/2 1 1 / 2 1 1 / 2 0 . 1 6 8 0 . 0 5 6 1 1 / 2 1 1 / 2 1 1 / 2 1 1 / 2 0 . 1 6 8 0 . 0 2 0 1 1 / 2 1 1 / 2 1 1 / 2 1 3 / 2 0 . 4 1 2 0 . 1 0 0 1 1 / 2 1 3 / 2 1 1 / 2 1 3 / 2 0 . 1 5 3 0 . 0 0 3 1 1 / 2 1 3 / 2 1 3 / 2 1 5 / 2 1 . 0 0 0 1. 0 0 0 1 1 / 2 1 1 / 2 1 3 / 2 1 3 / 2 0 . 7 7 9 0 . 7 5 5 1 1 / 2 9/2 1 3 / 2 1 1 / 2 0 . 7 1 8 0 . 6 7 4 1 1 / 2 1 3 / 2 1 3 / 2 1 3 / 2 0 . 0 7 6 0 . 0 0 9 1 1 / 2 1 1 / 2 1 3 / 2 1 1 / 2 0 . 1 0 7 0 . 0 1 1 1 1 / 2 1 3 / 2 1 3 / 2 1 1 / 2 0 . 0 0 0 0 3 a - Measured from band 9 9 data. The agreement i s by no means perfect (possibly because the corresponding e f f e c t i n the upper state has been ignored, being smaller) but the r e l a t i v e i n t e n s i t i e s are predicted c l o s e l y enough to aid i n the assignment. 130 APPENDIX III. Observed transitions (in cm"1) for the A^-X^TI, system of B O 2 . 000 - 000 *n2l2 (UB02) J R BRANCH P BRANCH Q BRANCH 1.5 1 8 3 1 6 . 7 5 6 4 9 18315 .20763 2 . 5 1 8 3 1 7 . 2 8 1 5 3 16313 .56427 18315 .11318 3 . 5 18317 .76921 18312.81264 18314 .98144 4 . 5 1 8 3 1 8 . 2 1 9 6 2 1 8 3 1 2 . 0 2 3 6 5 18314 .81192 5 . 5 1 8 3 1 8 . 6 3 0 7 8 18311 .19681 18314 .60465 6 . 5 1 8 3 1 9 . 0 0 6 2 7 18310 .33250 18314 .35949 7 . 5 1 8 3 1 9 . 3 4 2 9 3 1 8 3 0 9 . 4 3 0 1 0 18314 .07717 3 . 5 18319 .64391 18308.49074 18313 .75651 9 . 5 1 8 3 1 9 . 9 0 1 8 3 1 8 3 0 7 . 5 1 3 0 6 18313 .39819 1 0 . 5 1 8 3 2 0 . 1 3 0 8 7 1 8 3 0 6 . 4 9 9 8 5 1 1 . 5 18320 .31156 18305 .44382 1 2 . 5 18320.46324 18304 .35902 1 3 . 5 18320 .56978 18303 .22528 1 4 . 5 18320 .64081 1 5 . 5 18320 .67788 18300 .85533 1 6 . 5 16320.67301 1 8 2 9 9 . 6 1 3 3 5 1 7 . 5 1 8 3 2 0 . 6 3 5 2 3 18298 .33592 1 8 . 5 18320.55474 1 8 2 9 7 . 0 1 8 0 3 1 9 . 5 1 8 3 2 0 . 4 4 1 1 9 1 8 2 9 5 . 6 6 6 3 7 2 0 . 5 18320 .28651 18294.27304 2 1 . 5 18320 .09611 18292 .84567 2 2 . 5 18319 .86402 18291.37851 24 .5 1 8 3 1 9 . 2 9 7 4 5 1 8 2 8 8 . 3 3 0 0 5 2 3 . 5 18319 .60096 18289 .87431 2 5 . 5 18318 .95358 18286 .75349 2 6 . 5 18318 .57332 1 8 2 8 5 . 1 3 8 4 5 27 .5 18318.15811 18283 .48092 2 S . 5 18317 .70248 18281 .78972 2 9 . 5 1 8 3 1 7 . 2 1 2 5 3 1 8 2 8 0 . 0 6 1 3 5 3 0 . 5 18316 .68280 18278 .29520 3 1 . 5 18316 .11482 1 8 2 7 6 . 4 9 2 3 2 3 2 . 5 18315 .50658 18274 .65228 3 3 . 5 1 8 3 1 4 . 8 6 5 3 6 18272 .77190 34 .5 1 8 3 1 4 . 1 8 3 3 6 1 8 2 7 0 . 8 5 4 3 5 3 5 . 5 18313 .46868 18268.90021 3 6 . 5 1 8 3 1 2 . 7 0 6 7 5 18266.90964 3 7 . 5 18311 .91702 18264 .88217 3 8 . 5 18311.08194 18262 .81223 3 9 . 5 1 8 3 1 0 . 2 1 5 5 9 1 8 2 6 0 . 7 1 0 4 6 4 0 . 5 18309 .30482 18258 .56788 4 1 . 5 1 8 3 0 8 . 3 6 6 0 6 1 8 2 5 6 . 3 8 9 5 6 4 2 . 5 1 8 3 0 7 . 3 8 1 2 6 1 8 2 5 4 . 1 7 2 2 6 4 3 . 5 18306 .35590 1 8 2 5 1 . 9 2 1 9 5 4 4 . 5 18305 .30818 18249 .63118 4 5 . 5 18304 .21412 1 8 2 4 7 . 2 9 4 3 3 4 6 . 5 18303 .08137 18244 .94136 4 7 . 5 1 8 3 0 1 . 9 1 0 9 3 18242 .53620 4 8 . 5 18300 .70437 18240 .09972 4 9 . 5 1 8 2 9 9 . 4 5 2 7 6 18237 .61817 APPENDIX I I I , continued, 000 - 000 ^ 3 / 2 ("BOO) . 131 J R BRANCH P BRANCH Q BRANCH 5 C . 5 1 8 2 9 8 . 1 7 9 9 7 1 8 2 3 5 . 1 0 8 0 9 5 1 . 5 1 8 2 9 6 . 8 6 5 8 3 1 8 2 3 2 . 5 4 6 1 2 5 2 . 5 1 8 2 9 5 . 5 0 5 5 5 1 8 2 2 9 . 9 7 0 7 8 5 3 . 5 1 8 2 9 4 . 1 1 6 1 7 1 8 2 2 7 . 3 4 6 5 9 5 4 . 5 1 8 2 9 2 . 6 7 9 1 0 1 8 2 2 4 . 6 8 4 7 7 5 5 . 5 1 8 2 9 1 . 2 2 1 7 4 1 8 2 2 1 . 9 8 5 7 8 5 6 . 5 1 8 2 8 9 . 7 0 4 6 1 1 8 2 1 9 . 2 4 8 0 1 5 7 . 5 1 8 2 8 8 . 1 6 7 8 9 1 8 2 1 6 . 4 8 1 3 9 5 8 . 5 1 8 2 8 6 . 5 7 9 4 1 5 9 . 5 1 8 2 8 4 . 9 7 1 3 1 6 0 . 5 1 8 2 8 3 . 3 0 7 3 5 6 1 . 5 1 8 2 8 1 . 6 2 0 2 2 6 2 . 5 1 8 2 7 9 . 8 9 2 5 4 6 3 . 5 1 8 2 7 8 . 1 2 5 4 8 6 5 . 5 1 8 2 7 4 . 4 7 8 6 5 0 0 0 - 000 \ / 2 ( UB0 2) J R BRANCH P BRANCH Q BRANCH 1.5 1 8 2 6 9 . 4 4 3 0 5 1 8 2 6 7 . 8 5 6 2 1 2 . 5 1 8 2 6 9 . 9 4 8 6 2 1 8 2 6 6 . 2 2 0 0 4 1 8 2 6 7 . 8 0 6 6 0 3 . 5 1 8 2 6 5 . 4 7 6 7 1 4 . 5 1 8 2 7 0 . 8 9 5 0 7 1 8 2 6 4 . 6 7 2 5 9 5 . 5 1 8 2 7 1 . 3 5 9 7 4 1 8 2 6 3 . 8 6 7 7 9 6 . 5 1 8 2 7 1 . 6 9 3 7 8 1 8 2 6 2 . 9 7 8 0 0 7 . 5 1 8 2 7 2 . 0 9 7 8 8 1 8 2 6 2 . 1 1 1 6 8 8 . 5 1 8 2 7 2 . 3 4 5 5 0 1 8 2 6 1 . 1 3 6 2 2 9 . 5 1 8 2 6 0 . 2 0 8 4 3 1 0 . 5 1 8 2 7 2 . 8 4 9 7 8 1 8 2 5 9 . 1 4 7 6 6 1 1 . 5 1 8 2 7 3 . 1 3 C 7 4 1 8 2 5 8 . 1 5 8 1 3 1 2 . 5 1 8 2 7 3 . 2 0 6 5 8 1 8 2 5 7 . 0 1 1 6 5 1 3 . 5 1 8 2 5 5 . 9 6 0 9 8 1 4 . 5 1 8 2 7 3 . 4 1 6 9 7 1 8 2 5 4 . 7 2 8 7 1 1 5 . 5 1 8 2 7 3 . 5 7 3 0 7 1 8 2 5 3 . 6 1 6 3 9 1 6 . 5 1 8 2 7 3 . 4 7 8 4 8 1 8 2 5 2 . 2 9 8 6 2 1 7 . 5 1 8 2 7 3 . 5 7 3 0 7 1 8 2 5 1 . 1 2 5 4 1 1 8 . 5 1 8 2 7 3 . 3 8 4 8 6 1 8 2 4 9 . 7 2 2 9 8 1 9 . 5 1 8 2 7 3 . 4 2 4 6 1 1 8 2 4 8 . 4 8 7 2 2 2 0 . 5 1 8 2 7 3 . 1 5 9 8 0 1 8 2 4 6 . 9 9 0 3 8 2 1 . 5 1 8 2 7 3 . 1 3 0 7 4 1 8 2 4 5 . 6 9 9 6 8 2 2 . 5 1 8 2 7 2 . 7 7 8 4 9 1 8 2 4 4 . 1 2 7 4 4 2 3 . 5 1 8 2 7 2 . 6 8 8 6 1 1 8 2 4 2 . 7 6 8 7 1 2 4 . 5 1 8 2 7 2 . 2 4 8 3 2 1 8 2 4 1 . 1 0 8 8 5 2 5 . 5 1 8 2 7 2 . 0 9 7 8 8 1 8 2 3 9 . 6 8 9 9 3 2 6 . 5 1 8 2 7 1 . 5 7 6 0 5 1 8 2 3 7 . 9 4 1 2 1 2 7 . 5 1 8 2 7 1 . 3 5 9 7 4 1 8 2 3 6 . 4 6 2 3 0 2 8 . 5 1 8 2 7 0 . 7 5 4 2 5 1 8 2 3 4 . 6 3 2 8 4 2 9 . 5 1 8 2 7 0 . 4 8 7 9 3 1 8 2 3 3 . 0 8 7 9 7 3 0 . 5 1 8 2 6 9 . 7 8 0 3 6 1 8 2 3 1 . 1 7 5 7 5 3 1 . 5 1 8 2 6 9 . 4 4 3 0 5 1 8 2 2 9 . 5 8 1 5 6 3 2 . 5 1 8 2 6 8 . 6 6 4 6 4 1 8 2 2 7 . 5 6 6 8 0 APPENDIX I I I , c o n t i n u e d , 000 - 000 X t f ( U B0 2 ) . 132 R BRANCH 3 3 . 5 3 4 . 5 3 5 . 5 3 6 3 7 3 8 3 9 4 0 4 1 . 5 4 2 . 5 4 3 4 4 4 5 4 6 4 7 4 8 , 4 9 , 5 0 , 5 1 , 5 2 , 5 3 , 5 4 , 5 5 5 6 5 7 , 5 8 , 1 8 2 6 8 , 1 8 2 6 7 . 1 8 2 6 6 , 1 8 2 6 5 . 1 8 2 6 5 . 1 8 2 6 4 . 1 8 2 6 3 . 1 8 2 6 2 , 1 8 2 6 2 . 1 8 2 6 0 . 1 8 2 6 0 . 1 8 2 5 8 , 1 8 2 5 8 , 1 8 2 5 6 , 1 8 2 5 5 , 1 8 2 5 4 , 1 8 2 5 3 . 1 8 2 5 1 , 1 8 2 5 1 , 1 8 2 4 9 , 1 8 2 4 8 , 1 8 2 4 6 , 1 8 2 4 5 , 1 8 2 4 3 , 1 8 2 4 2 , 1 8 2 4 0 , 2 6 : , . 7 9 3 9 8 8 0 9 3 7 7 3 9 8 4 3 2 4 6 0 7 5 4 2 6 9 8 8 3 8 6 3 7 1 9 2 6 0 6 7 4 2 8 6 3 4 5 1 5 1 0 4 8 5 9 8 0 0 8 3 0 9 7 1 1 1 4 8 7 1 6 3 4 1 2 1 9 5 1 1 0 4 9 6 7 2 0 0 0 2 8 7 3 7 3 0 1 3 4 7 6 2 6 3 4 7 4 5 4 3 5 5 7 4 6 4 7 5 9 2 4 5 7 1 4 6 8 P BRANCH 1 8 2 2 5 . 9 0 2 1 9 1 8 2 2 3 . 8 1 7 5 5 1 8 2 2 2 . 0 8 8 1 5 1 8 2 1 9 . 9 1 8 8 0 1 8 2 1 8 . 1 3 1 6 7 1 8 2 1 5 . 8 7 1 6 4 Q BRANCH 1 8 2 0 2 . 8 6 0 7 0 R BRANCH 000 - 000 X / o ( 1 0BO 2) P BRANCH 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 , 1 0 . 1 1 , 1 2 . 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 1 9 2 0 2 1 , 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 8 3 3 8 , 1 8 3 3 8 . 1 8 3 3 9 , 1 8 3 3 9 , 1 8 3 4 0 , 1 8 3 4 0 , 1 8 3 4 0 , 1 8 3 4 1 , 1 8 3 4 1 , 1 8 3 4 1 , 1 8 3 4 1 , 1 8 3 4 2 , 1 8 3 4 2 , 1 8 3 4 2 , 1 8 3 4 2 , 1 8 3 4 2 1 8 3 4 2 1 8 3 4 2 1 8 3 4 2 1 8 3 4 1 , 1 8 3 4 1 , 3 6 0 0 5 8 8 5 5 9 3 7 3 1 8 8 2 3 2 3 2 3 5 3 7 6 0 9 9 0 9 4 6 8 2 2 4 5 8 7 5 0 7 4 3 7 3 1 1 2 9 1 7 3 7 0 6 5 3 1 1 7 6 4 0 2 4 9 0 8 2 8 4 6 3 2 8 1 9 9 2 4 1 7 3 1 6 4 2 8 0 4 9 5 7 8 9 5 8 4 7 0 5 6 6 1 8 3 3 5 . 1 8 3 3 4 , 1 8 3 3 3 , 1 8 3 3 2 , 1 8 3 3 1 , 1 8 3 3 1 , 1 8 3 3 0 1 8 3 2 9 1 8 3 2 8 1 8 3 2 7 , 1 8 3 2 5 , 1 8 3 2 4 1 8 3 2 3 1 8 3 2 2 1 8 3 2 1 1 8 3 1 9 1 6 8 4 9 4 1 7 0 2 6 2 8 0 5 8 0 0 8 2 9 3 6 4 3 0 3 4 5 0 0 9 4 7 1 1 1 7 3 8 1 0 2 4 2 0 4 9 7 9 9 5 9 6 3 8 3 1 7 2 6 6 6 5 0 4 6 3 3 7 2 2 2 6 9 9 4 4 3 1 Q BRANCH 1 8 3 3 6 . 8 1 1 4 9 1 8 3 3 6 . 7 1 7 3 9 1 8 3 3 6 . 5 8 5 3 8 1 8 3 3 6 . 4 1 5 9 6 1 8 3 3 6 . 2 0 8 6 3 1 8 3 3 5 . 9 6 3 7 5 1 8 3 3 5 . 6 8 0 8 6 1 8 3 3 5 . 3 6 0 9 8 1 8 3 3 5 . 0 0 2 6 0 1 8 3 3 4 . 6 0 7 6 3 1 8 3 3 4 . 1 7 3 8 9 1 8 3 3 3 . 7 0 3 6 9 1 8 3 3 3 . 1 9 3 6 1 1 8 3 1 7 . 2 7 4 8 6 1 8 3 1 5 . 8 8 4 3 1 1 8 3 1 4 . 4 5 5 8 8 APPENDIX I I I , c o n t i n u e d , 000 - 000 X /a ( 1 0B02) . 133 J R BRANCH P BRANCH Q BRANCH 2 2 . 5 1 8 3 4 1 . 4 7 6 6 4 1 8 3 1 2 . 9 8 9 7 0 2 3 . 5 1 8 3 4 1 . 2 1 1 6 6 1 8 3 1 1 . 4 8 5 9 2 2 4 . 5 1 8 3 4 0 . 9 0 6 8 8 1 8 3 0 9 . 9 4 4 9 9 2 5 . 5 1 8 3 4 0 . 5 6 6 2 9 1 8 3 0 8 . 3 6 6 0 6 2 6 . 5 1 8 3 4 0 . 1 8 6 4 6 1 8 3 0 6 . 7 5 0 3 6 2 7 . 5 1 8 3 3 9 . 7 7 1 3 4 1 8 3 0 5 . 0 9 6 3 9 2 8 . 5 1 8 3 3 9 . 3 1 6 2 9 1 8 3 0 3 . 4 0 5 5 4 2 9 . 5 1 8 3 3 8 . 8 2 5 4 7 1 8 3 0 1 . 6 7 6 7 1 3 0 . 5 1 8 3 3 8 . 3 0 0 6 8 1 8 2 9 9 . 9 1 1 7 3 3 1 . 5 1 8 3 3 7 . 7 2 8 4 2 1 8 2 9 8 . 1 0 7 4 3 3 2 . 5 1 8 3 3 7 . 1 2 1 9 0 1 8 2 9 6 . 2 7 2 8 5 3 3 . 5 1 8 3 3 6 . 4 8 1 4 1 1 8 2 9 4 . 3 8 7 6 7 3 4 . 5 1 8 3 3 5 . 8 0 0 3 8 1 8 2 9 2 . 4 7 1 8 2 3 5 . 5 1 8 3 3 5 . 0 8 4 8 7 1 8 2 9 0 . 5 1 8 7 0 3 6 . 5 1 8 3 3 4 . 3 2 6 7 9 1 8 2 8 8 . 5 2 8 9 6 3 7 . 5 1 8 3 3 3 . 5 3 6 9 6 1 8 2 8 6 . 5 0 1 2 0 3 8 . 5 1 8 3 3 2 . 7 0 3 5 8 1 8 2 8 4 . 4 3 4 9 5 3 9 . 5 1 8 3 3 1 . 8 3 9 1 0 1 8 2 8 2 . 3 3 3 1 9 4 0 . 5 1 8 3 3 0 . 9 3 0 4 6 1 8 2 8 0 . 1 9 2 6 9 4 1 . 5 1 8 3 2 9 . 9 9 2 1 2 1 8 2 7 8 . 0 1 6 3 3 4 2 . 5 1 8 3 2 9 . 0 0 6 0 6 1 8 2 7 5 . 8 0 0 8 8 4 3 . 5 1 8 3 2 7 . 9 9 4 4 5 1 8 2 7 3 . 5 5 1 4 4 4 4 . 5 1 8 3 2 6 . 9 3 4 2 0 1 8 2 7 1 . 2 5 9 2 3 4 5 . 5 1 8 3 2 5 . 8 4 5 8 2 1 8 2 6 8 . 9 3 6 3 4 4 6 . 5 1 8 3 2 4 . 7 1 0 5 5 1 8 2 6 6 . 5 7 0 7 4 4 7 . 5 1 8 3 2 3 . 5 4 8 1 1 1 8 2 6 4 . 1 7 1 3 7 4 8 . 5 1 8 3 2 2 . 3 3 8 1 4 1 8 2 6 1 . 7 3 1 8 3 4 9 . 5 1 8 3 2 1 . 1 0 0 1 5 1 8 2 5 9 . 2 5 8 4 2 5 0 . 5 1 8 2 5 6 . 7 4 4 9 9 5 1 . 5 1 8 2 5 4 . 1 9 6 9 1 5 2 . 5 1 8 2 5 1 . 6 0 9 9 1 5 3 . 5 1 8 2 4 8 . 9 8 9 4 3 5 4 . 5 1 8 3 1 4 . 3 2 0 0 3 1 8 2 4 6 . 3 2 4 7 3 5 5 . 5 1 8 3 1 2 . 8 6 0 7 6 1 8 2 4 3 . 6 3 1 2 5 5 6 . 5 1 8 2 4 0 . 8 9 2 0 3 5 8 . 5 1 8 2 3 5 . 3 1 1 2 8 5 9 . 5 1 8 2 3 2 . 4 6 8 7 7 000 - 000 X / a ( 1 0BO2) J R BRANCH P BRANCH 1. 5 18291 . 1 2 2 0 9 2 . 5 18291 . 6 2 6 6 1 18287 . 8 9 8 0 8 3 . 5 18292 . 1 5 3 9 0 18287 . 1 5 4 8 4 4 . 5 18292 . 5 7 2 9 9 18286 . 3 5 1 1 6 5 . 5 18293 . 0 3 8 3 0 18285 . 5 4 6 7 1 6 . 5 18293 . 3 7 2 4 6 18284 . 6 5 6 8 1 7 . 5 18293 . 7 7 6 6 8 18283 . 7 9 0 1 0 8 . 5 18294 . 0 2 4 5 8 18282 . 8 1 5 6 2 9 . 5 18294 . 3 6 7 3 1 18281 . 8 8 7 8 9 1 0 . 5 18294 . 5 1 9 7 4 18280 . 8 2 7 6 4 Q BRANCH APPENDIX I I I , c o n t i n u e d , 000 - 000 C^BO?.) . 134 J R BRANCH P BRANCH 11. 5 18294 .80838 18279 .83844 12. 5 18294 .88672 18278 .68261 13. 5 18295 .10554 18277 .64002 14. 5 18295 .09870 18276 .41011 15. 5 18295 .25066 18275 .29724 16. 5 18295 .16044 18273 .98235 17. 5 18295 .25632 18. 5 18295 .06901 18271 .40535 19. 5 18295 .10838 18270 .17082 20. 5 18294 .84166 18268 .67444 21. 5 18294 .81376 18267 .38487 22. 5 18294 .46206 18265 .81011 23. 5 18294 .37146 18264 .45267 24. 5 18293 .93316 18262 .79324 25. 5 18293 .78275 18261 .37320 26. c 18293 .26898 18259 .62730 27. 5 18293 .05563 18258 .14849 28. 5 18292 .45716 18256 .32731 29. 5 18292 .16310 18254 .78587 30. 5 18291 .46238 18252 .87999 31. 5 18291 .13120 18251 .25890 32. 5 18290 .35835 18249 .25101 33. 5 18289 .95270 18247 .59286 34. 5 18289 .08997 18245 .51312 35. 5 18288 .62756 18243 .78125 36. 5 18287 .67616 18241 .61197 37. 5 18287 .15484 18239 .82386 38. 5 18286 .12217 18237 .56641 39. 5 18285 .53228 18235 .71931 40. 5 18284 .41089 18233 .38107 41. 5 18283 .76110 18231 .4 6657 42. 5 18282 .55645 18229 .03959 43. 5 18281 .84832 18227 .06622 44. 5 18280 .55842 18224 .55674 45. 5 18279 .78551 18222 .52625 46. 5 18278 .41532 47. 5 18277 .58079 48. 5 18276 .11368 49. 5 18275 .20517 50. 5 18273 .66566 53. 5 18270 .05443 55. 5 18267 .25270 Q BRANCH 010 - 010 2 Ae / 2 ("B02) J R BRANCH 2.5 18350.18298 3.5 18350.66988 4.5 18351.11814 5.5 18351.52774 6.5 18351.89887 P BRANCH 18345.69942 18344.90510 18344.07291 18343.20157 Q BRANCH 18348.00873 18347.87396 18347.70128 18347.48961 18347.23931 APPENDIX I I I , c o n t i n u e d , 010 - 010 2 A s / 2 ( n B0 2 ) . 135 R BRANCH P BRANCH 7 . 5 8 . 5 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2 6 . 5 2 7 . 5 28 29 30 31 32 33 34 35 36 37 38 39 40 18352 18352 18352 18352 18353 18353 18353 18353 18353 18353 18353 18353 18353 18353 18352 18352 18352 18352 18351 18351 18350 18350 18349 18349 18348 18348 18347 18347 18345 18345 18344 18344 . 2 3 1 4 0 . 5 2 5 9 4 . 7 8 1 7 5 . 9 9 9 4 1 . 1 7 7 9 5 . 3 1 9 2 2 . 4 2 0 2 9 . 4 8 5 5 7 . 5 0 8 9 2 . 4 9 9 8 1 . 4 4 4 2 9 . 3 6 2 1 9 . 2 2 5 8 1 . 0 7 4 0 7 . 8 5 3 6 0 . 6 3 9 6 2 . 3 2 7 8 9 . 0 6 5 6 3 . 6 4 8 7 9 . 3 6 4 7 8 . 8 1 7 1 2 . 5 5 1 4 9 . 8 3 0 4 9 . 6 1 7 7 6 . 6 9 1 5 0 . 5 5 0 3 6 . 3 9 8 7 2 . 3 3 8 3 1 . 9 5 4 3 8 . 9 7 9 0 6 . 3 5 6 6 5 . 4 6 7 9 7 1 8 3 4 1 . 1 8 3 4 0 . 1 8 3 3 9 . 1 8 3 3 8 . 1 8 3 3 7 . 1 8 3 3 6 . 18334, 1 8 3 3 3 . 18332 . 1 8 3 3 1 . 18329 , 18328, 18326 , 18325, 18324, 18322, 18320, 18319, 18317, 18316, 18314, 18312, 18310, 18308, 18307, 18305, 18303, 18301, 18299, 18297, 18295, 18292, 18291 , 34399 35750 33255 26934 16813 02774 85014 63234 37870 08370 75585 38189 98142 52691 05737 51918 98730 35812 7 7 8 2 9 04447 44343 57831 99575 95833 42946 18723 72987 26226 88621 18582 89568 95904 75484 Q BRANCH 1 8 3 4 6 . 9 5 1 0 7 1 8 3 4 6 . 6 2 4 3 2 1 8 3 4 6 . 2 5 8 7 8 1 8 3 4 5 . 8 5 5 0 5 1 8 3 4 5 . 4 1 3 4 9 J 1. 2 . 3 . 4 . 5 . 6 . 5 7 . 5 8 . 5 9 . 5 1 0 . 5 1 2 . 5 1 3 . 5 1 4 . 5 1 5 . 5 R BRANCH 1 8 3 0 5 . 7 0 4 6 9 1 8 3 0 6 . 2 3 8 3 4 1 8 3 0 6 . 7 2 7 7 1 1 8 3 0 7 . 1 8 6 1 8 1 8 3 0 7 . 5 9 9 1 0 1 8 3 0 7 . 9 8 5 7 4 1 8 3 0 8 . 3 2 2 4 7 010 - 010 2 A 3 / 2 ("B02) P BRANCH 18308, 18309, 18309, 18309, 18309 , 18309 , 88528 12530 47145 57020 66400 68217 1 8 3 0 1 . 7 2 7 7 4 1 8 3 0 0 . 9 3 5 6 8 1 8 3 0 0 . 0 9 8 9 8 1 8 2 9 9 . 2 3 3 1 8 18297, 18296, 18295, 18293, 18292, 18290 18289, 37977 39153 37142 21753 07121 91336 68735 Q BRANCH APPENDIX I I I , c o n t i n u e d , 010 - 010 2 A 3 / 2 ( U B0 2 ) . 136 J R BRANCH P BRANCH Q BRANCH 1 6 . 5 1 8 3 0 9 . 7 1 1 2 1 1 8 2 8 8 . 4 5 6 1 1 1 7 . 5 1 8 3 0 9 . 6 4 6 6 9 1 8 . 5 1 8 3 0 9 . 6 0 6 0 5 1 8 2 8 5 . 8 5 3 5 6 1 9 . 5 1 8 3 0 9 . 4 5 8 3 6 1 8 2 8 4 . 4 6 5 0 6 2 0 . 5 1 8 3 0 9 . 3 3 7 8 3 1 8 2 8 3 . 0 9 9 8 6 2 1 . 5 1 8 3 0 9 . 1 0 8 6 6 1 8 2 8 1 . 6 2 8 0 2 2 2 . 5 1 8 3 0 8 . 9 2 6 7 4 1 8 2 8 0 . 1 8 3 2 3 2 3 . 5 1 8 3 0 8 . 6 2 4 5 0 1 8 2 7 8 . 6 2 9 9 5 2 4 . 5 1 8 3 0 8 . 3 6 6 0 6 1 8 2 7 7 . 1 2 4 2 9 2 5 . 5 1 8 3 0 7 . 9 7 9 8 8 1 8 2 7 5 . 4 9 9 1 9 2 6 . 5 1 8 3 0 7 . 6 5 5 2 8 1 8 2 7 3 . 9 1 5 2 9 2 7 . 5 1 8 3 0 7 . 1 8 6 1 8 1 8 2 7 2 . 2 0 6 9 4 2 8 . 5 1 8 3 0 6 . 7 9 1 5 6 1 8 2 7 0 . 5 5 9 0 9 2 9 . 5 1 8 3 0 6 . 2 4 3 3 5 1 8 2 6 8 . 7 6 6 9 5 3 0 . 5 1 8 3 0 5 . 7 8 1 0 1 1 8 2 6 7 . 0 4 9 4 8 3 1 . 5 1 8 3 0 5 . 1 2 3 9 2 1 8 2 6 5 . 1 7 9 0 5 3 2 . 5 1 8 3 0 4 . 6 1 4 3 1 1 8 2 6 3 . 3 9 4 3 8 3 3 . 5 1 8 3 0 3 . 8 9 1 5 9 1 8 2 6 1 . 4 1 5 2 3 3 4 . 5 1 8 3 0 3 . 2 9 9 2 0 1 8 2 5 9 . 5 8 3 7 5 3 5 . 5 1 8 3 0 2 . 4 8 6 1 8 1 8 2 5 7 . 5 3 9 2 2 3 6 . 5 1 8 3 0 1 . 8 3 4 1 0 1 8 2 5 5 . 6 2 5 7 5 3 7 . 5 1 8 2 5 3 . 4 9 1 9 0 3 8 . 5 1 8 3 0 0 . 2 1 8 4 3 1 8 2 5 1 . 5 1 9 1 8 4 0 . 5 1 8 2 9 8 . 4 5 3 5 4 1 8 2 4 7 . 2 6 2 6 4 4 2 . 5 1 8 2 9 6 . 5 3 7 9 0 1 8 2 4 2 . 8 5 8 1 4 4 4 . 5 1 8 2 3 8 . 3 0 3 7 2 4 6 . 5 1 8 2 9 2 . 2 5 5 3 6 1 8 2 3 3 . 5 9 8 6 2 4 8 . 5 1 8 2 2 8 . 7 4 8 2 4 010 - 010 2 S * - 2 S * ( U B0 2 ) J R BRANCH P BRANCH Q BRANCH 0 . 5 1 8 3 8 3 . 5 7 8 8 8 1 . 5 1 8 3 8 4 . 4 8 5 2 3 1 8 3 8 2 . 1 6 8 3 4 2 . 5 1 8 3 8 4 . 3 2 0 9 9 1 8 3 8 0 . 4 2 4 4 7 3 . 5 1 8 3 8 5 . 8 3 0 6 1 1 8 3 8 1 . 0 2 5 9 3 4 . 5 1 8 3 8 4 . 9 0 3 3 6 1 8 3 7 8 . 5 2 1 1 4 5 . 5 1 8 3 8 7 . 0 1 6 5 2 1 8 3 7 9 . 7 2 5 7 2 6 . 5 1 8 3 8 5 . 3 2 6 3 8 1 8 3 7 6 . 4 5 8 7 3 7 . 5 1 8 3 8 8 . 0 4 3 3 6 1 8 3 7 8 . 2 6 6 6 2 8 . 5 1 8 3 8 5 . 5 8 9 7 4 1 8 3 7 4 . 2 3 6 7 6 9 . 5 1 8 3 8 8 . 9 1 0 1 7 1 8 3 7 6 . 6 4 7 6 3 1 0 . 5 1 8 3 8 5 . 6 9 2 1 2 1 8 3 7 1 . 8 5 4 9 7 1 1 . 5 1 8 3 8 9 . 6 1 8 3 6 1 8 3 7 4 . 8 6 9 9 8 1 2 . 5 1 8 3 8 5 . 6 3 5 4 6 1 8 3 6 9 . 3 1 3 9 0 1 3 . 5 1 8 3 9 0 . 1 6 6 9 1 1 8 3 7 2 . 9 3 2 9 9 1 4 . 5 1 8 3 8 5 . 4 1 7 6 1 1 8 3 6 6 . 6 1 2 9 3 1 5 . 5 1 8 3 9 0 . 5 5 6 6 0 1 8 3 7 0 . 8 3 7 0 6 1 6 . 5 1 8 3 8 5 . 0 4 0 4 1 1 8 3 6 3 . 7 5 1 8 1 1 7 . 5 1 8 3 9 0 . 7 8 6 9 0 1 8 3 6 8 . 5 8 2 0 1 1 8 . 5 1 8 3 8 4 . 5 0 1 0 5 1 8 3 6 0 . 7 3 1 5 6 APPENDIX I I I , c o n t i n u e d , 010 - 010 ^ + - 2 S + ( n B0 2 ) . J R BRANCH P BRANCH Q BRANCH 1 9 . 5 1 8 3 9 0 . 8 5 7 3 7 1 8 3 6 6 . 1 6 8 7 6 2 0 . 5 1 8 3 8 3 . 7 9 8 1 8 1 8 3 5 7 . 5 4 9 7 9 2 1 . 5 1 8 3 9 0 . 7 6 9 1 5 1 8 3 6 3 . 5 9 6 3 8 2 2 . 5 1 8 3 8 2 . 9 2 8 5 2 1 8 3 5 4 . 2 0 5 2 8 2 3 . 5 1 8 3 9 0 . 5 2 2 7 5 1 8 3 6 0 . 8 6 5 7 1 2 4 . 5 1 8 3 8 1 . 8 8 5 0 5 1 8 3 5 0 . 6 9 4 7 0 2 5 . 5 1 8 3 9 0 . 1 1 6 2 9 1 8 3 5 7 . 9 7 6 2 0 2 6 . 5 1 8 3 8 0 . 6 5 4 4 9 1 8 3 4 7 . 0 1 0 9 5 2 7 . 5 1 8 3 8 9 . 5 5 0 9 2 1 8 3 5 4 . 9 2 8 6 7 2 8 . 5 1 8 3 7 9 . 2 2 4 8 7 1 8 3 4 3 . 1 4 1 0 0 2 9 . 5 1 8 3 8 8 . 8 2 7 0 8 1 8 3 5 1 . 7 2 2 2 9 3 0 . 5 1 8 3 7 7 . 6 0 1 8 3 1 8 3 3 9 . 0 7 2 9 0 3 1 . 5 1 8 3 8 7 . 9 4 4 3 4 1 8 3 4 8 . 3 5 8 6 2 3 2 . 5 1 8 3 3 4 . 8 1 2 2 4 3 3 . 5 1 8 3 8 6 . 9 0 2 6 8 1 8 3 4 4 . 8 3 6 7 6 3 4 . 5 1 8 3 3 0 . 3 7 1 8 8 3 5 . 5 1 8 3 8 5 . 7 0 3 7 1 1 8 3 4 1 . 1 5 7 1 0 3 6 . 5 1 8 3 2 5 . 7 6 6 6 4 3 7 . 5 1 8 3 8 4 . 3 4 5 4 2 1 8 3 3 7 . 3 1 9 5 6 3 9 . 5 1 8 3 8 2 . 8 2 8 1 9 1 8 3 3 3 . 3 2 4 6 1 4 1 . 5 1 8 3 8 1 . 1 5 3 3 3 1 8 3 2 9 . 1 7 1 6 2 4 3 . 5 1 8 3 7 9 . 3 1 9 9 3 1 8 3 2 4 . 8 6 1 5 8 010 - 010 ^ " - 2 Z + ( U B0 2 ) J R BRANCH P BRANCH Q BRANCH 0 . 5 1 8 4 8 7 . 8 3 9 8 5 1 8 4 8 7 . 1 5 4 4 4 1 . 5 1 8 4 8 6 . 9 2 5 9 1 2 . 5 1 8 4 8 8 . 2 7 1 3 9 1 8 4 8 4 . 6 8 6 2 7 3 . 5 1 8 4 9 1 . 3 0 2 0 8 1 8 4 8 6 . 1 3 4 9 4 4 . 5 1 8 4 8 8 . 5 4 9 7 5 1 8 4 8 2 . 4 7 1 3 3 5 . 5 1 8 4 9 2 . 8 6 4 7 5 1 8 4 8 5 . 1 9 7 1 9 6 . 5 1 8 4 8 8 . 6 8 7 1 0 1 8 4 8 0 . 1 0 5 3 3 7 . 5 1 8 4 9 4 . 2 8 1 5 1 1 8 4 8 4 . 1 1 4 8 4 8 . 5 1 8 4 8 8 . 6 8 0 0 6 1 8 4 7 7 . 5 9 7 9 0 9 . 5 1 8 4 9 5 . 5 5 2 1 7 1 8 4 8 2 . 8 8 6 5 5 1 0 . 5 1 8 4 8 8 . 5 2 6 9 2 1 8 4 7 4 . 9 4 5 3 5 1 1 . 5 1 8 4 9 6 . 6 7 6 6 7 1 8 4 8 1 . 5 1 2 3 1 1 2 . 5 1 8 4 8 8 . 2 2 9 4 9 1 8 4 7 2 . 1 4 8 2 5 1 3 . 5 1 8 4 9 7 . 6 5 4 9 7 1 8 4 7 9 . 9 9 1 9 2 1 4 . 5 1 8 4 8 7 . 7 8 8 2 2 1 8 4 6 9 . 2 0 7 2 0 1 5 . 5 1 8 4 9 8 . 4 8 7 1 7 1 8 4 7 8 . 3 2 5 2 2 1 6 . 5 1 8 4 8 7 . 1 9 8 7 9 1 8 4 6 6 . 1 2 2 3 8 1 7 . 5 1 8 4 9 9 . 1 7 1 6 1 1 8 4 7 6 . 5 1 3 4 6 1 8 . 5 1 8 4 8 6 . 4 6 8 9 5 1 8 4 6 2 . 8 9 0 1 1 1 9 . 5 1 8 4 9 9 . 7 0 9 3 5 1 8 4 7 4 . 5 5 3 9 9 2 0 . 5 1 8 4 8 5 . 5 9 3 0 0 1 8 4 5 9 . 5 1 7 8 8 2 1 . 5 1 8 5 0 0 . 1 0 1 5 6 1 8 4 7 2 . 4 4 9 4 5 2 2 . 5 1 8 4 8 4 . 5 7 3 1 6 1 8 4 5 6 . 0 0 0 5 8 . 2 3 . 5 1 8 5 0 0 . 3 4 6 7 4 1 8 4 7 0 . 1 9 8 2 3 2 4 . 5 1 8 4 8 3 . 4 0 9 7 0 1 8 4 5 2 . 3 3 9 9 2 APPENDIX I I I , c o n t i n u e d , 010 - 010 2 Z ~- 2 2 + ( U B0 2 ) . J R BRANCH P BRANCH Q BRANCH 2 5 . 5 1 8 5 0 0 . 4 4 4 9 3 1 8 4 6 7 . 8 0 1 1 0 2 6 . 5 1 8 4 8 2 . 1 0 2 6 5 2 7 . 5 1 8 5 0 0 . 3 9 6 2 8 1 8 4 6 5 . 2 5 7 8 6 2 8 . 5 1 8 4 8 0 . 6 5 1 1 0 2 9 . 5 1 8 5 0 0 . 2 0 0 2 7 1 8 4 6 2 . 5 6 8 3 4 3 0 . 5 1 8 4 7 9 . 0 5 4 9 0 3 1 . 5 1 8 4 9 9 . 8 5 6 9 4 1 8 4 5 9 . 7 3 1 9 9 3 2 . 5 1 8 4 7 7 . 3 1 9 2 2 3 3 . 5 1 8 4 9 9 . 3 6 6 9 9 1 8 4 5 6 . 7 4 9 6 0 3 4 . 5 1 8 4 7 5 . 4 3 6 9 6 3 5 . 5 1 8 4 9 8 . 7 3 1 3 5 1 8 4 5 3 . 6 2 1 0 4 3 6 . 5 1 8 4 7 3 . 4 1 6 1 3 3 7 . 5 1 8 4 9 7 . 9 4 4 5 8 1 8 4 5 0 . 3 4 6 7 5 3 8 . 5 1 8 4 7 1 . 2 2 4 8 1 3 8 . 5 1 8 4 7 1 . 2 6 8 4 9 3 9 . 5 1 8 4 9 7 . 0 1 8 7 3 1 8 4 4 6 . 9 2 3 5 9 4 0 . 5 1 8 4 6 8 . 9 4 2 9 2 4 2 . 5 1 8 4 6 6 . 4 8 7 6 5 4 4 . 5 1 8 4 6 3 . 8 9 2 8 6 4 6 . 5 1 8 4 6 1 . 1 3 1 3 9 4 8 . 5 1 8 4 5 8 . 2 0 7 0 3 5 0 . 5 1 8 4 5 5 . 0 8 6 0 0 010 - 010 2 2 + - 2 A 5 / 2 ( U B0 2 ) J R BRANCH P BRANCH Q BRANCH 2 4 . 5 1 8 3 5 1 . 6 1 4 2 0 2 6 . 5 1 8 3 5 0 . 9 7 0 1 2 2 8 . 5 1 8 3 5 0 . 1 3 2 1 9 1 8 3 1 4 . 0 4 8 6 2 3 0 . 5 1 8 3 4 9 . 1 0 5 5 6 1 8 3 1 0 . 5 7 6 5 6 3 2 . 5 1 8 3 4 7 . 9 0 2 9 7 1 8 3 0 6 . 9 1 6 9 5 3 4 . 5 1 8 3 4 6 . 5 3 8 8 4 1 8 3 0 3 . 0 8 1 3 7 3 6 . 5 1 8 3 4 5 . 0 1 4 3 7 1 8 2 9 9 . 0 8 6 3 1 3 8 . 5 1 8 3 4 3 . 3 3 2 3 0 1 8 2 9 4 . 9 3 1 0 3 4 0 . 5 1 8 3 4 1 . 4 9 4 5 2 1 8 2 9 0 . 6 1 9 0 9 4 2 . 5 1 8 3 3 9 . 5 0 1 0 2 1 8 2 8 6 . 1 5 2 7 9 4 4 . 5 1 8 3 3 7 . 3 5 2 1 5 1 8 2 8 1 . 5 3 1 5 4 4 6 . 5 1 8 3 3 5 . 0 4 8 9 3 1 8 2 7 6 . 7 5 5 9 3 4 8 . 5 1 8 3 3 2 . 5 9 0 1 7 1 8 2 7 1 . 8 2 6 9 5 5 0 . 5 1 8 2 6 6 . 7 4 3 3 7 010 - 010 'Ae/a- 2^* ("BOa) J R BRANCH P BRANCH Q BRANCH 2 4 . 5 1 8 3 8 2 . 3 3 6 4 1 2 6 . 5 1 8 3 8 1 . 0 4 9 5 3 1 8 3 4 7 . 4 6 2 5 8 2 8 . 5 1 8 3 7 9 . 6 4 4 3 3 1 8 3 4 3 . 5 3 5 8 3 3 0 . 5 1 8 3 7 8 . 1 1 4 4 7 1 8 3 3 9 . 4 9 2 1 7 3 2 . 5 1 8 3 7 6 . 4 4 6 0 3 1 8 3 3 5 . 3 2 4 6 4 3 4 . 5 1 8 3 7 4 . 6 2 8 8 3 1 8 3 3 1 . 0 1 9 1 4 3 6 . 5 1 8 3 7 2 . 6 5 9 4 7 1 8 3 2 6 . 5 6 5 9 4 APPENDIX I I I , c o n t i n u e d , 010 - 010 X / a - 2 ^ * ( " B O o ) . 139 J R BRANCH P BRANCH 3 8 . 5 18370 . 5 3 4 5 5 1 8 3 2 1 . 9 6 2 8 7 4 0 . 5 18368 . 2 5 4 1 9 4 2 . 5 18365 . 8 1 9 6 3 1 8 3 1 2 . 2 9 1 4 4 44 . 5 18363 . 2 2 4 3 2 1 8 3 0 7 . 2 2 6 1 5 4 6 . 5 18360 . 4 7 4 6 1 4 8 . 5 18357 . 5 6 9 4 9 1 8 2 9 6 . 6 2 3 0 4 5 0 . 5 18354 . 5 1 0 4 6 Q BRANCH 010 - 010 ^ ~ -'z ~ ("BO 2) J r R BRANCH P BRANCH 1 . 5 18260 . 3 8 5 8 7 2 . 5 18261 . 6 0 4 9 9 18258 . 0 2 0 3 0 3 .5 18261 .08204 18255 . 9 1 5 0 7 4 . 5 18262 . 9 0 5 1 8 18256 . 8 2 6 8 4 5 . 5 18261 . 6 3 5 9 0 18253 . 9 6 8 8 2 6 . 5 18264 . 0 6 5 7 8 18255 . 4 8 3 5 7 7 . 5 18262 . 0 4 6 4 7 18251 . 8 7 9 6 1 8 . 5 18265 .08344 18254 . 0 0 1 8 7 9 . 5 18262 . 3 1 3 9 3 18249 . 6 4 8 5 5 10 . 5 18265 .95864 18252 . 3 7 6 8 2 11 . 5 18262 . 4 3 8 2 2 18247 . 2 7 3 7 7 12 . 5 18266 . 6 9 0 1 9 18250 . 6 0 9 4 5 13 . 5 18262 . 4 1 9 5 0 18244 . 7 5 6 3 6 14 . 5 18267 . 2 7 9 8 2 18248 . 6 9 9 2 4 15 . 5 18262 . 2 5 7 9 6 18242 . 0 9 6 2 5 16 . 5 18267 . 7 2 3 0 7 18246 . 6 4 6 9 2 17 . 5 18261 . 9 5 2 4 5 18239 . 2 9 4 2 9 18 . 5 18268 . 0 2 7 1 2 18244 . 4 4 9 1 4 19 . 5 18261 . 5 0 4 1 2 18236 . 3 4 8 4 5 20 . 5 18268 . 1 8 6 5 9 18242 . 1 1 1 2 2 21 . 5 18260 . 9 1 2 8 2 22 . 5 18268 . 2 0 2 7 2 23 . 5 18260 . 1 7 8 2 0 18230 . 0 2 9 7 0 24 .5 18268 . 0 7 5 7 4 18237 . 0 0 5 9 5 25 . 5 18259 . 3 0 0 4 1 18226 . 6 5 6 4 4 26 . 5 18267 . 8 0 6 6 0 18234 . 2 3 9 6 5 27 . 5 18258 . 2 7 9 4 8 18223 . 1 4 1 0 4 28 . 5 18267 . 3 9 1 9 5 18231 . 3 3 0 7 0 29 . 5 18257 . 1 1 4 9 8 18219 . 4 8 3 1 7 30 . 5 18266 . 8 3 4 0 3 18228 . 2 7 7 9 7 31 . 5 18255 . 8 0 7 3 5 18215 . 6 8 2 2 4 32 . 5 18266 . 1 3 5 8 2 18225 . 0 8 2 0 5 33 . 5 18254 . 3 5 6 6 0 34 . 5 18265 . 2 9 1 5 1 18221 . 7 4 7 3 6 35 . 5 18252 . 7 6 4 6 1 36 . 5 18264 . 3 0 4 9 6 18218 . 2 6 6 5 1 37 . 5 18251 . 0 2 4 8 4 38 . 5 18263 . 1 5 2 0 6 38 . 5 18263 . 1 9 5 3 4 40 . 5 18261 . 9 0 5 2 2 140 APPENDIX IV. C a l c u l a t e d ground s t a t e energy l e v e l s of n B 0 2 i n wave numbers (cm"1) . J 0 . 5 148 .25574 1.5 0 .0 149 .24895 2 . 5 1 .64312 150 .88510 3 . 5 3 .94351 153 .21601 4 . 5 6 .90108 156 .16108 5 .5 10 .51594 159 .82385 6 . 5 14 .78789 164 .07779 7 . 5 19 .71720 169 .07233 8 . 5 2 5 . 3 0 3 4 0 174 .63510 9 . 5 31 .54710 180 .96126 1 0 . 5 3 8 . 4 4 7 4 6 187 .83279 1 1 . 5 46 .00544 195 .49039 1 2 . 5 54 .21964 203 .67062 1 3 . 5 6 3 . 0 9 1 9 6 212 .65943 1 4 . 5 7 2 . 6 1 9 8 3 222 .14827 1 5 . 5 82 .80634 232 .46803 1 6 . 5 93 .64762 243 .26537 1 7 . 5 105 .14821 254 .91579 18 .5 117 .30260 267 .02152 1 9 . 5 130 .11719 280 .00226 2 0 . 5 143 .58434 2 9 3 . 4 1 6 2 5 2 1 . 5 157 .71279 307 .72694 2 2 . 5 172 .49234 322 .44904 2 3 . 5 187 .93452 338 .08926 2 4 . 5 204 .02605 354 .11932 2 5 . 5 220 .78182 371 .08862 2 6 . 5 238 .18489 388 .42645 27 .5 256 .25408 406 .72436 2 8 . 5 274 .96821 425 .36977 2 9 . 5 2 9 4 . 3 5 0 6 5 444 .99575 3 0 . 5 314 .37532 464 .94855 3 1 . 5 335 .07083 485 .90204 3 2 . 5 356 .40548 507 .16200 3 3 . 5 378 .41386 529 .44240 34 .5 401 .05791 552 .00929 3 5 . 5 424 .37894 575 .61597 3 6 . 5 4 4 8 . 3 3 1 7 5 599 .48953 3 7 . 5 472 .96522 624 .42181 3 8 . 5 498 .22614 649 .60179 3 9 . 5 524 .17179 675 .85896 4 0 . 5 550 .74013 7 0 2 . 3 4 5 0 6 4 1 . 5 577 .99771 729 .92638 4 2 . 5 605 .87274 757 .71832 4 3 . 5 634 .44198 786 .62300 4 4 . 5 663 .62293 815 .72046 4 5 . 5 6 9 3 . 5 0 3 5 5 845 .94769 4 6 . 5 723 .98962 876 .35033 4 7 . 5 755 .18131 907 .89925 4 8 . 5 786 .97168 939 .60674 4 9 . 5 819 .47413 972 .47646 5 0 . 5 852 .56792 1005 .48843 5 1 . 5 886 .38081 1039 .67802 APPENDIX I V , c o n t i n u e d , ground s t a t e energy l e v e l s . 141 J l l 3 / 2 5 2 . 5 9 2 0 . 7 7 7 1 2 5 3 . 5 9 5 5 . 9 0 0 1 0 5 4 . 5 9 9 1 . 5 9 7 9 9 5 5 . 5 1 0 2 8 . 0 3 0 6 9 5 6 . 5 1 0 6 5 . 0 2 9 2 0 5 7 . 5 1 1 0 2 . 7 7 1 2 5 5 8 . 5 1 1 4 1 . 0 6 9 3 8 5 9 . 5 1 1 8 0 . 1 2 0 3 9 6 0 . 5 1 2 1 9 . 7 1 7 0 9 6 1 . 5 1 2 6 0 . 0 7 6 6 4 6 2 . 5 1 3 0 0 . 9 7 0 8 6 6 3 . 5 1 3 4 2 . 6 3 8 5 3 6 5 . 5 1 4 2 7 . 8 0 4 5 0 u l / 2 1 0 7 3 . 9 9 4 0 9 1 1 0 9 . 5 0 2 6 0 1 1 4 5 . 1 2 2 3 8 1 1 8 1 . 9 4 8 8 1 1 2 1 8 . 8 7 1 8 8 1 2 5 7 . 0 1 5 1 9 1 2 9 5 . 2 4 1 1 3 J V 0 . 5 4 0 5 . 5 4 0 3 2 6 3 3 . 2 2 4 9 0 1 . 5 4 0 5 . 7 6 8 6 9 6 3 4 . 9 7 6 4 8 2 . 5 4 0 8 . 6 9 4 7 9 6 3 5 . 3 6 0 6 9 3 . 5 4 0 9 . 2 2 7 7 5 6 3 9 . 4 4 7 6 1 4 . 5 414 . 4 9 4 5 2 6 4 0 . 1 3 9 3 1 5 . 5 4 1 5 . 3 3 2 2 3 6 4 6 . 5 6 1 3 3 6 . 5 4 2 2 . 9 3 9 3 2 647 . 5 6 0 7 3 7 . 5 4 2 4 . 0 8 2 0 8 6 5 6 . 3 1 7 4 5 8 . 5 4 3 4 . 0 2 8 9 7 6 5 7 . 6 2 4 8 5 9 . 5 4 3 5 . 4 7 7 1 6 668 . 7 1 5 7 5 1 0 . 5 4 4 7 . 7 6 3 1 8 670 . 3 3 1 5 4 1 1 . 5 4 4 9 . 5 1 7 3 0 6 8 3 . 7 5 5 8 9 1 2 . 5 464 . 1 4 1 6 2 6 8 5 . 6 8 0 6 2 1 3 . 5 4 6 6 . 2 0 2 2 5 701 . 4 3 7 7 5 1 4 . 5 4 8 3 . 1 6 3 8 8 7 0 3 . 6 7 1 8 6 1 5 . 5 4 8 5 . 5 3 1 7 4 721 . 7 6 0 7 5 1 6 . 5 5 0 4 . 8 2 9 5 2 724 . 3 0 4 9 8 17 . 5 5 0 7 . 5 0 5 4 0 744 . 7 2 4 5 3 1 8 . 5 5 2 9 . 1 3 8 0 4 7 4 7 . 5 7 9 6 4 1 9 . 5 5 3 2 . 1 2 2 8 5 7 7 0 . 3 2 8 6 3 2 0 . 5 5 5 6 . 0 8 8 8 6 7 7 3 . 4 9 5 4 8 2 1 . 5 5 5 9 . 3 8 3 6 2 7 9 8 . 5 7 2 5 2 2 2 . 5 5 8 5 . 6 8 1 3 8 802 . 0 5 2 0 4 2 3 . 5 5 8 9 . 2 8 7 2 0 8 2 9 . 4 5 5 6 2 2 4 . 5 6 1 7 . 9 1 4 9 2 8 3 3 . 2 4 8 8 6 2 5 . 5 6 2 1 . 8 3 3 0 1 862 . 9 7 7 3 0 2 6 . 5 6 5 2 . 7 8 8 7 4 8 6 7 . 0 8 5 3 9 2 7 . 5 6 5 7 . 0 2 0 4 3 8 9 9 . 1 3 6 8 8 2 8 . 5 6 9 0 . 3 0 2 0 5 9 0 3 . 5 6 1 0 4 2 9 . 5 6 9 4 . 8 4 9 7 7 9 3 7 . 9 3 3 6 3 3 0 . 5 7 3 0 . 4 5 4 0 1 9 4 2 . 6 7 5 1 7 3 1 . 5 7 3 5 . 3 1 7 2 8 9 7 9 . 3 6 6 7 4 3 2 . 5 7 7 3 . 2 4 3 7 0 984 . 4 2 7 0 7 3 3 . 5 7 7 8 . 4 2 5 1 6 1 0 2 3 . 4 3 5 3 7 3 4 . 5 8 1 8 . 6 7 0 1 6 1 0 2 8 . 8 1 5 9 9 3 5 . 5 8 2 4 . 1 7 1 5 5 1 0 7 5 . 8 4 1 1 0 APPENDIX I V , c o n t i n u e d , ground s t a t e energy l e v e l s . 142 3 6 . 3 7 . 3 8 . 3 9 . 40, 41 , 42 43 44 46 8 6 6 . 8 7 2 . 9 1 7 . 9 2 3 . 9 7 0 . 9 7 7 . 1 0 2 6 . 1 0 3 3 . 1 0 8 5 . 1 1 4 6 . 73276 55551 42921 57607 75955 23216 72217 52270 31579 53907 1070 . 1 3 8 6 1 1 1 1 9 . 4 7 5 4 9 1 1 2 5 . 5 0 1 5 4 1 1 7 1 . 4 4 4 9 7 1 1 7 7 . 7 9 6 3 6 J 2 a 5 / 2 1 . 5 2 . 5 4 4 5 . 04417 3 . 5 447 . 33312 4 . 5 4 5 0 . 32176 5 . 5 4 5 3 . 95006 6 . 5 4 5 8 . 23803 7 . 5 4 6 3 . 18562 8 .5 4 6 8 . 79283 9 .5 4 7 5 . 05962 10 . 5 4 8 1 . 78598 11 .5 4 8 9 . 57186 12 . 5 497 . 81723 13 . 5 5 0 6 . 72206 14 . 5 5 1 6 . 28633 15 .5 5 2 6 . 50992 16 . 5 5 3 7 . 39287 17 .5 5 4 8 . 93509 18 . 5 5 6 1 . 13654 19 . 5 5 7 3 . 99715 20 . 5 5 8 7 . 51688 21 . 5 6 0 1 . 69567 22 . 5 6 1 6 . 53343 23 . 5 6 3 2 . 03012 24 . 5 6 4 8 . 18566 25 .5 6 6 4 . 99997 26 . 5 6 8 2 . 47299 27 . 5 7 0 0 . 60463 28 . 5 7 1 9 . 39481 29 . 5 7 3 8 . 84344 30 . 5 7 5 8 . 95044 31 . 5 7 7 9 . 71572 32 . 5 8 0 1 . 13917 33 . 5 8 2 3 . 22072 34 . 5 8 4 5 . 96024 35 . 5 8 6 9 . 35766 36 . 5 8 9 3 . 41284 37 . 5 9 1 8 . 12569 38 . 5 9 4 3 . 49609 40 . 5 9 9 6 . 20909 589 591 593 596 599 604 609 614 621 628 635 644 653 662 672 683 695 707 720 734 748 763 778 795 812 829 847 866 886 906 927 948 971 993 1017 1041 *3/2 . 3 8 9 5 9 .04682 . 3 6 6 9 3 .34991 .99574 . 3 0 4 4 0 . 2 7 5 8 7 . 9 1 0 1 3 .20714 . 1 6 6 8 6 . 7 8 9 2 7 .07432 . 0 2 1 9 6 . 6 3 2 1 5 . 9 0 4 8 3 . 8 3 9 9 6 . 4 3 7 4 7 . 6 9 7 3 0 . 6 1 9 3 9 . 2 0 3 6 6 . 4 5 0 0 5 . 3 5 8 4 7 . 9 2 8 8 5 . 1 6 1 1 0 . 0 5 5 1 3 . 6 4 0 8 7 . 8 2 8 2 0 .70704 . 2 4 7 2 9 . 4 4 8 8 3 . 3 1 1 5 6 . 8 3 5 3 8 . 0 2 0 1 6 . 8 6 5 7 8 . 3 7 2 1 3 . 5 3 9 0 7 1 0 9 1 . 8 5 4 2 4 1 1 4 4 . 8 1 0 2 0 APPENDIX I V , c o n t i n u e d , ground s t a t e energy l e v e l s . 143 J %>/2 4 2 . 5 1 0 5 1 . 5 5 0 8 9 4 4 . 5 1 1 0 9 . 5 2 0 4 8 4 6 . 5 1 1 7 0 . 1 1 6 7 9 4 8 . 5 1 2 3 3 . 3 3 9 7 4 2 A A3/2 1 2 0 0 . 4 0 5 8 4 1 2 5 8 . 6 3 9 9 5 1 3 1 9 . 5 1 1 3 0 APPENDIX V . The upper s t a t e energy l e v e l s o f " B O 2 i n wave numbers (cm"1) . 1 J ^3/2 1 . 5 18315 . 2 0 7 3 9 18417 . 1 0 5 1 5 2 . 5 18316 . 7 5 6 3 2 18418 . 6 9 2 3 6 3 . 5 18318 . 9 2 4 6 9 18420 . 8 3 3 7 0 4 . 5 18321 . 7 1 2 7 3 18423 . 6 9 1 6 5 5 . 5 18325 . 1 2 0 5 5 18427 . 0 5 5 9 7 6 . 5 18329 . 1 4 7 0 1 18431 . 1 8 3 8 0 7 . 5 18333 . 7 9 4 1 5 18435 . 7 7 1 4 5 8 . 5 18339 . 0 6 0 1 5 18441 . 1 6 9 9 5 9 . 5 18344 . 9 4 7 2 8 18446 . 9 8 0 5 3 10 . 5 18351 . 4 4 9 1 0 18453 . 6 4 8 5 2 11 . 5 18358 . 5 7 8 4 7 18460 . 6 8 2 4 3 12 . 5 18366 . 3 1 7 1 2 18468 . 6 2 0 7 6 13 . 5 18374 . 6 8 2 8 8 18476 . 8 7 7 0 9 14 . 5 18383 . 6 6 1 7 0 18486 . 0 8 4 4 2 15 . 5 18393 . 2 6 0 8 0 18495 . 5 6 3 9 9 16 . 5 18403 . 4 8 4 1 7 18506 . 0 4 1 2 1 17 . 5 18414 . 3 2 0 6 2 18516 . 7 4 4 1 8 18 . 5 18425 . 7 8 3 5 0 18528 . 4 8 9 4 8 19 . 5 18437 . 8 5 7 3 6 18540 . 4 0 6 5 0 20 . 5 18450 . 5 5 8 4 2 18553 . 4 2 6 7 5 21 . 5 18463 . 8 7 0 8 5 18566 . 5 7 6 2 8 22 . 5 18477 . 8 0 8 8 6 18580 . 8 5 7 8 3 23 . 5 18492 . 3 5 6 2 3 18595 . 2 2 7 8 5 24 . 5 18507 . 5 3 5 4 0 18610 . 7 7 8 2 2 25 . 5 18523 . 3 2 3 4 3 18626 . 3 6 7 6 5 26 . 5 18539 . 7 3 5 2 0 18643 . 1 8 6 5 8 27 . 5 18556 . 7 5 8 0 7 18660 . 0 0 2 5 6 28 . 5 18574 . 4 1 2 1 0 18678 . 0 8 3 9 1 29 . 5 18592 . 6 7 0 6 1 18696 . 1 2 4 1 6 30 . 5 18611 . 5 6 3 1 8 18715 . 4 8 3 6 6 31 . 5 18631 . 0 5 7 9 4 18734 . 7 2 8 8 5 32 . 5 18651 . 1 8 5 7 1 18755 . 3 4 4 8 4 33 . 5 18671 . 9 1 2 1 6 18775 . 8 2 6 7 4 34 . 5 18693 . 2 7 9 1 9 18797 . 7 0 4 1 6 35 . 5 18715 . 2 4 1 3 4 18819 . 4 0 8 2 1 36 . 5 18737 . 8 4 7 5 1 18842 . 5 5 3 5 9 37 . 5 18761 . 0 3 8 4 4 18865 . 4 7 3 6 4 38 . 5 18784 . 8 8 2 2 5 18889 . 8 8 2 5 7 39 . 5 18809 . 3 0 8 0 4 18914 . 0 2 8 7 7 40 . 5 18834 . 3 8 7 3 2 18939 . 6 9 7 5 9 41 . 5 18860 . 0 4 4 9 7 18965 . 0 6 4 3 3 42 . 5 18885 . 3 6 3 8 5 18991 . 9 9 3 8 1 43 . 5 18913 . 2 5 4 0 5 19018 . 5 8 1 4 6 44 . 5 18940 . 7 9 7 8 7 19046 . 7 7 5 0 4 45 . 5 18968 . 9 3 1 0 5 19074 . 5 8 0 2 6 46 .5 18997 . 7 1 7 5 9 19104 . 0 3 0 7 8 47 . 5 19027 . 0 7 1 2 0 19133 . 0 6 1 4 8 48 . 5 19057 . 0 9 2 2 7 19163 . 7 7 0 8 8 49 . 5 19087 . 6 7 6 0 3 19194 . 0 1 8 9 3 50 . 5 19118 . 9 2 6 9 1 19225 . 9 8 7 5 0 51 . 5 19150 . 7 4 7 8 9 19257 . 4 5 5 6 3 APPENDIX V , c o n t i n u e d , upper s t a t e energy l e v e l s . 145 52, 53, 54 , 55, 5£ . 57, 58, 59, 60, 61, 62, 63, 64 , 65, J 1 3 / 2 1 9 1 8 3 . 2 4 6 6 6 1 9 2 1 6 . 2 8 2 7 1 1 9 2 5 0 . 0 1 6 3 6 1 9 2 8 4 . 2 7 7 1 5 1 9 3 1 9 . 2 5 2 5 4 1 9 3 5 4 . 7 3 3 8 2 1 9 3 9 0 . 9 3 9 1 5 1 9 4 2 7 . 6 4 8 7 9 1 9 4 6 5 . 0 9 1 7 0 1 9 5 0 3 . 0 2 4 4 4 1 9 5 4 1 . 6 9 6 8 7 1 9 5 8 0 . 8 6 3 4 0 1 9 6 2 0 . 7 6 4 0 2 1 9 6 6 1 . 1 4 6 7 4 %/2 1 9 2 9 0 . 6 8 0 8 9 1 9 3 2 3 . 3 6 7 1 0 1 9 3 5 7 . 8 5 0 2 2 1 9 3 9 1 . 7 5 7 1 2 1 9 4 2 7 . 4 9 2 3 6 1 9 4 6 2 . 6 1 8 3 5 1 9 4 9 9 . 6 0 7 6 5 1 9 5 3 5 . 9 5 5 8 1 2. 3 . 4 . 5 . 7 . 8. 9 . 10 . 11 . 12. 13 . 14 . 15 . 16 . 17 . 18. 19 . 20 . 2 1 . 22 . 2 3 . 24, 2 5 . 26, 27 , 28, 29, 30. 31 , 32, 33, 34 35 36 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 7 . 5 ^ 6 / 2 ^3/2 1 8 7 9 3 . 0 5 2 7 2 1 8 8 9 5 . 0 9 4 2 9 1 8 7 9 5 . 2 2 7 0 3 1 8 8 9 7 . 2 8 5 2 0 1 8 7 9 8 . 0 2 3 0 0 1 8 9 0 0 . 0 9 4 5 0 1 8 8 0 1 . 4 3 9 7 2 1 8 9 0 3 . 5 3 6 6 6 1 8 8 1 0 . 1 3 6 8 0 1 8 9 1 2 . 2 8 9 8 4 1 8 8 1 5 . 4 1 7 1 0 1 8 9 1 7 . 5 9 8 3 4 1 8 8 2 1 . 3 1 8 5 7 1 8 9 2 3 . 5 3 8 1 0 1 8 8 2 7 . 8 4 1 2 0 1 8 9 3 0 . 0 9 2 2 4 1 8 8 3 4 . 9 8 5 3 7 1 8 9 3 7 . 2 9 1 8 2 1 8 8 4 2 . 7 4 9 8 1 1 8 9 4 5 . 0 9 2 9 9 1 8 8 5 1 . 1 3 6 4 5 1 8 9 5 3 . 5 4 5 4 6 1 8 8 6 0 . 1 4 2 3 0 1 8 9 6 2 . 5 9 1 9 9 1 8 8 6 9 . 7 7 1 7 2 1 8 9 7 2 . 2 9 5 9 3 1 8 8 8 0 . 0 1 8 8 1 1 8 9 8 2 . 5 8 6 8 2 1 8 8 9 0 . 8 9 2 5 3 1 8 9 9 3 . 5 5 0 8 4 1 8 9 0 2 . 3 7 9 2 1 1 9 0 0 5 . 0 8 4 1 3 1 8 9 1 4 . 4 9 8 5 1 1 9 0 1 7 . 3 0 3 2 6 1 8 9 2 7 . 2 2 2 7 7 1 9 0 3 0 . 0 7 7 1 2 1 8 9 4 0 . 5 9 0 8 7 1 9 0 4 3 . 5 4 1 4 1 1 8 9 5 4 . 5 4 9 2 8 1 9 0 5 7 . 5 5 8 5 7 1 8 9 6 9 . 1 7 3 0 0 1 9 0 7 2 . 2 8 5 1 2 1 8 9 8 4 . 3 5 8 0 4 1 9 0 8 7 . 5 5 3 0 6 1 9 0 0 0 . 2 5 1 5 0 1 9 1 0 3 . 5 2 6 4 8 1 9 0 1 6 . 6 4 8 9 3 1 9 1 2 0 . 0 3 4 8 9 1 9 0 3 3 . 8 3 8 6 0 1 9 1 3 7 . 2 6 5 9 6 1 9 0 5 1 . 4 2 1 7 5 1 9 1 5 5 . 0 1 4 1 3 1 9 0 6 9 . 9 4 6 3 7 1 9 1 7 3 . 4 9 8 2 7 1 9 0 8 8 . 6 7 3 9 9 1 9 1 9 2 . 4 9 0 4 0 1 9 1 0 8 . 5 6 8 4 6 1 9 2 1 2 . 2 2 9 6 2 1 9 1 2 8 . 4 0 7 5 9 1 9 2 3 2 . 4 3 5 2 2 1 9 1 4 9 . 6 8 8 6 4 1 9 2 5 3 . 4 4 9 4 3 1 9 1 7 0 . 6 1 9 6 8 1 9 2 7 4 . 9 1 1 3 7 1 9 1 9 3 . 2 9 8 6 8 1 9 2 9 7 . 1 6 4 7 2 1 9 2 1 5 . 3 1 1 7 7 1 9 3 1 9 . 8 5 8 1 7 1 9 2 3 9 . 3 9 1 6 4 1 9 3 4 3 . 3 7 3 1 7 APPENDIX V , c o n t i n u e d , upper s t a t e energy l e v e l s . 146 ^ 6 / 2 A 3 / 2 3 8 . 5 1 9 2 6 2 . 4 8 2 6 6 3 9 . 5 1 9 2 8 7 . 9 6 3 8 6 1 9 3 9 2 . 0 7 2 5 7 4 1 . 5 1 9 4 4 3 . 2 6 3 6 8 4 3 . 5 1 9 3 9 2 . 5 4 0 0 9 1 9 4 9 6 . 9 4 3 5 3 4 5 . 5 1 9 4 4 8 . 8 3 7 2 1 1 9 5 5 3 . 1 0 9 7 4 4 7 . 5 1 9 5 0 7 . 0 1 3 8 7 1 9 6 1 1 . 7 6 6 5 7 4 9 . 5 1 9 5 6 7 . 9 5 8 2 9 J 4 0 . 5 18787 . 93703 18892 . 6 9 4 6 6 1 . 5 18789 . 11923 18893 . 3 8 0 7 4 2 . 5 1 8 7 9 0 . 25308 18895 . 3 6 2 5 7 3 . 5 1 8 7 9 3 . 01572 18896 . 9 6 5 9 7 4 . 5 1 8 7 9 5 . 05815 18900 . 5 2 9 7 6 5 . 5 1 8 7 9 9 . 39797 18903 . 0 4 4 4 1 7 . 5 1 8 8 0 8 . 26562 18911 . 6 2 6 6 3 8 . 5 1 8 8 1 2 . 12511 18918 . 3 6 3 8 8 9 . 5 1 8 8 1 9 . 61843 18922 . 7 0 8 5 5 10 . 5 18824 . 38730 18931 . 0 2 9 5 9 11 . 5 18833 . 45541 18936 . 2 9 0 0 5 12 . 5 1 8 8 3 9 . 13545 18946 . 1 9 4 1 1 13 . 5 1 8 8 4 9 . 77694 18952 . 3 7 1 0 3 14 . 5 1 8 8 5 6 . 36898 18963 . 8 5 7 1 1 15 . 5 1 8 8 6 8 . 58141 18970 . 9 5 1 9 0 16 . 5 1 8 8 7 6 . 08819 18984 . 0 1 8 8 3 17 . 5 1 8 8 8 9 . 86976 18992 . 0 2 8 3 2 18 . 5 1 8 8 9 8 . 46196 19006 . 6 7 6 9 8 19 . 5 1 8 9 1 3 . 63887 19015 . 6 0 6 8 0 20 . 5 1 8 9 2 2 . 98011 19031 . 8 3 2 6 7 21 . 5 1 8 9 3 9 . 88685 19041 . 6 8 1 9 6 22 . 5 1 8 9 5 0 . 15284 19059 . 4 8 5 3 2 23 . 5 1 8 9 6 8 . 60976 19070 . 2 5 4 7 4 24 . 5 1 8 9 7 9 . 80958 19089 . 6 3 3 9 0 25 . 5 1 8 9 9 9 . 79965 19101 . 3 2 4 7 0 26 . 5 1 9 0 1 1 . 94920 19122 . 2 7 7 9 6 27 . 5 1 9 0 3 3 . 44264 19134 . 8 9 1 7 0 28 . 5 1 9 0 4 6 . 57120 19157 . 4 1 6 7 5 29 . 5 1 9 0 6 9 . 52682 19170 . 9 5 3 0 9 30 . 5 1 9 0 8 3 . 67588 19195 . 0 4 8 9 7 31 . 5 1 9 1 0 8 . 05595 19209 . 5 0 9 0 9 32 . 5 1 9 1 2 3 . 26177 19235 . 1 7 4 3 6 33 . 5 1 9 1 4 9 . 04299 19250 . 5 6 3 0 5 34 . 5 1 9 1 6 5 . 32825 19277 . 7 9 2 2 4 35 . 5 1 9 1 9 2 . 49910 19294 . 1 0 7 4 4 36 . 5 1 9 2 0 9 . 87517 19322 . 9 0 2 7 9 37 . 5 1 9 2 3 8 . 42762 19340 . 1 4 7 2 7 38 . 5 1 9 2 5 6 . 90081 19373 . 5 0 0 0 3 39 . 5 19388 . 6 5 3 8 1 APPENDIX V , c o n t i n u e d , upper s t a t e energy l e v e l s . 147 J 2 Z + 39 . 5 1 9 2 8 6 . 82833 40 . 5 1 9 3 0 6 . 40402 41 . 5 1 9 3 3 7 . 70318 42 . 5 1 9 3 5 8 . 38488 43 . 5 1 9 3 9 1 . 05375 44 . 5 1 9 4 1 2 . 84263 45 . 5 1 9 4 4 6 . 87260 47 . 5 1 9 5 0 5 . 16551 2 S " 1 9 3 8 8 . 6 9 7 2 9 1 9 4 2 0 . 5 8 6 5 0 1 9 4 3 9 . 7 0 2 0 2 

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