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Optical spectroscopy of some simple free radicals 1981

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OPTICAL SPECTROSCOPY OF SOME SIMPLE FREE RADICALS by ALLAN SHI-CHUNG CHEUNG Sc. (Hon.), U n i v e r s i t y o f W a t e r l o o , 1977 A THESIS SUMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department o f C h e m i s t r y ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g to t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November 1981 0 A l l a n Shi-Chung Cheung, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o lumbia, I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the head o f my d e p a r t m e n t o r by h i s o r her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f cHints TRY The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, B.C. Canada V6T 1W5 Date %.L. tfjtfZ To my p a r e n t s - i i - A b s t r a c t T h i s t h e s i s r e p o r t s s t u d i e s o f t h e e l e c t r o n i c s p e c t r a o f some gaseous o x i d e m o l e c u l e s . The (0,0) band o f the C 4 z " - X 4 i ~ e l e c t r o n i c t r a n s i t i o n o f V0 has been r e c o r d e d by i n t e r m o d u l a t e d 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 t a r e s o l u t i o n o f about 100 MHz o v e r t h e range 17300 - 17427 cm" 1. The hyper- 51 f i n e s t r u c t u r e caused by the V n u c l e u s (I = 7/2) i s a l m o s t c o m p l e t e l y r e s o l v e d . I n t e r n a l h y p e r f i n e p e r t u r b a t i o n s between the and F^ e l e c - t r o n s p i n components (where N = J - j and J + 1, r e s p e c t i v e l y ) o c c u r i n both e l e c t r o n i c s t a t e s ; t h e s e a r e caused by h y p e r f i n e m a t r i x elements o f the type A J = ±1. The C E s t a t e has many l o c a l e l e c t r o n i c - r o t a t i o n a l p e r t u r b a t i o n s , and a l s o s u f f e r s from l a r g e s p i n - o r b i t p e r t u r b a t i o n s by d i s t a n t e l e c t r o n i c s t a t e s , f o r w hich i t has been n e c e s s a r y to i n t r o d u c e a second s p i n - r o t a t i o n p arameter, Y s > and the c o r r e s p o n d i n g i s o t r o p i c h y p e r f i n e p a r a m e t e r , b<.. The background t h e o r y f o r t h i s new h y p e r f i n e parameter and the c a l c u l a t i o n o f i t s m a t r i x elements a r e d e s c r i b e d . 4 4 - The A n - H e l e c t r o n i c t r a n s i t i o n o f V0 i n the n e a r i n f r a - r e d has been r e c o r d e d a t D o p p l e r - l i m i t e d r e s o l u t i o n by F o u r i e r t r a n s f o r m s p e c t r o s - copy, and r o t a t i o n a l a n a l y s e s p erformed f o r the (0,0) band a t 1 . 0 5 y a n d the (0,1) band i n 1.18 p. The h y p e r f i n e s t r u c t u r e i s prominent i n the 4 4 - n5/2 " x E subband, and i n many o f the s p i n s a t e l l i t e b r a n c h e s . As 4 shown by the v a l u e o f the Fermi c o n t a c t h y p e r f i n e parameter i n the A n i t s e l e c t r o n c o n f i g u r a t i o n i s (4SO) 1 ( S d S ) 1 HPTT) 1 i n the s i n g l e c o n f i - g u r a t i o n a p p r o x i m a t i o n . L a s e r - i n d u c e d f l u o r e s c e n c e s p e c t r a o f gaseous FeO have p r o v e d t h a t the bands whose P and R b ranches have been a n a l y s e d r o t a t i o n a l l y by H a r r i s and Barrow (and which a r e known to i n v o l v e the ground s t a t e ) a r e = 4 - - i i i - ft"" = 4 t r a n s i t i o n s . The e l e c t r o n c o n f i g u r a t i o n (4s.a)' (..3d6)° ( 3 d n ) ^ 5 A., i s the o n l y r e a s o n a b l e a s s i g n m e n t f o r the ground s t a t e o f FeO. The r o t a t i o n a l s t r u c t u r e o f the 000-000 band o f t h e 2490 K system o f N0 2 (.2 B 2 - \ A^) has been a n a l y s e d from h i g h d i s p e r s i o n g r a t i n g s p e c t r o g r a p h p l a t e s . The band i s found to be s l i g h t l y p r e d i s s o c i a t e d , e x a c t l y as i n t h e ^ N 0 2 i s o t o p e , which s u g g e s t s t h a t i t m i g h t be u s a b l e f o r l a s e r s e p a r a t i o n o f the i s o t o p e s o f n i t r o g e n . - i v- TABLE OF CONTENTS Cha p t e r 1 I n t r o d u c t i o n Page 1 C h a p t e r 2 T h e o r y o f M o l e c u l a r Energy L e v e l s o f Fr e e R a d i c a l s A. I n t r o d u c t i o n B. H a m i l t o n i a n s and e i g e n f u n c t i o n s ( i ) The g e n e r a l H a m i l t o n i a n ( i i ) Born-Oppenheimer s e p a r a t i o n o f n u c l e a r and e l e c t r o n i c m o t i o n s ( i i i ) R o t a t i o n a l w a v e f u n c t i o n ( i v ) E l e c t r o n s p i n f i n e s t r u c t u r e H a m i l t o n i a n ( v ) N u c l e a r s p i n h y p e r f i n e s t r u c t u r e H a m i l t o - n i a n ( v i ) E f f e c t i v e H a m i l t o n i a n and d e g e n e r a t e p e r t u r b a t i o n t h e o r y C. E v a l u a t i o n o f m a t r i x elements ( i ) A n g u l a r momenta I r r e d u c i b l e s p h e r i c a l t e n s o r Hund's c o u p l i n g c a s e s ( i i ) ( i i i ) ( i v ) M a t r i x elements i n case (b j ) and case 5 6 9 9 10 13 15 23 28 35 36 40 45 53 Chapter 3 H i g h e r Order S p i n C o n t r i b u t i o n s to the I s o t r o p i c H y p e r f i n e H a m i l t o n i a n i n H i g h M u l t i p l i c i t y E E l e c t r o n i c S t a t e s A. I n t r o d u c t i o n B. I s o t r o p i c h y p e r f i n e i n t e r a c t i o n i n the t h i r d - o r d e r e f f e c t i v e H a m i l t o n i a n C. T r a n s f o r m a t i o n t o case ( b g j ) c o u p l i n g D. C o n c l u s i o n -v- • Page Chapter 4 L a s e r Induced F l u o r e s c e n c e S p e c t r o s c o p y 90 A. I n t r o d u c t i o n 91 B. S a t u r a t i o n o f m o l e c u l a r a b s o r p t i o n l i n e s 92 C. S a t u r a t e d f l u o r e s c e n c e s p e c t r o s c o p y 97 D. 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 o s c o p y 102 E. R e s o l v e d f l u o r e s c e n c e s p e c t r o s c o p y 105 Chapter 5 L a s e r S p e c t r o s c o p y o f V0; A n a l y s i s o f the R o t a t i o n a l and H y p e r f i n e S t r u c t u r e o f the C 4 z - - X 4 E " (0,0) Band 109 A. I n t r o d u c t i o n 110 B. E x p e r i m e n t a l d e t a i l s 113 C. R o t a t i o n a l and h y p e r f i n e energy l e v e l e x p r e s s i o n s 118 D. A n a l y s i s o f the s p e c t r u m 129 ( i ) G e n e r a l d e s c r i p t i o n o f t h e band 129 ( i i ) I n t e r n a l h y p e r f i n e p e r t u r b a t i o n s 130 ( i i i ) The band c e n t r e 141 4 - E. E l e c t r o n i c p e r t u r b a t i o n i n t h e C E s t a t e 145 ( i ) The F 4 ( 2 6 ) p e r t u r b a t i o n 147 ( i i ) The F-](37) p e r t u r b a t i o n 152 F. L e a s t s q u a r e f i t t i n g o f the l i n e p o s i t i o n s 157 ( i ) D e p e r t u r b a t i o n o f t h e C ^ E ~ F^ l e v e l p o s i t i o n s 158 ( i i ) L e a s t s q u a r e s r e s u l t s 161 G. H y p e r f i n e parameters 162 H. D i s c u s s i o n 168 C h a p t e r 6 L a s e r - I n d u c e d F l u o r e s c e n c e and D i s c h a r g e E m i s s i o n S p e c t r a o f FeO; E v i d e n c e f o r a ^>A- Ground S t a t e A. I n t r o d u c t i o n 174 175 - v i - Page B. E x p e r i m e n t a l d e t a i l s 177 C. R e s u l t s 179 D. D i s c u s s i o n 183 Ch a p t e r 7 P r e d i s s o c i a t e d R o t a t i o n a l S t r u c t u r e i n t h e 2490 A Band o f 1 5 N 0 2 187 A. I n t r o d u c t i o n 188 B. E x p e r i m e n t a l d e t a i l s 188 C. A n a l y s i s o f t h e 2490 A band o f 1 5 N 0 2 189 D. C o n c l u s i o n 198 Chap t e r 8 F o u r i e r T r a n s f o r m S p e c t r o s c o p y o f V0; R o t a t i o n a l S t r u c t u r e i n the A ^ n - X ^ i - system near 10500 K 200 A. I n t r o d u c t i o n 201 B. E x p e r i m e n t a l d e t a i l s 202 C. Appearance o f t h e spectrum 202 4 4 - D. Energy l e v e l s o f n and i s t a t e s 204 E. A n a l y s i s o f br a n c h s t r u c t u r e 208 F. L e a s t s q u a r e s f i t t i n g o f the da t a 213 G. D i s c u s s i o n ( i ) S p i n - o r b i t c o u p l i n g c o n s t a n t s and i n d e t e r m i n a c i e s 218 ( i i ) A - d o u b l i n g p arameters 220 ( i i i ) H y p e r f i n e s t r u c t u r e o f the A 4 n s t a t e 221 B i b l i o g r a p h y 227 - v i i - page Appendices I . T r a n s f o r m a t i o n between c a r t e s i a n t e n s o r s and s p h e r i c a l t e n s o r s 240 I I . A d e r i v a t i o n o f the n u c l e a r s p i n - e l e c t r o n s p i n d i p o l a r i n t e r a c t i o n m a t r i x elements i n c a s e ( b g J ) c o u p l i n g 245 I I I . D e r i v a t i o n o f the m a t r i x elements o f t h e ope-r a t o r E T 1 ( I ) . T 1 [ T 3 { T 1 ( S ) , T 2 ( s s . ) } , c 2 ] / r . • .. . ~ ~ ~ i j i j i n c a s e ( b ^ j ) b a s i s 252 I V . Wigner 9-j symbols needed f o r I^.JS d i p o l a r i n t e r a c t i o n and the t h i r d o r d e r T s o t r o p i c h y p e r f i n e i n t e r a c t i o n 259 V . M o l e c u l a r o r b i t a l d e s c r i p t i o n o f the f i r s t - r o w t r a n s i t i o n metal o x i d e s 262 V I . R o t a t i o n a l l i n e assignments 265 - v i i i - LIST OF TABLES F4 l e v e l s page T a b l e 3.1 The f i v e t y p e s o f term c o n t r i b u t i n g t o the t h i r d -o r d e r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n 75 3.2 M a t r i x elements o f the t h i r d o r d e r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n i n a Hund's case (a.) b a s i s 82 p 5.1 M a t r i x elements o f t h e s p i n and h y p e r f i n e H a m i l t o n i a n f o r a 4 E s t a t e i n a c a s e ( b g j ) b a s i s 127 5.2 A n a l y s i s o f the C 4 E _ , F 4 ( 2 6 ) p e r t u r b a t i o n 149 5.3 A n a l y s i s o f t h e C 4 E - , F-|(37) p e r t u r b a t i o n 155 5.4 C a l c u l a t e d p e r t u r b a t i o n s h i f t s i n the VO C 4 £ ~ v=0 160 5.5 R o t a t i o n a l , s p i n and h y p e r f i n e c o n s t a n t s f o r t h e C 4E" and X 4 E - s t a t e s o f VO 163 5.6 R o t a t i o n a l and h y p e r f i n e e n e r g y l e v e l s o f the X 4 E " v=0 s t a t e o f VO f o r N < 5, c a l c u l a t e d from t h e c o n s t a n t s o f T a b l e 5.5 V a l u e s i n c m - 1 172 5.7 Ground s t a t e h y p e r f i n e c o m b i n a t i o n d i f f e r e n c e s , F 2 ( N ) - F 3 ( N ) , i n cm-1, f o r t h e X 4E", v=0 s t a t e o f VO i n t h e rang e N=8-20 173 7.1 R o t a t i o n a l c o n s t a n t s f o r the 2490 A band o f 1 5 N 0 9 (cm" 1) 1 8.1 M a t r i x elements o f the r o t a t i o n a l H a m i l t o n i a n f o r a 4 n s t a t e i n case (a) c o u p l i n g 4 - 8.2 M a t r i x elements f o r s p i n and r o t a t i o n i n a E s t a t e i n c a s e ( a ) c o u p l i n g 207 8.3 C o r r e c t i o n s a p p l i e d to t h e o b s e r y e d F2 and F3 l i n e p o s i t i o n s t o a l l o w f o r t h e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s h i f t s 215 8.4 Parameters d e r i v e d from r o t a t i o n a l a n a l y s i s o f t h e A 4 n - X 4 E " ( 0 , 0 ) and ( P , l ) bands o f V0 i n cm-1 217 197 206 - i x - T a b l e I R o t a t i o n a l l i n e s a s s i g n e d i n t h e C 4Z~-X 4E*" (0,0) band o f V0 II R o t a t i o n a l l i n e s a s s i g n e d i n the ST=4 - ft''=4 bands o f FeO o 15 I I I R o t a t i o n a l l i n e s a s s i g n e d i n 2491 A band o f N0 2 IV A s s i g n e d r o t a t i o n a l l i n e s o f the A 4 n - X 4 z " (0,0) and (0,1) bands o f V0 page 267 280 282 289 -X- LIST OF FIGURES page F i g . 2.1 The s t e p - w i s e development o f t h e t h e o r y employed i n the a n a l y s i s and i n t e r p r e t a t i o n o f m o l e c u l a r s p e c t r a 7 2.2 Hund's c o u p l i n g c a s e s (a) and (b) 47 2.3a M o l e c u l a r c o u p l i n g schemes i n c l u d i n g n u c l e a r s p i n c a s e (a ) and c a s e ( a 0 ) 49 2.3b M o l e c u l a r c o u p l i n g schemes c a s e ( h g S ) and c a s e (.b N ) 50 2.3c M o l e c u l a r c o u p l i n g scheme case ( b ^ j ) 51 4.1 Two l e v e l system 98 4.2 M o l e c u l a r v e l o c i t y d i s t r i b u t i o n s f o r both upper and l o w e r l e v e l s under the a c t i o n o f a l a s e r wave o f f r e q u e n c y co 99 4.3 Lamb d i p e x p e r i m e n t 100 4.4 V e l o c i t y d i s t r i b u t i o n c u r v e s 101 4.5 T o t a l f l u o r e s c e n c e i n t e n s i t y vs l a s e r f r e q u e n c y 101 4.6 E x p e r i m e n t a l s e t up f o 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 102 4.7 O r i g i n o f i n d u c e d f l u o r e s c e n c e l i n e s 106 5.1 E x p e r i m e n t a l s e t up f o 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 o s c o p y | 114 5.2 S c h e m a t i c diagram f o 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 d e t e c t i o n system 116 5.3 H y p e r f i n e s t r u c t u r e s o f l i n e s from t h e f o u r e l e c t r o n s p i n components o f the V0 C 4 £ - - X 4 - (0,0) band 131 5.4 E l e c t r o n s p i n f i n e s t r u c t u r e o f the V0 X4z.~ v=0 l e v e l p l o t t e d as a f u n c t i o n o f N. The r o t a t i o n a l and h y p e r f i n e s t r u c t u r e s a r e not shown 133 5.5 C a l c u l a t e d h y p e r f i n e energy l e v e l p a t t e r n s f o r t h e ?2 and F3 e l e c t r o n s p i n components, o f the X 4 E " y=0 s t a t e o f V0 i n t h e range N""=9-22. The c a l c u l a t i o n s a r e from t h e f i n a l l e a s t s q u a r e s f i t to the ground s t a t e h y p e r f i n e s t r u c t u r e s , and l e v e l s w i t h the same v a l u e s o f F " - N " a r e c o n n e c t e d 134 -XI- A A P a g e F i g . 5.6 The P3 branch l i n e s o f t h e VO C E " - X E " (0,0) band i n t h e r e g i o n N"=15-18, showing the h y p e r f i n e p a t t e r n s near the ground s t a t e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n . The F " quantum numbers f o r the h y p e r f i n e components a r e marked 136 5.7 The P 2 and P 3 b r a n c h l i n e s o f t h e MO C 4 E " - X 4 E ~ (.0,0) band i n the range W" =11-14. Numbers above the s p e c t r a a r e F - " v a l u e s f o r t h e h y p e r f i n e components o f t h e ?2 and P3 l i n e s ; o t h e r l i n e s b e l o n g i n g t o o v e r l a p p i n g branches a r e i n d i c a t e d below t h e s p e c t r a 138 5.8 C a l c u l a t e d h y p e r f i n e e n e r g y l e v e l p a t t e r n s f o r t h e F 2 and F3 e l e c t r o n s p i n components o f t h e C 4 z _ v=0 s t a t e o f VO i n the r e g i o n o f the i n t e r n a l h y p e r f i n e p e r -t u r b a t i o n (N'=2-13). L e v e l s w i t h the same v a l u e s o f F"-N." a r e c o n n e c t e d 139 5.9 The P 2 and P 3 b r a n c h l i n e s o f the VO C 4 E ~ - X V (0,0) band i n the r e g i o n N"=5-8; the F " quantum numbers o f the h y p e r f i n e components a r e marked above t h e s p e c t r a . O v e r l a p p i n g high-N R l i n e s and low-N Pi and P4 l i n e s a r e i n d i c a t e d below t h e s p e c t r a . A l l f o u r t r a c i n g s a r e to t h e same s c a l e 140 5.10 Two r e g i o n s o f t h e V0 C 4 z ~ - X 4 E ~ (0,0) band near t h e band o r i g i n . Low-N l i n e s a r e marked i n roman t y p e w i t h h y p e r f i n e quantum numbers i n d i c a t e d as F ' - F " ; high-N l i n e s a r e marked i n i t a l i c type w i t h o n l y the ? " quantum numbers o f t h e h y p e r f i n e components i n d i -c a t e d . C r o s s - o v e r s i g n a l s ( c e n t r e d i p s ) a r e marked ' c d 1 . The two t r a c i n g s a r e a t the same s c a l e 142 5.11 Energy l e v e l diagram i n d i c a t i n g the a s s i g n m e n t o f the f o u r h y p e r f i n e components o f t h e l i n e Q e f ( 3 s ) 144 4 - 5.12 R o t a t i o n a l e n e r g y l e v e l s o f the C E v=0 s t a t e o f V0 ( w i t h s c a l i n g as i n d i c a t e d ) p l o t t e d a g a i n s t J ( J + 1 ) . Dots i n d i c a t e r o t a t i o n a l p e r t u r b a t i o n s , and the p e r t u r b a t i o n m a t r i x e l e m e n t s , H-] 2, a r e g i v e n where they can be d e t e r m i n e d . The dashed l i n e i s p r o b a b l y a component o f a 2n s t a t e ( s e e t e x t ) w i t h B f f=0.482 cm-"1 146 - x i i - page F i g . 5.13 Two r e g i o n s o f t h e 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 u m o f the C 4 £ - - X 4 z - (.0,0) band o f VO. Upper t r a c i n g : t h e P 3 C 3 8 ) and p e r t u r b e d P i ( 3 8 ) l i n e s . Lower t r a c i n g : the u n p e r t u r b e d P-j(27) l i n e 153 5.14 U p p e r - s t a t e term v a l u e s (cm~^) o f t h e h y p e r f i n e l e v e l s o f the p e r t u r b e d F - | ( 3 7 ) r o t a t i o n a l l e v e l o f the C 4 i " , v=0 s t a t e o f VO p l o t t e d a g a i n s t F(F+1) 154 5.15 R e s i d u a l s ( O b s - c a l c ) f o r t h e R br a n c h l i n e s o f t h e VO C 4 £ - - X 4 £ - (.0,0) band, as compared to the p o s i t i o n s p r e d i c t e d from t h e c o n s t a n t s o f T a b l e 5 . 5 , p l o t t e d a g a i n s t N". 'Raw' d a t a have been u s e d , so t h a t the R 4 l i n e s from N"=9-22, which were d e p e r t u r b e d f o r the l e a s t s q u a r e s t r e a t m e n t ( t h i c k l i n e s ) have non-sero r e s i d u a l s . V e r t i c a l b a r s i n d i c a t e the s p r e a d o f the r e s i d u a l s f o r t h e h y p e r f i n e s t r u c t u r e 170 6.1 Upper p r i n t : Head o f the 5819 ft band o f FeO. Lower p r i n t : Head o f the 6180 ft band o f FeO 8.3 High N main b r a n c h Q and P l i n e s i n the t a i l o f t h e VO A 4 n - X 4 z - (0,0) band 180 6.2 Two r e g i o n s o f the 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 spectrum o f FeO: (a) the two A-components o f the R(15) l i n e o f the 5819 ft band, (b) the Q(4) and R(10) l i n e s o f the 58:19 A band; t h e A-doubling i s not r e s o l v e d f o r t h e s e l i n e s 181 6.3 R e s o l v e d f l u o r e s c e n c e s p e c t r a o f FeO produced by e x c i t a t i o n o f v a r i o u s l i n e s o f t h e 5819 A: e x c i t a t i o n o f Q ( 4 ) , Q ( 5 ) , Q(6) and R(16). The i n t e n s i t y o f the e x c i t e d l i n e i s a n o m a l o u s l y h i g h as a r e s u l t o f s c a t t e r e d l a s e r l i g h t 182 7.1 Heads o f the 2490 A bands (2 Z B ? - f t 2 A - |, 000-000) o f 1 4 N 0 2 (above) and 15NO 2 ( b e l o w ) . The l i n e a s s i g n m e n t s r e f e r t o t h e T5NO 2 spectrum 193 7.2 K=7 and 8 subbands o f t h e 2490 ft band o f 1 5 N 0 2 ( i n t h e r e g i o n 2502-2508 ft 195 8.1 F o u r i e r t r a n s f o r m spectrum o f VO i n the r e g i o n 9410-9570 cm-"1 showing the heads o f t h e A 4 n - X 4 z -(0,0) band o f VO 203 8.2 R o t a t i o n a l s t r u c t u r e i n the A 4 n - X 4 z ~ (0,0) band o f VO. showing t h e F 4 " branch s t r u c t u r e ( 4n5/ 2).. 210 212 - x i i i - F i g . 8.4 Reduced e n e r g y l e v e l s o f the A^n s t a t e o f ¥ 0 p l o t t e d a g a i n s t J ( J + 1 ) . The q u a n t i t y p l o t t e d i s t h e upper s t a t e term v a l u e l e s s (0.50865 + 0.00365ft) {j+h)2 - 6.7 x 10-7 (J+^)4 c m - l 214 8.5 H y p e r f i n e widths:, A E h . f S = E n f s C F = J - I ) , o f the f o u r s p i n components o f t h e A 4 n s t a t e o f VO, p l o t t e d a g a i n s t J . P o i n t s a r e w i d t h s c a l c u l a t e d from the ground s t a t e h y p e r f i n e s t r u c t u r e and t h e o b s e r v e d l i n e w i d t h s , w i t h o u t c o r r e c t i o n f o r t h e Doppler w i d t h 223 - x i v - ACKNOWLEDGEMENTS I would l i k e t o e x p r e s s my s i n c e r e g r a t i t u d e t o my r e s e a r c h d i r e c t o r , P r o f e s s o r Anthony J . Merer, f o r h i s g u i d a n c e , i n v a l u a b l e a d v i c e and c o n s t a n t encouragement t h r o u g h o u t my work and t h e p r e p a r a - t i o n o f t h i s t h e s i s . I am g r a t e f u l t o Dr. D a v i d M. M i l t o n , Dr. M a r j a t t a L y y r a , Dr. Y o s h i a k i Hamada, M i s s Rhoda M. Gordon, Mrs. Rhonda C. Hansen and Mr. A l a n W. T a y l o r f o r t h e many u s e f u l d i s c u s s i o n s and t h e f r i e n d l y atomsphere c r e a t e d by them. I w i s h to thank P r o f e s s o r R o b e r t F. S n i d e r f o r s t i m u l a t i n g d i s c u s s i o n s . I would a l s o l i k e t o thank P r o f e s s o r s G i a c i n t o S c o l e s , Donald E. I r i s h and T e r r y E. Gough ( U n i v e r s i t y o f W a t e r l o o ) f o r t h e i r c o n s t a n t i n t e r e s t and encouragement. T e c h n i c a l a s s i s t a n c e from the M e c h a n i c a l , E l e c t r o n i c and G l a s s - b l o w i n g Workshops i s a l s o much a p p r e c i a t e d . I am a l s o i n d e b t e d t o Mrs. T i l l y S c h r e i n d e r s f o r k i n d l y h e l p i n g w i t h t h e f o r m i d a b l e t a s k o f t y p i n g t h i s t h e s i s . A p p r e c i a t i o n i s e x t e n d e d to Mr. Rob Hubbard f o r h i s t e c h n i c a l a s s i s t a n c e i n o b t a i n i n g t h e F o u r i e r t r a n s f o r m s p e c t r a , and a l s o to K i t t Peak N a t i o n a l O b s e r v a t o r y f o r t h e used o f t h e F o u r i e r t r a n s f o r m s p e c t r o m e t e r . F i n a l l y , r e c e i p t o f a U n i v e r s i t y o f B r i t i s h Columbia Graduate F e l l o w s h i p (78-81) i s g r a t e f u l l y acknowledged. - 1 - C h a p t e r 1 I n t r o d u c t i o n -2- Free r a d i c a l s a r e m o l e c u l a r fragments o r u n s t a b l e m o l e c u l e s which u s u a l l y p o s s e s s one o r more u n p a i r e d e l e c t r o n s . T h i s d e f i n i t i o n o f C.H. Townes and h i s coworkers C l ) i n c l u d e s c l o s e d - s h e l l u n s t a b l e m o l e c u l e s such as CS, CF2, e t c . as w e l l as o p e n - s h e l l s t a b l e m o l e c u l e s l i k e C^, NC^, e t c . Most s p e c t r o s c o p i s t s would e n l a r g e the d e f i n i t i o n t o i n c l u d e any o p e n - s h e l l o r p a r a m a g n e t i c s p e c i e s r e g a r d l e s s o f s t a b i l i t y . In t h i s t h e s i s we s h a l l be d e a l i n g w i t h m o l e c u l e s w i t h o p e n - s h e l l ground and e x c i t e d e l e c t r o n i c s t a t e s . Free r a d i c a l s are o f i m p o r t a n c e i n a l m o s t e v e r y branch o f c h e m i s t r y , even though most branches a r e not d i r e c t l y c o n c e r n e d w i t h t h e i r s t u d y . A c h e m i c a l r e a c t i o n i n v o l v e s t h e b r e a k i n g and/or making o f c o v a l e n t bonds w h i l e the r e d i s t r i b u t i o n o f e l e c t r o n s i n v o l v e d i n e i t h e r o f t h e s e p r o c e s s e s can r e s u l t i n s p e c i e s w i t h o p e n - s h e l l e l e c t r o n c o n f i g u r a t i o n s . Thus the i n t e r m e d i a t e s i n c h e m i c a l r e a c t i o n s a r e o f t e n f r e e r a d i c a l s , which i s why r a d i c a l s a r e o f such i m p o r t a n c e i n c h e m i s t r y . R a d i c a l s can undergo v a r i o u s t y p e s o f r e a c t i o n , such as d e c o m p o s i t i o n , a b s t r a c t i o n and combin- a t i o n , and c o n s e q u e n t l y a wide v a r i e t y o f end p r o d u c t s can r e s u l t . A knowledge o f t h e s e r e a c t i o n pathways i s i m p o r t a n t f o r both k i n e t i c and p h o t o c h e m i c a l s t u d i e s , a l t h o u g h f o r d i f f e r e n t r e a s o n s - whereas a k i n e t i - c i s t d e t e r m i n e s r e a c t i o n r a t e s f o r i n d i v i d u a l s t e p s i n v o l v e d i n t h e t o t a l r e a c t i o n , a p h o t o c h e m i s t i s c o n c e r n e d w i t h t h e way i n which t h e s e i n t e r - m e d i a t e s , when formed i n e x c i t e d s t a t e s , l o s e t h e i r e xcess e n e r g y . N e v e r t h e l e s s , t h e i n f o r m a t i o n p r o v i d e d by such s t u d i e s i s complementary i n t he s e n s e t h a t both i n d i c a t e the r e a c t i o n mechanism. An u n d e r s t a n d i n g o f t h e p r o c e s s e s I n v o l v e d i n t h e s e complex gas phase r e a c t i o n s i s a l s o -3- e s s e n t i a l to t h o s e i n t e r e s t e d i n t h e c h e m i s t r y o f the upper atmosphere. As a f u r t h e r example o f t h e r o l e p l a y e d by r a d i c a l s i n gas phase r e a c t i o n s , i t has been s u g g e s t e d t h a t many r e a c t i o n s o c c u r r i n g i n i n t e r - s t e l l a r gas c l o u d s p r o c e e d v i a f r e e r a d i c a l i n t e r m e d i a t e s . The e v i d e n c e f o r such r e a c t i o n s i s s u p p l i e d by t h e d e t e c t i o n o f a b s o r p t i o n and e m i s s i o n s i g n a l s by r a d i o a s t r o n o m e r s . The i n t e r s t e l l a r r a d i c a l s d e t e c t e d so f a r have been m a i n l y o r g a n i c m o l e c u l e s , b ut i n view o f t h e h i g h cosmic abun- dances o f t r a n s i t i o n m e t a l s and oxygen, t r a n s i t i o n metal monoxides a r e p o s s i b l e i n t e r s t e l l a r m o l e c u l e s . These r a d i c a l s a r e , t h e r e f o r e , o f g r e a t a s t r o n o m i c a l s i g n i f i c a n c e , s i n c e i n any case s e v e r a l o f them a r e i m p o r t a n t c o n s t i t u e n t s o f t h e atmospheres o f c o o l s t a r s ( 2 ) . A r a t h e r d i f f e r e n t a s p e c t o f o p e n - s h e l l m o l e c u l e s a r i s e s from the pr e s e n c e o f e l e c t r o n s p i n and/or o r b i t a l a n g u l a r momenta w i t h i n t he m o l e c u l e s . I n t e r a c t i o n s between t h e s e a n g u l a r momenta, a l t h o u g h o f t e n making t h e a n a l y s i s o f t h e s p e c t r a more c o m p l i c a t e d , u l t i m a t e l y y i e l d f a r more i n f o r m a t i o n about t h e e l e c t r o n i c s t r u c t u r e o f t h e m o l e c u l e than can be o b t a i n e d f o r c l o s e d - s h e l l s y s t e m s . An a c c u r a t e d e t e r m i n a t i o n o f t h e parameters d e s c r i b i n g the i n t r a - m o l e c u l a r i n t e r a c t i o n s i s i n v a l u a b l e i n e v a l u a t i n g t h e o r e t i c a l models f o r the e l e c t r o n i c s t r u c t u r e : t h e e x p e r i m e n t a l parameters a r e compared w i t h t h o s e computed u s i n g ab i n i t i o w a v e f u n c t i o n s . The m a g n e t i c h y p e r f i n e parameters a r e p a r t i c u l a r l y u s e f u l i n t h i s r e s p e c t s i n c e they a r e s e n s i t i - ve to t h e d i s t r i b u t i o n o f u n p a i r e d e l e c t r o n s , and hence p r o v i d e a r i g o r o u s t e s t f o r proposed w a v e f u n c t i o n s . Measurements o f t h i s o r d e r o f a c c u r a c y c a l l f o r h i g h r e s o l u t i o n e x p e r i m e n t a l t e c h n i q u e s such, as s u b - D o p p l e r l a s e r s p e c t r o s c o p y ( 3 ) , m o l e c u l a r beam methods ( 4 ) , o r m i c r o w a v e - o p t i c a l d o u b l e - 4 - resonance (.5). Our particular interest lies in high resolution studies giving information on the electronic structures of gas-phase radicals. The ex- periments described here employ the high resolution techniques of conven- tional grating spectroscopy and sub-Doppler laser spectroscopy in the visible region, and also Fourier transform spectroscopy in the near infrared region. The parameters obtained in these experiments are inter- preted through an effective Hamiltonian which is restricted to operate only within the particular electronic and vibrational state from which the spectra arise. Chapter 2 deals in detail with the theory of molecular energy levels, including the construction of an effective Hamiltonian. The derivation of the third order isotropic Fermi contact interaction (a higher order effect appearing in the energy levels of high multiplicity states) is detailed in chapter 3. Chapter 4 describes briefly the technique of laser induced fluorescence spectroscopy. Chapters 5, 6 and 7 describe studies of free radicals by means of different experimental 4 - 4 - techniques. Laser-induced fluorescence studies of the C E - X z system of VO and the ground state of FeO are presented in chapters 5 and 6; chapter 7 2 ~2 is concerned with conventional grating spectroscopy of the 2 B 2 - X A-| system of N02- Finally chapter 8 gives the analysis of the Fourier 4 4 - transform spectra of the A n(b) - X £ system of VO. -5- Chapter 2 Theory o f M o l e c u l a r Energy L e v e l s o f Free R a d i c a l s -6- A. I n t r o d u c t i o n The energy l e v e l s o f a m o l e c u l e a r e g i v e n hy t h e e i g e n v a l u e s o f the t i m e - i n d e p e n d e n t S c h r o d i n g e r e q u a t i o n where H i s t h e t o t a l H a m i l t o n i a n which may be w r i t t e n as H=V H ™ t + H e l + H h f s ( 2 - 2 ) H Q r e p r e s e n t s t h e n o n r e l a t i v i s t i c H a m i l t o n i a n o f t h e n o n - r o t a t i n g mole- c u l e , i .e. t h e k i n e t i c and p o t e n t i a l e n e r g i e s o f t h e e l e c t r o n s and n u c l e i o t h e r than t h e n u c l e a r r o t a t i o n a l e n e r g y , H r Q t s y m b o l i z e s t h e r o t a t i o - n a l m o t i o n o f the n u c l e i , H e-j c o n t a i n s m a g n e t i c terms t h a t c a u s e t h e e l e c t r o n s p i n f i n e s t r u c t u r e and H n f s i n c l u d e s a l l n u c l e a r s p i n and n u c l e a r moment terms t h a t cause t he h y p e r f i n e s t r u c t u r e , v i s t h e e i g e n f u n c t i o n a s s o c i a t e d w i t h a s t a t i o n a r y s t a t e and t h e e i g e n v a l u e E i s t h e energy o f t h i s s t a t e . I t i s i m p o s s i b l e t o s o l v e Eq. (2.1) a n a l y t i c a l l y . In p r a c t i c e , one chooses a c o n v e n i e n t f i n i t e b a s i s s e t <j>̂  and expands the e i g e n - f u n c t i o n s i n terms o f <j>̂ ¥ =E.ai4>i ( 2 - 3 ) T h i s then r e d u c e s t h e s o l u t i o n o f Eq. (.2.1) t o f i n d i n g t h e r o o t s o f t h e s e c u l a r d e t e r m i n a n t | H f j - E ^ l - O ( 2 . 4 ) -7- where t h e q u a n t i t i e s H. . a r e the m a t r i x elements o f H, d e f i n e d as H i j = y * i H * o d T (2.5) The c h o i c e o f b a s i s s e t i s , o f c o u r s e , a r b i t r a r y and any complete b a s i s s e t would s u f f i c e p r o v i d e d t h e c a l c u l a t i o n s a r e c a r r i e d o u t t o s u f f i c i e n t a c c u r a c y . However by making a w i s e c h o i c e o f the i n i t i a l b a s i s s e t , H may be r o u g h l y p a r t i t i o n e d i n t o d i a g o n a l b l o c k s ( s u b - m a t r i c e s ) between which t h e r e a r e o n l y s m a l l o f f - d i a g o n a l m a t r i x e l e - ments. In t h e g e n e r a l c a s e the d i a g o n a l b l o c k s r e f e r t o Born-Oppenheimer o r a d i a b a t i c s t a t e s . T h i s t h e s i s i s c o n c e r n e d w i t h the r o t a t i o n a l and s p i n s t r u c t u r e o f i n d i v i d u a l v i b r o n i c s t a t e s . A c o n v e n i e n t way t o o b t a i n t h e r e q u i r e d energy l e v e l e x p r e s s i o n i s i l l u s t r a t e d i n F i g . 2.1. In the f i r s t s t e p d e g e n e r a t e p e r t u r b a t i o n t h e o r y i s used t o i n c l u d e m a t r i x elements l i n k i n g a p a r t i c u l a r v i b r o n i c s t a t e w i t h nearby v i b r a t i o n a l and/or e l e c t r o n i c s t a t e s . T h i s l e a d s to an e f f e c t i v e H a m i l t o n i a n o p e r a t o r which o p e r a t e s o n l y w i t h i n the r o t a t i o n a l sub-space o f t h a t v i b r o n i c s t a t e . H T o t Degenerate p e r t u r b a t i o n H e f f M a t r i x element r <H > e f f M a t r i x d i a q o n a l i z a t i o n r Ei gen-v a l u e s + Ei gen-f u n c t i ons t h e o r y e v a l u a t i o n by computer F i g . 2.1 The s t e p - w i s e development o f the t h e o r y employed i n t h e a n a l y s i s and i n t e r p r e t a t i o n o f m o l e c u l a r s p e c t r a . -8- The s e c o n d s t e p i n F i g . 2.1 i s the e v a l u a t i o n o f a m a t r i x r e p r e - s e n t a t i o n o f t h . e e f f e c t i v e H a m i l t o n i a n . I r r e d u c i b l e t e n s o r methods have been used i n t h e s e e v a l u a t i o n s because t h e y a r e p a r t i c u l a r l y c o n v e n i e n t f o r d e a l i n g w i t h c o u p l i n g o f t h r e e o r more a n g u l a r momenta. The f i n a l s t e p i s the d i a g o n a l i z a t i o n o f the H a m i l t o n i a n m a t r i x and the d e t e r m i n a t i o n o f the e i g e n v a l u e s and e i g e n f u n c t i o n s . E l e c t r o n i c computers make t h i s nowadays a r o u t i n e m a t t e r . In s e c t i o n B ( i ) t h e g e n e r a l H a m i l t o n i a n H Q i s d i s c u s s e d . T h i s i s f o l l o w e d by a d e s c r i p t i o n o f the Born-Oppenheimer s e p a r a t i o n o f n u c l e a r and e l e c t r o n i c m o t i o n i n s e c t i o n B ( i i ) . S e c t i o n B ( i i i ) i s c o n c e r n e d w i t h the e i g e n f u n c t i o n s o f the r o t a t i o n a l H a m i l t o n i a n , ^r0^- S e c t i o n s B ( i v ) and B(v) d e r i v e the o p e r a t o r s f o r t h e e l e c t r o n s p i n f i n e s t r u c t u r e and the n u c l e a r h y p e r f i n e s t r u c t u r e from p h y s i c a l p r i n c i p l e s . F i n a l l y , B ( v i ) d e a l s w i t h t h e i m p o r t a n t c o n c e p t s and d e r i v a t i o n o f an e f f e c t i v e H a m i l t o n i a n o p e r a t o r . S e c t i o n C d e s c r i b e s the e v a l u a t i o n o f m a t r i x elements o f t h e e f f e c - t i v e H a m i l t o n i a n . In s e c t i o n s C ( i ) and ( i i ) , some i m p o r t a n t r e s u l t s from the t h e o r y o f a n g u l a r momentum and some u s e f u l r e l a t i o n s h i p s i n s p h e r i a l t e n s o r a l g e b r a a r e s t a t e d . These w i l l be needed i n the s u b s e - quent c a l c u l a t i o n s . Hund's c o u p l i n g c a s e s ( a ) and ( b ) , and t h e i r i n t e r - c o n v e r s i o n , a r e d i s c u s s e d i n d e t a i l i n s e c t i o n C ( i i i ) . M a t r i x element e x p r e s s i o n s a r e l i s t e d i n s e c t i o n C ( i v ) , f o r both c a s e ( b g j ) and c a s e ( a . ) , i n terms o f Wigner's 3 - j , 6 - j and 9-j symbols. The a b s e n c e o f the t h i r d E u l e r a n g l e i n l i n e a r m o l e c u l e s , which l e a d s to problems i n the c o m p u t a t i o n o f "matrix e l e m e n t s , i s a l s o d i s c u s s e d . -9- B. H a m i l t o n i a n s and e i g e n f u n c t i o n s ( i ) The General M o l e c u l a r H a m i l t o n i a n A m o l e c u l e i s an assembly o f n u c l e i and e l e c t r o n s , which i s , i n C a r r i n g t o n ' s w o r d s ( l ) , " p r e p a r e d t o c o e x i s t i n a c e r t a i n c o n f i g u r a t i o n w i t h c o n s i d e r a b l e s t a b i l i t y " . To u n d e r s t a n d t he energy l e v e l s o f a m o l e c u l e i t i s n e c e s s a r y t o b e g i n w i t h t h e H a m i l t o n i a n o p e r a t o r c o r r e s - ponding t o the t o t a l e n e r g y ( 2 , 3 ) . i t has been shown by Howard and Moss (4,5) t h a t , s t a r t i n g from a r e l a t i v i s t i c many-body H a m i l t o n i a n , w i t h terms c o r r e c t to o r d e r c , i t i s p o s s i b l e to o b t a i n a l l t h e f a m i l i a r e n e r g y l e v e l e x p r e s s i o n s used by e x p e r i m e n t a l s p e c t r o s c o p i s t s , and i n a d d i t i o n , some not r e a d i l y d e t e c t a b l e new terms s u c h as mass p o l a r i z a t i o n and s p i n - v i b r a t i o n i n t e r a c t i o n s . F o r our p u r p o s e , however, a n o n - r e l a t i v i s t i c many-body H a m i l t o n i a n , i n which s p i n i s added as an a d d i t i o n a l h y p o t h e s i s , i s adequate to d e s c r i b e t he s y s t e m . We t h e r e f o r e omit t he r e l a t i v i s t i c e f f e c t s , and d e f i n e t h e p o t e n t i a l energy o f the H a m i l t o n i a n as dep e n d i n g o n l y on t h e p o s i t i o n s o f t h e n u c l e i (Q) and the p o s i t i o n s o f the e l e c t r o n s ( q ) . The H a m i l t o n i a n o p e r a t o r f o r the t o t a l e n e r g y o f a m o l e c u l e c o n s i s t i n g o f N n u c l e i and n e l e c t r o n s i s the g i v e n by H ( e l e c t r o n k i n e t i c energy) ( n u c l e a r k i n e t i c energy) + V(q,Q) ( p o t e n t i a l energy) (2.6) - 1 0 - where V(q,Q)=-Z Z a l ~ i a ( . e l e c t r o n - n u c l e a r a t t r a c t i o n s ) N N , +z. z a ( n u c l e a r - n u c l e a r r e p u l s i o n s ) (2.7) a b>a r -ab n n 2 +z z f-i j > i ~ i j ( e l e c t r o n - e l e c t r o n r e p u l s i o n s ) In eq. (2.6) P and P denote the l i n e a r momenta o f e l e c t r o n i (mass e l e c t r o n c h a r g e , Z e i s the ch a r g e on n u c l e u s a and r i s the r a d i u s from p a r t i c l e x t o p a r t i c l e y . The m a g n e t i c i n t e r a c t i o n s , such as e l e c t r o n i c and n u c l e a r depen- dent terms, a r e much s m a l l e r t h e n t h e e l e c t r o s t a t i c i n t e r a c t i o n s which c h a r a c t e r i z e d V, and f o r the p r e s e n t purpose may be n e g l e c t e d . They w i l l be added l a t e r i n a p e r t u r b a t i o n t r e a t m e n t . ( i i ) Born-Oppenheimer S e p a r a t i o n o f N u c l e a r and E l e c t r o n i c M o t i o n J u s t as the c l a s s i c a l p roblem o f the r e l a t i v e motions o f t h r e e b o d i e s cannot be s o l v e d e x a c t l y , t h e quantum m e c h a n i c a l problem o f f i n d i n g t h e e x a c t s o l u t i o n s o f the f u l l S c h r O d i n g e r e q u a t i o n i s a l s o i m p o s s i b l e f o r a n y t h i n g e x c e p t the hydrogen atom. The a p p r o x i - m a t i o n i n t r o d u c e d by Born and Oppenheimer ( 6 ) , i n which t h e e l e c t r o n i c p a r t o f t h e problem i s s o l v e d f i r s t , forms a good b a s i s f o r f i n d i n g m ) and n u c l e u s « (mass M p) r e s p e c t i v e l y , and i n eq. (2.7) e i s the (.2.8) -n- a p p r o x i m a t e s o l u t i o n s . The t o t a l w a v e f u n c t i o n v i s assumed t o be expand- a b l e i n a complete s e t o f f u n c t i o n s which a r e p r o d u c t s o f an e l e c t r o n i c p a r t ^ei.(..q,Q) and a n u c l e a r p a r t T(.Q), t h a t i s W ( q , Q ) = ^ e i ( q ' Q ) 4 r ( Q ) ( 2' 9 ) where i j j e 1 - ( q » Q ) i s an e i g e n f u n c t i o n o f the " e l e c t r o n i c H i a m i l t o n i a n " V i T ^ P e 2 + (2.10) e e a c c o r d i n g to H e ^ e i ( q , Q ) = E e \ e i ( q , Q ) (2.11) F o r eq. (2.9), w h i c h i m p l i e s t h a t t h e t o t a l energy £ i s the sum o f the e l e c t r o n i c e n e r g y , E g , and t h e n u c l e a r e n e r g y , E y r , t h a t i s £ = E + E e e v r (2.12) on s u b s t i t u t i n g eq. (2.9) i n t o eq. (2.8) one has P 2 ( izP p 2 +V(q .0>l j :^-)z* e 1(q ,Q)* v r 1 (Q) - g ^ e l ^ K r ' W (2.13) e e n n i i or i ( e e 1 Pn2 . v(rq 3Q)^ p i(q,Q)+^ e i(q»Q)(^ :ir)^ v r e i n n -12- * -(q.QHkJ-TP t q . Q ) ] [ P n E ^ e i ( q , Q ) ^ v r 1 ( Q ) (2.14) S i n c e the ^ e 1-(q,Q) a r e e i g e n f u n c t i o n s o f H g , the f i r s t two terms g i v e ^ v r 1 ( Q ) E e 1 ( Q ) ^ e i ( q , Q ) - M u l t i p l y i n g from t h e l e f t by ̂ e k * ( q , Q ) and i n t e - g r a t i n g o v e r the e l e c t r o n c o o r d i n a t e s q (which e s s e n t i a l l y p i c k s o u t the s t a t e k from t h e e l e c t r o n i c m a n i f o l d ) the e q u a t i o n becomes 2 2 (\ n n' N n n/ ) ( Q ) + , i J ^ e k ( ^ ^ ( ^ e i ^ ' Q ) d c l ^ n ] v i ( Q ) =^C(Q) { 2 J 5 ) k N e g l e c t i n g t h e c r o s s - t e r m which c o n t a i n s ^ v y t 1 ( Q ) , eq. (2.15) i s a new d i f f e r e n t i a l e q u a t i o n t h a t d e f i n e s t h e v i b r a t i o n a l and r o t a t i o n a l f u n c t i o n s o f t h e e l e c t r o n i c s t a t e k. The e q u a t i o n has t h e form o f a S c h r b d i n g e r e q u a t i o n where the H a m i l t o n i a n c o n s i s t s o f the n u c l e a r k i n e t i c e n e r g y , t h e e l e c t r o n i c energy as an e f f e c t i v e p o t e n t i a l e n e r g y , and a s m a l l mass-dependent term which c o n t r i b u t e s a s m a l l e l e c t r o n i c i s o t o p e e f f e c t . The c r o s s - t e r m s n e g l e c t e d i n eq. (2.15) a r e t h o s e which c o u p l e the e l e c t r o n i c and v i b r a t i o n a l motions i n d i f f e r e n t e l e c t r o n i c s t a t e s . I f t he e l e c t r o n i c s t a t e s a r e d e g e n e r a t e (meaning t h a t t h e r e a r e two o r more o r t h o g o n a l e l e c t r o n i c f u n c t i o n s c o r r e s p o n d i n g t o s t a t e s o f t h e same energy) s e v e r e breakdowns, o f the Born-Oppenheiraer a p p r o x i m a t i o n can o c c u r : t h e s e were f i r s t d e s c r i b e d f o r l i n e a r m o l e c u l e s by Renner ( 7 ) , and f o r symmetric t o p m o l e c u l e s ( p o s s e s s i n g a 3 - f o l d o r -13- h i g h e r a x i s o f symmetry) by Jahn and T e l l e r (_8). I f t h e e l e c t r o n i c s t a t e s a r e non-degenerate t he c r o s s - t e r r a s g i v e r i s e t o the phenomenon o f v i b r a t i o n a l momentum c o u p l i n g , which i s one o f the p r i n c i p a l causes o f t h e c o m p l e x i t y o f t h e e l e c t r o n i c s p e c t r a o f N0 2 and S 0 2 . ( i i i ) R o t a t i o n a l w a v e f u n c t i o n s The s e p a r a t i o n o f v i b r a t i o n and r o t a t i o n i s i n g e n e r a l d i f f i c u l t b ecause c e r t a i n c o m b i n a t i o n s o f v i b r a t i o n s produce motions i n d i s t i n - g u i s h a b l e f r o m r o t a t i o n s , w i t h accompanying v i b r a t i o n a l a n g u l a r momen- tum. T h i s s e p a r a t i o n has been d i s c u s s e d i n many e x c e l l e n t t e x t s ( 9 , 1 0 ) , and i t i s n o t n e c e s s a r y to r e p e a t i t h e r e . The s u c c e s s o f t h i s s e p a r a - t i o n depends o f c o u r s e on the magnitude o f the v i b r a t i o n - r o t a t i o n c o u p l i n g terms. I f t h e s e can be n e g l e c t e d one has e s s e n t i a l l y i n d e p e n - dent H a m i l t o n i a n s which d e s c r i b e t h e v i b r a t i o n a l and r o t a t i o n a l motions s e p a r a t e l y . We r e s t r i c t o u r s e l v e s i n t h i s t h e s i s to an e x t r e m e l y b r i e f d i s c u s s i o n o f the r o t a t i o n a l H a m i l t o n i a n and i t s e i g e n f u n c t i o n s . The c l a s s i c a l r o t a t i o n a l H a m i l t o n i a n (10) i s 0 2 J 2 J , 2 Hrot=2r + 2 f + 217 <2'16' x y z where J , J and J a r e the components o f t h e r o t a t i o n a l a n g u l a r momen-x y z turn J a l o n g t h e p r i n c i p a l axes o f t h e m o l e c u l e , and I , I and I a r e t h e ~ x y z. p r i n c i p a l moments o f i n e r t i a ( t h a t i s , where t h e a x i s system i s chosen such t h a t the o f f - d i a g o n a l elements o f t h e moment o f i n e r t i a t e n s o r I v a n i s h ) . -14- I f t h e moments o f i n e r t i a do not change d u r i n g t h e r o t a t i o n a l m o t i o n eq. (2.16) i s known as t h e r i g i d r o t a t o r H a m i l t o n i a n . The c o r r e s p o n d i n g S c h r b d i n g e r e q u a t i o n H ip = E IJJ (2.17) r o t y r r y r can o n l y be s o l v e d a n a l y t i c a l l y f o r s p h e r i c a l and symmetric t o p s . The m o t i o n s o f a r o t a t i n g body (a 'top') a r e u s u a l l y d e s c r i b e d by s p e c i f y - i n g t h e E u l e r a n g l e s (CX,3,Y ) (.11) a b o u t the p r i n c i p a l moments o f i n e r t i a which a r e a n g l e s d e s c r i b i n g how much t h e body has r o t a t e d from an i n i t i a l r e f e r e n c e c o n f i g u r a t i o n . The symmetric t o p e i g e n f u n c t i o n s o f "th eq. (2.17) a r e s i m p l y r e l a t e d to t h e elements o f the J rank r o t a t i o n m a t r i x (12) h ( J ) * 2 J+1 8TT Dm (a . e.v) (2-18) where J i s t h e r o t a t i o n a l a n g u l a r momentum quantum number, K i s t h e p r o j e c t i o n o f t h e r o t a t i o n a l a n g u l a r momentum v e c t o r on t h e m o l e c u l e - ( J ) * f i x e d z - a x i s , M i s i t s p r o j e c t i o n on the s p a c e - f i x e d Z - a x i s , and ' i s an element o f Wigner's r o t a t i o n m a t r i x . The c h o i c e o f phase f a c t o r i m p l i c i t i n eq. (2.18) i s e q u i v a l e n t t o d e f i n i n g t h e m a t r i x element o f J , the component r e f e r r e d t o t h e m o l e c u l e - f i x e d x - a x i s , as r e a l and -X p o s i t i v e (13) . The e i g e n f u n c t i o n s o f a r i g i d asymmetric t o p m o l e c u l e (where no axes o f symmetry h i g h e r than 2 - f o l d a r e p r e s e n t ) a r e more c o m p l i c a t e d than t h o s e j u s t c o n s i d e r e d ; they may n e v e r t h e l e s s be e x p r e s s e d as -15- l i n e a r c o m b i n a t i o n s o f symmetric top w a v e f u n c t i o n s (14) where <Jj^ i s an a p p r o p r i a t e n u m e r i c a l c o e f f i c i e n t , t h e s u b s c r i p t K -j and K-j a r e t h e v a l u e s o f t h e component K f o r t h e l i m i t i n g p r o l a t e symme- t r i c t o p and t h e l i m i t i n g o b l a t e symmetric t o p , r e s p e c t i v e l y . ( i v ) E l e c t r o n S p i n Fine S t r u c t u r e H a m i l t o n i a n The e f f e c t s o f s p i n i n m o l e c u l a r s p e c t r a a r i s e from the i n t e r - a c t i o n o f each o f t h e e l e c t r o n s p i n m a g n e t i c moments w i t h : (a) t h e m a g n e t i c moments g e n e r a t e d by t h e o r b i t a l motions o f t h e e l e c - t r o n s ( i n t e r a c t i o n w i t h i t s own o r b i t a l m o t i o n b e i n g the most im- p o r t a n t ) ; t h i s i s known as s p i n - o r b i t i n t e r a c t i o n . (b) t h e m a g n e t i c moments g e n e r a t e d by t h e r o t a t i o n a l m otions o f the n u c l e i ; t h i s i s c a l l e d s p i n - r o t a t i o n i n t e r a c t i o n . ( c ) t h e s p i n m a g n e t i c moments o f t h e o t h e r e l e c t r o n s which i s known as s p i n - s p i n i n t e r a c t i o n . T h e r e f o r e t h e e l e c t r o n s p i n f i n e s t r u c t u r e H a m i l t o n i a n o p e r a t o r , He-|, i s t h e sum o f t h e s e i n t e r a c t i o n s , H e l = H s o + H s r + H s s < 2- 2 0> where H s Q , H s r and H g s a r e r e s p e c t i v e l y the energy o p e r a t o r s f o r s p i n - o r b i t , s p i n - r o t a t i o n and s p i n - s p i n i n t e r a c t i o n s , (a) S p i n - o r b i t I n t e r a c t i o n An e l e c t r o n possesses an i n t r i n s i c s p i n a n g u l a r momentum, ŝ . T h i s g i v e s r i s e t o a s p i n m a g n e t i c moment y^, whose v a l u e i s -16- ~m = " ^ B s (2.21) In t h i s e q u a t i o n g i s the r e l a t i v i s t i c " g - f a c t o r " , 2.0023, i s the u n i t o f m a g n e t i c moment ( t h e Bohr magneton, efi/2m, which e q u a l s -24 -1 % 9.274 x 10 JT i n SI u n i t s ) , and s i s t h e s p i n a n g u l a r momentum such t h a t <s 2>* = ( s f s + l ) ) 3 * fi (2.22) A c c o r d i n g t o Maxwell's e q u a t i o n s (15) an e l e c t r o n moving w i t h u n i - form l i n e a r v e l o c i t y v r e l a t i v e to a c o o r d i n a t e system where a s t a t i c e l e c t r i c f i e l d E e x i s t s , e x p e r i e n c e s a mag n e t i c f i e l d , B, g i v e n by B = E ^ ~ (2.23) cc as a r e s u l t o f i t s m o t i o n . Now e l e c t r i c f i e l d i s t h e g r a d i e n t o f po- t e n t i a l , s o t h a t i f t h e e l e c t r i c f i e l d r e s u l t s from t h e p r e s e n c e o f a cha r g e d n u c l e u s we have E = -vV = - ( £ \ dV (2.24) d r where ~ i s t h e u n i t v e c t o r from the p o s i t i o n o f the n u c l e u s towards r the p o s i t i o n o f the e l e c t r o n , and dV i s t h e r a t e o f change o f t h e dr (Coulomb) p o t e n t i a l o f t h e n u c l e u s w i t h d i s t a n c e . Then B = - - 4 — (&\r*v (2.25) r~ L \ d r / ~ ~ c r \ / -17- The o p e r a t o r f o r t h e i n t e r a c t i o n energy o f a m a g n e t i c d i p o l e w i t h a m a g n e t i c f i e l d i s H = -JJ.B. C2.26) T h i s becomes the o p e r a t o r f o r t h e s p i n - o r b i t i n t e r a c t i o n , on s u b s t i t u t i n g the v a l u e s o f v and B; the r e s u l t o f t h e s u b s t i t u t i o n i s H,„ = " 9 P B /dV\ r „ v . s (2.27) When the e x p l i c i t meanings o f r, and v a r e c o n s i d e r e d , r becomes r , the d i s t a n c e o f t h e e l e c t r o n from t he n u c l e u s , and v becomes v g n , t h e v e l o c i - t y o f the e l e c t r o n w i t h r e s p e c t t o t h e n u c l e u s . F o r completeness the e l e c t r o n s p i n s i s w r i t t e n s . When the f u r t h e r r e l a t i v i s t i c c o r r e c t i o n i n t r o d u c e d by Thomas (16) i s i n c l u d e d i n the e x p r e s s i o n , v ^ e q u a l s ^ X e " X n ' ^ n i s 'Thomas p r e c e s s i o n ' r e p r e s e n t s t h e f a c t t h a t time appears to be slowed down f o r t h e f a s t - m o v i n g e l e c t r o n as seen by the n u c l e u s , and i t happens t h a t i t appears ( t o the n u c l e u s ) t o be s p i n n i n g o n l y h a l f as f a s t as i f i t were s t a t i c . As a r e s u l t we have hi 9 J JB /dV \ r „ , f e - v j . s ; (2.28) ( " e n ) so " 1 ^ — i - e n ^ ^ e ~ n ' ~ e c r en The s p i n - o r b i t i n t e r a c t i o n i s a d d i t i v e f o r the v a r i o u s e l e c t r o n s and n u c l e i , so t h a t f o r a m o l e c u l e H 0 = E E _ J — / dv \ ^ - C ^ - v n ) . s e (2.29) • ° " n r e n l f r e n j -18- I f eq. (2.29) i s w r i t t e n as t h e d i f f e r e n c e o f two ter r a s , t h e s e c o n d term may be t a k e n as the i n t e r a c t i o n between t h e r o t a t i o n o f t h e n u c l e i , r e p r e s e n t e d by v , and the e l e c t r o n s p i n : i t i s the s p i n - r o t a t i o n i n t e r a c t i o n . The secon d terra w i l l be dropped a t t h i s p o i n t and c o n s i - d e r e d a g a i n i n the n e x t s e c t i o n . The f i r s t term becomes r e c o g n i s a b l y the s p i n - o r b i t i n t e r a c t i o n when we s u b s t i t u t e l = r , * m v„ (2.30) *en ~ e v ~ e as the o r b i t a l a n g u l a r momentum v e c t o r o p e r a t o r f o r e l e c t r o n e moving a r o u n d n u c l e u s n: H S 0 = " ^ B E Z j _ / d L k n 'le ( 2 ' 3 1 ) ° " n r e n Znrnc2 6 T h i s may be a p p r o x i m a t e d as H s o = E ^ e ( r ) £e • ie » ( 2 - 3 2 ) e where 2*mc 2 n r en K n , i s a parameter f o r e l e c t r o n e, which suras o v e r a l l n n u c l e i . Eq. (2.32) i s t h e s o - c a l l e d " m i c r o s c o p i c s p i n - o r b i t H a m i l t o n i a n " ( 1 7 , 1 8 ) , d e s c r i b - i n g t h e i n t e r a c t i o n o f e l e c t r o n s p i n s w i t h t h e f i e l d due t o e l e c t r o n s and n u c l e i , and c o n t a i n i n g s p i n - o r b i t , s p i n - o t h e r o r b i t and e l e c t r o n i c s c r e e n i n g e f f e c t s . U s i n g e i t h e r e q. (.2.31) o r (2.32) t h e c o r r e c t p a r a - m e t r i c dependence o f t h e i n t e r a c t i o n s i s o b t a i n e d but the i n t e r p r e t a t i o n s -19- o f t h e c o n s t a n t s d i f f e r . A c o n v e n i e n t s i m p l e i s o t r o p i c form o f eq. (.2.32), (2.33) where A(.r) i s an r-dependent p a r a m e t e r , i s o f t e n used f o r t h e c a l c u l a - t i o n o f m a t r i x elements d i a g o n a l i n S. In eq. (2.33) L i s t h e t o t a l e l e c t r o n i c o r b i t a l a n g u l a r momentum o f t h e m o l e c u l e o r r a d i c a l , and S i s t h e t o t a l e l e c t r o n i c s p i n a n g u l a r momentum o f the m o l e c u l e o r r a d i c a l , The m i c r o s c o p i c s p i n - o r b i t o p e r a t o r , Ea.(r)£..s. , i s needed i f m a t r i x elements o f t h e s p i n - o r b i t o p e r a t o r o f f - d i a g o n a l i n S a r e t o be c a l c u l a t e d . (b) S p i n - R o t a t i o n H a m i l t o n i a n The s o - c a l l e d s p i n - r o t a t i o n i n t e r a c t i o n a r i s e from two c a u s e s . One o f t h e s e i s the d i r e c t i n t e r a c t i o n o f the e l e c t r o n s p i n w i t h t he m a g n e t i c f i e l d o f the m o l e c u l a r r o t a t i o n . I t can be shown (.18) t h a t the s e c o n d term i n eq. (2.29) i s e q u i v a l e n t to (2.34) S = (2.35) E (2.36) -20- where the v e c t o r £ denotes t h e n u c l e a r r o t a t i o n a l a n g u l a r momentum, and the c o e f f i c i e n t s e' a r e c o m p l i c a t e d f u n c t i o n s o f t h e .moments o f i n e r t i a , t h e i n t e r n u c l e a r d i s t a n c e s , and the a v e r a g e d i s t a n c e s o f t h e e l e c t r o n s to t h e v a r i o u s n u c l e i o f t h e m o l e c u l e . B e s i d e s t h e d i r e c t c o u p l i n g between e l e c t r o n i c s p i n and m o l e c u l a r r o t a t i o n , t h e c o m b i n a t i o n o f o f f - d i a g o n a l s p i n - o r b i t and o r b i t - r o t a t i o n m a t r i x e l e m e n t s , t r e a t e d by secon d o r d e r p e r t u r b a t i o n t h e o r y , g i v e s r i s e t o much l a r g e r e f f e c t i v e o p e r a t o r s w i t h t h e same a n g u l a r momentum dependence ( 1 9 , 2 0 ) . The two c o n t r i b u t i o n s cannot be d i s t i n g u i s h e d , and a r e t h e r e f o r e added t o produce the d e t e r m i n a b l e s p i n - r o t a t i o n c o e f f i c i e n t the t e n s o r z, so t h a t , s t r i c t l y s p e a k i n g , i t s h o u l d be w r i t t e n as t h e h e r m i t i a n a v e r a g e : ( c ) S p i n - S p i n H a m i l t o n i a n S i n c e e l e c t r o n s p o s s e s s e s m a g n e t i c moments, a l l t h e e l e c t r o n s o f a m o l e c u l e must i n t e r a c t w i t h each o t h e r t h r o u g h Coulomb's law o f mag- n e t i c i n t e r a c t i o n . T h i s mechanism g i v e s r i s e t o t h e d i p o l a r e l e c t r o n s p i n - s p i n i n t e r a c t i o n . A s p i n n i n g e l e c t r o n a t a p o i n t j i n space can be c o n s i d e r e d as a bar magnet w i t h m a g n e t i c moment y.. The mag n e t i c v e c t o r p o t e n t i a l (15) t h a t i s pro d u c e d a t c o o r d i n a t e i i s g i v e n by Eq. (2.36) r e l a t e s two non-commuting o p e r a t o r s , 1̂1 and S, t h r o u g h H (2.37) s r (2.38 -21- By M a x w e l l ' s e q u a t i o n s t h i s c o r r e s p o n d s to a mag n e t i c f i e l d a t p o i n t i B. = V ̂  A. (2.39) The H a m i l t o n i a n f o r t h e i n t e r a c t i o n o f a second e l e c t r o n p l a c e d a t i (w i t h m a g n e t i c moment V i ) and the e l e c t r o n a t j ( w i t h m a g n e t i c moment v5) i s v A s . r . . Using t h e r e l a t i o n s 6 K (2.40) (2.41) and v /s v ~ V = v (v.V) - v V (2.42) f o r any v e c t o r V, t h i s can be w r i t t e n .2 ~2 n The f i r s t term o f eq. (2.43) can be f u r t h e r s i m p l i f i e d w i t h G a u s s ' d i v e r - gence theorem which g i v e s H .. = g 2 pB [ s , . s.) v 2 - ( s r v ) ( S j . v (2.43) 4u 5 ( T j | ) (2.44) -22- where <5(.r...) i s a D i r a c d e l t a f u n c t i o n . I t i s d e f i n e d such t h a t i t p i c k s o u t the s q u a r e o f t h e a m p l i t u d e o f the e l e c t r o n w a v e f u n c t i o n f o r e l e c t r o n j a t the c o o r d i n a t e o r i g i n , i . e . a t the p o s i t i o n o f e l e c t r o n i < « ( r j i ) > = 1^(0) | 2 (2.45) T h e r e f o r e , t h e f i r s t term becomes H s s ' i j ( c o n t a c t ) = - g 2 p B 2 4^ | ^ . ( o ) | 2 ^ (2.46) The second term i n eq. (2.43) i s more i n v o l v e d , b u t , i t can be shown t h a t , a f t e r some a l g e b r a , i t becomes 2 2 J 4 f U.(o)| 2 + £ ( s . . s . ) r . . 2 - 3 ( s . . r . , ) ( s . . r . . ) ] r . , ~ 5 i (2.47) ft2 | 3 | V j v 1 L ~ i ~ j ' j i ~ j i ~ j ^ j i ' J j i \ ' F i n a l l y , c o l l e c t i n g terms and summing o v e r a l l e l e c t r o n s , the s p i n - s p i n i n t e r a c t i o n H a m i l t o n i a n i s (2.48) The f i r s t term t u r n s o u t t o g i y e a c o n s t a n t c o n t r i b u t i o n to the e n e r g y , and i s i n c l u d e d w i t h t h e Born-Oppenheimer p o t e n t i a l ; the s e c o n d term i s the d i p o l a r e l e c t r o n s p i n - s p i n i n t e r a c t i o n . -23- (v) N u c l e a r S p i n H y p e r f i n e S t r u c t u r e H a m i l t o n i a n H y p e r f i n e i n t e r a c t i o n s , caused by non- z e r o n u c l e a r s p i n s , a r e the l a s t terms we c o n s i d e r i n the m o l e c u l a r H a m i l t o n i a n . H y p e r f i n e s t r u c t u r e r e s u l t s from t h e i n t e r a c t i o n o f the m a g n e t i c and e l e c t r i c moments o f the n u c l e i w i t h t h e o t h e r e l e c t r i c and-magnetic moments i n the m o l e c u l e . We s h a l l w r i t e H h f s = H m a g . h f s + HQ ( 2- 4 9 ) where H m a g ^ i s the m a g n e t i c h y p e r f i n e H a m i l t o n i a n , which a r i s e s from the i n t e r a c t i o n o f t h e n u c l e a r - s p i n m a g n e t i c moments w i t h o t h e r m a g n e t i c moments i n t h e m o l e c u l e , and HQ i s t h e e l e c t r i c q u a d r u p o l e H a m i l t o n i a n , which a r i s e s when n u c l e i w i t h s p i n 151 (which possess e l e c t r i c q u a d r u p o l e moments) i n t e r a c t w i t h t he n o n - s p h e r i c a l e l e c t r o n charge d i s t r i b u t i o n i n the f i n i t e volume o f each n u c l e u s . In m o l e c u l e s w i t h e l e c t r o n i c a n g u l a r momentum, t h e m a g n e t i c hyper- f i n e s t r u c t u r e i s u s u a l l y much l a r g e r t h a n t h a t due t o e l e c t r i c q u a d r u p o l e moments. We b e g i n by d i s c u s s i n g the magnetic h y p e r f i n e H a m i l t o n i a n f o r a s i n g l e s p i n n i n g - n u c l e u s . (a) M a g n e t i c h y p e r f i n e s t r u c t u r e h a m i l t o n i a n The t h e o r y o f m a g n e t i c h y p e r f i n e s t r u c t u r e i n d i a t o m i c m o l e c u l e s has been g i v e n by F r o s c h and F o l e y ( 2 1 ) , who d e r i v e d t h e H a m i l t o n i a n from t he D i r a c e q u a t i o n f o r the e l e c t r o n . An a l t e r n a t i v e and s i m p l i f i e d d e r i v a t i o n o f the same h y p e r f i n e i n t e r a c t i o n e x p r e s s i o n i s g i v e n by Dousmanis ( 2 2 ) . The mechanism g i v i n g r i s e t o m a g n e t i c h y p e r f i n e s t r u c t u r e i s e x a c t l y -24- t h e same as t h e e l e c t r o n s p i n i n t e r a c t i o n , s o w i t h t h e r e p l a c e m e n t o f one o f t h e e l e c t r o n s p i n m a g n e t i c moments, JJ , by t h e n u c l e a r s p i n mag- ~ s n e t i c moment, j j j , Hi - Vnl (2'5°) Eq. (2.48) becomes 2 2 H u.r = 9" P" £ j 8TT U , ( o ) | 2 I,.S. m a 9 - h f s I T »>J ! ^ 1 ~ J (2.5u- - [ ( l r s j ) r j 1 2 - 3 U t . I j l ) < . , . r j 1 ) ] r j ( - 5 | In t h i s c a s e t h e f i r s t term does not g i v e a c o n s t a n t c o n t r i b u t i o n ; i t i s c a l l e d the Fermi c o n t a c t i n t e r a c t i o n and i s a measure o f t h e e x t e n t t o which t h e u n p a i r e d e l e c t r o n s have non-zero p r o b a b i l i t y a m p l i t u d e a t t h e s p i n n i n g n u c l e u s . F o r the a m p l i t u d e t o be non-ze r o an u n p a i r e d e l e c t r o n must occupy a m o l e c u l a r o r b i t a l d e r i v e d from an a t o m i c s o r b i t a l . U n l i k e t h e d i p o l a r c o u p l i n g , the Fermi c o n t a c t i n t e r a c t i o n i s i s o t r o p i c , and i s r e p r e s e n t e d by a term o f t h e form c , i 1 (2.52) where a . i s t h e i s o t r o p i c c o u p l i n g c o n s t a n t , c, 1 The second term i s t h e d i p o l a r ( I , s ) i n t e r a c t i o n between t h e n u c l e a r m a g n e t i c moment and t h e m a g n e t i c f i e l d p r oduced a t t h e n u c l e u s by the v a l e n c e e l e c t r o n s . Eq. (2.51) g i v e s the c o n t a c t and d i p o l a r p a r t s o f t h e m a g n e t i c i n t e r - -25- a c t i o n ; on a d d i t i o n o f the term z a. ( 2 3 ) , w h i c h g i v e s t h e i n t e r - a c t i o n o f the n u c l e a r m a g n e t i c moment w i t h t h e o r b i t a l a n g u l a r momentum o f the u n p a i r e d e l e c t r o n o r e l e c t r o n s , t h e t o t a l m a g n e t i c h y p e r f i n e H a m i l t o n i a n i s o b t a i n e d as H i ^ = z a. I.£. + z a .. I .s. raag.hfs . l ~ ~ i . c , i ~ ~ i 1 (.2.53) - E W n [ ( I . s . ) r 2 - 3 ( I . r ) ( s . . r ) ] r " 5 . 7.— ~ ~ i ~ ~ ~ i ~ I t i s u s u a l l y s u f f i c i e n t t o c o l l e c t t h e sums o v e r e l e c t r o n s i n t o a s i n g l e p a r a m e t e r , g i v i n g H ., = a I.L + b I . S + c I _ S 7 (2.54) mag,hfs ~ ~ ~ ~ z z where a = W n / J v (2.55) -t2 \ „ 3 / r . l a = Sri g V B y n U (o)|2 (2.56) C 3 * 2 c =39WV(3 cosVlK (2.57) and b = a c - l c (2.58) The terms a , b and c a r e d e t e r m i n a b l e c o e f f i c i e n t s i n t h e ma g n e t i c hyper- f i n e H a m i l t o n i a n ; a i s t h e n u c l e a r s p i n - o r b i t i n t e r a c t i o n , b i s a com- b i n a t i o n o f c w i t h t h e Fermi c o n t a c t i n t e r a c t i o n , a c , and c i s t h e d i p o - l a r e l e c t r o n s p i n - n u c l e a r s p i n i n t e r a c t i o n . - 2 6 - (b). E l e c t r i c q u a d r u p o l e H a m i l t o n i a n A l t h o u g h t h e m a g n e t i c d i p o l e i n t e r a c t i o n i s r e s p o n s i b l e f o r the l a r g e s t c o n t r i b u t i o n t o t h e o b s e r v e d h y p e r f i n e s t r u c t u r e o f a m o l e c u l e i n a m u l t i p l e t e l e c t r o n i c s t a t e , e l e c t r i c q u a d r u p o l e e f f e c t s a r e p r e s e n t whatever t h e m u l t i p l i c i t y i s , p r o v i d e d I L 1. These a r e caused by the i n t e r a c t i o n between the n u c l e a r e l e c t r i c q u a d r u p o l e moment and t h e e l e c t r i c f i e l d g r a d i e n t produced by the s u r r o u n d i n g e l e c t r o n s ( 2 4 ) . The e l e c t r o s t a t i c i n t e r a c t i o n between a n u c l e u s a t p o i n t r . and an e l e c t r o n a t t h e p o i n t i s g i v e n by Coulomb's law as H c = ^ z e 2 , = - z e £ (2.59) r . . I r . - r . I ~ i j ' ~ i ~ j 1 R e c a l l i n g t h a t e l e c t r o n s a r e p o i n t charges b u t n u c l e i have a f i n i t e s i z e f o r t h e i r e l e c t r i c c h a r g e d i s t r i b u t i o n , we s h a l l n e g l e c t any e l e c t r o n i c charge l y i n g w i t h i n the n u c l e a r r a d i u s , then r* > r . and we can expand eq. (2.59) i n a s c e n d i n g powers o f r-/r- g i v i n g (25) 2 00 r k H = - z e ' z l i _ P j c o s e . .) (2.60) c k=o k+1 K 1 J r i where P. ( c o s e . . ) i s the Legendre p o l y n o m i a l o f o r d e r k and e . • i s the a n g l e between r, and r . . The f i r s t term i n the summation o f eq. (2.60) r e p r e s e n t s a monopole i n t e r a c t i o n and when summed o v e r a l l e l e c t r o n s g i v e s t h e f a m i l i a r coulomb i n t e r a c t i o n . The secon d term r e p r e s e n t s a n u c l e a r e l e c t r i c d i p o l e i n t e r - a c t i o n w h i c h , by a p p l i c a t i o n o f p a r i t y and time r e v e r s a l symmetry arguments ( 2 6 ) , can be shown t o be i d e n t i c a l l y z e r o , as a r e the h i g h e r e l e c t r i c m u l t i p o l e moments o f odd o r d e r . F i n a l l y , t h e term w i t h k=2 -27- c o r r e s p o n d s to an e l e c t r i c q u a d r u p o l e i n t e r a c t i o n . The s e p a r a t i o n o f e l e c t r i c and n u c l e a r c o o r d i n a t e s i n e q . (2.60). can be compl e t e d By a p p l y i n g the s p h e r i c a l harmonic a d d i t i o n theorem: k z 2k+l q=-k k where Y q (e,<|>) i s t h e q t n component o f th e s p h e r i c a l harmonic o f o r d e r k, and th e a n g l e s e and <t> a r e s p h e r i c a l p o l a r c o o r d i n a t e s . The e l e c t r i c q u a d r u p o l e i n t e r a c t i o n t h e n becomes Pk(cos 6 i .) - l (-Dq Y k (.V*f) Y.k ( 6 ( 2 . 6 1 ) (2.62) We o b s e r v e t h a t the above e x p r e s s i o n has th e form o f the s c a l a r p r o d u c t o f a n u c l e a r e l e c t r i c q u a d r u p o l e t e n s o r and an e l e c t r i c f i e l d g r a d i e n t t e n s o r , each o f rank two. (The p r o p e r t i e s o f s p h e r i c a l t e n s o r s w i l l be d i s c u s s e d i n s e c t i o n ( 2 . G ) ) . T h e r e f o r e : HQ = e T 2 ( Q ) . T 2(/VE) (2.63) For a l i n e a r m o l e c u l e t h e t e n s o r T (vE) has one i n d e p e n d e n t component, so t h a t t h e r e i s o n l y one " q u a d r u p o l e c o u p l i n g c o n s t a n t " w hich w i l l be d e f i n e d i n s e c t i o n C2.C). -28- ( v i ) E f f e c t i v e H a m i l t o n i a n and Degenerate P e r t u r b a t i o n T h e o r y The a n a l y s i s o f m o l e c u l a r s p e c t r a u s i n g the t r u e J D O I e c u ! a r e i g e n - f u n c t i o n s i s i m p r a c t i c a l , s i n c e i t would r e q u i r e t h e d i a g o n a l i z a t i o n o f an i n f i n i t e m a t r i x . Even i f t h i s m a t r i x were s u i t a b l y t r u n c a t e d t h e problem would s t i l l be v e r y d i f f i c u l t t o h a n d l e . I d e a l l y a m a t r i x r e - p r e s e n t a t i o n i s r e q u i r e d t h a t c o n t a i n s no terms o f f - d i a g o n a l i n v i b r a - t i o n a l o r e l e c t r o n i c s t a t e ; a l t h o u g h the m a t r i x r e p r e s e n t a t i o n i s s t i l l i n f i n i t e i t c o n s i s t s o n l y o f s u b m a t r i c e s , each c o n t a i n i n g o n l y elements p e r t a i n i n g t o a s i n g l e e l e c t r o n i c - V i b r a t i o n a l l e v e l . E i g e n v a l u e s can be o b t a i n e d f o r the v a r i o u s s u b m a t r i c e s , from w h i c h i t i s p o s s i b l e t o d e t e r m i n e the t r a n s i t i o n f r e q u e n c i e s . The c o n s t r u c t i o n o f such sub- m a t r i c e s r e q u i r e s t h a t the e f f e c t s o f a l l elements o f t h e f u l l H a m i l t o - n i a n t h a t a r e o f f - d i a g o n a l i n v i b r a t i o n a l o r e l e c t r o n i c s t a t e be r e d u c e d to a n e g l i g i b l e l e v e l . A s i m p l e p r a c t i c a l way to s e t up t h e s e m a t r i c e s employs an e f f e c t i v e H a m i l t o n i a n t h a t o n l y o p e r a t e s w i t h i n the m a n i f o l d o f a p a r t i c u l a r e l e c t r o n i c - v i b r a t i o n a l s t a t e . There a r e two commonly- used methods f o r d e r i v i n g the e f f e c t i v e H a m i l t o n i a n , namely c o n t a c t t r a n s f o r m a t i o n s and p e r t u r b a t i o n t h e o r y . In the c o n t a c t t r a n s f o r m a t i o n method (27) a c a r e f u l l y - c h o s e n u n i t a r y t r a n s f o r m a t i o n i s a p p l i e d to t h e H a m i l t o n i a n to e l i m i n a t e s p e c i f i c o f f - d i a g o n a l e l e m e n t s ; s i n c e t h i s method has n o t been used i n t h i s t h e s i s i t w i l l n o t be d i s c u s s e d f u r t h e r . The t e c h n i q u e o f d e g e n e r a t e p e r t u r b a - t i o n t h e o r y has been used e x t e n s i v e l y i n the next c h a p t e r and w i l l t h e r e f o r e be d i s c u s s e d i n more d e t a i l . M e s s i a h (25) has d e s c r i b e d t h e t e c h n i q u e s o f non-degenerate and d e g e n e r a t e p e r t u r b a t i o n theorymost c o m p r e h e n s i v e l y , b u t a r a t h e r more -29- r e a d a b l e a c c o u n t o f t h e d e r i v a t i o n o f an e f f e c t i v e H a m i l t o n i a n has been g i v e n by S o l i v e r e z ( 2 8 ) , u s i n g t h e f o r m a l i s m s e t up by B l o c h (.29). The e i g e n f u n c t i o n s |i> , o f t h e t o t a l H a m i l t o n i a n which o p e r a t e s o v e r a l l v e c t o r s p a c e , form a complete o r t h o n o r m a l s e t . We want a H a m i l t o n i a n t h a t o p e r a t e s o n l y w i t h i n a p a r t i c u l a r m a n i f o l d o f the t o t a l H i l b e r t s p a c e . In o t h e r words we w i s h t o p r o j e c t t h e e f f e c t s o f the t o t a l H a m i l t o n i a n o p e r a t o r onto a c h o s e n v e c t o r s p a c e which i s o f d i m e n s i o n l e s s than t h a t o f the t o t a l v e c t o r s p a c e , and hence to c o n s t r u c t an e f f e c t i v e H a m i l t o n i a n t h a t o p e r a t e s o n l y w i t h i n t h i s chosen v e c t o r s p a c e , and w i t h t h e e q u i v a l e n t o p e r a t o r form w i t h i n t h i s m a n i f o l d o f t h e t o t a l H a m i l t o n i a n . The o p e r a t o r which b r i n g s about t h i s p r o j e c t i o n o f t h e t o t a l H a m i l t o n i a n i s a p r o j e c t i o n o p e r a t o r , P . S o l i v e r e z (28) shows t h a t i t i s p o s s i b l e t o s e t up an e f f e c t i v e H a m i l t o n i a n which has t h e f o l l o w i n g p r o p e r t i e s : (a) I t o p e r a t e s o n l y w i t h i n a m a n i f o l d o f d i m e n s i o n l e s s than t h a t o f the t o t a l v e c t o r s p a c e . (b) I t s e i g e n v a l u e s a r e i d e n t i c a l to t h o s e c o r r e s p o n d i n g e i g e n v a l u e s o f t h e t o t a l H a m i l t o n i a n . ( c ) I t s e i g e n v e c t o r s a r e s i m p l y r e l a t e d t o t h e c o r r e s p o n d i n g e i g e n - v e c t o r s o f t h e t o t a l H a m i l t o n i a n . (d) I t can be expanded as a power s e r i e s i n terms o f a p e r t u r b a t i o n V, and i s H e r m i t i a n to a l l o r d e r s o f the e x p a n s i o n . We s h a l l i n d i c a t e b r i e f l y h e r e how such a H a m i l t o n i a n i s s e t up. The t o t a l H a m i l t o n i a n o f t h e s y s t e m under s t u d y i s s p l i t i n t o two p a r t s -30- H = H Q + V (2.64) where the e i g e n v a l u e s and e i g e n v e c t o r s o f H a r e known: H 0 l J > 0 = E. | j > o (2.65) and the e i g e n f u n c t i o n s | j > Q from a complete o r t h o n o r m a l s e t o v e r a l l v e c t o r s p a c e . V i s a p e r t u r b a t i o n t o t h i s H a m i l t o n i a n and we a r e i n t e r e s t e d i n i t s e f f e c t s on the e i g e n v e c t o r s and e i g e n v a l u e s o f H Q . In p a r t i c u l a r , we a r e c o n c e r n e d w i t h the e i g e n v e c t o r s s p a n n i n g the p a r t i c u l a r m a n i f o l d onto which the t o t a l H a m i l t o n i a n i s p r o j e c t e d ; t h e s e w i l l have a p a r t i c u l a r e i g e n v a l u e , E Q . The p r o j e c t i o n o p e r a t o r i s d e f i n e d as P 0 = S |1><i| (2.66) where the e i g e n v e c t o r s | i> span t h e m a n i f o l d under c o n s i d e r a t i o n . I t f o l l o w s a t once t h a t H „ p „ = P~ H. = EJ> (2.67) 0 0 0 0 0 0 I't i s supposed t h a t E Q can be d e g e n e r a t e , and t h a t t h e p e r t u r b a t i o n y l i f t s t h i s d e g e n e r a c y . The e i g e n v a l u e s c o r r e s p o n d i n g to the p e r t u r b e d energy l e v e l s a r e g i v e n by ( H Q + M) |k> = ( E 0 + Ak) |k> (2.68) -31- which can be r e a r r a n g e d to g i v e ( H Q - E Q ) |k> = ( A k - M) |k> (2.69) A K a r e t h e s h i f t s i n the energy l e v e l s caused by the p e r t u r b a t i o n y. Using eq. (.2.67) i t can e a s i l y be shown t h a t P QV |k> = A K P Q |k> = A K |k> 0 (2.70) where the |k> Q a r e e i g e n f u n c t i o n s o f H Q , and i n p a r t i c u l a r a r e t h o s e e i g e n f u n c t i o n s |i> s p a n n i n g the v e c t o r space under c o n s i d e r a t i o n . T h e r e i s a complementary p r o j e c t i o n o p e r a t o r Q Q which f o l l o w s from t h e c l o s u r e r e l a t i o n s h i p H o o = l\a><i\ (2.71) where the e i g e n v e c t o r s \i> have been e x c l u d e d from eq. (2.66) s i n c e t h e y do n o t span t h e m a n i f o l d we a r e i n t e r e s t e d i n . Q Q a l s o has the p r o p e r t y " 0 = ( ^ K - Eo> -Uo-vtr) (2-72) \ 3 / W H 6 R E - I J ^ l L (2.73) From eq. (2.69) and e q . (.2.72) i t f o l l o w s t h a t k> =(%\(U - V) |k> 12.74) Q 0 |k> = ( ^ ) ( A k -32- We a r e now a b l e t o f i n d a r e l a t i o n s h i p f o r | k> i n terras o f the un- p e r t u r b e d e i g e n v e c t o r s |k> Q: |k> = ( P 0 + Q 0) |k> = |k> +/QOUA, - V) |k> (2.75) 0 W • The terms can be e l i m i n a t e d from e q . (.2.75) by r e p e a t e d use o f eq. (2.70) t o g i v e an e x p a n s i o n o f | k> i n terms o f |k>0> V, P Q a n d ^ c ^ , t h a t i s s u b s t i t u t i n g L.H.5. o f eq. (2.75) i n t o R.H.S. The e i g e n v e c t o r s f o r t h e p e r t u r b e d and u n p e r t u r b e d H a m i l t o n i a n s a r e thus r e l a t e d by an i d e n t i t y o f g e n e r a l form |k> = U |k> Q (2.76) where U i s an o p e r a t o r i n v o l v i n g V, P n and _°_ which can be expanded as a i n f i n i t e s e r i e s i n terms o f t h e s e o p e r a t o r s . S u b s t i t u t i o n o f eq. an (2.76) i n t o eq. (2.70) l e a d s t o t h e f o l l o w i n g e i g e n v a l u e e x p r e s s i o n P o V U l k > o = A k l k > o ( 2 ' 7 7 > I f we i d e n t i f y ( P Q V |J) w i t h the e f f e c t i v e H a m i l t o n i a n we see t h a t i t does p o s s e s s t h e p r o p e r t i e s l i s t e d by S o l i v e r e z ( 2 8 ) , a l t h o u g h i t s h e r m i t i a n n a t u r e has n o t been d e m o n s t r a t e d . Note t h a t t he A^'s denote the e n e r g y s h i f t s from an o r i g i n E Q . Eq. (.2.77) i s a l s o p a r t i c u l a r l y c o n v e n i e n t i n t h a t i t uses e i g e n f u n c t i o n s o f t h e u n p e r t u r b e d H a m i l t o n i a n as b a s i s f u n c t i o n and t h e s e a r e by d e f i n i t i o n known. -33- We can now r e t u r n t o t h e d e r i v a t i o n o f an e f f e c t i v e .molecular Ha- m i l t o n i a n . The b a s i s f u n c t i o n s Ik> a r e t a k e n to be the e l e c t r o n i c - . v i . b r a - 1 o t i o n a l s t a t e s o f i n t e r e s t , which a r e e i g e n f u n c t i o n s o f H Q , and V i s the p e r t u r b a t i o n t h a t mixes them. As d e s c r i b e d above (28,30) t h e t o t a l H a m i l t o n i a n i s d i v i d e d i n t o two p a r t s : H , = H Q + A V (2.78) The parameter A i s a s m a l l number because V i s assumed s m a l l compared t o V The p r o j e c t i o n o p e r a t o r s P 0 and Q Q a r e g i v e n the more e x p l i c i t d e f i n i t i o n s P o = E l 4 o i > < £ o ( 2' 7 9 ) i -)= E I l& i x f t i 1 (2.80) a7 lH0 i ( E o - E A ) n where I r e f e r s to t h e e l e c t r o n i c - v i b r a t i o n a l s t a t e o f i n t e r e s t , l r e f e r s o to t h e e l e c t r o n i c - v i b r a t i o n a l s t a t e s o t h e r than £ Q and i r e f e r s t o t h e s e t o f quantum numbers w i t h i n t h e m a n i f o l d such as o r K e t c . The e f f e c t i v e H a m i l t o n i a n i s g i v e n by H e f f = A P 0 V U (2.81) As has a l r e a d y been n o t e d , U can be expanded as an i n f i n i t e s e r i e s -34- U = £ A" U" (2.82) n=o where U n i s g i v e n by the g e n e r a l f o r m u l a U n = z V l V S K 2 V . . . S K n V Pn (2.83) o e x c e p t t h a t U° = P (2.84) o K can t a k e t he v a l u e s 0, 1, 2 ... such t h a t n K 1 + K 2 + . . . K n = n K 1 + K 2 + . . . K j l j ( j = 1,2 ...n-1) (2.85) Y." i s the c o n d i t i o n t h a t a l l K i a r e n o n - n e g a t i v e . In a d d i t i o n n s" f o r n f o (2-86) As n o t e d by F r e e d ( 3 1 ) , c e r t a i n terms i n t h e e x p a n s i o n may be non- H e r m i t i a n , b u t by t a k i n g t h e H e r m i t i a n a v e r a g e o f such terms t h e e f f e c - t i v e H a m i l t o n i a n i s made H e r m i t i a n t o a l l o r d e r s . By t h e use o f eq. (2.82) and ( 2 . 8 3 ) , e q . (2.81) can be expanded as f o l l o w s : •35- H e f f - * P 0 V U = * P o V P o + A P Q V ( Q 0 / a ) V P 0 ,3, +. r ( P 0 V (Q 0/a) V (Q 0 / a ) V P Q - I P 0 V ( 0 / a 2 ) V P 0 V P / } + A 0 4 where the dagger means t h a t the H e r m i t i a n a v e r a g e o f t h e term i a s q u a r e b r a c k e t s i s t o be t a k e n . The c o e f f i c i e n t o f A n r e p r e s e n t s t h e n t h o r d e r c o n t r i b u t i o n t o t h e e f f e c t i v e H a m i l t o n i a n . The e x p a n s i o n o f t h e e f f e c - t i v e H a m i l t o n i a n i s e x p e c t e d t o c o n v e r g e f a i r l y r a p i d l y , a l t h o u g h the r a t e o f co n v e r g e n c e w i l l depend on how the H a m i l t o n i a n was o r i g i n a l l y p a r t i t i o n e d . In p r a c t i c e the t o t a l H a m i l t o n i a n i s p a r t i t i o n e d i n suc h a way t h a t t h e dominant i n t e r a c t i o n s a r i s e i n f i r s t o r d e r o f p e r t u r b a t i o n t h e o r y . S m a l l e r i n t e r a c t i o n s a r e i n c l u d e d i n the e f f e c t i v e H a m i l t o n i a n by a p p e a l i n g t o h i g h e r o r d e r s u n t i l t h e r e q u i r e d p r e c i s i o n o f the e i g e n - v a l u e s , a l i m i t u s u a l l y imposed by e x p e r i m e n t , i s r e a c h e d . The e f f e c t i v e H a m i l t o n i a n w r i t t e n i n terms o f o p e r a t o r e q u i v a l e n t s w i l l be c o n s i d e r e d i n t h e next s e c t i o n a f t e r a b r i e f d i s c u s s i o n o f a n g u l a r momentum o p e r a t o r s and some s t a n d a r d s p h e r i c a l t e n s o r t e c h n i q u e s . C. C a l c u l a t i o n o f M a t r i x Elements The c o n c e p t s o f a n g u l a r momentum and r o t a t i o n a l i n v a r i a n c e p l a y an i m p o r t a n t p a r t i n the a n a l y s i s o f m o l e c u l a r s p e c t r a . U s i n g t h e g e n e r a l t h e o r y o f a n g u l a r momentum 0 1 , 1 2 ) , e x p r e s s i o n s which depend o n l y on the r o t a t i o n a l p r o p e r t i e s o f v a r i o u s o p e r a t o r s and s t a t e v e c t o r s can be s e p a r a t e d from q u a n t i t i e s which a r e i n v a r i a n t under r o t a t i o n s . -36- I t i s wo r t h n o t i n g t h a t t h e s t r u c t u r e o f t h e s e e x p r e s s i o n s i s p r i m a r i l y a f u n c t i o n o f t h e c o m p l e x i t y o f t h e system b e i n g s t u d i e d s u c h a s , f o r i n s t a n c e , t h e number o f a n g u l a r momenta i n t h e c o u p l i n g scheme. When t h e s e i d e a s a r e to be c a r r i e d o u t m a t h e m a t i c a l l y , s p h e r i c a l t e n s o r a l g e b r a has pr o v e d e x t r e m e l y u s e f u l and a l s o o f f e r s g r e a t p h y s i c a l i n - s i g h t . I t i s n o t i n t e n d e d t o g i v e a t h r o u g h a c c o u n t o f a n g u l a r momentum and s p h e r i c a l t e n s o r t h e o r y i n t h i s s e c t i o n j the aim i s m e r e l y t o g i v e some i m p o r t a n t r e s u l t s and u s e f u l r e l a t i o n s h i p s t h a t w i l l be c a l l e d upon i n s u b s e q u e n t c a l c u l a t i o n s . ( i ) A n g u l a r Momenta A n g u l a r momentum o p e r a t o r s a r e d e f i n e d as t h o s e quantum m e c h a n i c a l o p e r a t o r s t h a t obey t h e commutation r u l e s (.11, 12). [ P x , P Y J = i P Z (2.88) (and c y c l i c p e r m u t a t i o n s o f X, Y, Z) where Px, Py and P^ a r e c a r t e s i a n components o f t h e o p e r a t o r £ . S i n c e t h e s e components do n o t commute i t i s n o t p o s s i b l e t o d e t e r m i n e them a l l s i m u l t a n e o u s l y . However, the an- g u l a r momentum and the e n e r g y a r e both c o n s t a n t s o f t h e m o t i o n , s o t h a t [P, Hj = 0 (2.89) where H i s t h e H a m i l t o n i a n o p e r a t o r . Because they commute P and H must po s s e s s s i m u l t a n e o u s e i g e n f u n c t i o n s . T h i s i s an i m p o r t a n t p o i n t because -37- m a t r i x elements o f t h e H a m i l t o n i a n o p e r a t o r can be c a l c u l a t e d u s i n g the a n g u l a r momentum e i g e n f u n c t i o n s a s b a s i s f u n c t i o n s . F o r t h i s purpose i t i s c o n v e n i e n t t o w r i t e t h e H a m i l t o n i a n o p e r a t o r i n terms o f a n g u l a r mo- mentum o p e r a t o r s and t h e i r components, r a t h e r than i n terms o f d i f f e r e n - t i a l o p e r a t o r s . The b a s i s f u n c t i o n s a r e d e f i n e d i n terms o f quantum numbers r e l e v a n t t o t h e i n d i v i d u a l a n g u l a r momenta r a t h e r than the ex- p l i c i t forms o f t h e w a v e f u n c t i o n s . There a r e v a r i o u s k i n d s o f a n g u l a r momenta t h a t can a r i s e i n a m o l e c u l e . F i r s t l y , t h e r e i s t h e e l e c t r o n i c o r b i t a l a n g u l a r momentum L. which i s the sum o f t h e o r b i t a l a n g u l a r momenta o f each o f t h e e l e c t r o n s where r . and p. a r e r e s p e c t i v e l y t h e p o s i t i o n and momentum o p e r a t o r s f o r th e i n d i v i d u a l e l e c t r o n s . S and I a r e the e l e c t r o n i c s p i n and n u c l e a r s p i n a n g u l a r momenta r e s p e c t i v e l y . F i n a l l y we s h a l l c o n s i d e r t h e a n g u l a r momentum due t o r o t a t i o n o f t h e n u c l e i , R. A c c o r d i n g to t h e r u l e s o f v e c t o r c o u p l i n g 1 r e s u l t a n t s £ and £ can be c o n s t r u c t e d : L = ? * i = ? ^ £i (2.90) J = R + L + S (2.91) F = J + I (2.92) 1 The p o s s i b i l i t y o f d i f f e r e n t c o u p l i n g schemes and the d i f f e r e n t s e t s o f w e l l - d e f i n e d quantum numbers t h a t emerge w i l l be d e a l t w i t h i n s e c t i o n C ( i i i ) . -38- J i s the t o t a l a n g u l a r momentum In the absence o f n u c l e a r s p i n s , w h i l e F i s the grand t o t a l a n g u l a r momentum. A p a r t i a l sum N = J - S (2.93) '^w rs^i r++s w i l l a l s o be c o n s i d e r e d . C o n s e r v a t i o n o f a n g u l a r momentum a p p l i e s t o the t o t a l a n g u l a r momentum ( F o r J) b u t not n e c e s s a r i l y t o t h e component a n g u l a r momenta. T h i s i s e q u i v a l e n t t o s a y i n g t h a t o n l y t h e c o n s e r v e d a n g u l a r momenta p o s s e s s w e l l d e f i n e d e i g e n f u n c t i o n s . I n g e n e r a l , f o r any c o n s e r v e d a n g u l a r momentum P we have t h e w e l l known r e l a t i o n s P 2 | P M > = P ( P + l ) f i 2 | P M > (2.94) r r Pz | P M p > = M p n | P M p > (2.95) P i s the quantum number o f t h e a n g u l a r momentum^, and can t a k e i n t e g r a l o r h a l f - i n t e g r a l n o n - n e g a t i v e v a l u e s . M p i s t h e quantum number r e f e r r i n g t o t h e p r o j e c t i o n o f t h e o p e r a t o r JP a l o n g the z-axis (as y e t u n d e f i n e d ) and t a k e s t h e (2P+1) v a l u e s P, P - l , -P. The symbol "n i s P l a n c k ' s c o n s t a n t d i v i d e d by 2-rr; however i n what f o l l o w s i t w i l l be assumed t h a t the a n g u l a r momenta a r e d i m e n s i o n l e s s , and the in u n i t s w i l l be m a i n l y d r o p p e d . I t i s p o s s i b l e t o d e f i n e two new o p e r a t o r s P + and P_ f o r which P +|PM p> = ( P x + i P Y ) | P M p > - exp C i * ) [P(.P+l)-^p(Mp+ 1 ) : i 5 5 l p M p + 1 > ( 2 - 9 6 > p.|pM p> - ( p x - i p Y ) | p y = exp ( i * ) l P ( P + l ) - M p C M p - l ) ] 3 5 |PM p-l> (2.97) -39- where P + and P_ a r e c a l l e d r a i s i n g and l o w e r i n g o r s h i f t o p e r a t o r s , as th e y r a i s e and l o w e r by u n i t y t he p r o j e c t i o n quantum number Mp. In eqs. (.2.9.6) and (.2.97) $ i s an a r b i t r a r y phase a n g l e : t h e commutation r e l a - t i o n s h i p s o f eq. (_2.88) do not f t x the m a g n i t u d e o f Condon and S h o r t l e y (24) t a k e $=0 which f i x e s t he r e l a t i v e phases o f t h e C.2P+1) s t a t e s | P Mp> o f d i f f e r e n t ^ . T h i s phase c o n v e n t i o n i s used t h r o u g h o u t t h i s t h e s i s . I t i s w e l l known t h a t t he o p e r a t o r P 2 i s t h e " g e n e r a t o r o f i n f i n i t e - s i m a l r o t a t i o n s a b o u t the z - a x i s " ( 3 2 ) , because by s u c c e s s i v e i n f i n i t e s i m a l r o t a t i o n s about t h i s a x i s i t i s p o s s i b l e t o ge n e r a t e an o p e r a t o r f o r r o t a t i o n t h r o u g h a f i n i t e a n g l e a about t h i s z - a x i s . T h i s o p e r a t o r w i l l be c a l l e d D ( a ) , and d e f i n e d as D(a) = 1 - i a P - a_ P 2 4 n 2 = exp (-ia P2.) ( 2 > 9 8 ) In g e n e r a l , f o r a r o t a t i o n o f a p h y s i c a l system i n which t h e c o o r d i n a t e s o f p o i n t s a f t e r r o t a t i o n a r e r e l a t e d to the o r i g i n a l c o o r d i n a t e s by the E u l e r a n g l e s a,B,y ( 1 1 ) . D(CI,S,Y) = exp ( - i a P z / ' h ) e x p ( - i 6 P y / ' n ) e x p ( - i Y P z / ' n ) (2.99) The m a t r i x elements o f t h i s r o t a t i o n o p e r a t o r a r e d e f i n e d as <P M | D(a,B,Y) I P '>= 4 " P i ' k » B ' T } ( 2 ' 1 0 ° ) P P P U n f o r t u n a t e l y the phase c o n v e n t i o n s u s e d by v a r i o u s a u t h o r s (Rose ( 3 3 ) , -40- B r f n k and S a t c h l e r (.12), and Edmonds O U ) d i f f e r ; t h e d e f i n i t i o n s f o r DCa.B.-y) g i v e n by Rose w i l l be f o l l o w e d i n t h i s t h e s i s . I t has been s t a t e d i n s e c t i o n B ( i i i ) t h a t t h e r o t a t i o n m a t r i x (P) elements # M M ' Ca , 6,y) a r e the e i g e n f u n c t i o n s o f a symmetric t o p , and P P i t can a l s o be shown t h a t t h e y a r e e i g e n f u n c t i o n s o f t h e a n g u l a r momentum o p e r a t o r s (11).; t h e r e f o r e t h e a n g u l a r momentum ei g e n f u n c t i o n s |P ,M > i n g e n e r a l can be d e f i n e d i n terms o f r o t a t i o n m a t r i x e l e m e n t s . ( i i ) I r r e d u c i b l e S p h e r i c a l T e n s o r s An i r r e d u c i b l e s p h e r i c a l t e n s o r o p e r a t o r o f rank k w i l l be d e f i n e d as c o n s i s t i n g o f (2k+l)functions T (£) (q = -k, -k+1, ...,-k) w h i c h t r a n s - form under t h e 2k+l - d i m e n s i o n a l r e p r e s e n t a t i o n o f t h e r o t a t i o n group a c c o r d i n g t o D(a3y) T^(P) D"1 U Y ) = I (P) D[}J (aBy) (2.1C1) q ~ q q q T h i s means t h a t under a c o o r d i n a t e r o t a t i o n D(aBy) t h e o p e r a t o r T (£) i s t r a n s f o r m e d i n t o l i n e a r c o m b i n a t i o n s o f t h e 2k+l o p e r a t o r s T q i p ) » where th e e x p a n s i o n c o e f f i c i e n t s a r e the elem e n t s o f t h e Wigner r o t a t i o n m a t r i c e s , D^(a&y). T h i s d e f i n i t i o n can be shown (32) t o be e q u i v a l e n t q q ' to Racah's o r i g i n a l e x p r e s s i o n (34) . T k ( £ ) ] = T k , (P) [ ( k + q H k i q + O J 3 2 (2-102) and [ J - , i q v ^ ; j - i q ± 1 U z. T$(£>] - q TjCP) ( 2- 1 0 3 ) -41- which a r e the a l t e r n a t i v e d e f i n i t i o n s o f i r r e d u c i b l e t e n s o r o p e r a t o r s . A s i m p l e i l l u s t r a t i o n o f Racah.'s e x p r e s s i o n s i s p r o v i d e d by the s e t o f a n g u l a r momentum o p e r a t o r s ; f o r i n s t a n c e , i f k=l th e n t h e sphe- r i c a l components T^i.P). a r e r e l a t e d to the c a r t e s i a n components o f a v e c t o r P ( a v e c t o r i s a f i r s t rank t e n s o r ) a c c o r d i n g t o T l , { P ) = + — (P-y ± i P Y) (-2.104) ±1 ~ JI Y T\P) = P 7 (2.105) A n g u l a r momentum o p e r a t o r s a r e v e c t o r s , and can t h e r e f o r e be w r i t t e n i n s p h e r i c a l t e n s o r form. T e n s o r o p e r a t o r s o f rank 2 o r more can a l s o a r i s e i n t h e H a m i l t o n i a n o p e r a t o r s , f o r i n s t a n c e , t h e d i p o l a r s p i n - s p i n i n t e r a c t i o n o p e r a t o r and t h e e l e c t r i c q u a d r u p o l e o p e r a t o r both i n v o l v e s e c o n d rank t e n s o r s . We choose to w r i t e a l l t h e terms i n t h e H a m i l t o n i a n o p e r a t o r i n s p h e r i c a l t e n s o r n o t a t i o n . We a r e then a b l e to use the e x t r e m e l y use- f u l , and p o w e r f u l , s p h e r i c a l t e n s o r t e c h n i q u e i n the c a l c u l a t i o n o f m a t r i x e l e m e n t s . P r o d u c t s o f s p h e r i c a l t e n s o r o p e r a t i o n s can be t r e a t e d w i t h o u t much d i f f i c u l t y , which i s o f p a r t i c u l a r v a l u e when p r o d u c t s o f m a t r i x elements a r e to be w r i t t e n i n an e q u i v a l e n t o p e r a t o r f o r m . Some p a r t i c u l a r e q u a t i o n s t h a t w i l l be o f i m p o r t a n c e a r e as f o l l o w s ; t h e s e and o t h e r s t a n d a r d e x p r e s s i o n s can be found i n s t a n d a r d t e x t s on angu- l a r momentum ( 1 1 , 12, 32, 33, 3 5 ) . (a) T e n s o r p r o d u c t o f two t e n s o r o p e r a t o r s with |krk2|<k<|k1+k2| ( 2 - 1 0 6 ) -42- (b) S c a l a r p r o d u c t o f two t e n s o r o p e r a t o r s o f t h e same rank T k ( A ) . T k ( B ) = Z ( - l ) q T k ( A ) T k ( B ) (2.107) q 4 ~ ' - q x ~ / q=-k M ( c ) W i g n e r - E c k a r t Theorem <P M | T k ( P ) | P'M;>= ( - 1 ) P " M P / P k P-\<P||T R(P)||P'> (2.108) / P The symbol <P||T ( P ) | | P ' > i s c a l l e d a r e d u c e d m a t r i x element b e c a u s e i t c o n t a i n s no r e f e r e n c e t o a c o o r d i n a t e s y s t e m . I t i s the W i g n e r - E c k a r t theorem t h a t e n a b l e s terms which depend on t h e o r i e n t a t i o n o f the c o o r - d i n a t e system, m a i n l y terms i n v o l v i n g Mp, to be f a c t o r e d o u t . In t h i s t h e s i s Edmonds' d e f i n i t i o n (11) o f r e d u c e d m a t r i x elements i s used, which i m p l i e s e x p l i c i t l y <P| |T(P) | |P> = [P(P+1)(2P+1)] ! 2 6 p p ^ (2.109) (d) R e l a t i o n between t e n s o r o p e r a t o r s i n d i f f e r e n t c o o r d i n a t e systems - t h e s e systems can be t r a n s f o r m e d i n t o each o t h e r by means o f r o t a t i o n s t h r o u g h t he a p p r o p r i a t e E u l e r a n g l e s . I f p and q a r e the components o f t h e t e n s o r i n the two d i f f e r e n t c o o r d i n a t e s y s t e m s , where D^*{a8y) i s t h e complex c o n j u g a t e o f the (p,q) element o f the k t n rank r o t a t i o n matrix"/3^ k^(aBy) • The phase c o n v e n t i o n i m p l i c i t i n eq. (2.110) i s t h a t a d o p t e d by B r i n k and S a t c h l e r , and Rose ( o p p o s i t e t o Edmonds). The symbol / J - , J 2 J 3 \ i n e q u a t i o n s (2.106) and (2.108) i s the I m-j m 2 .m-j -43- Wigner 3-j symbol, which i s a c o e f f i c i e n t r e l a t i n g t h e e i g e n v e c t o r s c o r r e s p o n d i n g to t h e a n g u l a r momenta J-j and j J 2 t o those c o r r e s p o n d i ng to t h e a n g u l a r momentum t h a t r e s u l t s from c o u p l i n g w i t h ^ Wigner 3 - j symbols a r e s i m p l y r e l a t e d t o the C l e b s c h - G o r d a n c o e f f i c i e n t s t h a t a r i s e i n t h e c o u p l i n g o f two a n g u l a r momenta. We s h a l l a l s o have o c c a s i o n to use Wigner's 6-j and 9-j s y m b o l s , which a r e needed to d e s c r i b e the c o u p l i n g o f t h r e e and f o u r a n g u l a r momenta, r e s p e c t i v e l y . Wigner symbols a r e used h e r e because t h e y have g r e a t e r symmetry than t h e c o r r e s p o n d i n g C l e b s c h - G o r d a n and Racah c o e f f i c i e n t s ; i n a d d i t i o n , t h e y a r e e a s i e r to m a n i p u l a t e . The symmetry p r o p e r t i e s o f t h e s e symbols a r e g i v e n i n s t a n d a r d t e x t s ( 1 1 , 1 2 ) . The r e l a t i o n s o f p a r t i c u l a r r e l e v a n c e i n s u b s e q u e n t c a l c u l a t i o n s a r e t h o s e by which 3-j (and 6-j) symbols can be r e e x p r e s s e d i n terms o f 6 - j ( a n d 9-j) symbols; a p r o d u c t o f two 3 - j symbols can be c o n t r a c t e d i n t h e f o l l o w i n g manner = E £ | J- j m ^ l J 2 m 2 > ( - 1 ) J 1 " J 2 m 3 (2.111) E (2 J.+1) J 3 (2.112) -44- where P = £-| + £ 2 + £3 + y-j + y 2 + y 3 ' a n d t n e l a s t c o l l e c t' i o n o f symbols i n b r a c e s i s a 6-j symbol. S i m i l a r l y , by the use o f t h e B i e d e n h a r n - E l l i o t r e l a t i o n s h i p ( 1 1 ) , a p r o d u c t o f two 6-j symbols can be r e w r i t t e n 1 J 2 J 1 2 ) j J 2 3 J l J 1 2 3 | = j 3 J 1 2 3 J23S <J4 0 J 1 4 ' E ( - l ) b ( 2 J 1 2 4 + 1 ) 124 l J 3 J 2 J 2 3 ( J 1 4 J J 1 2 4 i J 2 J l J12 J 4 J124 J 1 4 J 3 J 1 2 J 1 2 3 J 4 J J124) (2.113) where S = J-, + J 2 + J3 + J 4 + J 1 2 + J 2 3 + J 1 4 + 23 + J124 + J A p r o d u c t o f f o u r o r f i v e 3 - j symbols summed o v e r a p p r o p r i a t e i n d i c e s can be c o n t r a c t e d as f o l l o w s : E /B E H d e f g h i \ b e h l\ /C F I \ / A D G\/D E F\ /G H I ./\c f ij\a d g/Vd e f) \g h i / A B C \ \a b c j A B C D E F G H I S / C F I W A D G\ /D E F\ /G H l \ d f 9 1 \c f ijVa d gHd e fJVg h 1/ = E (2B+1) B /A B C \ /B E H\ \a b c / \ b e f) IA B C D E F G H I (2.114) (2.115) The symbols c o n t a i n i n g a 3 x 3 a r r a y o f l e t t e r s a r e Wigner 9-j symbols. In t h i s s e c t i o n , o n l y the p r o p e r t i e s o f i r r e d u c i b l e s p h e r i c a l tensors have been d i s c u s s e d . The t r a n s f o r m a t i o n from c a r t e s i a n t e n s o r s T ^ j -45- t o i r r e d u c i b l e s p h e r i c a l t e n s o r s i s c o n s i d e r e d i n Appendix I . In t h e n e x t s e c t i o n , we w i l l c o n s i d e r d i f f e r e n t Hund's c o u p l i n g cases and t h e d i f f e r e n t s e t s o f w e l l - d e f i n e d quantum numbers t h a t a r e i m p l i e d . A f t e r t h a t we can c a l c u l a t e t h e m a t r i x elements we r e q u i r e by a p p l y i n g i r r e d u c i b l e s p h e r i c a l t e n s o r t e c h n i q u e s . ( i i i ) Hund's C o u p l i n g Cases P r o v i d e d we use b a s i s f u n c t i o n s t h a t a r e c o m b i n a t i o n s o f e i g e n - f u n c t i o n s o f t h e a p p r o p r i a t e a n g u l a r momentum o p e r a t o r s , we have e s s e n - t i a l l y c o mplete freedom o f c h o i c e i n how to s e t up t h e s e c o m b i n a t i o n s when we c a l c u l a t e m a t r i x elements o f the H a m i l t o n i a n . Sometimes i t may be advantageous to use a b a s i s where t h e c a l c u l a t i o n o f t h e m a t r i x e l e - ments i s easy b u t the m a t r i x i t s e l f i s f a r from d i a g o n a l , s i n c e d i g i t a l computers make t h e d i a g o n a l i z a t i o n s r o u t i n e . A t o t h e r times we may w i s h t o use a b a s i s t h a t g i v e s t h e most n e a r l y d i a g o n a l r e p r e s e n t a t i o n , s i n c e t h e d i a g o n a l elements w i l l a l r e a d y be a good a p p r o x i m a t i o n t o the o b s e r v e d energy l e v e l p a t t e r n . The b a s i s g i v i n g t h e most n e a r l y d i a g o n a l r e p r e - s e n t a t i o n depends on t h e r e l a t i v e magnitudes o f t h e c o u p l i n g s o f t h e v a r i o u s a n g u l a r momenta. In a l l known c a s e s , c o u p l i n g between t h e n u c l e a r s p i n and o t h e r a n g u l a r momenta by h y p e r f i n e i n t e r a c t i o n s i s much s m a l l e r t h a n o t h e r c o u p l i n g s , so i t i s r e a s o n a b l e t o b e g i n the c o n s i d e r a t i o n s by e x c l u d i n g n u c l e a r s p i n . Hund i n v e s t i g a t e d t h e v a r i o u s c o u p l i n g schemes f o r e l e c - t r o n i c m o t i o n and showed t h a t t h e r e a r e f i v e p o s s i b i l i t i e s , known as Hund's c o u p l i n g s c a s e s (a) to (e) ( 3 6 ) . L i g h t d i a t o m i c and symmetric t r i a t o m i c m o l e c u l e s ( f r e e r a d i c a l s ) , s u c h as d e s c r i b e d i n t h i s t h e s i s , -46- n o r m a l l y b e l o n g t o e i t h e r c a s e ( a ) o r c a s e ( b ) , a l t h o u g h c a s e ( c ) i s o c c a s i o n a l l y met. F i g u r e 2.2 i l l u s t r a t e s c o u p l i n g schemes (a) and ( b ) . Case (a) a p p l i e s when the s p i n - o r b i t i n t e r a c t i o n i s q u i t e l a r g e r e l a t i v e t o the r o t a t i o n a l e n e r g y . The o r b i t a l a n g u l a r momentum JL (which r e s u l t s from the c i r c u l a t i o n o f t h e e l e c t r o n s around t h e i n t e r n u c l e a r a x i s ) p r e c e s s e s about t h i s a x i s , and I . the p r o j e c t i o n o f L onto the i n t e r n u c l e a r z - a x i s , remains a c o n s t a n t o f the mo t i o n b ut L i t s e l f i s n o t c o n s e r v e d . S i s c o u p l e d s t r o n g l y t o L by t h e s p i n - o r b i t i n t e r a c t i o n . The quantum num- bers E and A r e f e r to the p r o j e c t i o n s o f £ and JL a l o n g t h e i n t e r n u c l e a r - l a r a x i s , r e s p e c t i v e l y , j. and £ a r e c o u p l e d to t h e n u c l e a r r o t a t i o n a l a n g u l a r momentum, IR, to produce the t o t a l a n g u l a r momentum e x c l u d i n g n u c l e a r s p i n , R + L + S (2.116) The quantum number f o r t h e p r o j e c t i o n o f J a l o n g the m o l e c u l a r a x i s i s c a l l e d ft, and i s g i v e n by A + E = ft (2.117) The b a s i s s e t i n t h i s c a s e i s c o m p l e t e l y d e f i n e d i n terms o f t h e above-mentioned quantum numbers; | n A ; S E ; J ft > (2.118) The symbol n r e f e r s t o the o t h e r e l e c t r o n c o o r d i n a t e s needed t o d e s c r i b e the e l e c t r o n i c s t a t e f u l l y . -47- ( i i ) case (b) N + S = J ~ ~ n A N (K) S J > K „ A F i g . 2.2 Hund's c o u p l i n g c ases (a) and (b) -48- Th e c a s e (b) scheme i s i m p o r t a n t when s p i n - o r b i t c o u p l i n g i s s m a l l , so t h a t t h e r e i s no r e a s o n f o r JS t o be c o u p l e d t o t h e i n t e r n u c l e a r a x i s . JL has t h e same s i g n i f i c a n c e as f o r case ( a ) ; however i n c a s e (b) no d i s t i n c t i o n i s made between t h e a n g u l a r momenta f o r t h e motions o f the n u c l e i and the e l e c t r o n s i n o b t a i n i n g t h e t o t a l r o t a t i o n a l a n g u l a r momen- tum o f the m o l e c u l e , JN. F o r m a l l y one can w r i t e R + L = N (2.119) T h i s i s then c o u p l e d w i t h £ to form the t o t a l a n g u l a r momentum e x c l u d i n g n u c l e a r s p i n J : N + S = J (2.120) <-ŝ  />J I-*/ The quantum numbers ft and z a r e u n d e f i n e d i n t h i s scheme; the o n l y 'good' p r o j e c t i o n quantum number i s Kj c o r r e s p o n d i n g to the component o f a l o n g t h e z a x i s . K i s c a l l e d A f o r a l i n e a r m o l e c u l e , where R i s p e r p e n d i c u - l a r t o t h e m o l e c u l a r a x i s so t h a t t h e p r o j e c t i o n o f N, a l o n g t h e a x i s i s due t o t h e e l e c t r o n o r b i t a l m otion o n l y . The b a s i s s e t f o r c a s e (b) t a k e s the form | n A ; N K S J > (2.121) The t r a n s f o r m a t i o n o f b a s i s f u n c t i o n s from c a s e (b) to c a s e (a) i s g i v e n by Brown and Howard ( 1 3 ) : | n A ; N K S J > = E (-1) N ~ S + ^ (2N + 1 )Vj S N \ | n A ; S Z ; J f t > E,ft \ft - I -K/ (2.122) -49- z z A M— Z —• I n A S z J f i l F > F i g . 2.3a M o l e c u l a r c o u p l i n g schemes i n c l u d i n g n u c l e a r s p i n Case (a ) and Case ( a a ) -50- Case ( b g S ) K .r A I + S = G & G + N = F A (I S) G N F > N + I = F n & F-, + S = F | n A (N I) F 1 S F > F i g . 2.3b M o l e c u l a r c o u p l i n g schemes Case ( b g S -51- I z | n A (N S) J I F > F i g . 2.3c M o l e c u l a r c o u p l i n g scheme Case ( b 5 j ) -52- We now c o n s i d e r t h e c o u p l i n g s i n v o l v i n g n u c l e a r s p i n a n g u l a r momen- tum. I t w i l l be assumed t h a t o n l y one s p i n n i n g n u c l e u s i s p r e s e n t . The n u c l e a r s p i n may be c o u p l e d w i t h v a r y i n g s t r e n g t h to the s e v e r a l molecu- l a r a n g u l a r momenta, p r o v i d i n g a d d i t i o n a l c o u p l i n g p o s s i b i l i t i e s . The commonly e x p e c t e d c o u p l i n g schemes a r e shown i n F i g . 2.3 . They a r e c l a s s i f i e d a c c o r d i n g t o Hund's scheme, w i t h the s u b s c r i p t a i n d i c a t i n g t h a t the n u c l e a r s p i n i s most s t r o n g l y c o u p l e d t o the m o l e c u l a r a x i s ( a s i s £ i n Hund's c a s e ( a ) ) and a s u b s c r i p t 3 i n d i c a t i n g t h a t t h e n u c l e a r s p i n i s n o t c o u p l e d to t h e m o l e c u l a r a x i s b u t t o some o t h e r a n g u l a r mo- mentum (as i n case ( b ) ) . F o r Hund's case ( a ) , one may e x p e c t t h e n u c l e a r s p i n t o be c o u p l e d e i t h e r t o t h e m o l e c u l a r a x i s ( c a s e ( a ^ ) ) o r t o J ( c a s e ( a ^ ) ) . F i g . 2.3a. However, f o r Hund's case ( b ) , where the e l e c t r o n s p i n i s not c o u p l e d t o t h e m o l e c u l a r a x i s , i t i s v e r y u n l i k e l y t h a t t he n u c l e a r s p i n w i l l be c o u p l e d t o t h e m o l e c u l a r a x i s s i n c e t he i n t e r a c t i o n o f i t s s m a l l n u c l e a r m a g n e t i c moment w i t h t h e m o l e c u l a r f i e l d s s h o u l d be c o n s i d e r a b l y l e s s t h a n t h a t between the e l e c t r o n moment and t h e m o l e c u l a r f i e l d s . Hence o n l y t h e v a r i o u s ( b D ) cases a r e e x p e c t e d t o o c c u r . The v e c t o r p models and b a s i s s e t s f o r d i f f e r e n t c o u p l i n g schemes are w r i t t e n e x p l i c i t - l y i n F i g . 2 . 3 b & c . C a s e ( b p N ) where the n u c l e a r s p i n i s c o u p l e d to t h e m o l e c u l a r r o t a t i o n more s t r o n g l y t h a n t h e e l e c t r o n s p i n i s , i s a l s o not e x p e c t e d t o o c c u r . Case ( b g S ) i s found when t h e Fermi c o n t a c t i n t e r a c t i o n a I.S i s l a r g e r than any o f t h e e l e c t r o n s p i n i n t e r a c t i o n s , b ut case c ~ ~ ( b f i 1 ) i s t h e most common s i t u a t i o n . -53- i v ) M a t r i x Elements i n case ( b ^ j ) a n d ( a £ ) c o u p l i n g Quantum m e c h a n i c a l c a l c u l a t i o n s f o r systems h a v i n g symmetry may u s u a l l y be d i v i d e d f a i r l y c o m p l e t e l y i n t o two p a r t s . One p a r t c o n s i s t s o f d e r i v i n g as much i n f o r m a t i o n as p o s s i b l e from t h e symmetry a l o n e . The o t h e r i s t h e e v a l u a t i o n o f c e r t a i n i n t e g r a l s , t h e e s t i m a t i o n o f p a r a m e t e r s , o r t h e s o l u t i o n o f e q u a t i o n s which have no symmetry o r f o r which symmetry c o n s i d e r a t i o n s can p r o v i d e no i n f o r m a t i o n . The i r r e d u c i - b l e t e n s o r methods d e s c r i b e d above a r e d e s i g n e d to s e p a r a t e t h e s e two p a r t s and then to p r o v i d e a w e l l d e v e l o p e d and c o n s i s t e n t way o f c a l c u - l a t i n g m a t r i x elements i n v o l v i n g t h e a n g u l a r momentum o p e r a t o r s . S i n c e the f r e e r a d i c a l s d i s c u s s e d i n t h i s t h e s i s a r e d i a t o m i c and t r i a t o m i c , m a t r i x elements f o r both Hund's cases (a^) and ( b ^ j ) a r e g i v e n . Case (a) e x p r e s s i o n s a r e p a r t i c u l a r l y u s e f u l i n d e a l i n g w i t h d i a t o m i c o r l i n e a r f r e e r a d i c a l s . In t h e d e r i v a t i o n , we f o l l o w Brown and Howard's p r o c e d u r e s ( 1 3 ) . They do n o t use Van V l e c k ' s r e v e r s e d a n g u l a r momentum method (18) but i n s t e a d e v a l u a t e m a t r i x elements d i r e c t l y i n s p a c e - f i x e d (p) components, r a t h e r than i n a m o l e c u l e - f i x e d c o o r d i n a t e s y s t e m ( q ) . O p e r a t o r s t h a t a r e n a t u r a l l y d e f i n e d i n t h e m o l e c u l e - f i x e d a x i s s y s t e m , s u c h as the e l e c t r o n o r b i t a l a n g u l a r momentum I, are r e f e r r e d back from the s p a c e - f i x e d a x i s s y s t e m by the use o f the r o t a t i o n m a t r i x e q. ( 2 . 1 1 0 ) . In t h i s way t h e anomalous commutation r e l a t i o n s a r e c o m p l e t e l y a v o i d e d and s p h e r i c a l t e n s o r methods can be a p p l i e d i n t h e i r s t a n d a r d form. M a t r i x elements i n the case ( b D 1 ) c o u p l i n g scheme w i l l be p r e s e n t e d f i r s t . The d e t a i l s o f t h e d e r i v a t i o n s a r e m a i n l y o m i t t e d , but as an -54- i l l u s t r a t i o n t h e m a t r i x elements o f t h e d i p o l a r h y p e r f i n e i n t e r a c t i o n a r e d e r i v e d i n Appendix I I . The r o t a t i o n a l H a m i l t o n i a n f o r a near- p r o l a t e asymmetric t o p can be w r i t t e n i n the f a m i l i a r c a r t e s i a n form o f e q. (2.16 ) , where the I r a s s o c i a t i o n (10) o f the i n e r t i a l a n g u l a r momentum axes has been made: H r o t = A N z 2 + B N x 2 + C N y 2 { 2 J 2 3 ) The e q u i v a l e n t i r r e d u c i b l e t e n s o r form o f H r Q t i s 2 k k H r o t = E T ( B ) • T ( M ) ( 2- 1 2 4) k=o where t h e T (B) a r e i r r e d u c i b l e t e n s o r s o f rank k. There a r e o n l y t h r e e o 2 2 2 d i s t i n c t non-zero components, namely T Q ( B ) , T Q (B) and T 2 ( B ) = T _ 2 ( B ) (Appendix I ) . T (N,N) i s an i r r e d u c i b l e t e n s o r o p e r a t o r o f k t h rank o b t a i n e d by c o u p l i n g T"1(N) w i t h i t s e l f , a c c o r d i n g to eq. ( 2 . 1 0 6 ) . The m a t r i x elements o f the r o t a t i o n a l H a m i l t o n i a n H r Q t i n eq. (2.124) a r e g i v e n by < n N K' S J I F | H r o t | n N K S J I F > 2 N k / N k N\ . = I I (-DN'K C(B) ( ^ n J< N||T K(N,N)||N> (2.125) k=o q q \ - K q K/ where t he r e d u c e d m a t r i x elements a r e < N| |T k(N,N)||N > = ( - l ) k I ( 2 k + 1 ) ] 5 5 [N(N+1)(2N+1)] (N N 1) Ik 1 N (2.126) -55- Th e q u a r t i c c e n t r i f u g a l d i s t o r t i o n terms f o r an asymmetric top can be d e s c r i b e d by an o p e r a t o r o f t h e form (37) H c d = - V ^ S K N 2 N 2 - A K N 4 V 2 ( N x 2 " N y 2 ) - 6 K { ( N x 2 - N y 2 ) N z 2 + N z 2 ( N x 2 - N y 2 ) } (2.127) T h i s e x p r e s s i o n i s v a l i d p r o v i d e d t h a t t h e e f f e c t i v e H a m i l t o n i a n i s s o u n d l y based ( i . e . t h e r e a r e no s t r o n g i n t e r a c t i o n s between v i b r a t i o n a l s t a t e s ) and t h a t t h e m o l e c u l e i s n o t an a c c i d e n t a l symmetric t o p . When c a s t i n i r r e d u c i b l e form t he g e n e r a l e x p r e s s i o n f o r i s Hcd = " k=o kl!k2 Tk(v2> • <2-128> k^k 2) i s an i r r e d u c i b l e t e n s o r o p e r a t o r formed by c o u p l i n g t h e v e c t o r T'(N) w i t h i t s e l f f o u r times Tp<4lk2> • ( - I » P , K R K 2 {2M)\y\ T P ! ( W ) il^ G £ -p) (2.129) The f i v e d e t e r m i n a b l e c e n t r i f u g a l d i s t o r t i o n parameters can be chosen i n s e v e r a l d i f f e r e n t ways; eq. (2.127) r e p r e s e n t j u s t one such c h o i c e . The m a t r i x elements o f t h e c e n t r i f u g a l d i s t o r t i o n H a m i l t o n i a n a r e -56- < n N K S J I F | H c d | n. N K S J I F > = -AN N 2(N+1) 2-A N KN(N+1)K 2-A KK 4 (2.130) and <n N K + 2 S J I F | H , | n N K S J I F > = { - 6 N N(N+1) - \ 6 K [ K 2 + ( K ± 2 ) 2 ] } (2.131) [N(N+1 )-K(K±l )2h [ N ( N + 1 ) - ( K ± l ) ( K ± 2 ) ] % The e l e c t r o n s p i n f i n e s t r u c t u r e H a m i l t o n i a n c o n s i s t s o f s p i n - o r b i t , s p i n - r o t a t i o n and s p i n - s p i n d i p o l a r i n t e r a c t i o n s . We w i l l f i r s t c o n s i d e r the s p i n - o r b i t i n t e r a c t i o n f o r a n o n - l i n e a r m o l e c u l e . The s p i n - o r b i t H a m i l t o n i a n t a k e s the g e n e r a l form so where A" i s a 3x3 m a t r i x c o n t a i n i n g t h e s p i n - o r b i t c o u p l i n g c o e f f i c i e n t s . S i n c e m o l e c u l e s do n o t p o s s e s s s p h e r i c a l symmetry, L i s f a r from b e i n g a good quantum number. T h i s makes i t n e c e s s a r y t o c o n s i d e r t h e p r o d u c t L + . A S ° as a v e c t o r V ( 1 9 ) . The i n t e r a c t i o n H a m i l t o n i a n i s t h e r e f o r e a s c a l a r p r o d u c t o f t h i s v e c t o r w i t h !S, o r , i n i r r e d u c i b l e t e n s o r n o t a t i o n , H so S (2.132) so H so = T ^ V ) . T ^ S ) (2.133) The m a t r i x elements (38) are < n' N' K' S J I F | H s o | n N K S J I F > = ( - 1 ) N + S + J [S(S+1)(2S+1)] 1' £ I(2N+1)(2N'+1)] 1'S ( J S IT) (_!)N'-K- I I N' 1 N \ < n' 1 T 1 (J (2.134) -57- The s p i n - r o t a t i o n H a m i l t o n i a n i n c a r t e s i a n t e n s o r n o t a t i o n (39) i s H c = 1 E' e a ( N S _ + S _ N ) (2.135) s r 2 a,B aB ~ a ~B ~B ~ a v '• where a and g run s e p a r a t e l y o v e r t h e m o l e c u l e - f i x e d c o o r d i n a t e s . In i r r e d u c i b l e t e n s o r n o t a t i o n , t h i s H a m i l t o n i a n i s 2 E k=o H = A lr l T k ( e ) . T k ( N , S ) + T k ( N , S ) . T k ( e ) ] (2.136) i n which T (N,!S) i s the t e n s o r o p e r a t o r o b t a i n e d by c o u p l i n g the two f i r s t rank t e n s o r s T^jJ) and T^S,) i n a manner d e f i n e d i n eq. ( 2 . 1 0 6 ) . T (e) i s an i r r e d u c i b l e t e n s o r (k= 0,1,2 ) and i n g e n e r a l a l l n i n e compo- nents can be n o n - z e r o ; t h e s e components a r e u s u a l l y d e f i n e d i n t h e mole- c u l e f i x e d i n e r t i a l a x i s s y s t e m . The r e l a t i o n s h i p s between t h e s e n i n e components and t^, van V l e c k ' s s p i n - r o t a t i o n parameters (18) as extended by Raynes ( 4 0 ) ) , and Curl and K i n s e y ' s parameters (41) a r e g i v e n by Bowater et_ a l _ ( 3 9 ) . The m a t r i x elements o f the s p i n - r o t a t i o n H a m i l t o n i a n a r e g i v e n by the e x p r e s s i o n < n NVs J I F | H s r | n N K S J I F > = Z ( 2 k + l ; % [ S ( S + 1 ) ( 2 S + 1 ) ] 3 5 [(2N+1)(2N^+1)] J 5 k=o ( - 1 ) J + S + N ( N S J ) 1 I ( - D k [N(N+1)(2N+1)] 3 I s N - i r 2 •1 1 k l + [ N ' ( N ' + l ) ( 2 N ' + l ) ] 3 5 j l 1 k j i K/ N' k N \ V-K' q K / I ( - 1 ) N "K/ N T k ( e ) ( 2 J 3 7 ) -58- Th e s p i n - s p i n d i p o l a r i n t e r a c t i o n g i v e n i n eq. (.2.48) can be ex- p r e s s e d i n t e n s o r n o t a t i o n e i t h e r as t h e t e n s o r p r o d u c t (.32) H s s = - v T 0 g 2 y B 2 i R.f ^ ( ^ ( T ^ s . ) , ! 1 ^ ) ^ 2 ) (2.138) o r the s c a l a r p r o d u c t H =/f0 g 2 y 2 l R, -3 T ^ s J . T 1 ^ . , C 2) (2.139) " i > j ~ J where R. • i s t h e d i s t a n c e between the u n p a i r e d e l e c t r o n s and C (e,<f>) i s c l o s e l y r e l a t e d t o the second rank s p h e r i c a l h a r m o n i c , C2(e,<J>) = (^j2 Y 2(e,d>) (2.140) e and <f> a r e t h e s p h e r i c a l p o l a r a n g l e s d e f i n i n g t h e p o s i t i o n o f one e l e c t r o n r e l a t i v e t o t h e o t h e r . The m a t r i x elements o f the s p i n - s p i n i n t e r a c t i o n a r e r 2 2 < TI N' K' S J I F I H s s | n N K S J I F > = -/6 g y£ ( - 1 ) S + J + N ) J S N | I < S M T ^ . s , ) ||S > J 2 N s t i > J E ( - 1 ) N " K [ ( 2 N + 1 ) ( 2 N ' + 1 ) ] ^ K - q KJ r ( C ) (2.141) where t h e r e d u c e d m a t r i x element z <S || T 2 ( ^ - \ | S > has t h e i > j J v a l u e l i s t e d i n r e f . (42) and Appendix H I f o r d i f f e r e n t s p i n m u l t i p l i c i - t i e s . The components -59- (2.142) a r e the p arameters d e s c r i b i n g the d i p o l a r i n t e r a c t i o n . The n u c l e a r s p i n terms c o n s i s t o f two kind s o f i n t e r a c t i o n s , namely the m a g n e t i c h y p e r f i n e and t h e e l e c t r i c q u a d r u p o l e . We s h a l l c o n f i n e o u r s e l v e s w i t h t h e c a s e i n which o n l y one o f th e n u c l e i has non-zero s p i n . Thus the m a g n e t i c h y p e r f i n e i n t e r a c t i o n s i n eq. (2.54) can be w r i t t e n as (39) where t h e t h i r d term i s t h e n u c l e a r s p i n - e l e c t r o n s p i n d i p o l a r i n t e r a c t i o n and R i s d i s t a n c e between t h e u n p a i r e d e l e c t r o n and the n u c l e u s w i t h s p i n I . The m a t r i x elements o f H h f s a r e H mag.hfs = a T ^ D . T ^ L ) + a r T^.D.T^S) (2.143) < n N' K"S I F | H mag.hfs n N K S J I F > = (-1) J+I+F F I J] I ia+l)(2I+l)(2J+l)(.2j'+l)J^ 1 J l i i {2n+m.zn'+mh (-D N +S+J+1 N J S J N 1 -60- + y N V K is(s+i).(.2s+i)] % c -n N + s + J ' + 1 IS J Nj a { J S 1 C30) 3 sgn Bg rpa N [S(S+1 )(.2S+1 )(2N+1 }(.2N'+1 ) ] ' ,'N N 2 S S 1 j ' J 1 q N'-K7n 2 N\ \-K q Kj TJ<C> (2.144) where the q u a n t i t i e s • <n|Tq(L)|n>.a and a c a r e e x p e r i m e n t a l l y d e t e r m i n a b l e r a m e t e r s . The components T (C) a r e t h e parameters d e s c r i b i n g t h e d i p o l a r i n t e r a c t i o n . In g e n e r a l , t h e r e a r e f i v e i n d e p e n d e n t components o f T (C) but f o r a p l a n a r m o l e c u l e t h e s e a r e r e d u c e d to t h r e e ( 3 9 ) . A d e t a i l e d d e r i v a t i o n o f t h e n u c l e a r s p i n - e l e c t r o n s p i n d i p o l a r i n t e r a c t i o n m a t r i x elements i s worked out i n Appendix I I . I f t h e n u c l e u s has a s p i n g r e a t e r than we must i n c l u d e the n u c l e a r q u a d r u p o l e i n t e r a c t i o n . The o p e r a t o r f o r t h i s i n t e r a c t i o n has been w r i t t e n i n t e n s o r n o t a t i o n i n eq. ( 2 . 6 3 ) . The m a t r i x elements o f HQ a r e g i v e n by (39) . < n N' K' S J ' I F | H N | n. N K S J I F > eg 2 I 2 I V 1 ( - 1 ) J + I + F j i J ' F / I j I 2 J ( - l ) r + S + J \N' J S j [ ( 2 J + l ) ( 2 J % l ) ( 2 N + l ) ( 2 N ' + l ) ] i 2 J N 2 z ( - 1 ) N " K / H2 N\ T*(.vE) q \-K q K/1 (2.145) -61- where t h e e l e c t r i c q u a d r u p o l e moment Q i s d e f i n e d by l Q = < HI T?, CQ) | I I > 2 I Q I 2 I \ < I || T 2 CQ) || I > (2.146) / I 2 I \ \ - I 0 I ) and T 2 ( v E ) i s t h e e l e c t r i c f i e l d g r a d i e n t t e n s o r . F i n a l l y , t h e i n t e r a c t i o n o f t h e n u c l e a r s p i n w i t h t h e m a g n e t i c f i e l d o f the m o l e c u l a r r o t a t i o n g i v e s r i s e to an o p e r a t o r H I N c T T ] ( I ) . T ^ N ) (2.147) 1 ^ ~ where Cj i s the n u c l e a r s p i n - r o t a t i o n a l c o u p l i n g parameter. The m a t r i x elements a r e < n N K S j ' l F | H I N | r , N K S J I F > - ( - 1 ) J + I + F (F I j ' | ( - 1 ) N + S + J + 1 \S N J ' (1 J I ) j l J N [I ( I + 1 ) ( 2 I + 1 ) (2J+1)(2J'+1) N(N+1)(2N+1)]^ Cj (2.148) T h i s completes t h e m a t r i x element e x p r e s s i o n s f o r case ( b g J ) c o u p l i n g For n o n - l i n e a r m o l e c u l e s a l l t h r e e E u l e r a n g l e s a re needed to s p e c i f y t h e o r i e n t a t i o n o f the m o l e c u l e , but t h i s i s a problem f o r l i n e a r m o l e c u l e s b e c a u s e t he o r i e n t a t i o n round the m o l e c u l a r a x i s i s u n d e f i n e d so t h a t one o f t h e E u l e r a n g l e s i s m i s s i n g . The 'absence' o f t h e t h i r d E u l e r a n g l e l e a d s t o problems l a t e r on i n t h e t h e o r e t i c a l t r e a t m e n t when one comes t o -62- compute and use m a t r i x e l e m e n t s , s i n c e much o f the t h e o r y , and i n p a r t i - c u l a r i r r e d u c i b l e t e n s o r .methods, r e q u i r e t h e r e t o be t h r e e r o t a t i o n a l c o o r d i n a t e s . Hougen (43) and l a t e r Watson ( 4 4 ) , have p r o v i d e d a s o l u t i o n to t h i s problem by i n t r o d u c i n g t h e t h i r d E u l e r a n g l e as a r e d u n d a n t c o o r d i n a t e i n an i s o m o r p h i c H a m i l t o n i a n . C o n s i d e r a l i n e a r m o l e c u l e which i s m o d i f i e d by the a d d i t i o n o f an o f f a x i s , n e a r l y m a s s l e s s p a r t i c l e , which i s bound to t h e m o l e c u l e , but which does n o t a f f e c t the m o t i o n o f t h e n u c l e i and e l e c t r o n s o f the mole- c u l e . The m i s s i n g t h i r d E u l e r a n g l e must be r e s t o r e d i n o r d e r to s p e c i f y the o r i e n t a t i o n o f t h i s n o n - l i n e a r p s e u d o - m o l e c u l e . The v i s u a l i z a t i o n o f such a m o l e c u l a r system i m m e d i a t e l y s u g g e s t s t h a t the f o r m a l i s m d e v e l o p e d e a r l i e r w i l l be a p p l i c a b l e t o l i n e a r m o l e c u l e s . T h i s i s o m o r p h i c H a m i l - t o n i a n can be h a n d l e d i n the normal way, e x c e p t t h a t o n l y c e r t a i n o f i t s e i g e n v a l u e s and e i g e n f u n c t i o n s a r e a c c e p t a b l e , the o t h e r s o l u t i o n s b e i n g a consequence o f t h e redundancy i n t r o d u c e d i n t o the H a m i l t o n i a n . L i n e a r m o l e c u l e s o r f r e e r a d i c a l s where A and S a r e g r e a t e r than z e r o have f i r s t o r d e r s p i n - o r b i t e f f e c t s . The d i a g o n a l elements i n Hund's c o u p l i n g c a s e (a) a r e then a c l o s e r a p p r o x i m a t i o n t o t h e energy l e v e l p a t t e r n . We now d i s c u s s t h e o p e r a t o r s and m a t r i x elements f o r l i n e a r m o l e c u l e s o r f r e e r a d i c a l s i n c a s e (a^) c o u p l i n g . F i r s t o f a l l , the r o t a t i o n a l k i n e t i c energy and c e n t r i f u g a l d i s t o r - t i o n energy a r e d e s c r i b e d by t h e o p e r a t o r s (45) H r Q t = B T 2(.R) (2.149) and -63- H.cd = " D T (S) • T'CR) (2.150) where R = J - L - S cv <*>̂  rsw> T h e i r m a t r i x elements a r e r e s p e c t i v e l y . < n A S £ J ft' I F | H r Q t | n A S Z J B l F > = B|VE V, U ( ^ ) - ^ ^ ( S + 1 ) - E 2 + < L 2 + L 2 > ] -2 E (-1) q=±l J-ft +S-E \ - f t q ft / \-E q S  E [ J ( J + 1 ) ( 2 J + 1 ) S(S+1)(2S+1)] (2.151) and < n A s J a ' I F I HCD h A s U a I F > " D{Vl Vft [J(J+D - ft2 + s(s+i) - E2 ] 2 + 4 E E / J 1 J \2 / S 1 S \' q = ± 1 fi"'E"U q n'7 W q z'V rJ(J+D(2J+l) S(S+1).(2S+1) ] 1 J \ / S 1 s \ q ft/ \-E' q E/ -2 * , ( - 1 ) J " / J q=±l \-ft q U(J+1)(2J+1) S( S + 1 ) ( 2 S + 1 ) ] 3 5 [2J(J+1) - ( f t ' ) 2 - ft2 + 2S(S+1) - (E') 2 - E2] -64- + 4 E E (-1) q=±l fi"E" Q +E +Q"+E" / J 1 J \ / S 1 S \ / J 1 J \ / s ' i s \ U ( J + 1 ) ( 2 J + 1 ) S(S+1)(2S+1)J | (2.152) The s p i n f i n e s t r u c t u r e H a m i l t o n i a n c o n s i s t s o f t h r e e d i f f e r e n t con- t r i b u t i o n s . H c = A TV|_) T ^ S ) r e p r e s e n t s t h e s p i n - o r b i t SO 0 o ^ i n t e r a c t i o n s (2.153) 2 7 4 /6 A T„(S , S) t h e s p i n - s p i n i n t e r a c t i o n (2.154) H s s " 3 u ~ ~ H = Y T ^ J - S) . T ^ S ) t h e s p i n - r o t a t i o n i n t e r a c t i o n (2.1 55) where A, \ and y a r e s p i n - o r b i t , s p i n - s p i n and s p i n - r o t a t i o n p arameters r e s p e c t i v e l y . T h e i r m a t r i x elements a r e < n A S E J n I F | H S Q | r, A S E J fi I F > = A A E (2.156) < n A S E J f i I F | H s s | n A S E J n I F > = | x [3E 2 - S(S+1)] (2.157) < n A SZ'JQ'I F I H s r | n A S £ J n I F > = V J 6 E . Z 6 ^ [ , E - S ( S + 1 ) ] + ^ ( - 1 ) ^ ^ J , 1 J \ / S 1 S \ £J(J+1)(2J+1) S ( S + 1 ) ( 2 S + 1 ) J ^ ( 2 - 1 5 8 ) / J .1 J \ / s 1 S\ q ft/ q E / The m a g n e t i c h y p e r f i n e i n t e r a c t i o n terms a p p r o p r i a t e to Hund's c a s e (ag) c o u p l i n g a r e H , . = a T 1 ( I ) . T 1 (L) + a T ] ( l ) . T ] ( S ) mag.hfs q=o ~ q=o ~ c ~ ~ + c T 2 (I , S) (2.159) -65- where a and c a r e t h e h y p e r f i n e parameters d e f i n e d by p r o s c h and F o l e y (21) and a c i s t h e Fermi c o n t a c t p a r a m e t e r . The m a t r i x elements a r e g i v e n by < n A S J:' j ' n ' I F | H r a a g > h f s | n A S E J ft I F > = ( - D J + I + F J F J I j [1(1+1) ('21+1). (2 J+1 ) . ( 2 j ' + l ) ] * jl I J ) E ( - 1 ) ^ / j ' l J \ [ a A 6 ^ 6 f l. n q q ft y-ft q + a , ( - 1 ) S " Z / S 1 S \ [S(S+1)(2S+1)J 1 7   \ [S + 1 ( 3 0 ) h c ( - l ) q ( - 1 ) S _ E [S(S+1)(2S+1)] ! 2 / S 1 S\ / 1 2 1 \ y-E' q E ) \ - q 0 q / (2.160) In a l i n e a r m o l e c u l e , the charge d i s t r i b u t i o n i s symmetric around the m o l e c u l a r a x i s ; the e l e c t r i c q u a d r u p o l e i n t e r a c t i o n becomes HQ = e T 2 ( Q ) . T 2 ( v E ) (2.161) and t he m a t r i x elements a re -66- < n, A S E J ft I F | HQ | n. A S E J ft I F > = 1 eq„Q / I 2 ( - 1 ) J + I + F JF J I 0 1 / 2 I J l(2j+-i)(2j%i);] J l c-i) J " n / j ' 2 J \ where q Q = e q = e < V E Z Z > which i s t h e e x p e c t a t i o n v a l u e o f t h e zz com- ponent o f the e l e c t r i c f i e l d g r a d i e n t t e n s o r a t the n u c l e u s produced by t h e e l e c t r o n s . -67- C h a p t e r 3 High Order S p i n C o n t r i b u t i o n s to the I s o t r o p i c H y p e r f i n e H a m i l t o n i a n i n High M u l t i p l i c i t y I E l e c t r o n i c S t a t e s -68- A. I n t r o d u c t i o n The n u c l e a r h y p e r f i n e s t r u c t u r e o f an o p e n - s h e l l m o l e c u l e :is domina- t e d by i n t e r a c t i o n s between the n u c l e a r s p i n m a g n e t i c moments and the e l e c t r o n s p i n and o r b i t a l m a g n e t i c moments ( 1 ) . The two p r i n c i p a l i n t e r - a c t i o n s between an e l e c t r o n s p i n £ and a n u c l e a r s p i n I are the i s o t r o p i c , o r Fermi c o n t a c t , i n t e r a c t i o n , which has the o p e r a t o r form I-S ( 2 ) , and 2 -5 the d i p o l a r i n t e r a c t i o n , w i t h o p e r a t o r form [3(Ij-r )(£•£)-(I-S)r ] r , which c o r r e s p o n d s t o the i n t e r a c t i o n between the two p a r t i c l e s t r e a t e d as t i n y bar magnets whose s e p a r a t i o n i s g i v e n by the v e c t o r r. The Fermi c o n t a c t i n t e r a c t i o n i s p r o p o r t i o n a l t o ^ ( 0 ) , t he p r o b a b i l i t y o f f i n d i n g the e l e c t r o n a t the n u c l e u s , and i s t h e r e f o r e p a r t i c u l a r l y l a r g e when u n p a i r e d e l e c t r o n s occupy a m.o.'s d e r i v e d from s a t o m i c o r b i t a l s i n the l . c . a . o . d e s c r i p t i o n . When the u n p a i r e d e l e c t r o n s p o s s e s s o r b i t a l angu- l a r momentum t h e i r o r b i t a l m a g n e t i c moments a l s o i n t e r a c t w i t h the n u c l e a r s p i n m a g n e t i c moments, p r o d u c i n g an o p e r a t o r o f the form l^-l ( 1 , 3 ) . The much s m a l l e r h y p e r f i n e e f f e c t s t h a t a r e f a m i l i a r i n c l o s e d s h e l l m o l e c u l e s , and which do n o t depend on the p r e s e n c e o f u n p a i r e d e l e c t r o n s , a r e o f c o u r s e s t i l l p r e s e n t i n o p e n - s h e l l m o l e c u l e s ; t h e s e i n c l u d e e l e c t r i c q u a d r u p o l e i n t e r a c t i o n s f o r n u c l e i w i t h I 5-1, n u c l e a r s p i n - r o t a t i o n i n t e r a c t i o n s and c o u p l i n g s between two o r more n u c l e i . So f a r the h y p e r f i n e s t r u c t u r e i n gaseous o p e n - s h e l l m o l e c u l e s has o n l y been s t u d i e d e x t e n s i v e l y f o r d o u b l e t and t r i p l e t e l e c t r o n i c s t a t e s . S t a t e s o f q u a r t e t and h i g h e r m u l t i p l i c i t y a r e q u i t e r a r e , and h i g h r e s o l u t i o n s p e c t r a have o n l y been o b t a i n e d f o r such s t a t e s i n d i a t o m i c m o l e c u l e s ; as a r e s u l t not much i n f o r m a t i o n on t h e i r h y p e r f i n e s t r u c t u r e i s a v a i l a b l e ( 4 ) . The aim o f t h i s c h a p t e r i s t o p o i n t o u t t h a t h i g h e r -69- o r d e r m a g n e t i c h y p e r f i n e i n t e r a c t i o n s a r e r e q u i r e d f o r a f u l l d e s c r i p t i o n o f s t a t e s o f q u a r t e t and h i g h e r m u l t i p l i c i t y , and t h a t the l a r g e s t o f t h e s e i s a t h i r d - o r d e r c r o s s term between the s p i n - o r b i t i n t e r a c t i o n and the i s o t r o p i c h y p e r f i n e o p e r a t o r . A second i n d e p e n d e n t l y - d e t e r m i n a b l e i s o t r o p i c h y p e r f i n e parameter a r i s e s , whose e x i s t e n c e i s r e q u i r e d by group t h e o r y arguments. Recent s u b - D o p p l e r o p t i c a l s p e c t r a o f the C E s t a t e o f VO (4) have shown t h e need f o r t h i s s e c o n d i s o t r o p i c p a r a m e t e r , and i t s h o u l d o b v i o u s - l y be i n c l u d e d i n a c c u r a t e work on a l l z s t a t e s where S >y 3/2. T h e r e i s a c l o s e p a r a l l e l between the new h y p e r f i n e parameter and the s e c o n d s p i n - 4 r o t a t i o n i n t e r a c t i o n p a r a m e t e r , o r i g i n a l l y i n t r o d u c e d f o r E s t a t e s by Hougen ( 6 ) , and d i s c u s s e d i n more d e t a i l by Brown and M i l t o n ( 7 ) ; i t w i l l be shown t h a t t h e mechanisms f o r t h e i r appearance a r e v e r y s i m i l a r , and t h a t t h e i r q u a l i t a t i v e e f f e c t s on t h e l e v e l s t r u c t u r e a r e a n a l o g o u s . B. I s o t r o p i c h y p e r f i n e i n t e r a c t i o n i n the t h i r d - o r d e r e f f e c t i v e H a m i l - t o n i a n When unseen e l e c t r o n i c s t a t e s are c a u s i n g p e r t u r b a t i o n s t h a t a f f e c t e v e r y l e v e l o f a v i b r o n i c s t a t e whose s t r u c t u r e i s t o be a n a l y s e d , i t i s o f t e n c o n v e n i e n t to s e t up an e f f e c t i v e H a m i l t o n i a n (8) w h i c h has m a t r i x elements a c t i n g o n l y w i t h i n t h e s t a t e o f i n t e r e s t . A l l t h e parameters d e t e r m i n e d by l e a s t s q u a r e s a r e e f f e c t i v e , but t h e problem o f d e t e r m i n - i n g them i s s e p a r a t e d from the problem o f i n t e r p r e t i n g them. A conve- n i e n t p r o c e d u r e f o r s e t t i n g up the e f f e c t i v e H a m i l t o n i a n , based on dege- n e r a t e p e r t u r b a t i o n t h e o r y , has been d e s c r i b e d by M i l l e r ( 9 ) . The Hamil -70- tom'an i s • d i v i d e d i n t o a z e r o - o r d e r p a r t t h a t i s i n d e p e n d e n t o f the s p i n c o n t r i b u t i o n s , and a p e r t u r b i n g H a m i l t o n i a n , which f o r the purposes o f t h i s work can be t a k e n as (7). V = H + H + H + H r o t a t i o n s p i n - o r b i t s p i n - s p i n s p i n - r o t a t i o n + H mag n e t i c h f s (3.1) The e f f e c t i v e H a m i l t o n i a n c o n s i s t s o f t h e z e r o - o r d e r p a r t , p l u s a d d i t i o - nal terms, w h i c h , up t o t h i r d o r d e r , from eq. ( 2 . 8 7 ) , r e a d H e f f ( 1 ) = P o V P o H e f f U ) = P o V < V a> V po (3.2) H e f f = P o V <Va>V <Va)V P o - ? L P 0 V ( Q 0 / a 2 ) V P QV P Q + P QV P oV ( Q 0 / a 2 ) V P Q ] where t h e p e r t u r b i n g H a m i l t o n i a n V i s t h a t p a r t o f t h e t o t a l H a m i l t o n i a n g i v i n g m a t r i x elements o f f d i a g o n a l i n v i b r o n i c s t a t e , and P = I II k><! k l o , o o 1 k (3.3) (Q / a n ) = z z l l k x l k j o " " l In eq. (3.3) the symbol 1 r e f e r s t o any v i b r o n i c s t a t e , i n c l u d i n g t h e s t a t e o f i n t e r e s t (which i s g i v e n t h e s p e c i a l symbol 1 Q ) , and k s t a n d s f o r a l l the r o t a t i o n a l and s p i n quantum numbers f o r t h e s u b - l e v e l s making up a v i b r o n i c s t a t e . -71- S i n c e t h i s s t u d y i s c o n c e r n e d w i t h h y p e r f i n e e f f e c t s i n E e l e c t r o n i c s t a t e s we s h a l l w r i t e t h e p e r t u r b i n g H a m i l t o n i a n as v-B ( a ) 2 + » i l i - * i M > l 3 s « 2 - s 2 ) + Y t * ' y ' 4 • + ( 3 ' 4 > 1 1 where the terms c o r r e s p o n d to the way i n which eq. (3.1) i s w r i t t e n . A l l the c o e f f i c i e n t s i n eq. (3.4) a r e assumed t o be f u n c t i o n s o f i n t e r n u c l e a r d i s t a n c e , and i t i s f u r t h e r assumed t h a t i e l e c t r o n s a r e p r e s e n t , but o n l y one s p i n n i n g n u c l e u s , which has s p i n I . When the a d d i t i o n a l terms i n the e f f e c t i v e H a m i l t o n i a n , eq. ( 3 . 2 ) , a r e computed, t h e l a r g e s t o f the h i g h e r o r d e r terms a r e 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 s to the r o t a t i o n a l energy and the s p i n e n e r g i e s , and c o r r e c t i o n s t o a l l o f the main parameters r e s u l t i n g from t r a n s f o r m a - t i o n o f o f f - d i a g o n a l elements o f the s p i n - o r b i t i n t e r a c t i o n . For t h e h y p e r f i n e s t r u c t u r e the l a r g e s t c o r r e c t i o n s can be shown, by o r d e r - o f - magnitude c o n s i d e r a t i o n s , t o be c r o s s - t e r m s between t h e s p i n - o r b i t i n t e r - a c t i o n and the i s o t r o p i c h y p e r f i n e i n t e r a c t i o n , because i n g e n e r a l the s p i n - o r b i t parameters a^ f o r the v a r i o u s e l e c t r o n s a r e l a r g e r than t h e r o t a t i o n a l c o n s t a n t B, and the i s o t r o p i c h y p e r f i n e p arameters b^ a r e l a r g e r than t h e d i p o l a r parameters c^ when s e l e c t r o n s a r e c o n s i d e r e d . From here on we can omit the s p i n - s p i n and s p i n - r o t a t i o n i n t e r a c t i o n s ( t h e terms i n \ and y, r e s p e c t i v e l y ) because they a re v e r y much s m a l l e r than t h e e f f e c t s o f t h e s p i n - o r b i t i n t e r a c t i o n . -72- Q u a l i t a t i v e arguments based on p e r t u r b a t i o n t h e o r y show t h a t the f i r s t c r o s s - t e r m s t h a t g i v e a d d i t i o n a l h y p e r f i n e i n t e r a c t i o n s i n i e l e c - t r o n i c s t a t e s a r i s e i n t h i r d o r d e r . F o r i n s t a n c e , i f t h e o f f - d i a g o n a l elements o f V a r e t r e a t e d d i r e c t l y by second o r d e r p e r t u r b a t i o n t h e o r y , and the s p i n - o r b i t and i s o t r o p i c h y p e r f i n e o p e r a t o r s a r e a p p r o x i m a t e d as L.-!S and S-J. r e s p e c t i v e l y , the second o r d e r c r o s s - t e r m e s s e n t i a l l y has the form o ~ ~ 1 — —̂' o ' 1 1 0 u s i n g the n o t a t i o n o f eq. ( 3 . 3 ) . Even w i t h o u t w r i t i n g e x p l i c i t m a t r i x elements i t can be seen t h a t t h i s i s e q u i v a l e n t to t h e r e s u l t s o f h a v i n g (2) an e f f e c t i v e o p e r a t o r H g ^ f v ' o f t h e type J_-^ a c t i n g w i t h i n the s t a t e o f i n t e r e s t , 1 . An o p e r a t o r J . - ^ , o r l^-l, must have z e r o m a t r i x elements w i t h i n a I e l e c t r o n i c s t a t e , because the v a l u e o f A i s z e r o . In t h i r d o r d e r one o f t h e c r o s s - t e r m s g i v e n by p e r t u r b a t i o n t h e o r y has t h e form <1 k | S - L | l k ' x l k > | L - S | l ' k " > < l ' k " | S - I | l k ' " > / A E 2 (3) which has the same e f f e c t as i f an o p e r a t o r ' o f t h e t y p e S/I^ ( o r J^-Sj were a c t i n g w i t h i n t h e s t a t e o f i n t e r e s t , 1 . I t w i l l be shown below t h a t t h i s t h i r d o r d e r e f f e c t i v e o p e r a t o r H g f f v ' c o n t a i n s a p a r t w h i c h i s e x a c t l y e q u i v a l e n t to the f i r s t o r d e r i s o t r o p i c h y p e r f i n e i n t e r - a c t i o n , and a p a r t which has a s l i g h t l y d i f f e r e n t dependence on the s p i n quantum numbers; i n q u a l i t a t i v e terms t h e d i f f e r e n c e between t h e two p a r t s i s c o n n e c t e d to the r e l a t i o n s h i p between k and k " ' . The f i r s t p a r t o f H g f ^ 3 ^ i s i n c o r p o r a t e d i n t o the f i r s t o r d e r h y p e r f i n e i n t e r a c t i o n , -73- but the second p a r t gives, t h e new h y p e r f i n e e f f e c t w hich appears f o r s t a t e s w i t h S>3/2. The e x a c t form o f t h i s t h i r d - o r d e r i n t e r a c t i o n i s most e a s i l y d e r i v e d u s i n g s p h e r i c a l t e n s o r a l g e b r a . We choose a Hund's case ( a ^ ) b a s i s , |nASZjftIF>, because i t i s an advantage t o have as many o f the e l e c t r o n i c a n g u l a r momenta as p o s s i b l e w i t h w e l l - d e f i n e d e i g e n v a l u e s f o r t h e i r mole- c u l e - f i x e d z-components. The symbols i n t h i s b a s i s a r e well-known; n s t a n d s f o r t h e v i b r o n i c s t a t e , and t h e o t h e r s a r e a l l f a m i l i a r d i a t o m i c m o l e c u l e quantum numbers ( 1 0 ) . T r a n s l a t i n g t h e o p e r a t o r s o f eq. (3.4) i n t o s p h e r i c a l t e n s o r form, we have Hspin-orbit = f^ili)- T 1 ^ . ) (3.5) "isotropic hfs = JV U>- t1 -̂) T h e i r m a t r i x e l e m e n t s , i n Hund's c a s e ( a D ) , a r e P < n A S U n l F | H s p . n _ o r b . t | n ' A ' S ' i ' J n l F > = z(-l ) q ( - l ) s " ^ j s-Z/S 1 S q i " X Z<S| i T ^ . H I S - x n A l T 1 ( a . l . ) | n ' A ' > (3.6) i -q ^ n A S l J Q l F l H . . . , , |n'AS'Z'J'fi'IF> = ( - 1 ) I + J + F i F J I } 1 i s o t r o p i c h f s 1 ( 1 1 J ) X [ I ( I + 1 ) ( 2 I + 1 ) ( 2 J + 1 ) ( 2 J ' + 1 ) ] * Z ( - l ) J _ n / J 1 j A q ^-ft q P.J X (-DS"Z/s 1 sA Z<S| IT1 (s .) J |S'><nAS|b .|n'AS'> ^-Z q Z J i (3.7) -74- The f i n a l terms i n each o f t h e s e e x p r e s s i o n s a r e parameters t h a t i n p r i n - c i p l e can be e v a l u a t e d e x p e r i m e n t a l l y , and can be computed by ab i n i t i o methods. Both e x p r e s s i o n s obey the s e l e c t i o n r u l e AS = 0, ± 1 , as a r e s u l t o f the m i c r o s c o p i c form o f t h e e l e c t r o n s p i n o p e r a t o r s ; i n a d d i t i o n t h e i s o t r o p i c h y p e r f i n e o p e r a t o r i s d i a g o n a l i n A, but t h e s p i n - o r b i t o p e r a - t o r f o l l o w s AA = 0, ±1. When we s u b s t i t u t e the m a t r i x elements o f eq. (3.6) i n t o t h e ex- (3) p r e s s i o n f o r H g f ^ g i v e n i n eq. (3.2) we g e t n i n e t e r m s , because t h e r e (3) a r e t h r e e p a r t s to H e ^ f , and t h r e e ways o f permuting t h e o p e r a t o r s o f eq. (3.6) remembering t h a t H . ,.. must be taken t w i c e . There i s no ^ 3 s p i n - o r b i t need t o w r i t e o u t any o f t h e s e terms because t h e s u b s t i t u t i o n i s e n t i r e l y s t r a i g h t f o r w a r d . Many common f a c t o r s o c c u r i n a l l n i n e because they a r e c o n s t r u c t e d s i m i l a r l y , and c l o s e r e x a m i n a t i o n shows t h a t they can be c o l l a p s e d to f i v e d i f f e r e n t types o f term, w h i c h must be e v a l u a t e d s e p a - r a t e l y . T a b l e 3.1 summarizes t h e p r o p e r t i e s o f th e s e f i v e t y p e s . I t i s a p p a r e n t t h a t t h e quantum numbers S and z f o r t h e d i s t a n t p e r - t u r b i n g e l e c t r o n i c s t a t e s o c c u r i n t h e m a t r i x elements o f the t h i r d o r d e r e f f e c t i v e H a m i l t o n i a n , but t h e y must not appear e x p l i c i t l y i n t h e f i n a l e x p r e s s i o n s b e c a u s e t h e e f f e c t i v e H a m i l t o n i a n i s assumed to a c t o n l y w i t h i n the v i b r o n i c s t a t e o f i n t e r e s t . I t i s t h e r e f o r e n e c e s s a r y t o use r e l a t i o n s h i p s between t h e Wigner c o e f f i c i e n t s t o sum o v e r t h e s e quantum numbers as f a r as p o s s i b l e , and to c a s t them i n t o t h e form o f an e x p e r i - mental p a r a m e t e r or p a r a m e t e r s . We f o l l o w Brown and M i l t o n ( 7 ) , who en- c o u n t e r e d a s i m i l a r problem i n t h e i r d i s c u s s i o n o f h i g h e r o r d e r s p i n - r o t a t i o n i n t e r a c t i o n s i n m u l t i p l e t I s t a t e s , and s o l v e d i t by a p p l y i n g the r e l a t i o n i n eq. (2.112). T a b l e 3.1 The f i v e types o f term c o n t r i b u t i n g to the t h i r d - o r d e r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n . F i r s t Second I n i t i a l O p e r a t o r i n t e r m e d i a t e O p e r a t o r i n t e r m e d i a t e O p e r a t o r F i n a l Energy s t a t e s t a t e s t a t e s t a t e d e nominator 1 SA S . 0 . S'A' i s o S'A' S . 0 . SA S . 0 . S'A' i s o SA' S .0 SA S . 0 . SA' i so S'A' S .0 . SA 2 SA S . 0 . S'A' s .0 . S'A' i s o SA i s o S'A' S .0 . S'A' S .0 . SA 3 SA S . 0 . SA' S . 0 . S'A i s o SA i s o S'A s.o. SA' S . 0 . SA 4 SA S . 0 . S'A' S . 0 . SA i s o SA 5 SA i s o SA s.o. S'A' S . 0 . SA (E0-E1><E0-E25 ( E 0 " E 1 > - 0 - t 2 j The o p e r a t o r s are s.o. = s p i n - o r b i t and i s o = i s o t r o p i c h y p e r f i n e ( s e e eq ( 3 . 6 ) ) -76- For example, i n the terms o f t y p e 1 from T a b l e 3.1 we haye t h e p r o d u c t / s i s ' \ (- i) s ' -Ws- i s '\(-i) s'-E'/ s i s \ where the s i n g l e and do u b l e primes on S and I r e f e r to the i n t e r m e d i a t e s t a t e s , and 1 " ' r e f e r s t o the f i n a l s t a t e . A f t e r two a p p l i c a t i o n s o f eq. (2.112) t h i s p r o d u c t becomes ( - 1 ) S " _ S " C « I ( 2 k + l ) ( 2 K + l ) JS S K j / S S K\/ 1 k K\ k ' K i i k s " f \ i -E"-q7\-q q+q'- 'q/ j S " S k I / 1 1 k \ X 1 1 1 S ' t \ - q -q' q+q7 In t h i s e x p r e s s i o n k and K ar e t e n s o r i n d i c e s t h a t a r i s e i n the s u c c e s s i v e a p p l i c a t i o n s o f eq. ( 2 . 1 1 2 ) . I t w i l l be seen t h a t nowhere do the i n t e r - m e d i a t e s t a t e E v a l u e s ( E ' and 1 " ) a p p e a r . E v e n t u a l l y t h e g e n e r a l m a t r i x element o f the t h i r d o r d e r e f f e c t i v e H a m i l t o n i a n can be o b t a i n e d as < n A S E j f t I F | H . s o ( 3 ) | n A S E " ' j ^ ' s ? ' " l F > = ( - 1 ) J ' " + I + F ) F J I ( 1 I J ' " XLKI+1X2I + 1 ) ] 1 1 [ ( 2 J + 1 ) ( 2 J ' - + 1 ) J 1 5 I ( - l ) J ~ r / J l J ' " \ y ( - D S " Z / S K S q' Uq- - ' i k U x \ £ ( - l ) S ' ~ S ( - l ) q ( 2 k + l ) ( 2 K + l ) / 1 1 k \ / l k K \ { 1 k K j ( J } Vq -q' Q+q7\q -q- q ' q'/ls S S " ( -77- X I < ^ | T ] q ( a i l i ) | n ' A ' > < n ' A ' | T q ( a i ^ i ) | n A > < S ' 1 | T 1 ( s i ) | | S > 1 J S S " S ' j x ( E ^ , - E ^ . f l . ^ J • 1 { < n A ' S 1 b . | n ' ^ ^ ' S - > ( - l ) U k + K + < n ' ' A S - | b i i n A S > [ l + ( - ^ ) K + 1 *• nAS n A b ' i 1 [ S ( S + 1 ) ( 2 S + 1 ) ] J 5 y < S | ! T 1 ( s i ) | | S " > j l 1 k j ( E ° A S - E ° _ A . s _ ) 1 Js S " s( -2 x <nAS!b.|nAS>xin+(-l) 1 (3.8) The s e p a r a t e c o n t r i b u t i o n s o f the f i v e terms from T a b l e 3.1 can be d i s t i n - g u i s h e d i n t h e b r a c k e t f o r m i n g t h e second h a l f o f eq. ( 3 . 8 ) . The t r i a n g l e r u l e s from t h e 3 - j symbols l i m i t t h e v a l u e s t h a t k and K can assume, such t h a t k can be 0, 1 and 2, and K i s then r e s t r i c t e d , a c c o r d i n g t o t h e v a l u e o f k, t o k = 0 1 2 K = 1 0,1,2 1,2,3 -78- K+1 I t i s e v i d e n t t h a t t h e c o e f f i c i e n t s 1+C.-1). i n e q . (_3.8) cause most o f i t t o y a n i s h f o r even K v a l u e s , and o n l y the term w i t h c o e f f i c i e n t (-1). coming from t h e t y p e 1 terms i n T a b l e 3.1 i s l e f t . However i t can be shown, by arguments s i m i l a r to t h o s e used by Brown and M i l t o n ( 7 ) , t h a t t h i s a l s o v a n i s h e s f o r K even. The p r o c e d u r e , i n e s s e n c e , i s t o p r o v e , by t h e B i e d e n h a r n - E l l i o t r e l a t i o n s h i p ( e q . 2.113), t h a t the terms 1+K w i t h q=l and -1 i n the sum o v e r q i n e q . (3.8) d i f f e r by a f a c t o r (-1) ; t h e r e f o r e t h ey a r e equal and o p p o s i t e f o r even K. The q=0 term i s e a s i l y shown t o be z e r o f o r even K, so t h a t the whole sum v a n i s h e s . One o f the s t e p s i n the p r o o f r e q u i r e s t h e e q u a l i t y < n A | T q ( a i l i ) | n'A'> = < n ' A ' | T ^ ( a i l i ) h A > (3.9) which t h e r e f o r e l i m i t s t h e r e s u l t s t o A=0, i . e . z s t a t e s o n l y . The v a l u e o f K can be 1 or 3 o n l y , i n consequence. C o n s i d e r K=l f i r s t . The two 3-j symbols i n v o l v i n g k i n eq. (3.8) can be c o n t r a c t e d : 2 I I 1 1 k \/l k 1 \ l(\ 1 k \ n-q -q -q + q 7 W ^ q 7 = " W W ( 3 J 0 ) and, u s i n g the o r t h o g o n a l i t y p r o p e r t i e s o f 3 - j symbols [ 1 1 ] , become W 1 1 k \ ? = 4- (3.11) q \ q q- - q - q ' / The g e n e r a l m a t r i x element r e d u c e s to -79- < n A S E 0 n I F | H i s ^ 3 ) , K = 1 | n A S Z ' " J ' - n ' " I F > = ( - l ) J ' " + I + F JF J I }[I(1+1)(21+1)]' 1 I J ( * y ( - i ) J " 7 j i j - \ ( - i ) s " 7 s 1 S \l I I ( - i ) s " - s ( - D q ( 2 k + i ) X H M T ] I n ' A - x n - A ' l T j t a ^ ^ |nA> \<S | 1T1 ( s i ) | |S"> x j l k 1 i s S S " I <S I IT 1 ( ^ ) 1 IS - > <S -1 IT 1 ( ^ ) 1 IS - > j l 1 k j U ° A S - E n ' A ' S ' r l S S " S ' \ X ( E ° A S - E ° _ A . s . J ' 1 { < n A ' S ' | b i | n A ' S " > ( - l ) k + 1 + 2 < n " A S " | b i | n A S > ) (3.12) - lS(S+l)(2S+l)] l 5I<S!|T 1(s u i)]|S-> j l 1 k | ( E ° A S - E ° _ ; i . s . J _ 2 < n A S | b i J n A S > 1 IS S " s which can be s e e n , by comparison w i t h eq. ( 3 . 7 ) , to be e x a c t l y s i m i l a r i n form to the normal i s o t r o p i c h y p e r f i n e m a t r i x e l e m e n t s . For K=l the t h i r d o r d e r s p i n - o r b i t i n t e r a c t i o n t h e r e f o r e g i v e s a h i g h e r o r d e r c o n t r i b u t i o n to t h e Fermi c o n t a c t p a r a m e t e r . The f i n a l r e m a i n i n g term i n eq. (3.8) has K=3 and k=2. U s i n g eq. (2,112) a g a i n , t he p a i r o f 3-j symbols i n v o l v i n g k i n eq. (3.8) can be r e c a s t i n t o t h e form -80- '3 1 2 vq' q -q-q 1 1 2 j = ( - l ) q _ q ' •q -q' q'+q (5) f 3 1 2 ) C '2\ j 3 1 2 ( (3.13) Vq'-q' 0/\q -q 0/ h 1 2J (-1)' ( 1 2 \ ( - l ) q -q' 07 C 1 \q -q 2 q 0 (3.14) where we have s u b s t i t u t e d t he v a l u e ^3 1 2 ) 1 ( «- (3.15) 1 1 2 b I t t u r n s o ut t h a t the 3 - j symbol w i t h q ' i n i t i s i m p o r t a n t when we c a r r y o u t the t r a n s f o r m a t i o n from c a s e (a^) c o u p l i n g t o c a s e ( b g j ) > s ° "that i t must not be i n c o r p o r a t e d i n t o t h e sum o v e r d i s t a n t s t a t e s c o m p r i s i n g the e x p e r i m e n t a l parameter. The K=3 term then becomes < n A S U f i I F | H i s J ) 3 ) , K = 3 ! p 1 A S l " ' J " T r " I F > = ( - 1 ) J ' " + I + F J F J I j | l I J " - i n(i+D(2i+i)]- X [ ( 2 J + 1 ) ( 2 J ' " + 1 ) ] ^ I ( - l ) J " n / J 1 J ' " \ ( - 1 ) S - E / S 3 S q '-!) q ' fl"7 \-E q 7 E S"~S / t T 0 \ , I T l x (-DH/3 1 2^ W -q' o, 35 I U-lf ° / I 1 2\ <nA|T q(a il i) |n'A'> q -q oj \ 5 I I ( - 1 ) J "J j |_ S'A'S" q \̂ < T>' A'|T] q(a.l ( i)|nA> J < S | | T 1 ( ^ i ) | | S - > j l 2 3 ( [J<S | | T 1 ) | |S '> 1 I s S S " ! 1 -81- •1 /̂ O ro \ - l x < S - M T 1 ( l i ) i i S - > jl 1 2 j (E° A S- E:-A-S-» <Cs-En- Is S " S'j { < n A ' S'|b i | n " A S ' ->+2 < n " A S " | bi |nAS>] - [S(S+1) (2S+1) ]** 1<S | IT1 ( s . ) | |S " > (3.16) Eq. ( 3 . 6 ) r e p r e s e n t s a new type o f h y p e r f i n e i n t e r a c t i o n m a t r i x element, w h i c h , as can be seen from the p r o p e r t i e s o f t h e s e c o n d 3 - j symbol, i s non- v a n i s h i n g f o r e l e c t r o n i c s t a t e s where S ̂  3/2. The form o f eq. (3.16) i s somewhat s i m i l a r to t h e i s o t r o p i c h y p e r f i n e m a t r i x element g i v e n i n e q . ( 3 . 6 ) ; the d i f f e r e n c e s a r e t h a t t h e r e d u c e d s p i n m a t r i x e l e m e n t appears i n a n o t h e r p l a c e , and t h a t t h e s i m p l e m a t r i x element o f b^ i s r e p l a c e d by t h e c o m p l i c a t e d e x p r e s s i o n between t h e l a r g e b r a c k e t s which becomes t h e e x p e r i m e n t a l p a r a m e t e r . Our d e f i n i t i o n o f the new e x p e r i m e n t a l parameter has been chosen w i t h t h e a n a l o g y between i t and Brown and M i l t o n ' s s e c o n d s p i n - r o t a t i o n p a r a - meter -ye- (7) i n mind. Not s u r p r i s i n g l y , s i n c e t h e mechanisms f o r t h e i r appearance a r e s i m i l a r , t h e a n a l o g y between the two parameters i s v e r y c l o s e . To make t h e a n a l o g y as c l o s e as p o s s i b l e we name t h e new parameter b^, and d e f i n e i t as b s=-4( 3 / 3 5 ) 3 5 [ ( 2 S - 2 ) ( 2 S - l ) 2 S ( 2 S + l ) ( 2 S + 2 ) ( 2 S + 3 ) ( 2 S + 4 ) ] ~ ^ t (3.17) where t i s t h e c o m p l i c a t e d e x p r e s s i o n i n l a r g e b r a c k e t s i n eq. ( 3 . 1 6 ) . The rea s o n f o r t h e p e c u l i a r n u m e r i c a l f a c t o r w i l l become a p p a r e n t when we con- s i d e r the m a t r i x elements i n c a s e ( b ^ j ) c o u p l i n g i n t h e n e x t S e c t i o n . T a b l e 3.2 M a t r i x elements o f the t h i r d o r d e r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n i n a Hund's c a s e ( a g ) b a s i s < T)ASEJMF|H. ^ |nASEjftIF>= - b c f l E [ F ( F + l ) - ! ( I +1) - J( J+1) ] [ 3 S ( S + 1 ) - 5 E 2 - 1 ] i s o i 2J(J+1) <nASE JftIF|H. S J ) 3 ) |DASE , J - l ,«I F>= M ( J 2 - Q 2 ) h l ( F+I+J+1)(I+J-F) ( F+J-I) ( F+I-J+1) ] h [3S(S+1 )-5g2-T]_ 2 J ( 4 J 2 - l ) i a <nASEJnIF|H. ^ |nAS,E±l ,j,n±l ,IF>= - b P [ J ( J+1)-^(^±1 ) ]^ [ s ( S+1)-E(E±1)]^ i s o b 4J(J+1) x [F(F+l)-I(I+1)-J(J+1)][S(S+l)-5E(E±1)-2] < n A S E j f i I F | H . s ^ J ; | n A S , E ± 1 ,J-1 ,n±1 ,IF>= ± b s [( JT£>) ( JTB-1 ) ] rc( £ j i ) £ ( z ± i ) ] ' / z 4 J ( 4 J 2 - ! ) ' 5 x [( F+I+J+1) (I + J - F ) ( F+J-I) ( F+I-J+1) ]' 5[S(S+1 )-5E(E±1 )-2] The phase c h o i c e f o r the r o t a t i o n a l wave f u n c t i o n s f o l l o w s t h a t o f Brown and Howard (14) o r C a r r i n g t o n , Dyer and Levy ( l 9 ) , based on Condon and S h o r t l e y ' s c o n v e n t i o n s ( 2 0 ) . -83- On s u b s t i t u t i n g e q . (.3.17) i n t o e q . (.3.16), and w r i t i n g e x p l i c i t ex- p r e s s i o n s f o r the Wigner c o e f f i c i e n t s , t h e m a t r i x elements o f t h e t h i r d o r d e r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n i n c a s e ( a D ) c o u p l i n g can be o b t a i n e d ; P t h e y a r e l i s t e d i n T a b l e 3.2. C. T r a n s f o r m a t i o n t o case ( b ^ j ) c o u p l i n g Of t h e v a r i o u s s t a t e s known, o n l y two, t h o s e i n GeF (12) and SnH ( 1 3 ) , show marked d e p a r t u r e s from c a s e (b) c o u p l i n g . T h e r e f o r e , d e s p i t e t h e l o g i c a l p r e f e r e n c e among d i a t o m i c s p e c t r o s c o p i s t s f o r c a l c u - l a t i n g e n e r g y l e v e l s i n a Hund's c a s e (a) b a s i s , i t i s i n s t r u c t i v e t o l o o k a t the form o f the m a t r i x elements i n case (b) c o u p l i n g , because t h e d i a g o n a l elements show d i r e c t l y how the parameter a f f e c t s t h e l e v e l s t r u c - t u r e i n a r e a l s i t u a t i o n . With the c a s e (b) f u n c t i o n s g i v e n i n terms o f case (a) f u n c t i o n s (14) by |NASJ>= I ( - l ) N " S + n ( 2 N + l ) ' a S |ASEJQ> (3.18) • n Vft -1 -A we o b t a i n < NASJIF|H i s o ( 3 ),K=3|N'A'SJ'IF>= t . ( - l ) I + J ' + F JF J I j [I (1+1) ( 21+1) l* 5 x I I ( - l ) N " S + n + N ' - S + f i [ ( 2 N + l ) ( 2 N ' + l ) ] l j / J S N \ / J ' S N'\ i , n r $ r \p -z -A] y r - r A'/ -84- X q [(2j+i)(2j'+i)] , s (3-19) / J 1 J A ( - 1 ) S - Z / S 3 S i ( - D q 7 3 1 2\ (where we have r e p l a c e d t h e t r i p l e primes o f eq. (3.16) by s i n g l e p r imes) A f t e r some r e a r r a n g e m e n t (which by happy chance e l i m i n a t e s v i r t u a l l y a l l the phase f a c t o r s ) we can c o n t r a c t t h e sum o v e r t h e p r o d u c t o f f i v e 3-j symbols t o an e x p r e s s i o n i n v o l v i n g a 9 - j symbol ( 1 5 ) ; remembering t h a t A and A ' a r e r e s t r i c t e d t o the v a l u e 0 we f i n a l l y get < N S J I F i H i s | ) 3 ) , K = 3 | N ^ J ' I F > = 1 ( 3 5 / 3 ) % . ( - 1 ) I + J ' + F j F J I ! Il I J') X [ I ( I + 1 ) ( 2 I + 1 ) . ( 2 J + 1 X 2 J ' + D .(2N+l)(2N'+l).(2S-2)(2S-l)2S(2S+l)(2S+2) (2S+3)(2S+4 ) r (-ir /N 2 N'\ |N N' « ( 3 2 0 ) . 1 ) N /N 2 N'\ |N N \p o o M s s which i s the d e s i r e d r e s u l t . These 9-j symbols a r e u n f o r t u n a t e l y not l i s t e d i n s t a n d a r d t a b u l a t i o n s ( 1 6 ) , and g i v e v e r y cumbersome a l g e b r a i c e x p r e s s i o n s , but v a s t amounts o f c a n c e l l i n g o c c u r i n the e v a l u a t i o n o f a c t u a l m a t r i x e l e m e n t s , so t h a t q u i t e s i m p l e e x p r e s s i o n s a r e f i n a l l y ob- t a i n e d . Our a l g e b r a i c forms f o r t h e 9-j symbols a r e g i v e n i n r e f . ( 5 ) . -85- We do not l i s t g e n e r a l forms f o r t h e m a t r i x elements o f eq. (.3.20), but g i v e m e r e l y the d i a g o n a l elements f o r a 4 i s t a t e . I t i s u s e f u l to i n c l u d e t h e d i a g o n a l elements, o f the i s o t r o p i c h y p e r f i n e i n t e r a c t i o n b I.S^ i n t h e s e e x p r e s s i o n s : F 1 ( J = N + | ) F 2 ( J = N + ^ ) F 3 ( J = N 4 ) F 4 ( J = N - | ) C[b-b.N/(2N+3)]/(2N+3) ic[b(2N+9)+3b s{(3N+2)+3/(2N+3))]/[(2N+l)(2N+3)] jC[b(2N-7)+3b s{(3N+l)+3/(2N-l)}]/[(2N-l)(2N+l)] (3.21) • | C [ b - b . ( N + l ) / ( 2 N - l ) ] / ( 2 N - l ) where C=F(F+1)-I(I+1)-J(J+1) (3.22) I t can be seen how when N i s l a r g e , so t h a t s i m i l a r powers o f N can be c a n c e l l e d , t h e s e e x p r e s s i o n s s i m p l i f y so t h a t t h e r e i s one e f f e c t i v e b p a r a - meter f o r t h e F-| and F^ l e v e l s and a n o t h e r e f f e c t i v e b parameter f o r the F 2 and F 3 l e v e l s : F 1 and F 4 : b e f f ^ - 2 b c ; F 2 a n d F 3 : b e f f b + 2 b S (3.23) The 3 - j symbol and i t s phase f a c t o r i n e q . ( 3 . 2 0 ) , i f t h e v a l u e s o f A and A' a r e l e f t u n s p e c i f i e d , i r e a c t u a l l y ( - 1 ) N " A / N 2 N \ , w h i c h , when the {-A q K') n o r m a l i z a t i o n f a c t o r [(2N+1)(2N'+1) i s i n c l u d e d , i s t h e r e d u c e d m a t r i x element o f the second rank r o t a t i o n m a t r i x ( 1 7 ) : -86- < N A | | P ( 2 ) * ( a ) ) | | N ' 0 = ( - l ) N - A [ ( 2 N + 1 ) ( 2 N ' + l ) ] 1 5 / N 2 N'\ ( g \-A q A'j T h i s s u g g e s t s t h a t i t i s p o s s i b l e t o d e v i s e an e q u i v a l e n t o p e r a t o r , a c t i n g w i t h i n the m a n i f o l d o f a g i v e n v i b r a t i o n a l l e v e l o f a m u l t i p l e t l e l e c t r o n i c s t a t e , which has the same m a t r i x e l e m e n t s as eq . (3.20) b u t which g i v e s a d i f f e r e n t p e r s p e c t i v e on how the new h y p e r f i n e i n t e r a c t i o n o p e r a t o r i s con- s t r u c t e d . A f t e r some e x p e r i m e n t a t i o n the e q u i v a l e n t o p e r a t o r was found to be H i , i 3 ) . ^ 1 j / U ) . T , [ T 3 n , ( y . T ^ t l ^ , ) . c 2 ] / r 1 3 ( 3 . 2 5 ) 2 where i and j a r e e l e c t r o n s , r ^ i s t h e i r s e p a r a t i o n and C i s r e l a t e d to the s p h e r i c a l harmonic g i v i n g t h e i r r e l a t i v e p o l a r c o o r d i n a t e s , ^ = ( 4 ^ / 5 ) ^ ^ ( 6 ^ ) (3.26), The m a t r i x elements o f eq. (3.25) a r e i d e n t i c a l to t h o s e o f eq. (3.20) 2 e x c e p t t h a t t h e y are g i v e n i n terms o f a parameter T Q ( C ) , which must be e x p r e s s e d i n terms o f b^, a c c o r d i n g t o b s = ( 3 / 1 0 ) T 2 ( C ) / ( 1 4 ) 3 2 ( 3 - 2 7 ) The d e r i v a t i o n o f the m a t r i x elements o f eq. (3.25) i s i n t e r e s t i n g because i t i n v o l v e s a number o f w i d e l y - o c c u r r i n g e l e c t r o n s p i n r e d u c e d m a t r i x e l e m e n t s , s e v e r a l o f which a p p e a r n o t t o have been g i v e n i n g e n e r a l form (though some e x p l i c i t e x p r e s s i o n s have been g i v e n by Brown and Merer ( 1 8 ) . -87- In o r d e r t o i n t e r p r e t the parameter a c c o r d i n g t o e q . ( 3 . 1 6 ) , o r to ob- t a i n e x p r e s s i o n s f o r the A - d o u b l i n g p a r a m e t e r s , o, p and q, i n terms o f the s p i n - o r b i t m a t r i x elements f o r h i g h m u l t i p l i c i t y s t a t e s ( 1 8 ) , i t i s u s e f u l t o have t h e s e g e n e r a l forms. A c c o r d i n g l y t h e d e r i v a t i o n o f t h e m a t r i x elements o f eq. (3.25) i s g i v e n i n Appendix H I . The o p e r a t o r form o f eq. (.3.25) shows e x a c t l y how the e f f e c t i v e o p e r a - t o r f o r the new t h i r d - o r d e r c r o s s - t e r m i s c o n s t r u c t e d . In C a r t e s i a n t e n s o r n o t a t i o n i t c o n s i s t s o f a sum o f terms o f t h e type I S.S S •, the a d vantage o f t h e s p h e r i c a l t e n s o r form i s r e a d i l y a p p r e c i a t e d . E x a c t l y s i m i l a r e x p r e s s i o n s t o eq. (3.21) a r e found to h o l d f o r Brown and M i l t o n ' s Y$ parameter ( 7 ) . The t r a n s f o r m a t i o n o f t h e c a s e (a) m a t r i x g i v e n i n r e f . (7) to c a s e (b) c o u p l i n g i s r a t h e r more messy than the t r a n s - f o r m a t i o n o f eq. (3.19) to eq. (3.20) because now t h e r e i s o n l y a p a r t i a l sum o v e r the i n d e x q" (which c a n n o t t a k e the v a l u e z e r o s i n c e t h e s p i n - u n - c o u p l i n g o p e r a t o r i s - 2 B ( J x s x + J y s y ) "rather than -2B J.S so t h a t the q'=0 component i s m i s s i n g ) . A f t e r some a l g e b r a we f i n d < N S J i H s p i n - r o t a t i o n ' K = 3 l N ' S J ^ X [ ( 2 S - 2 ) ( 2 S - l ) 2 S ( 2 S + l ) ( 2 S + 2 ) ( 2 S + 3 ) ( 2 S + 4 ) ] l s X I (22+1) / 3 z 1 \ ( - 1 ) N /N z N ' \ N z j 2=2,4 1-10 1 / \0 0 0 / S S 3 (3.28) J J 1 1 As e x p l a i n e d i n r e f , ( 5 ) the m a t r i x elements are more e a s i l y o b t a i n e d by an a l g e b r a i c t r a n s f o r m a t i o n o f the c a s e (a) m a t r i x r a t h e r than d i r e c t l y from eq. (3.28) because o f the c o m p l e x i t y o f the 9-j symbols. C o r r e s p o n d - i n g to eq. (3.21) we have -88- F 1 ( 0 = N + | ) : |[Y-Y S(N+1)/(2N+3)]N F 2 ( J = N + 2 - ) : |[Y(N-3)+3Y SN(3N+5)/(2N+3)J F 3 ( J = N - ^ ) : -J[Y(N+4)+3Y S(N+1)(3N-2)/(2N-1)] F 4 ( J = N + | ) : -|[Y-Y SN/(2N-1)](N+1) which s i m p l i f i e s f o r high N t o F] and F4: Y F I F F ^ Y - ^ $ ; F 2 a n d F3: Y f i f f , Y+1Y$ (3-30) The p o i n t s we make i n t h i s s e c t i o n a r e ( i ) t h a t by c h o o s i n g t h e nu m e r i c a l f a c t o r as i n eq. (3.17) we can d e f i n e t h e new h y p e r f i n e parameter b s so t h a t eqs. (3.23) and (3.30) have e x a c t l y the same form, and ( i i ) t h a t the case (b) e x p r e s s i o n s show how the t h i r d o r d e r i s o t r o p i c h y p e r f i n e term and the t h i r d - o r d e r s p i n - r o t a t i o n term b o t h a c t i n t h e same way, which i s to g i v e t h e F-| and F^ l e v e l s d i f f e r e n t e f f e c t i v e parameters from the F 2 and F 3 l e v e l s i n a 4 £ s t a t e . In a d d i t i o n i t i s p o s s i b l e to d e r i v e t h e form o f t h e e f f e c t i v e o p e r a t o r , a c t i n g e n t i r e l y w i t h i n t h e m a n i f o l d o f t h e £ e l e c t r o n i c s t a t e , w hich i s e q u i v a l e n t t o the t h i r d - o r d e r i s o t r o p i c h y p e r f i n e term. D. C o n c l u s i o n T h i s c h a p t e r g i v e s t h e background t h e o r y f o r the new h y p e r f i n e para- meter b s which had to be i n t r o d u c e d by Cheung e t aj_ (.5) to e x p l a i n t h e -89- 4 -h y p e r f i n e s t r u c t u r e o f t h e C z s t a t e o f VO ltieasiired by s u b - D o p p l e r l a s e r - i n d u c e d f l u o r e s c e n c e . The new. parameter i s a t h i r d - o r d e r c r o s s - t e r m be- tween the s p i n - o r b i t i n t e r a c t i o n and the f a m i l i a r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n , a nd, l i k e the c o r r e s p o n d i n g s p i n - r o t a t i o n e f f e c t ( 7 ) , must be i n c l u d e d i n a c c u r a t e work on a l l e l e c t r o n i c z s t a t e s o f q u a r t e t and h i g h e r m u l t i p l i c i t y . The new e f f e c t w i l l be e s p e c i a l l y l a r g e i f t h e r e a r e nearby s t a t e s t h a t i n t e r a c t s t r o n g l y t h r o u g h t he s p i n o r b i t o p e r a t o r w i t h the s t a t e o f i n t e r e s t ; t h e r e f o r e i t w i l l p r o b a b l y be more i m p o r t a n t i n t h e e x c i t e d e l e c t r o n i c s t a t e s o f h i g h m u l t i p l i c i t y m o l e c u l e s than i n t h e i r ground s t a t e s , s i n c e ground s t a t e s a r e o f t e n w e l l s e p a r a t e d from o t h e r i n t e r a c t i n g e l e c t r o n i c s t a t e s . The new term i n b^ i s i n f a c t r e q u i r e d f o r a l 1 e l e c t r o n i c s t a t e s o f q u a r t e t and h i g h e r m u l t i p l i c i t y , n o t j u s t Z s t a t e s . The r e a s o n i s t h a t no a p p r o x i m a t i o n s have been made i n i t s d e r i v a t i o n w hich l i m i t t h e v a l u e o f A ( o r K f o r p o l y a t o m i c m o l e c u l e s ) , so t h a t e q s . (.3.16) and (3.17) f o r case (a) c o u p l i n g , o r e q s . (3.20) and (3.24) f o r case (b) c o u p l i n g , a r e e n t i r e l y g e n e r a l . The o n l y r e s t r i c t i o n t o z e l e c t r o n i c s t a t e s i s i n eq. ( 3 . 9 ) , which was i n v o k e d to prove t h a t t he terms i n v o l v i n g t he t e n s o r ranks K=0 and 2 v a n i s h f o r A=0. We have n o t i n v e s t i g a t e d t h e consequences o f n o t i n v o k i n g eq. ( 3 . 9 ) , but q u a l i t a t i v e l y i t seems t h a t t h e K=2 terms s h o u l d g i v e r i s e to a h i g h e r o r d e r c o n t r i b u t i o n to one o f the o t h e r h y p e r f i n e o p e r a t o r s , p r o b a b l y a J^.J., which i s n o n - v a n i s h i n g when A/0 -90- Chapter 4 L a s e r Induced F l u o r e s c e n c e S p e c t r o s c o p y -91- A. I n t r o d u c t i o n The advent o f h i g h power monochromatic t u n a b l e l a s e r s o u r c e s has s t i m u l a t e d i m p o r t a n t advances i n o p t i c a l s p e c t r o s c o p y as documented by s e v e r a l r e c e n t r e v i e w s (1 - 4 ) . L a s e r i n d u c e d f l u o r e s c e n c e (which w i l l be taken h e r e t o mean the p r o c e s s whereby a m o l e c u l e a b s o r b s l a s e r l i g h t a t one w a v e l e n g t h and emits some f r a c t i o n o f the energy as l i g h t a t t h e same and o t h e r wave- l e n g t h s ) i s a much more s e n s i t i v e t e c h n i q u e than a b s o r p t i o n s p e c t r o s c o p y ( 5 ) . The r a t i o o f ( S / N ) a b s and ( s / N ) f i u o r > where S/N means s i g n a l - t o - n o i s e r a t i o , i s p r o p o r t i o n a l t o t h e n o i s e e q u i v a l e n t power o f t h e d e t e c t o r , NEP, and the f o u r t h power o f t h e f l u o r e s c e n c e w a v e l e n g t h . As the f l u o r e s c e n c e wavelength d e c r e a s e s , f l u o r e s c e n c e d e t e c t i o n i s i n c r e a s i n g - l y f a v o r e d . In r e g i o n s where p h o t o m u l t i p l e r tubes can be u s e d , t h e NEP drops c o n s i d e r a b l y and f l u o r e s c e n c e d e t e c t i o n becomes even more f a v o r a b l e . C o n s e q u e n t l y , most l a s e r e x p e r i m e n t s done i n t h o s e r e g i o n s use f l u o r e s c e n c e d e t e c t i o n t e c h n i q u e s . The h i g h e s t r e s o l u t i o n i n o p t i c a l s p e c t r o s c o p y i s a c h i e v e d by e l i m i n a t i n g the Doppler b r o a d e n i n g o f a t o m i c and m o l e c u l a r s p e c t r a l l i n e s . The h i g h i n t e n s i t y o f l a s e r l i g h t has l e d t o t h e development o f a v a r i e t y o f new n o n l i n e a r s p e c t r o s c o p i c t e c h n i q u e s which p e r m i t D o p p l e r - f r e e o b s e r v a t i o n s o f a s i m p l e gas sample. S a t u r a t e d a b s o r p t i o n s p e c t r o s c o p y o r Lamb d i p s p e c t r o s c o p y (6,7) was t h e e a r l i e s t d e v e l o p e d and perhaps t h e most w i d e l y used o f t h e s e methods; here the s p r e a d o f a t o m i c v e l o c i t i e s a l o n g the d i r e c t i o n o f o b s e r v a t i o n i s e f f e c t i v e l y r e d u c e d by v e l o c i t y - s e l e c t i v e b l e a c h i n g and p r o b i n g w i t h two c o u n t e r - p r o p a g a t i n g monochromatic l a s e r beams. S a t u r a t e d f l u o r e s c e n c e s p e c t r o s - -92- copy (8) and i n p a r t i c u l a r t h e s e n s i t i v 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 (9) extended the p o t e n t i a l o f t h i s j o e t h o d t o o p t i c a l l y v e r y t h i n f l u o r e s c e n t s a m p l e s . In t h i s c h a p t e r , v a r i o u s e f f e c t s r e l a t e d t o l a s e r experiments w i l l be d i s c u s s e d ; t h e y i n c l u d e ( i ) n o n - l i n e a r i n t e r a c t i o n s o f mole- c u l e s w i t h a v e r y i n t e n s e l a s e r beam to produce s a t u r a t i o n e f f e c t s , ( i i ) o b s e r v a t i o n o f e x c i t a t i o n s p e c t r a (which a r e e s s e n t i a l l y a b s o r p t i o n s p e c t r a p r o v i d e d no r a d i a t i o n l e s s p r o c e s s i s o c c u r r i n g ) by m o n i t o r i n g the t o t a l f l u o r e s c e n c e , and ( i i i ) t h e use o f s a t u r a t i o n e f f e c t s t o s t u d y a t o m i c or m o l e c u l a r l i n e s w i t h o u t D o p p l e r b r o a d e n i n g . Moreover, the e x p e r i m e n t a l t e c h n i q u e s 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 and r e s o l v e d f l u o r e s c e n c e w i l l a l s o be c o n s i d e r e d ; the f o r m e r y i e l d s l i n e p o s i t i o n s p t o v e r y h i g h p r e c i s i o n (1 p a r t i n 10 ) , w h i l e the l a t t e r g i v e s r e l a t i o n - s h i p s between l i n e s which p e r m i t unambiguous r o t a t i o n a l a s s i g n m e n t . B. S a t u r a t i o n o f M o l e c u l a r A b s o r p t i o n L i n e s In l i g h t a b s o r p t i o n experiments a t " c o n v e n t i o n a l " low power l e v e l s B e e r ' s law s t a t e s t h a t t h e a b s o r b e d power i s a c o n s t a n t f r a c t i o n o f t h e i n c i d e n t power. T h i s l i n e a r r e l a t i o n s h i p h o l d s o n l y i f t h e i n c i d e n t power i s low. In c o n t r a s t , i n t e n s e c o h e r e n t l i g h t s o u r c e s , which a r e c a p a b l e o f s u p p l y i n g v e r y h i g h power d e n s i t i e s , and hence a v e r y s t r o n g o p t i c a l e l e c t r i c f i e l d , can g e n e r a t e a w e a l t h o f n o n - l i n e a r phenomena as a r e s u l t o f t h e d i e l e c t r i c p o l a r i z a t i o n o f the a b s o r b i n g medium by the i n t e n s e f i e l d . The o r d e r o f magnitude o f the o p t i c a l e l e c t r i c f i e l d £ r e q u i r e d t o produce n o n - l i n e a r e f f e c t s can be o b t a i n e d from H e i s e n b e r g ' s u n c e r t a i n t y pri.nci.pl e -93- jje-T»f|. (.4.1) In t h i s e x p r e s s i o n JJ i s the e l e c t r i c d i p o l e m a t r i x element f o r the t r a n s i t i o n , T the r e l a x a t i o n time and fi (= P l a n c k ' s c o n s t a n t h, d i v i d e d by 2-n) ̂  1 0 " 3 4 J . s . A l a s e r power o f 1 W th r o u g h a c r o s s - s e c t i o n o f 2 5 - 2 0.1 cm ( t y p i c a l beam s i z e ) g i v e s a power d e n s i t y o f 10 Wm and 4 -1 produces an e l e c t r i c f i e l d o f e ^ 10 NC • , so t h a t a power o f 1 pW g i v e s E ^ 10 NC"^ ( s i n c e power d e n s i t y i s p r o p o r t i o n a l t o t h e s q u a r e o f the e l e c t r i c f i e l d ) . T h e r e f o r e , a l a s e r power o f 10"^ to 1 W i s adequate t o o b s e r v e n o n - l i n e a r e f f e c t s i n m o l e c u l e s w i t h p = 1 D and a r e l a x a t i o n time o f 10"^ t o 1 0 " 8 s. A r o t a t i o n a l r e l a x a t i o n time o f -3 t h i s m agnitude i s t y p i c a l a t a gas p r e s s u r e o f 10" t o 1 t o r r ( 1 0 ) . Let us c o n s i d e r t he b e h a v i o r o f an i s o l a t e d system w i t h two energy l e v e l s , E-| and Eg, under t h e a c t i o n o f an e l e c t r o m a g n e t i c f i e l d . The b e h a v i o u r o f the syste m , t h a t i s i t s w a v e f u n c t i o n i s d e s c r i b e d by the time-dependent S c h r o d i n g e r e q u a t i o n , ifi dv_ = (4.2) 3t where H i s the t o t a l h a m i l t o n i a n o f the s y s t e m , composed o f t h e unper- t u r b e d h a m i l t o n i a n H Q and t h e energy o f the quantum s y s t e m - f i e l d i n t e r - a c t i o n , o r , more s p e c i f i c a l l y i n t h i s c a s e , t h e e l e c t r i c d i p o l e i n t e r - a c t i o n between p a r t i c l e and f i e l d , which has the form H' H = -p "E = H + H' 0 = H - p-e cos I D t (4.3) -94- In eq. (4.3) JJ i s the component o f the jnol e c u l a r d i p o l e moment a l o n g t h e d i r e c t i o n o f t h e f i e l d , e i s t h e , s t r e n g t h o f the e l e c t r i c f i e l d o f the l i g h t wave, and u i t s f r e q u e n c y . The w a v e f u n c t i o n ¥ can be e x p r e s s e d i n term o f e i g e n f u n c t i o n s o f the H Q o p e r a t o r , * = £a n(t)<|> n (4.4) i . e . as a s u p e r p o s i t i o n o f t h e w a v e f u n c t i o n s $ o f the quantum system w i t h o u t a l i g h t f i e l d , where $ i s d e f i n e d as i f i ^ n = H 0 <j»n (4.5) " a t Then t h e e q u a t i o n t o d e t e r m i n e t h e c o e f f i c i e n t s i n the e x p a n s i o n i s I t 3 n = t k H ' n k 3 k 6 X P [ f T ( E n " E k ) t ] ( 4 - 6 ) from eq. (4.2) . In a t w o - l e v e l s y s t e m , n = 1 and 2, and the s i n g l e t r a n s i t i o n f r e q u e n c y i s (JJq = (E 2 - E -j) /ti. Eq. (4.6) g i v e s a p a i r o f c o u p l e d e q u a t i o n s f o r a-j(t) and a ^ ( t ) da-i= i _ K E a ? {exp[i(w-u> ) t ] + exp[-i(w+w ) t ] } dt 2 L 0 0 (4.7) dao= i K e a, { e x p [ - i ( to-w ) t ] + exp £i(w+w ) t ] } d T 2 1 0 0 where K = 2 y - ^ / t i , i n which p-| 2 i s t n e d l * p o l e m a t r i x element between the s t a t e s 1 and 2. As lon g as t h e Rabi f r e q u e n c y ^ = Ke i s very much -95- l e s s than w , we may n e g l e c t the h i g h - f r e q u e n c y t e r r a s , exp£ i(o)+co 0)t], i n t h e r o t a t i n g wave a p p r o x i m a t i o n ; then e q . (4.7) g i v e s d 2 a ? + i ( a ) - a i j dao + ( K e ) 2 a = 0 (4.8) d t 2 ° dt 4 2 The g e n e r a l s o l u t i o n to eq. (4.8) i s a 2 ( t ) = [A exp(iftt/2) + B ex p ( - i f t t / 2 ) ] exp(-iAt/2) M*) = - ] _ [ ( A - n ) A exp(iftt/2) + (A+ft)B e x p ( - i f t t / 2 ) ] exp ( i A t/2) ' Ke w i t h A = 0 3 - a > o and ft = .[A + (KE) t h e c o n s t a n t s A and B a r e de t e r m i n e d from t h e i n i t i a l c o n d i t i o n o f the sy s t e m . Assuming t h a t t he m o l e c u l e i s i n i t i a l l y i n t h e lower s t a t e 1, a l ^ t o ^ = e x P ( i e ) ( i - e - o n e m u l t i p l i e d by an a r b i t r a r y phase f a c t o r ) and a 2 ( t Q ) = 0. T h i s g i v e s t h e c o e f f i c i e n t a , ( t ) = f c o s ft ( t - t ) - i A s i n ft ( t - t ) ' lexp Ei e + i A ( t - t J / 2 ] 1 L 2 ° f t 2 ° J a 2 ( t ) = i Ke s i n p ( t - t 0 ) J e x p [ i e - i A ( t - t Q ) / 2 ] (4.10) The s q u a r e s o f t h e c o e f f i c i e n t s which c o r r e s p o n d t o r e l a t i v e p o p u l a t i o n s o f the two s t a t e s a r e N n ( t ) = | a . ( t ) | 2 = A 2 + ( K e ) 2 C O S 2 ft(t-tj ft^ ^ d (4.11) No(t) = | a 9 ( t ) | 2 = ( K e ) 2 s f n 2 f t ( t - t n ) ft C In a gas a t low p r e s s u r e , t h e c o h e r e n t o s c i l l a t i o n o f t h e m o l e c u l a r d i p o l e moment i s i n t e r r u p t e d by c o l l i s i o n s between m o l e c u l e s and by the -96- K f e time o f t h e e i g e n s t a t e . The e f f e c t o f c o l l i s i o n s can be i n c o r p o r a t e d i n t h i s t r e a t m e n t by a v e r a g i n g eq. (.4.11) o v e r a P o i s s o n d i s t r i b u t i o n o f d e p h a s i n g c o l l i s i o n s w i t h r e l a x a t i o n time T. The p r o b a b i l i t y t h a t t he m o l e c u l e has s u r v i v e d under c o h e r e n t i n t e r a c t i o n w i t h t h e f i e l d i n the i n t e r v a l t = t Q i s dN(t) = 1 e x p [ - ( t - t 0 ) / x ] d t (4.12) T The t r a n s i t i o n p r o b a b i l i t y f o r an ensemble o f m o l e c u l e s i n the gas i s then o b t a i n e d by t a k i n g an avera g e o v e r t Q , to g i v e <|a^i"> = i i ia„it.i: II e x D i " - ( t - t . ) / x ] d t . i 2 | 2 > = 1 f l^'VI exp[-(t-t0)/x] 0 = 1 ( K e ) 2 (4.13) 2 ( w - o > o ) 2 + T " 2 + ( . K e ) 2 T h i s a v e r a g e i n c r e a s e s m o n o t o n i c a l l y w i t h t he i n t e n s i t y o f the r a d i a t i o n f i e l d and approaches 0 . 5 as t h e l i m i t e ->-«>. T h i s means t h a t a ve r y i n t e n s e f i e l d w i l l e v e n t u a l l y e q u a l i z e p o p u l a t i o n s between upper and lower l e v e l s o f a t r a n s i t i o n . The power a b s o r b e d , which i s t h e o b s e r v a b l e i n t h i s s y s t e m , can be o b t a i n e d as AP = dW = ( N r N 2 ) fiw ( K e ) 2 d t 2T L>-U)o)2 + x-2 + ( K e ) 2 ] where N-j and N 2 a r e the numbers o f m o l e c u l e s i n the s t a t e s 1 and 2, r e s p e c - t i v e l y . The power f l o w p e r u n i t a r e a i n SI u n i t s P = l « „ . c e 2 (4.14) 2 -97- where €c i s t h e d i e l e c t r i c c o n s t a n t and c i s t h e speed o f l i g h t . As E -><*>, AP becomes a c o n s t a n t , the power a b s o r p t i o n c o e f f i c i e n t o f a gas o f t w o - l e y e l m o l e c u l e s i s g i v e n by a = AP = ( N , - N J 2fiwK 2 -> 0 (4.15) p ! r. €„ T I(o)-a.o) + T~ 2 + ( K E ) 2 ] and the medium s a t u r a t e s . With co = ID , t h i s can be r e w r i t t e n as t h e p h e n o m e n o l o g i c a l e x p r e s s i o n a = o a- (4.16) 1 + I / I s where a l l the a p p r o p r i a t e f a c t o r s a r e i n c o r p o r a t e d i n t o a Q and 1^. The s a t u r a t i o n parameter 1^ i s t h e power per u n i t a r e a t h a t a wave on r e s o - nance must c a r r y i n o r d e r to r e d u c e the p o p u l a t i o n d i f f e r e n c e to one- h a l f i t s u n s a t u r a t e d v a l u e ( 1 1 ) . With a moderate i n t e n s i t y (I < I s ) , we have a = a n (1 - I + . . . ) (4.17) T A s i m i l a r d e r i v a t i o n , by t h e use o f d e n s i t y - m a t r i x e q u a t i o n s , i s d i s c u s s e d by Letokhov and Chebotayev ( 3 ) . S a t u r a t i o n o f D o p p l e r - broadened a b s o r p t i o n l i n e s has been c o n s i d e r e d by Shimoda and S h i m i z u ( 1 0 ) . C. S a t u r a t e d F l u o r e s c e n c e S p e c t r o s c o p y A D o p p l e r - b r o a d e n e d s p e c t r a l l i n e i s a sum o f a g r e a t number o f much nar r o w e r l i n e s c o r r e s p o n d i n g t o m o l e c u l e s w i t h d i f f e r e n t thermal -98- v e l o c i t y , v. T h i s i s why the D o p p l e r e f f e c t on s p e c t r a l l i n e s i s o f t e n c a l l e d inhomogeneous b r o a d e n i n g . A c o h e r e n t l i g h t wave o f wave v e c t o r k i n t e r a c t s o n l y w i t h p a r t i c l e s i t r e s o n a t e s w i t h , t h a t i s , w i t h p a r t i - c l e s f o r which t h e D o p p l e r s h i f t i n the a b s o r p t i o n f r e q u e n c y , k.v^, com- pe n s a t e s p r e c i s e l y f o r the d e t u n i n g o f the f i e l d f r e q u e n c y , w, w i t h The e x c i t a t i o n o f p a r t i c l e s w i t h a c e r t a i n v e l o c i t y changes the e q u i l i b r i u m d i s t r i b u t i o n o f p a r t i c l e v e l o c i t i e s i n each l e v e l o f the t r a n s i t i o n . In the lo w e r l e v e l t h e r e i s a l a c k o f p a r t i c l e s whose v e l o - c i t y c o m p l i e s w i t h t h e re s o n a n c e c o n d i t i o n , t h a t i s , a h o l e appears i n t h e v e l o c i t y d i s t r i b u t i o n , an e f f e c t i - known as B e n n e t t h o l e b u r n i n g ( 1 2 ) , F i g . 4 . 2 . By c o n t r a s t , i n the upper l e v e l t h e r e i s an excess o f p a r t i c l e s w i t h r e s o n a n c e v e l o c i t i e s o r a peak i n t h e v e l o c i t y d i s t r i b u t i o n . The h o l e depth and peak h e i g h t depend on t h e degree o f a b s o r p t i o n s a t u r a t i o n by the l i g h t f i e l d . The wi d t h o f the h o l e d e t e r - mines t h e homogeneous l i n e w i d t h , which can be thousands o f times l e s s than t h e Do p p l e r w i d t h . r e s p e c t to the t r a n s i t i o n f r e q u e n c y , w , o f a f i x e d m o l e c u l e , ( F i g . 4.1) 0) = w + k.v (4.18) o 7\ co =(E2-Eil/fi F i g . 4.1 Two l e v e l s y s t e m -99- F i g . 4.2 M o l e c u l a r y e l o c i t y d i s t r i b u t i o n s f o r b o t h upper and l o w e r l e v e l s under t he a c t i o n o f a l a s e r wave o f f r e q u e n c y u>. A r e l a t e d phenomenon known as t h e Lamb d i p (7) forms t he b a s i s f o r many e x p e r i m e n t s i n s a t u r a t i o n s p e c t r o s c o p y ( 6 ) . A l t h o u g h Lamb has shown, i n h i s g a s - l a s e r t h e o r y ( 7 ) , t h a t the i n t e r a c t i o n o f a D o p p l e r - broadened l i n e w i t h a s t a n d i n g wave produces t h i s phenomenon, i n f a c t , the l i g h t wave need not be a s t a n d i n g wave: a s t r o n g t r a v e l l i n g wave i s s u f f i c i e n t t o produce t h e same e f f e c t ( 9 ) . A l s o t h i s s i g n a l can be d e t e c t e d by m o n i t o r i n g t he t o t a l f l u o r e s c e n c e , which i s j u s t a c o n s t a n t f r a c t i o n o f the t o t a l a b s o r b e d power. C o n s i d e r the s i t u a t i o n t h a t two s t r o n g t r a v e l l i n g waves from t h e same l a s e r s o u r c e pass through a c e l l i n o p p o s i t e d i r e c t i o n s . A photo- m u l t i p l i e r tube i s mounted next t o t h e c e l l s o t h a t f l u o r e s c e n c e l i g h t -100- frora t h e c e l l , p e r p e n d i c u l a r t o t h e l a s e r p r o p a g a t i o n d i r e c t i o n , can be m o n i t o r e d , F i g . 4.3. F i g . 4.3 Lamb d i p e x p e r i m e n t Each t r a v e l l i n g wave burns i t s own h o l e i n t h e v e l o c i t y d i s t r i b u t i o n . Because t h e s e two waves run i n o p p o s i t e d i r e c t i o n s , t h e r e a r i s e two h o l e s s y m m e t r i c a l about the c e n t r e o f t h e Dop p l e r p r o f i l e , f i g . 4.4a In t h i s c a s e , t h e t o t a l f l u o r e s c e n c e i n t e n s i t y i s t h e sum o f t h e c o n t r i - b u t i o n s f r o m each beam. As t h e l a s e r f r e q u e n c y i s tuned near t o t h e c e n t r e o f t h e D o p p l e r p r o f i l e , t h e two h o l e s g e t c l o s e r and c l o s e r ; a l s o because o f t h e g a u s s i a n d i s t r i b u t i o n o f the m o l e c u l a r v e l o c i t i e s , t h e t o t a l f l u o r e s c e n c e i n t e n s i t i e s i n c r e a s e s . When the l a s e r f r e q u e n c y i s t uned t o the c e n t r e o f t h e D o p p l e r p r o f i l e , t h o s e two h o l e s c o i n c i d e and t he t r a v e l l i n g l i g h t wave i n t e r a c t s w i t h o n l y one group o f p a r t i c l e s , F i g . 4.4b. T h i s r e s u l t s i n a r e s o n a n t d e c r e a s e o f a b s o r b e d power, which i n t u r n , d e c r e a s e s t he t o t a l f l u o r e s c e n c e i n t e n s i t y , F i g . 4.5. T h i s -101- e f f e c t i s known as a 'Lamb d i p ' . E x p e r i m e n t a l o b s e r v a t i o n s o f t h i s e f f e c t were r e p o r t e d i n (6,8). (a) (b) P o p . P o p . 0 V V F i g . 4.4 V e l o c i t y d i s t r i b u t i o n c u r v e s F i g . 4.5 T o t a l f l u o r e s c e n c e i n t e n s i t y vs l a s e r f r e q u e n c y -102- D. 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 o s c o p y The d e c r e a s e i n f l u o r e s c e n c e i n t e n s i t y from the Lamb d i p phenomenon has been used t o d e t e c t s a t u r a t i o n peaks and i s p a r t i c u l a r l y u s e f u l when the t o t a l a b s o r p t i o n i s s m a l l . Sorem and Schawlow (9) d e v e l o p e d a s e n - s i t i v e m o d u l a t i o n method f o r i s o l a t i n g a s m a l l change i n f l u o r e s c e n c e i n t e n s i t y . C o n s i d e r an e x p e r i m e n t a l s e t up as F i g . 4.6. The m o l e c u l e s a r e exposed to l i g h t o f two o p p o s i t e l y - d i r e c t e d beams from t h e same l a s e r which a r e chopped a t d i f f e r e n t f r e q u e n c i e s , w-j and u^. The modulated f l u o r e s c e n c e s i g n a l i s d e t e c t e d by phase s e n s i t i v e d e t e c t i o n w i t h r e f e r e n c e s i g n a l s e t a t t h e sum f r e q u e n c y , + u^. Io PMT F i g . 4.6. E x p e r i m e n t a l s e t up f o 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 -103- The b a s i c c o n c e p t s b e h i n d t h i s s a t u r a t e d f l u o r e s c e n c e t e c h n i q u e can be u n d e r s t o o d i n terms Df a v e r y s i m p l e two l e v e l m o d el. In the l i m i t o f t h e D o p p l e r w i d t h b e i n g much g r e a t e r than t h e homogeneous l i n e w i d t h and I < I s , from eq. (4.17) t h e a b s o r p t i o n c o e f f i c i e n t a t f r e q u e n c y co i s g i v e n by a = a (1 - I ) (4.18) is The t o t a l power o f the beam i s A l where A i s the c r o s s - s e c t i o n o f t h e beam and the t r a n s m i t t a n c e i s g i v e n by A l exp [ - a L ] where L i s the sample l e n g t h . S i n c e t he f l u o r e s c e n c e power, F, i s some f r a c t i o n o f the ab- s o r b e d power, F = KAI { 1 - expI-aL] \ (4.19) where K i s a p r o p o r t i o n a l i t y c o n s t a n t . When aL < 1, th e e x p o n e n t i a l term i n eq. (4.19) becomes 1-aL, such t h a t F = KAIuL (4.20) S i n c e the beams a r e modulated a t w-, and u 2 > the powers c o n t a i n e d i n beams 1 and 2 a r e I.| = I Q COS co-jt, *2 = 7 *o C 0 S " 2 * ' r e s P e c t i v e l v > w i t h I = I-] + I 2 = - I (cos W l t + cos u 9 t ) (4.21) 2 0 I £• where I„ i s t h e t o t a l power b e f o r e s p l i t t i n g . S u b s t i t u t i n g eq. (4.17) -104- and eq. (4.19) i n t o eq. (.4.20) w i t h r e a r r a n g e m e n t g i v e s •i F = -5-KALI„a„-(cos to, t + COS 0) ot 2 o 01 1 2 - I 0 I I + COS (w-| + a) 2)t + COS (u-| - u 2 ) t + j cos 2 ( 0 ^ + |- cos 2a> 2t]| (4.22) T h e r e f o r e t h e f l u o r e s c e n c e power a t the sum f r e q u e n c y w-j + w2 i s K A L I o 2 ° o c o s ^ w-| + oo 2)t (4.23) I t s h o u l d be not e d t h a t t h e r e w i l l a l s o be narrow r e s o n a n t terms modu- l a t e d a t the f r e q u e n c i e s w-j, co2, w-j - w2 and z e r o , b u t t h a t each o f t h e s e w i l l be accompanied by a l a r g e background because o f low f r e q u e n c y a m p l i t u d e n o i s e i n t h e l a s e r power. E q u a t i o n s (4.22) and (4.23) show t h a t t h e r a t i o o f t h e power a t t h e r e f e r e n c e f r e q u e n c y to t h e d c f l u o r e s c e n - ce background i s i n d e p e n d e n t o f a QL. Comparing t o a s i m i l a r c a l c u l a t i o n f o r t h e s a t u r a t e d a b s o r p t i o n (.13) shows t h a t t h i s r a t i o i s p r o p o r t i o n a l t o a QL i n t h e l i m i t o f a Q L < l . Thus t h i s method, which i s c a l l e d 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 , has a s t r o n g advantage f o r e x p e r i m e n t a l i s t s w o r k i n g w i t h v e r y weak t r a n s i t i o n s , v e r y low p a r t i c l e d e n s i t i e s , o r a p o o r l y p o p u l a t e d l o w e r s t a t e . T h i s t e c h n i q u e has been employed by a few workers i n s p e c t r o s c o p y t o s t u d y h y p e r f i n e s p l i t t i n g s (9,14 - 16). The h i g h e s t r e s o l u t i o n a c h i e v a b l e by t h i s method, where t h e s i g n a l s have a f u l l w i d t h a t h a l f maximum (FWHM) o f a few hundred k i l o h e r t z i s o b t a i n e d when t h e gas p r e s s u r e i s l e s s than 0.1 t o r r . The s i g n a l s u f f e r s from s i z e a b l e p r e s s u r e b r o a d e n i n g i f t h e p r e s s u r e i s s u b s t a n t i a l l y h i g h e r than 0.1 t o r r . -105- E. R e s o l v e d F l u o r e s c e n c e S p e c t r o s c o p y Heavy m o l e c u l e s with, h i g h s p i n m u l t i p l i c i t y always e x h i b i t s e v e r e l y b l e n d e d s p e c t r a i n t h e o p t i c a l r e g i o n . The o v e r l a p o f d i f f e r - e n t subbands or h o t bands makes the a n a l y s i s d i f f i c u l t s i m p l y because i t i s no l o n g e r p o s s i b l e t o r e c o g n i z e t h e p a t t e r n s o f br a n c h s t r u c t u r e , and some " l i n e s " have unexpected i n t e n s i t y due t o t h e b l e n d i n g o f many l i n e s . In a d d i t i o n , r o t a t i o n a l p e r t u r b a t i o n s by d i f f e r e n t e l e c t r o n i c s t a t e s c a u s e s h i f t s and s p l i t t i n g s o f the l i n e s . Under such c i r c u m s t a n - c e s , t he r o t a t i o n a l a n a l y s i s would be c o m p l e t e l y i m p o s s i b l e w i t h o u t some knowledge o f the quantum numbers i n v o l v e d i n the i n d i v i d u a l l i n e s o r t h e i r r e l a t i o n s h i p t o o t h e r l i n e s . R e s o l v e d f l u o r e s c e n c e i s o f v e r y g r e a t v a l u e i n s o l v i n g t h i s problem. I f a m o l e c u l e i s i r r a d i a t e d w i t h l a s e r l i g h t h a v i n g the wavelength o f a s i n g l e r o t a t i o n a l l i n e , the a b s o r b i n g m o l e c u l e s w i l l be b r o u g h t i n t o t h e upper s t a t e o f t h i s p a r t i c u l a r a b s o r p t i o n l i n e o n l y . The e x c i t e d m o l e c u l e s can then emit l i g h t , f a l l i n g to d i f f e r e n t r o t a t i o n a l l e v e l s i n t h e ground s t a t e a c c o r d i n g t o t h e s e l e c t i o n r u l e s f o r e l e c - t r o n i c t r a n s i t i o n s (17) w i t h e m i s s i o n o f r a d i a t i o n , and g i v i n g r i s e t o o t h e r f l u o r e s c e n c e wavelengths b e s i d e s the e x c i t i n g w a v e l e n g t h . The f i r s t s t e p i n the e x p e r i m e n t i s t o tune t he l a s e r t o a p a r t i - c u l a r l i n e o f an e l e c t r o n i c t r a n s i t i o n . F l u o r e s c e n c e , i n d u c e d by the pump l a s e r , i s m o n i t o r e d , p e r p e n d i c u l a r t o the l a s e r p r o p a g a t i o n d i - r e c t i o n , by a monochromator w i t h a p h o t o m u l t i p l e r t u b e . A r e s o l v e d f l u o r e s c e n c e s p e c t r u m i s o b t a i n e d by s c a n n i n g t h e monochromator. There a r e two i m p o r t a n t p i e c e s o f i n f o r m a t i o n c o n c e r n i n g t he l i n e a s s i g n m e n t t h a t can be o b t a i n e d from a r e s o l v e d f l u o r e s c e n c e s p e c t r u m : -106- i ) the AJ s e l e c t i o n r u l e f o r t h e l i n e which i s e x c i t e d by the l a s e r . When a s i n g l e energy l e v e l i s o p t i c a l l y e x c i t e d t h e f l u o r e s c e n c e spectrum c o n s i s t s o f two o r t h r e e l i n e s . I f X^ i s t h e e x c i t i n g wave- l e n g t h , one o f t h e l i n e s i n t h e f l u o r e s c e n c e s p ectrum w i l l always appear a t X^. I f o n l y two l i n e s appear and the o t h e r l i n e i s a t a s h o r t e r w a v e l e n g t h , the e x c i t i n g l i n e b e l o n g s t o an P b r a n c h ; i f the o t h e r l i n e i s a t a l o n g e r w a velength the e x c i t i n g l i n e b e l o n g s to an R b r a n c h . A Q l i n e may a p p e a r , i n between t he RP d o u b l e t , depending on t h e s e l e c t - i o n r u l e s and t h e t y p e o f t r a n s i t i o n . When a Q l i n e i s b e i n g e x c i t e d i n a p a r a l l e l t r a n s i t i o n , most o f t h e i n t e n s i t y i s r e - e m i t t e d i n the P and R l i n e s , F i g . 4.7. 7 ^ J ' = J R J " + I ) Q ( J " ) GO R(J"-1) J"+1 • i J ± J"-1 F i g . 4.7 O r i g i n o f i n d u c e d f l u o r e s c e n c e l i n e s -107- i i ) . The s e p a r a t i o n between P(_J"+1) and R ( J " - l ) f l u o r e s c e n c e l i n e s , c a l l e d A 2 F " ( J " ) , i s g i v e n f o r a l i n e a r m o l e c u l e by A 2F"(.J") = (4BJ - 6 D;)(J" + - 8DJ(.J" + (4.24) where B" and D" a r e t h e e f f e c t i v e r o t a t i o n a l and c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s f o r t h e lo w e r s t a t e ( 1 7 ) . Measurement o f the v a r i o u s s e p a - r a t i o n s a l l o w s t h e as s i g n m e n t o f the J numberings t o be made and s i m u l - t a n e o u s l y g i v e s a rough B^ v a l u e f o r t h i s ground s t a t e . In an a c t u a l c a s e , because t h e r e s o l u t i o n o f the monochromator i s an o r d e r o f magni- tude l o w e r than t h a t o f the l a s e r e x c i t a t i o n s p e c t r u m , i t i s o f t e n not p o s s i b l e t o i d e n t i f y e x a c t l y which l i n e i n a crowded e x c i t a t i o n s p ectrum goes w i t h t h e l i n e b e i n g e x c i t e d , t o w i t h i n 1 c m - 1 . I t may be n e c e s s a r y t o d e t e r m i n e t h i s by mea s u r i n g t h e r e s o l v e d f l u o r e s c e n c e s f o r v a r i o u s l i n e s i n the range i n d i c a t e d by the f i r s t e x p e r i m e n t . F u r t h e r m o r e , t h e e x c i t e d m o l e c u l e s can a l s o emit to d i f f e r e n t v i - b r a t i o n a l l e v e l s o f the lo w e r s t a t e . We c o u l d thus o b t a i n i n f l u o r e s c e n - ce a p r o g r e s s i o n w i t h V = c o n s t a n t b u t d i f f e r e n t v" i n t h e lower s t a t e , each band c o n s i s t i n g o f a P l i n e , an R l i n e and p o s s i b l y a Q l i n e . V i b r a t i o n a l a s s i g n m e n t o f the upper s t a t e can sometimes be made by c o u n t i n g t h e minima i n the i n t e n s i t y p a t t e r n o f a v i b r a t i o n a l l y r e s o l v e d f l u o r e s c e n c e s p e c t r u m ( 1 8 ) ; t h e number o f minima c o r r e s p o n d s to the number o f nodes i n t h e v i b r a t i o n a l w a v e f u n c t i o n . Combining t h e powerful t e c h n i q u e s o f i n t e r m o d u l a t e d and r e s o l v e d f l u o r e s c e n c e , i t becomes p o s s i b l e i n p r i n c i p l e t o a n a l y s e any spectrum o f any c o m p l e x i t y - f o r i n s t a n c e h i g h r e s o l u t i o n s p e c t r a where s m a l l -108- h y p e r f i n e s p l i t t i n g s , a r e p r e s e n t can be a n a l y s e d t o g i y e the d e t a i l s o f the e l e c t r o n s p i n and h y p e r f i n e c o u p l i n g c o n s t a n t w i t h g r e a t p r e c i s - i o n , and p e r t u r b e d e l e c t r o n i c band systems can be unambiguously a s s i g n e d no m a t t e r how fearsome t h e p e r t u r b a t i o n s may be . -109- C h a p t e r 5 L a s e r S p e c t r o s c o p y o f VO; A n a l y s i s o f the R o t a t i o n a l and H y p e r f i n e S t r u c t u r e o f the cV - xV (0,0) Band -110- A. I n t r o d u c t i o n Vanadium monoxide, VO, i s an i m p o r t a n t c o n s t i t u e n t o f t h e atmos- pheres o f c o o l r e d s t a r s , i t s band systems b e i n g u s e d f o r t h e s p e c t r a l c l a s s i f i c a t i o n o f s t a r s o f s p e c t r a l types M7-M9. ( 1 ) . There a r e t h r e e band systems o f VO i n t h e - v i s i b l e and near i n f r a - r e d . Near 1.05 y i s t h e A-X s y s t e m d i s c o v e r e d by K u i p e r , W i l s o n and Cashman ( 2 ) , and l a t e r s t u d i e d i n t h e l a b o r a t o r y by L a g e r q v i s t and S e l i n (3); f o l l o w i n g r e c e n t F o u r i e r t r a n s f o r m work by Cheung, T a y l o r and Merer (4) t h i s i s 4 4 - 4 a s s i g n e d as A n--X Z , where the A n s t a t e has q u i t e s m a l l s p i n - o r b i t c o u p l i n g . At 7900 A i s t h e B 4 n - X 4 Z ~ syst e m (5,6) where the B 4 n s t a t e 4 - ° i s e x t e n s i v e l y p e r t u r b e d by an unseen z s t a t e ( 4 ) , and near 5700 A 4 - 4 - / v i s t h e C z -X z system (6-9) whose d e t a i l e d a n a l y s i s i s d e s c r i b e d i n t h i s c h a p t e r . A p a r t i a l r o t a t i o n a l a n a l y s i s o f bands o f t h e C-X s y s t e m was f i r s t p e r f o r m e d by L a g e r q v i s t and S e l i n ( 8 ) . T h e i r s p e c t r a were o b t a i n e d u s i n g an a r c between vanadium e l e c t r o d e s , which g i v e s a v e r y h i g h tem- p e r a t u r e , and c o r r e p o n d i n g l y wide l i n e s . T h e i r l i n e a s s i g nments were c o r r e c t , but t h e y were o n l y a b l e t o a n a l y s e p a r t s o f the R 2, R3, P 2 and 2 2 P3 b r a n c h e s , and they s u g g e s t e d t h a t t h e t r a n s i t i o n was p o s s i b l y A- A. 2 4 - The ground s t a t e was l a t e r e s t a b l i s h e d as 06 z by t h e e . s . r . work o f K a s a i ( 1 0 ) , and s h o r t l y a f t e r w a r d s R i c h a r d s and Barrow ( 6 ) , r e i n v e s - t i g a t i n g t h e B-X and C-X s y s t e m s , found t h a t the two systems c o n t a i n l i n e s o f d i f f e r e n t w i d t h s as a r e s u l t o f h y p e r f i n e s t r u c t u r e i n t h e ground s t a t e caused by t h e 5 1 V n u c l e u s , which has 1=7/2. R i c h a r d s and Barrow c o u l d not r e s o l v e the h y p e r f i n e s t r u c t u r e i n t h e i r f u r n a c e s p e c t r a , but t h e y d i s c o v e r e d a v e r y unusual i n t e r n a l h y p e r f i n e p e r t u r b - a t i o n i n t h e ground s t a t e . What happens i s t h a t the F 2 and F-j l e v e l s (J=N± 3s) w i t h t h e same N v a l u e happen t o c r o s s n e a r N=15 because o f the p a r t i c u l a r v a l u e s o f t h e r o t a t i o n a l and e l e c t r o n s p i n p a r a m e t e r s . M a t r i x elements o f t h e h y p e r f i n e H a m i l t o n i a n o f t h e t y p e AN=0, A J = ± 1 a c t between them, and cause an a v o i d e d c r o s s i n g o f the h y p e r f i n e l e v e l s making up the two r o t a t i o n a l l e v e l s . At medium r e s o l u t i o n the p e r t u r b - a t i o n appears as a s m a l l d o u b l i n g o f the l i n e s n e a r N " > = 1 5 , and t h e minimum s e p a r a t i o n , which i s r e l a t e d to t h e i s o t r o p i c h y p e r f i n e p a r a - meter b ( 1 1 ) , was found t o be c o n s i s t e n t w i t h t h e e . s . r . work. The h y p e r f i n e s t r u c t u r e s o f t h e R-|, R^, P-| and P^ l i n e s were r e - s o l v e d by H o c k i n g , Merer and M i l t o n (12) i n h i g h r e s o l u t i o n g r a t i n g e m i s s i o n s p e c t r a . The h y p e r f i n e p a t t e r n s g i v e the d i f f e r e n c e o f the b parameters i n t h e C and X s t a t e s , and i t was fo u n d t h a t t h e r e a r e s i z e a b l e h y p e r f i n e s p l i t t i n g s i n the C s t a t e as w e l l . A n o t h e r i n t e r n a l h y p e r f i n e p e r t u r b a t i o n , s i m i l a r t o t h a t i n the ground s t a t e , was d i s - c o v e r e d a t N"=5 i n t h e C4Z~ s t a t e . A rough v a l u e c o u l d be o b t a i n e d f o r t h e i s o t r o p i c parameter b , but the d i p o l a r i n t e r a c t i o n p a rameter c c o u l d n o t be e x t r a c t e d from t h e i r D o p p l e r - l i m i t e d s p e c t r a . In t h i s c h a p t e r we d e s c r i b e an a n a l y s i s o f the C4E~-X4Z" (0,0) band from s u b - D o p p l e r s p e c t r a r e c o r d e d by 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 ( 1 3 ) . The l i n e w i d t h i s l i m i t e d by p r e s s u r e b r o a d e n i n g e f f e c t s t o about 100 MHz, but t h i s i s s u f f i c i e n t f o r t h e h y p e r f i n e s t r u c t u r e t o be e s s e n t i a l l y c o m p l e t e l y r e s o l v e d , b a r r i n g t h e r e g i o n a t the R 2 head. E x c e p t f o r t h e p l a c e s where t he upper s t a t e s u f f e r s from e l e c t r o n i c p e r t u r b a t i o n s t h e l i n e s can be f i t t e d by l e a s t sq u a r e s w i t h -112- a s t a n d a r d d e v i a t i o n o f b e t t e r than 0.0008 c m - 1 . A c c u r a t e v a l u e s f o r the r o t a t i o n a l , e l e c t r o n s p i n and n u c l e a r s p i n c o n s t a n t s have been ob- t a i n e d f o r both s t a t e s . -113- B. E x p e r i m e n t a l D e t a i l s VO was p r e p a r e d i n a fl o w system by p a s s i n g VOCI3 mixed w i t h argon through a 2450 MHz e l e c t r o d e l e s s d i s c h a r g e o p e r a t i n g a t 75 W. The m i x t u r e was pumped through a s t a i n l e s s s t e e l f l u o r e s c e n c e c e l l f i t t e d w i t h q u a r t z windows, and VO f l u o r e s c e n c e was e x c i t e d by l i g h t from a Coherent I n c . Model CR-599-21 t u n a b l e dye l a s e r . The s t r o n g e s t f l u o r e s - cence o c c u r r e d when t h e microwave d i s c h a r g e was a p u r p l i s h p i n k c o l o u r w i t h a p a l e b l u e ' t a i l ' ; t h e f l u o r e s c e n c e i n d u c e d by e x c i t a t i o n o f the C 4E~-X 4E~ (0,0) band ( a t 5738 A) was a y e l l o w orange c o l o u r . S p e c t r a o f VO were r e c o r d e d a t s u b - D o p p l e r r e s o l u t i o n by 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 ( 1 3 ) . F i g u r e 5.1 i l l u s t r a t e s t h e o p t i c a l a r r a n g e - ment f o r t h i s e x p e r i m e n t . A Coh e r e n t R a d i a t i o n CR-10 A r + l a s e r o p e r a t - i n g a t 514.5 nm w i t h 2.2 W o u t p u t i s used t o pump a Coherent R a d i a t i o n CR-599-21 dye l a s e r w i t h rhodamine 6 G dye. Dye l a s e r o u t p u t i s t y p i - -5 1 c a l l y 30 mW s i n g l e f r e q u e n c y ( A v ^ ^ ^ 3x10 cm ) , and i s m o n i t o r e d u s i n g a 1.5 GHz f r e e s p e c t r a l range (FSR) spe c t r u m a n a l y z e r , a 299-MHz FSR f i x e d l e n g t h s e m i 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 , and an I 2 c e l l . I 2 f l u o r e s c e n c e e x c i t e d by the dye l a s e r i s d e t e c t e d p e r p e n d i - c u l a r to the l a s e r p r o p a g a t i o n d i r e c t i o n by an RCK IP28 p h o t o m u l t i p l i e r tube (PMT 1) o p e r a t e d a t -870 VDC. The dye l a s e r beam was s p l i t by a 50-50% beam s p l i t t e r (B S ) , and the r e s u l t i n g two beams were chopped m e c h a n i c a l l y a t 582 Hz and 784 Hz. The l a s e r power was about 15 mW i n each beam, and t h e f l u o r e s c e n c e s i g n a l was r e c o r d e d through a s h a r p - c u t y e l l o w f i l t e r u s i n g an RCA C31025C (PMT 2 ) . A narrow-band e l e c t r i c a l f i l t e r s e l e c t e d t h e sum o f t h e ch o p p e r f r e q u e n c i e s , and the i n t e r m o d u l - a t e d s i g n a l was e x t r a c t e d w i t h a P r i n c e t o n A p p l i e d Research model 128A PMT 2 12 c e l l 299 MHz FSR I n t e r f e r o m e t e r 99% R e f l 1.5 GHz FSR CR599-21 L a s e r Dye L a s e r Spectrum A n a l y z e r Fig. 5.1 Experimental set up f o r intermodulated fluorescence spectroscopy. -115- l o c k - i n - a m p l i f i e r . A l l t h e n e c e s s a r y e l e c t r o n i c s were c o n n e c t e d as i n F i g . 5.2 t o m i n i m i z e t h e ground l o o p problem. The r e s u l t s were d i s - p l a y e d on a t h r e e - p e n c h a r t r e c o r d e r . As t h e l a s e r f r e q u e n c y was scanned one pen p l o t t e d the i n t e r m o d u l a t e d s i g n a l , t h e s e c o n d pen gave f r e q u e n c y markers s p a c e d a t 299 MHz i n t e r v a l s from t h e s e m i - 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 , and the t h i r d pen r e c o r d e d t h e f l u o r e s c e n c e s p e c t r u m o f \̂  f o r a b s o l u t e c a l i b r a t i o n . T h i s system i s s i m i l a r t o t h a t used by F i e l d e t a l (14) i n t h e i 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 ex- p e r i m e n t s , e x c e p t t h a t we use a narrow band e l e c t r i c a l f i l t e r r a t h e r than a s e c o n d l o c k - i n - a m p l i f i e r ( 1 5 ) . F o r i n t e n s i t y r e a s o n s we were n o t a b l e to run the microwave d i s - c h a r ge such t h a t the t o t a l p r e s s u r e i n t h e f l u o r e s c e n c e c e l l was l e s s t han about 1 mm Hg i f we were t o r e c o r d 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 . As a r e s u l t t h e l i n e w i d t h s i n o u r s p e c t r a a r e p r e s s u r e b r o a d e n e d , and were n e v e r l e s s than a b o u t 80 MHz even though t h e l a s e r l i n e w i d t h i s 1-2 MHz. For the weaker h i g h N l i n e s we had t o i n c r e a s e t h e t o t a l p r e s s u r e ; t h e l i n e w i d t h s r o s e to about 130 MHz, but f o r t u n a t e l y t h e s e l i n e s a r e o n l y r a r e l y b l e n d e d so t h a t the o n l y a d v e r s e e f f e c t was l o w e r p r e c i s i o n i n t h e i r c a l i b r a t i o n . We e n c o u n t e r e d no problems w i t h r e l a t i v e c a l i b r a t i o n o f t h e s p e c t r a o v e r a 1 cm"^ s c a n o f the dye l a s e r ; f o r example the ground s t a t e h y p e r f i n e c o m b i n a t i o n d i f f e r e n c e s i n the P^ and P 3 l i n e s were r o u t i n e l y r e p r o d u c e d to w i t h i n ±0.0005 c m - 1 (15 MHz); t h i s i s because t h e i n t e r - f e r o m e t e r markers a r e s h a r p compared t o the V0 l i n e s , and because t h e t e m p e r a t u r e o f t h e room (which a f f e c t s t he p o s i t i o n s o f the markers Ref. s i g n a l PD Ammeter PMT E x p t . J High V o l t a g e Power Li ne Fig. j I n t e r f e r o m e t e r r . 5.2 Schematic diagram f o r intermodulated fluorescence d e t e c t i o n system. \ , H R i CTl BS L a s e r Beam BS -117- though not t h e i r s p a c i n g ) remained s u f f i c i e n t l y c o n s t a n t d u r i n g t h e few minutes r e q u i r e d f o r a s c a n . However the a b s o l u t e c a l i b r a t i o n was always much l e s s c e r t a i n . As e x p l a i n e d we used D o p p l e r - l i m i t e d I 2 f l u o r e s c e n c e l i n e s , e x c i t e d by a p o r t i o n o f t h e l a s e r beam p i c k e d o f f by a b e a m - s p l i t t e r . The wavenumbers o f t h e I 2 l i n e s have been l i s t e d t o 0.0001 cm" 1 (3 MHz) by G e r s t e n k o r n and Luc ( 1 6 ) , b u t , s i n c e t h e I 2 l i n e s a r e a b o u t 1 GHz wide because o f u n r e s o l v e d h y p e r f i n e s t r u c t u r e , t h e i r a b s o l u t e u n c e r t a i n t y i s about 0.002 cm - 1 (60 MHz). F o r t u n a t e l y the V0 s p e c t r u m i s s u f f i c i e n t l y dense n e a r t h e band head t h a t we were a b l e t o e s t a b l i s h t h e r e l a t i v e s h i f t s o f t h e " l a d d e r s " o f i n t e r f e r o - meter markers between s u c c e s s i v e 1 cm - 1 l a s e r s c a n s , u s i n g l i n e s d u p l i c a t e d i n t h e o v e r l a p r e g i o n s . In t h i s way we c o u l d p l o t a c a l i - b r a t i o n graph f o r t h e I 2 l i n e s r e l a t i v e t o the i n t e r f e r o m e t e r markers o v e r ranges o f up to 10 c m - 1 . T h i s gave us t h e marker s p a c i n g w i t h g r e a t a c c u r a c y , and e n a b l e d us t o use 40-50 I 2 l i n e s to e s t a b l i s h t h e a b s o l u t e c a l i b r a t i o n o f the " l a d d e r " . The c a l i b r a t i o n graphs c o n s i s t e d o f a s c a t t e r o f p o i n t s , one f o r each I 2 l i n e , s p r e a d randomly o v e r ±0.002 cm - 1 a l o n g a s t r a i g h t l i n e . T h i s p r o c e d u r e i s v e r y l a b o r i o u s , but we c o n s i d e r i t w o r t h w h i l e because i t improves the s t a n d a r d d e v i a t i o n i n a l e a s t s q u a r e s f i t t o t h e l i n e p o s i t i o n s by n e a r l y a f a c t o r o f two: t h e f i n a l s t a n d a r d d e v i a t i o n f o r 1300 low N l i n e s c a l i b r a t e d i n t h i s way was 0.00076 cm" 1 (23 MHz). -118- C. R o t a t i o n a l and h y p e r f i n e energy l e v e l e x p r e s s i o n s S i n c e t h i s work on VO i s t h e f i r s t d e t a i l e d s t u d y o f t h e h y p e r f i n e s t r u c t u r e i n a e l e c t r o n i c s t a t e , we d i d not know a t f i r s t which terms t o i n c l u d e i n the H a m i l t o n i a n . A f t e r some e x p e r i m e n t a t i o n we found i t n e c e s s a r y t o v a r y 12 parameters f o r each e l e c t r o n i c s t a t e . The H a m i l t o n i a n was taken (17,18) as H = H . + H , + H. ,c + H -j . + H p ) , (5.1) r o t e l h f s e l , c . d . s.o. where H r o t = * ^ H e l = TN . S + f A O S ^ - S 2 ) (5.2) H h f Q - b l . S + c l S + e ' q q ( 3 I 2 f c - I f c ) + l n hfs ~ ~ z z 41(21-1) !~ ~ H e l , c . d . = + T V ( 3 S z 2 - i 2 ) N 2 + H2^z2-iZK (3) which a r e b a s i c a l l y d e s c r i b e d i n c h a p t e r 2, and H^ ' r e p r e s e n t s t h i r d - o r d e r s p i n - o r b i t c o n t r i b u t i o n s t o the parameters y and b, which a r e d e s c r i b e d l a t e r . The r o t a t i o n a l e n e r g y , g i v e n by H t, r e q u i r e s no e x p l a n a t i o n ; t h e terms i n H g^ a r e t h e e l e c t r o n s p i n - r o t a t i o n i n t e r - a c t i o n and the e l e c t r o n s p i n - s p i n d i p o l a r i n t e r a c t i o n , w h i l e t h e terms i n ^Y\fs a r e t n e d i r e c t e l e c t r i c and mag n e t i c h y p e r f i n e i n t e r a c t i o n s . The terms i n b and c a r e t h e d e t e r m i n a b l e c o e f f i c i e n t s i n t h e mag n e t i c h y p e r f i n e H a m i l t o n i a n f o r a i s t a t e ; c i s t h e d i p o l a r e l e c t r o n s p i n - n u c l e a r s p i n i n t e r a c t i o n , and b i s a c o m b i n a t i o n o f c w i t h the Fermi c o n t a c t i n t e r a c t i o n , a c > A more fundamental way o f w r i t i n g t h e s e I ,S ma g n e t i c h y p e r f i n e terms ( 1 9 ) , which i s c o n v e n i e n t f o r t h e c a l c u l a t i o n -119- o f m a t r i x e l e m e n t s , i s H m a „ hfc = ar I.S + c(I,S, - h.S) (5.3) mag.hfs c ~ ~ z z 3^ ~ The e l e c t r i c q u a d r u p o l e i n t e r a c t i o n (e Qq) and t h e n u c l e a r s p i n - r o t a t i o n i n t e r a c t i o n ( c ^ ) a r e f a m i l i a r from microwave s p e c t r a o f s i n g l e t s t a t e s ( 1 8 , 2 0 ) , w h i l e t h e 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 s to H -j s h o u l d be s e l f - e x p l a n a t o r y . In t h e z s t a t e s o f VO under d i s c u s s i o n t h e e l e c t r o n s p i n - s p i n i n t e r a c t i o n •(x) i s l a r g e compared to t h e h y p e r f i n e e f f e c t s , so t h a t the b a s i s g i v i n g t h e most n e a r l y d i a g o n a l r e p r e s e n t a t i o n i s c a s e ( b ^ j ) c o u p l i n g (11,18) where N + S = J - , J + i = F (5.4) The b a s i s f u n c t i o n s are then |NASJIF>, where A can be s u p p r e s s e d be- cause i t i s equal t o z e r o . However the m a t r i x elements i n c a s e ( b ^ j ) c o u p l i n g a r e much more c o m p l i c a t e d a l g e b r a i c a l l y than t h o s e i n c a s e ( a ^ ) c o u p l i n g so t h a t t h e use o f c a s e ( b ^ j ) i s l o g i c a l f o r VO o n l y b e c a u s e o f t h e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s mentioned i n the I n t r o - d u c t i o n . The m a t r i x elements r e s p o n s i b l e f o r the i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s a r e o f f - d i a g o n a l i n both b a s i s s e t s , but i n c a s e (a^) the s p i n - u n c o u p l i n g i s a l s o o f f - d i a g o n a l . The s p i n - u n c o u p l i n g m a t r i x e l e m e n t s , which a r i s e from t h e x and y components o f t h e o p e r a t o r -2B J.S, a r e v e r y much l a r g e r than the i n t e r n a l h y p e r f i n e p e r t u r b a t i o n e l e m e n t s , and we found t h a t t h e y gave t r o u b l e w i t h the energy o r d e r i n g -120- o f the e i g e n v a l u e s i n t h e r e g i o n s o f the i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s . T h i s l e d t o d i s a s t e r i n o u r att e m p t s to f i t the l i n e p o s i t i o n s by l e a s t s q u a r e s , so t h a t we r e t u r n e d t o c a s e ( b ^ j ) , though n o t w i t h o u t m i s g i - v i n g s . In r e t r o s p e c t a two s t e p d i a g o n a l i z a t i o n s t a r t i n g from c a s e ( a ^ ) might have s a v e d much l e n g t h y a l g e b r a : t h e f i r s t s t e p would have been e s s e n t i a l l y a n u m e r i c a l t r a n s f o r m a t i o n t o case ( b ^ j ) , and the second s t e p would have completed the d i a g o n a l i z a t i o n . We q u i c k l y d i s c o v e r e d t h a t i t was n e c e s s a r y t o use a f u l l m a t r i x t r e a t m e n t f o r a c o r r e c t d e s c r i p t i o n o f the magn e t i c h y p e r f i n e e f f e c t s ; f o r i n s t a n c e t h e AN= ± 2 , AJ=±1 elements o f t h e d i p o l a r i n t e r a c t i o n have a s i g n i f i c a n t e f f e c t on the c o u r s e o f t h e energy l e v e l s a t the i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s . T h e r e f o r e the o n l y s i m p l i f i c a t i o n we have made was t o omit t h e AJ=±2 elements o f the e l e c t r i c q u a d r u p o l e i n t e r - a c t i o n , a f t e r c a l c u l a t i n g t h a t t h e s e were l e s s than 1 MHz. The m a t r i x elements o f the terms i n B, D, Y and y n a r e d i a g o n a l i n case ( b ^ j ) : < N S J I F l H r o t + s p i n - r o t l N S J I F > = B N ( N + 1 ) - D N 2 ( N + 1 ) 2 ^ - )S[N(N+1)+S(S+1)-J(J+1)J[Y+Y I/I(N+1)] M a t r i x elements o f t h e o t h e r terms i n eq. (5.2) a r e most c o n v e n i e n t l y c a l c u l a t e d u s i n g s p h e r i c a l t e n s o r f o r m a l i s m as shown i n c h a p t e r 2 ( 1 9 , 2 1 ) . F o r e l e c t r o n i c Z s t a t e s o f any m u l t i p l i c i t y , where a s i n g l e s p i n n i n g n u c l e u s i s p r e s e n t , we have •121- 2,, -, N N + S + J < N - S J I F | H I N _ I N | N S J I F > - f x t - i r - " ( J S N - 2 N S x [ S ( S + l ) ( 2 S + l ) ( 2 S - l ) ( 2 S + 3 ) ] J 5 ( - 1 ) N ' /N' 2 N\[( 2N+1) (2N'+1) ] /N'  N\ l \0 o oj (5.6) < N ' S J ' I F | H |NSJ IF> = ( - 1 ) J + I + F ( F I J')[(2J+1)(2J'+1) 1(1+1) 1 mag.hfs 1 J ( , 4 L I J I (21+ur ( _ 1 } N + S + J'+1 - ^ N . [ s ( S + l)(2S+ l ) ] 3 2 a ( J S 1 -±c[30(2N+l)(2N'+l) S(S+1)(2S+1)J 3 5 ,N' N 2\ ( - 1 ) N ' / N ' 2 N' S S 1 J ' J 1 N' \ \0 0 0 / (5.7) +(-D N+S+J+l N J ' S) [N(N+1)(2N+1)]^ C j J N l ( where a ( t h e t r u e Fermi c o n t a c t i n t e r a c t i o n ) i s b +-j c , and < r s j ^ F | H q u a d n j p o l e NSJIF> = he2 -1 Qq / I 2 I \ (-1) V I 0 1/ J + I + F F I J ' 2 J I (5.8) c ( - D N " + S + J [(2J+1)(2J'+1) (2N+1)(2N'+1) j S J ' ) ( - l ) /N' 2 N 2 J N j ( - l ) r /N' 2 N \ } VO 0 0/ I t i s s t r a i g h t - f o r w a r d t o o b t a i n e x p l i c i t e x p r e s s i o n s e x c e p t f o r t h e d i p o l a r term c d ^ - ^ . S ) , where t h e 9-j symbol i s not l i s t e d i n s t a n d a r d t a b u l a t i o n s ( 2 2 ) , and the a l g e b r a i c e x p r e s s i o n s g i v e n by -122- Mizushima (23) c o n t a i n y a r i o u s s m a l l b u t i m p o r t a n t e r r o r s . F o r c o m p l e t e - ness we l i s t t h e c o r r e c t e d forms o f t h e r e l e v a n t 9-j symbols i n Appen- d i x IV. Our phase c h o i c e i n e q s . ( 5 . 5 ) - ( 5 . 8 ) c o r r e s p o n d s t o t h a t o f Bowater, Brown and C a r r i n g t o n , where N i s t r e a t e d as a s p a c e - f i x e d o p e r a t o r ( 1 9 ) , and t h e o r d e r o f c o u p l i n g the v e c t o r s i n eq. (5.4) i s a l s o the same as t h e i r s . The t h i r d - o r d e r s p i n - o r b i t c o n t r i b u t i o n s w i l l be u n f a m i l i a r s i n c e 3 they o n l y o c c u r f o r S ^ , t h a t i s f o r e l e c t r o n i c q u a r t e t s t a t e s o r worse. In t h i s study we have had to use t h e t h i r d - o r d e r c o n t r i b u t i o n to t h e s p i n - r o t a t i o n i n t e r a c t i o n , y s, i n t r o d u c e d by Brown and M i l t o n ( 2 5 ) , and t h e c o r r e s p o n d i n g c o r r e c t i o n to t h e i s o t r o p i c h y p e r f i n e i n t e r a c t i o n , which we c a l l b s ( 2 6 ) . The term i n y<. has a c o m p l i c a t e d h i s t o r y . The o r i g i n a l f o r m u l a e f o r 4 E s t a t e s d e r i v e d by Budo (27) and Kovacs (28) a c c o u n t e d n i c e l y f o r N e v i n ' s d a t a f o r ( 2 9 ) , but not f o r the d a t a on GeH (30) and S i F ( 3 1 ) . Hougen (32) a t t e m p t e d to f i n d t h e s o u r c e o f the d i s c r e p a n c y , and extended t h e t h e o r y t o i n c l u d e t h e s e c o n d s p i n - 4 r o t a t i o n p a r ameter r e q u i r e d by group t h e o r y arguments f o r a I s t a t e i n t h e g e n e r a l c a s e . L a t e r work by M a r t i n and Merer (33) showed t h a t t h e o r i g i n a l S i F s p e c t r u m has been m i s a s s i g n e d , and t h a t t h e r e were no d i s - c r e p a n c i e s , b u t t h e i r t h e o r e t i c a l t r e a t m e n t was s t i l l i n c o m p l e t e . F i n a l l y Brown and W i l t o n (25) gave a f u l l d i s c u s s i o n o f t h e h i g h e r o r d e r s p i n dependence o f t h e 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 I s t a t e s . T h e i r c o n c l u s i o n was t h a t t h e s i n g l e s p i n - r o t a t i o n term i n y g i v e n i n eq. (5.2) w i l l u s u a l l y s u f f i c e , e x c e p t i n cases where v e r y h i g h r e s o l u t - i o n i s a v a i l a b l e , o r where a n o t h e r nearby e l e c t r o n i c s t a t e i n t e r a c t s -123- 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 ; i n t h i s c a s e a second s p i n - r o t a t i o n terra w i l l be needed. T h i s s e c o n d term r e s u l t s from a t h i r d o r d e r i n t e r a c t i o n where the s p i n - u n c o u p l i n g o p e r a t o r - 2 B ( J S +J S ) i s t a k e n x x y y w i t h t h e s p i n - o r b i t o p e r a t o r za 1..s- t w i c e : t h e m a t r i x elements i n i i c a s e ( a ) , w i t h z and n t a k e n as s i g n e d q u a n t i t i e s , a r e <SZ,Jft|H^ )|S^±'I ,Jfi±l> = - J5Yc[S(S+l)-5z(z±l)-2] x [jfj+D-nCnii)]15 [sis+D-zCz+l)]15 (5.9) The e x p e r i m e n t a l parameter y$ i s a c o m p l i c a t e d f u n c t i o n o f t h e s p i n - o r b i t m a t r i x e l e m e n t s , t h e energy s e p a r a t i o n t o the i n t e r a c t i n g s t a t e s and the d i f f e r e n c e s AB between the B v a l u e s o f the i n t e r a c t i n g s t a t e s and t h e s t a t e o f i n t e r e s t . Barrow (34) has r e p o r t e d t h a t the C 4 z " and X 4 z " s t a t e s o f VO both r e q u i r e two s p i n - r o t a t i o n parameters ( p r e s u m a b l y i n Hougen's f o r m a l i s m ) . The p r e s e n t work c o n f i r m s t h i s c o n c l u s i o n , a l t h o u g h we have used Brown and M i l t o n ' s d e f i n i t i o n s f o r t h e p a r a m e t e r s . These s t a t e s o f VO ar e t h e f i r s t 4 Z s t a t e s known where two y's a r e d e f i n i t e l y needed. The h i g h p r e c i s i o n o f our d a t a has r e q u i r e d t h a t we c o n s i d e r t h e c o r r e s p o n d i n g e f f e c t i n t h e i s o t r o p i c h y p e r f i n e H a m i l t o n i a n . The mechanism f o r i t s appearance i s e n t i r e l y a n a l o g o u s : i n s t e a d o f th e s p i n - u n c o u p l i n g o p e r a t o r - 2 B ( J S +J S ) we ta k e t h e i s o t r o p i c h y p e r f i n e x x y y o p e r a t o r z b . I . s - w i t h t h e s p i n - o r b i t o p e r a t o r t w i c e ( 2 6 ) . The r e s u l t i s as i f t h e r e were an e f f e c t i v e o p e r a t o r = £ T V M ^ T V t S ) , T 2 ( S . . S . ) ) , T 2 ( C ) ) (5.10) i U i > j ~ -124- a c t i n g w i t h i n the m a n i f o l d o f the \ s t a t e o f i n t e r e s t . As g i v e n i n r e f . ( 2 5 ) , the m a t r i x elements o f eq. (5.10) i n c a s e ( b g J ) c o u p l i n g f o r A=0 a r e < N ' S J - I F | H W | N S J I F > 4 ( -D J + I + F j F I J ' J [ t 2 J + l ) ( 2 J ' + l ) . I ( I + D ( 2 I + l ) ] 1 S ° j l J I ) x ( - l ) N V 2 N\ [ ( 2 N + l ) ( 2 N ' + l ) ] J s /N' N 2\ t 5 - 1 1 ) \0 0 0/ S S 3] l J - J 1 x[ 3 5 ( 2 S - 2 ) ( 2 S - l ) 2 S ( 2 S + l ) ( 2 S + 2 ) ( 2 S + 3 ) ( 2 S + 4 ) / 3 ] 2 b $ The e x p e r i m e n t a l parameter T Q ( C ) , which would r e s u l t from eq. ( 5 . 1 0 ) , has been put e q u a l t o -y- (14) 3* b^, i n o r d e r to d e f i n e a para m e t e r b<- which i s as s i m i l a r as p o s s i b l e to Brown and M i l t o n ' s YS• C a l c u l a t i o n o f t h e e x p l i c i t m a t r i x elements from eq. (5.11) i n c a s e ( b D l ) c o u p l i n g was a l o n g p r o c e s s because the 9-j symbols g i v e u n c o m p r o m i s i n g l y i n t r a c t a b l e a l g e b r a i c e x p r e s s i o n s . F o r r e f e r e n c e they a r e i n c l u d e d i n Appendix IV. No doubt t h e r e a r e s i m p l e r forms f o r t h e 9-j symbols w i t h N'=N, but we have not t r i e d t o s e a r c h f o r them. The f i n a l m a t r i x e l e m e n t s , on the o t h e r hand, a r e q u i t e s i m p l e because o f e x t e n s i v e c a n c e l l i n g . A n o t h e r a p p r o a c h t o t h e m a t r i x elements o f t h e s e t h i r d - o r d e r terms i s to s e t up t h e m a t r i c e s i n case ( a Q ) c o u p l i n g and t r a n s f o r m them a l - p g e b r a i c a l l y t o c a s e ( b g J ) u s i n g the e i g e n v e c t o r s o f t h e c a s e (a^) r o t a t i o n a l m a t r i c e s . T h i s i s no r e a l advantage f o r t h e h y p e r f i n e term -125- because t h e case (a.) m a t r i c e s a r e a l r e a d y q u i t e c o m p l i c a t e d , h a v i n g P elements o f t h e t y p e Aft=0,±l, A J = 0 , ± 1 ; we d i d however t r a n s f o r m t h e d i a g o n a l b l o c k s i n t h i s way i n o r d e r t o check t h e c a l c u l a t i o n o f t h e 9-j symbols. On the o t h e r hand t he s p i n - r o t a t i o n term has a s i m p l e m a t r i x r e p r e s e n t a t i o n i n case (a) c o u p l i n g , c o n s i s t i n g o n l y o f t h e elements g i v e n i n e q . ( 5 . 9 ) , b u t a fearsome form i n case ( b ) : - < N ' S J I F | H ^ ] n _ r o t | N S J I F > = { [ ( 2 N + 1 ) ( 2 N ' + 1 ) . J ( J + 1 ) ( 2 J + 1 ) ] % x[2( 2S-2) (2S-1) 2S( 2S+1) ( 2S+2) (2S+3) ( 2S+4)/2>fz Yc; b (5.12) x I ( 2 x + l ) / 3 x 1\ ( - 1 ) N ' /N' x N\ ' "• x = 2' 4 \-l 0 1 / \0 0 o) J J 1 A c c o r d i n g l y we c o n v e r t e d eq. (5.9) t o c a s e (b) a l g e b r a i c a l l y u s i n g a Wang t r a n s f o r m a t i o n f o l l o w e d by the s i m i l a r i t y t r a n s f o r m a t i o n H ( b ) = s - l H ( a ) s (5.13) I f t h e element c o r r e s p o n d s t o |n|=3/2 and the H ^ ) element t o F 3 o r F^, the e i g e n v e c t o r m a t r i x £ , g i v e n by c -s" s c (5.14) has elements f o r a £~ s t a t e as f o l l o w s : - •126- e l e v e l s ( F , and F , ) : c = h{3{J-h, s = - ^ [ ( J + l j / J j * (5.15) f l e v e l s ( F 2 and F 4 ) : % [ ( 0 - 1 ) / ( J + 1 ) ^ - ^ [ 3 ( J+|)/( J+DJ5 The two t h i r d - o r d e r terms have q u i t e s i m i l a r e f f e c t s on the energy l e v e l s t r u c t u r e . T a b l e 5.1 g i v e s t h e a l g e b r a i c forms o f t h e m a t r i x e l e - ments we have u s e d . I t can be seen t h a t when s i m i l a r powers o f N a r e c a n c e l l e d , t h e s p i n - r o t a t i o n and i s o t r o p i c h y p e r f i n e e n e r g i e s f o l l o w t h e same t y p e o f e x p r e s s i o n : F, and F 4 (J=N±f): E $ r + E ^ * ±§N(Y-* S) ±§C(bJ,b s)/2N - Q q (5.16) F 2 and F 3 (J=N±^): E $ r + E . S Q = ± 3 2N( y+|Y s) ± C ( b + | b s ) / 2 N where C = F(F+1) - J(J+1) - I(I+D (5.17) The r e s u l t i s t h a t the F-| and F 4 l e v e l s have d i f f e r e n t e f f e c t i v e y and b parameters from the F 2 and F^ l e v e l s . The e n e r g y l e v e l c a l c u l a t i o n s based on the m a t r i x elements o f T a b l e 5.1 r e q u i r e t h a t two 16 x 16 m a t r i c e s be s e t up and d i a g o n a l i z e d f o r each F v a l u e . One o f t h e s e m a t r i c e s has b a s i s f u n c t i o n s e x t e n d i n g from N=F+5 t o N=F-5 i n s t e p s o f 2, and the o t h e r has b a s i s f u n c t i o n s from N=F+4 t o N=F-4, a l s o i n s t e p s o f 2. U n f o r t u n a t e l y t h e s t r u c t u r e s o f t h e two m a t r i c e s a r e not t h e same, and t h e r e a r e problems w i t h m i s s i n g l e v e l s a t low F v a l u e s : t h e complete 16x16 m a t r i c e s f i r s t a p p e a r f o r F=5. We have r e d u c e d t h e g e n e r a l forms o f th e m a t r i x elements -127- Table 5.1 Matrix elements of the spin and hyperfine Hamiltonian for a Z state i n a case (b ) basis. Diagonal elenents F l (>,»3) : + [ - ^ K C S f D ^ X K + c , * + jctb + ) - ^ g j ^ ^ ] /(2,+3) F 2 ( J = N + | ) : ir ( N-3) + [-|YSN(3N+5) + 2X(N+3) ]/(2N+3) + [ClC-;2K(K+l) + (N-3)} + -|<:{b(2M-9) + 3bg(3N+2+ ) + c( JL^- + 7)} . e2QSX(^3)(2N-3) ] / [ ( 2 N + 1 ) ( 2 N f 3 ) ] 21(21-1) .NQN+3) F3(J=S-|): + [-|ys(N+l)(3N-2) + 2A(N-2) ]/(2N-1) + [c C{2N(N+1)-(N+A) } - -|c{b(2N-7) + 3bg(3N+l+ ) + c( - 7)} . e% X(N-2)(2N+S) ] / [ ( 2 s _ 1 ) ( 2 W f a ) ] 21(21-1) .(N+1)(2N-1) F 4(>K-|): - yy(N+l) + [yr£N(N+l) - 2X(N+1) + CjCCN+l) - |c{b c+b (N+1) 2 t 2K-1 2 r e Qq>:CN--̂> ]/(2s-i) 21(21-1) .N(2N-1) X - TC(C+1)-1(1+1) J ( J+1) , C - ? ( F+1 ) - J ( J+1 ) - I ( I+1), and b « a -c/3 A u 2 : The rotational Hamiltonian has only diagonal elements, equal to BN(N+1)-DN (N+1)' <N-2 J F | H ! N J F > - j [3 (J-i ) (J+|)] > 5/(2N-l)} [2X + { 5( J + | ) (J-N+l)-2} Y g + i c C + 3e^2s_X + i c {2(5N+3) ( J - N + l ) - | > } /( 2 J ( J+1) ) ] \ 1(21-1) (2J-1M2J+3) 2S 2 J where X and C have been defined in the diagonal elements -128- Table 5.1 Continued. <K J - l F|HiKJT>- [(F+J+I+D (J-F+I) (F+J-I) (F-J+I+l) 3̂  [R(«/AJ] [b-Cj + b g Q ( N ) ( n ( n + 1 H J 2 _ 1 9 / M I 2 { F ( F 4 1 ) - I ( I + 1 ) - ( J - 1 ) ( ^ 1 ) 1 I 3 + ( 2 N - l ) ( 2 N + 3 ) " 4 1 ( 2 1 - 1 ) ( J - D (J+D / where R(N) and Q(K) a r e g i v e n by J=N+§: R(K)- [3N/(N+1)]N Q ( » - - C ^ ) / ( 4 N + 6 ) i s • [ ( 2 N - l ) ( 2 N + 3 ) / ( N ( . + l ) } ] i i [ 1 2 K ( » » -3]/[2( 2K-1) ( 2 M 3 ) 1 J=N+2 = 1- [ 3 ( N + 1 ) / N ] l i - ( 4 N + l ) / ( 4 N - 2 ) <K-2 J+1 F IH j K JF>" - | [ ( F + N + I + | ) ( N - F + I - | ) ( F + N - 4 ) ( F - N + 1 + | ) / { N ( N - 1 ) ) ] H <>^2 J - l F ,K jNJF>= 1 ( 2 1 - 1 ) ( 2 N - 3 M 2 N + 1 ) [ ( F + J + I + D ( J - F + I ) ( F + J - D ( F - J + I 4 1 ) ] ^ [ w ( K ) / { 8 J ( 2 N - l ) } ] « r , 2 . J F ( F + 1 ) - I ( I + 1 ) - ( J - D ( J + 1 ) ^ _ b A d I ] l c + 3 e Q<5 4 1 ( 2 1 - 1 ) ( J - D ( J + D s 2 S J > N _ 2 where 6 1 i s t h e K r o n e c k e r d e l t a , and W(N) i s g i v e n by J » « — J«N+i: W(N)« [ 3 ( K - 1 ) ( 2 N - l ) ( 2 N + 3 ) / N ] ! s > N - | : 4 [ ( W - l ) ( N - 2 ) ] ' 1 J - N - | : [ 3 N ( 2 N - l ) ( 2 N - 5 ) / ( N - l ) ] l i -129- t o a l g e b r a i c e x p r e s s i o n s , r a t h e r than programming t h e computer to c a l l s u b r o u t i n e s f o r t h e Wigner a n g u l a r momentum c o u p l i n g c o e f f i c i e n t s , i n o r d e r to save computing t i m e ; immense amounts o f c a n c e l l i n g o c c u r i n the c a l c u l a t i o n o f t h e a l g e b r a i c e x p r e s s i o n s from t h e Wigner c o e f f i - c i e n t s , and even w i t h o u t t h i s i t t a k e s 20 seconds o f CPU time on the U n i v e r s i t y ' s Amdahl 470 V/8 computer f o r one l e a s t s q u a r e s i t e r a t i o n to N=25 u s i n g t h e a l g e b r a i c e x p r e s s i o n s . D. A n a l y s i s o f the spectrum ( i ) General d e s c r i p t i o n o f the band 4 - 4 - The (0,0) band o f t h e C I - X T sy s t e m o f V0 i s q u i t e s t r o n g l y r e d - d e g r a d e d , and has two w e l l - m a r k e d heads a t 17426.4 cm - 1 (R-j and R^) and 17424.2 cm - 1 ( R 2 and R 3 ) . The s p e c t r u m i s v e r y crowded down t o abou t 17400 c m - 1 , w i t h t y p i c a l l i n e d e n s i t i e s o f the o r d e r o f 50 per wavenumber. To the r e d o f t h i s t h e band opens out and the r o t a t i o n a l l i n e s become w e l l s e p a r a t e d ; t h e e i g h t h y p e r f i n e components o f each 51 l i n e r e s u l t i n g from t h e 1=7/2 s p i n o f t h e V n u c l e u s a r e c l e a r l y r e - s o l v e d , and t h e l i n e s can r e a d i l y be a s s i g n e d t o t h e i r r e s p e c t i v e e l e c - t r o n s p i n components by t h e i r d i s t i n c t i v e h y p e r f i n e p a t t e r n s . Some t y p i c a l h y p e r f i n e p a t t e r n s a r e i l l u s t r a t e d i n F i g . 5.3. The h y p e r f i n e p a t t e r n s o f the F-j and F^ branches a r e t h r e e t i m e s as wide as t h o s e o f the ?2 a n c i F3 b r a n c h e s , as can be u n d e r s t o o d from t h e d i a g o n a l m a t r i x elements o f t h e h y p e r f i n e H a m i l t o n i a n g i v e n i n T a b l e 5.1. There i s no problem w i t h t h e r o t a t i o n a l a s s i g n m e n t s i n t h e t a i l o f th e band because t h e c o n s t a n t s g i v e n by Barrow (34) u s u a l l y r e p r o d u c e -130- the l i n e p o s i t i o n s to w i t h i n 0.2 cm" 1. We have r e c o r d e d t h e band o u t t o 17288 cm" 1 ( t h e P(41) g r o u p ) , where t h e branches have n e a r l y d i e d o u t . A l l f o u r e l e c t r o n s p i n components s u f f e r from r o t a t i o n a l p e r t u r b a - t i o n s by o t h e r e l e c t r o n i c s t a t e s , which m a i n l y a p p e a r as d i s c o n t i n u i t i e s i n t h e branch s t r u c t u r e ; we have found e x t r a l i n e s i n o n l y two o f the p e r t u r b a t i o n s , though L a g e r q v i s t and S e l i n (8) have i d e n t i f i e d e x t r a l i n e s i n v a r i o u s h i g h e r N p e r t u r b a t i o n s i n t h e i r a r c s p e c t r a . The a s s i g n m e n t o f t h e h y p e r f i n e F quantum numbers i s e a s y f o r the F-| and F^ e l e c t r o n s p i n components because t h e h y p e r f i n e s t r u c t u r e f o l l o w s the Lands i n t e r v a l - t y p e p a t t e r n d e s c r i b e d by eq. (5.17) and t h e d i a g b n a l e l e m e n t s o f T a b l e 5.1: the h y p e r f i n e p a t t e r n s open out a t the h i g h F s i d e , and the h i g h e r F l i n e s have g r e a t e r i n t e n s i t y . For t h e F 2 and F 3 branches t h e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s cause the p a t t e r n s to be i r r e g u l a r o v e r t h e complete range o f N v a l u e s we have s t u d i e d , even though t h e maxima i n the i n t e r n a l p e r t u r b a t i o n s a r e a t N'=5 and 15. I t can be seen i n F i g . 5.3 how the most i n t e n s e h y p e r f i n e l i n e s (which have t h e h i g h e s t F v a l u e s ) a r e c l u s t e r e d t o g e t h e r , i n con- t r a s t to t h e p a t t e r n s f o r the F-j and F^ b r a n c h e s . The r e a s o n i s t h a t t h e F 2 and F 3 s p i n components o f t h e ground s t a t e a r e o n l y 0.5 cm - 1 a p a r t a t N=30, and t h e h y p e r f i n e m a t r i x elements between them, which a r e F-dependent and o f t h e o r d e r o f 0.1 cm" 1, a r e a b l e to r e v e r s e t h e Lande - p a t t e r n s . ( i i ) I n t e r n a l h y p e r f i n e p e r t u r b a t i o n s As t h e F 2 and F 3 branches a r e f o l l o w e d t o lower N v a l u e s e x t r a l i n e s i n d u c e d by t h e h y p e r f i n e p e r t u r b a t i o n s s t a r t t o appear a t N=21. -131- Fig. 5.3 Hyperfine s t r u c t u r e s of l i n e s from the four e l e c t r o n s p i n components of the VO c V - x V (0,0) band. -132- These e x t r a l i n e s , though, n o t r e s o l v e d i n t o i n d i v i d u a l h y p e r f i n e compo- n e n t s , had been o b s e r v e d by R i c h a r d s and Barrow (9) and H o c k i n g , M e r e r and M i l t o n (.12). The p a t t e r n s o f h y p e r f i n e l i n e s become v e r y c o m p l i c a - t e d because t h e F o r d e r o f the h y p e r f i n e components i n y e r t s a t an i n t e r - n a l h y p e r f i n e p e r t u r b a t i o n , p r o d u c i n g a k i n d o f band-head i n t h e hyper- f i n e s t r u c t u r e f o r b o t h t h e main l i n e s and the e x t r a l i n e s . The p e r t u r - b a t i o n - i n d u c e d e x t r a l i n e s can be seen f o r a b o u t f i v e N v a l u e s on each s i d e o f the maximum, so t h a t t h e e f f e c t s o f t h e upper and lo w e r s t a t e i n t e r n a l p e r t u r b a t i o n s run i n t o each o t h e r and produce e x t r a l i n e s o v e r t h e complete r a n g e N=4-21. These l i n e s have proved t o be v e r y v a l u a b l e i n d e t e r m i n i n g t h e s p i n and h y p e r f i n e c o n s t a n t s , as w i l l be d e s c r i b e d i n S e c t i o n F. The i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s a r e b e s t u n d e r s t o o d from p l o t s o f t h e energy l e v e l s a g a i n s t N. F i g . 5.4 shows t h e q u a r t e t e l e c t r o n s p i n s t r u c t u r e o f t h e ground s t a t e w i t h t h e h y p e r f i n e e f f e c t s o m i t t e d . The F-| and F^ l e v e l s c r o s s near N=10, b u t t h e i r J v a l u e s d i f f e r by 3 u n i t s , so t h a t t h e y do not i n t e r a c t . The F 2 and F 3 l e v e l s c r o s s i n z e r o o r d e r n e a r N=15, but must a v o i d each o t h e r because o f the AF=AN=0, AJ=±1 m a t r i x elements o f t h e h y p e r f i n e H a m i l t o n i a n . The a v o i d e d c r o s s i n g i s shown m a g n i f i e d i n F i g . 5.5 w i t h t h e h y p e r f i n e s t r u c t u r e drawn i n . Only seven o f the e i g h t h y p e r f i n e components o f each l e v e l a c t u a l l y a v o i d each o t h e r . The r e a s o n i s t h a t t h e range o f F v a l u e s i s d i f f e r e n t i n the two l e v e l s ; F 2(J=N+ 32) has F=N-3 t o N+4 w h i l e F 3(J=N-*i) has F=N-4 t o N+3. H y p e r f i n e components w i t h F=N-3 t o N+3 o c c u r i n both e l e c t r o n s p i n l e v e l s , and t h e r e f o r e p e r t u r b each o t h e r , b u t the F=N+4 component o f F 2 and the F=N-4 component o f F 3 pass through t h e a v o i d e d VO X 4 I spin splitting pin f i n e structure of the VO X*E v = 0 l e v e l plotted as a t a t i o n a l and hyperfine structures are not shown. N - 4 N + 3 CO I Fig. 5.5 C a l c u l a t e d hyperfine energy l e v e l patterns f o r the F 2 and F 3 e l e c t r o n s p i n components of the X^Z~ v = 0 s t a t e of VO i n the range N" = 9 - 22. The c a l c u l a t i o n s are from the i i n a l l e a s t squares f i t to the ground s t a t e h y p e r f i n e s t r u c t r u e s , and l e v e l s w i t h the same values of F"-N" are connected. -135- c r o s s i n g r e g i o n u n a f f e c t e d . They g i v e r i s e t o t h e i s o l a t e d s t r o n g l i n e s i n t h e c e n t r e s o f some o f t h e h y p e r f i n e p a t t e r n s which a r e v e r y c h a r a c t e r i s t i c , as can be seen i n F i g . 5.6. F i g u r e 5.6 shows t h e P3 l i n e s f o r N " = 15-18. T h i s i s a l m o s t the o n l y r e g i o n where t h e p e r t u r b e d l i n e s a r e n o t o v e r l a p p e d by o t h e r b r a n c h s t r u c t u r e . The P3O8) l i n e has most o f i t s i n t e n s i t y i n t h e s h o r t wave- l e n g t h components ( l e f t hand s i d e ) , which a r e the z e r o o r d e r P^ t r a n s i - t i o n s . The i n t e n s i t y t r a n s f e r r e d t o the i n d u c e d l i n e s (on t h e r i g h t ) depends on two f a c t o r s , t h e s e p a r a t i o n o f t h e z e r o - o r d e r h y p e r f i n e com- ponents , and t h e v a l u e o f F. F o r P 3 ( 1 8 ) t h e two f a c t o r s a p p r o x i m a t e l y b a l a n c e f o r F=15-18, b u t the h i g h e r F components, which a r e s t a r t i n g t o form a h y p e r f i n e 'head', a r e much weaker. The c e n t r a l u n p e r t u r b e d F=14 l i n e (F=N-4) i s v e r y d i s t i n c t . W i t h d e c r e a s i n g N t h e i n t e n s i t y o f t h e P3 l i n e s i s p r o g r e s s i v e l y t r a n s f e r r e d to t h e l o n g w a v e l e n g t h components, and t h e 'heads' i n the h y p e r f i n e s t r u c t u r e become v e r y pronounced. As m ight be e x p e c t e d from F i g . 5.5 the e f f e c t s pass t h r o u g h a maximum a t N"=15. An i n t e r e s t i n g e f f e c t o f t h e r e v e r s a l o f t h e h y p e r f i n e energy o r d e r a t the p e r t u r b a t i o n i s t h a t the d i f f e r e n t h y p e r f i n e ' b r a n c h e s ' ( w i t h the same v a l u e o f F-N) have t h e i r minimum s e p a r a t i o n and most n e a r l y equal i n t e n s i t i e s a t d i f f e r e n t N v a l u e s : f o r i n s t a n c e the F=N+3 h y p e r f i n e components have minimum s e p a r a t i o n a t N=14, b u t t h e F=N-3 components have minimum s e p a - r a t i o n a t N=17. T h i s c a u s e d us some d i f f i c u l t y i n t h e e a r l y s t a g e s o f the l e a s t s q u a r e s f i t t i n g , b ecause we needed t o e s t a b l i s h the e x a c t p a r e n t a g e o f a h y p e r f i n e component i n o r d e r t o match i t w i t h an e i g e n - v a l u e from the d i a g o n a l i z a t i o n . -136- hternal hyperfine perturbations in VO, X4Z O 01 0-2 cm-' Fig. 5.6 The P 3 branch l i n e s of the C 4 I ~ - X 4 Z " (0,0) band in the region n" = 15-18, showing the hyperfine patterns near the ground state i n t e r n a l hyperfine perturbation. The F M quantum numbers for the hyperfine components are marked. -137- Below N"=15 t h e p a t t e r n s a r e u n f o r t u n a t e l y b l e n d e d because t h e upper s t a t e s p i n s p l i t t i n g s a r e s m a l l e r than t h e ground s t a t e p e r t u r - b a t i o n d o u b l i n g s ; a l s o o t h e r branches i n t e r f e r e . F i g u r e 5.7 shows t h e ?2 and P3 l i n e s f o r N " = l l - 1 4 . At t h i s s t a g e t h e N v a l u e s a r e low enough f o r t h e energy ' s p r e a d ' o f t h e h y p e r f i n e s t r u c t u r e o f the F 2 " ( N ) and F 3 " ( N ) l e v e l s t o be n o t i c e a b l y d i f f e r e n t , as can be seen i n F i g . 5.5. T h i s d i f f e r e n c e governs many o f t h e f e a t u r e s o f t h e low N h y p e r f i n e p a t t e r n s , and r e s u l t s from t h e f a c t o r s (2N+9) and (2N- 7 ) , r e s p e c t i v e l y , i n t h e d i a g o n a l elements o f b l . S f o r t h e F 2 and F3 components. The r e s u l t i n t h e spectrum i s t h a t the F 2 branches a r e v e r y open w h i l e t h e F3 branches b e g i n to c o l l a p s e i n t o s h a r p s p i k e s where t h e h y p e r f i n e s t r u c t u r e i s o f t e n n o t f u l l y r e s o l v e d . The f a c t o r (2N-7) f o r t h e F3 l e v e l s i n f a c t causes t h e h y p e r f i n e energy o r d e r t o i n v e r t between N=3 and 4. The upper s t a t e has a s i m i l a r h y p e r f i n e p e r t u r b a t i o n c e n t r e d n e a r N'=5. The en e r g y l e v e l p a t t e r n i s shown i n F i g . 5.8. P a r t s o f t h i s p a t t e r n a r e anomalous because t h e N v a l u e s a r e so low t h a t the f u l l complement o f e i g h t h y p e r f i n e components i s not p r e s e n t . A l s o t h e i n - v e r s i o n o f t h e h y p e r f i n e energy o r d e r f o r t h e lower s e t o f i n t e r a c t i n g components does not o c c u r : t h e re a s o n i s t h a t the i n v e r s i o n i n the F3 components between N=3 and 4 c a n c e l s t h e i n v e r s i o n caused by t h e f a c t t h a t t h e y t u r n i n t o F 2 l e v e l s a t the p e r t u r b a t i o n . The o n l y seeming i r r e g u l a r i t y i n t h e lower s e t i s t h a t t h e F=N+3 components l i e above t h e F=N+2 f o r N'=4 and 5. The P 2 and P3 branches i n t h e r e g i o n N"=5-8 a r e shown i n F i g . 5.9. The P 2 l i n e s a r e more than 0.5 cm" 1 t o t h e b l u e o f the P3 l i n e s w i t h t h e -138- 0.1 cm-' Fig. 5.7 The P 2 and P 3 branch l i n e s of the V0 C*T, - X*Z (0,0) band i n the range N"= 11-14. Numbers above the spectra arc F" values f o r the hype r f i n e components of the P 2 and P^ l i n e s ; other l i n e s belonging to overlapping branches are i n d i c a t e d below the sp e c t r a . -139- Fig. 5 . 8 C a l c u l a t e d hyperfine energy l e v e l p a t t e r n s f o r the a n c* ^3 4 - e l e c t r o n s p i n components of the C £ v=0 s t a t e of VO i n the reg i o n of the i n t e r n a l h y p e r f i n e p e r t u r b a t i o n (N' = 2-13). L e v e l s w i t h the same values of F'-N1 are connected. P , ( 8 ) P 3 ( 8 ) R 4 ( 2 9 ) R»(7) .::~»..-:.:.?.B:»- D(7) 2 9 2 8 2 7 ~ <™-\2t a 30 rlv'' 2D 263 24 23 2  3 _ ,oc-i 25 24 23 2 RJilO) R4(27) ^ 4 23 5 24 23 2  21-̂ 26) R,f24) P4(6) o i Fig. 5.9 The P 2 and P 3 branch l i n e s of the VO c V - x V (0,0) band i n the re g i o n N" = 5-8 ; the F" quantum numbers of the hyperfine components are marked above the spe c t r a . Overlapping high-N R l i n e s and low-N P, and P, l i n e s are i n d i c a t e d below the spectra. A l l four t r a c i n g s are to the same s c a l e . 1 4 -141 - same W" v a l u e , r e f l e c t i n g the l a r g e s p i n s p l i t t i n g i n the ground s t a t e , and t h e i r appearance i s a l t o g e t h e r d i f f e r e n t because o f t h e d i f f e r e n t o v e r a l l h y p e r f i n e e n e r g y 'spread' d e s c r i b e d above. The P 3 l i n e s a r e o n l y p a r t i a l l y r e s o l v e d even a t o u r r e s o l u t i o n o f 100 MHz, and i t i s f o r t u n a t e t h a t t h e i n d u c e d l i n e s i n t h e P 2 branches a r e so w e l l r e s o l v e d , o t h e r w i s e i t would n o t be p o s s i b l e t o f o l l o w t h e upper s t a t e h y p e r f i n e energy p a t t e r n . The h y p e r f i n e a s s i g n m e n t s a r e v e r y d i f f i c u l t to make i n t h i s r e g i o n , because t h e l i n e p o s i t i o n s depend c r i t i c a l l y on the s p i n and h y p e r f i n e c o n s t a n t s o f both s t a t e s ; t h i s was i n f a c t the l a s t r e g i o n o f t h e band t o be a s s i g n e d . ( i i i ) The band c e n t r e The c e n t r e o f t h e band c o n t a i n s R l i n e s w i t h N"=15-20 ( c o r r e s p o n d - i n g to the ground s t a t e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n ) t o g e t h e r w i t h the v e r y low N l i n e s . The R l i n e s c o n f i r m the h y p e r f i n e p a t t e r n s g i v e n by t h e P l i n e s , b u t b l e n d i n g l i m i t s t h e i r u s e f u l n e s s . The low N l i n e s on the o t h e r hand a r e v e r y i n t e r e s t i n g because t h e y c a r r y most o f the i n f o r m a t i o n about t h e d i p o l a r I,S i n t e r a c t i o n and t h e qu a d r u p o l e con- s t a n t s . O f t e n they a r e q u i t e d i f f i c u l t t o a s s i g n because t h e h y p e r f i n e p a t t e r n s a r e f r a g m e n t a r y , and d e t a i l e d c a l c u l a t i o n s o f t h e energy l e v e l s a r e needed. T y p i c a l p a t t e r n s a r e shown i n F i g . 5.10. The upper t r a c i n g , which c o v e r s t h e r e g i o n j u s t t o t h e b l u e o f the band o r i g i n , shows the P 2 ( l ) , R 3 ( 2 ) and Qefik) l i n e s , s u perimposed on t h e p e r t u r b e d R 2 ( 1 7 ) and R 3 ( 1 7 ) l i n e s . The l i n e s t r e n g t h s i n o u r s p e c t r a a r e such t h a t h y p e r f i n e compo- nents w i t h F"-S2 a r e u s u a l l y not s e e n . However i n t h e range F"=3-7 we 20 19 V P,(4) R,(22) P2(3) 1 t i Fig. 5.10 Two regions of the VO C^E ~ - X^ E - (0,0) band near the band o r i g i n . Low-N l i n e s are marked i n roman type with hyperfine quantum numbers i n d i c a t e d as F'-F"; high-N l i n e s are marked i n i t a l i c type with only the F" quantum numbers of the hy p e r f i n e components i n d i c a t e d . Cross-over s i g n a l s (centre dips) are marked 'cd' . The two t r a c i n g s are at the same s c a l e . -143- f r e q u e n t l y o b s e r v e l i n e s w i t h AF/AJ, and, where l i n e s w i t h a common lower l e v e l l i e w i t h i n t h e same Dop p l e r p r o f i l e , we o b s e r v e c e n t r e d i p s . These two e f f e c t s a r e well-known i n s u b - D o p p l e r s p e c t r o s c o p y , p a r t i c u - l a r l y f o r 1 2 ( 3 5 ) , and r e q u i r e no f u r t h e r e x p l a n a t i o n . The advantage o f the a d d i t i o n a l l i n e s t h a t a r i s e i s t h a t they g i v e d i r e c t h y p e r f i n e com- b i n a t i o n d i f f e r e n c e s , which break t h e c o r r e l a t i o n between the upper and lower s t a t e h y p e r f i n e c o n s t a n t s r e s u l t i n g from t h e p a r a l l e l s e l e c t i o n r u l e s o f the e l e c t r o n i c t r a n s i t i o n . For example, i n t h e l i n e Q e ^(%) p ( o r Q-|2(0), t o g i v e i t i t s c a s e (b) d e s i g n a t i o n ) we o b s e r v e a l l f o u r o f the p o s s i b l e h y p e r f i n e components, and t h e r e f o r e o b t a i n d i r e c t l y t he s e p a r a t i o n s o f the F=3 and F=4 components o f the two combining J=% l e v e l s . An energy l e v e l diagram i l l u s t r a t i n g t h i s i s g i v e n i n F i g . 5.11. The l o w e r t r a c i n g o f F i g . 5.10 shows the P-j (4) and ?2(3) l i n e s , a g a i n s t the background o f R 3 ( 2 1 ) and R.j(22). The P-|(4) l i n e has par- t i c u l a r l y c l e a r AF=AJ h y p e r f i n e components, and a l s o c e n t r e d i p s between them and the c a s e (b) a l l o w e d AF=AJ=-1 components. An i n t e r e s t i n g c e n t r e d i p i n v o l v e s t h e s t r o n g F'=7-F"=8 component and the unobserved F'=8-F"=8 component; t h i s c e n t r e d i p i s q u i t e weak, because the s t r e n g t h o f a c e n t r e d i p i s p r o p o r t i o n a l t o the s q u a r e r o o t o f t h e p r o d u c t o f t h e s t r e n g t h s o f t h e two c o n t r i b u t i n g t r a n s i t i o n s ( 3 5 ) . A l t o g e t h e r about f o r t y AF^AJ h y p e r f i n e components haye been i d e n t i - f i e d i n t h e low N l i n e s . We had not a n t i c i p a t e d them i n our o r i g i n a l l e a s t s q u a r e s programme f o r f i t t i n g t h e o b s e r v e d t r a n s i t i o n s , and had to i n c l u d e them as s p e c i a l c a s e s . S i m i l a r l y we had not a n t i c i p a t e d t h a t the Q branches would be so c o m p a r a t i v e l y s t r o n g . Twelve h y p e r f i n e com- ponents b e l o n g i n g t o f o u r Q l i n e s have been a s s i g n e d ; the o b s e r v e d Q -144- F,(-1) F2(0) F 3 o O CD CO CD O CP d CD O LO LO CO LO LO 00 CO LO CT) ro • O LO LO cr> ro E exile / c m _ 1 - 17419.6774 17419.6165 -0.8193 -0.9341 Fig. 5.11 Energy l e v e l diagram i n d i c a t i n g the assignment of the four hyperfine components of the l i n e Q e f ( ^ ) - -145- l i n e s a r e Qê (%) and Q êC%)> which form the A - d o u b l i n g components o f 4 - 4 - the Q{J=h) l i n e o f the EX - ET sub-band i n a case (a) d e s c r i p t i o n , 4 - 4 - and t he c o r r e s p o n d i n g f i r s t Q l i n e s o f the E ^ 2 ~ ^3/2 s u ' : ) - l : ) a n d , Q e f ( | ) arid Q f e ( | ) . A t t h e s e low J v a l u e s the case (b) d e s c r i p t i o n o f the l e v e l s and t r a n s i t i o n s breaks down, and some a p p a r e n t l y i m p o s s i b l e l i n e s a r i s e . p The Q(%) l i n e s a r e good examples: Q e f ( } s ) and Qfe(h) become Q-] 2(0) an<^ Q 2 - | ( - l ) , r e s p e c t i v e l y . We have k e p t t he case (b) n o t a t i o n f o r the main branches s i n c e they show no d i s c o n t i n u i t i e s when f o l l o w e d down from h i g h N. T h i s 4 breakdown i n n o t a t i o n f o r a E s t a t e i n f a c t o n l y happens when the Q=J component i n case (a) c o r r e s p o n d s to the F-| and F 2 l e v e l s , t h a t i s when t h e s p i n - s p i n parameter X i s g r e a t e r than t he r o t a t i o n a l c o n s t a n t B ( 3 3 ) ; both t h e C 4E" and X 4z" s t a t e s o f VO have X>B. The i n n e r band head formed by t h e R 2 and R 3 branches i s v e r y complex because i t c o n t a i n s many e x t r a l i n e s caused by t h e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s , t o g e t h e r w i t h o v e r l a p p i n g R-| and R^ l i n e s . N e v e r t h e l e s s a l l t h e f e a t u r e s have been a s s i g n e d w i t h t h e a i d o f com- p u t e r c a l c u l a t i o n s . The a s s i g n e d l i n e s o f the band a r e l i s t e d i n Appendix VI , T a b l e I . 4 _ E . E l e c t r o n i c p e r t u r b a t i o n s i n the C E s t a t e 4 _ N i n e e l e c t r o n i c p e r t u r b a t i o n s have been f o u n d i n the C z v=0 l e v e l . Seven o f t h e s e a r e shown i n F i g . 5.12, which i s a p l o t o f t h e energy l e v e l s a g a i n s t J ( J + 1 ) ; two o t h e r s , a t F 2 ( ^ 4 ) and F 3 ( 8 5 ) , which 3 b i 17800 17600 r 17400 0 H,2 = 0 50 cm-' 1000 >V 0.069 cm-' 2000 J ( J+1) 3000 i Fig. 5.12 R o t a t i o n a l energy l e v e l s of the C 4£ v = 0 s t a t e of V0 (with s c a l i n g as in d i c a t e d ) p l o t t e d against J(J+1). Dots i n d i c a t e r o t a t i o n a l p e r t u r b a t i o n s , and the p e r t u r b a t i o n matrix elements, 2 H 1 2, are given where they can be determined. The dashed l i n e i s probably a component of a II state (see te x t ) w i t h B e f f= 0.482 cm -1 -147- were d i s c o v e r e d by L a g e r q v i s t and S e l i n (8) l i e beyong t h e range o f the f i g u r e . The f i g u r e a l s o i n c l u d e s the p e r t u r b a t i o n m a t r i x elements where t h e y can be d e t e r m i n e d from t h e i n d u c e d e x t r a l i n e s . The o n l y r e g u l a r i t y we can r e c o g n i z e i s t h a t t h e p e r t u r b a t i o n s a t F^(26) and F 3 ( 4 5 ) a r e caused by the two A-components o f an o r b i t a l l y - d e g e n e r a t e s t a t e , which c o u l d p o s s i b l y be n, as we now show. ( i ) The F^C26) p e r t u r b a t i o n The p e r t u r b a t i o n a t F 4 ( 2 6 ) i s p a r t i c u l a r l y a n n o y i n g because t h e 4 - p e r t u r b i n g s t a t e has a l m o s t t h e same B v a l u e as C I ; i t s e f f e c t s t h e r e - f o r e do n o t d i e away r a p i d l y t o z e r o on e i t h e r s i d e o f the maximum o f th e a v o i d e d c r o s s i n g . I t can be shown t h a t the p e r t u r b a t i o n s h i f t a t the o r i g i n o f C 4E~ i s s t i l l 0.006 cm" 1, so t h a t s i n c e we can d e t e r m i n e t h e l i n e p o s i t i o n s t o b e t t e r than 0.0005 cm" 1 we must a l l o w f o r t h e e f f e c t s o f t h e p e r t u r b a t i o n o v e r t h e complete r a n g e o f F 4 l e v e l s . The d e t a i l s o f how t h i s was done a r e g i v e n i n S e c t i o n F. We can a s s i g n t h e p e r t u r b a t i o n s a t F^(26) and F 3 ( 4 5 ) t o the same p e r t u r b i n g s t a t e because i t i s p o s s i b l e t o i n t e r p r e t t h e F^(26) p e r t u r - b a t i o n i n d e t a i l from the l a s e r s p e c t r a . I g n o r i n g t h e h y p e r f i n e s t r u c - t u r e i n i t i a l l y , we f i t t e d the upper s t a t e term v a l u e s w i t h N'=24-28 ( i n c l u d i n g t h e s i n g l e o b s e r v e d ' e x t r a ' l e v e l ) to the e i g e n v a l u e s o f t h e H a m i l t o n i a n m a t r i x H = T c + B CN(N+1) '12 112 T ^ + B .N(N+1) p e r t p e r t (5.18) •148- I t was n e c e s s a r y t o f i x ( f o r t h e C 4z" s t a t e ) a t an e f f e c t i v e y a l u e f o r t h i s N range c a l c u l a t e d from p r e l i m i n a r y l e a s t s q u a r e s work, and to assume t h a t t h e p e r t u r b a t i o n m a t r i x element can be t r e a t e d as c o n s t a n t o v e r s u c h a s m a l l N r a n g e . The r e s u l t s a r e g i v e n i n T a b l e 5.2.The f i t i s good, and t h e p a r a m e t e r TQ comes o u t t o w i t h i n 0.1 cm - 1 o f the C s t a t e o r i g i n ; a l s o t h e p e r t u r b a t i o n m a t r i x e l e m e n t H-^ i s g i v e n t o an a c c u r a c y o f ± 0 . 0 0 5 ^ c m - 1 . Eq. (5.18) assumes t h a t t h e r o t a t i o n a l energy o f the p e r t u r b i n g s t a t e i s p r o p o r t i o n a l t o N(N+1); however the N range o f t h e f i t i s so s m a l l t h a t we can c o n v e r t the r e s u l t s , w i t h o u t l o s s o f a c c u r a c y , t o the c a s e where the energy o f the p e r t u r b i n g s t a t e i s l i n e a r i n J ( 0 + 1 ) . When t h i s i s done i t i s f o u n d t h a t B p g r t a g r e e s t o w i t h i n 0.0007 cm" 1 w i t h what i s o b t a i n e d i f t h e ^ ( 2 6 ) and F 3 ( 4 5 ) p e r t u r b a t i o n s a r e assumed to be c a u s e d by t h e same p e r t u r b i n g s t a t e . T h i s i s e x c e l l e n t agreement, and unambiguously proves a c o n n e c t i o n between t h e two p e r t u r b a t i o n s . Because the F^ and F 3 l e v e l s have d i f f e r e n t e / f symmetries the p e r t u r b - i n g s t a t e must have r o t a t i o n a l l e v e l s w i t h d o u b l e p a r i t y ( i . e . i t must be o r b i t a l l y - d e g e n e r a t e ) ; the two p e r t u r b a t i o n s a r e t h e r e f o r e c a u s e d by d i f f e r e n t A-components o f a p e r t u r b i n g o r b i t a l l y - d e g e n e r a t e s t a t e . 4 4 E x t r a p o l a t i o n o f t h e v i b r a t i o n a l s t r u c t u r e s o f the B n and A n s t a t e s (34) r u l e s them ou t as c a n d i d a t e s f o r the p e r t u r b i n g s t a t e . Of c o u r s e a t h i r d 4 n s t a t e c o u l d be r e s p o n s i b l e , though we see no e v i d e n c e f o r such a s t a t e i n o u r F o u r i e r t r a n s f o r m s p e c t r a , which extend down to 6000 cm" 1: the e m i s s i o n t r a n s i t i o n t o X 4 £ ~ would be s p i n and o r b i - t a l l y a l l o w e d . On t h e o t h e r hand the h y p e r f i n e s t r u c t u r e s u g g e s t s t h a t -149- Table 5.2 A n a l y s i s of the C Z , F A(26) p e r t u r b a t i o n . Upper s t a t e energy l e v e l s _Nj w i t h F = N-1 (cm - 1) Obs-calc (cm" 1) 24 17716.011 -0.003 25 17740.641 0.003 26 17766.039 17767.084 0.000 -0.000 27 17793.169 0.002 28 17820.725 -0.001 Least squares r e s u l t s : ( l o ) T = 17420.055 ± 0.018 cm"1 c B = 0.49336 ( f i x e d ) c T = 17447.313 + 0.020 pert B = 0.4550 ± 0.0003 pert H = 0.496 ± 0.005 5 -150- the p e r t u r b i n g s t a t e has o n l y moderate s p i n - o r b i t c o u p l i n g , so t h a t a p o s s i b l e c a n d i d a t e would be a n s t a t e from the same e l e c t r o n c o n f i g u r a - t i o n as A 4 n ( p r o b a b l y 4sa 13dS 14p-iT 1). The sum o f h y p e r f i n e e n e r g i e s o f the d o u b l e d F (̂26) l e v e l s i s found t o be l i n e a r i n F(F+1), so t h a t t h e p e r t u r b i n g s t a t e must f o l l o w c a s e (a„) o r (b ',) c o u p l i n g ; a l s o t h e s p a c i n g o f i t s h y p e r f i n e l e v e l s P 3d i s found t o be a l m o s t e x a c t l y the same as t h a t o f C ^ i " , F^(26). I f we w r i t e the h y p e r f i n e energy e x p r e s s i o n f o r a r o t a t i o n a l l e v e l (N ,J) as E h f s = T o + k F ( p + 1 ) ( 5 J 9 ) where k i s a f u n c t i o n o f N, J and the h y p e r f i n e c o n s t a n t s , t h e deper- t u r b e d v a l u e s a r e k ( C 4 z " , N=26, J=24i) = 0.000234 cm" 1, \ , (5.20) k ( p e r t u r b i n g , J=242) = 0.000205 cm Now case ( b n l ) s t a t e s have w i d e r h y p e r f i n e s p a c i n g s a t h i g h J than c a s e 3d ( a ) s t a t e s f o r the same h y p e r f i n e p a r a m e t e r s , as can be seen from t h e 3 d i a g o n a l elements o f b l .£: t h e case ( b g J ) e x p r e s s i o n <NASJIF IbKSJNASJIF> = -b[N( N+1)-S(S+1)- J( J+1)]£F( F+l) -1( 1+1)-J( J+1)] 4J(J+1) (5.21) has e s s e n t i a l l y an e x t r a f a c t o r o f J compared to the case (a^) e x p r e s s i o n (11,36) -151- <JftSEAlF|bI .S| JfiSEAlF> = bnzlF(F+l)-I(I+l)-J(J+lU — 2 J t W ) ( 5 ' 2 2 ) T h e r e f o r e t h e c o m p a r a t i v e l y l a r g e v a l u e o f k f o r the o r b i t a l l y - d e g e n e - r a t e p e r t u r b i n g s t a t e i n d i c a t e s a c o n s i d e r a b l e t e n d e n c y t o c a s e ( b ^ j ) c o u p l i n g , o r i n o t h e r words t h a t i t has c o m p a r a t i v e l y s m a l l s p i n - o r b i t c o u p l i n g . T h i s i s what we e x p e c t f o r a n s t a t e from t h e same c o n f i - g u r a t i o n as A 4 n (where A = 30 c m - 1 ) , b u t not what we e x p e c t f o r a 2 n 4 -1 s t a t e from t h e same c o n f i g u r a t i o n as B n (where A^70 cm ). I f t h e 2 p e r t u r b i n g s t a t e i s i n d e e d a component o f t h e n s t a t e c o r r e s p o n d i n g to A 4 n , t h e p o s i t i v e s i g n o f t h e h y p e r f i n e p a rameter k s u g g e s t s t h a t i t i s t h e F-j component, though i t s magnitude i s o n l y a q u a r t e r o f what we c a l c u l a t e f o r c a s e (b c o u p l i n g , i n d i c a t i n g t h a t t h e s p i n - u n c o u p l i n g i s o n l y q u i t e p a r t i a l 1 . 1 4 The argument runs as f o l l o w s . The h y p e r f i n e p a rameter b f o r A n i s 4 - -1 known to be v i r t u a l l y i d e n t i c a l to t h a t o f X E , namely 0 .027 3 cm . 2 4 The n s t a t e from t h e same c o n f i g u r a t i o n as A n s h o u l d , i n f i r s t a p p r o x i m a t i o n , have a b - v a l u e t h r e e times as l a r g e , because the i s o t r o p i c h y p e r f i n e o p e r a t o r i s s t r i c t l y E b. I . s . r a t h e r t h a n b i .S. T h e r e f o r e "' . , l ~ ~i ~ ~ I e l e c t r o n s from eq. (5.21) we c a l c u l a t e , f o r c a s e ( b ^ j ) c o u p l i n g , k ( 2 n , F - , , J=24%) = 0.00084 cm" 1 K ( 2 J T , F 2 , J=24*s) =-0.00080 cm" 1 F u r t h e r e v i d e n c e t h a t t h e s p i n - u n c o u p l i n g has not p r o g r e s s e d v e r y f a r comes from t h e s p i n - o r b i t m a t r i x elements g i v e n by Kovacs (37) : i n pure c a s e (b) c o u p l i n g 0 » so t h a t no p e r t u r b a t i o n would have been o b s e r v e d . -1 52- ( i i ) The F-j(37) p e r t u r b a t i o n The v e r y s m a l l p e r t u r b a t i o n a t F-j(37) has been r e p o r t e d a l r e a d y (.38) I t forms an i n s t a n c e where an a v o i d e d c r o s s i n g o c c u r s w i t h i n t h e hyper- f i n e s t r u c t u r e o f a s i n g l e r o t a t i o n a l l e v e l , and t h e a n a l y s i s can be c a r r i e d out by t r e a t i n g the h y p e r f i n e s t r u c t u r e as a f r a g m e n t o f r o t a - t i o n a l b ranch s t r u c t u r e . Two r e g i o n s o f t h e s p ectrum a r e shown i n F i g . 5.13. The l o w e r t r a c i n g i s t h e P-j(27) l i n e n e a r 17353 cm" 1, w hich i s u n p e r t u r b e d and shows t h e Lande-type p a t t e r n ; t h e l o w e r s t a t e F v a l u e s a r e g i v e n u n d e r n e a t h . The upper t r a c i n g shows two l i n e s , t h e P-j(38) and P 3 ( 3 8 ) l i n e s . The P-j(38) l i n e c o n s i s t s o f 13 components, r a t h e r than e i g h t , and t h e i n t e n s i t y p a t t e r n i s anomalous. The P 3 ( 3 8 ) l i n e has been i n c l u d e d t o g i v e the i n t e n s i t y s c a l e ; n e v e r t h e l e s s i t s h y p e r f i n e p a t t e r n i s found t o be i r r e g u l a r as w e l l , as a r e s u l t o f t h e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n i n the ground s t a t e d e s c r i b e d e a r l i e r . I n t e n s i t y c o n s i d e r a t i o n s a l l o w t h e F" quantum numbers t o be a s s i g n e d t o t h e components o f P ^ ( 3 8 ) , as g i v e n i n F i g . 5.3. I t i s e v i d e n t t h e r e has to be a p e r t u r b a t i o n w i t h i n t h e h y p e r f i n e s t r u c t u r e o f t h e F-j(37) r o t a t i o n a l l e v e l o f the upper s t a t e . The lower s t a t e r o t a t i o n a l - h y p e r f i n e e n e r g i e s can be c a l c u l a t e d from the r o t a t i o n a l c o n s t a n t s got by f i t t i n g the ground s t a t e c o m b i n a t i o n d i f f e r e n c e s The upper s t a t e term v a l u e s can then be o b t a i n e d by c o m bining t h e s e w i t h the l i n e p o s i t i o n s , as i n T a b l e 5.3. When the upper s t a t e energy l e v e l s a r e p l o t t e d a g a i n s t F ( F+1), t h e c l a s s i c p a t t e r n o f an a v o i d e d c r o s s i n g (41) emerges ( s e e F i g . 5.14): t h e r e a r e two s e t s o f energy l e v e l s which have minimum s e p a r a t i o n where the i n t e n s i t i e s o f t h e c o r r e s p o n d i n g l i n e s a r e e q u a l , and t h e a v e r a g e d energy l e v e l s a r e P, (38) F = 4 1 4 0 3 8 3 6 3 4 3 6 3 7 3 8 3 9 4 0 L 3 6 3 7 3 8 3 9 4 0 41 S P, (27) u u u u u 4 2 4 3 F= 2 5 2 6 2 7 2 8 2 9 3 0 31 3 2 on 00 I Fig. 5.13 Two regions of the intermodulated fluorescence spectrum of the C 4£ - X 4S (0,0) band of VO. Upper tracxn ing: the P_(38) and perturbed P ^ S ) l i n e s . Lower t r a c i n g : the unperturbed P 1(27) l i n e . 18111.55 h 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 F(F*1) Fig-5.14 Upper-state term values (cm" 1) of the h y p e r f i n e l e v e l s of the perturbed F ^ ) r o t a t i o n a l l e v e l of the c V , v=0 s t a t e of V0 p l o t t e d " against F(F+1) . -155- Table 5.3 A n a l y s i s of the C E , ¥^37) p e r t u r b a t i o n . P l (3B) lines 1^(36) lines T l '(37) levels 104(0-C) F' lower upper F^'OB) energv lower upper Fj-oe) energv lower upper lover upper 35 17 304, ,0117 4.1577 807.3564 17385.7700* 5.9101 725.6015 18111. ,3674 1.5129 1 2 36 3. ,9807 4.1195 7.3933 5. 7339 5.8716 5.63S3 1. .3731 1.5114 0 0 37 3. ,9̂ 51 4.0824 7.4314 5. ,6954 5.6360 5.6764 1, .3757 1.5131 -1 -3 38 3, .9044 4.0479 7.470E 5. .6587 5.8034 5.7158 1 .3749 1.5190 -1 0 39 3. .8601 4.0173 7.5115 5. .6131 5.7565 1 .3706 1.5231 _2 1 40 3 .6116 7.5534 5 .5654 5.7986 1 .3645 6 41 3 .7585 7.5968 5 .5116 5.6420 1 .3545 -3 42 3 .7031 7.6414 5 .4568 5.8E69 1 .3441 1 Values in cn" 1; * means blended line. Allowance has beer, made in the averaging for an absolute calibration shift of 0.0007 CE" 1 between the FjOS) and K^C36) lines. Least squares results: 18111.6888 i 0.0020 c r T 1 (lo) -0.000178 t 0.0000012 18111.1122 t 0.0032 0.000241 * 0.000002 0.0685 * 0.0001 Standard deviation • 0.00028 etc"1 C V : T ( 1 ) o k l pert : T ( 2 ) o K 2 H12 -156- l i n e a r i n F ( F + 1 ) . The p e r t u r b a t i o n m a t r i x element i s v e r y s m a l l , b u t a n a l y s i s i s n o n e t h e l e s s p o s s i b l e because the k parameters a r e v e r y d i f - f e r e n t f o r t h e two i n t e r a c t i n g l e v e l s . We have now d i s e n t a n g l e d t h e R-|(36) l i n e s from t h e s t r o n g o v e r - l a p p i n g P 2 ( 1 7 ) l i n e s and can r e f i n e the parameters r e p o r t e d p r e v i o u s l y ( 3 8 ) . The e f f e c t o f a v e r a g i n g t h e (36) amd P-j(38) d a t a has been to improve t h e s t a n d a r d d e v i a t i o n o f t h e l e a s t squares, f i t c o n s i d e r a b l y . The r e s u l t s a r e g i v e n i n T a b l e 5.3. The model used was a s i m p l e 2x2 ' m a t r i x f o r each F v a l u e , a k i n t o eq. ( 5 . 1 8 ) : H = T ^ + k 1 F ( F + l ) H 1 2 H 1 2 i f } + k 2 F ( F + l ) (5.23) Nothing was h e l d f i x e d i n t h e l e a s t sq u a r e s t r e a t m e n t , and the a c c u r a c y o f t h e model and the f i t must be a s s e s s e d by comparing the o b s e r v e d and c a l c u l a t e d k v a l u e s f o r C 4z" , F-j(37) : k ( c V , N=37, J = 3 8 ^ ) c a 1 c = -0.000185 cm" 1 (5.24) k ( C , V , N=37, J=3835)obs = -0.000178±0.00004 c m _ 1 ( 3 a ) 4 - The p e r t u r b i n g s t a t e has a lower B v a l u e t h a n C z , which means 4 - t h a t f u r t h e r p e r t u r b a t i o n s i n the o t h e r s p i n components o f C z might be e x p e c t e d a t lower N v a l u e s . We have n o t i d e n t i f i e d any such per- t u r b a t i o n s , and can u n f o r t u n a t e l y s ay n o t h i n g about the n a t u r e o f the p e r t u r b i n g s t a t e e x c e p t t h a t i t s h y p e r f i n e s p l i t t i n g i s l a r g e . An -157- T n t e r e s t i n g e f f e c t o f the p e r t u r b a t i o n i s t h a t t h e h y p e r f i n e s t r u c t u r e s o f t h e l e v e l s w i t h i n about f o u r u n i t s o f N on e i t h e r s i d e o f the a v o i d e d c r o s s i n g a r e n o t i c e a b l y i r r e g u l a r ; t h i s r e f l e c t s t h e f a c t t h a t t h e un- p e r t u r b e d l e v e l s e p a r a t i o n s show a s t r o n g dependence on F. A secon d s m a l l p e r t u r b a t i o n o c c u r s i n the F-| component a t N=36. The ?2 component i s a l s o p e r t u r b e d a t t h i s p o s i t i o n . U n f o r t u n a t e l y e x t r a l i n e s do not o c c u r and we can s a y n o t h i n g about t h e s e p e r t u r b a t i o n s e x c e p t t h a t t h e y appear t o be u n r e l a t e d t o each o t h e r o r t o t h e o t h e r p e r t u r b a t i o n s d e s c r i b e d . F. L e a s t s q u a r e s f i t t i n g o f the l i n e p o s i t i o n s Ht has been a f o r m i d a b l e problem a c h i e v i n g a l e a s t s q u a r e s f i t t o the o b s e r v e d d a t a t h a t r e f l e c t s t h e i r p r e c i s i o n a d e q u a t e l y . We d i s - c o v e r e d a t an e a r l y s t a g e t h a t a f u l l m a t r i x t r e a t m e n t o f the h y p e r f i n e s t r u c t u r e was r e q u i r e d , and the o n l y a p p r o x i m a t i o n we have made has been to o m i t the AJ=±2 elements o f the e l e c t r i c q u a d r u p o l e i n t e r a c t i o n ; t h e s e a r e the o n l y elements which do not add t o m a t r i x elements o f the ma g n e t i c i n t e r a c t i o n s , and i n any case a r e c a l c u l a t e d t o be v e r y s m a l l . We a l s o q u i c k l y found t h a t the a b s o l u t e c a l i b r a t i o n o f the i s o l a t e d l i n e s i n the t a i l o f t h e band was l e s s p r e c i s e than t h a t o f t h e crowded l i n e s i n the head o f t h e band where t h e o v e r l a p p i n g o f VO l i n e s between s u c c e s s i v e 1 c m - 1 s c a n s o f the l a s e r p e r m i t s s e v e r a l scans to be c a l i - b r a t e d a t once ( s e e S e c t i o n B ) . 4 - However, the main d i f f i c u l t y has been the f a c t t h a t the C E F^ l e v e l s a r e s h i f t e d by e l e c t r o n i c p e r t u r b a t i o n s t h r o u g h o u t t h e N range -158- 4 -o f our s p e c t r a ; a l s o we cannot t r u s t t h e o t h e r C E s p i n components not to have been s h i f t e d s i m i l a r l y a f t e r about N'= 25. When we at t e m p t e d t o f i t t h e raw d a t a we were u n a b l e t o o b t a i n a s a t i s f a c t o r y c onverged f i t u n l e s s we r e s t r i c t e d o u r s e l v e s t o N<20, and even then the upper s t a t e c e n t r i f u g a l d i s t o r t i o n parameter D was u n r e a l i s t i c a l l y low (>6.2xl0"^ cm" 1, compared t o t h e K r a t z e r r e l a t i o n v a l u e o f 6.62 x 10"^ c m " 1 ) . A f t e r some e x p e r i m e n t i n g w i t h h i g h e r o r d e r terms we r e a l i s e d t h a t i t would be n e c e s s a r y to a l l o w f o r the e f f e c t s o f t h e s t a t e c r o s s i n g t h e l e v e l s a t N=26, and not t o attempt t o f i t t h e upper s t a t e beyond N"=25; the ground s t a t e h^F" c o m b i n a t i o n d i f f e r e n c e s , a l t h o u g h l e s s p r e c i s e l y d e t e r m i n e d b e c a u s e o f c a l i b r a t i o n p r o b l e m s , c o u l d be i n c l u d e d f o r the f u l l r ange o f our d a t a , t o N=40. 4 - ( i ) D e p e r t u r b a t i o n o f t h e C z l e v e l p o s i t i o n s The ' d e p e r t u r b a t i o n ' p r o c e d u r e o b v i o u s l y depends c r i t i c a l l y on the n a t u r e o f the p e r t u r b i n g s t a t e (hence the d e t a i l e d d i s c u s s i o n i n the p r e v i o u s S e c t i o n ) . To summarize, we a r e c e r t a i n t h a t t h e p e r t u r b i n g s t a t e i s o r b i t a l l y - d e g e n e r a t e , has an e f f e c t i v e r o t a t i o n a l energy ex- p r e s s i o n E p e r t = 0 , 4 8 2 J ( J + 1 ) 0 1 1 - 1 (5.25) and t h a t t h e i n t e r a c t i o n m a t r i x element < J p e r t = 2 4 |H|C 4z~, N=26, J=24 > i s 0.496±0.006 cm" 1. I t seems u n l i k e l y t h a t A s t a t e s , o r s t a t e s w i t h 2 4 2S+1>4 w i l l be i m p o r t a n t , so t h a t we a r e r e s t r i c t e d to n and n s t a t e s . P r o v i s i o n a l l y we f a v o u r n because we m i g h t e x p e c t t o see a p e r t u r b i n g -159- 4 4 -n s t a t e d i r e c t l y i n e m i s s i o n t o the ground z s t a t e , and p a r t i c u l a r l y b e c a u s e o f arguments b a s e d on the h y p e r f i n e s t r u c t u r e . F o r t u n a t e l y , as can be seen from the t a b l e s o f Kovcics ( 3 7 ) , t h e m a t r i x elements between F 4 and any component o f ^n(a) o r ^n(a) a r e e s s e n t i a l l y i n d e p e n d e n t o f J , i n a s f a r as m a t r i x elements o f BL can be n e g l e c t e d compared to t h o s e o f E a ^ K ; t h e s i n g l e e x c e p t i o n i s A l - ^ ' ^ n 1 s 1 S r u^ e c' o u t ^y h y p e r f i n e arguments s i n c e the h y p e r f i n e s p l i t t i n g s o f t h e p e r t u r b i n g s t a t e a r e e i t h e r much t o o b i g o r o f t h e wrong s i g n . We can t h e r e f o r e t a k e the i n t e r a c t i o n m a t r i x e l e m e n t as b e i n g i n - dependent o f J , and can c a l c u l a t e t h e downward s h i f t s c a u sed i n the \~ F^ l e v e l s a c c o r d i n g t o eq. ( 5 . 1 8 ) , w i t h the z e r o o r d e r p e r t u r b i n g s t a t e energy w r i t t e n as i n e q . ( 5 . 2 5 ) . The s h i f t s a r e g i v e n i n T a b l e 5 . 4 , where t h e y a r e seen t o r i s e from 0.0060 c m - 1 a t N=3 t o 0.0300 cm - 1 a t N=22. For t h e f i n a l l e a s t s q u a r e s work we r a i s e d t he F 4 l e v e l s by t h e amount f r o m T a b l e 5 . 4 i n excess o f 0.0060 c m - 1 ; the q u a n t i t y 0.0060 cm" 1 4 - i s thus i n c o r p o r a t e d i n t o t he e f f e c t i v e A p arameter f o r C z . S i m i l a r c o r r e c t i o n s s h o u l d be needed i n t h e F3 l e v e l s (as F i g , 5.12 shows), b u t i t i s easy to prove t h a t t h e y a r e an o r d e r o f magnitude s m a l l e r . We have n o t i n c l u d e d them s p e c i f i c a l l y , so t h a t t h e y a r e taken up i n t h e e f f e c t i v e s p i n and c e n t r i f u g a l d i s t o r t i o n p a r a m e t e r s . The p r i n c i p a l j u s t i f i c a t i o n o f t h i s d e p e r t u r b a t i o n p r o c e d u r e i s t h a t i t works - i t removes the s y s t e m a t i c t r e n d s i n t h e l e a s t s q u a r e s r e s i d u a l s f o r t h e r o t a t i o n a l s t r u c t u r e , and i t makes the c e n t r i f u g a l d i s t o r t i o n parameters more r e a l i s t i c : f o r i n s t a n c e the upper s t a t e D v a l u e , d e t e r m i n e d from l e v e l s up t o N'=24 o n l y , r i s e s t o 6.44 x 1 0 " 7 -160- Table 5.4 C a l c u l a t e d p e r t u r b a t i o n s h i f t s i n the VO C E v=0 F 4 l e v e l N Av/cm 3 0.0060 4 0.0062 5 0.0065 6 0.0067 7 0.0071 8 0.0074 9 0.0078 N Av/cm 10 0.0082 11 0.0087 12 0.0093 13 0.0099 14 0.0107 15 0.0116 16 0.0126 N Av/cm 17 0.0139 18 0.0156 19 0.0177 20 0.0204 21 0.0243 22 0.0300 23 0.0393 -161- -1 2 -7 -1 cm , compared t o the K r a t z e r r e l a t i o n v a l u e o f 6.62 .x 10 cm We emphasize t h a t t h e ground s t a t e r o t a t i o n a l c o n s t a n t s a r e u n a f f e c t e d : t h e y a r e d e t e r m i n e d p r i n c i p a l l y from the A 2 F " measurements up t o N=40. S i m i l a r l y the h y p e r f i n e c o n s t a n t s o f b o t h s t a t e s a r e u n a f f e c t e d ; a l e a s t s q u a r e s f i t o f t h e u n c o r r e c t e d d a t a up to N=6 g i v e s e s s e n t i a l l y the same h y p e r f i n e c o n s t a n t s as the d e p e r t u r b e d d a t a up t o N=24, though the p r e c i s i o n o f the l a t t e r i s g r e a t e r . ( i i ) L e a s t s q u a r e s r e s u l t s The l e a s t s q u a r e s f i t t i n g was c a r r i e d out i n two s t e p s . In a f i r s t s t e p a l l t h e l i n e s t o N""=23 and the h i g h N A 2 F " ' S up to N=40 were f i t t e d s i m u l t a n e o u s l y . T h i s g i v e s a good d e t e r m i n a t i o n o f t h e s p i n and r o t a t i o n a l c o n s t a n t s , though t h e lower a c c u r a c y o f t h e h i g h N A 2 F " ' ' S a f f e c t s t h e s t a t i s t i c s f o r t h e h y p e r f i n e c o n s t a n t s . In a second s t e p the ground s t a t e r o t a t i o n a l c o n s t a n t s were h e l d f i x e d , and o n l y the more a c c u r a t e l y c a l i b r a t e d d a t a up t o N"=23 were f i t t e d . The s p i n and h y p e r f i n e c o n s t a n t s do not change, b ut t h e i r s t a n d a r d e r r o r s improve c o n s i d e r a b l y . Anomalous D y a l u e s i n a p p a r e n t l y u n p e r t u r b e d e x c i t e d s t a t e s o f t r a n s i - t i o n metal o x i d e s a r e not uncommon. For i n s t a n c e the A 6 z + v=l s t a t e o f MnO (24) has a D v a l u e t h r e e times h i g h e r than e x p e c t e d , and many o f the upper l e v e l s o f t h e 'orange system" o f FeO have u n u s u a l l y l a r g e D v a l u e s (A.S-C. Cheung, A.M. L y y r a and A . J . M e r e r , work i n p r o g r e s s ) . -162- The model used was t h e f u l l m a t r i x o f Table 5.1 i n each c a s e . The o n l y c o n s t r a i n t s a p p l i e d were t h a t Ap and b^ were s e t t o z e r o f o r t h e ground s t a t e , and t h a t t h e n u c l e a r s p i n - r o t a t i o n parameters Cj were f i x e d . I t was found t h a t c o n v e r g e n c e was v e r y slow w i t h t h e c T parame- t e r s f l o a t i n g , b u t t h a t t h e d i f f e r e n c e A C T = C J ' - C t " was w e l l d e t e r m i n e d and remained c o n s t a n t i n s u c c e s s i v e i t e r a t i o n s . A c c o r d i n g l y , s i n c e t h e a p p a r e n t n u c l e a r s p i n - r o t a t i o n i n t e r a c t i o n , c T I . N ^ a r i s e s p r i n c i p a l l y from s e c o n d - o r d e r s p i n - o r b i t e f f e c t s ( 1 8 ) , as does t h e e l e c t r o n s p i n - r o t a t i o n i n t e r a c t i o n y S ^ , we made the a r b i t r a r y c h o i c e cT(cV) _ Y(cV) {5 2 6 ) Cj(xV) YuV) E f f e c t i v e l y t h i s p o r t i o n s o u t t h e c o n t r i b u t i o n s t o ACj from t he two s t a t e s i n the r a t i o o f t h e two Y'S, on t h e as s u m p t i o n t h a t t h e s p i n - o r b i t terms a r e s i m i l a r . The r e s u l t s a r e g i v e n i n T a b l e 5.5. The e r r o r l i m i t s f o r the s p i n and r o t a t i o n a l c o n s t a n t s a r e 3a v a l u e s , t a k e n from t he f i r s t f i t , i n - c l u d i n g t h e A 2 F ' ' " s , where t h e o v e r a l l s t a n d a r d d e v i a t i o n , n o r m a l i z e d t o u n i t w e i g h t , i s 0.00092 cm" 1 (28 MHz). The e r r o r l i m i t s f o r the hyper- f i n e c o n s t a n t s a r e from t h e secon d f i t , u s i n g o n l y t h e 1363 l i n e s up to N'"=23, where t h e n o r m a l i z e d s t a n d a r d d e v i a t i o n i s 0.00076 cm" 1 (23 MHz). G. H y p e r f i n e parameters The e l e c t r o n s p i n r e s o n a n c e spectrum o f V0, xV, i n an argon m a t r i x a t 4 K has been measured by Kasa i ( 1 0 ) . He d e r i v e d v a l u e s f o r the i s o - Table 5.5 R o t a t i o n a l , spin 4 and hyperfine constants f o r the C 4 - Z and X Z s t a t e s of V0. c V '. v=0 X< 'z"r v=0 T 0 17420.1025 7 ± o. o o o i 7 0.0 B 0.493789 6 + 0.000003 3 0.546383 3 + 0.000002 9 10 7 D 6.44 ± 0.03 6.50 9 ± o . o i A Y -0.018444 ± 0.000069 0.022516 ± 0.000066 X 0.7469 ? ± 0.0003 2.0308 ? ± 0.0002^ l O 7 Y D 5.43 ± 0.50 0.56 ± 0.32 i o 6 x D -4.3 ± 0.5 0.0 f i x e d l O 5 T g -23.1 ± 1.4 -1.0 ± 1.5 b -0.00881 + 0.00003 0.02731 + 0.00004 c -0.00114 ± 0.00009 -0.00413 ± 0.00008 2n e Qq 0.00139 + 0.00023 0.00091 ± 0.00088 10 6 c T -3.9 f i x e d 4.7 f i x e d i- i o 5 b s 4.5 + 1.8 0.0 f i x e d Values i n cm" 1; e r r o r l i m i t s are three standard d e v i a t i o n s ; o = 0.00076 -1 cm The bond lengths ( r Q ) are: c V 1.6747 A, X AZ 1.5920 A. •164- t r o p i c and d i p o l a r i n t e r a c t i o n s which a r e c l o s e l y s i m i l a r t o o u r gas phase v a l u e s . With t h e c o n v e r s i o n s b - a c 4 c = A i = A , s o - A d i p ; c-A,, - A x = 3 A d i p (5.27) the v a l u e s a r e b = 0.02731 0.00004 cm" 1 (gas) = 0.02792 0.00002 cm" 1 ( m a t r i x ) and c =-0.00413 0.00008 cm" 1 (gas) =-0.00408 0.00003 cm" 1 ( m a t r i x ) (5.28) (5.29) There i s e x c e l l e n t agreement f o r the d i p o l a r c o n s t a n t c, b u t t h e r e i s a sm a l l though d e f i n i t e d i f f e r e n c e between t h e gas and m a t r i x v a l u e s o f the i s o t r o p i c p arameter. As p o i n t e d out by Kasai ( 1 0 ) , t h e s e parameters p r o v i d e s t r o n g e v i d e n c e f o r t h e ground s t a t e e l e c t r o n c o n f i g u r a t i o n 4S0 1 3d6 : the s i g n o f b f o r t r a n s i t i o n m e t a l d e l e c t r o n r a d i c a l s i s n e g a t i v e because o f s p i n p o l a r i z a t i o n e f f e c t s u n l e s s s e l e c t r o n s a r e a l s o p r e s e n t ( 3 9 ) , and c w i l l be n e g a t i v e a l s o . The par a m e t e r c i s a sum o v e r t h e v a l e n c e e l e c t r o n s o f t h e terms 3 1 2 = 3 g J j B g I j J N < n ] r " .^(3cos e-l)|n> (5.30) where r i s t h e d i s t a n c e between an e l e c t r o n c a r r y i n g s p i n a n g u l a r momen- tum and the vanadium n u c l e u s . I f we make the a p p r o x i m a t i o n t h a t t h e s t a t e s n c l o s e l y resemble V atomic o r b i t a l s , t h e sum becomes -165- c=3x(.2/3)gu Bg Ijj N<3d6 | r " 3 . J 5 ( 3 c o s 2 e - l ) 13d<S> (.5.31) where t h e f a c t o r 2/3 a r i s e s because o n l y t h e two 3d6 e l e c t r o n s , o ut o f the t h r e e v a l e n c e e l e c t r o n s , g i v e n o n - v a n i s h i n g a v e r a g e v a l u e s o f 3cos 6-1. Fo r a t o m i c - l i k e o r b i t a l s t h e a v e r a g e v a l u e e x p r e s s i o n i s (.18,20) -3 2 2 -3 <n£m|r .3j(3cos e-l)|njyn> = ̂ m ^ c o s e-1|an><n£|r |n£> = _ [ 3 m 2 - i i ( £ + l ) ] < r - 3 > n o ( 5' 3 2 ) (2£-l)(2£+3) w h i c h , f o r the ground s t a t e o f V0, g i v e s c i n cm - 1 u n i t s as c = - 7 W i y N < r " 3 V h c ( 5- 3 3 ) The o b s e r v e d v a l u e c = -0.00413 cm"1 t h e r e f o r e g i v e s <r" 3>,. = 3.0 x 1 0 2 4 cm" 3 (5.34) which i s 85% o f the v a l u e g i v e n by t h e H a r t r e e - F o c k c a l c u l a t i o n s o f Freeman and Watson (40) f o r t h e f r e e V atom. I t i s i n t e r e s t i n g t h a t t h i s s i m p l e model a l s o a c c o u n t s f o r the v a l u e o f the ground s t a t e e l e c t r i c q u a d r u p o l e parameter e Qq. Assuming t h a t o n l y t h e two 3d6 e l e c t r o n s a r e r e s p o n s i b l e f o r t h e q u a d r u p o l e parameter we f i n d e 2 Q q = - ( 4 . 8 0 3 x 1 0 ' 1 0 e s u ) 2 x 0 . 2 7 x 1 0 " 2 4 c m 2 x 2 x - y < r " 3 > 3 d / h c (.5.35) -166- 2 -1 from which t h e e x p e r i m e n t a l v a l u e e Qq = 0.00091 cm g i v e s < r " 3 > 3 d = 2.5 x 1 0 2 4 cm" 3 (5.35) However the e r r o r l i m i t s on e 2Qq a r e so v e r y l a r g e t h a t t h e agreement >3d i n t h e v a l u e s o f <i""3>od i s p r o b a b l y m a i n l y f o r t u i t o u s . I t i s n o t so easy t o a p p l y t h e s e agreements t o t h e C 4E~ e x c i t e d s t a t e because the 4sa e l e c t r o n has been r e p l a c e d by a 4pa e l e c t r o n . As e x p e c t e d t h e i s o t r o p i c p a rameter b i s n e g a t i v e because o f s p i n p o l a r i z a - t i o n , and the d i p o l a r parameter c i s much s m a l l e r than i n t h e ground s t a t e . The d e c r e a s e o f c on e l e c t r o n i c e x c i t a t i o n can be u n d e r s t o o d from t h e model g i v e n i n eq. (5.30). F o r the C 4 i " s t a t e t h e e x p r e s s i o n becomes 3 2 3 2 1 3 2 I t i s not easy to o b t a i n i n d e p e n d e n t e s t i m a t e s o f <r >^^, but w i t h t h e -3 -3 v e r y c r u d e a p p r o x i m a t i o n t h a t <r > 3 C | = : < r >4p> w e o b t a i n , 4 - 3 , 2 - 3 » c = g P B g I y N ( - 7 < r > 3 d + 3-<r > 4 p ) / h c = - 0 . 1 7 1 g p B g I u N < r " 3 > 3 d / h c = -0.00126 cm" 1 Somewhat s u r p r i s i n g l y , t h i s number a g r e e s , a l m o s t t o w i t h i n t h e e x p e r i - m e n t a l e r r o r , w i t h o u r o b s e r v e d v a l u e s . The upper s t a t e q u a d r u p o l e parameter does not f i t t h i s model . As f o r t h e o t h e r two h y p e r f i n e p a r a m e t e r s , b s and c T , we have d i f f i - c u l t y d e t e r m i n i n g t h e parameters s e p a r a t e l y f o r t h e two e l e c t r o n i c s t a t e s , -167- though t he d i f f e r e n c e between t h e parameters on e l e c t r o n i c e x c i t a t i o n i s w e l l d e t e r m i n e d . T h i s i s a consequence o f t h e s e l e c t i o n r u l e AF=AN, which a p p l i e s e x c e p t a t the l o w e s t N v a l u e s ; o b v i o u s l y , d i r e c t measure- ments o f t h e h y p e r f i n e l e v e l s e p a r a t i o n s w i l l be needed t o break t h e c o r r e l a t i o n . We e s t i m a t e t h a t t h e measured d i f f e r e n c e ACj=Cj'-Cj" i s a c c u r a t e t o about 10%, o r , i n f i g u r e s A c I = c I , - c I " = ( - 8 . 6 ± 0 . 9 ) x l O - 6 cm' 1 (5.39) The s t a n d a r d d e v i a t i o n i n the l e a s t s q u a r e s f i t was improved by about 2% when we i n c l u d e d t h e Cj t e r m s , though as e x p l a i n e d above we had t o f i x t h e Cj parameters i n the r a t i o o f the y parameters i n t h e f i n a l l e a s t s q u a r e s f i t t i n g . The symptom showing t h a t t h e Cj terms were needed was t h a t the l e a s t s q u a r e s r e s i d u a l s f o r the h y p e r f i n e s t r u c t u r e s o f a l l f o u r e l e c t r o n s p i n components showed a s y s t e m a t i c t r e n d from p o s i t i v e t o n e g a t i v e w i t h i n c r e a s i n g F; t h e e f f e c t i s o n l y about 0.002 cm" 1, b u t i t d i s a p p e a r e d a t once on i n c l u s i o n o f t h e Cj terms. 4 - The t h i r d - o r d e r c r o s s term bg i s w e l l d e t e r m i n e d f o r t h e C E s t a t e p r o v i d e d we s e t b s " equal t o z e r o . T h i s i s n o t an u n r e a s o n a b l e a p p r o x i - m a t i o n because the v a l u e o f b s r e f l e c t s t h e p o s i t i o n s o f nearby e l e c t r o - n i c s t a t e s c o u p l e d t h r o u g h s p i n - o r b i t i n t e r a c t i o n , and we know t h a t t h e C 4E" s t a t e s u f f e r s many l o c a l p e r t u r b a t i o n s . A l s o t h e r e l a t e d parameter Ys i s e s s e n t i a l l y u ndetermined f o r the ground s t a t e b u t w e l l - d e t e r m i n e d f o r t h e C4z~ s t a t e . I t i s u n f o r t u n a t e l y n o t s i m p l e t o i n t e r p r e t the parameter b<., so t h a t a l l we can do i s p o i n t out the need t o i n c l u d e -168- t>s i n p r e c i s e work on e x c i t e d e l e c t r o n i c s t a t e s o f q u a r t e t and h i g h e r m u l t i p l i c i t y . H. D i s c u s s i o n 4 - 4 -, T h i s a n a l y s i s o f t h e C E -X E (.0,0) band o f VO a t s u b - D o p p l e r r e s o l u t i o n i s the most d e t a i l e d a c c o u n t so f a r o f an e l e c t r o n i c band i n v o l v i n g q u a r t e t s t a t e s . The most i n t e r e s t i n g a s p e c t s a r e t h e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n s i n the two e l e c t r o n i c s t a t e s , w hich we can f i t 4 i n d e t a i l u s i n g t h e complete E s p i n and h y p e r f i n e H a m i l t o n i a n . 4 _ E l e c t r o n i c p e r t u r b a t i o n s a r e u n f o r t u n a t e l y w i d e s p r e a d i n the C E s t a t e , which means t h a t we cannot o b t a i n a f i t t o t h e h i g h e r N data t h a t does j u s t i c e to t h e i r p r e c i s i o n . In f a c t we have had to a p p l y c o r r e c t i o n s t o a l l t h e F4 l e v e l s o f the C4E~ s t a t e t o a l l o w f o r t h e c o m p a r a t i v e l y l a r g e p e r t u r b a t i o n a t N = 26. W i t h o u t t h e s e c o r r e c t i o n s t h e l e a s t s q u a r e s r e s u l t s show s y s t e m a t i c r e s i d u a l s even a t low N, and the c e n t r i - f u g a l d i s t o r t i o n parameters a r e u n r e a l i s t i c . The f i n a l f i t was p e r f o r m e d w i t h ' d e p e r t u r b e d ' d a t a o n l y up t o N' = 2 4 , p l u s ground s t a t e A 2F" com- b i n a t i o n d i f f e r e n c e s t o N = 40. The upper s t a t e term v a l u e s beyond N = 24 a r e c a l c u l a t e d from t h i s f i t t o w i t h i n a b o u t 0.1 c m - 1 , as can be seen from F i g . 5.15. T h i s f i g u r e shows the r e s i d u a l s f o r the R b ranch l i n e s up to N 1 = 42, c a l c u l a t e d u s i n g t h e f i n a l c o n s t a n t s o f T a b l e 5.5. The c h a o t i c c o u r s e s o f a l l f o u r s p i n components a r e r e a d i l y a p p r e c i a t e d . 4 - The C E s t a t e a l s o s u f f e r s f r o m ' g l o b a l ' p e r t u r b a t i o n s a f f e c t i n g a l l l e v e l s , b e s i d e s the 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 . These r e q u i r e the i n t r o d u c t i o n o f a second s p i n - r o t a t i o n parameter ys» f o l l o w i n g t h e f o r m a l i s m o f Brown and M i l t o n ( 2 5 ) . T h i s p a r ameter i s a t h i r d - o r d e r -169- s p i n - o r b i t e f f e c t , and g i v e s e v i d e n c e f o r a c l o s e - l y i n g e l e c t r o n i c s t a t e t h a t i n t e r a c t s s t r o n g l y t h r o u g h t h e s p i n - o r b i t o p e r a t o r . We have a l s o had t o i n t r o d u c e t he c o r r e s p o n d i n g i s o t r o p i c h y p e r f i n e parameter b^ ( 2 6 ) , which a r i s e s i n s i m i l a r f a s h i o n as a c r o s s - t e r m between the I .S h y p e r f i n e o p e r a t o r and the s p i n - o r b i t i n t e r a c t i o n . The parameters y$ and b<. o n l y o c c u r f o r s t a t e s o f q u a r t e t and h i g h e r m u l t i p l i c i t y . The m a g n e t i c h y p e r f i n e c o n s t a n t s f o r the two s t a t e s have been a c c u r a t e l y d e t e r m i n e d . F o r t h e ground s t a t e t h e r e i s good agreement w i t h t h e e . s . r . v a l u e s o f Kasai ( 1 0 ) , i n a s f a r as the gas phase and m a t r i x v a l u e s can be compared. The m a g n e t i c c o n s t a n t s f o r t h e upper s t a t e show 2 1 c l e a r l y t h a t i t s e l e c t r o n c o n f i g u r a t i o n i s 3d6 4pa . The e l e c t r i c qua- d r u p o l e parameters a r e u n f o r t u n a t e l y n o t w e l l d e t e r m i n e d , though t h e r e appears to be c o n s i s t e n c y between t h e ground s t a t e e l e c t r i c q u a d r u p o l e and m a g n e t i c d i p o l e p a r a m e t e r s , which i n v o l v e r o u g h l y t h e same averag e s o v e r e l e c t r o n c o o r d i n a t e s . The e l e c t r o n s p i n - s p i n parameters A f o r the two s t a t e s have been a c c u r a t e l y d e t e r m i n e d , d e s p i t e the p a r a l l e l s e l e c t i o n r u l e s , because o f th e o b s e r v a t i o n o f Q branches a t low N v a l u e s . The Q l i n e s e s s e n t i a l l y 4 p e r m i t an e x a c t d e t e r m i n a t i o n , a c c o r d i n g t o the E r e l a t i o n 4 A - 2 Y = F 2 ( N ) + F 3 ( N ) - F 1 ( N ) - F 4 ( N ) (5.40) (which f o l l o w s from T a b l e 5.1) because t h e y can be combined w i t h t h e main branch R and P l i n e s to p r o v i d e t he s e p a r a t i o n s F 2(N)-F-j(N) and F 4 ( N ) - F 3 ( N ) . Less p r e c i s e v a l u e s c o u l d s t i l l have been o b t a i n e d w i t h o u t t h e Q b r a n c h measurements because t h e low N l i n e p o s i t i o n s a r e q u i t e s e n s i - -170- Fig- 5.15 Residuals (obs-calc) f o r the R branch l i n e s of the VO 4 - 4 - C £ -XI (0,0) band, as compared to the p o s i t i o n s p r e d i c t e d from the constants of Table 5.5, p l o t t e d against N'.'Raw' data have been used, so that the R^ l i n e s from N' = 9-22,.which were deperturbed f o r the l e a s t squares treatment ( t h i c k l i n e s ) have i-zero r e s i d u a l s . V e r t i c a l bars i n d i c a t e the spread of the non- hyper f i n e s t r u c t u r e . -171- t i ve t o t h e e x a c t v a l u e s o f t h e two A p a r a m e t e r s . 4 - The low N energy l e v e l s o f VO .X E , v=0 w i l l be o f i n t e r e s t t o a s tronomers c o n c e r n e d w i t h t h e d e t e c t i o n o f VO i n i n t e r s t e l l a r s p a c e and to s p e c t r o s c o p i s t s c o n c e r n e d w i t h microwave and f a r i n f r a - r e d s t u - d i e s o f the ground s t a t e . We t h e r e f o r e l i s t t h e ground s t a t e r o t a t i o n a l and h y p e r f i n e e n e r g y l e v e l s (as c a l c u l a t e d from t h e c o n s t a n t s o f T a b l e 5.5)up to N" = 5 i n T a b l e 5.6. The e x p e r i m e n t a l ground s t a t e h y p e r f i n e c o m b i n a t i o n d i f f e r e n c e s , between l e v e l s w i t h the same F v a l u e i n t h e F 2 and F 3 e l e c t r o n s p i n components o v e r t h e range N" = 8-20, a r e l i s t e d i n T a b l e 5.7. 4 - 4 - To summarize, t h e VO C E - X E e l e c t r o n i c t r a n s i t i o n p r o v i d e s ' t e x t b o o k ' examples o f t h e e f f e c t s o f e l e c t r o n s p i n and h y p e r f i n e s t r u c t u r e i n a q u a r t e t s t a t e i n c a s e ( b g J ) c o u p l i n g ; i t i s f r u s t r a t i n g t h a t e l e c t r o n i c p e r t u r b a t i o n s p r e v e n t the h i g h e r N l i n e s from b e i n g f i t t e d t o an a c c u r a c y t h a t matches t h e i r p r e c i s i o n . .4 - Table 5.6 R o t a t i o n a l and hyperfine energy l e v e l s of the X E -1 from the constants of Table 5.5. Values i n cm . N J F^.T-7/2 P=J-5/2 P-J-3/2 F=J-l/2 F-J+l/2 v=0 s t a t e of VO f o r N >5, c a l c u l a t e d F=J+3/2 F=J+5/2 F-J+7/2 0 1 1.5 -2.8982 -2.8391 -2.7595 -2.6576 2.5 -1.6426 -1. 6160 -1.5759 -1.5220 -1.4541 -1.3716 1.5 1.2592 1.2536 1.2434 1.2264 0.5 a -3.1255 -2.9492 3.5 0.6665 0.6766 0.6969 0. 7273 0.7681 0.8194 0.8814 0.9545 2.5 4.0231 4. 0275 4.0340 4.0425 4.0527 4.0642 1.5 5.9068 5.9379 5.9782 6.0269 0.5 a -0.8199 -0.9343 4.5 4.0522 4.0685 4.0930 4. ,1258 4.1670 4.2167 4.2751 4.3424 3.5 7.6362 7.6389 7.6442 7 . ,6525 7 .6638 7.6785 7.6967 7.7190 2.5 9.0674 9. .0741 9 .0835 9 .0948 9.1070 9.1187 1.5 6.2261 6.2351 6.2490 6.2694 5.5 8.5121 8.5326 8.5599 8, .5943 8.6357 8.6842 8.7401 8.8035 4.5 12.2171 12.2219 12.2294 12 .2396 12.2531 12.2703 12.2918 12.3183 3.5 13.3492 13.3505 13.3531 13 .3565 13.3601 13.3632 13.3647 13.3636 2.5 10.0328 10 .0224 10.0072 9.9875 9.9638 9.9369 6.5 14.0522 14.0756 14.1050 14 .1405 14.1820 14.2297 14.2838 14.3443 5.5 17.8219 17.8279 17.8364 17 .8476 17 .8621 17 .8803 17.9032 17.9318 4.5 18.7437 18.7449 18.7463 18 .7475 18.7479 18.7468 18 .7430 18.7354 3.5 15.1594 15.1529 15.1400 15 .1208 15.0955 15.0643 15.0275 14.9855 J - h and F^(2), J = h must be t r e a t e d as F^( -1) and F 2 ( 0 ) , r e s p e c t i v e l y , s i n c e \ > Table 5.7 Gronud s t a t e hyperfine combination d i f f e r e n c e s , F 2 ( N ) - F 3 ( N ) , -1 , i n cm , t o r the X 4E", v=0 s t a t e of VO i n the range N=8-20. N F=N+3 F=N+2 F=N+1 F=N F=N-1 F=N-2 F=N-3 8 -0.4193 -0.4606 -0.4896 -0.5101 9 -0.3325 -0.3770 -0.4066 -0.4272 -0.4411 -0.4510 -0.4559 10 -0.2638 -0.3103 -0.3411 -0.3629 -0.3735 -0.3804 11 -0.2091 -0.2581 -0.2871 -0.3057 -0.3163 12 -0.1699 -0.2190 -0.2461 -0.2615 -0.2692 -0.2700 -0.2677 13 -0.1484 -0.1947 -0.2184 -0.2300 -0.2330 -0.2287 -0.2213 14 0.1455 -0.1829 -0.2021 -0.2080 -0.2056 -0.1958 -0.1795 15 0.1589 0.1854 0.1977 -0.1993 -0.1909 -0.1750 -0.1501 16 0.1813 0.1992 0.2045 0.2001 -0.1859 -0.1628 -0.1281 17 0.2101 0.2197 0.2195 0.2104 0.1913 0.1631 -0.1202 18 0.2413 0.2400 0.2248 0.2040 0.1723 0.1252 19 0.2619 0.2466 0.2222 0.1888 0.1406 20 0.2694 0.2438 0.2095 0.1618 The tabulated values are observed q u a n t i t i e s corresponding to the d i f f e r e n c e s betw hyper f i n e l e v e l s w i t h the same F value shown i n F i g 5.3. -174- C hapter 6 L a s e r - I n d u c e d F l u o r e s c e n c e and D i s c h a r g e E m i s s i o n s p e c t r a o f FeO; E v i d e n c e f o r a 5 A.j Ground S t a t e -175- A. I n t r o d u c t i o n F e r r o u s o x i d e , FeO, i s p r o b a b l y the most i m p o r t a n t o f the d i a t o m i c o x i d e m o l e c u l e s whose s p e c t r a have so f a r d e f i e d d e t a i l e d i n t e r p r e t a t - i o n . I t i s o f i n t e r e s t i n a s t r o p h y s i c s , as w e l l as m o l e c u l a r s p e c t r o s - copy, because o f the h i g h cosmic abundances o f both i r o n and oxygen. The d i f f i c u l t i e s w i t h FeO have been i t s low d i s s o c i a t i o n energy (which means t h a t i t i s q u i t e d i f f i c u l t t o p r e p a r e i n d i s c h a r g e s y s t e m s ) , i t s i n v o l a t i l i t y , and t h e tremendous c o m p l e x i t y o f i t s s p e c t r u m . C o n s i d e r a b l e argument has s u r r o u n d e d t h e n a t u r e o f t h e ground s t a t e o f FeO. Q u i t e r e c e n t l y E n g e l k i n g and L i n e b e r g e r ( 1 ) have i n t e r - 5 p r e t e d the p h o t o e l e c t r o n spectrum o f FeO i n terms o f a A ground s t a t e f o r FeO, w i t h a v i b r a t i o n a l f r e q u e n c y o f 970 ± 60 cm" 1, and De Vore and G a l l a h e r (2) i d e n t i f i e d a band a t 943.4 ± 2.0 cm" 1 i n i n f r a - r e d e m i s s i o n e x p e r i m e n t s on FeO, but i t i s now c l e a r , from t h e m a t r i x i s o l a t i o n work o f Green e t a l . ( 3 ) , t h a t FeO has a g r o u n d - s t a t e v i b r a - t i o n a l f r e q u e n c y o f about 875 cm" 1. T h i s number i s a l s o found f o r t h e l o w e r s t a t e o f t h e well-known e l e c t r o n i c band s y s t e m i n t h e orange r e g i o n ( 4 - 6 ) , which must t h e r e f o r e i n v o l v e t h e ground s t a t e . The o r a n g e band system i s u n u s u a l l y complex ( 7 , 8 ) , but has been f o u n d t o c o n t a i n a few s u r p r i s i n g l y s i m p l e p a r a l l e l bands c o n s i s t i n g o f s i n g l e P and R branches ( 4 - 6 ) , a p p a r e n t l y o f t y p e 1z- 1 i . H a r r i s and Barrow (6) r e c o g n i z e t h a t t h e bands must be more complex t h a n t h i s , s i n c e t h e o r e t i c a l p r e d i c t i o n s g i v e , v a r i o u s l y , 5 A ( 9 , 1 0 ) and 5 z + (11) f o r the ground s t a t e . -176- The p u r p o s e o f t h i s c h a p t e r i s to r e p o r t new e m i s s i o n and l a s e r - i n d u c e d f l u o r e s c e n c e s p e c t r a o f FeO which proye t h a t t h e s i m p l e bands i n t h e orange system are Q' = 4 - Q" = 4 bands. T h i s p r o v i d e s s t r o n g e v i d e n c e t h a t t h e ground s t a t e i s A . . , s i n c e n = 4 components do not 5 + 5 7 + a r i s e i n t h e o t h e r p o s s i b l e c a n d i d a t e s t a t e s , E , n, and E . T h i s i d e n t i f i c a t i o n a g r e e s a l s o w i t h W e l t n e r ' s r e p o r t (12) t h a t m a t r i x i s o l a t e d FeO g i v e s no ESR s p e c t r u m under c o n d i t i o n s where o r b i t a l l y n o n d egenerate s p e c i e s n o r m a l l y g i v e s t r o n g s i g n a l s . -177- B. E x p e r i m e n t a l D e t a i l s E m i s s i o n s p e c t r a o f FeO were e x c i t e d by a 2450-MHz e l e c t r o d e l e s s d i s c h a r g e i n a m i x t u r e o f f l o w i n g a r g o n , oxygen, and f e r r o c e n e ( d i c y - c l o p e n t a d i e n y l i r o n ) a t low p r e s s u r e . The d i s c h a r g e i s u n f o r t u n a t e l y n o t v e r y s t a b l e , and g i v e s m a i n l y CO s p e c t r a i f t h e b u i l d u p o f s o l i d r u s t - l i k e p r o d u c t s becomes e x c e s s i v e , because t h e s e i n t e r f e r e w i t h t h e t r a n s m i s s i o n o f the microwave power. As a r e s u l t , p h o t o g r a p h i c expo- s u r e s l o n g e r t h a n 1 hr were o f t e n u n s a t i s f a c t o r y , but t h i s t i m e was s u f f i c i e n t t o g i v e good s p e c t r a i n the r e g i o n 5500 - 6300 K u s i n g Kodak I l a - D p l a t e s i n a 7-m Ebert-mounted p l a n e g r a t i n g s p e c t r o g r a p h . The t e m p e r a t u r e o f t h e e m i t t i n g m o l e c u l e s , as e s t i m a t e d from the de- velopment o f r o t a t i o n a l b r a n c h s t r u c t u r e , i s about 500°C. L a s e r - i n d u c e d f l u o r e s c e n c e o f FeO was produced u s i n g a Coherent I n c . CR-599-21 t u n a b l e dye l a s e r o p e r a t i n g w i t h rhodamine 6G, and pumped by an argon i o n l a s e r . The o p t i c a l arrangement f o r t h i s e x p e r i - ment was the same as i n c h a p t e r 5. The l a s e r beams were s e n t t h r o u g h t h e end o f t h e f l a m e o f t h e microwave d i s c h a r g e s y s t e m d e s c r i b e d above, and o b s e r v e d a t r i g h t a n g l e t o the stream o f m o l e c u l e s . Broadband and s i n g l e f r e q u e n c y l a s e r e x c i t a t i o n s p e c t r a , and Sub-Doppler 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 (.13) o f c e r t a i n r e g i o n s o f the 5819^ band were r e c o r d e d , as w e l l as r e s o l v e d f l u o r e s c e n c e s p e c t r a . The p h o t o g r a p h i c " s u r v e y " s p e c t r a were measured on a Grant a u t o - m a t i c comparator, and r e d u c e d t o vacuum wavenumbers u s i n g a f o u r - t e r m p o l y n o m i a l . C a l i b r a t i o n s p e c t r a were p r o v i d e d by an i r o n - n e o n h o l l o w c a t h o d e lamp, f o r which t h e wavelengths have been l i s t e d by C r o s s w h i t e ( 1 4 ) . The l a s e r s p e c t r a were c a l i b r a t e d by means o f the i o d i n e s p e c - -178- trum a t l a s of G e r s t e n k o r n and Luc (H) , w i t h t he c o r r e c t i o n o f 0.0056 cm" 1 t o g i v e a b s o l u t e wavenumbers a p p l i e d . -179- C. R e s u l t s H a r r i s and Barrow (6) have i d e n t i f i e d t h r e e bands i n v o l v i n g the l e v e l v" = 0 o f the s t a t e which appears a l s o i n the m a t r i x i s o l a t i o n e x p e r i m e n t s o f Green e t a l . (_3); t h e y o c c u r a t 5583, 5819 and 5911 K. The 5583- and 5911 A bands l i e i n crowded s p e c t r a l r e g i o n s where b l e n d i n g i s s e v e r e , b u t t h e 5819-A band i s i n a c o m p a r a t i v e l y c l e a r r e g i o n and we have s e l e c t e d i t f o r s t u d y . I t s head i s i l l u s t r a t e d i n F i g . 6.1. The band i s a v e r y s t r o n g l y r e d - d e g r a d e d p a r a l l e l band, and the a s s i g n m e n t s o f t h e P and R l i n e s a r e t h o s e o f H a r r i s and Barrow ( 6 ) . A f e a t u r e o f t h i s band i s t h a t where s m a l l r o t a t i o n a l p e r t u r b a t i o n s o c c u r they u s u a l l y appear as two l i n e s o f e q u a l i n t e n s i t y . Because t h i s i m p l i e s t h a t t h e r e a r e s e v e r a l e x a c t c o i n c i d e n c e s o f p e r t u r b e d and p e r - t u r b i n g l e v e l s i f the s t a t e s i n v o l v e d have Q = 0 (as p o s t u l a t e d by H a r r i s and Barrow (6) we s u s p e c t e d t h a t t h e r e c o u l d be A-doubling p r e - s e n t which i s n o t r e s o l v e d i n t h e g r a t i n g s p e c t r a . S u b - D o p p l e r i n t e r - m o dulated f l u o r e s c e n c e s p e c t r a o f t h e u n p e r t u r b e d l i n e R(15) a t a r e s o l u t i o n o f about 75 MHz showed at once t h a t t h i s i s t r u e ( s e e F i g . 6.2); i n t h i s f i g u r e t h e l i n e i s seen to c o n s i s t o f two e q u a l l y i n t e n s e c l o s e l y s p a c e d components s e p a r a t e d by a b o u t 120 MHz. G i v e n t h a t ft' = ft" f 0, we s e a r c h e d f o r p o s s i b l e Q l i n e s i n t h e g r a t i n g s p e c t r a , because t h e J numbering o f the f i r s t Q l i n e would g i v e the ft v a l u e . Q u i t e a number o f l i n e s o c c u r i n t h e e x p e c t e d r e g i o n , but a branch c o u l d be p i c k e d o u t . I t s numbering was e s t a b l i s h e d by p l o t t i n g the l i n e p o s i t i o n s a g a i n s t n(n + 1 ) , where n i s an a r b i t r a r y r u n n i n g number, and c h o o s i n g t h e b e s t s t r a i g h t l i n e . The r e s u l t gave ft = 4. •180- The Q(4) l i n e o f the 5819-A band runs: i n t o t h e R(..10) l i n e i n F i g . 6.1 because o f the s t r o n g e x p o s u r e ; t h e l i n e s a r e seen r e s o l v e d i n t h e s u b - D o p p l e r spectrum shown i n F i g . 6.2b. In v iew o f t h e f a c t t h a t t h e r e a r e many weak background l i n e s u n d e r l y i n g the 5819-A band which c o u l d e a s i l y be m i s t a k e n f o r Q l i n e s ( s e e F i g . 6.1) c o n f i r m a t i o n was s o u g h t from r o t a t i o n a l l y r e s o l v e d l a s e r - i n d u c e d f l u o r e s c e n c e e x p e r i m e n t s . These t u r n e d o u t t o be a b s o l u - t e l y c o n c l u s i v e and l e a v e no doubt about t h e Q b r a n c h and i t s numbering. Some o f the p a t t e r n s o b s e r v e d a r e i l l u s t r a t e d i n F i g . 6.3. As e x p e c t e d , e x c i t a t i o n a t t h e w a v e l e n g t h o f t h e f i r s t l i n e , Q(4), g i v e s o n l y Q(4) and P(5) e m i s s i o n , w h i l e h i g h e r J l i n e s o f the Q branch g i v e R-, Q-, and P-branch e m i s s i o n . E x c i t a t i o n o f the R-branch l i n e s g i v e s c o n s i s - t e n t p a t t e r n s , and somewhat s u r p r i s i n g l y , weak Q-branch e m i s s i o n can even be seen when the l i n e R(16) i s e x c i t e d . In the end i t was p o s s i b l e to f o l l o w t h e Q branch on the g r a t i n g s p e c t r a from i t s f i r s t l i n e , Q ( 4 ) , up t o Q ( 2 6 ) , where i t becomes l o s t i n the background o f weak l i n e s . The a s s i g n e d l i n e s o f the 5819-A band and t h e o t h e r bands we have s t u d i e d a r e l i s t e d i n Appendix VI T a b l e I I . o F i g u r e 6.1 a l s o shows the head r e g i o n o f the 6180-A band. T h i s band was shown by H a r r i s and Barrow (6) t o have t h e same u " v a l u e as the 5819-$ band, but w i t h v" = 2 r a t h e r t h a n v" = 0. Q u i t e a s t r o n g Q branch can be s e e n , which can be numbered unambiguously by means o f th e A-|F" c o m b i n a t i o n d i f f e r e n c e s f o r t h e l e v e l -v" = 2. A g a i n t h e f i r s t l i n e i s J = 4, which i s c o n s i s t e n t w i t h the l a s e r - i n d u c e d f l u o r e s c e n c e e x p e r i m e n t s d e s c r i b e d aboye. S i m i l a r Q b r a n c h e s have been i d e n t i f i e d i n t h e 5583- and 5866-A bands ( v " = 0 and 1 ) . .181. 4 5 10 i iiiiniiiiiMiiiuiiiiiii R. 4 5 10 5819 A 6180 A 10 4 10 12 Q 4 16174.03 Cfn - 1 j I I I I I | I | I 16152.15 cm - 1 • iiiiiiiiiiiiHiMiipm ILLS ' » " , i i* I I I I 4°. tot Rz Fig 6.1 Head of the 5819-A band of FeO. Lower p r i n t : head of the 6180-A band of FeO. (a) (b) Q(4) R(10) 17167.6050 cm-' I I I I I 1 I 1 I I 300 MHz morkers Iodine fluorescence spectrum 17176.1197 cm-' F i g 6.2 Two regions of the intennodulated fluorescence spectrum o f s FeO:(a) The twoA components of the R(15) l i n e of the 5819-A band, (b) The Q(4) and R(10) l i n e s of the 5819-A band; the A doubling i s not resolved f o r these l i n e s . -182- 5823 5829 5821 5830 Q(6) R06) excitation 5823 5829 5832 5846 Wavelength markers 6.3 Resolved f k o r s e c e n c e spectra of FeO produced by e x c i t a t i o n of v a r i o u s l i n e s of the 5819-Aband: e x c i t a t i o n of Q(4),Q(5}, Q(6) and R(16). The i n t e n s i t y of the e x c i t e d l i n e i s anomalousl high as a r e s u l t of sc a t t e r e d l a s e r l i g h t . -183- D. D i s c u s s i o n The work p r e s e n t e d h e r e proves, t h a t the c o m p a r a t i v e l y s i m p l e bands a n a l y z e d by Barrow and h i s co-workers (5,6) have Q" = 4. Some o f t h e s e bands ( i n c l u d i n g t h e 6180-$ band i l l u s t r a t e d i n F i g . 6.1) form a lower- s t a t e p r o g r e s s i o n which g i v e s v i b r a t i o n a l c o n s t a n t s (.6). t h a t a r e a l m o s t i d e n t i c a l t o t h o s e o b t a i n e d from t h e i n f r a r e d spectrum o f m a t r i x - i s o l a t e d FeO by Green e t a l . ( 3 ) , v i z . , Gas: C J 0 = 880.61 cm""1, W Q X 0 = 4.64 c m - 1 ; 6 i 1 ( 6 J ) M a t r i x : w e = 880.2 cm w e * e = 3.47 cm . T h e r e f o r e t h e ground s t a t e o f FeO c o n t a i n s an n = 4 s p i n - o r b i t compo- nent . The ground e l e c t r o n c o n f i g u r a t i o n s o f th e t r a n s i t i o n o x i d e s i m m e d i a t e l y b e f o r e FeO a r e known (15,16) t o be (Appendix V) T i O ( 4 s a ) 1 ( 3 d 6 ) 1 3 A r VO ( 4 s a ) 1 ( 3 d 6 ) 2 V (6.2) CrO ( 4 s a ) 1 ( 3 d 6 ) 2 ( 3 d ^ ) 1 5 n r MnO ( 4 s o V ( 3 d 6 ) 2 ( 3 d ^ ) 2 6 z + where t h e energy o r d e r o f 4sa and 3d6 i s not c e r t a i n . In FeO th e e x t r a e l e c t r o n c o u l d go i n t o the next u n o c c u p i e d m.o., 3d C T , g i v i n g a 7 + 5 + 5 E ground s t a t e , o r i n t o the 4sa o r 3d6 m.o.'s, g i v i n g z o r A as th e ground s t a t e . T h e o r e t i c a l c o m p u t a t i o n s (9-11) a r e d i v i d e d between 5 Z + and 5 A , t h r o u g h t h e CI c a l c u l a t i o n s o f Bagus and P r e s t o n (9) con- 184- 5.+ e l u d e t h a t t h e ground s t a t e i s not b z . The f a c t t h a t t h e r e i s an 5 n = 4 component i n the ground s t a t e i s o n l y c o n s i s t e n t w i t h where 3 + 7 + t h e " f i v a l u e s run from 0. t o 4; t h e h i g h e s t n v a l u e s i n z and z s t a t e s a r e 2 and 3, r e s p e c t i v e l y . 5 I t i s i n t e r e s t i n g t h a t t h e A s t a t e under d i s c u s s i o n , which comes 1 3 2 from the c o n f i g u r a t i o n (4sa) (3d6) (3C1TT) , must be i n v e r t e d , w i t h i t s ft = 4 component as t h e l o w e s t i n e n e r g y . T h i s i s p r o b a b l y t h e r e a s o n why t h e m a t r i x - i s o l a t i o n v i b r a t i o n a l c o n s t a n t s (3) a g r e e so e x a c t l y w i t h the gas-phase c o n s t a n t s (5,6) because a t the low t e m p e r a t u r e o f the m a t r i x o n l y the Q, = 4 component i s l i k e l y t o be a p p r e c i a b l y popu- l a t e d , assuming s p i n - o r b i t i n t e r v a l s o f about 100 cm" 1 (by a n a l o g y w i t h T i O ( 1 7 ) , where t h e r e i s a l s o an u n p a i r e d 3dS e l e c t r o n ) . The o t h e r 5 ii components o f t h e A s t a t e w i l l have s l i g h t l y d i f f e r e n t e f f e c t i v e v i b r a t i o n a l f r e q u e n c i e s because o f t h e v a r i a t i o n o f t h e s p i n - o r b i t c o u p l i n g w i t h v i b r a t i o n . The subbands so f a r a n a l y z e d c a r r y no d i r e c t i n f o r m a t i o n about t h e s p i n - o r b i t c o u p l i n g o f t h e ground s t a t e . For a s t a r t the subbands a r e a l l p a r a l l e l - p o l a r i z e d ( f t 1 = ft"), and as y e t o n l y one s p i n component has been i d e n t i f i e d . However, t h e s e ft = 4 subbands, though prominent i n t he s p e c t r u m , a c c o u n t f o r o n l y a s m a l l f r a c t i o n o f t h e t o t a l e m i s s i o n i n t e n s i t y , and subbands i n v o l v i n g the o t h e r s p i n components must a l s o be p r e s e n t . The prominence o f the ft = 4 subbands p r o b a b l y r e s u l t s from the f a c t t h a t the A - d o u b l i n g i s u n r e s o l v e d , so t h a t t h e i r l i n e s have a p p a r e n t l y t w i c e the s t r e n g t h o f o t h e r l i n e s b e l o n g i n g t o subbands i n the same r e g i o n w i t h r e s o l v e d A - d o u b l i n g ; t h i s e f f e c t i s 5 5 a l s o pronounced f o r t h e ft = 3 subbands o f the A n-X n system o f CrO (18) •185- In the CrO spectrum the f i v e subband heads form a r e g u l a r s e r i e s , which i s o b v i o u s i n l o w - d i s p e r s i o n s p e c t r a , b u t t h e same i s not t r u e i n t h e FeO sp e c t r u m . I t appears, t h a t t h e r e a r e e x t e n s i v e i n t e r a c t i o n s between two o r more e x c i t e d e l e c t r o n i c s t a t e s i n FeO which produce an al m o s t random d i s t r i b u t i o n o f Q, s u b s t a t e s , as i f t h e s p i n c o u p l i n g were case ( c ) . Because t he l v a l u e o f each band has t o be d e t e r m i n e d i n d i v i d u a l l y i n t h e FeO s p e c t r u m , i t w i l l be a l e n g t h y p r o c e s s a s s e m b l i n g d a t a f o r a l l f i v e s p i n - o r b i t components o f t h e ground s t a t e . At p r e s e n t even the bond l e n g t h i s not a c c u r a t e l y g i v e n by the a v a i l a b l e B v a l u e f o r 5 b e c a u s e o f t h e s p i n - u n c o u p l i n g . In p r i n c i p l e , i t would be p o s s i b l e t o use t h e d i f f e r e n c e between t h e a p p a r e n t c e n t r i f u g a l d i s t o r t i o n con- s t a n t f o r t h e component and the v a l u e g i v e n by the K r a t z e r r e l a t i o n t o e s t i m a t e t h e s p i n - o r b i t s e p a r a t i o n s and then c o r r e c t t h e B v a l u e f o r s p i n - u n c o u p l i n g . In p r a c t i c e we f i n d t h a t t h e e r r o r l i m i t s on ^apparent^ A 4 ^ a r e t o ° ^ a r 9 e f o r tni's a P P r o a c n t o s u c c e e d . A l e a s t - s q u a r e s f i t t i n g o f t h e A2F"'s we have measured g i v e s B . ( 5 i . ) = 0.51089 ± 0.00003r cm" 1 ( l a ) ((- ,A a p p a r e n t 4' 5 15.3) D + ( 5 A „ ) = 6.6 n x 1 0 " 7 ± 0.2, x 1 0 ' 7 cm" 1 apparent^ 4' 0 3 ( i n c l o s e agreement w i t h t h e r e s u l t s o f Barrow e t a l . ( 5 , 6 ) ) . A comparison o f the m o l e c u l e s FeO and FeF (19) i s i n s t r u c t i v e . FeF i s known to have a A I ground s t a t e w h i c h a r i s e s from t he e l e c t r o n 1 3 2 1 c o n f i g u r a t i o n ( 4 s c ) (3dS) (3dir) (3da) ; i n o t h e r words t h e e x t r a e l e c t r o n i n FeF goes i n t o t h e 3da m.o. r a t h e r than i n t o 3d6 . T h i s -186- shows the c l o s e a n a l o g y between FeO and F e F , because t h e l i g a n d f i e l d e f f e c t o f an F atom i s not as g r e a t as t h a t o f an 0 atom so t h a t t h e s p l i t t i n g o f the i r o n 3d m a n i f o l d Is s m a l l e r . In FeF Hund's r u l e s a p p l y to t h e t h r e e 3d o r b i t a l s and the 4s o r b i t a l as a g r o u p , p r o d u c i n g a h i g h - s p i n s i t u a t i o n , w h i l e i n FeO Hund's r u l e s a p p l y o n l y to 3d6, 3dir, and 4 s a . In f u r t h e r l a s e r - i n d u c e d f l u o r e s c e n c e e x p e r i m e n t s which are not r e p o r t e d h e r e , we have a n a l y z e d v a r i o u s sub-bands w i t h ft" = 0, 1, 2 and 3 i n the orange system o f FeO. The r e g u l a r i t y o f the B" v a l u e s l e a v e s no doubt t h a t the l o w e r l e v e l s form the o t h e r s p i n - o r b i t compo- 5 ° nents o f the X A., s t a t e . The ground s t a t e bond l e n g t h i s 1.619 A, and the s p i n - o r b i t i n t e r v a l s a r e about 190 cm" 1. -187- C h a p t e r 7 P r e d i s s o c i a t e d R o t a t i o n a l S t r u c t u r e i n t h e 2490-A Band o f 1 5 N 0 2 -188- A. I n t r o d u c t i o n S l i g h t l y p r e d i s s o c i a t e d j m o l e c u l a r band systems where t h e r e i s a s i z a b l e i s o t o p e e f f e c t o f f e r t h e p o s s i b i l i t y o f s e l e c t i v e d i s s o c i a t i o n o f one member o f a m i x t u r e o f i s o t o p e s . The e x p e r i m e n t a l r e q u i r e m e n t s a r e a s u i t a b l e s o u r c e o f r a d i a t i o n which can be tuned t o an a p p r o p r i a t e wavelength and a s c a v e n g i n g system ( c h e m i c a l o r o t h e r ) which can c o l l e c t t h e d i s s o c i a t e d p r o d u c t s w i t h o u t i n t e r f e r e n c e from the u n d i s s o c i a t e d compound. V a r i o u s e x p e r i m e n t s o f t h i s type have been s u c c e s s f u l l y c a r r i e d out u s i n g n a r r o w - l i n e l a s e r s , f o r example, on s - t e t r a z i n e by K a r l and Innes (1) and by H o c h s t r a s s e r and King ( 2 ) , and on ICfc by L i u e t a l . ( 3 ) . The 2490-A a b s o r p t i o n t r a n s i t i o n o f N 0 2 ( 2 2 B 2 - X2A-,) i s s l i g h t l y p r e d i s s o c i a t e d (4) and t h e r e f o r e a l l o w s t h e p o s s i b i l i t y o f l a s e r - i n d u c e d i s o t o p e e n r i c h m e n t . F o r t h i s r e a s o n we have s t u d i e d t h e 15 c o r r e s p o n d i n g band o f N0 2 w i t h a view t o i d e n t i f y i n g t h o s e wavelengths where i r r a d i a t i o n would s e l e c t i v e l y d i s s o c i a t e one i s o t o p e and p e r m i t 14 15 t h e s e p a r a t i o n o f N and N. o 15 S p e c t r o s c o p i c a l l y , t h e 2490-A band o f N0 2 i s v e r y s i m i l a r t o t h a t o f 1 4 N 0 2 , though i t s o r i g i n i s s h i f t e d 14.5 cm" 1 t o h i g h e r e n e r g y . An a n a l y s i s o f the q u a r t i c c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s o f t h e upper s t a t e has been c a r r i e d o u t . B. E x p e r i m e n t a l D e t a i l s The a b s o r p t i o n s p e c t r u m o f 1 5 N 0 2 i n t h e r e g i o n 2480 - 2520 K was photographed i n t h e 23rd o r d e r o f a 7-m p l a n e g r a t i n g s p e c t r o g r a p h , u s i n g Kodak SA-1 p l a t e s . The a b s o r p t i o n path was 1.85 m, and photo- - 1 8 9 - graphs were t a k e n w i t h t h e c e l l a t room te m p e r a t u r e and y a r i o u s tempe- r a t u r e s up t o 2 0 0 ° C . D o p p l e r b r o a d e n i n g o f the l i n e s becomes a p p r e c i - a b l e a t the h i g h e r t e m p e r a t u r e s , and t h e h i g h e s t t e m p e r a t u r e where u s e f u l s p e c t r a c o u l d be o b t a i n e d was 1 2 0 ° C . The background continuum was s u p p l i e d by a 1000-W xenon a r c , and exposure t i m e s , w i t h t h e s p e c - trograph s l i t s e t t o 2 5 -jim, were a b o u t 1 h r . C a l i b r a t i o n l i n e s were s u p p l i e d by an i r o n - n e o n h o l l o w cathode lamp, t h e r e f e r e n c e w a v e l e n g t h s f o r which have been g i v e n by C r o s s w h i t e ( 5 ) . The p l a t e s were measured on a Grant a u t o m a t i c c o m p a r a t o r , and reduced t o vacuum wave numbers w i t h a f o u r - t e r m p o l y n o m i a l . C. A n a l y s i s o f t h e 2 4 9 0 f l Band o f 1 5 N0 Q The t h e o r y o f the energy l e v e l s o f asymmetric t op m o l e c u l e s i n m u l t i p l e t e l e c t r o n i c s t a t e s i s now f a i r l y w e l l u n d e r s t o o d ( 6 - 9 ) . F o r r e f e r e n c e we g i v e t h e m a t r i x elements r e q u i r e d f o r t h e NC^ s p e c - trum i n t h e absence o f h y p e r f i n e e f f e c t s : < NK |H | NK >= ]-(B + C)N(N + 1) + [A - h(B + C ) ] K 2 - % [ 0 ( J + 1) - N(N + 1) - 3 / 4 ] [ a o + a { 3 K 2 / ( N ( N + 1 ) ) - 1} - n K 4 / ( N ( N + 1 ) ) ] - A K K 4 - A N K N ( N + 1 ) K 2 - A N N 2 ( N + l ) 2 + + H K i K 6 + H K N K 4 N ( N + 1 ) < NK ± 2 | H | N K >= {^(B - C) -h b [ J ( J + 1 ) - N( N + 1) - 3 / 4 ] / [ N ( N + 1 ) ] - 6 N N ( N + 1) - % 6 K . [ K 2 + ( K ± 2 ) 2 ] } ( 7 . 1 ) x [ N ( N + 1) - K ( K ± 1)J 1 ' 2 [N(N + 1) - ( K ± 1 ) ( K ± 2)f 2; <N - 1 K | H | N K > = ( | a - 1 n K 2 ) K [ N 2 - £ Zf2/H- <N - I K ± 2 | H [ N K > = ± | b [ N ( N + 1) - K ( K ± I ) ] 5 * x [N + K - 1 ) ( N + K - 2 ) ] % / N . -190- The b a s i s s e t f o r E q . ( 7 . 1 ) i s t h e type I r r e p r e s e n t a t i o n c a s e ( b ) b a s i s |NJSK>, where t h e quantum numbers J and S have been s u p p r e s s e d because the elements a r e d i a g o n a l w i t h r e s p e c t t o them. The c o u p l i n g scheme (8) N + S = J (7.2) has been u s e d , and t h e phases, a r e a p p r o p r i a t e f o r the d e f i n i t i o n o f t h e r o t a t i o n a l a n g u l a r momentum v e c t o r N as a s p a c e - f i x e d o p e r a t o r r a t h e r t h a n a m o l e c u l e - f i x e d o p e r a t o r . The q u a r t i c c e n t r i f u g a l d i s t o r t i o n terms a r e c o m p l e t e , b ut o n l y the l a r g e s t s e x t i c terms have been i n c l u d e d . The s p i n - r o t a t i o n c o n s t a n t s i n C a r t e s i a n form (10) a r e r e l a t e d t o th o s e o f Eq. (7.1) by a o = " T ( e a a + e b b + £ c c ) ; a = " 1 ( 2 e a a ' e b b " e c c ) ; b = " J ( e b b ' ecc> < 7- 3> and n i s t h e l e a d i n g 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 t h e s p i n - r o - t a t i o n i n t e r a c t i o n parameter e ( c a l l e d n „ = , by Dixon and Duxbury ( 1 1 ) , a a a a a a n K by Brown and Se a r s (12) and A ^ by Cook e t a l . ( 1 3 ) ) . The ground s t a t e o f ^ N 0 2 has not been s t u d i e d as c o m p r e h e n s i v e l y 14 as t h a t o f N0 2. The microwave spectrum o f N0 2 i s v e r y s p a r s e because o f t h e l a r g e A r o t a t i o n a l c o n s t a n t , and a l t h o u g h i t has been c a r e f u l l y measured by B i r d e t a l . (14) and Lees e t a l . (15) the a v a i l a b l e l i n e s do n o t c a r r y enough i n f o r m a t i o n t o d e t e r m i n e a l l t h e c e n t r i f u g a l d i s t o r t i o n parameters r e q u i r e d t o d e s c r i b e t h e energy l e v e l s t o the p r e c i s i o n o f t h e o p t i c a l s p e c t r u m . The m i s s i n g d a t a have been s u p p l i e d f o r 1 4 N 0 9 by t h e h i g h - r e s o l u t i o n i n f r a r e d spectrum ( 1 6 ) , b u t as y e t the -191- 15 i n f r a r e d d a t a f o r NC^ 07-19) a r e l e s s c o m p l e t e . Even s o , the b e s t 15 c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s f o r N 0 2 a r e i n f a c t t h o s e from t h e microwave s p e c t r u m ( 1 4 ) , t h o u g h , as r e c o g n i z e d by Lees e t a l . ( 1 5 ) , they a r e not p a r t i c u l a r l y good because so few l i n e s a r e a v a i l a b l e . In a n a l y z i n g t h e 2490-A band o f N0 2 we proceeded as f o l l o w s . F i r s t we c o n v e r t e d t h e g r o u n d - s t a t e c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s (14) from K i v e l s o n and W i l s o n ' s T'S to Watson's f o r m a l i s m a c c o r d i n g to t h e r e c i p e o f Yamada and W i n n e w i s s e r ( 2 0 ) . Next the g r o u n d - s t a t e energy l e v e l s were c a l c u l a t e d employing t h e s e c o n d - o r d e r a p p r o x i m a t i o n f o r m u l a o f Cabana e t a l . (21) f o r the s p i n c o r r e c t i o n s t o t h e r o t a t i o n a l energy l e v e l s . With t h e s p i n parameters w r i t t e n i n s p h e r i c a l t e n s o r n o t a t i o n t h i s f o r m u l a i s E S p 1 n ( J = N ± 1/2) = ±l/2{(a - a Q ± l / 2 b 6 1 K ) N ( N ± 1) - 3 a K 2 + n K 4 - 9 a 2 K 2 [ l - K 2 / ( N + 1/2 ± 1/2) 2]/4B" }/(N + 1/2 ± 1/2). (7.4) where t h e term ± l b 6 i „ r e f e r s t o the two asymmetry components o f 2 1 >̂ K = 1 , not t o t h e two J components o f an (N,K) l e v e l . T h i s a p p r o x i m a t - i o n works v e r y w e l l , s i n c e N 0 2 i s so c l o s e to b e i n g a p r o l a t e symmetric t o p , and b r e a k s down (22) o n l y where, by a c c i d e n t , a n e a r - d e g e n e r a c y oc- c u r s between l e v e l s (N,K) and (N - 1, K + 2 ) . I t s advantage i s t h a t i t p e r m i t s t h e r o t a t i o n a l energy l e v e l s to be c a l c u l a t e d u s i n g t h e forma- l i s m f o r a s i n g l e t e l e c t r o n i c s t a t e , where the m a t r i c e s a r e h a l f the s i z e o f t h o s e r e q u i r e d f o r an e x a c t t r e a t m e n t o f a d o u b l e t s t a t e . A t t h i s s t a g e t h e l i n e s c o u l d be a s s i g n e d by s t a n d a r d c o m b i n a t i o n d i f f e r e n c e t e c h n i q u e s , s i n c e t h e r e a r e no p e r t u r b a t i o n s . The band i s a -192- normal asymmetric top type A Cparallel), band, with the N and K structures 15 ? both strongly degraded to the red. The head.of the N 0 2 249.0-A band is 14 illustrated in F i g . 7.1 with the corresponding band of N0 2 printed alongside in its correct relative position. The line assignments 15 14 refer to the lower print, which is the N 0 2 spectrum; the N0 2 assignments are not repeated, having been given in Ref. ( 4 ) . I t can be seen that the bands of the two isotopes are quite similar, particularly in the head region, but that they are sufficiently different that i t 15 is not possible to assign the N0 2 lines by simply comparing the two spectra. The two spin components of the ^P^(2) line are clearly resolved, 14 and their relative intensities are found to be exactly as in the N0 2 spectrum; this confirms the assignment of the ordering of the F-j( J = N + 1/2) and F 2 ( J = N - 1/2) spin component lines made in Ref. ( 4 ) . The argument goes as follows. The case (b) selection rule AN = AJ forbids any satellite branch transitions of the type that can be used in case (a) coupling situations to identify which spin component is which. At very low N, however, the relative intensities of the two spin components of a given line will be noticeably unequal, because the intensities are governed essentially by the number of components in the combining states, or in other words the J values. F o r the q P 1 ( 2 ) line the two com- ponents are J* = ~ - J " = 1-1 ( F £ - F 2 ) and J' = l l - J " = ll ( F ] - F ^ , so that the F-j component is expected to be stronger. I t can be seen in Fig. 7.1 that the stronger line is the long wavelength component. A very weak j " = ]~ - J " = ITJ- satellite transition is predicted to occur; this should fall between the F-j and F-, main branch lines, but has not been observed. 7.1 Head of the 2490-A bands ( 2 B, - X A. ) of N0 2 (above) and NC>2 (below). The l i n e assignments r e f e r to the ^N0„ spectrum. -194- F i g u r e 7.2 shows t h e K = 7 and 8 subbands, p r i n t e d from a p l a t e taken w i t h t h e gas a t 1 20 oC, where t h e branches r u n t o N v a l u e s r a t h e r h i g h e r than i n F i g . 7.1. The spectrum i s seen to be s t i l l q u i t e com- p l e x , d e s p i t e the wide s e p a r a t i o n o f t h e subbands,' but the c h a r a c t e r i s t i c l a r g e s p i n d o u b l i n g s o f the branches a r e c l e a r i n t h e r i g h t - h a n d p a r t o f t h e f i g u r e . The K dependence o f the s p i n s p l i t t i n g f o r c o n s t a n t N can be seen when t h e s p l i t t i n g s f o r K = 7 and 8 a r e compared. The u p p e r - s t a t e r o t a t i o n a l c o n s t a n t s were d e t e r m i n e d by a d d i n g the e n e r g i e s o f t h e unblended l i n e s to the l o w e r - s t a t e e n e r g i e s , t o o b t a i n the u p p e r - s t a t e term v a l u e s , and f i t t i n g t h e s e by l e a s t s q u a r e s . The H a m i l t o n i a n used f o r the upper s t a t e was the same as t h a t f o r the lower s t a t e . The r e s u l t s a r e g i v e n i n T a b l e 7.1. The q u a r t i c c e n t r i - f u g a l d i s t o r t i o n c o n s t a n t s o f t h e upper s t a t e a r e v e r y much what would be e x p e c t e d by a n a l o g y w i t h 1 4 N 0 2 ( 4 ) , but t h e s e x t i c c o n s t a n t s a r e c o m p l e t e l y d i f f e r e n t . The r e a s o n i s t h a t the g r o u n d - s t a t e H a m i l t o n i a n i n c l u d e s no 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 , s i n c e the microwave data do not a l l o w t h e s e c o n s t a n t s t o be d e t e r m i n e d . We t h e r e f o r e s e t a l l t h e s e x t i c c o n s t a n t s t o z e r o f o r the ground s t a t e , w h i c h means t h a t the u p p e r - s t a t e s e x t i c c o n s t a n t s i n T a b l e 7.1 a r e s t r i c t l y the d i f f e r e n c e s between t h e upper- and l o w e r - s t a t e s e x t i c d i s t o r t i o n c o n s t a n t s . When t h i s i s a p p r e c i a t e d t h e c o n s t a n t s a r e found t o be v e r y much as e x p e c t e d . For i n s t a n c e , i n 1 4 N 0 2 , Hal 1 i n and M e r e r (4) c o u l d o n l y d e t e r m i n e the constants H V K and H 1 ^ , a l l t h e o t h e r s b e i n g t o o s m a l l to measure, and t h e y found H' K - H" K = (-3.26 ± 0.46) * 1 0 " 6 cm" 1, H ' K N - H " K N = (-1.85 ± 1.30) x 1 0 " 8 cm' 1. (7.5) F i g . 7.1 K = 7 and 8 subbands of the 2490-A band of N0 ? ( i n the region 2502 - 2508 A). -196- A l l o w i n g f o r t h e mass d i f f e r e n c e t h e s e q u a n t i t i e s compare f a v o r a b l y w i t h t h o s e i n T a b l e 7.1. We d i d n o t a t t e m p t t o r e f i n e the g r o u n d - s t a t e q u a r t i c c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s i n t h i s work, even though" they c l e a r l y need impro- vement. The r e a s o n i s t h a t the most p o o r l y d e t e r m i n e d microwave con- s t a n t s a r e p r e c i s e l y t h o s e which the p a r a l l e l s e l e c t i o n r u l e s o f the e l e c - t r o n i c t r a n s i t i o n p r e v e n t us from i m p r o v i n g . S p e c i f i c a l l y , t h e c o n s t a n t i s p r o b a b l y t h e l e a s t w e l l d e t e r m i n e d b e c a u s e the microwave r e s u l t s go o n l y to K = 3. The e l e c t r o n i c s p e c t r u m a l s o c o n t a i n s i n f o r m a t i o n on the K - s t a c k s e p a r a t i o n s up t o K = 3 i n t h e form o f S R b r a n c h e s , but n a t u r a l l y i t i s f a r l e s s p r e c i s e ; f o r K > 4 t h e r e a r e no t r a n s i t i o n s o b s e r v e d i n the e l e c t r o n i c s p e c t r u m e x c e p t = 0 b r a n c h e s . A l s o the c o n s t a n t i s d i f f i c u l t t o o b t a i n from t h e e l e c t r o n i c s p e c t r u m because i t i s d e t e r m i n e d from d i f f e r e n c e s between the asymmetry components o f the low K s t a c k s , where t h e l i n e s i n v o l v e d l i e i n the most crowded r e g i o n o f the spectrum and b l e n d i n g i s most s e v e r e . A s e c o n d r e a s o n i s t h a t b e c a u s e o f the p r e d i s s o c i a t i o n i n the upper s t a t e the l i n e s a r e w i d e r than t h e D o p p l e r w i d t h a l o n e . Because o f the d i f f i c u l t i e s w i t h t h e ground s t a t e we have n o t been 15 a b l e t o o b t a i n as good a f i t t o t h e d a t a f o r N0 2 as H a l l i n and Merer (4) c o u l d f o r 1 4 N 0 2 , eyen though t h e d a t a t h e m s e l v e s a r e o f comparable -1 15 q u a l i t y . The o v e r a l l s t a n d a r d d e v i a t i o n i s 0.032 cm f o r N0 2, which -1 14 1 s h o u l d be compared t o 0.025 cm f o r N0 2- The a s s i g n e d r o t a t i o n a l l i n e s have been c o l l e c t e d i n Appendix VI , ^In o t h e r words we c o u l d perhaps have l o w e r e d t h e s t a n d a r d d e v i a t i o n f o r 1 5 N 0 2 by r e f i n i n g t h e g r o u n d - s t a t e c e n t r i f u g a l d i s t o r t i o n b ut because o f the problems d e s c r i b e d above t h e improvement would a l m o s t c e r t a i n l y have been a r t i f i c i a l . -197- TABLE 7.1 R o t a t i o n a l C o n s t a n t s f o r t h e 2490-$ Band o f 1 5 N 0 , ( c m " 1 ) 2 B 9 , 000 (upper s t a t e ) 000 ( g r o u n d s t a t e ) T o 40140. .339 ± 24 0.00 A 3, .9266 19 7.63047 B 0, .403300 94 0.433735 C 0, .363946 102 0.409440 1 0 6 A K 460 51 2297 io 6 ANK 13 .17 158 -17.69 io 6 AN 0 .439 31 0.2817 1 0 6 A K 8 .8 86 2.704 1 0 5 A N 0, .098 34 0.03189 1 0 6 H K -1 .93 42 - io 8 HKN -1 .9 20 - £ aa -0 .1724 40 0.1718 e b b 0 0.000256 e c c 0 -0.003163 n. - -0.000103 a 0 .032 - N o t e s : Ground s t a t e r o t a t i o n a l c o n s t a n t s from r e f . ( 1 4 ) , w i t h t h e quar-t i c c e n t r i f u g a l d i s t o r t i o n c o n v e r t e d from the T v a l u e s , which a r e T a a a a = (-9.12 ± 0 . 1 3 ) x 1 0 ' 3 T L k , . = (-1.382 ± 0.005) x 1 0 " 5 cm"1 aaaa j . bbbD , T a a b b = C 5 . 8 7 ± 0 . 0 6 ) x 1 0 " b x a b a b = ( 8 . 1 6 ± 0 . 0 3 ) x 10 Ground s t a t e s p i n c o n s t a n t s from r e f . ( 1 5 ) e x c e p t n which was c a l c u l a t e d a s T a a a a - eaa/2A (.11). U n c e r t a i n t i e s a r e t h r e e s t a n d a r d d e v i a t i o n s , i n u n i t s o f t h e l a s t s i g n i f i c a n t f i g u r e q u o t e d . -198- D. C c n c l us i o n The 2 4 9 0 4 bands o f both 1 4 N 0 2 (.4) and 1 5 N 0 2 a r e found t o be s l i g h t l y p r e d i s s o c i a t e d , and t o have e s s e n t i a l l y t h e same l i n e w i d t h s . The p r e d i s s o c i a t i o n l i f e t i m e was d e t e r m i n e d i n Ref. (4) to be 42 ± 5 p s e c . 14 The p o s s i b i l i t y o f s e l e c t i v e d i s s o c i a t i o n o f N0 2 i n t h e p r e s e n c e o f 15 N0 2, and v i c e v e r s a , has been e x p l o r e d i n t h i s work. The two s p e c t r a a r e compared i n F i g . 7.1 I t i s seen t h a t t h e r e would be no d i f f i c u l t y 15 14 d i s s o c i a t i n g N0 2 i n t h e p r e s e n c e o f N0 2 because the i s o t o p e s h i f t o f 14.40 cm" 1 f o r t h e q R Q heads means t h a t many s t r o n g ^ N 0 2 l i n e s l i e i n a r e g i o n where o n l y weak s p a r s e S R l i n e s o f 1 4 N 0 2 f a l l . On the 14 15 o t h e r hand, d i s s o c i a t i n g N0 2 i n t h e p r e s e n c e o f N0 2 would r e q u i r e n a r r o w - l i n e l a s e r s s p e c i f i c a l l y tuned t o c e r t a i n w a v e l e n g t h s . The most p r o m i s i n g r e g i o n s appear t o be i n t h e heads o f the qR-| and q R „ branches 14 14 o f N0 9, where many c l o s e - l y i n g s t r o n g N0 2 l i n e s f a l l i n gaps between the 1 5 N 0 2 , qP.| , q R 2 , and q P 2 l i n e s . The e x a c t wavelengths can be c a l - 15 c u l a t e d from the t a b l e s o f a s s i g n m e n t s g i v e n i n T a b l e I I I f o r N0 2 and the Appendix t o Ref. (4) f o r 1 4 N 0 2 - c 15 S p e c t r o s c o p i c a l l y , the 2490-A band o f N0 2 c o n f i r m s the a n a l y s i s 14 o f t h e c o r r e s p o n d i n g band o f N0 2 i n d e t a i l , and poses no q u e s t i o n s . The need f o r a more d e t a i l e d e x a m i n a t i o n o f t h e i n f r a r e d s p e c t r u m o f 15 NO <2 i s p o i n t e d o u t . I t i s l i k e l y t h a t the r e s u l t i n g changes i n the g r o u n d - s t a t e c o n s t a n t s can be t r a n s f e r r e d , d i r e c t l y to t h e u p p e r - s t a t e c o n s t a n t s r e p o r t e d i n t h i s work, s i n c e we have e s s e n t i a l l y d e t e r m i n e d t h e d i f f e r e n c e s between the upper- and l o w e r - s t a t e c o n s t a n t s i n t h i s work. However, i t i s not i m p o s s i b l e t h a t t h e u p p e r - s t a t e c o n s t a n t s may- need to be reworked i f t h e g r o u n d - s t a t e changes a r e c o n s i d e r a b l e , s i n c e •199- the r e l a t i o n s h i p between t h e two s.ets i s n o t one-to-one because o f the l a r g e changes i n the r o t a t i o n a l c o n s t a n t s on e l e c t r o n i c e x c i t a t i o n . -200- C h a p t e r 8 F o u r i e r T r a n s f o r m S p e c t r o s c o p y o f VO; 4 4 - R o t a t i o n a l S t r u c t u r e i n the A J I - X E System n e a r 10500 & -201- A. I n t r o d u c t i o n Vanadium monoxide, V0, i s p r e s e n t i n c o n s i d e r a b l e amounts i n t h e atmospheres o f c o o l s t a r s , t o the e x t e n t t h a t i t s two e l e c t r o n i c band systems i n the n e a r i n f r a - r e d a r e used f o r t h e s p e c t r a l c l a s s i f i c a t i o n o f s t a r s o f t y p e s M7-M9 .(1). Both o f t h e s e s y s t e m s , A-X near 10500 A and B 4 n - X 4 E _ near 7900 ft, were i n f a c t f i r s t found i n s t e l l a r s p e c t r a ('2,3) b e f o r e l a b o r a t o r y work, r e s p e c t i v e l y by L a g e r q v i s t and S e l i n (4,) and Keenan and S c h r o e d e r ( 5 ) , p r o v e d t h a t V0 i s the c a r r i e r . The purpose o f t h i s chapter i s to r e p o r t r o t a t i o n a l a n a l y s e s o f the (0,0) and (0,1) bands o f t h e A-X system from h i g h d i s p e r s i o n F o u r i e r t r a n s f o r m e m i s s i o n 4 4 - s p e c t r a ; the A-X system i s shown to be a n o t h e r n- E t r a n s i t i o n . The A 4 n s t a t e o f V0 i s f o u n d to have q u i t e s m a l l s p i n - o r b i t c o u p l i n g , so t h a t the r o t a t i o n a l and h y p e r f i n e s t r u c t u r e f o l l o w s case (a^) c o u p l i n g a t low r o t a t i o n a l quantum numbers, but i s a l m o s t t o t a l l y u n c o u p l e d to c a s e ( b Q 1 ) c o u p l i n g a t the h i g h e s t o b s e r v e d quantum numbers. The hyper- 51 f i n e s t r u c t u r e caused by t h e V n u c l e u s (I = 7/2) i s n o t r e s o l v e d i n t h e s p e c t r a r e p o r t e d h e r e , but an i n t e r e s t i n g r e s u l t i s t h a t the h y p e r f i n e p a r a meter b f o r t h e A 4 n s t a t e can be e s t i m a t e d from the l i n e shapes a t h i g h N v a l u e s and i s f o u n d to be e s s e n t i a l l y t h e same as i n the ground X 4E" s t a t e . The c o n c l u s i o n i s t h a t the A 4 n s t a t e comes from an e l e c t r o n c o n f i g u r a t i o n c o n t a i n i n g an u n p a i r e d 4so e l e c t r o n , as does t h e ground s t a t e . 4 In c o n t r a s t t o t h e o t h e r e x c i t e d s t a t e s o f V0 the A n,v = 0 l e v e l i s u n p e r t u r b e d r o t a t i o n a l l y ; i t t h e r e f o r e p r o v i d e s one o f the v e r y few examples known where the energy f o r m u l a e f o r 4 n s t a t e s can be checked d i r e c t l y a g a i n s t o b s e r v a t i o n . -202- B. E x p e r i m e n t a l d e t a i l s The n ear i n f r a - r e d e l e c t r o n i c t r a n s i t i o n s o f V0 i n the r e g i o n 6000-14000 cm" 1 were r e c o r d e d i n e m i s s i o n u s i n g the 1 m e t e r F o u r i e r T r a n s f o r m s p e c t r o m e t e r c o n s t r u c t e d by Dr. J.W. B r a u l t f o r the McMath S o l a r T e l e s c o p e a t K i t t Peak N a t i o n a l O b s e r v a t o r y , T u c s o n , U.S.A. The s o u r c e was a microwave d i s c h a r g e through f l o w i n g VOCil-j and h e l i u m a t low p r e s s u r e s , which was f o c u s e d d i r e c t l y i n t o t h e a p e r t u r e o f t h e s p e c t r o - m e t e r . An i n d i u m a n t i m o n i d e d e t e c t o r c o o l e d by l i q u i d n i t r o g e n was used, and the r e s o l v i n g power o f t h e s p e c t r o m e t e r was s e t t o a p p r o x i m a t e - l y 800,000. F o r t y - t w o i n t e r f e r o g r a m s , each t a k i n g s i x minutes to r e c o r d , were co-added f o r the f i n a l t r a n s f o r m . The r e s u l t i n g s p e c t r u m , c o n s i s t - i n g o f t a b l e s o f e m i s s i o n i n t e n s i t y a g a i n s t wave number f o r e v e r y 0.013608 cm" 1, was p r o c e s s e d by a t h i r d d e gree p o l y n o m i a l f i t t i n g p ro- gramme to e x t r a c t the p o s i t i o n s o f t h e l i n e peaks. C. Appearance o f the s p e c t r u m The s p e c t r u m o f V0 i n the n e a r i n f r a - r e d down to 6000 cm" 1 c o n s i s t s 4 4 - 4 4 -o f the two e l e c t r o n i c t r a n s i t i o n s B Ji-X z and A n-X z . The B-X system i s v e r y much s t r o n g e r than t h e A-X system under our d i s c h a r g e c o n d i t i o n s , so t h a t the B-X p r o g r e s s i o n s and sequences mask most o f t h e A-X system e x c e p t f o r the (0,0) and (0,1) bands.. Even the (0,0) band o f the A-X s y s t e m (which i s by f a r t h e s t r o n g e s t band) i s n o t f r e e from o v e r l a p p i n g B-X s t r u c t u r e , which causes some d i f f i c u l t y i n the a n a l y s i s . The main heads o f the A-X (0,0) band a r e i l l u s t r a t e d i n F i g . 8.1; each o f the S R f o u r sub-bands produces one s t r o n g head ( R 4 3 , R 3, Q 2 1 and R-j), and t h e r e VO A 4n-X 4E\(0,0) 4 n 5/2 4 n V2 4 n 1 / 2 F l . 8.1 Fourier t r a n s f e r , spectrum of VO In the re g i o n 9410-9570 cm'1 shoving the heads A 4H - X 4E (0,0) band of VO. -204- 4 4 - i s a l s o a l e s s p r o m i n e n t Q-j head i n t h e Ji - z sub-hand. Two o t h e r heads, b e l o n g i n g t o the B-X (.1,4) band, appear i n t h e r e g i o n o f t h e 4 4 - n 5 / 2 " E sub-band; they have n o t been i d e n t i f i e d i n the F i g u r e , though t h e i r branch, s t r u c t u r e i s r e a d i l y p i c k e d o u t a t h i g h e r d i s p e r s i o n . The A-X (0,1) band i s q u a l i t a t i v e l y s i m i l a r , though s i n c e i t i s weaker t h e background o f B-X l i n e s i s more t r o u b l e s o m e . The A-X (1,0) band i s so h e a v i l y o v e r l a p p e d by B-X s t r u c t u r e t h a t we have not been S _ i a b l e to a n a l y s e i t ; t h e R ^ head appears t o be a t 10503.3 cm"' but even t h i s i s not d e f i n i t e . 4 4 D. Energy l e v e l s o f JI and z s t a t e s 4 Energy l e v e l s f o r n e l e c t r o n i c s t a t e s have been c o n s i d e r e d by a number o f a u t h o r s (6-11"). The most d e t a i l e d t r e a t m e n t i s t h a t o f Femenias (.9), who has g i v e n a f u l l e x p l a n a t i o n o f how t o c a l c u l a t e the m a t r i x elements f o r t h e h i g h e r o r d e r c e n t r i f u g a l d i s t o r t i o n terms. D e t a i l e d a n a l y s e s o f 4 n s t a t e s , a g a i n s t which t o t e s t the f o r m u l a e , a r e l e s s common; t h e b e s t examples come from t h e s p e c t r a o f 0^ + ( 1 0 ) and NO ( 1 2 ) . 4 z s t a t e s , on t h e o t h e r hand, a r e much more numerous, and have been e x t e n s i v e l y t r e a t e d ( 6 , 7 , 9 , 1 0 , 1 3 - 1 7 ) r I t w i l l t h e r e f o r e o n l y be n e c e s s a - r y t o s k e t c h t h e H a m i l t o n i a n and i t s d e r i v a t i o n , and to g i v e t h e m a t r i c e s we have used. F o l l o w i n g van y l e c k ( l 8 ) w e t a k e the r o t a t i o n a l H a m i l t o n i a n , the f i r s t and s e c o n d - o r d e r s p i n - o r b i t i n t e r a c t i o n s and the s p i n - r o t a t i o n i n t e r a c t i o n , r e s p e c t i v e l y , as -205- H = B ( r ) ( J - L - S ) 2 + A ( r ) L . S + |A( r) ( 3 S 2 - S 2 ) . + Y(r)(.J-S) .S (.8.1). The e x p a n s i o n o f the parameters A, B, A and y» which a r e f u n c t i o n s o f the i n t e r n u c l e a r d i s t a n c e r , i n terms o f t h e normal c o o r d i n a t e , produces c e n t r i f u g a l d i s t o r t i o n t e r m s , which a r e c o n v e n i e n t l y w r i t t e n i n o p e r a t o r form as H r d = - D ( J - L - S ) 4 + W \ D I ( J - L - S ) 2 , L z S z ] + 'cd, + \ h^3SU2)' + * D I C O . - L - : U 2 . { & ^ ] + ( 8 - 2 ) where [ x , y ] + means t h e a n t i - c o m m u t a t o r xy + y x , which i s n e c e s s a r y t o 4 p r e s e r v e H e r m i t i a n form f o r the m a t r i c e s . The A - d o u b l i n g o f the n s t a t e 4 was c a l c u l a t e d by s e t t i n g up t h e 12x12 m a t r i x f o r a n s t a t e i n t e r a c t i n g w i t h a s i n g l e 4 z~ s t a t e a c c o r d i n g t o t h e f i r s t two terms o f eq. (8.1,), a p p l y i n g a Wang t r a n s f o r m a t i o n t o c o n v e r t t o a p a r i t y b a s i s and t r e a t i n g t h e elements o f f - d i a g o n a l i n A by sec o n d o r d e r p e r t u r b a t i o n t h e o r y . The e f f e c t i s as i f t h e r e were an o p e r a t o r H L D = ^ o + p + q ) ( S 2 + S 2 M ( p + 2 q ) ( j + S + + J _ S _ ) + ^ ( j J + J 2 ) (8.3) a c t i n g o n l y w i t h i n t h e m a n i f o l d o f t h e 4 n s t a t e ( l 1 , 1 9 ) - The A - d o u b l i n g parameters (o+p+q), (p+2q) and q a r e r e l a t e d t o m a t r i x elements o f the s p i n - o r b i t o p e r a t o r , as g i v e n i n r e f . ( l l ) . The 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 s to eq. (8.3) a r e o b t a i n e d i n the same way t h a t eq. (8.2) i s 1 2 2 c o n s t r u c t e d from eq. ( 8 . 1 ) . The s p i n - s p i n o p e r a t o r s -g a(.r)(S ++S_) and 4 A ^ ( r ) ( 3 S 2 - S 2 ) a r e i n c o r p o r a t e d i n t o t h e terms i n (o+p+q) and A, r e s p e c t i v e l y . T a b l e 8.1 Matrix elements of the r o t a t i o n a l Hamiltonian f o r a \ s t a t e i n case (a) coupling. |-S> T . L + ( B - ^ A 0 + 2 A d ) (z+1) -D(z +5z+l) i 3(J+'-i)D o+p+q - / 3 l [ B - V f - A D - 2 D ( z + 2 ) ] */3[(o+p+q) + U + 2 ) D 0 + p + q + ' 5 ( 2 z - D D p + 2 q ] \ + ( B - ' 5 A 0 - 2 v 0 ) ( z + 3 ) - D ( z 2 + 13z + 5 ) ' ( J + ' , ) [ ( p + 2 q ) +30 + D o (z +3)+D (z -1 )] o+p+q p+2q q -/Slz^T} [ZD(J+b) « i(p*Zq)» D Q + p + q ^ ( z + D D p + 2 q * 4 ( z - 2 ) D q ] - 2 / r i [ B - ' - 5 Y - 2 A D - 2 0 ( z + 2 ) l ^ ( J + g < q i D p + 2 q +D ( z + 2 ) ) ] q Symmetr ic T 3 / y + ( B * ' s A D - 2 » D ) ( z + l ) - D ( z 2 + 9 z - 1 5 ) •(z- l)(J+»i)D n * / f z - l ) ( z - 4 ) [!,q 3 q 3°P+2q + , - i 0 " ( Z - 2 ) ] / 3 ( z - 1 ) ( z - 4 ) [ - 2 0 -+4D ( J + 4 ) ] - / 3 ( z - 4 ) [B - ' - i Y+A D - 2 0 ( z - 2 ) ] Tr, / : ,+(B+^A D+2A 0) ( z - 5 ) - 0 ( z 2 - 7 z + l 3 ) z - (J+'-s) 2 . Upper and lower s i g n s r e f e r to e and f r o t a t i o n a l l e v e l s r e s p e c t i v e l y . The b a s i s f u n c t i o n s | Jsi • have been a b b r e v i a t e d to |si> -207- Table 8.2 M a t r i x elements f o r s p i n and r o t a t i o n i n a Z i n case (a) coupling . I!> 2 2x + Bx - D(x +3x) -/3X[B-1 2Y-Y s- 12Y D(X+7=I-{2J+1}) - \ i - 3 Y [ )x -2D(x+2)*{J+**})] -2\ + B(x+4)-D[(x+4) 2+7x+4] symmetri c - 7 y - Y d ( 7 X + 1 6 ) * 2 [ B - i 5 Y - ! s Y D ( x + n ) + | Y s - 2D(x+4)](J+is) x = ( j + 4 ) 2 - l . Upper and lower s i g n s g i v e t h e e( F-j and F 3 ) and f ( F 2 and F 4 ) l e v e l s r e s p e c t i v e l y . The b a s i s f u n c t i o n s |JZ> have been w r i t t e n |z> -208- The r e s u l t i n g H a m i l t o n i a n m a t r i c e s w hich we have u s e d a r e gi.yen i n T a b l e s 8.1 and 8.2 ( f o r 4 n and 4 i s t a t e s r e s p e c t i v e l y ) . The X 4E~,v=0 parameters were not v a r i e d i n t h i s work s i n c e they have been d e t e r m i n e d 4 - 4 - w i t h g r e a t p r e c i s i o n from the C z -J z t r a n s i t i o n u s i n g s u b - D o p p l e r t e c h n i q u e s ( 1 7 ) . The parameter y$ i n the Z m a t r i x r e p r e s e n t s t h e t h i r d - o r d e r s p i n - o r b i t c o n t r i b u t i o n t o the s p i n - r o t a t i o n i n t e r a c t i o n ( 1 6 , 1 7 ) ; n e i t h e r y^ n o r the 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 y^ appears i n the 4 H m a t r i x because they a r e not needed. H y p e r f i n e e f f e c t s have n o t been c o n s i d e r e d i n T a b l e s 8.1 and 8.2 be- c a u s e t h e h y p e r f i n e s t r u c t u r e i s not r e s o l v e d . However, w i t h t h e l a r g e 51 s p i n and n u c l e a r m a g n e t i c moment o f V (I = 7/2), the h y p e r f i n e s t r u c t u r e i s i m p o r t a n t i n d e t e r m i n i n g the d e t a i l s o f t h e b r a n c h s t r u c t u r e , as w i l l be shown below. E. A n a l y s i s o f t h e b ranch s t r u c t u r e ft n _ R a t h e r s u r p r i s i n g l y , t h e a n a l y s i s o f the h'Ti-Vz bands o f VO p r o v e d to be r e m a r k a b l y d i f f i c u l t b ecause o f u n r e s o l v e d h y p e r f i n e s t r u c t u r e e f f e c t s and o v e r l a p p i n g sequence bands from the B-X t r a n s i t i o n . The p r o - blem w i t h t h e h y p e r f i n e s t r u c t u r e i s t h a t o n l y when the h y p e r f i n e 'widths' o f t h e c o m b i n i n g l e v e l s making up a r o t a t i o n a l l i n e a r e t h e same does t h e s p e c t r u m c o n s i s t o f s h a r p r o t a t i o n a l l i n e s (where the e i g h t h y p e r f i n e t r a n s i t i o n s l i e on t o p o f each o t h e r ) . S i n c e t h e f o u r e l e c t r o n s p i n components o f t h e ground s t a t e have h y p e r f i n e w i d t h s t h a t d i f f e r from one t o t h e n e x t by about 0.2 cm" 1, r o t a t i o n a l l i n e s w i t h t h e same upper s t a t e which go t o d i f f e r e n t e l e c t r o n s p i n components o f the ground s t a t e -209- have n o t i c e a b l y d i f f e r e n t l i n e - w i d t h s . The b r o a d e r the l i n e - w i d t h s t h e more the i n t e n s i t y i s s p r e a d o u t , and t h e more t h e l i n e tends to g e t l o s t i n the background o f o v e r l a p p i n g B-X s t r u c t u r e . T h e r e f o r e a l - though a n- E t r a n s i t i o n s h o u l d have 48 b r a n c h e s , m o s t o f them a r e broadened beyond r e c o g n i t i o n by t h e h y p e r f i n e s t r u c t u r e i n t h i s c a s e . There are o n l y two r e g i o n s o f c l e a r branch s t r u c t u r e i n t h e (0,0) band. One o f t h e s e , shown i n F i g . 8.2, l i e s between the two s h o r t e s t w a v e l e n g t h heads. The o b v i o u s b r a n c h , l a t e r i d e n t i f i e d as Q 4 3 , c o u l d be a s s i g n e d a t once t o t h e F 3 s p i n component o f t h e ground s t a t e because i t c o n t a i n s the c h a r a c t e r i s t i c i n t e r n a l h y p e r f i n e p e r t u r b a t i o n p a t t e r n a t N" = 15 d i s c o v e r e d by R i c h a r d s and Barrow (20 ) i n t h e B--X and C-X s y s t e m s . T h i s i n t e r n a l h y p e r f i n e p e r t u r b a t i o n i s a r e m a r k a b l e o c c u r r e n - c e , where t h e F 2 and F 3 e l e c t r o n s p i n components (N = J - ^ and N = J+h r e s p e c t i v e l y ) would c r o s s a t N = 15, because o f t h e p a r t i c u l a r v a l u e s o f the r o t a t i o n a l and s p i n p a r a m e t e r s , were i t n o t f o r the f a c t t h a t they d i f f e r by one u n i t i n J , and t h e r e f o r e i n t e r a c t through m a t r i x elements o f t h e h y p e r f i n e H a m i l t o n i a n c f the t y p e AiN = AF = 0, AJ = ± 1 . E x t r a l i n e s a r e i n d u c e d , and, s i n c e the d e t a i l e d c o u r s e o f the ground s t a t e l e v e l s i s known ( 1 7 ) , t h e i r p o s i t i o n s t e l l whether a branch c o n t a i n i n g them has F 2 " o r F 3 " , and a l s o g i v e i t s N-numbering. Given t h e numbering o f the o b v i o u s F 3 " b r a n c h , the o t h e r t h r e e F 4 ' b r anches marked i n F i g . 8.2 c o u l d be numbered e a s i l y u s i n g ground' s t a t e s p i n and 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 , The R^ and Q 4 branches a r e h y p e r f i n e - b r o a d e n e d , and even though t h e y a r e i n t r i n s i c a l l y s t r o n g t h e y a r e by no means o b v i o u s i n t h e s p e c t r u m . At t h i s s t a g e the lower s t a t e s o f the b ranches were known, b u t the n a t u r e o f t h e upper s t a t e was s t i l l  -211- u n c l e a r . The o t h e r r e g i o n o f o b v i o u s b r a n c h s t r u c t u r e i s t h e t a i l o f the band, p a r t o f which i s i l l u s t r a t e d i n F i g . 8.3. T h e r e a r e a t l e a s t t e n s h a r p b r a n c h e s i n t h i s r e g i o n , but o n l y e i g h t o f them a c t u a l l y b e l o n g t o t h e A-X (0,0) band. A f u r t h e r c o m p l i c a t i o n i s t h a t t h e r e a r e no ground s t a t e c o m b i n a t i o n d i f f e r e n c e s c o n n e c t i n g any o f t h e s e e i g h t . The a n a l y - s i s was p e r f o r m e d by comparing the (0,0) and(0,1) bands, s i n c e the s e p a r a t i o n s between c o r r e s p o n d i n g ( N , J ) l e v e l s o f the )( 4E~ v=0 and 1 l e v e l s a r e known from the a n a l y s i s o f t h e C-X s y s t e m ( 2 l ) . T h i s method g i v e s a t most two p o s s i b l e N-numberings f o r t h e b r a n c h e s , but i t i s l e s s e asy t o d e t e r m i n e t h e ground s t a t e s p i n component s i n c e t h e i n t e r v a l s a r e v e r y n e a r l y the same f o r t h e f o u r s p i n components. E v e n t u a l l y a l l e i g h t o f t h e s e branches were i d e n t i f i e d , and a s s i g n e d to t h e i r r e s p e c - t i v e ground s t a t e s p i n components. The r e s u l t i n g p a t t e r n can be i n t e r - p r e t e d as t h e Q and P main branches o f a 4n-4z t r a n s i t i o n where the 4n s t a t e i s c l o s e t o c a s e (b) c o u p l i n g a t t h e s e h i g h N v a l u e s , and a l l f o u r components show A - d o u b l i n g . The a n a l y s i s i s c o n f i r m e d by the i d e n t i f i c a - t i o n o f t h e f o u r R b r a n c h e s , and v a r i o u s weak h y p e r f i n e - b r o a d e n e d s p i n s a t e l l i t e b r a n c h e s . The b r a n c h i s i n t e r e s t i n g b e c a u s e i t i s a s h a r p b r a n c h a t the h i g h N v a l u e s o f F i g . 8.3, b u t h y p e r f i n e - b r o a d e n e d a t the l o w e r N v a l u e s o f F i g . 8.2. I t i s p o s s i b l e t o f o l l o w the - b r a n c h o v e r t h e complete range o f N v a l u e s , and to see how i t changes from b r o a d t o narrow f a i r l y q u i c k l y i n t h e r e g i o n N=40-50. The r e a s o n f o r t h e sudden d i s a p p e a r a n c e R R o f t h e Q 4 3 b r a n c h near N=35 ( s e e F i g . 8.2) i s then c l e a r - the Q43 1 cm-' -213- branch i s p r o m i n e n t a t low N because the h y p e r f i n e s t r u c t u r e o f the 4 4 - Jl F 4 l e v e l i s i n i t i a l l y t he same as t h a t o f t h e J. z l e v e l , b ut w i t h i n c r e a s i n g N s p i n - u n c o u p l i n g changes t h e 4 n h y p e r f i n e l e v e l p a t t e r n u n t i l 4 - "R a t h i g h N i t becomes the same as X E F^; as a r e s u l t t h e b r a n c h TO becomes bro a d e n e d . In a d d i t i o n t h e i n t e n s i t y o f Q ^ , w h i c h i s a s p i n s a t e l l i t e b r a n c h t h a t becomes f o r b i d d e n i n a 4 n ( b ) - 4 E ( b ) t r a n s i t i o n , must d i m i n i s h as s p i n - u n c o u p l i n g s e t s i n . 4 4 What emerges f i n a l l y i s a ' t e x t - b o o k ' example o f a n - E t r a n s i t i o n 4 where t h e n s t a t e has q u i t e s m a l l s p i n - o r b i t c o u p l i n g so t h a t i t changes f a i r l y q u i c k l y from case (a) t o c a s e (b) c o u p l i n g . The 4 n s t a t e i s shown to be r e g u l a r ( w i t h a p o s i t i v e s p i n - o r b i t c o u p l i n g c o n s t a n t ) because t h e r e i s no d e t e c t a b l e A-doubling i n t h e F^ component ( 4 n^ 2 ) b e f o r e about N=45, whereas t h e o t h e r t h r e e s p i n components show A - d o u b l i n g e f f e c t s a l m o s t from t h e i r f i r s t l e v e l s . The A-doubling and s p i n - u n c o u p l - i n g p a t t e r n s a r e shown q u a l i t a t i v e l y i n F i g . 8.4, where the upper s t a t e e n e r g y l e v e l s , s u i t a b l y s c a l e d , are p l o t t e d a g a i n s t J ( J + 1 ) . The c u r v a t u r e i n t he p l o t s o f F i g . 8.4 i s a consequence o f the s p i n - u n c o u p l i n g . The a s s i g n e d l i n e s o f the (0,C) and (0,1) bands o f the A-X system a r e g i v e n i n t he A ppendix; o n l y the s h a r p l i n e s a r e l i s t e d , because t h e y are s u f f i - c i e n t to d e t e r m i n e the upper s t a t e c o n s t a n t s , and i n any c a s e i t i s o f t e n q u i t e d i f f i c u l t t o o b t a i n t h e e x a c t l i n e c e n t r e s f o r the h y p e r f i n e - broadened b r a n c h e s . F. L e a s t s q u a r e s f i t t i n g o f the data One o f the u n e x p e c t e d e f f e c t s o f t h e ground s t a t e i n t e r n a l h y p e r f i n e p e r t u r b a t i o n i s t h a t the F 2 " and F 3 " l e v e l s a r e a p p r e c i a b l y s h i f t e d from 2000 4000 6000 8000 J ( J + 1 ) F l g 8 . A Reduced energy leve!s o£ the A** s t a t e of VO p l o t t e d against J O + l ) . The ,ua„tity p l o the upper s t a t e t e r - value l e s s (0.50865 + 0.00365U ) C J * 2 - 6.7 « 10 O H ) « Table 8.3 Corrections applied to the observed F 2 and F 3 l i n e p o s i t i o n s to a l l o w f o r the i n t e r n a l hyperfine p e r t u r b a t i o n s h i f t s . N F2 F 3 N F2 F 3 N F2 F 3 4 -0.030 -0.003 14 -0.079 +0.055 24 +0.029 -0.026 5 -0.031 +0.008 15 ±0. .080 25 0.027 -0.025 6 -0.031 0.012 16 +0.075 -0.086 26 0.026 -0.024 7 -0.033 0.017 17 0.065 -0.075 27 0.025 -0.022 8 -0.034 0.022 18 0.051 -0.060 28 0.023 -0.021 9 -0.036 0.025 19 0.047 -0.058 29 0.023 -0.020 10 -0.053 0.031 20 0.043 -0.043 30 0.022 -0.019 11 -0.060 0.033 21 0.038 -0.039 31 0.021 -0.018 12 -0.065 0.034 22 0.035 -0.031 32 0.021 -0.018 13 -0.070 0.043 23 0.032 -0.028 33 0.020 -0.017 I ro i The c o r r e c t i o n s were o b t a i n e d by s u b t r a c t i n g t h e r o t a t i o n a l e nergy c a l c u l a t e d i n t h e absence o f h y p e r f i n e e f f e c t s from a weighted a v e r a g e o f the r o t a t i o n a l - h y p e r f i n e e n e r g i e s g i v e n by a f u l l c a l c u l a t i o n o f t h e h y p e r f i n e s t r u c t u r e . -216- the p o s i t i o n s t h a t t h e y would have i n the absence o f h y p e r f i n e s t r u c t u r e . T h e r e f o r e i t i s n e c e s s a r y t o c o r r e c t a l l t h e l i n e p o s i t i o n s i n t h e branches i n v o l v i n g F 2 o r F 3 l o w e r l e v e l s f o r t h i s e f f e c t . I t may seem s u r p r i s i n g t h a t a h y p e r f i n e e f f e c t can s h i f t t h e p o s i t i o n s o f r o t a t i o n a l l e v e l s , but the h y p e r f i n e m a t r i x e l e m e n t a c t i n g between F 2 and F 3 l e v e l s w i t h t h e same N v a l u e i s ab o u t 0.08 cm" 1, w h i l e the z e r o - o r d e r s e p a r a t i o n o f t h e F 2 and F3 l e v e l s (which depends cn the s p i n - r o t a t i o n p a r a m e t e r y) remains l e s s t h a n 1 cm" 1 even some d i s t a n c e from t he N-value o f the i n t e r n a l p e r t u r b a t i o n . The c a l c u l a t e d s h i f t s a r e g i v e n i n T a b l e 8.3. A f t e r a p p l y i n g t h e s e c o r r e c t i o n s to the F 2 " and F^" branches we f i t t e d t h e l i n e s d i r e c t l y t o the a p p r o p r i a t e d i f f e r e n c e s between e i g e n - 4 4 - v a l u e s o f the n and 1 m a t r i c e s . No attempt was made to v a r y t h e 4 - X 1 , v=0 parameters i n the p r e s e n t work s i n c e t h e y have been d e t e r m i n e d w i t h h i g h p r e c i s i o n by the s u b - D o p p l e r s p e c t r a o f ( 1 7 ) , where t h e r e s o - l u t i o n i s a f a c t o r o f t e n h i g h e r . Our p r o c e d u r e i s t h e r e f o r e e q u i v a l e n t 4 t o f i t t i n g t h e term v a l u e s o f t h e A n, v=0 s t a t e t o the e i g e n v a l u e s o f T a b l e 8.1. The (0,1) band was then f i t t e d s i m i l a r l y , but w i t h the A 4 n upper s t a t e parameters f i x e d a t t h e v a l u e s d e r i v e d from the (0,0) band; the r e s u l t s g i v e e s s e n t i a l l y the d i f f e r e n c e s between t h e parameters f o r 4 - X I v=0 and v = l . The f i n a l parameters a r e assembled i n T a b l e 8.4. The o v e r a l l s t a n - d a r d d e v i a t i o n s l i s t e d c o r r e s p o n d to u n i t w e i g h t i n g o f a l l t h e d a t a ; they a r e n o t as low as we had e x p e c t e d , b u t i n view o f t h e b l e n d i n g and the unusual l i n e shapes produced by u n r e s o l v e d h y p e r f i n e s t r u c t u r e e f f e c t s i n some o f the branches we see no r e a s o n f o r c o n c e r n . 4 4 - Table 8.4 Parmeters derived from r o t a t i o n a l a n a l y s i s of the A II - X S (0,0) and (1,0) bands of VO i n cm -1 A 4n, v = 0 \ l 2 T 3 / 2 9555.500 ±0. 011 (3o) T 0 9512.432 ±0. 017 B 7 \ l 2 9477.830 ±0. 023 10 D T - . / 2 9449.710 + 0. 021 A B 0.516932 ±0. 000006 If 107D 6.782 ±0. .010 10 5v s q -0.000151 + 0, .000012 10 8 Y D p+2q -0.01349 ±0 .00027 o+p+q 2.107 ±0 .008 Y 0.00383 ±0 .00010 10?D Q 0.023 ±0 .022 1 0 ?V2q -2.32 i O .68 1 0 5°O+ P+q -4.95 ±0 .42 XD 0.000050 + 0 .000004 xV, v 0 1001 . 812 ±0.011 (3<J) 0.5463833 0. .542864 ±0.000013 6.509 6. .54 ±0.03 2.03087 2, .028 ±0.002 0.022516 0 .0226 fixed -1 -1 fixed 5.6 5 .6 fixed f i xed I ro i Standard deviations (unit weight):- A % , v = 0: 0.024 cnf 1 ; xV. v - 1: 0.024 cm' Bond lengths: A 4n. r = 1.6368 A; XV , rQ - 1.5920 A, r 1 .5894 A , = 0.548143, a e = 0.00351g cm" ) -218- G. D i s c u s s i o n (i.) S p i n - o r b i t c o u p l i n g C o n s t a n t s and i n d e t e r m i n a c i e s 4 S i n c e n s t a t e s a r e c o m p a r a t i v e l y uncommon i t i s i n s t r u c t i v e to see what p a r a m e t e r s can be d e t e r m i n e d i n t h i s c a s e , and what happens t o the problem o f t h e i n d e t e r m i n a c y o f some o f t h e parameters i n t h e g e n e r a l c a s e . V e s e t h ( 2 1 ) has p o i n t e d o u t how y and [ t h e s p i n - r o t a t i o n i n t e r a c - t i o n and the 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 t h e s p i n - o r b i t c o u p l i n g ) c a n n o t be d e t e r m i n e d s e p a r a t e l y i n a 2 n s t a t e , and Brown e t a l ( 1 9 ) have pr o v e d t h i s r i g o r o u s l y . Brown e t a l have a l s o shown t h a t an i n d e t e r m i n a - cy e x i s t s between B, A D , A D and y f o r case ( a ) n s t a t e s , e s s e n t i a l l y because t h e r e a r e o n l y t h r e e e f f e c t i v e B - v a l u e s f o r the t h r e e s p i n - o r b i t components, but f o u r parameters t o be d e t e r m i n e d from them. The i n d e t e r - minacy can be a v o i d e d i f t h e l e v e l s can be f o l l o w e d to h i g h J v a l u e s , where c a s e (b) c o u p l i n g a p p l i e s , because t h e r e i s a d d i t i o n a l i n f o r m a t i o n i n t he e f f e c t i v e D-values o f t h e t h r e e s p i n - o r b i t components. No s u c h i n d e t e r m i n a c y o c c u r s f o r 4 n s t a t e s because t h e r e a r e now f o u r e f f e c t i v e B - v a l u e s t o d e t e r m i n e the same f o u r p a r a m e t e r s ; o n l y i f h i g h e r - o r d e r terms s u c h as y$ ( t h e t h i r d - o r d e r s p i n - o r b i t c o r r e c t i o n t o t h e s p i n - r o t a - t i o n i n t e r a c t i o n (16,17.)) a r e needed w i l l f u r t h e r i n d e t e r m i n a c i e s a r i s e . ' I t i s v e r y c l e a r from o u r data t h a t A D i s e f f e c t i v e l y z e r o f o r the A 4 J I s t a t e o f VO. I f A N i s f l o a t e d t h e s t a n d a r d d e v i a t i o n i n c r e a s e s m a r g i n a l l y , and A N i s g i y e n as ( 4 ±12) .x 1 0 ~ 6 cm" 1. N e v e r t h e l e s s i f i t were not so s m a l l i t would i n p r i n c i p l e have been d e t e r m i n a b l e from t h e d a t a . -219- A n o t h e r i n d e t e r m i n a c y may a r i s e i n t h e s u b - s t a t e o r i g i n s f o r the components o f a m u l t i p l e t 71 s t a t e . These o r i g i n s can be e x p r e s s e d , i n terms o f t h e s p i n - o r b i t and s p i n - r o t a t i o n p a r a m e t e r s , as T f i = T 0 + A A E + | A l 3 E 2 - S ( S + l ) J ( 8 < 4 ) + yIftZ -SCS+l)J + nAlz3-(;3S2+3S-l).E/5] where n i s t h e t h i r d - o r d e r s p i n - o r b i t i n t e r a c t i o n (.22,23). From t h e p r e v i o u s d i s c u s s i o n i t i s seen t h a t f o r a n(a) s t a t e o n l y e f f e c t i v e y a l u e s o f T 0 , A and A can be d e t e r m i n e d , but t h a t a l l f i v e parameters can be d e t e r m i n e d f o r a n s t a t e , because y can be o b t a i n e d from the r o t a t i o - nal s t r u c t u r e . Because y has t o be de t e r m i n e d s e p a r a t e l y we have w r i t t e n the sub- s t a t e o r i g i n s i n T a b l e s 8.1 and 8.4 i n t h e form o f T^ v a l u e s . However, i t would be e n t i r e l y e q u i v a l e n t t o use e x p r e s s i o n s d e r i v e d from eq. (8.4) i n the l e a s t s q u a r e s work. C o n v e r t i n g from the T f i v a l u e s g i v e n i n T a b l e 8.4 we have T = 9498.878 cm" 1 ; A = 35.193 cm" 1 (8.5) A = 1.867 cm" 1 ; r, = 0.331 cm" 1 I t i s i n t e r e s t i n g to se e how c o m p a r a t i v e l y l a r g e the s e c o n d - o r d e r p a r a m e t e r A i s compared to A. As i s well-known (18) the s e c o n d - o r d e r pa- rameter A i n c l u d e s t h e d i a g o n a l s p i n - s p i n i n t e r a c t i o n , b u t s i n c e the l a t t e r cannot be e s t i m a t e d e a s i l y i t i s n o t p o s s i b l e to say how much o f the o b s e r v e d A i s caused by i t . The o b s e r y e d A f o r the A^n s t a t e i s -220- s i m i l a r t o t h a t f o r the X 4 E~ s t a t e ( s e e T a b l e 8 . 4 ) , s o t h a t i t s l a r g e s i z e i s not u n e x p e c t e d . To our knowledge an a c c u r a t e v a l u e o f t h e t h i r d - o r d e r parameter n has o n l y p r e v i o u s l y been o b t a i n e d f o r t h e l e v e l y = 4 o f the 4 i i ^ s t a t e o f 0 2 + ( 2 3 ) , though e s t i m a t e s have been made f o r t h e A 5 n and X 5n s t a t e s o f CrO ( 2 2 ) . ( i i ) A - d o u b l i n g parameters In t h e a p p r o x i m a t i o n where a s i n g l e V s t a t e causes the A - d o u b l i n g i n a 4n s t a t e t h e parameters o, p and q a r e g i v e n by O = -3 5 < 4 n!AL + | 4 r> 2/AE ] i E p = -2< 4 n!AL + | 4 z" x 4 n|BL + | 4 z">/AE ] i z (8.6) q = -2<4n|BL+| V> 2/AE n z Two a p p r o x i m a t e r e l a t i o n s between the A - d o u b l i n g parameters f o l l o w a t once: and (8.7) p/q = A/B 2 „ (8-8) p = 4oq E q u a t i o n (8.7) s h o u l d i n f a c t be obeyed q u i t e w e l l no m a t t e r what the s t a t e s c a u s i n g t h e A - d o u b l i n g a r e because i t assumes o n l y t h a t the m a t r i x elements o f A L + and B L + a r e i n t h e r a t i o o f A t o B; from T a b l e 8.4 we f i n d ( p / q ) / ( A / B ) = 1.26 -221- (8.9) which i s n o t f a r from u n i t y . E q u a t i o n (8.8) on the o t h e r hand i s not obeyed a t a l l , and the e x p e r i m e n t a l r a t i o p /4oq i s -0.13. There a r e two p o s s i b l e r e a s o n s . One i s t h a t t h e o f f - d i a g o n a l s p i n - s p i n i n t e r a c t i o n p arameter a (which s h o u l d be s u b t r a c t e d from the e x p r e s s i o n f o r o i n eq. (8.6)), i s i m p o r t a n t ; the o t h e r , which i s r a t h e r more l i k e l y , i s t h a t t h e r e i s a n e a r b y s t r o n g l y i n t e r a c t i n g e l e c t r o n i c s t a t e o f d i f f e r e n t m u l t i p l i c i t y . Assuming t h a t t h e s p i n - o r b i t o p e r a t o r i s r e s p o n s i b l e , 4 such a s t a t e w i l l have r o t a t i o n - i n d e p e n d e n t m a t r i x elements w i t h A n, so t h a t i t w i l l c o n t r i b u t e t o the parameter o, b u t not t o p o r q. 4 As f a r as we can t e l l from o u r s p e c t r a t h e A U, v=0 l e v e l i s unper-4 t u r b e d r o t a t i o n a l l y , and the p r i n c i p a l p e r t u r b a t i o n s i n B I I a r e by 4 - < \ 2 a n o t h e r E s t a t e ; however, t h e r e i s e v i d e n c e \\7) f o r a n s t a t e p e r t u r b i n g C 4 E " , V = 0 ( a t 17420 c m " 1 ) , which p o s s i b l y comes from the same e l e c t r o n c o n f i g u r a t i o n as A^n and i s a good c a n d i d a t e f o r c a u s i n g t h e e f f e c t s d e s c r i b e d . ( i i i ) H y p e r f i n e s t r u c t u r e o f the A 4 n s t a t e S e c t i o n E d e s c r i b e d how t h e main branches ( A N = A J ) i n a l l f o u r 4 n - 4 E _ sub-bands become 'sharp' a t h i g h N v a l u e s (where t h e s p i n c o u p l i n g a p p r o x i m a t e s c a s e ( b g J ) i n both s t a t e s ) a l t h o u g h t h e y a r e o f t e n h y p e r f i n e - broadened a t low N . I t has been p o s s i b l e to o b t a i n t h e a p p r o x i m a t e hyper- f i n e w i d t h s o f t h e f o u r components o f A 4 n from d e t a i l e d measurements o f the l i n e shapes i n t h e v a r i o u s b r a n c h e s , t o g e t h e r w i t h t h e known h y p e r - f i n e s t r u c t u r e o f t h e ground s t a t e ( 1 7 ) ; t h e r e s u l t s a r e shown i n -222- F i g . 8.5. T h i s f i g u r e s h o u l d be c o n s i d e r e d o n l y as an " a r t i s t ' s i m p r e s s - 4 4 - i o n " because t h e h y p e r f i n e s t r u c t u r e i s n e v e r r e s o l v e d i n the A n-X z t r a n s i t i o n , and the d e c o n v o l u t i o n o f the D o p p l e r and h y p e r f i n e p r o f i l e s has n o t been a t t e m p t e d . The e r r o r b a r s g i v e n f o r the F 2 and F^ components show t h a t i t i s r e l a t i v e l y f u t i l e to t r y to o b t a i n v a l u e s f o r any o f the h y p e r f i n e parameters e x c e p t b, but on the o t h e r hand the v a l u e o f b can be o b t a i n e d w i t h r e a s o n a b l e a c c u r a c y . To u n d e r s t a n d why o n l y t h e h y p e r f i n e parameter b i s d e t e r m i n a b l e we c o n s i d e r the m a g n e t i c h y p e r f i n e H a m i l t o n i a n [24] i n d e t a i l : H m a n h f c = a I-L + b I-S + c I 7 S 7 + j d ( e 2 l ' * I S + e " 2 i * I + S . ) (8.10) nmag.hfs ~~ z z 2 -- + + In t h i s e q u a t i o n the f i r s t term i s t h e i n t e r a c t i o n between the e l e c t r o n o r b i t a l m o t i o n and t h e n u c l e a r s p i n , t h e second term i s a c o m b i n a t i o n o f t h e Fermi c o n t a c t i n t e r a c t i o n and t h e d i p o l a r i n t e r a c t i o n , and t h e l a s t two terms a r e d i p o l a r i n t e r a c t i o n s , r e s p e c t i v e l y d i a g o n a l and o f f - d i a g o n a l i n A i n a s i g n e d quantum number b a s i s . The term i n d g i v e s r i s e to 4 d i f f e r e n t h y p e r f i n e s t r u c t u r e s i n the two A - d o u b l i n g components o f n ^ 2 > and i t s e f f e c t s can be seen i n F i g . 8.5, where t h e r e i s a d e f i n i t e d i f f e - r e n c e between t h e h y p e r f i n e w i d t h s o f t h e F 2 e and F 2 f l e v e l s up to about J = 50. T h i s d i f f e r e n c e can be measured f a i r l y a c c u r a t e l y because the l i n e w i d t h s i n t h e P 2 and Q 2 branches a r e q u i t e o b v i o u s l y d i f f e r e n t , though the a b s o l u t e v a l u e s o f t h e h y p e r f i n e w i d t h s a r e u n c e r t a i n to t h e e x t e n t o f the e r r o r bars i n F i g . 8.5. In c a s e ( a ^ ) c o u p l i n g the d i a g o n a l m a t r i x elements (25) o f the f i r s t t h r e e terms o f e q . (8.10) a r e ro r-o • F i g 8.5 Hyperfine widths, A E h f s = E h f g ( F = J + I ) - E h f s ( F - J - I ) , of the four s p i n components of the A4n s t a t e of VO, p l o t t e d against J . P o i n t s are widths c a l c u l a t e d from the ground s t a t e hyperfine s t r u c t u r e and the observed l i n e widths, without c o r r e c t i o n f o r the Doppler width. •224- < J f i A I F | H h f s | J f l A I F > = l F ( F + l ) - I ( I + l ) - j ( J + l ) M a A + ( b + c ) E ] / [ 2 j ( J+1)] (8.11) w h i l e the d term c o n t r i b u t e s ± d ( S+ 3 s ) ( J + 3 5 ) [ F ( F + 1 ) - I ( I + 1 ) - J ( J + 1 ) ] / [ 4 J ( J + 1 ) ] to t h e d i a g o n a l elements f o r = h when S i s h a l f - i n t e g r a l . The h y p e r - f i n e w i d t h s ( i n o t h e r words t h e s e p a r a t i o n s o f t h e h y p e r f i n e components w i t h F = J + I and F = J - I) f o r a 4 n s t a t e where I = 7/2 a r e t h e r e f o r e A Eh+- c = 7 ( J + J 2 M a + ( b + c ) E ] / [ J ( J + l ) ] h t S  9 (8.12) ± 7 ( J + ^ d 6 ^ / [ J ( J + l ) ] E q u a t i o n (8.12) i m p l i e s t h a t t h e h y p e r f i n e w i d t h s s h o u l d d e c r e a s e as 1/J e x c e p t t h a t t h e r e i s a J - i n d e p e n d e n t c o n t r i b u t i o n o f ±7d i n t h e two 4 A-components o f n^. In c a s e ( b ^ j ) . on the o t h e r hand, t h e d i a g o n a l m a t r i x elements o f th e m a g n e t i c h y p e r f i n e H a m i l t o n i a n a r e <NASJIF|H. - I :NASJIF> = [F( F + 1 ) - I ( I + 1 ) - J ( J+1)] ( a A 2 X ( N J S ) n t S 4J(J+1) \ N(N+1) + bX(JSN) - c [ 3 A 2 - N ( N + l ) ] [ 3 X ( S N J ) X ( N J S ) + 2X( JSN) N(N+1) ] 3N(N+l)(2N-l)(2N+3) ± d[3X(SNJ)X(NJS) + 2X(JSN)N(N+1)] 6,., -, I l* W 2(2N-l)(2N+3) l A l * - | J / where X(xyz) = x ( x + l ) + y ( y + l ) - z ( z + l ) . I t i s n o t so easy to see the J-dependence i n t h e s e f o r m u l a e , b ut o r d e r - o f - m a g n i t u d e c o n s i d e r a t i o n s show t h a t t he c o e f f i c i e n t s o f a and c d e c r e a s e as 1 / J , w h i l e t h e c o e f f i - c i e n t s o f b and d a r e a l m o s t i n d e p e n d e n t o f J . The h y p e r f i n e energy -225- e x p r e s s i o n s f o r 4 n ( b ) s t a t e s a r e r o u g h l y F ^ J = N+3/2) E h f s = - | (b±hd) X(JIF)/(2N+3) F 9 ( J = N+l/2) - i (b±>»d) X(JIF)(2N+9)/.[(2N+l)(2N+3)] 2 \ (8.14) F 3 ( J - N - l / 2 ) 2 (b±%d) X ( J I F ) ( 2 N - 7 ) / I ( 2 N - l ) ( 2 N + l ] F 4 ( J = N-3/2) | (b±*jd) X ( J I F ) / ( 2 N - 1 ) where the terms i n ±kd r e f e r t o the A - d o u b l i n g components; f o r I = 7/2 the a p p r o x i m a t e h y p e r f i n e w i d t h s i n t h e f o u r s p i n components, i n u n i t s o f 7 ( b ± ^ d ) / 2 , a r e 3,1,-1 and -3, r e s p e c t i v e l y . 4 F i g . 8.5 shows t h a t t h e h y p e r f i n e p a t t e r n s i n the A n s t a t e o f VO, o v e r t h e r a n g e J = 10-80, c o r r e s p o n d t o a s p i n c o u p l i n g i n t e r m e d i a t e between c a s e s (a„) and ( b „ , ) - As d e s c r i b e d above, t h e d i f f e r e n t h y p e r f i n e P p O w i d t h s i n the F 2 e and F 2 f components r e p r e s e n t the d i p o l a r d term, b u t the o b s e r v e d d i f f e r e n c e i s a c o m p l i c a t e d f u n c t i o n o f how f a r t h e s p i n - u n c o u p l i n g has p r o c e e d e d . The d term s h o u l d show up a g a i n as a s m a l l d i f f e r e n c e between the Q and P b r a n c h w i d t h s f o r t h e h i g h N F-j and F^ l i n e s , b ut t h i s i s not o b s e r v a b l e a t o u r r e s o l u t i o n . The h i g h N p a t t e r n c o r r e s p o n d s t o a l m o s t pure case ( b ^ j ) c o u p l i n g , w i t h the parameter b b e i n g v e r y n e a r l y t h e same as i n t h e ground s t a t e (hence t h e 'sharp' main b r a n c h l i n e s where t h e h y p e r f i n e components a l l f a l l on t o p o f one a n o t h e r ) . The e x p e r i m e n t a l v a l u e o f b i s b ( A 4 n ) = +0.026 ± 0.002 cm" 1 (8.15) compared to t h e ground s t a t e v a l u e 0.02731 ± 0.00004 cm - 1 ( 1 7 ) . •226- We have not at t e m p t e d t o o b t a i n v a l u e s f o r a, c and d from F i g . 8.5, s i n c e t h e p a t t e r n i s c l e a r l y dominated by t h e parameter b, w i t h t h e e x a c t d e t a i l s b e i n g g overned by the e x t e n t o f t h e s p i n - u n c o u p l i n g . The f a c t t h a t b ( A 4 n ) i s c l o s e l y s i m i l a r to b ( X 4 E ~ ) i n d i c a t e s t h a t t h e same 4sa e l e c t r o n r e s p o n s i b l e f o r the Fermi c o n t a c t i n t e r a c t i o n i n 4 the ground s t a t e i s a l s o p r e s e n t i n the A n s t a t e . In s i n g l e c o n f i g u r - a t i o n a p p r o x i m a t i o n t h e e l e c t r o n c o n f i g u r a t i o n s must t h e r e f o r e be xV : ( 4 s a ) 1 ( 3 d 6 ) 2 (8.16) A 4 n : ( 4 s a ) 1 ( 3 d 6 ) 1 ( 4 p 7 i ) 1 4 4 The c o n f i g u r a t i o n g i v e n f o r A n a l s o produces a E s t a t e , which s h o u l d l i e a t s t i l l l o w e r e n e r g y ; t h e chances o f o b s e r v i n g i t a ppear s l i m a t p r e s e n t s i n c e i t s A - v a l u e d i f f e r s by a t l e a s t 2 u n i t s from a l l t h e o t h e r known s t a t e s o f VO. -227- B i b l i o g r a p h y -228- Chapter 1 (1) G.C. Dousmanis, T.N. S a n d e r s , J r . , and C.H. Townes, Phys. Rev. 100, 1735-1754 ( 1 9 5 5 ) . (2) H. S p i n r a d and R.F. Wing, Ann. Rev. A s t r o n . and A s t r o p h y s . 7, 249-302 ( 1 9 6 9 ) . (3) P. 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Barrow, Nature 219_, 1244-1245 (1968). 21. L. V e s e t h , J . M o l . S p e c t r o s c o p y 38, 228-242 ( 1 9 7 1 ) . 22. W.H. H o c k i n g , A . J . Merer, D.J. M i l t o n , W.E. Jones and G. K r i s h n a m u r t y , Canad. J . Phys. 58, 516-533 ( 1 9 8 0 ) . 23. J.M. Brown, p r i v a t e communication; J.M. Brown, D.J. M i l t o n , J.K.G. Watson, R.N. Z a r e , D.L. A l b r i t t o n , M. H o r a n i and J . R o s t a s , J . M o l . S p e c t r o s c o p y , s u b m i t t e d f o r p u b l i c a t i o n . 24. R.A. F r o s c h and H.M. F o l e y , Phys. Rev. 88, 1337-1349 ( 1 9 5 2 ) ; G.C. Donsmanis, Phys. Rev. 97 , 967-970 TT955). 25. A. C a r r i n g t o n , P.N. Dyer and D.H. Levy, J . Chem. Phys. 47, 1756-1763 ( 1 9 6 7 ) ; J.M. Brown, I . Kopp, C. Malmberg and B. Rydh, P h y s i c a S c r i p t a 17, 55-67 ( 1 9 7 8 ) ; C.H. Townes and A.L. Schawlow "Microwave S p e c t r o s c o p y " M c G r a w - H i l l , New York (1955). 26. I.C. Bowater, J.M. Brown and A. C a r r i n g t o n , P r o c . Roy. S o c . (London) A 333, 265-288 ( 1 9 7 3 ) . Appendices ( A l ) J.H. van V l e c k , Rev. Mod. Phys., 23, 213-227 ( 1 9 5 1 ) . (A2) J.A.R. Coope, J . Math. Phys. Y\_, 1591-1612 ( 1 9 7 0 ) . (A3) A. C a r r i n g t o n , D.H. Levy and T.A. M i l l e r , Advan. Chem. Phys. 1_8, 149-248 ( 1 9 7 0 ) . (A4) J . Jer p h a g u o n , D. Chemla and R. B o n n e v i l l e , Advan. i n Phys., 27, 609-650 ( 1 9 7 1 ) . (A5) A.R. Edmonds, " A n g u l a r Momentum i n Quantum M e c h a n i c s " , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n ( 1 9 7 4 ) . (A6) B.L. S i l v e r , " I r r e d u c i b l e T e n s o r Methods", Academic P r e s s , London (1976). (A7) M.D. B r i n k and G.R. S a t c h l e r , " A n g u l a r Momentum", Clarendon P r e s s , O x f o r d ( 1 9 6 8 ) . (A8) T.M. Dunn, i n " P h y s i c a l C h e m i s t r y , An Advanced T r e a t i s e , V o l . V", (H. E y r i n g , E d . ) , C h a p t e r 5, Academic P r e s s , New York ( 1 9 7 2 ) . -239- (A9) C . J . Cheetham and R.F. Barrow, Adv. H i g h Temp. Chem., 1, 7 -41(1967) (A10) R M Van Zee, C M . Brown, K . J . Z e r i n g n e , and W. W e l t n e r , J r . , A c c t s . Chem. Res. 13, 237 -242(1980). -240- Appendix I T r a n s f o r m a t i o n between C a r t e s i a n T e n s o r s and S p h e r i c a l T e n s o r s -241- . T h i s a p p e n d i x g i v e s t h e t r a n f o r m a t i o n between c a r t e s i a n and s p h e r i - c a l t e n s o r components f o r o p e r a t o r s o f up t o s e c o n d r a n k , and o u t l i n e s some problems t h a t a r i s e when a n g u l a r momenta r e f e r r e d t o d i f f e r e n t a x i s systems commute i n d i f f e r e n t ways. In p r a c t i c e , i t i s not always n e c e s s a r y t o f o r m u l a t e e v e r y , i n t e r - a c t i o n i n s p h e r i c a l form; f o r example, t h e e l e c t r o n s p i n - r o t a t i o n i n t e r - a c t i o n can be w r i t t e n c o n v e n i e n t l y i n c a r t e s i a n form ( A l ) . However f o r u n i f o r m i t y a l l terms i n t h e H a m i l t o n i a n w i l l be w r i t t e n i n s p h e r i c a l t e n s o r form i n t h i s t h e s i s . T e n s o r s o f rank 0 a r e s c a l a r s , and a r e t h e same i n c a r t e s i a n and s p h e r i c a l form. F i r s t rank t e n s o r s ( i . e . v e c t o r s ) can be e x p r e s s e d e i t h e r i n a c a r t e s i a n b a s i s |i> w i t h i={x,Y,Z} o r i n a s p h e r i c a l b a s i s |l,u> w i t h p = {-1,0,1}. The u n i t a r y t r a n s f o r m a t i o n <i|£,u> between the two bases i s |1,1> |i,o> p,-l> |X> -(2)'h 0 (2)~h |Y> 0 ( A . I . I ) |Z> 0 1 0 i . e . f o r the components o f a v e c t o r T. rl - • ( 2 ) " l 5 ( T y + i T Y ) (A.I.2) -242- I t i s i m p l i c i t i n ( A . I . I ) t h a t c a r t e s i a n components s a t i s f y t h e commutation r e l a t i o n s w i t h the "normal" s i g n o f i , t T X ' T Y ] = T X T Y " T Y T X = + i T Z (A.1.3) The t r a n s f o r m a t i o n o f sec o n d - r a n k t e n s o r s from c a r t e s i a n t o s p h e r i - c a l form can be pe r f o r m e d i n a two s t e p p r o c e s s . S t a r t i n g from a second- rank c a r t e s i a n t e n s o r (whose elements a r e T.- ^one f i r s t changes t he V2 c a r t e s i a n c o o r d i n a t e s i n t o s p h e r i c a l c o o r d i n a t e s t h r o u g h a p r o d u c t o f u n i t a r y t r a n s f o r m a t i o n s <i|s,,.y> . T h i s r e s u l t s i n a r e d u c i b l e s p h e r i c a l t e n s o r , which i s r e d u c e d by t h e well-known p r o p e r t i e s o f t h e C l e b s c h - G o r d a * c o e f f i c i e n t s to g i v e t h e i r r e d u c i b l e s p h e r i c a l t e n s o r s T^ (A 2,A 4 ) . Tjj = <J m I 1 1 u i 3J2 > < 1 ' y l I h > < 1 ' y 2 I 1-2 > T i n i"2 ^ A J - 4 ^ where < J m | 1 1 ^ ^ > ^ t h e u s u a l C l e b s c h - G o r d a n c o e f f i c i e n t , which i s r e l a t e d t o the Wigner 3-j symbols by < J m I 1 1 y , u 2 > = ( - l ) m ( 2 J + l f 2 n 1 0 \ (A.1.5) VI  v2 ~ m) S u b s t i t u t i n g n u m e r i c a l v a l u e s f o r the 3 - j sym b o l s , and u s i n g ( A . I . I ) , a second rank c a r t e s i a n t e n s o r can be decomposed i n t o a sum o f s p h e r i c a l t e n s o r s o f ranks 0, 1 and 2, T 1 X T 1 = T° + T 1 + T 2 (A.1.6) -243- The e x p l i c i t e x p r e s s i o n s a r e i = - <3>"* { T X X + TYY + V T j - 1 ( 2 ) - * ( T x y - T Y X ) T l _ 1 ' + W { ( T Z X - T X Z ) ± 1 ^ TZY " T Y Z ) } (A.I .7;) T 0 = ( 6 ) " ' { 2 T Z Z " TXX " T Y Y } i l = * <?> « T X Z + TZX> ± 1 ( T Y Z + T Z Y ' } T ± 2 = l { ( T X X - T Y Y ) 1 1 < TXY + T Y Z ) } In m o l e c u l a r s p e c t r o s c o p y we must d i s t i n g u i s h between s p a c e - f i x e d and m o l e c u l e - f i x e d o p e r a t o r s . T e n s o r components d e f i n e d i n terms o f axes mounted on the m o l e c u l e , which a r e denoted by x, y and z, have the s i g n o f i i n the commutation r e l a t i o n s r e v e r s e d (A 1 ) , i . e . [T , T ] = - i T (A.I.8) x y z Components o f m o l e c u l e - f i x e d t e n s o r o p e r a t o r s w i l l t r a n s f o r m d i f f e - r e n t l y from c a r t e s i a n to s p h e r i c a l form. F o r components o f a m o l e c u l e - f i x e d v e c t o r T (A 3 ) : l] = - ( 2 ) _ 3 5 ( T - i T ) 1 ' x y T j = T z (A.I.9) = ( 2 ) " 3 5 ( T X + i T y ) -244- For second rank t e n s o r s , o n l y TJ and a r e d i f f e r e n t from (A.1.7 ) 0 XY Y (A.I.10) ' ̂  - TXZ) ± 1 (TZY-TYZ)} -245- A p pendix I I The D e r i v a t i o n o f t h e N u c l e a r s p i n - E l e c t r o n s p i n D i p o l a r I n t e r a c t i o n M a t r i x Elements i n c a s e ( b g J ) c o u p l i n g -246- T h i s Appendix g i v e s a d e r i v a t i o n o f t h e m a t r i x elements o f - - i ^ 9 » B 9 N i H . T 1 l I > - T 1 < S . C i ! > t A - I I ' 1 ) U ns-es i n the c a s e ( b D l ) b a s i s ( i . e . | n N K S J I F > ) g i v e n as eq. (2.144) i n C h a p t e r 2. M a t r i x elements can o n l y be e v a l u a t e d when the o p e r a t o r and the r e l e v a n t p a r t s o f t h e w a v e f u n c t i o n s a r e i n the same r e f e r e n c e f r a m e , i.e. e i t h e r both i n the m o l e c u l e - f i x e d a x i s system (q) o r i n t h e sp a c e f i x e d a x i s system ( p ) . Our p r o c e d u r e w i l l be t o expand the H a m i l t o n i a n i n t h e f i r s t i n s t a n c e as a s p a c e - f i x e d o p e r a t o r , so t h a t i r r e d u c i b l e t e n s o r methods can be a p p l i e d i n t h e i r s t a n d a r d form-, and then t h o s e p a r t s t h a t a r e p h y s i c a l l y a p p r o p r i a t e a r e r e f e r r e d to t h e m o l e c u l e - f i x e d a x i s system by means o f t h e r o t a t i o n m a t r i x eq. ( 2 . 1 1 0 ) . The o p e r a t o r H n s _ e s i s a s c a l a r p r o d u c t o f two commuting t e n s o r o p e r a t o r s b e c a u s e ^ and £ a r e i n d i f f e r e n t s p i n s p a c e s . T h e r e f o r e , from Edmonds' eq. (7.1.6) (A5) < ri N' K ' S ' r r r M F I H N S _ E S | N N K S J I F M F > = - (lo)3* g P b g N y N V F 6 M / M ( 1 ) I + J + F ( F J " i l i J < \ II T ] ( I ) || I > < n N' S' || T ^ C 2 ) || n N K S J > (A. I I . 2 ) R 3 where the r e d u c e d m a t r i x elements a r e w i t h r e s p e c t to the s p a c e - f i x e d a x i s system. <\ \ \ T ^ I ) | | I > can now be e v a l u a t e d d i r e c t l y , because, -247- i n c a s e ( b ^ j ) , t he n u c l e a r s p i n I i s q u a n t i z e d i n the s p a c e - f i x e d a x i s s y s t e m , as i s the o p e r a t o r ( I ) . T h e r e f o r e by eq. (2.109) < V || T ](0 || I > = 6 r i [ I ( I + 1 ) ( 2 I + 1 ) ] 3 5 ( A . I I . 3 ) 1 7 The o p e r a t o r T (!S,C ) , i n t h e s e c o n d r e d u c e d m a t r i x e l e m e n t , i s a compound t e n s o r o p e r a t o r c o n s t r u c t e d from s i m p l e r commuting o p e r a t o r s ; by u s i n g Edmonds' eq. (7.1.5) ( A 5 ) , < n N' K- S" || T ^ S . C 2 ) II n N K'S 0 , - <3 ) W)<2J' +1>3' R3 < S ' ! I T 1CS> | | S X T , N- K' 1 | C i 11 n N K - I N ' H 2 FT j S S 1 j ' 0 1 ( A . I I . 4 ) A g a i n f o r a c a s e ( b g j ) c o u p l i n g scheme t h e e l e c t r o n s p i n S i s q u a n t i z e d i n t h e s p a c e - f i x e d a x i s s y s t e m , so t h a t , f o r t h e m a t r i x elements < S' || T ^ S ) || S > = 6 $ - s [ S ( S + 1 ) ( 2 S + 1 ) ] 1 5 ( A . I I . 5 ) The s e c o n d - r a n k s p h e r i c a l harmonic C i s d e f i n e d i n t h e m o l e c u l e - f i x e d a x i s s y s t e m , b u t so f a r i t s m a t r i x elements have been r e d u c e d w i t h r e s p e c t t o the s p a c e - f i x e d s y s t e m . The t r a n s f o r m a t i o n from s p a c e - f i x e d to m o l e c u- l e - f i x e d axes i s c a r r i e d o u t by means o f t h e r o t a t i o n m a t r i x D, ( 2 ) * C 2 = I D „ U) C 2 ( A . I I . 6 ) .q Q q -248- where co r e p r e s e n t s the t h r e e E u l e r a n g l e s (agy). (.2)* D (co)is t h e r o t a t i o n m a t r i x w i t h no r e f e r e n c e t o t h e s p a c e - f i x e d • q' components, and i t s r e d u c e d m a t r i x element i s d e f i n e d by eq. ( A . I I . 1 7 ) . When t h i s i s s u b s t i t u t e d i n t o the r e d u c e d m a t r i x element we o b t a i n < n N K n N K > = E < n | C , n > ( 2 ) * < rf K' | | D (co) || N K > (A . I I .7) where < n | Cn | n > i s an e x p e r i m e n t a l l y d e t e r m i n a b l e p a r a m e t e r , which 7 w i l l be r e d e f i n e d as T 2 ( C ) = < n | C j j j n > R 3 ( A .II.8) ( 2 ) * L a s t l y , we must c a l c u l a t e < N' K' | | D (u)| | N K > ( 2 ) * w hich i s t h e r e d u c e d m a t r i x element o f < N K M | DnC[ (co) | N K M >. We w i l l f i r s t e v a l u a t e < l T K' M' | D$* (CO) | N K M > and then a p p l y the W i g n e r - E c k a r t Theorem (A5) to g e t i t s r e d u c e d m a t r i x e l e m e n t . S i n c e ^ N K M > = "2N+1 ( A . I I . 9 ) and i t s complex c o n j u g a t e i s < N' K' M' I 2N +1 8TT2 J ( A .II.10) -249- the r e l a t i o n s h i p between D* and D i s (A7) D<£* W - (-D P" q^p )- q M (A.II .n) T h e r e f o r e , < N' K' M | P ( K )* | N K M > = £(2N +1)(2N+1)]^ (A . I I .12) U s i n g the r e l a t i o n s h i p i n S i l v e r (A6) p. 43, we Ijave < M | Z^V) | N K M > = ( - 1 ) P " q ( - 1 ) M ~ K [ ( 2 N ^ 1 ) ( 2 N + 1 ) ] ^ pq V k N\ /V k N X -P - i y \K' -q -Kj ( A . I I . 1 3 ) The p a i r o f 3 - j symbols i n ( A . I I . 1 3 ) w i l l be non-zero o n l y i f t h e y s a t i s f y the c o n d i t i o n s NT + (-pl + (-M) = 0 and K' + (-q) + (-K) = 0 ; (A . I I . 1 4 ) A l s o w i t h the use o f symmetry p r o p e r t i e s o f t h e 3 - j s y m b o l , we f i n a l l y o b t a i n -250- < N ' K ' l T \D']* (CO) I N K.M > 1 Pq = (-1) M^ K^[(2N^1).(2N+1)] 17 N' kN\/N' k N> V ' N\  N  N\ \-M' p M ] \ - K ' q K J A p p l y i n g the W i g n e r - E c k a r t Theorem, we f i n d < N' K' M' \DM* (CO) I N K M > = (-1) rH / N k N\ l-M' p M j < N' K.' M D{K)* (co) || N K T h e r e f o r e , < IT r || || N K > = ( - l ) r - K ' [ ( 2 N ' + l ) ( 2 N + l ) f 2 w i t h t h e s p e c i a l c a s e o f k=2 ( 2 ) * - -< IT K' | | D | | N K > = ( - 1 ) N " K £(2N +1) (2N+1 )jV l\T 2 \K q Combining eqs ( A . I I . 2 ) , ( A . I I . 3 ) , ( A . I I . 4 ) , ( A . I I . 5 ) , ( A . I I . 7 ) and ( A . I I . 1 8 ) we f i n a l l y g e t -251- n K' S J " I F | H ns-es N K S J I F > •(30f2g u B g N ^ N (-D J + I + F j U ' F J I 1 I I ( I + l ) ( 2 I + l ) S ( S + l ) ( 2 S - r l ) ] 3 5 [(2J'+l)(2J+l)(2N'+l)(2N+l ) r N N 2 S S 1 j ' J 1 I (-D q r-r / N ' 2 N \ T 2 ( c ) \ - K ' q K J q ( A . I I . 1 9 ) which completes the d e r i v a t i o n . -252- Appendix I I I D e r i v a t i o n o f t h e m a t r i x elements o f the o p e r a t o r i T 1 { I ) . T 1 [ T 3 | T 1 ( S ) . T 2 ( £ i , s j ) } , C 2 ] / r ? . i>j i n a Hund's case ( b g j ) b a s i s i -253- We b e g i n by a p p l y i n g eqs. ( 7 . 1 . 5 ) and ( 7 . 1 . 6 ) o f Edmonds (A5) t o re d u c e t h e g e n e r a l form o f t h e m a t r i x elements o f eq. (3 .25) i n Ch a p t e r 3 , E T ^ D - T ^ T ^ T ^ S ) , ASLSJ)}, C 2 ] / r 3 . i > j J s e t up i n a Hund's case ( b c 1 ) b a s i s : < n N A S J I F I E T 1 ( I ) - T 1 [ T 3 { T 1 ( S ) , T 2 ( s 1 - , s i ) } , C 2 J / r ? . | n ' N ' A' S j ' l F > i>J J = M ) I + J + F (F J I )< I || T](X) || I > x (3)h [ ( 2 J + l ) ( 2 J % l ) f 2 (1 I J ' l N f f 2 j S S 3 ( E < S || T 3 { T ] f S ) - T 2 ( s . , s .)} || S> < n N A | | C 2 | | n V h > ( A . I I I . l ) The s c a l e d s p h e r i c a l harmonic C i s d e f i n e d i n m o l e c u l e - f i x e d a x e s , so we need t o p r o j e c t i t i n t o m o l e c u l e - f i x e d axes a c c o r d i n g t o C 2 = E D ( 2 ) * (co) C 2 ( A . I I I . 2 ) q , q q where cu s t a n d s f o r t h e E u l e r a n g l e s o f t h e r o t a t i o n . The r e d u c e d m a t r i x 2 3 element o f C /r^. then becomes N A I! C 2 / r 3 , || n ' N' A > - I < N A || D\Z) (U.) II IT A > E < n  A | | t / r . j  | n n ^ -> - u - » * M 1 > j x z < n 1 Cl/r3 | n' > i > j = E (-1)N-AI(2N+1)(2N%1)]3'Y N 2 N'\ T 2 ( C ) ( A . I l l .3) \-A q A / -254- In o u r case A = A= 0, so t h a t q = 0. Th e r e w i l l be j u s t one e x p e r i m e n t a l parameter TQ(.C), which w i l l be p r o p o r t i o n a l t o b<-. Next we break up the r e d u c e d m a t r i x element o f the e l e c t r o n s p i n t e n s o r p r o d u c t i n eq. ( A . I I I . l ) u s i n g Edmonds' eq. ( 7 . 1 . 1 ) : I < S || T W s J . T 2 ^ , * . ) } II S > = ( 7 ) * ( - D 2 S + 3 j l 2 3} [S( S + 1 ) ( 2 S + 1 ) ] ^ x E < S | | T 2 ( s . , s . ) || S > ( A . I l l .4) i > j ~' ~J where we have e l i m i n a t e d the: sum o v e r s t a t e s w i t h t o t a l s p i n S" because < S || T ] ( S ) || S' > = [S( S + 1 ) ( 2 S + 1 ) ] 3 5 6 S S' ( A . I l l .5) The t e n s o r p r o d u c t i n eq. ( A . I I I . 4 ) appears i n t h e m a t r i x elements o f the d i p o l a r e l e c t r o n s p i n - s p i n i n t e r a c t i o n and the A - d o u b l i n g parameter o ;. w i t h Edmonds' eq. (7.1.1) a g a i n we f i n d < S || T 2 ( s , , s . ) || S > = ( 5 ) 1 5 E ( - 1 ) 2 S + 2 (1 1 2|< S || T 1 ^ ) || S' > * (S S S ) < S' I I T \ S , ) | | S > ( A . I I I . 6 ) The t r i a n g l e r u l e s on the 6-j symbol l i m i t s ' t o S o r S ± l , but S = S+l i s p h y s i c a l l y i m p o s s i b l e , so t h a t the t e n s o r p r o d u c t becomes -255- < S || T 2 ( S i , s . ) || S > = (2) (-1) 2 s r jl 1 2 )< S || T ^ ( s . ) || S - l Is S S - l i < S - l r T l ( s . ) || S > + j l 1 2) <S HT 1 ^) ||S ~ J s s s < s || r ( s . ) || s > ( A . I I I . 7 ) To e v a l u a t e the scheme r e d u c e d m a t r i x elements o f ^ and s^ we d e f i n e the c o u p l i n g S = I s. = s 1 + s,2 + • • • £ . = s T + 5 a ( A . I l l .8) Then, by Edmonds 1 eq. ( 7 . 1 . 7 ) < S- || T 1 ^ ) || S > - (-D Sl + Sa + S + 1 [ ( 2 S + 1 ) ( 2 S j s , S' Saj < S T || T 1 ^ ) || ST > ( A v I I I .9) where < s, | | T 1 ^ ) II s i > = l^Ws^)V' 0/2)*, f o l l o w i n g eq. ( A . I I I . 5 ) , s i n c e S ] - \. Usi n g t h e d e f i n i t i o n ^ - S - s , , and s u b s t i t u t i n g f o r the Wigner c o e f f i c i e n t , we o b t a i n < S || T 1 ^ ) || S > = • 2-[(S+l)(2S+l)/S] 1" 2 which i s a g e n e r a l e x p r e s s i o n h o l d i n g f o r a l l s,. and < S - l || T 1 ^ ) || S > = -\ [ ( 2 S-l ) ( 2 S+l ) / s f 2 ( A . I I I .10) = _ < s || r ^ ) II s-i > ( A . I I I . 1 1 ) -256- For t h e o f f - d i a g o n a l r e d u c e d m a t r i x elements o f t h e o t h e r s p i n s we need to e x t e n d the c o u p l i n g scheme: S = - s,l + *2 + h > 0 T h = ^ 2 + h ( A . I I I . 1 2 ) Then < S - l || T ' l s , ) II S > - ( -1)W S + 1 R 2 S + D ( 2 S - l ) ] ^ S a S - l s , l ~̂  (S S_ 1 ) < S . II T 1 ^ ) | | S , , ( A - > " - 1 3 > a S s t i l l r e p r e s e n t s a c o u p l e d b a s i s as f a r as e l e c t r o n 2 i s concerned, ~ a so t h a t Now < S a || T ] ( s 2 ) || % > = ( - l ) S a + S b + s Z + l . . ( 2 S a + l ) j s 2 S a Sbj |S a s 2 1 j < s 2 || T ] ( s 2 ) || s 2 > ( A . I I I . 1 4 ) which g i v e s the r e s u l t s < S - l || T ] ( , s 2 ) |1 S > = ^[(2S+1)/{S(2S-1)}] 1' 2 = - < S || T 1 ^ ) | | S - l > ( A . I I I . 15) We can now r e t u r n to eq. ( A . I I I . 7 ) , and s u b s t i t u t i n g t he new e x p r e s s i o n s j u s t o b t a i n e d , we g e t < S || T 2 ( s . , s . ) || S > = [ ( S + l ) ( 2 S + l ) ( 2 S + 3 ) / . { 2 4 S ( 2 S - l ) } ] % ( A n i - 1 6 ) -257- We need t h e sum, E < S j | T 2 ( s . ,s,) | | S >, f o r eq. ( A . I I I . 4 ) , which i s e q u i v a l e n t to m u l t i p l y i n g eq. (.A.III .16) by S(.2S-1) s i n c e each p a i r o f e l e c t r o n s i s co u n t e d once o n l y ; t h i s g i v e s E < S I I T 2 ( s . , S . ) II S > = [(2 S - l ) 2 S ( 2 S + l ) ( 2 S + 2 ) ( 2 S + 3 y 2 4 ] i s = 1 < S || T 2 ( S , S ) || S > ( A . I I I . 1 7 ) F i n a l l y , s u b s t i t u t i n g i n t o eq. ( A . I I I . 4 ) we f i n d E < S || T 3 il\s), T ^ s . , ^ ) ) || S > = [ ( 2 S - 2 ) ( 2 S - l ) 2 S ( 2 S + l ) ( 2 S + 2 ) ( 2 S + 3 ) ( 2 S + 4 ) / 6 4 0 ] l s ( A . I I I . 1 8 ) so t h a t I F| E ^ m ^ a ^ S ) - ! 2 ^ ^ ) , C 2 ] / r ^ | V N - * - S'J- I F > < n N A S J (jjf ( - D i + j ' + f i F J 1 ( [ i ( i + i ) ( 2 i + l ^ ' \ 6 4 0 J J! i j ' \ [ ( 2 J + 1 ) ( 2 J ^ 1 ) ( 2 N + 1 ) ( 2 N ' + 1 ) ] 1 5 x E ( - D N - V N 2 N \ | q V-A q A ' / N - A / M 0 M'\ / N M " 2 S S 3| J J ' 1 [( 2 S - 2 ) ( 2 S - l ) ( 2 S ) ( 2 S + l ) ( 2 S + 2 ) ( 2 S + 3 ) ( 2 S + 4 ) ] 3 s T 2 ( C ) ( A . I I I . 1 9 ) Where q, A and A ' a r e z e r o t h i s becomes e q u i v a l e n t to eq. (3,20) i n c h a p t e r 3 w i t h T 2 ( C ) - ^ ( 1 4 ) 3 5 b , ( A . I I I . 2 0 ) 0 6 5 -258- The u s e f u l r e s u l t s from t h e d e r i v a t i o n a r e t h e g e n e r a l forms f o r the e l e c t r o n s p i n r e d u c e d m a t r i x e l e m e n t s , ( A . I I I . 1 0 ) , (A.111.11), ( A . I I I . 1 5 ) and ( A . I l l .17). -259- Appendix IV Wigner 9-j symbols needed f o r the US d i p o l i n t e r a c t i o n and t h e t h i r d - o r d e r i s o t r o p i c h y p e r f i n e i n t e r a c t i o n 3X(NSJ)X(NJS) + 2X(SJN) N(N+1) — " % [30(2N-l)2N(2N+l)(2N+2)(2N+3) .S(S+1) ( 2S+1) .J(J+1) (2J+1) ] where X(abc) - a(a+l) + b(b+l) - c(c+l) [N(N+1) + 3S(S+1) - 3J )[ (N+J-S) (N+S-J+1)(J+S-N) (J+S+N+l)] - •—• — !7 (15(2N-l)2N(2N+l)(2N+2)(2N+3).S(S+l)(2S+1).(2J-1)2J(2J+1) ] [ (J+S+N+l)(N+S-J)(N+J-S)(J+S-N+l)(N+S-J-2)(N+S-J-l)(J+S-N+2)(J+S-N+3)] |10(2N-3) (2N-2) (2N-t)2N(2N+l) .S(S+1) (2S+1) . (2J+1) (2J+2)GJ+3) l * * [(J+S+N+l)(N+S-J)(N+J-S)(J+S-N+l)(N+S+J)(N+J-S-l)(N+S-J-l)(J+S-N+2)] [20(2N-3)(2N-2)(2N-1)2N(2N+1).S(S+1)(2S+1).J(J+1)(2J+1)^ [(j+S+N+1) (N+S-J) (N+J-S) (J+S-N+l) (N+S+J) (N+J-S-l) (N+S+J-l) ( N+J-S-2) ] (10(2N-3)(2N-2)(2N-1)2N(2N+1).S(S+1)(2S+1).(2J-1)2J(2J+1)]5 12(2X(JSN)X(SNJ)[5N(N+l)-2] + 2X(NJS)S(S*1)[2X(SNJ)-l] - SX(NJS) X(SNJ) X(JSNj -4X(SNJ)J(J*1) - 4X(JSN)N(N+1)S(S*1) } [lOS(2N-l)2N(2N+l) (2N+2) (2N*3) .2J(2J* 1) (2J+ 2) . (2S-2) (2S-1) 2S(2S+1) (2S+2) (2S+3) (2S+4)]11 where X(abc) = a(a+l) + b(b*l) - c(c*l) 2 [6(N+S+J+1) (J+S-N) (N+J-S) (N+S-J+l) )** x [35(2N-l)2N(2N+l)(2N+2)(2N+3).(2J-1)2J(2J+1).(2S-2)(2S-1)2S(2S+1)(2S+2)(2S+3)(2S+4)J- x {[5N(N+1) + S(S+1) - 5(J-1)(J+1) - 13/2](N(N+l) + S(S+1) - (J-1)(J+1) - 3/2] - 4N(N+1)S(S+1) + 3(J-l/2)(J+l/2)} 2] (N+S+J) (N+S+J+l) (N+S+J-l) (N+J-S-2) (N+J-S-l) (N+J-S) (N+S-J) (J+S-N+l) J*5 (35(2N-3)(2N-2)(2N-l)2N(2N+l).(2J-1)2J(2J+1). (2S-2) (2S-1) 2S( 2S+1) (2S+2) (2S+3) (2S+4)l*5 x (5J(J+1) + 5N(N+1) - S(S+1) - 10N(J+1) + 2] M(N+S+J)(N+S+J+l)(N+S-J)(N+S-J-l)(N+J-S-l)(N+J-S)(J+S-N+t)(J+S-N+2) — — — — r « (70(2N-3)(2N-2)(2N-1)2N(2N+1) . 2J(2J+1) (2J+2) . (2S-2) ( 2S-1) 2S(2S+1) (2S+2) (2S+3) ( 2S+4) P x [5J(J+D - 5N(N+1) + S(S+l) + 2(5N-1)] 2[(J+S-N+3)(J+S-N+2)(J+S-N+l)(N+S-J)(N+J-S)(N+J+S+l)(N+S-J-2)(N+S-J-l)] 7 ^ 3 ^ 2 7 ^ ^ 1 x [5J(J+D + 5N(N+1) - S(S+1) + 10NJ + 2] -262- Appendix V M o l e c u l a r O r b i t a l D e s c r i p t i o n o f the F i r s t - r o w T r a n s i t i o n Metal Monoxides -263- T h i s a p p e n d i x g i v e s a m o l e c u l a r o r b i t a l d e s c r i p t i o n o f t h e f i r s t - r o w t r a n s i t i o n m e t a l o x i d e s . In t h i s a p p r oach t h e m o l e c u l e s a r e f o r m a l l y 2+ 2 r e p r e s e n t e d as M 0 and t h e s p l i t t i n g o f t h e d e g e n e r a t e d o r b i t a l s o f 2+ 2-the N i o n by the l i g a n d f i e l d o f the 0 ~ i o n i s c o n s i d e r e d ( A 8 ) . The r e l a t i v e s t a b i l i t i e s o f h i g h - s p i n o r low s p i n s t a t e s w i l l depend on the s i z e o f t h i s l i g a n d f i e l d s p l i t t i n g (A9.). A t y p i c a l m o l e c u l a r - o r b i t a l e n e r g y l e v e l diagram f o r a f i r s t - r o w t r a n s i t i o n metal monoxide i s p r e s e n t e d i n F i g . A.V.I. The r e l a t i v e p o s i t i o n s o f t h e 3ds and 4so o r b i t a l s i n t h e s e o x i d e s v a r y as t h e atomi c number i n c r e a s e s , as a r e s u l t o f the s c r e e n i n g e f f e c t c r e a t e d by t h e e l e c - t r o n s . T h e r e f o r e F i g . A.V.I r e p r e s e n t s o n l y one way o f w r i t i n g t h i s mo- l e c u l a r - o r b i t a l e n e r g y l e v e l s e q u e n ce, as found i n ScO and T i O , f o r example. The ground s t a t e e l e c t r o n i c c o n f i g u r a t i o n s and term symbols t h a t have been d e t e r m i n e d f o r t h e t r a n s i t i o n metal o x i d e s (A8, A10) a r e 1 2 + ScO ( 4 s o ) ' E T i O VO CrO MnO FeO CuO ( 4 5 a ) 1 (3d&)] 3 A r ( 4 3 a ) 1 ( 3 d 6 ) 2 V ( 4 s a ) ] ( 3 d 6 ) 2 ( 3 d * ) 1 5 n ( 4 s a ) ] ( 3 d 6 ) 2 ( 3 d ^ ) 2 ( 4 s a ) ] ( 3 d & ) 3 (3d7r) 2 5 A i ( 4 s a ) 2 ( 3 d S ) 4 ( 3 d 7 i ) 3 2 H . r -264- M o r b i t a l s MO o r b i t a l s 0 o r b i t a l s 4p 3d 4s / ~7- / / / ' ^Y i / // \ \ \\ w \ \ \ \ \ \ \ \ \ = \ \ \ x \ \ \ \ \_ \ \ \ 4po / 4pir 3do 3 d TT 3d6 4sc 2pn 2po 2so \ \ \ 2p 2s F i g A. V . 1 R e l a t i v e o r b i t a l e n e r g i e s i n a g e n e r a l t r a n s i t i o n - m e t a l monoxide m o l e c u l e , MO. -265- Appendix VI T a b l e s o f a s s i g n e d l i -266- T h i s Appendix c o n t a i n s t h e r o t a t i o n a l l i n e a ssignments on which the r e s u l t s o f t h i s t h e s i s a r e b a s e d . The a b b r e v i a t i o n s used have meanings as f o l l o w s : * i n d i c a t e s a b l e n d e d l i n e } i n d i c a t e s m u l t i p l e l i n e a s s i g n m e n t s f o r a g i v e n r o t a t i o n a l t r a n s i t i o n ( ) i n d i c a t e s t h e l i n e a s s i g n m e n t i s u n c e r t a i n - i n d i c a t e s a l i n e i s e x p e c t e d b u t has n o t been o b s e r v e d i n d i c a t e s t h e l i n e i s p e r t u r b e d . T A B L E I R O T A T I O N A L L I N E S A S S I G N E D O F T H E C 4 2 ~ X 4 2 ~ ( 0 . 0 ) B A N D O F B R A N C H J " F ' - F N = - 1 R 1 ( - 1 ) 0 . 5 1 0 R Q 2 1 ( - 1 ) F E 0 . 5 1 0 - 1 N = 0 R K 0 ) 1 . 5 0 P K 0 ) 1 . 5 - 1 0 4 R 2 ( 0 ) 0 . 5 1 1 P 0 1 2 ( 0 ) E F 0 . 5 1 0 - 1 N = 1 R K 1 ) 2 . 5 1 P K 1 ) 2 . 5 - 1 0 R 2 ( 1 ) 1 . 5 1 P 2 ( 1 ) 1 . 5 - 1 0 N = 2 R K 2 ) 3 . 5 1 0 P K 2 ) 3 . 5 - 1 0 R 2 ( 2 ) 2 . 5 1 P 2 ( 2 ) 2 . 5 - 1 R 3 ( 2 ) 1 . 5 1 0 R 0 4 3 ( 2 ) F E 1 . 5 0 - 1 N = 3 R K 3 ) 4 . 5 1 P K 3 ) 4 . 5 - 1 0 R 2 ( 3 ) 3 . 5 1 P 2 ( 3 ) 3 . 5 - 1 0 R 3 ( 3 ) 2 . 5 1 0 P 3 ( 3 ) 2 . 5 - 1 R 4 ( 3 ) 1 . 5 1 P Q 3 4 ( 3 ) E F 1 . 5 0 F " = J " - 7 / 2 F = J - 5 / 2 F = d - 3 / 2 F = J - 1 / 2 F = J + 1 / 2 F = J + 3 / 2 1 7 4 2 2 . 5 7 5 8 1 7 4 2 1 . 0 9 5 6 1 7 4 2 4 . 6 4 4 1 * 1 7 4 2 4 . 6 2 8 8 * 1 7 4 2 4 . 5 9 9 4 * 1 7 4 2 4 . 5 5 4 5 * 1 7 4 2 4 . 5 0 2 0 4 1 9 . 7 5 6 9 4 1 9 . 7 2 6 3 4 1 9 . 7 1 1 3 4 1 9 . 6 6 9 9 4 1 9 . 6 5 0 0 1 7 4 2 5 . 1 7 0 8 1 7 4 2 5 . 1 4 7 8 ' 1 7 4 2 5 . 1 1 2 2 4 1 8 . 2 7 6 2 4 2 1 . 6 6 1 5 - 1 7 4 2 5 . 0 6 7 9 4 1 8 . 2 4 3 7 4 1 8 . 2 2 8 R 4 1 6 . 2 9 7 4 4 2 1 . 6 4 8 3 * 4 2 1 . 6 5 2 1 1 7 4 2 5 . 0 1 2 7 4 1 B . 1 8 7 6 F = J + 5 / 2 1 7 4 2 2 . 6 4 3 5 * 4 2 4 . 7 9 4 6 4 2 4 . 7 6 5 7 * 1 7 4 2 3 . 2 7 6 0 1 7 4 2 2 . 4 3 5 7 * 4 2 2 . 5 1 7 0 4 2 2 . 3 7 5 1 * 4 2 0 . 4 3 3 1 4 2 0 . 4 9 5 2 * 1 7 4 2 3 . 8 5 7 1 4 2 1 . 0 0 9 6 4 2 2 . 6 8 5 4 4 2 0 . 3 8 1 4 * 1 7 4 2 4 . 4 3 3 6 4 2 4 . 4 5 1 4 4 1 9 . 5 9 8 7 4 1 9 . 5 7 5 7 * 4 2 0 . 8 0 4 6 4 1 9 . 1 9 2 7 1 7 4 2 4 . 9 4 6 1 4 1 8 . 1 1 9 1 4 1 6 . 2 6 2 2 ' 4 1 6 . 2 4 4 1 1 7 4 2 3 . 7 6 4 6 * 4 2 0 . 9 4 1 3 4 2 0 . 9 0 1 5 * 4 2 2 . 6 7 7 9 4 2 0 . 3 9 14 4 2 0 . 4 2 4 6 * 1 7 4 2 4 . 3 5 0 8 4 1 9 . 5 1 1 1 4 1 9 . 4 8 1 5 * 4 2 2 . 7 9 5 5 4 1 7 . 9 7 3 7 4 2 0 . 7 6 0 8 4 2 0 . 7 6 3 7 * 4 1 9 . 2 2 6 0 1 7 4 2 4 . 8 6 8 2 4 18 0 3 8 6 4 2 3 . 0 8 4 4 * 4 1 6 . 2 2 5 5 F = J + 7 / 2 1 7 4 2 2 . 3 9 6 0 4 2 4 . 6 1 6 1 4 2 4 . 5 8 4 7 * 1 7 4 2 2 . 2 7 3 5 4 2 2 . 9 2 9 4 4 2 0 . 5 5 0 7 4 2 0 . 6 1 1 6 1 7 4 2 3 . 6 5 8 9 4 2 0 . 8 1 7 7 * 4 2 2 . 6 7 1 8 4 2 0 . 4 4 2 0 1 7 4 2 4 . 2 5 4 0 4 1 9 . 4 0 7 6 4 2 2 . 7 6 0 5 4 1 7 . 9 3 2 4 4 2 0 . 7 0 8 4 4 1 9 . 2 1 2 4 1 7 4 2 4 . 7 7 9 5 4 1 7 . 9 4 5 1 4 2 3 . 0 3 1 4 4 1 6 . 1 7 9 6 4 2 1 . 6 0 5 5 * 4 2 1 . 5 9 3 5 * 4 1 4 . 6 4 8 9 * 4 2 3 . 0 1 9 3 4 2 3 . 0 2 1 8 * 4 1 4 . 5 9 5 0 * 4 2 3 . 0 2 1 8 * 4 1 7 . 4 4 4 2 I ro cn — i i T A B L E I ( C O N T I N U E D ) B R A N C H J " F ' - F " F " = d " - 7 / 2 F = J - 5 / 2 N = 4 R 1 ( 4 ) 5 . 5 1 1 7 4 2 5 . 6 1 3 1 * 1 7 4 2 5 . 5 8 4 7 P K 4 ) 5 . 5 - 1 4 16 . 7 9 8 6 * 4 16 . 7 7 3 4 * O - 4 1 6 . 7 6 2 8 * R 2 ( 4 ) 4 . 5 1 - - R 0 3 2 ( 4 ) 4 . 5 1 - - P 2 ( 4 ) 4 . 5 - 1 - 4 1 4 . 6 5 9 3 * R 3 ( 4 ) 3 . 5 1 - - P 3 ( 4 ) 3 . 5 - 1 P 0 2 3 ( 4 ) 3 . 5 - 1 R 4 ( 4 ) 2 . 5 1 P 4 ( 4 ) 2 5 - 1 N = 5 R K 5 ) 6 5 1 1 7 4 2 5 9 6 4 8 1 7 4 2 5 9 3 1 3 * P K 5 ) 6 5 - 1 4 1 5 1 7 0 2 4 1 5 1 3 9 5 R 2 ( 5 ) 5 5 1 R 0 3 2 ( 5 ) 5 5 1 P 2 ( 5 ) 5 5 - 1 4 12 9 8 4 8 * 4 12 9 7 7 6 P 0 3 2 ( 5 ) 5 5 - 1 R 3 ( 5 ) 4 5 1 R S 2 3 ( 5 ) 4 5 1 P 3 ( 5 ) 4 5 - 1 4 1 1 9 8 7 3 P 0 2 3 ( 5 ) 4 5 - 1 R 4 ( 5 ) 3 5 1 4 2 5 0 2 0 2 4 2 5 0 3 2 3 P 4 ( 5 ) 3 5 - 1 N = 6 R 1 ( 6 ) 7 . 5 1 1 7 4 2 6 . 2 2 0 7 1 7 4 2 6 1 8 2 8 * P K 6) 7 . 5 - 1 4 1 3 . 4 4 9 5 4 1 3 4 1 5 6 R 2 ( 6) 6 . 5 1 4 2 4 . 0 3 2 6 P 2 ( 6) 6 . 5 - 1 4 1 1 . 2 4 9 7 * 4 1 1 . 2 4 5 4 * P 0 3 2 ( 6) 6 . 5 - 1 4 1 1 . 1 9 9 4 * R 3 ( 6 ) 5 . 5 1 - R S 2 3 ( 6 ) 5 . 5 1 P 3 ( 6 ) 5 . 5 - 1 4 1 0 . 4 4 7 1 4 1 0 . 4 3 9 8 P 0 2 3 ( G ) 5 . 5 - 1 4 1 0 . 4 8 1 2 R 4 ( 6 ) 4 . 5 1 4 2 5 . 6 3 9 3 * 4 2 5 . 6 5 5 8 * P 4 ( G ) 4 . 5 - 1 - - F = J - 3 / 2 1 7 4 2 5 . 5 4 7 4 4 1 6 . 7 3 6 6 4 1 6 . 7 2 4 5 4 1 4 . 6 4 8 9 * 4 1 3 . 4 0 4 2 4 1 3 . 3 9 9 5 F = J - 1 / 2 1 7 4 2 5 . 5 0 0 1 4 1 6 . 6 9 0 5 4 1 6 . 6 7 6 5 * 4 2 3 . 4 9 3 4 * 4 1 4 . 6 3 4 9 * 4 1 3 . 3 9 4 6 * 4 1 3 . 3 9 2 0 1 7 4 2 5 4 1 5 8 9 2 2 1 0 0 4 4 1 2 . 9 6 8 9 4 1 1 . 9 8 2 0 4 2 5 . 0 5 3 9 * 1 7 4 2 6 4 13 4 2 4 4 1 1 4 1 1 4 2 3 4 1 0 4 2 5 4 12 1 4 2 7 . 3 7 4 3 * . 0 0 1 3 . 2 4 0 2 * . 1 7 5 4 . 2 5 8 2 . 4 3 2 9 . 6 8 2 3 * . 7 6 0 6 F = d + 1 / 2 1 7 4 2 5 . 4 4 5 2 4 1 6 . 6 3 4 6 4 1 6 . 6 1 7 6 4 2 3 . 4 7 9 6 4 1 4 . 6 1 6 7 * 4 1 3 . 3 8 9 8 4 1 3 . 3 8 2 7 1 7 4 2 5 4 1 5 8 4 4 2 0 5 3 1 4 1 2 . 9 5 7 7 4 1 2 . 9 5 2 8 4 1 1 . 9 7 6 9 * 4 2 5 . 0 7 9 4 1 7 4 2 6 4 13 4 2 3 4 1 1 4 1 1 0 9 5 4 3 2 6 0 9 8 2 6 2 2 6 8 1 5 7 3 4 2 5 4 12 4 1 2 7 1 7 3 7 9 3 1 8 0 4 0 F - J + 3 / 2 1 7 4 2 5 . 3 8 0 2 4 1 6 . 5 6 8 6 4 1 6 . 5 4 8 7 * 4 2 3 . 3 8 7 2 4 2 3 . 4 5 7 3 4 1 4 . 5 8 8 9 * 4 1 3 . 3 8 0 7 * 4 1 3 . 5 0 9 5 1 7 4 2 5 4 14 7 8 8 7 9 9 7 0 4 1 2 . 9 3 9 4 4 12 . 9 3 2 5 * 4 1 1 . 9 7 0 7 * 4 2 5 . 4 1 4 . 4 1 4 . 1 7 4 2 6 . 4 1 3 4 2 3 4 1 1 4 1 1 4 2 3 1 1 7 8 1 4 4 1 1 5 7 4 0 3 9 3 2 7 0 2 9 6 2 1 2 1 3 5 * 1 3 7 7 3 3 8 3 4 1 0 . 4 2 5 4 * 4 1 0 . 4 2 0 4 * 4 2 5 4 12 F = d + 5 / 2 1 7 4 2 5 . 3 0 5 7 4 1G . 4 9 2 0 4 2 3 . 3 6 5 2 4 2 3 . 4 2 6 1 4 1 4 . 5 5 5 1 * 4 1 3 . 3 7 4 3 * 4 1 3 . 4 9 8 9 4 2 4 . 3 0 7 3 4 1 5 . 2 5 4 6 * F = J + 7 / 2 1 7 4 2 5 . 4 14 4 2 3 7 2 5 5 9 3 2 3 6 9 6 6 4 12 . 9 1 3 7 4 1 2 . 9 0 0 2 * 4 2 2 4 1 1 4 2 5 4 14 1 7 4 2 5 4 13 4 2 3 8 3 3 0 * 9 6 7 7 * . 1 6 2 3 . 1 8 9 1 9 7 7 1 2 0 7 2 9 4 0 0 * . 7 5 9 9 . 8 3 3 7 * 1 7 4 2 5 4 16 1 7 4 2 5 4 14 4 2 3 4 2 3 4 12 6 5 2 9 * 8 5 9 4 6 7 2 3 . 7 4 8 8 * . 8 7 7 6 4 1 2 . 8 1 0 2 4 2 3 . 3 4 4 0 4 1 0 . 4 2 0 4 * 4 2 5 . 8 0 8 8 * 4 1 2 . 8 8 4 5 * 4 1 2 . 9 0 0 2 * 4 2 2 4 1 1 4 1 2 4 2 5 4 14 1 7 4 2 5 4 13 4 2 3 4 1 1 4 1 1 4 1 0 4 2 5 4 1 2 4 1 2 8 3 3 0 * 9 6 8 9 * 0 5 1 3 2 1 4 6 . 2 4 2 4 9 0 6 9 1 3 6 4 * 9 1 2 6 * 0 9 3 0 . 1 5 3 6 . 2 2 15 . 4 0 5 3 4 2 3 . 3 5 0 4 * 4 1 4 4 2 2 4 1 3 . 5 0 5 9 . 3 5 3 4 . 3 7 1 3 4 1 3 . 4 8 5 0 4 1 3 . 4 6 1 4 . 4 2 4 . 3 5 2 7 4 1 5 . 3 0 1 6 1 7 4 2 5 4 14 4 2 3 5 7 3 8 7 7 8 1 6 4 6 3 4 1 2 . 8 1 9 0 4 2 2 4 2 2 4 1 1 4 1 2 4 2 5 4 14 9 1 5 3 8 3 8 3 9 7 6 9 * . 0 4 5 3 . 2 7 5 4 . 3 0 4 5 1 7 4 2 5 . 8 3 0 0 4 1 3 . 0 5 7 4 4 2 3 . 8 8 1 9 * 4 1 1 . 0 7 5 6 4 2 4 0 8 6 6 8 9 4 2 7 * 9 6 0 8 4 1 0 . 4 9 5 8 * 4 1 0 . 4 3 5 5 4 2 5 . 9 3 1 3 * 4 1 3 . 0 0 9 0 I ro CD co I T A B L E I ( C O N T I N U E D ) B R A N C H J " - F » F " = J " - 7 / 2 F = J - 5 / 2 F = J - 3 / 2 F = J - 1 / 2 F = J + 1 / 2 F = J + 3 / 2 F = J * 5 / 2 F - J + 7 / 2 N - 7 R 1 ( 7 ) 8 . 5 P 1 ( 7 ) 8 . 5 R 2 ( 7 ) 7 . 5 P 2 ( 7 ) 7 . 5 P 0 3 2 ( 7 ) 7 5 R 3 ( 7 ) 6 5 P 3 ( 7 ) 6 5 P 0 2 3 ( 7 ) 6 5 R 4 ( 7 ) 5 5 P 4 ( 7 ) 5 5 N = 8 R 1 ( 8 ) 9 5 P 1 ( 8 ) 9 5 R 2 ( 8 ) 8 5 R 0 3 2 ( 8 ) 8 . 5 P 2 ( 8 ) 8 . 5 P 0 3 2 ( 8 ) 8 . 5 R 3 ( 8 ) 7 . 5 P 3 ( 8 ) 7 . 5 P 0 2 3 ( 8 ) 7 . 5 R 4 ( 8 ) 6 . 5 P 4 ( 8 ) 6 . 5 1 7 4 2 6 . 3 8 0 5 4 1 1 . 6 3 1 2 4 2 4 . 1 9 4 6 4 0 9 . 4 0 5 1 • 4 0 9 . 4 4 1 8 * 4 2 3 . 6 2 6 7 4 0 8 . 7 8 7 7 4 2 6 . 0 8 2 8 1 7 4 2 6 . 3 4 4 6 * 4 1 1 . 5 9 5 0 4 2 4 . 1 8 0 0 * 4 0 9 3 9 0 3 * 4 0 9 . 4 4 1 8 * 4 2 3 . 6 3 2 4 4 0 8 . 8 0 1 3 4 0 8 . 7 6 6 0 4 2 6 . 1 0 7 7 1 7 4 2 6 . 4 3 7 6 4 0 9 . 7 1 5 4 4 2 4 . 2 6 6 6 4 0 7 . 5 0 5 1 * 4 0 7 . 5 6 0 5 * 4 0 7 . 0 1 2 2 4 2 6 . 3 8 0 8 * 4 0 9 . 5 3 1 0 * 1 7 4 2 6 . 3 9 8 5 4 0 9 . 6 7 6 4 4 2 4 . 2 5 4 0 4 0 7 . 4 9 0 2 4 0 7 . 5 6 0 5 * 4 0 7 . 0 1 8 8 4 2 6 . 4 0 7 9 * 4 0 9 . 5 5 6 0 1 7 4 2 6 . 2 9 9 1 4 1 1 . 5 5 2 9 * 4 2 4 . 1 6 3 9 4 0 9 . 3 7 3 5 4 0 9 . 4 3 8 4 4 2 3 . 6 3 7 7 4 0 8 . 8 0 8 5 4 0 8 . 7 5 5 4 * 4 2 6 . 1 3 5 8 4 1 1 . 2 4 9 7 1 7 4 2 6 . 3 5 4 0 4 0 9 . 6 3 3 4 4 2 4 . 2 3 8 9 4 0 7 . 4 7 4 4 4 0 7 . 5 5 4 9 4 0 7 . 0 2 6 1* 4 2 6 . 4 4 2 8 4 0 9 . 5 8 8 6 1 7 4 2 6 4 1 1 4 2 4 4 0 9 4 0 9 4 2 3 4 0 8 4 0 8 4 2 6 4 1 1 2 4 9 1 5 0 3 2 1 4 7 1 * 3 5 5 2 4 2 9 2 . 6 4 2 2 . 8 1 3 6 . 7 4 8 8 * . 1 7 4 0 . 2 8 6 9 1 7 4 2 6 4 1 1 4 2 4 4 0 9 4 0 9 4 2 3 4 0 8 4 0 8 4 2 6 4 1 1 1 9 3 6 4 4 7 5 * 1 2 5 3 3 3 4 7 4 1 5 2 6 4 8 0 8 1 8 9 . 7 4 4 9 * . 2 2 0 3 . 3 3 2 5 1 7 4 2 6 . 1 3 1 8 4 1 1 . 3 8 5 0 * 4 2 4 . 1 0 1 8 4 0 9 . 3 1 1 7 4 2 3 . 6 5 5 2 * 4 0 8 . 8 2 4 9 4 0 8 . 7 4 4 9 * 4 2 6 . 2 7 2 2 4 1 1 . 3 8 5 0 * 1 7 4 2 6 . 3 0 3 8 4 0 9 . 5 8 2 7 4 2 4 . 2 1 9 2 * 4 0 7 . 4 5 6 0 4 0 7 . 5 4 7 6 4 0 7 . 0 3 0 1 * 4 2 6 . 4 8 4 2 4 0 9 . 6 2 9 0 * 1 7 4 2 6 . 2 4 8 9 4 0 9 . 5 2 7 3 4 2 4 . 2 0 1 0 * 4 2 4 . 3 4 2 8 4 0 7 . 4 3 5 7 4 0 7 . 5 3 4 9 4 2 3 . 8 4 6 5 4 0 7 . 0 3 7 5 4 2 6 . 5 3 1 1 4 0 9 . 6 7 6 4 * 1 7 4 2 6 . 4 0 9 . 4 2 4 . 4 2 4 . 4 0 7 . 4 0 7 . 4 2 3 . 4 0 7 . 4 0 6 . 4 2 6 4 0 9 1 8 7 2 4 6 5 1 1 7 5 6 3 2 5 1 4 1 2 1 . 5 1 5 1  8 5 5 6 * 0 4 5 3 9 4 3 2 * . 5 8 4 2 . 7 2 9 7 1 7 4 2 6 4 1 1 4 2 4 4 0 9 4 0 9 4 2 3 4 0 8 4 0 8 4 2 6 4 1 1 1 7 4 2 6 . 4 0 9 . 4 2 4 . 4 2 4 . 4 0 7 . 4 0 7 . 4 2 3 . 4 0 7 . 4 0 6 . 4 2 6 4 0 9 F R O M H E R E O N A L L T H E L I N E S H A V E D E L T A F E Q U A L T O D E L T A N B R A N C H E S L A B E L L E D R Q 3 2 . P 0 3 2 . R S 2 3 A N D P 0 2 3 A R E I N D U C E D B Y I N T E R N A L H Y P E R F I N E P E R T U R B A T I O N S 0 6 3 4 3 1 6 0 0 7 4 3 2 8 5 8 3 6 3 6 . 6 6 4 5 . 8 3 1 2 . 7 4 8 8 * . 3 3 2 1 . 4 4 5 1 * . 1 1 9 8 3 9 8 9 * 1 4 7 1 3 0 0 5 . 3 8 4 2 . 4 8 6 9  8 6 6 0 * 0 5 4 5 9 4 9 6 * . 6 4 4 5 . 7 8 9 6 1 7 4 2 5 . 9 8 9 0 4 1 1 . 2 4 0 2 4 2 4 . 0 3 9 6 4 0 9 . 2 5 7 1 4 2 3 . 6 7 7 4 4 0 8 . 8 3 8 6 * 4 0 8 . 7 6 1 8 4 2 6 . 3 9 8 6 * 4 1 1 . 5 1 2 0 1 7 4 2 6 . 0 4 7 1 4 0 9 . 3 2 4 0 4 2 4 . 1 1 0 2 4 0 7 . 3 4 9 5 4 2 3 . 8 8 1 9 4 0 7 . 0 6 7 7 4 0 6 . 9 6 4 7 4 2 6 . 7 0 9 8 * 4 0 9 . 8 5 6 3 I cn to i TABLE I (CONTINUED) BRANCH F"=J" -7 /2 F = J - 5 / 2 F = J - 3 / 2 F=J- 1/2 F = J * 1/2 F = cH 3/2 F=J+5/2 F=J+7/2 N= 9 R K P K R2( P2( P032( R3( P3( P023( R4( P4( 9) 9) 9) 9) 9 ) 9 ) 9) 9) 9) 9 ) 10 10 9 9 9 8 8 8 7 7 17426.3957 17426.3542 407.6953 407 .6570* 424.2448 424.2333 405.5092 405.4956 405 .5876* 405 .5876* 405.1270 426.5492 407.7252 405.1317 426.5802 407.7555 17426.3091 407.6107 424 . 2 192* 405.4809 405.5831 423.9205 405.1366 426 .6174 407.7913 N=10 R1(10) 1 1 . 5 17426 . 2491 17426 2083 17426. 1627 P1(10 ) 1 1 5 405 5765* 405 5349 405 4894 R2(10) 10 5 424 1253* 424 1 150 424 1018* R032(10) 10 5 3894 P2(10 ) 10 5 403 4 155 403 4035 403 P032(10) 10 5 403 5190 403 5 167 R3(10) 9 5 423 8959 P3(10 ) 9 5 403 1321* 403 1343 403 1386 P023(10) 9 5 R4(10) 8 5 426 5960 426 6301 426 6689 P4(10 ) 8 5 405 8008 405 834 1 405 87 19 N= 1 1 R1 ( 1 1 ) 12 5 17426 0025 17425 9602 17425 9 136 P K 1 1 ) 12 5 403 3551* 403 3 125 403 2651 * R2( 1 1 ) 1 1 5 423 9091 4 23 9002* 423 8878 R032( 1 1 ) 1 1 5 2017 P2( 1 1 ) 1 1 5 401 2233 401 2 132* 401 P032( 1 1 ) 1 1 5 401 3506 R3( 1 1 ) 10 5 423 7610* 423 7610* 423 7610* P3( 1 1 ) 10 5 401 0289* 401 .0289* 401 0313* P023( 1 1 ) 10 . 5 603 1 R4( 1 1 ) 9 . 5 426 .5266 426 . 5629 426 P4( 1 1 ) 9 .5 403 . 7575 403 . 7923 403 .8321 17426.2584 407.5G06 424.2010 405.4640 405 .5763* 423.9249 405.1420 426.6603 407.8334 17426.1113 405.4 386 4 2 4 . 0 8 6 9 ' 403.3747 403.5092 423.9002* 403.1432 426.7130 405.9161 17425.8609* 403.2 147 423.8725 401.1872 401.3455 423 . 7646* 401.0343 400 .8881* 426.6488 403.8775 17426 407 424 405 405 423 405 405 426 407 . 2031 .5051 .18 13 .4436 .5639 .9314* .1491 .0368* . 7098* .8824 17426. 407 . 424 . 405 . 405 . 423 405 405 426 407 17426.0554 405.3828 424.0674* 403 403 423 403 403 426 405 . 3548* . 4994 . 9067 . 1506 .0118 .7631* . 9655 17425.8065 403 .1583* 423 .8556* 401 .1695 401.3356 423 .7705 401.0398 400 .8825* 426 .6989* 403 .9279 1427 . 4432 . 1579 4203 5457 . 9400* . 1573 .0368* . 763 1 * . 9368 17425.9963 405.3216 424.0437 403.3317 403.4813 423 .9149* 403 .1583* 403 .0165* 426.8168 406 .0200 17426 407 424 405 405 423 405 405 426 407 17425 405 424 424 403 403 423 403 403 426 406 17425. 403 . 423. 424 401 401 423 401 400 426 403 747 1 0993 . 8322* .0509 . 1467 . 3 192 . 7782 .0483* . 8825* . 7540 . 9827 .0762 . 3768 . 1272* . 3907 .5180* .9527 . 1687 .0424 .823 1 .9970 .9313 . 2566 .0135 .2 132 .3015 .4537 .9314* . 1707 .02 18 .8776* .0807 17425.6823 403.0344 423 .8015 401 401 423 401 400 426 404 1 153 29 19 7928 0622* .8879* .8140 .0429 17426.0052* 407.3041 424 .0869* 405 .3516 423 .97 14 405 .1855* 405 .0578* 426 .8886 408.0636 17425.8609 405.1855 423.9678 403 .2570 423.9493 403 .1899 403 .0373 426.9419 406.1457 17425.6131 402.9647 423.7488 401 .0628* 423.8146 401 .0825* 400.9062 4 2 6 . 8 7 7 6 * 404 .1076 I o i T A B L E I ( C O N T I N U E D ) B R A N C H N = 1 2 R 1 ( 1 2 ) P K 1 2 ) R 2 ( 1 2 ) R 0 3 2 ( 1 2 ) P 2 ( 1 2 ) P 0 3 2 ( 1 2 ) R 3 ( 1 2 ) R S 2 3 ( 1 2 ) P 3 ( 1 2 ) P 0 2 3 ( 1 2 ) R 4 ( 1 2 ) P 4 ( 12 ) N = 1 3 R 1 ( 1 3 ) P K 1 3 ) R 2 ( 1 3 ) R 0 3 2 ( 1 3 ) P 2 ( 1 3 ) P 0 3 2 ( 1 3 ) R 3 ( 1 3 ) R S 2 3 ( 1 3 ) P 3 ( 1 3 ) P 0 2 3 ( 1 3 ) R 4 ( 1 3 ) P 4 ( 1 3 ) N = 1 4 R 1 ( 1 4 ) P 1 ( 14 ) R 2 ( 1 4 ) R 0 3 2 ( 14 ) P 2 ( 14 ) P 0 3 2 ( 1 4 ) R 3 ( 14 ) R S 2 3 ( 14 ) P 3 ( 1 4 ) P Q 2 3 ( 14 ) R 4 ( 14 ) P 4 ( 14 ) 1 3 . 13 . 12 . 12 . 12 . 12 . 1 1 . 1 1 . 1 1 . 1 1 . 1 0 1 0 14 14 1 3 1 3 1 3 1 3 1 2 1 2 1 2 1 2 1 1 . 1 1 . 1 5 . 1 5 . 14 14 14 14 1 3 1 3 1 3 1 3 1 2 1 2 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 5 5 F " = J " - 7 / 2 F = J - 5 / 2 F = J - 3 / 2 F = J - 1 / 2 1 7 4 2 5 . 6 5 2 9 1 7 4 2 5 . 6 0 8 9 1 7 4 2 5 . 5 6 18 1 7 4 2 5 . 5 1 0 3 4 0 1 . 0 2 9 0 4 0 0 . 9 8 6 3 4 0 0 . 9 3 8 9 4 0 0 . 8 8 8 0 * 4 2 3 . 5 9 0 0 * 4 2 3 . 5 8 3 4 4 2 3 . 5 7 4 2 4 2 3 . 5 6 1 8 3 9 8 . 9 3 3 1 * 3 9 8 . 9 2 5 8 3 9 8 . 9 1 6 3 3 9 8 . 9 0 3 8 3 9 9 . 0 8 6 9 * 3 9 9 . 0 8 3 9 * 4 2 3 . 5 1 9 5 * 4 2 3 . 5 1 9 5 * 4 2 3 . 5 1 9 5 * 4 2 3 . 5 1 9 5 * 3 9 8 . 8 1 7 4 * 3 9 8 . 8 1 7 4 * 3 9 8 . 8 1 7 4 * 3 9 8 . 8 1 7 4 * 3 9 8 . 6 6 5 5 * 3 9 8 . 6 5 5 6 3 9 8 . 6 4 8 1 * 4 2 G 3 4 4 7 * 4 2 6 . 3 8 0 8 * 4 2 6 . 4 2 4 0 4 2 6 . 4 6 9 9 4 0 1 . 6 0 5 9 4 0 1 6 4 2 8 4 0 1 . 6 8 4 0 4 0 1 . 7 2 9 B 1 7 4 2 5 . 1 9 7 4 1 7 4 2 5 . 1 5 3 5 1 7 4 2 5 . 1 0 5 8 1 7 4 2 5 . 0 5 3 9 3 9 8 . 6 0 3 6 3 9 8 . 5 6 0 0 3 9 8 . 5 1 2 4 3 9 8 . 4 6 0 7 4 2 3 . 1 6 1 2 * 4 2 3 . 1 5 2 5 4 2 3 . 3 9 8 0 3 9 6 . 5 4 17 3 9 6 . 5 3 7 5 * 3 9 6 . 5 3 1 6 * 3 9 6 . 5 2 0 5 3 9 6 . 7 2 3 5 * 3 9 6 . 7 2 3 5 * 4 2 2 . 9 4 9 2 * 3 9 6 . 4 9 9 5 * 3 9 6 . 4 9 2 2 * 3 9 6 . 4 9 2 2 * 3 9 6 . 3 2 2 0 3 9 6 . 3 0 8 8 * 3 9 6 . 2 9 9 2 * 4 2 6 . 0 4 9 2 4 2 6 . 0 9 0 0 4 2 6 . 1 3 17 4 2 6 . 1 7 9 4 3 9 9 . 3 4 1 0 3 9 9 . 3 7 9 2 3 9 9 . 4 2 17 3 9 9 . 4 6 8 7 F = J + 1 / 2 F = J + 3 / 2 F = J + 5 / 2 F = d + 7 / 2 1 7 4 2 5 4 0 0 4 2 3 4 2 3 3 9 8 3 9 9 4 2 3 4 2 3 3 9 8 3 9 8 4 2 6 4 0 1 1 7 4 2 4 . 6 3 9 9 3 9 6 . 0 7 2 1 4 2 2 . 6 5 0 2 * 4 2 2 8 8 7 5 3 9 4 . 0 5 0 4 * 3 9 4 . 2 4 5 5 4 2 2 . 7 1 7 9 3 9 4 . 0 7 3 8 4 2 5 . 6 4 6 5 3 9 6 . 9 6 6 4 1 7 4 2 4 3 9 6 4 2 2 4 2 2 3 9 4 3 9 4 4 2 2 4 2 2 3 9 4 3 9 3 4 2 5 3 9 7 . 5 9 4 7 . 0 2 7 7 . 6 5 0 2 * . 9 0 0 6 * . 0 5 0 4 * . 2 5 5 7 . 7 0 8 9 . 4 7 0 8 . 0 6 5 1 . 8 6 9 5 . 6 8 6 1 . 0 0 5 9 1 7 4 2 4 3 9 5 4 2 2 4 2 2 3 9 4 3 9 4 4 2 2 4 2 2 3 9 4 3 9 3 4 2 5 3 9 7 . 5 4 7 0 . 9 7 9 8 . 6 4 7 3 . 9 0 5 5 * . 0 4 6 5 . 2 6 1 3 * . 7 0 3 4 . 4 5 4 4 . 0 5 9 0 * . 8 5 2 7 . 7 3 0 5 . 0 4 9 6 1 7 4 2 4 . 3 9 5 . 4 2 2 . 4 2 2 . 3 9 4 . 3 9 4 4 2 2 4 2 2 3 9 4 3 9 3 4 2 5 3 9 7 4 9 5 0 9 2 8 2 . 6 4 0 1 9 0 8 1 0 3 9 4 . 2 6 3 7 * . 6 9 9 9 * . 4 4 2 0 * . 0 5 5 8 * . 8 4 0 4 * . 7 7 8 0 . 0 9 6 4 . 4 5 4 6 . 8 3 2 4 . 5 4 5 9 . 7 7 8 4 * . 8 8 7 6 . 0 7 6 7 . 5 2 5 1 * . 3 0 0 0 * . 8 2 2 5 * . 6 4 2 6 * . 5 2 0 9 . 7 8 0 9 1 7 4 2 5 . 4 0 0 . 4 2 3 . 4 2 3 . 3 9 8 . 3 9 9 . 4 2 3 4 2 3 3 9 8 3 9 8 4 2 6 4 0 1 1 7 4 2 4 . 9 9 8 3 3 9 8 . 4 0 5 1 4 2 3 . 1 3 7 3 * 4 2 3 . 3 9 1 5 * 3 9 6 . 5 0 6 1 3 9 6 . 7 1 7 8 * 3 9 6 . 4 9 4 1 * 3 9 6 . 2 9 1 5 * 4 2 6 . 2 3 0 2 3 9 9 . 5 1 9 9 1 7 4 2 4 3 9 5 4 2 2 4 2 2 3 9 4 3 9 4 4 2 2 4 2 2 3 9 4 3 9 3 4 2 5 3 9 7 . 4 3 9 7 . 8 7 17 . 6 2 9 4 . 9 0 5 5 * . 0 2 8 2 . 2 6 1 3 * . 6 9 9 9 * . 4 3 2 3 . 0 5 5 8 * . 8 3 0 8 * . 8 3 0 0 . 1 4 8 2 3 9 5 3 . 7 7 3 0 5 2 5 1  7 6 4 6 * . 8 6 5 2 * . 0 6 1 1 . 5 3 2 7 . 3 0 0 0 * . 8 3 0 4 * . 6 4 2 6 * . 5 7 6 1 . 8 3 5 8 1 7 4 2 4 . 3 9 8 . 4 2 3 . 4 2 3 . 3 9 6 . 3 9 6 . 4 2 3 . 4 2 2 . 3 9 6 . 3 9 6 4 2 6 3 9 9 9 3 9 2 . 3 4 5 7 , 1 1 6 9 * 3 7 9 3 4 8 7 9 * 7 0 3 9 1 7 1 7 * 9 1 8 7 4 9 9 5 . 2 8 9 0 * . 2 8 5 1 . 5 7 5 0 1 7 4 2 5 . 4 0 0 . 4 2 3 . 4 2 3 . 3 9 8 . 3 9 9 4 2 3 4 2 3 3 9 8 3 9 8 4 2 6 4 0 1 1 7 4 2 4 3 9 8 4 2 3 4 2 3 3 9 6 3 9 6 4 2 3 4 2 2 3 9 6 3 9 6 4 2 6 3 9 9 1 7 4 2 4 . 3 9 5 . 4 2 2 . 4 2 2 . 3 9 4 . 3 9 4 . 4 2 2 4 2 2 3 9 4 3 9 3 4 2 5 3 9 7 3 8 0 9 8 1 2 7 . 6 1 3 0 8 9 6 1 0 1 1 2  2 5 0 9 . 7 0 3 4 . 4 2 9 3 * . 0 5 9 0 * . 8 2 6 8 * . 8 8 4 9 . 2 0 2 9 3 3 12 . 7 0 8 5 4 9 3 4  7 3 8 6 8 3 4 5 . 0 3 4 1 . 5 4 5 9 * . 3 0 5 8 . 8 4 2 6 * . 6 4 8 1 . 6 3 5 4 . 8 9 5 5 . 8 7 5 7 * . 2 8 2 3 . 0 8 7 5 3 5 5 9 * 4 5 7 6 * 6 7 9 8 1 8 4 9 9 2 3 6 5 1 0 8 2 9 3 7 * 3 4 4 7 * . 6 3 4 5 1 7 4 2 4 . 3 9 5 . 4 2 2 . 4 2 2 . 3 9 3 . 3 9 4 . 4 2 2 . 4 2 2 3 9 4 3 9 3 4 2 5 3 9 7 3 1 7 7 7 4 9 8  4 4 2 0  7 3 0 9 . 8 4 0 4 * . 0 8 5 7 . 7 1 3 0 . 4 3 0 1 . 0 6 8 3 . 8 2 7 9 " . 9 4 3 6 . 2 6 2 2 1 7 4 2 5 . 2 6 3 0 4 0 0 . 6 4 0 0 4 2 3 . 4 2 9 7 3 9 8 . 7 7 14 4 2 3 . 5 6 8 4 4 2 3 . 3 2 2 9 3 9 8 . 8 6 5 2 * 3 9 8 . 6 6 5 5 * 4 2 6 . 6 9 8 9 * 4 0 1 . 9 5 9 4 1 7 4 2 4 . 8 0 9 0 3 9 8 . 2 1 4 5 4 2 3 . 0 0 7 7 3 9 6 . 3 7 7 1* 4 2 3 . 2 0 7 5 4 2 2 . 9 3 9 0 3 9 6 . 5 3 1 3 * 3 9 6 . 3 0 8 8 * 4 2 6 . 4 0 7 9 * 3 9 9 . 6 9 8 2 1 7 4 2 4 . 2 5 1 7 3 9 5 . 6 8 3 2 4 2 2 . 4 8 3 1 3 9 3 . 8 8 1 5 4 2 2 4 2 2 3 9 4 3 9 3 4 2 6 3 9 7 . 8 7 6 5 . 5 8 7 5 . 2 3 1 1 . 9 8 5 9 . 0 0 5 2 . 3 2 5 2 T A B L E I ( C O N T I N U E D ) - B R A N C H J " N = 1 5 R 1 ( 1 5 ) 1 6 . 5 P 1 ( 1 5 ) 1 6 . 5 R 2 ( 1 5 ) 1 5 . 5 R 0 3 2 ( 1 5 ) 1 5 . 5 P 2 ( 1 5 ) 1 5 . 5 P 0 3 2 ( 1 5 ) 1 5 . 5 R 3 ( 1 5 ) 1 4 . 5 R S 2 3 ( 1 5 ) 1 4 . 5 P 3 ( 1 5 ) 1 4 . 5 P Q 2 3 ( 1 5 ) 1 4 . 5 R 4 ( 1 5 ) 1 3 . 5 P 4 ( 1 5 ) 1 3 . 5 N = 1 6 R 1 ( 1 6 ) 1 7 . 5 P 1 ( 1 6 ) 1 7 . 5 R 2 ( 1 6 ) 1 6 . 5 R Q 3 2 ( 1 6 ) 1 6 . 5 P 2 ( 1 6 ) 1 6 . 5 P 0 3 2 ( 1 6 ) 1 6 . 5 R 3 ( 1 6 ) 1 5 . 5 R S 2 3 ( 1 6 ) 15 5 P 3 ( 1 6 ) 1 5 . 5 P 0 2 3 ( 1 6 ) 1 5 . 5 R 4 ( 1 6 ) 1 4 . 5 P 4 ( 1 6 ) 1 4 . 5 N = 17 F " = J " - 7 / 2 F = J - 5 / 2 F = J - 3 / 2 F = J - 1 / 2 1 7 4 2 3 . 9 7 7 4 1 7 4 2 3 . 9 3 1 4 * 1 7 4 2 3 . 8 8 1 9 * 1 7 4 2 3 . 8 3 1 8 * 3 9 3 . 4 3 7 3 3 9 3 . 3 9 2 3 3 9 3 . 3 4 4 0 3 9 3 . 2 9 2 6 4 2 2 . 0 3 0 7 * 4 2 2 . 0 3 4 6 * 4 2 2 . 0 3 4 6 * 4 2 2 . 0 3 0 7 * 4 2 2 . 2 9 2 8 4 2 2 . 3 0 5 1 • 4 2 2 . 3 1 4 5 * 4 2 2 . 3 2 0 1 * 3 9 1 . 4 5 7 1 * 3 9 1 . 4 6 1 3 * 3 9 1 . 4 6 1 3 * 3 9 1 . 4 5 7 1 * 3 9 1 . 6 7 4 1 3 9 1 . 6 8 8 6 3 9 1 . 6 9 7 9 * 3 9 1 . 7 0 4 2 * 4 2 2 . 1 5 7 2 4 2 2 . 1 4 0 9 * 4 2 2 . 1 3 0 4 * 4 2 2 . 1 2 4 1 4 2 1 . 8 8 0 8 4 2 1. 8 6 0 1 4 2 1. 8 4 4 1 3 9 1 . 5 4 14 3 9 1 5 2 5 1 * 3 9 1 . 5 1 4 4 3 9 1 . 5 0 7 9 3 9 1 . . 3 0 8 0 3 9 1 2 8 7 2 3 9 1 2 7 1 0 4 2 5 . 1 3 3 9 4 2 5 . 1 7 4 5 4 2 5 . 2 1 9 0 4 2 5 . . 2 6 7 0 3 9 4 . 4 8 2 4 3 9 4 . 5 2 2 9 3 9 4 . 5 6 7 2 3 9 4 . 6 1 5 0 F = J + 1 / 2 F = J + 3 / 2 F = d + 5 / 2 F = d + 7 / 2 1 7 4 2 3 . 3 9 3 . 4 2 1 . 4 2 2 . 3 9 1 . 3 9 1 . 4 2 2 . 4 2 1 3 9 1 3 9 1 4 2 5 3 9 4 1 7 4 2 3 . 2 1 0 3 3 9 0 . 6 9 9 4 4 2 1 . 3 1 0 2 4 2 1 . 5 9 2 6 * 3 8 8 . 7 6 4 3 3 8 9 . 0 0 3 5 4 2 1 . 4 9 0 1 3 8 8 . 9 0 3 0 4 2 4 . 5 1 2 7 3 9 1 . 8 9 1 0 1 7 4 2 3 . 3 9 0 . 4 2 1 . 4 2 1 . 3 8 8 . 3 8 9 4 2 1 4 2 1 3 8 8 3 8 8 4 2 4 3 9 1 R K 1 7 ) 1 8 . 5 1 7 4 2 2 . 3 3 8 9 1 7 4 2 2 P K 1 7 ) 1 8 . 5 3 8 7 . 8 5 5 6 3 8 7 R 2 ( 1 7 ) 17 . 5 4 2 0 . 4 9 2 0 4 2 0 R 0 3 2 ( 1 7 ) 17 . 5 4 2 0 . 7 9 3 0 4 2 0 P 2 ( 17 ) 17 . 5 3 8 5 . 9 7 4 4 3 8 5 P 0 3 2 ( 17 ) 17 . . 5 3 8 6 . . 2 3 4 8 3 8 6 R 3 ( 1 7 ) 1 6 . 5 4 2 0 . 7 1 7 1 4 2 0 R S 2 3 ( 17 ) 16 . 5 4 2 0 P 3 ( 1 7 ) 1 6 . 5 3 8 6 . 1 5 8 6 3 8 6 P 0 2 3 ( 17 ) 16 . 5 3 8 5 R 4 ( 1 7 ) 1 5 . 5 4 2 3 . 7 8 3 1 * 4 2 3 P 4 ( 1 7 ) 1 5 . 5 3 8 9 . 1 9 0 6 3 8 9 1 6 5 0 * . 6 5 3 7 3 1 8 7 * 6 1 0 1 7 7 3 7 * . 0 2 2 8 . 4 6 2 5 . 1 8 16 . 8 7 5 5 . 6 3 6 5 . 5 5 4 5 * . 9 3 2 6 . 2 9 2 8 * . 8 0 9 4 . 3 4 1 9 . 6 5 3 7 . 8 2 5 1 . 0 9 5 4 . 6 7 3 0 . 3 7 12 . 1 1 4 5 . 8 5 4 7 . 8 2 5 3 . 2 3 2 6 1 7 4 2 3 3 9 0 4 2 1 4 2 1 3 8 8 3 8 9 4 2 1 4 2 1 3 8 8 3 8 8 4 2 4 3 9 1 1 7 4 2 2 3 8 7 4 2 0 4 2 0 3 B 5 3 8 6 4 2 0 4 2 0 3 8 6 3 8 5 4 2 3 3 8 9 . 1 1 6 1 * . 6 0 5 1 . 3 2 2 5 . 6 2 3 4 * . 7 7 7 1 * . 0 3 5 9 * . 4 4 7 2 . 1 5 6 7 . 8 6 0 1 . 6 1 1 1 . 5 9 9 4 . 9 7 7 9 . 2 4 3 4 . 7 6 1 0 . 3 2 0 3 . 6 4 1 5 . 8 0 3 4 . 0 8 3 3 * . 8 1 7 7 * . 5 0 5 0 . 2 5 8 4 . 9 8 7 7 * . 8 7 2 5 * . 2 7 8 8 1 7 4 2 3 3 9 0 4 2 1 4 2 1 3 8 8 3 8 8 4 2 1 4 2 1 3 8 8 3 8 8 4 2 4 3 9 2 . 0 6 3 6 . 5 5 2 0 . 1 2 13 . 4 3 2 1* . 5 7 8 1 . 8 4 5 1 * . 4 3 7 5 * . 1 3 6 6 . 8 5 0 6 * . 5 9 13 . 6 4 8 2 . 0 2 5 8 1 7 4 2 2 . 3 8 7 . 4 2 0 . 4 2 0 . 3 8 5 . 3 8 6 . 4 2 0 . 4 2 0 . 3 8 6 . 3 8 5 . 4 2 3 . 3 8 9 . 1 9 1 6 * 7 0 8 2 3 0 3 5 6 3 3 3 7 8 6 4 0 7 5 6 * 8 3 3 2 5 1 1 2 * 2 7 4 2 9 9 4 4 * 9 2 0 5 * 3 2 7 4 7 7 8 2 * 2 3 6 7 8 2 5 4 * . 1 2 2 1* . 2 5 1 7 * . 5 0 8 2 * . 1 2 2 1 * . 8 3 1 7 * . 5 0 6 2 * . 2 5 8 4 * . 3 1 9 4 . 6 6 6 2 1 7 4 2 3 . 3 9 0 . 4 2 1 . 4 2 1 . 3 8 8 . 3 8 8 . 4 2 1 . 4 2 1 3 8 9 3 8 B 4 2 4 3 9 2 0 0 7 7 * 4 9 6 7 . 1 1 2 4 . 4 3 1 2 * . 5 6 7 4 . 8 4 5 1 * . 6 3 2 4 * . 3 2 2 5 . 0 4 5 1 * . 7 7 7 1 * . 7 0 0 0 . 0 7 7 6 1 7 4 2 3 . 3 9 3 . 4 2 1 . 4 2 2 . 3 9 1 . 3 9 1 . 4 2 2 . 4 2 2 3 9 1 3 9 1 4 2 5 3 9 4 1 7 4 2 2 3 9 0 4 2 1 4 2 1 3 8 8 3 8 8 4 2 1 4 2 1 3 8 9 3 8 8 4 2 4 3 9 2 1 7 4 2 2 . 3 8 7 . 4 2 0 . 4 2 0 . 3 8 5 . 3 8 6 . 4 2 0 . 4 2 0 3 8 6 3 8 5 4 2 3 3 8 9 1 3 6 0 * 6 5 3 2 2 9 1 4 6 3 0 2 * . 7 7 3 0 * . 0 7 1 7 * . 8 4 5 2 * . 5 1 2 8 * . 2 8 6 0 * . 9 9 4 4 * . 9 7 14 . 3 7 9 4 7 1 7 8  1 7 8 0  8 2 5 4 *  1 3 0 4 * 2 5 1 7 * 5 1 2 0 *  3 2 0 1 * . 0 2 3 1 . 7 0 4 2 * . 4 4 9 6 . 3 7 4 6 . 7 2 1 4 . 9 4 9 2 . 4 3 7 9 . 1 0 8 9 . 4 3 3 4 . 5 6 3 4 . 8 4 5 1 * . 6 3 3 9 * . 3 1 6 8 . 0 4 5 1 * . 7 7 2 0 * . 7 5 4 8 . 1 3 2 7 1 7 4 2 3 3 9 3 4 2 1 4 2 2 3 9 1 3 9 1 4 2 2 4 2 2 3 9 1 3 9 1 4 2 5 3 9 4 1 7 4 2 2 3 9 0 4 2 1 4 2 1 3 8 8 3 8 8 4 2 1 4 2 1 3 8 9 3 8 8 4 2 4 3 9 2 1 7 4 2 2 . 3 8 7 . 4 2 0 . 4 2 0 . 3 8 5 . 3 8 6 . 4 2 0 4 2 0 3 8 6 3 8 5 4 2 4 3 8 9 0 7 7 2 5 9 4 4 . 2 8 4 4 . 6 3 0 2 * . 7 6 7 2 * . 0 7 1 7 * . 8 4 9 5 * . 5 1 1 2 * . 2 9 1 0 * . 9 9 4 4 * . 0 2 7 8 . 4 3 4 3 . 6 5 5 2 . 1 1 5 4 . 8 3 1 7 * . 1 4 0 9 * . 2 5 8 4 * . 5 2 4 8 * . 3 1 4 5 * . 0 1 0 8 . 6 9 7 9 * . 4 3 6 3 . 4 3 3 3 . 7 8 0 2 . 8 8 7 5 * . 3 7 6 0 . 1 1 1 4 . 4 4 2 2 . 5 6 6 8 * . 8 5 0 6 * . 6 3 2 4 * . 3 0 7 1 . 0 4 5 1 * . 7 6 3 0 * . 8 1 3 1 . 1 9 1 4 1 7 4 2 3 . 5 9 0 0 3 9 3 . 0 4 9 1 4 2 1 . 8 5 5 3 3 9 1 . 2 8 1 9 4 2 2 . 3 0 0 8 4 2 1 . 9 8 9 9 3 9 1 . 6 8 3 5 3 9 1 . 4 1 4 0 * 4 2 5 . 4 9 5 5 3 9 4 . 8 4 2 3 1 7 4 2 2 . 8 2 2 2 3 9 0 . 3 1 0 5 4 2 1 . 1 2 3 9 3 8 8 . 5 7 8 6 " 4 2 1 4 2 1 3 8 9 3 8 8 4 2 4 3 9 2 6 2 3 4 * 2 9 2 9 . 0 3 6 1 * . 7 4 7 8 * . 8 7 5 7 * . 2 5 2 8 1 7 4 2 2 . 0 1 5 8 3 8 7 . 5 3 3 0 4 2 0 . 2 8 3 5 4 2 0 . 6 3 4 7 * 3 8 5 . 7 6 7 2 * 3 8 6 . 0 7 5 6 * 4 2 0 . 8 4 9 5 * 3 8 6 . 2 9 1 0 * 4 2 4 . 0 8 6 9 * 3 8 9 . 4 9 2 6 1 7 4 2 1 . 9 5 0 9 3 8 7 . 4 6 7 7 4 2 0 . 2 9 0 0 3 8 5 . 7 7 3 0 * 4 2 0 . 8 4 5 2 * 3 8 6 . 2 8 6 0 * 4 2 4 . 1 4 7 1 3 8 9 . 5 5 3 9 I P O ro T A B L E I ( C O N T I N U E D ) B R A N C H J " N = 1 8 R K 1 8 ) P 1 ( 1 8 ) R 2 ( 1 8 ) R 0 3 2 ( 1 8 ) P 2 ( 18 ) P 0 3 2 ( 1 8 ) R 3 ( 1 8 ) R S 2 3 ( 1 8 ) P 3 ( 1 8 ) P 0 2 3 ( 1 8 ) R 4 ( 1 8 ) P 4 ( 1 8 ) N = 1 9 R 1 ( 1 9 ) P K 1 9 ) R 2 ( 1 9 ) R 0 3 2 ( 1 9 ) P 2 ( 1 9 ) P 0 3 2 ( 1 9 ) R 3 ( 1 9 ) R S 2 3 ( 1 9 ) P 3 ( 1 9 ) P 0 2 3 ( 1 9 ) R 4 ( 1 9 ) P 4 ( 1 9 ) N = 2 0 R K 2 0 ) 2 1 P 1 ( 2 0 ) 2 1 R 2 ( 2 0 ) 2 0 P 2 ( 2 0 ) 2 0 P 0 3 2 ( 2 0 ) 2 0 R 3 ( 2 0 ) 1 9 R S 2 3 ( 2 0 ) 1 9 P 3 ( 2 0 ) 1 9 P 0 2 3 ( 2 0 ) 1 9 R 4 ( 2 0 ) 1 8 P 4 ( 2 0 ) 1 8 F " = J " - 7 / 2 F - J - 5 / 2 F = d - 3 / 2 F = d - 1 / 2 F = d + 1 / 2 F = J + 3 / 2 F = J + 5 / 2 F = J + 7 / 2 1 9 . 5 1 7 4 2 1 . 3 6 2 8 1 7 4 2 1 . 3 1 5 8 1 9 . 5 3 8 4 . 9 1 1 2 3 8 4 . 8 6 4 7 1 8 . 5 4 19 . 4 4 8 5 4 1 9 . 4 1 7 6 1 8 . 5 4 1 9 . 7 7 1 1 4 1 9 . 7 5 0 1 1 8 5 3 8 2 . 9 6 2 7 3 8 2 . 9 3 1 3 1 8 . 5 3 8 3 . 2 4 4 2 3 8 3 . 2 2 3 0 17 . 5 4 19 . 8 3 6 8 4 1 9 . 8 9 5 7 * 17 . 5 4 1 9 . 5 7 3 8 17 . 5 3 8 3 . 3 1 0 1 3 8 3 . 3 6 9 5 17 . 5 3 8 3 . 0 8 7 9 1G . . 5 4 2 2 9 4 5 4 4 2 2 . 9 8 8 9 16 . 5 3 8 6 . . 3 8 4 6 3 8 6 . 4 2 7 4 2 0 . 5 1 7 4 2 0 . 2 8 3 5 * 1 7 4 2 0 . 2 3 5 9 2 0 . 5 3 8 1 . 8 7 4 1 3 8 1 . 8 2 7 3 1 9 . 5 4 1 8 . 4 1 5 9 4 1 8 . 3 8 4 5 1 9 . 5 4 1 8 . 7 5 9 5 4 1 8 . 7 3 6 3 1 9 . 5 3 7 9 . 9 5 9 7 3 7 9 . 9 2 7 8 1 9 . 5 3 8 0 . 2 6 16 3 8 0 . 2 3 9 7 1 8 . 5 4 1 8 . 8 5 1 1 4 1 8 . 8 9 9 1 1 8 . 5 4 1 8 . 5 5 6 6 1 8 . 5 3 8 0 . 3 5 3 7 3 8 0 . 4 0 2 2 1 8 . 5 3 8 0 . 1 0 0 4 17 . 5 4 2 1 . 9 9 9 9 4 2 2 . 0 4 3 9 17 . 5 3 8 3 . 4 7 3 3 * 3 8 3 . 5 1 7 3 ' 1 7 4 2 1 3 8 4 4 19 4 1 9 3 8 2 3 8 3 4 1 9 4 1 9 3 8 3 3 8 3 4 2 3 3 8 6 1 7 4 1 9 . 0 9 8 0 3 7 8 4 1 7 3 7 6 3 7 7 4 1 7 7 0 6 6 * 2 7 4 0 * 8 5 0 2 1 7 2 3 . 7 5 9 1 * 3 7 7 . 2 9 2 2 4 2 0 . 9 4 7 6 3 8 0 . 4 5 3 0 1 7 4 1 9 . 0 5 0 6 3 7 8 . 6 5 9 4 * 4 1 7 . 2 4 3 3 3 7 6 . 8 1 8 7 3 7 7 . 1 5 0 8 4 1 7 . B O O 1 3 7 7 . 3 3 3 7 3 7 7 . 0 1 2 4 4 2 0 . 9 9 2 0 3 8 0 . 4 9 7 5 2 6 6 8 . 8 1 5 7 . 3 9 4 1 . 7 3 6 2 . 9 0 8 1 . 2 0 8 9 . 9 2 1 9 . 5 8 9 9 . 3 9 5 2 . 1 0 4 1 . 0 3 5 0 . 4 7 3 9 1 7 4 2 0 . 3 8 1 . 4 1 8 . 4 1 8 . 3 7 9 . 3 8 0 . 4 1 8 . 4 1 8 . 3 8 0 . 3 8 0 4 2 2 3 8 3 1 8 6 1 7 7 8 1 3 6 0 3 7 2 1 1 . 9 0 3 8 . 2 2 5 5 . 9 2 5 5 . 5 7 2 5 . 4 2 8 7 . 1 1 6 6 . 0 9 0 7 . 5 6 3 7 < 1 7 4 2 1 . 3 8 4 . 4 1 9 . 4 19 . 3 8 2 . 3 8 3 . 4 19 . 4 19 . 3 8 3 . 3 8 3 4 2 3 3 8 6 1 7 4 1 9 . 3 7 8 . 4 1 7 . 3 7 6 . 3 7 7 . 4 1 7 . 4 1 7 3 7 7 3 7 7 4 2 1 3 8 0 0 0 0 7 6 0 9 4 * . 2 1 9 0 . 7 9 4 5 . 1 3 5 5 . 8 2 6 5 . 4 5 3 1 . 3 5 9 8 . 0 2 8 6 . 0 3 8 2 * . 5 4 4 6 2 1 4 3 7 6 3 0 3 7 6 1 7 2 6 3 * 8 9 0 0 . 1 9 9 8 . 9 3 9 7 . 5 9 8 7 * . 4 1 3 3 . 1 1 2 3 * . 0 8 4 4 . 5 2 3 1 1 7 4 2 1 . 3 8 4 . 4 1 9 . 4 1 9 . 3 8 2 . 3 8 3 . 4 19 . 4 1 9 . 3 8 3 . 3 8 3 . 4 2 3 . 3 8 6 1 7 4 2 0 . 1 3 3 6 3 8 1 . 7 2 4 7 •1 18 . 3 4 12 3 7 9 3 8 0 4 18 4 18 3 8 0 3 8 0 4 2 2 3 8 3 8 8 4 6 2 1 4 7 9 4 4 2 5 8 2 3 4 4 6 9 1 2 6 0 * 1 4 0 9 * 6 1 3 3 * 1 7 4 1 8 . 9 4 7 7 3 7 8 5 5 6 7 * 4 1 7 . 1 9 9 3 3 7 6 . 7 7 4 7 3 7 7 . 1 2 4 6 4 1 7 . 8 4 5 8 4 1 7 . 4 6 2 6 3 7 7 . 3 7 9 2 3 7 7 . 0 3 8 4 4 2 1 . 0 8 9 1 3 8 0 . 5 9 4 2 1 5 9 0 7 0 8 1 3 6 2 3 7 2 0 9 8 7 6 0 1 9 3 7 * 9 5 2 4 6 0 1 4 * 4 2 4 6 1 1 5 9 * 1 3 7 3 5 7 5 0 1 7 4 2 1 3 8 4 4 1 9 4 19 3 8 2 3 8 3 4 1 9 4 1 9 3 8 3 3 8 3 4 2 3 3 8 6 1 7 4 2 0 . 0 7 8 4 3 8 1 . 6 7 0 2 4 1 8 . 3 2 5 0 3 7 9 . 8 6 9 7 3 8 0 . 2 0 8 6 4 1 8 . 9 5 7 9 4 1 8 . 5 8 7 9 3 8 0 . 4 6 1 3 3 8 0 . 1 3 1 2 * 4 2 2 . 1 9 2 4 * 3 8 3 . 6 6 5 6 * 1 7 4 1 8 . 8 9 3 0 3 7 8 . 5 0 2 0 * 4 1 7 . 1 8 3 2 3 7 6 . 7 5 8 5 3 7 7 . 1 1 7 5 * 4 1 7 . 8 6 0 3 1 0 0 9 6 4 9 5 3 5 2 5 * . 7 1 9 6 . 8 6 6 4 * . 1 9 3 7 * . 9 6 1 1 * . 6 0 1 4 * . 4 3 4 6 * . 1 1 5 9 * . 1 9 2 0 . 6 2 9 9 1 7 4 2 1 . 0 3 8 2 * 3 8 4 . 5 8 8 0 * 4 1 9 . 3 5 0 8 * 1 7 4 2 0 . 0 2 0 4 3 8 1 . 6 1 2 1 4 1 8 . 3 1 5 1 3 7 9 . 8 5 8 8 3 8 0 . 2 0 4 1* 4 1 8 . 9 6 7 4 3 8 0 . 4 7 0 5 3 8 0 4 2 2 1 3 1 2 * 2 4 7 7 3 8 3 . 7 2 0 3 * 1 7 4 1 8 . 8 3 5 0 3 7 8 . 4 4 4 2 * 4 1 7 . 1 7 1 0 3 7 6 . 7 4 6 7 3 7 7 . 1 1 7 5 * 4 1 7 . 8 7 0 8 3 8 2 3 8 3 4 1 9 4 1 9 3 8 3 3 8 3 4 2 3 3 8 6 8 6 6 4 * 1 9 3 7 * 9 6 1 1* 5 9 8 7 * 4 3 4 6 * 1 1 2 3 * 2 5 0 2 * . 6 8 8 5 1 7 4 2 0 . 9 7 5 7 3 8 4 . 5 2 3 5 4 1 9 . 3 5 0 8 * 3 8 2 . 8 6 6 4 " 4 1 9 4 1 9 3 8 3 3 8 3 4 2 3 3 8 6 9 6 1 1* 5 8 9 9 * 4 3 4 6 * 1 0 4 1 * . 3 1 1 6 * . 7 4 9 4 1 7 4 1 9 . 9 6 1 1* 3 8 1 . 5 5 0 9 4 1 8 . 3 0 8 3 * 3 7 9 . 8 5 2 0 * 3 8 0 . 2 0 4 1 • 4 1 8 . 9 7 3 2 * 3 8 0 . 4 7 5 9 * 3 8 0 . 1 3 1 2 * 4 2 2 . 3 0 5 1 * 3 8 3 . 7 7 8 5 * 3 7 7 . 3 9 4 0 3 7 7 . 4 0 4 7 3 7 7 . 0 4 5 3 * 3 7 7 . 0 4 5 3 * 4 2 1 . 1 4 1 2 4 2 1 . 1 9 6 2 3 8 0 . 6 4 6 5 3 8 0 . 7 0 1 8 1 7 4 1 9 . 8 9 5 7 * 3 8 1 . 4 8 6 9 4 1 8 . 3 0 8 3 * 3 7 9 . 8 5 2 0 * 4 1 8 . 9 7 3 2 * 3 8 0 . 4 7 5 9 * 3 B 0 . 1 3 1 2 * 4 2 2 . 3 6 6 3 3 8 3 . 8 3 9 0 * 1 7 4 1 8 . 7 7 4 5 1 7 4 1 8 . 7 1 1 4 * 3 7 8 . 3 8 3 3 * 3 7 8 . 3 1 9 5 * 4 1 7 . 1 6 2 7 4 1 7 . 1 5 9 7 3 7 6 . 7 3 8 2 3 7 6 . 7 3 4 9 3 7 7 . 1 1 7 5 * 4 1 7 . 8 7 7 6 4 1 7 . 8 8 0 4 3 7 7 . 4 1 1 9 3 7 7 . 0 4 5 3 * 4 2 1 . 2 5 3 8 3 8 0 . 7 5 9 6 3 7 7 . 4 1 3 9 3 7 7 . 0 4 5 3 * 4 2 1 . 3 1 4 0 3 8 0 . 8 1 9 9 ro T A B L E I ( C O N T I N U E D ) B R A N C H J - F - J - 7 / 2 F - J - 5 / 2 F = U - 3 / 2 F - J - , / 2 F = U + . / 2 F = U + 3 / 2 F = U + 5 / 2 F - J - 7 / 2 N = 2 1 R K 2 1 ) P 1 ( 2 1 ) R 2 ( 2 1 ) P 2 ( 2 1 ) P 0 3 2 ( 2 1 ) R 3 ( 2 1 ) P 3 ( 2 1 ) P 0 2 3 ( 2 1 ) R 4 ( 2 1 ) P 4 ( 2 1 ) N = 2 2 N = 2 3 N = 2 4 R K 2 2 ) P K 2 2 ) R 2 ( 2 2 ) P 2 ( 2 2 ) R 3 ( 2 2 ) P 3 ( 2 2 ) R 4 ( 2 2 ) P 4 ( 2 2 ) R K 2 3 ) P K 2 3 ) R 2 ( 2 3 ) P 2 ( 2 3 ) R 3 ( 2 3 ) P 3 ( 2 3 ) R 4 ( 2 3 ) P 4 ( 2 3 ) R K 2 4 ) P 1 ( 2 4 ) R 2 ( 2 4 ) P 2 ( 2 4 ) R 3 ( 2 4 ) P 3 ( 2 4 ) R 4 ( 2 4 ) P 4 ( 2 4 ) 2 2 . 5 1 7 4 1 7 . 8 0 8 4 1 7 4 1 7 . 7 6 0 4 1 7 4 1 7 . 7 1 0 8 1 7 4 1 7 . 6 5 8 5 2 2 . 5 3 7 5 . 4 4 8 3 3 7 5 . 4 0 0 5 3 7 5 . 3 5 0 8 3 7 5 . 2 9 8 0 2 1 . 5 4 16 . 0 2 6 1 * 4 1 5 . . 9 9 5 4 4 15 9 7 0 7 4 1 5 . 9 5 0 8 2 1 . 5 3 7 3 . 6 3 0 4 3 7 3 . 6 0 0 7 3 7 3 5 7 6 5 3 7 3 5 5 6 1 2 1 . . 5 3 7 3 . 9 7 4 5 * 3 7 3 . 9 5 4 7 * 3 7 3 9 3 8 6 * 2 0 . 5 4 16 . 5 5 9 8 4 16 . 5 9 6 7 4 16 . 6 2 2 1 4 16 . 6 4 14 2 0 . 5 3 7 4 . 1 2 5 4 3 7 4 . 1 6 2 0 3 7 4 . 1 8 6 4 3 7 4 . 2 0 6 6 2 0 . 5 3 7 3 . 8 3 5 7 3 7 3 . 8 4 4 5 * 1 9 . 5 4 1 9 . 7 8 6 4 4 1 9 . 8 3 1 5 4 1 9 . 8 7 9 0 4 19 . 9 2 8 8 1 9 . 5 3 7 7 . 3 2 6 9 3 7 7 . 3 7 16 3 7 7 . 4 1 8 8 3 7 7 . 4 6 8 9 2 4 . 2 4 2 3 2 3 2 2 2 2 2 1 2 1 2 5 2 5 2 4 2 4 2 3 2 3 2 2 2 2 1 7 4 1 7 . 6 0 2 7 3 7 5 . 2 4 2 9 4 1 5 . 9 3 3 2 3 7 3 . 5 3 9 3 2 3 . 5 1 7 4 1 6 . 4 1 3 1 1 7 4 1 6 . 3 6 5 5 * 1 7 4 1 6 . 3 1 4 9 2 3 . 5 3 7 2 . 0 8 6 6 3 7 2 . 0 3 9 0 3 7 1 9 8 8 8 2 2 5 4 14 . 6 7 0 1 4 1 4 . 6 4 0 7 4 14 . 6 1 6 7 2 2 5 3 7 0 . 3 0 8 4 3 7 0 . 2 7 8 3 3 7 0 . 2 5 4 0 2 1 . 5 4 1 5 . 2 5 4 6 * 4 1 5 . 2 8 8 4 4 15 . 3 1 3 0 2 1 . 5 3 7 0 . 8 5 7 0 3 7 0 . 8 9 0 7 3 7 0 . 9 1 5 8 2 0 . 5 4 1 8 . 5 1 5 5 4 18 . 5 6 1 1 4 1 8 . 6 0 9 3 2 0 . 5 3 7 4 . 0 9 3 0 3 7 4 . 1 3 7 6 3 7 4 . 1 8 6 4 * 1 7 4 14 3 6 8 4 1 3 . 3 6 6 . 4 1 3 . 3 6 7 . 4 1 7 . 3 7 0 . 1 7 4 1 3 . 3 6 5 4 11 3 6 3 4 12 3 6 3 4 1 5 3 6 7 9 1 2 7 6 1 7 4 2 1 0 1 * 8 7 5 6 8 4 4 0 . 4 7 0 4 . 1 3 0 1 . 7 5 2 2 . 3 0 9 1 . 0 4 6 7 . 6 4 4 4 . 3 4 8 6 . 3 2 8 5 9 9 3 8 6 1 0 2 2 9 7 3 1 7 4 14 . 8 6 4 9 3 6 8 . 5 7 0 0 4 1 3 . 1 8 1 5 3 6 6 . 8 4 7 3 4 1 3 . 8 7 4 4 3 6 7 . 5 0 1 8 4 1 7 . 1 7 5 8 3 7 0 . 8 0 2 1 1 7 4 1 4 . 8 1 4 8 3 6 8 . 5 1 9 3 4 1 3 . 1 5 7 6 * 3 6 6 . 8 2 2 7 4 1 3 . 8 9 9 0 3 6 7 . 5 2 6 1 4 1 7 . 2 2 3 9 3 7 0 . 8 5 0 0 1 7 4 1 3 3 6 4 4 1 1 3 6 3 4 12 3 6 4 4 15 3 6 7 2 6 1 2 9 9 7 4 6 1 5 8 3 2 1 1 3 5 7 8 0 2 3 7 6 5 6 8 3 4 3 7 1 7 4 1 3 3 6 4 4 1 1 3 6 3 4 12 3 6 4 4 1 5 3 6 7 . 2 1 0 1 * . 9 4 7 3 . 5 9 2 3 . 2 9 6 7 . 3 8 1 1 . 0 4 8 4 . 7 0 5 2 . 3 9 1 5 2 6 2 2 9 3 6 3 5 9 5 0 2 3 3 6 3 3 2 3 3 7 0 . 9 3 4 9 4 1 8 . 6 5 9 2 3 7 4 . 2 3 5 6 1 7 4 16 3 7 1 4 14 3 7 0 4 15 1 7 4 1 4 . 3 6 8 . 4 1 3 . 3 6 6 . 4 1 3 . 3 6 7 4 17 3 7 0 1 7 4 1 3 3 6 4 4 1 1 3 6 3 4 12 3 6 4 4 15 3 6 7 7 6 17 4 6 6 8 1 3 6 4 * 8 0 2 1 9 1 7 8 * 5 4 5 3 2 7 4 0 + 9 0 0 4 1 5 7 6 * 8 9 4 1 . 5 7 1 8 * . 2 7 6 3 . 4 0 0 5 . 0 6 6 7 . 7 5 5 7 . 4 4 2 3 1 7 4 1 7 . 5 4 5 2 3 7 5 . 1 8 5 0 4 1 5 . 9 2 0 9 * 3 7 3 . 5 2 6 1 4 16 3 7 4 6 5 7 2 2 2 17 4 1 9 . 9 8 16 3 7 7 . 5 2 1 6 1 7 4 16 3 7 1 4 14 3 7 0 4 1 5 3 7 0 4 1 8 3 7 4 1 7 4 1 4 . 3 6 8 . 4 1 3 . 3 6 6 . 4 1 3 . 3 6 7 . 4 1 7 . 3 7 0 . 1 7 4 1 3 . 3 6 4 4 1 1 3 6 3 4 1 2 3 6 4 4 1 5 3 6 7 2 0 7 1 8 8 1 0 5 7 8 7 2 1 6 4 3 4 8 0 9 5 1 2 7 1 1 4 * . 2 8 8 3 . 7 0 6 7 4 1 1 2 1 1 9 3 7 8 5 0  9 3 4 3 . 5 6 1 0 . 3 2 7 1 . 9 5 2 8 1 0 2 9 8 3 9 0 5 5 2 9 * 2 5 9 0 4 1 6 5 0 8 2 9 8 0 8 4 4 9 4 8 4 16 3 7 4 . 6 6 8 7 . 2 3 3 3 4 2 0 . 0 3 6 7 3 7 7 . 5 7 6 7 1 7 4 1 6 . 1 4 9 3 3 7 1 . 8 2 3 1 4 1 4 . 5 6 5 2 3 7 0 . 2 0 2 9 4 1 5 . 3 6 0 1 3 7 0 . 9 6 3 3 4 1 B . 7 6 6 9 3 7 4 . 3 4 3 2 1 7 4 1 4 . 6 4 9 0 3 6 8 . 3 5 3 6 4 1 3 . 1 0 4 7 3 6 6 . 7 7 0 2 4 1 3 . 9 4 7 1 3 6 7 . 5 7 3 6 4 1 7 . 3 8 2 2 3 7 1 . 0 0 7 8 1 7 4 1 3 . 0 4 4 8 3 6 4 . 7 8 17 4 1 1 . 5 3 9 5 3 6 3 . 2 4 3 9 4 1 2 . 4 3 0 0 3 6 4 . 0 9 5 2 4 1 5 . 8 6 3 3 3 6 7 . 5 4 9 8 1 7 4 1 7 3 7 5 4 1 5 3 7 3 4 8 4 0 1 2 3 9 . 9 1 1 7 . 5 1 7 0 * 4 1 6 . 6 7 6 6 3 7 4 . 2 4 0 7 4 2 0 . 0 9 4 1 3 7 7 . 6 3 4 3 1 7 4 1 7 . 4 2 1 4 3 7 5 . 0 6 0 9 4 1 5 . 9 0 6 0 3 7 3 . 5 1 1 7 * 4 1 6 . 6 8 0 8 3 7 4 . 2 4 4 9 4 2 0 . 1 5 3 8 3 7 7 . 6 9 3 9 1 7 4 16 3 7 1 . 4 1 4 . 3 7 0 . 4 1 5 . 3 7 0 . 4 1 8 . 3 7 4 . 1 7 4 1 4 . 3 6 8 4 1 3 3 6 6 4 1 3 3 6 7 4 17 3 7 1 1 7 4 1 2 3 6 4 4 1 1 3 6 3 4 1 2 3 6 4 4 1 5 3 6 7 0 8 9 0 7 6 2 7 5 5 5 1 * . 1 9 2 1 . 3 6 9 0 . 9 7 2 1 . 8 2 4 0 4 0 0 8 . 5 8 8 9 . 2 9 3 6 . 0 9 4 1 . 7 5 9 4 . 9 5 7 0 . 5 8 4 7 . 4 3 9 2 . 0 6 5 2 . 9 8 4 8 . 7 2 14 . 5 2 7 2 2 3 16 4 4 0 1 1 0 6 4 9 2 0 9 * 6 0 7 0 1 7 4 16 3 7 1 . 4 1 4 . 3 7 0 . 4 1 5 . 3 7 0 . 4 1 8 . 3 7 4 . 1 7 4 1 4 . 3 6 8 . 4 1 3 . 3 6 6 . 4 1 3 3 6 7 4 1 7 3 7 1 1 7 4 1 2 3 6 4 4 1 1 3 6 3 4 12 3 6 4 4 15 3 6 7 0 2 6 1 • 7 0 0 1 5 4 7 8 1 B 5 2 . 3 7 4 7 . 9 7 6 8 8 8 3 4  4 6 0 1 5 2 6 1 2 3 0 7 0 8 5 8 7 5 13 9 6 3 2 5 9 0 0 4 9 8 6 1 2 4 8 9 2 2 5 6 5 9 7 5 1 8 1 2 2 3 3 4 4 8 6 1 1 3 1 . 9 7 9 3 . 6 6 6 5 I ro -4 4 ^ T A B L E I ( C O N T I N U E D ) N = 25 N = 26 N = 27 B R A N C H R1 ( 2 5 ) P 1 ( 2 5 ) R 2 ( 2 5 ) P 2 ( 2 5 ) R 3 ( 2 5 ) P 3 ( 2 5 ) R 4 ( 2 5 ) P 4 ( 2 5 ) R 1 ( 2 6 ) P 1 ( 26 ) R 2 ( 2 6 ) P 2 ( 2 6 ) R 3 ( 2 6 ) P 3 ( 2 6 ) R 4 ( 2 6 ) P 4 ( 2 6 ) R 1 ( 2 7 ) P K 2 7 ) R 2 ( 2 7 ) P 2 ( 2 7 ) R 3 ( 2 7 ) P 3 ( 2 7 ) R 4 ( 2 7 ) P 4 ( 2 7 ) F " = J " - 7 / 2 F = J - 5 / 2 F = J - 3 / 2 F = J - 1 / 2 F - J + 1 / 2 F = J + 3 / 2 F = J + 5 / 2 F = J + 7 / 2 N = 2 8 26 . 26 . 25 . 25 . 24 . 24 . 23 23 27 27 26 26 25 25 24 24 R 1 ( 2 8 ) P 1 ( 2 8 ) R 2 ( 2 8 ) P 2 ( 2 8 ) R 3 ( 2 8 ) P 3 ( 2 8 ) R 4 ( 2 8 ) P 4 ( 28 ) 5 5 5 5 5 5 5 5 5 5 . 5 . 5 . 5 . 5 . 5 . 5 174 1 1 36 1 4 0 9 3 5 9 4 10 3 6 0 4 13 3 6 3 6 0 0 3 3 6 8 5 9 7 2 6 . 7 1 4 0 . 7 0 5 3 . 4 0 8 6 . 77 16 . 7 3 9 8 174 1 1 361 4 0 9 3 5 9 4 10 3 6 0 4 13 3 6 3 5 4 9 6 * 3 2 0 3 9 4 5 7 6 8 7 1 7 3 5 0 4 3 6 4 . 8 1 8 3 . 7 8 6 4 1 7 4 0 9 . 7 8 6 7 3 5 7 . 5 8 8 0 4 0 8 . 1 9 4 0 3 5 5 . 9 6 9 3 4 0 8 . 9 7 8 6 3 5 6 . 7 1 3 2 4 1 2 . 5 7 4 8 3 6 0 . 0 4 6 4 1 7 4 0 9 . 7 3 7 6 3 5 7 . 5 3 9 3 4 0 8 . 1678 3 5 5 . 9 4 3 2 4 0 9 . 0 0 6 0 3 5 6 . 7 3 9 7 4 1 2 . 6 1 8 9 3 6 0 . 0 9 3 2 174 1 1 36 1 4 0 9 359 4 10 3 6 0 4 13 3 6 3 28 . 5 1 7 4 0 7 . 8 6 5 4 1 7 4 0 7 . 8 178 28 . 5 3 5 3 . 7 0 4 8 3 5 3 . 6 5 5 4 27 . 5 4 0 6 . 3 112 4 0 6 . 2 8 4 7 27 . 5 352 . 1217 3 5 2 . 0 9 5 5 26 . 5 407 . 1 4 2 9 * 4 0 7 . 1683 26 . 5 352 . 9 1 4 3 3 5 2 . 9 4 0 9 25 . 5 4 1 0 . 7 125 4 1 0 . 7 6 1 3 25 . 5 356 . 0 2 9 6 ) 3 5 6 . . 0 7 5 5 ) 357 . . 0 7 7 4 ) 357 . 1 2 3 5 ) 29 . 5 17405 . B4 12 1 7 4 0 5 . 7 9 2 5 29 . 5 3 4 9 . 7 179 3 4 9 . 6 6 7 3 28 . 5 4 0 4 . 3 2 3 7 4 0 4 . 2 9 6 8 28 . 5 3 4 8 . 1768 3 4 8 . 1 5 1 1 27 . 5 4 0 5 . 2 0 1 6 4 0 5 . 2 2 7 6 27 . 5 3 4 9 . 0 1 4 0 3 4 9 . 0 4 0 2 26 . 5 4 0 8 . 7 9 1 5 4 0 8 . 8 3 8 8 * 26 . 5 352 . 6 5 6 0 352 . 7 0 3 9 5 0 1 3 2 6 9 6 92 16 6 6 3 1 7 5 8 2 4 6 0 1 . 8 6 6 4 . 8 3 5 5 1 7 4 0 9 . 6 8 7 0 3 5 7 . 4 8 8 9 4 0 8 . 1 4 3 9 3 5 5 . 9 1 9 2 4 0 9 . 0 2 7 8 3 5 6 . 7 6 2 2 4 1 2 . 6 6 7 9 3 6 0 . 1 4 2 2 17407 353 4 0 6 352 407 352 4 10 3 5 6 357 7 6 5 9 6 0 4 4 26 14 0 7 2 0 1904 . 9 6 3 5 . 8 0 8 6 . 1244 . 1 7 18 1 7 4 1 1 . 4 4 7 5 * 3 6 1 . 2 1 6 9 4 0 9 . 9 0 1 0 3 5 9 . 6 4 1 2 4 1 0 . 7 7 7 0 3 G O . 4 7 8 8 4 1 3 . 9 1 7 8 * 3 6 3 . 8 8 5 5 1 7 4 0 5 . 7 4 18 3 4 9 . 6 1 8 4 4 0 4 . 2 7 4 3 348 . 1279 4 0 5 . 2 4 9 7 3 4 9 . 0 6 1 9 4 0 8 . 8 8 8 0 3 5 2 . 7 5 2 3 174 1 1 361 4 0 9 3 5 9 4 10 3 6 0 4 13 3 6 3 1 7 4 0 9 357 4 0 8 3 5 5 4 0 9 3 5 6 4 12 3 6 0 6 3 3 4 * 4 3 6 3 1222 8 9 7 9 0 4 7 0 * 78 18 . 7 1 8 1 . 1 9 16 1 7 4 0 7 . 7 1 2 9 3 5 3 . 55 10 4 0 6 . 2 4 0 5 3 5 2 . 0 5 2 1 4 0 7 . 2 0 9 7 352 . 9 8 2 4 4 1 0 . 8 5 9 8 3 5 6 . 1 7 5 1 ) 3 5 7 . 2 2 0 3 ) 1 7 4 0 5 . 6 8 8 7 3 4 9 . 5 6 5 1 4 0 4 . 2 5 3 8 3 4 8 . 1 0 6 9 4 0 5 . 2 6 8 0 3 4 9 . 0 8 1 3 4 0 8 . 9 3 9 2 3 5 2 . 8 0 3 9 3 9 3 0 1617 8 8 3 4 6 2 4 1 7 9 3 3 4 9 4 6 9 6 7 5 . 9 3 8 5 1 7 4 0 9 . 5 7 9 3 3 5 7 . 3 8 0 8 4 0 8 . 1 0 4 8 3 5 5 . 8 8 0 2 4 0 9 . 0 6 3 3 3 5 6 . 7 9 7 4 4 12 . 77 10 3 6 0 . 2 4 5 5 1 7 4 0 7 . 6 5 7 0 * 3 5 3 . 4 9 6 0 4 0 6 . 2 2 2 1 3 5 2 . 0 3 3 3 4 0 7 . 2 2 6 1 3 5 2 . 9 9 8 5 4 1 0 . 9 1 2 5 3 5 6 . 2 2 7 8 ) 3 5 7 . 2 7 2 2 ) 1 7 4 0 5 . 6 3 3 6 3 4 9 . 5 1 0 5 4 0 4 . 2 3 4 5 3 4 8 . 0 8 9 3 4 0 5 . 2 8 4 8 3 4 9 . 0 9 7 4 4 0 8 . 9 9 2 2 3 5 2 . 8 5 6 4 174 1 1. 3 3 4 2 * 36 1 . 1043 4 0 9 . 8 6 8 1 3 5 9 . 6 0 8 9 4 1 0 . 8 0 6 9 3 6 0 . 5 0 8 3 4 1 4 . 0 2 3 1 3 6 3 . 9 9 3 8 1 7 4 0 9 . 5 2 1 9 3 5 7 . 3 2 4 7 4 0 8 . 0 8 9 4 3 5 5 . 8 6 5 4 4 0 9 . 0 7 7 3 3 5 6 . 8 1 2 0 4 12 . 8 2 5 6 3 6 0 . 3 0 0 9 1 7 4 0 7 . 6 0 1 0 3 5 3 . 4 3 8 7 4 0 6 . 2 0 6 8 3 5 2 . 0 1 8 4 4 0 7 2 3 9 9 3 5 3 . 0 1 2 2 4 1 0 . 9 6 7 1 3 5 6 . 2 8 1 9 ) 3 5 7 . 3 2 5 1 ) * 174 1 1 36 1 4 0 9 3 5 9 4 1 0 3 6 0 4 14 364 2 7 4 4 0 4 4 4 8 5 6 3 * 5 9 6 2 8 168 . 5 1 8 7 . 0 7 9 0 . 0 5 0 4 1741 1 3 6 0 4 0 9 3 5 9 4 10 3 6 0 4 14 364 2 1 3 5 * 9 8 3 2 8 4 6 2 5 8 6 8 8 2 5 5 . 5 2 6 0 . 1377 . 1097 1 7 4 0 5 3 4 9 4 0 4 3 4 8 4 0 5 3 4 9 4 0 9 3 5 2 5 7 6 3 4 5 3 4 2 193 0 7 2 7 . 2 9 9 1 . 1 1 1 7 . 0 4 7 0 * . 9 1 1 4 1 7 4 0 9 . 4 6 2 3 3 5 7 . 2 6 3 2 4 0 8 . 0 7 5 8 3 5 5 . 8 5 1 9 4 0 9 . 0 8 8 7 3 5 6 . 8 2 3 2 4 1 2 . 8 8 2 3 3 6 0 . 3 5 6 7 1 7 4 0 7 . 5 4 1 7 3 5 3 . 3 7 9 5 4 0 6 . 1 9 3 4 3 5 2 . 0 0 4 1 4 0 7 . 2 5 2 9 3 5 3 . 0 2 4 6 4 1 1 . 0 2 3 6 3 5 6 . 3 3 6 8 ) 3 5 7 . 3 8 0 8 ) 1 7 4 0 5 . 5 1 8 0 3 4 9 . 3 9 4 3 4 0 4 . 2 0 5 0 3 4 8 . 0 5 8 4 4 0 5 . 3 1 1 3 3 4 9 . 1 2 3 5 4 0 9 . 1 0 3 3 3 5 2 . 9 6 6 8 1 7 4 0 9 . 3 9 8 9 * 3 5 7 . 2 0 2 5 4 0 8 . 0 6 5 8 3 5 5 . 84 14 4 0 9 . 0 9 7 5 3 5 6 . 83 10 4 1 2 . 9 4 0 7 3 6 0 . 4 1 5 9 1 7 4 0 7 . 4 7 9 6 3 5 3 . 3 1 8 1 4 0 6 . 1 8 2 6 3 5 1 . 9 9 3 4 4 0 7 . 2 6 1 1 3 5 3 . 0 3 2 5 4 1 1 . 0 8 2 2 * 3 5 6 . 3 9 7 5 ) 3 5 7 . 4 3 6 5 ) * 1 7 4 0 5 . 4 5 6 0 3 4 9 . 3 3 2 6 4 0 4 . 1 9 4 0 3 4 8 . 0 4 7 8 4 0 5 . 3 2 1 6 * 3 4 9 . 1 3 3 2 4 0 9 . 1619 3 5 3 . 0 2 4 6 r o — I t n i T A B L E I ( C O N T I N U E D ) B R A N C H d - F - J - - 7 / 2 F = d - 5 / 2 F = d - 3 / 2 F = J - 1 / 2 F = d + 1 / 2 F = d + 3 / 2 F = d + 5 / 2 F = d + 7 / 2 N = 2 9 N = 3 0 N = 3 1 R 1 ( 2 9 P 1 ( 2 9 R 2 ( 2 9 P 2 ( 2 9 R 3 ( 2 9 P 3 ( 2 9 R 4 ( 2 9 P 4 ( 2 9 R 1 ( 3 0 P 1 ( 3 0 R 2 ( 3 0 P 2 ( 3 0 R 3 ( 3 0 P 3 ( 3 0 R 4 ( 3 0 P 4 ( 3 0 R K 3 1 P 1 ( 3 1 R 2 ( 3 1 P 2 ( 3 1 R 3 ( 3 1 P 3 ( 3 1 R 4 ( 3 1 P 4 ( 3 1 R 1 ( 3 2 R 2 ( 3 2 R 3 ( 3 2 R 4 ( 3 2 P 4 ( 3 2 N = 3 3 R 1 ( 3 3 P 1 ( 3 3 R 2 ( 3 3 P 2 ( 3 3 R 3 ( 3 3 P 3 ( 3 3 R 4 ( 3 3 N = 3 2 3 0 3 0 2 9 2 9 2 8 2 8 2 7 2 7 3 1 3 1 3 0 3 0 2 9 2 9 2 8 2 8 3 2 . 3 2 . 3 1 . 3 1 . 3 0 . 3 0 . 2 9 . 2 9 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 3 . 5 3 2 . 5 3 1 . 5 3 0 . 5 3 0 . 5 3 4 3 4 3 3 3 3 3 2 3 2 3 1 1 7 4 0 3 . 3 4 5 . 4 0 2 . 3 4 4 . 4 0 3 3 4 5 4 0 6 3 4 8 1 7 4 0 1 3 4 1 7 1 2 9 6 2 4 9 2 2 9 0 1 1 3 7 1 5 5 9 * 0 0 5 2 * 7 7 2 9 6 2 4 0 4 7 8 1 4 2 9 0 1 7 4 0 3 . 6 6 3 7 3 4 5 . 5 7 5 4 4 0 2 . 2 0 3 4 3 4 4 . 0 8 8 8 4 0 3 . 1 8 0 6 3 4 5 . 0 2 8 9 * 4 0 6 . 8 2 0 6 3 4 8 . 6 7 0 7 4 0 0 . 0 3 0 5 3 3 9 . 9 5 6 1 4 0 1 3 4 0 4 0 4 3 4 4 0 0 2 7 . 8 9 0 8 . 6 5 5 6 . 5 2 7 8 1 7 3 9 9 . 1 4 1 9 3 3 7 . 1 2 7 7 3 9 7 . 7 2 8 2 3 3 5 . 6 9 0 0 3 9 8 . 7 4 5 7 3 3 6 . 6 7 17 4 0 2 . 4 3 1 3 3 4 0 . 3 3 7 6 1 7 4 0 1 3 4 1 . 4 2 9 6 . 3 7 8 4 4 0 0 . 0 0 4 8 3 3 9 . 9 3 0 8 4 0 1 . 0 2 7 4 * 3 4 0 . 9 1 5 3 4 0 4 . 7 0 2 9 3 4 4 . 5 7 5 1 1 7 3 9 9 . 0 9 2 2 3 3 7 . 0 7 8 7 3 9 7 . 7 0 2 9 3 3 5 . 6 6 5 5 3 9 8 . 7 7 1 4 * 3 3 6 . 6 9 6 7 4 0 2 . 4 7 9 1 3 4 0 . 3 8 4 4 1 7 3 9 6 . 7 0 0 0 * 1 7 3 9 6 . 6 5 0 0 3 9 5 . 3 1 7 2 3 9 5 . 2 9 1 5 3 9 6 . 3 8 1 4 * 3 9 6 . 4 0 3 7 4 0 0 . 1 0 5 8 4 0 0 . 1 5 3 7 3 3 6 . 0 4 6 8 3 3 6 . 0 9 4 4 1 7 4 0 3 . 6 1 2 6 3 4 5 . 5 2 4 0 4 0 2 . 1 8 0 2 3 4 4 . 0 6 5 3 4 0 3 . 2 0 2 1 3 4 5 . 0 5 1 0 * 4 0 6 . 8 6 9 8 3 4 8 . 7 2 0 0 1 7 4 0 1 . 3 7 7 7 3 4 1 . 3 2 7 5 3 9 9 . 9 8 2 5 3 3 9 . 9 0 7 4 4 0 1 . 0 4 8 1 * 3 4 0 . 9 3 6 0 4 0 4 . 7 5 2 3 3 4 4 . 6 2 5 2 1 7 3 9 9 . 0 4 0 7 3 3 7 . 0 2 7 2 3 9 7 . 6 8 0 2 3 3 5 . 6 4 2 4 3 9 8 . 7 9 0 5 3 3 6 . 7 1 7 6 4 0 2 . 5 2 9 0 3 4 0 . 4 3 4 6 1 7 3 9 6 . 5 9 9 2 3 9 5 . 2 6 8 9 3 9 6 . 4 2 4 6 4 0 0 . 2 0 3 3 3 3 6 . 1 4 6 6 1 7 4 0 3 3 4 5 4 0 2 3 4 4 4 0 3 3 4 5 4 0 6 3 4 8 1 7 3 9 4 . 1 5 5 8 1 7 3 9 4 . 1 0 6 4 1 7 3 9 4 . 0 5 7 0 3 2 8 . 2 1 9 3 3 9 2 . 8 0 3 4 3 2 6 . 8 4 8 7 3 9 3 . 9 1 14 3 2 7 . 9 2 2 5 3 9 7 . 6 7 6 3 3 2 8 . 1 7 0 6 3 9 2 . 7 7 8 6 3 2 6 . 8 2 4 5 3 9 3 . 9 3 4 3 3 2 7 . 9 4 6 1 3 9 7 . 7 2 4 2 3 2 8 . 1 1 8 8 3 9 2 . 7 5 6 0 3 2 6 . 8 0 1 9 3 9 3 . 9 5 5 3 3 2 7 . 9 6 6 5 3 9 7 . 7 7 3 9 5 5 9 5 4 7 17 1 5 9 9 0 4 4 3 2 2 0 8 0 7 0 5 * 9 2 13 . 7 7 15 1 7 4 0 3 . 5 0 5 0 3 4 5 . 4 1 6 4 4 0 2 . 1 4 2 0 3 4 4 . 0 2 6 5 4 0 3 . 2 3 7 9 3 4 5 . 0 8 6 9 * 4 0 6 . 9 7 4 3 3 4 8 . 8 2 3 4 1 7 4 0 1 . 3 2 4 0 3 4 1 . 2 7 4 6 3 9 9 . 9 6 14 3 3 9 . 8 8 6 9 4 0 1 . 0 6 3 0 * 3 4 0 . 9 5 5 5 4 0 4 . 8 0 4 0 3 4 4 . 6 7 5 3 1 7 3 9 8 . 9 8 7 6 3 3 6 . 9 7 4 3 3 9 7 . 6 5 9 8 3 3 5 . 6 2 14 3 9 8 . 8 1 0 1 * 3 3 6 . 7 3 4 7 4 0 2 . 5 8 0 1 3 4 0 . 4 8 6 1 1 7 3 9 6 . 5 4 5 3 * 3 9 5 . 2 4 8 8 3 9 6 . 4 4 4 2 4 0 0 . 2 5 4 6 3 3 6 . 1 9 5 5 17 3 9 4 . 0 0 2 0 3 2 8 . 0 6 6 0 3 9 2 . 7 3 5 2 3 2 6 . 7 8 1 3 3 9 3 . 9 7 4 0 3 2 7 . 9 8 5 7 3 9 7 . 8 2 4 9 1 7 4 0 3 . 4 4 7 1 3 4 5 . 3 5 9 2 4 0 2 . 1 2 5 9 3 4 4 . 0 0 8 8 4 0 3 . 2 5 2 6 * 3 4 5 . 1 0 1 1 * 4 0 7 . 0 3 0 1 3 4 8 . 8 7 9 0 1 7 4 0 1 3 4 1 3 9 9 3 3 9 4 0 1 3 4 0 4 0 4 3 4 4 2 6 9 4 2 1 9 8 * 9 4 3 2 8 6 8 7 . 0 8 2 5 * . 9 7 17 . 8 5 7 0 . 7 2 9 1 1 7 4 0 1 3 4 1 3 9 9 3 3 9 4 0 1 3 4 0 4 0 4 3 4 4 2 1 3 3 * 1 6 3 0 9 2 6 8 8 5 2 2 0 9 7 6 9 8 5 6 9 1 1 3 . 7 8 3 9 1 7 3 9 8 . 9 3 2 9 * 3 3 6 . 9 1 9 7 3 9 7 . 6 4 1 6 3 3 5 . 6 0 3 4 3 9 8 . 8 2 5 3 * 3 3 6 . 7 5 19 4 0 2 . 6 3 4 0 3 4 0 . 5 3 9 5 1 7 3 9 8 8 7 6 0 3 3 6 . 8 6 3 0 3 9 7 . 6 2 4 9 3 3 5 . 5 8 6 9 3 9 8 . 8 4 0 1 * 3 3 6 . 7 6 6 8 4 0 2 . 6 8 7 6 3 4 0 . 5 9 4 5 1 7 3 9 6 . 4 8 7 9 * 1 7 3 9 6 . 4 3 3 4 3 9 5 . 2 3 0 0 3 9 5 . 2 1 3 6 3 9 6 . 4 6 1 1 * 3 9 6 . 4 7 6 3 * 4 0 0 . 3 0 8 0 4 0 0 . 3 6 3 0 3 3 6 . 2 4 9 4 3 3 6 . 3 0 4 6 1 7 3 9 3 . 9 4 6 8 1 7 3 9 3 . 8 8 9 9 3 2 B . 0 1 0 1 3 9 2 . 7 1 6 9 3 2 6 . 7 6 2 8 3 9 3 . 9 9 0 7 3 2 8 . 0 0 1 2 3 9 7 . 8 7 7 9 3 2 7 . 9 5 3 3 3 9 2 . 6 9 9 4 3 2 6 . 7 4 6 2 3 9 4 . 0 0 5 6 3 2 8 . 0 1 6 7 3 9 7 . 9 3 2 6 1 7 4 0 3 . 3 8 8 8 * 3 4 5 . 3 0 0 2 4 0 2 . 1 1 1 5 3 4 3 . 9 9 5 8 4 0 3 . 2 6 5 2 * 3 4 5 . 1 1 2 5 4 0 7 . 0 8 5 0 3 4 8 . 9 3 4 8 1 7 4 0 1 . 1 5 3 5 3 4 1 . 1 0 4 3 3 9 9 . 9 1 2 9 3 3 9 . 8 3 8 2 4 0 1 . 1 1 0 8 3 4 0 . 9 9 9 1 4 0 4 . 9 6 8 3 3 4 4 . 8 4 0 1 1 7 3 9 8 . 8 1 8 0 * 3 3 6 . 8 0 2 9 3 9 7 . 6 1 0 6 3 3 5 . 5 7 1 9 3 9 8 . 8 5 3 3 3 3 6 . 7 7 9 6 4 0 2 . 7 4 4 0 3 4 0 . 6 4 9 9 1 7 3 9 6 . 3 7 6 5 * 3 9 5 . 1 9 8 4 3 9 6 . 4 8 7 9 * 4 0 0 . 4 1 9 2 3 3 6 . 3 6 0 5 1 7 3 9 3 . 8 2 7 8 3 2 7 . 8 9 4 7 3 9 2 . 6 8 4 8 3 2 6 . 7 3 1 1 3 9 4 . 0 1 9 0 3 2 8 . 0 2 9 8 i 3 9 7 . 9 8 8 5 1 7 4 0 3 . 3 2 7 0 3 4 5 . 2 3 9 7 4 0 2 . 1 0 0 1 3 4 3 . 9 8 4 7 4 0 3 . 2 7 4 5 3 4 5 . 1 2 3 3 4 0 7 . 1 4 2 9 * 3 4 8 . 9 9 2 9 1 7 4 0 1 . 0 9 2 6 3 4 1 . 0 4 3 0 * 3 9 9 . 9 0 1 1 3 3 9 . 8 2 6 3 4 0 1 . 1 2 1 3 3 4 1 . 0 0 8 7 4 0 5 . 0 2 5 9 3 4 4 . 8 9 8 8 1 7 3 9 8 . 7 5 6 4 3 3 6 . 7 4 2 4 3 9 7 . 5 9 8 5 3 3 5 . 5 5 9 5 3 9 8 . 8 6 5 2 * 3 3 6 . 7 8 9 9 4 0 2 . 8 0 1 7 3 4 0 . 7 0 8 7 1 7 3 9 6 . 3 1 4 0 * 3 9 5 . 1 8 5 6 3 9 6 . 4 9 9 5 4 0 0 . 4 7 7 1 3 3 6 . 4 1 8 6 1 7 3 9 3 . 7 7 12 3 2 7 . 8 3 3 8 3 9 2 . 6 7 2 0 3 2 6 . 7 1 9 3 3 9 4 . 0 3 0 3 * 3 2 8 . 0 4 1 4 3 9 8 . 0 4 6 6 PO (Tl I T A B L E I ( C O N T I N U E D ) N = 34 N=35 B R A N C H R K 3 4 ) P 1 ( 3 4 ) R 2 ( 3 4 ) P 2 ( 3 4 ) R 3 ( 3 4 ) P 3 ( 3 4 ) R 4 ( 3 4 ) P 4 ( 3 4 ) R 1 ( 3 5 ) P 1 ( 3 5 ) R 2 ( 3 5 ) P 2 ( 3 5 ) R 3 ( 3 5 ) P 3 ( 3 5 ) R 4 ( 3 5 ) P 4 ( 3 5 ) F " = d " - 7 / 2 F = d - 5 / 2 F = d ~ 3 / 2 F = d - 1/2 F = d + 1 / 2 F = d + 3 / 2 F = d + 5 / 2 F = d + 7 / 2 N = 37 3 5 . 35 . 34 . 34 . 33 . 33 . 32 . 32 . 36 . 36 . 3 5 . 35 . 34 . 34 33 33 R 1 ( 3 7 ) P 1 ( 3 7 ) R 2 ( 3 7 ) P 2 ( 3 7 ) R 3 ( 3 7 ) P 3 ( 3 7 ) R 4 ( 3 7 ) P 4 ( 3 7 ) N = 3 6 R 1 ( 3 6 ) 3 7 . 5 P K 3 6 ) 3 7 . 5 R 2 ( 3 6 ) 3 6 . 5 P 2 ( 3 6 ) 3 6 . 5 R 3 ( 3 6 ) 3 5 . 5 P 3 ( 3 6 ) 3 5 . 5 R 4 ( 3 6 ) 3 4 . 5 P 4 ( 3 6 ) 3 4 . 5 1 7 3 9 1 . 5 1 4 5 * 3 2 3 . 6 0 8 6 3 9 0 . 1 8 2 6 3 2 2 . 2 7 1 6 391 . 3 3 4 7 3 2 3 . 3 8 8 7 395 . 1 3 7 2 327 . 1 6 2 0 1 7 3 9 1 . 4 6 7 8 * 3 2 3 . 5 5 9 8 3 9 0 . 1584 3 2 2 . 2 4 7 5 391 . 3 5 7 4 3 2 3 . 4 1 2 7 3 9 5 . 1856 3 2 7 . 2 1 0 6 1 7 3 8 8 . 3 1 8 . 387 . 3 17 388 318 392 322 17385 385 314 384 312 385 314 389 317 6 9 5 1 8 9 6 2 4 7 6 4 . 5 8 7 9 . 6 5 1 1 . 7 5 0 9 . 4 9 3 8 . 5 6 1 1 .7 7 0 0 ) . 9 1 0 1 ) . 1003 . 6 1 4 3 . 8 1 2 0 . 86 16 . 0 1 6 5 . 7 4 2 8 . 8 5 5 5 * 17391 . 4 1 6 7 * 3 2 3 . 5 0 7 2 3 9 0 . 1358 3 2 2 . 2 2 5 2 391 . 3 7 8 1 3 2 3 . 4 3 1 8 3 9 5 . 2 3 4 5 3 2 7 . 2 6 0 4 17388 3 1 8 3 8 7 3 1 7 3 8 8 3 1 8 3 9 2 3 2 2 6 4 7 1 8 4 5 6 * 4 5 1 8 5 6 3 9 . 6 7 3 8 . 7 7 3 1 . 5 4 2 3 . 6 0 9 8 1 7 3 8 8 3 1 8 387 317 3 8 8 3 18 392 322 1 7 3 8 5 . 7 3 3 9 ) 3 8 5 . 8 7 1 6 ) 3 1 4 . 0 4 5 7 3 8 4 . 5 8 8 0 3 1 2 . 7 8 7 4 3 8 5 . 8 8 4 0 3 1 4 . 0 3 7 6 3 8 9 . 7 9 1 4 3 1 7 . 9 0 4 2 * 38 . 5 1 7 3 8 2 . 9 0 0 5 1 7 3 8 2 . 8 5 2 3 * 38 5 3 0 9 . 1084 3 0 9 . 0 6 0 5 37 . 5 381 . . 6 9 3 4 38 1 . 6 7 0 2 37 5 307 . 9 3 3 9 * 3 0 7 . 9 0 8 9 36 . 5 382 . 9 6 6 8 3 8 2 . 9 8 9 0 3 6 . 5 3 0 9 . 1624 3 0 9 . 1845 35 . 5 386 . 8 8 8 7 3 8 6 . 9 3 6 6 3 5 . 5 313 . 0 5 5 8 3 1 3 . 1056 5 9 7 0 7 9 4 3 * . 4 2 8 8 . 54 10 . 6 9 5 1 . 7 9 4 3 * . 5 9 2 4 . 6 6 0 5 1 7 3 8 5 . 6 9 8 4 ) 3 8 5 . 8 3 6 0 ) 3 1 3 . 9 9 8 8 3 8 4 . 5 6 7 9 3 1 2 . 7 6 4 7 3 8 5 . 9 0 4 0 3 1 4 . 0 5 0 5 3 8 9 . 84 19 3 1 7 . 9 5 3 9 * 1 7 3 8 2 . 7 9 9 3 3 0 9 . 0 0 9 5 381 . 6 4 7 6 3 0 7 . 8 8 5 6 3 8 3 . 0 0 9 3 3 0 9 . 2 0 3 4 3 8 6 . 9 8 7 1 3 1 3 . 1 5 5 2 17391 323 3 9 0 322 39 1 323 395 327 17388 3 18 . 387 . 3 1 7 . 388 . 3 18 392 322 1 7 3 8 5 3 8 5 313 384 3 12 3 8 5 3 14 389 3 1 8 3 6 5 1 4 5 2 2 * 1 145* . 2 0 4 4 . 3 9 6 7 . 4 5 2 2 . 2 8 6 2 .31 19 . 5 4 4 9 7 4 2 4 . 4 0 7 4 . 52 1 1 . 7 128 . 8 1 2 1 . 6 4 4 1 . 7 1 2 1 6 5 8 7 ) 8 0 3 4 ) 9 4 7 9 5 4 7 6 7 4 5 3 9 2 2 6 0 7 7 8 8 9 3 3 . 0 0 6 5 * 1 7 3 9 1 . 3 1 0 7 3 2 3 . 3 9 9 8 3 9 0 . 0 9 6 2 3 2 2 . 1866 3 9 1 . 4 140 3 2 3 . 4 6 7 6 3 9 5 . 3 3 9 7 3 2 7 . 3 6 5 2 1 7 3 8 8 . 4 9 14 3 1 8 . 6 8 6 6 3 8 7 . 3 8 7 0 3 1 7 . 5 0 1 6 3 8 8 . 7 2 9 7 3 1 8 . 8 2 9 0 3 9 2 . 6 9 7 3 3 2 2 . 7 6 4 5 1 7 3 8 2 . 7 4 6 1 3 0 8 . 9 5 7 1 3 8 1 . 6 2 7 4 3 0 7 . 8 6 4 6 3 8 3 . 0 2 7 9 3 0 9 . 2 2 2 8 3 8 7 . 0 3 8 7 3 1 3 . 2 0 7 0 1 7 3 9 1 . 2 5 4 4 3 2 3 . 3 4 2 8 3 9 0 . 0 7 9 5 3 2 2 . 1 6 9 4 39 1 . 4 2 8 6 3 2