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High field Stark effects in diatomic hydrides Scarl, Ethan Adam 1973

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HIGH FIELD STARK EFFECTS ' IN DIATOMIC MOLECULES .BY ETHAN ADAM SCARL B.A., Reed C o l l e g e , 1961 M.S., Washington U n i v e r s i t y , 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF .THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1973 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission fo r extensive copying of t h i s thesis fo r scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P h y s i c s  The University of B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT . • i A LoSurdo d i s c h a r g e was used to. apply e l e c t r i c f i e l d s of more than 300 kV/cm to water vapor, a mixture of cyanogen and hydrogen, and ammonia, p e r m i t t i n g the o b s e r v a t i o n of S t a r k e f f e c t s i n the o p t i c a l s p e c t r a o f the OK, CH, and NH m o l e c u l e s , r e s p e c t i v e l y . These experiments y i e l d e d the f o l l o w i n g v a l u e s of the m o l e c u l a r e l e c t r i c d i p o l e moment i n the ground v i b r a t i o n a l s t a t e s : u(OH, A 2 E + ) •= 1.94 g ± 0.08 5 D y(OD, A 2 Z + ) = 2 - 1 4 2 ± ° - 0 3 o D y(CH, A 2A ) = 0.88? ±-0.04 5 D Vi (NH, X 3 E ~ ) = 1.38g ± 0. 04 3 D . The d i p o l e moments of the X 2n s t a t e s i n both OH and OD were a l s o e v a l u a t e d f o r v=0, 1, and 2, and were found t o decrease as a f u n c t i o n of v. i i i TABLE OF CONTENTS CHAPTER I : I n t r o d u c t i o n 1 CHAPTER I I : The Molecu-lar D i p o l e Moment 5 1. Formulation of the St a r k P e r t u r b a t i o n 5 2. C l a s s i c a l Formulation 6 3. Quantum Mechanical Formulation 8 4. P a r i t y and I n v e r s i o n 10 5. The T r a n s i t i o n Moment 13 CHAPTER I I I : The M a t r i x Elements of n 17 z 1. The 1l S t a t e 17 2. The lR S t a t e 2 0 3. The 2 E S t a t e 21 4. The 2n S t a t e 22 5. The 3E St a t e 24 6. The 3n S t a t e 25 CHAPTER IV: P e r t u r b a t i o n Theory 2 8 1. Nondegenerate P e r t u r b a t i o n Theory 2 8. 2. Degenerate L e v e l s 31 3. A-doublets 33 4. Stark E f f e c t s i n a A-doublet 36 5. The S e c u l a r Equation f o r E S t a t e M u l t i p l e t s . 4 8 6. 2 £ S t a t e s , I n t e r a c t i o n of L e v e l s w i t h N=l. . 50 7. 3E S t a t e s , I n t e r a c t i o n of N=l L e v e l s . . . . 53 CHAPTER V: Intermediate Coupling 55 1. I n t r o d u c t i o n 55 i v 2. The 2n S t a t e 56 2.1. The Harniltonian 56 2.2. J=£ 57 2.3. J>£ 57 2.4. Hund's Cases as L i m i t s 5 9 2.5. M a t r i x Elements of n , Example 62 3. The 3n S t a t e 63 3.1. Hamiltonian 63 3.2. J=0 64 3.3. J=l 65 3.4. J=2 67 3.5. M a t r i x Elements of n z , Example 68 3.6. J=3 69 3.7. Hund's Case (b) 69 4. The 2 A S t a t e 71 CHAPTER VI: Apparatus 74 1. E a r l y Work 74 2. Mechanism 76 3. The F l u s h Cathode and Older Tube Design. . . . 77 4. The Sunken Cathode 7 9 5. D e t a i l s of Tube Assembly 81 5.1. Ground Glass vs. Rubber, and Other F a i l -ures 81 5.2. The Cleanable Tube 83 5.3. The Window 83 5.4. The S h i e l d 84 5.5. The Anode 84 V 6. D e t a i l s of Other Apparatus 85 6.1. Sample Input C o n t r o l 85 6.2. Pumping Rate Reduction and Pressure Measurements 8 8 6.3. Cold Traps and Pump 89 6.4. Power Supply 90 6.5. The Spectrograph 93 6.6. P l a t e Handling i n the Spectrograph . . . 94 6.7. The Lens 95 6.8. Alignment 96 7. Cathode Manufacture 97 7.1. M a t e r i a l s • 97 7.2. P o l i s h i n g 100 7.3. Running-In 103 7.4. Cathode F a i l u r e and S p u t t e r i n g 107 8. Experiments I l l 8.1. Hydrogen I l l 8.2. Helium 114 8.3. Argon 115 8.4. N i t r o g e n , Oxygen, A i r , and N i t r i c Oxide.. 115 8.5. Aluminum, Trimethylaluminum 115 8.6. Water and Heavy Water 116 8.7. Methanol 117 8.8. Dimethyl Mercury 117 8.9. D i e t h y l Zinc 118 8.10. Phosphorous O x y c h l o r i d e 118 8.11. S i l a n e 119 v i 8.12. B e r y l l i u m w i t h Hydrogen, Oxygen, or Water 12 0 8.13. Boron T r i c h l o r i d e 121 8.14. Cyanogen 121 8.15. Ammonia 123 9. Other Means of E x c i t a t i o n 12 5 CHAPTER V I I : The Measurement of the P l a t e s and the C a l i b r a t i o n of E l e c t r i c F i e l d s 126 1. P l a t e Measurement 126 2. C a l i b r a t i o n o f E l e c t r i c F i e l d s 128 CHAPTER V I I I : The 2 J I S t a t e s of OH and OD 136 1. The L i n e s 136 2. The Methods 139 3. The R e s u l t s 149 CHAPTER IX: The A 2 I + S t a t e s of OH and OD . . . . 157 1. I n t r o d u c t i o n 157 2. Measurement of the L i n e and Slope F i t t i n g . . . 160 3. The Z e r o - F i e l d P o s i t i o n 162 4. Higher Order C o r r e c t i o n s 171 5. C o r r e c t i o n s f o r Intermediate Coupling 172 6. R e s u l t s 175 7. The P 1 2 ( 2 ) L i n e i n 0 0 1 8 2 CHAPTER X: The A 2A S t a t e o f CH 187 1. The P r o d u c t i o n of the Spectrum 187 2. Energy L e v e l S t r u c t u r e 188 3. The Assignment of M 190 4. The Diagonal M a t r i x Elements of n z 194 5. The S p l i t t i n g Formulae 196 v i i 6. R e s u l t s 198 CHAPTER XI: The X 3 E ~ S t a t e of NH 201 1. Observed S p e c t r a 201 2. Energy L e v e l S t r u c t u r e and Constants 203 3. Choice of S p e c t r a l L i n e 206 4. Stark E f f e c t s i n the 3n State 209 5. Stark E f f e c t s i n the 3 I ~ S t a t e 212 6. Experimental R e s u l t s 215 7. Other C o n s i d e r a t i o n s 220 8. The 3n S t a t e 224 CHAPTER X I I : Summary 226 APPENDIX A: U n i t s 228 APPENDIX B: Quantum Numbers, Hund's Cases, and S e l e c t i o n Rules 229 APPENDIX C: Glow Discharges 234 1. I n t r o d u c t i o n 234 2. General C h a r a c t e r i s t i c s 234 3. S i m i l a r i t y R e l a t i o n s 238 4. Normal and Abnormal Glow Discharges 239 5. The Aston Dark Space and the Cathode Glow. . . 242 6. The Cathode Dark Space 242 7. I o n i c C o l l i s i o n s and the Shape of the Cathode F a l l 247 8. More on the Cathode Glow 256 9. The R a d i a l Current D i s t r i b u t i o n 258 10. The Negative Glov; 261 11. The Faraday Dark Space 263 12. The P o s i t i v e Co.lumn_, P l a i n 2 63 v i i i 13. The P o s i t i v e Column, Deluxe 2 64 14. The Anode 266 APPENDIX D: Some Background i n M o l e c u l a r Quantum Mechanics. 267 The M o l e c u l a r Hamiltonian 269 The Diagonal M a t r i x Element o f the Hamiltonian . . 272 The O f f - D i a g o n a l M a t r i x Element of the Hamiltonian 274 APPENDIX E: C a l c u l a t i n g the M a t r i x Elements of n . . 277 1. Summary of Related Formulae 277 2. l l S t a t e s 281 3. 1J[ S t a t e s 282 4. 2 E S t a t e s 283 5. 2n S t a t e s 286 6. 3E S t a t e s 288 7. 3n S t a t e s 290 i x LIST OF TABLES I: Experimental C o n d i t i o n s 112 I I : Stark E f f e c t C o e f f i c i e n t s f o r the F i r s t Three Balmer L i n e s of Atomic Hydrogen 130 I I I : Stark E f f e c t C o e f f i c i e n t s f o r Deuterium . . . . 131 IV: Helium C a l i b r a t i o n Data 135 V: S p l i t t i n g s and F i e l d s Used f o r C a l c u l a t i n g the Di p o l e Moment of the 2n S t a t e i n OH f o r v=0 150 VI: Data f o r C a l c u l a t i n g the 2n D i p o l e Moment i n OH v=l 151 V I I : Data f o r C a l c u l a t i n g the 2n. D i p o l e Moment i n OH v=2 151 V I I I : Data f o r C a l c u l a t i n g the 2n D i p o l e Moment i n OD v=0 152 IX: Data f o r C a l c u l a t i n g the 2n Dipo l e Moment of OD v=l 152 X: Data f o r C a l c u l a t i n g the 2rt St a t e D i p o l e Moment i n OD v=2 153 XI: Experimental Values o f the D i p o l e Moment i n the 2n S t a t e 153 X I I : C o r r e c t i o n s to the Z e r o - F i e l d P o s i t i o n s of the P12(D 167 X I I I : Data f o r P^/^X1 Adjusted.Absolute C and Mean Measured Values of U and L, wit h e r r o r s 168 XIV: Data f o r Pp / v x 1 U a n < ^ L a s C o r r e c t e < ^ f ° r High Order Terms, x and y as Used by QUAFF 17 6 XV: D i p o l e Moments i n the 2 z State 181 XVI: Measurements and R e s u l t s f o r the 2A St a t e o f CH 199 X XVII: Some Term Values of NH . . . 204 XVIII: Measured Stark S h i f t s i n the Q 2, (0) and P 3 1 ( 0 ) L i n e s i n NH A + X (0,07 217 XIX: Comparison w i t h P u b l i s h e d C a l c u l a t i o n s o f the D i p o l e Moment of NH (X 3E ) 221 XX: D i p o l e Moments of First-Row Diatomic Hydrides . 226 E . I : The Squares of the M a t r i x Elements of a 2 E State 284 E . I I : Formulae of Table E.I w r i t t e n i n terms of J . . 285 E . I I I : The Squares of the M a t r i x Elements of n i n a 3Z St a t e Z. . . . 289 x i LIST OF FIGURES 1. Energy L e v e l s of a 1 E + S t a t e 18 2. Energy L e v e l s of a 2 £ + S t a t e 18 3. Energy L e v e l s of the 2 I I S t a t e of OH 23 4. Energy L e v e l s of a 3 E State 23 5. P e r t u r b a t i o n s i n I n t e r a c t i n g Doublets 4 0 6. Stark E f f e c t s i n E S t a t e M u l t i p l e t s with N=l . . 40 7. 2 n Energy L e v e l s as a F u n c t i o n of Y 61 8. 3 n Energy L e v e l s as a F u n c t i o n of Y 61 9. The Discharge Tube 6 8 10. P h y s i c a l Schematic of Vacuum System 86 11. Power Supply 91 12. E l i m i n a t i o n o f Cathodic I r r e g u l a r i t i e s o f P o s i t i v e Curvature 105 13. Stark E f f e c t s I n v o l v i n g the J = l Energy L e v e l of OH 137 14. Schematic D e t a i l of the T r a n s i t i o n s of F i g . 13. . . 140 15. The A -* X Bands of OH a t low f i e l d 141 16. The A -> X Bands of OD a t low f i e l d 142 17. S' v s . E f o r OH, v=0 146 18. S' v s . E f o r OH, v=l 14 6 19. S' v s . E f o r OH, v=2 147 20. S 1 vs. E f o r OD, v=0 147 21. S 1 vs. E f o r OD, v=l 148 22. S \ v s . E f o r OD, v=2 148 23. D i p o l e Moments of OH and OD as a F u n c t i o n of v 155 24. Stark E f f e c t s i n a T r a n s i t i o n Between S u b l e v e l s of a 2 E -> 2 n band. x i i 25. The (0,0) A -»• X band of OH at hi g h f i e l d . . . . 163 26. The (0,0) A •* X band of OD at high f i e l d . . . . 164 27. The (1,1) A X band of OH at hi g h f i e l d . . . . 165 28. Determination o f u A / y x f o r OH, v=0 179 29. Determination of P^/v x f o r 0 0 / v = 0 1 8 0 30. Determination of V A / v x f o r OH, v=l 180 31. The (1,1) A -»• X band of OD at h i g h f i e l d . . . . 183 32. Stark E f f e c t s i n the P 1 2 ( 2 ) L i n e of OD 185 33. I n t e n s i t i e s i n the P12(2) L i n e of OD 185 34. The (0,0) A •+ X band of CH 189 35. A 2A •* X 2 n T r a n s i t i o n s i n CH 191 36. P r e d i c t e d Components of the 0^(2) and 0^(3) L i n e s 193 37. Appearance of the Stark E f f e c t i n a A -> II L i n e . . 197 38. D i p o l e Moment of the A 2A S t a t e 19 7 39. Energy L e v e l s of the A 3 n i -»• X 3I~ Band of NH. . 205 40. The (0,0) A -> X band of NH 216 41. Determination of p^ from the $ 2 1 ^ L i - n e • • • 2 1 9 42. Determination of the R P 3 1 ( 0 ) L i n e from p^. . . . 223 C l . C u r r e n t - V o l t a g e C h a r a c t e r i s t i c of a Glow Discharge. 235 C2. F i e l d and I n t e n s i t y v s. P o s i t i o n 235 x i i i ACKNOWLEDGEMENTS T h i s i s t o express my g r a t i t u d e to the two people whose p a t i e n c e and f a i t h made t h i s work p o s s i b l e : P r o f e s s o r F.W. Dalby, my r e s e a r c h d i r e c t o r , and Doha J . S c a r l , my w i f e . I am i n the debt of P r o f e s s o r Dalby f o r h i s many h e l p f u l d i s c u s s i o n s and h i s p h y s i c a l i n s i g h t , and i n the debt of Ms. S c a r l f o r the e f f o r t and perseverance which were necessary to t u r n the work of p h y s i c s i n t o the p h y s i c a l work of t h i s document. T h i s work was supported by the N a t i o n a l Research C o u n c i l of Canada. x i v t I He 4922 A (b) H. 4861 A «- X (c) Fr o n t e s p i e c e (a) The discharge tube, showing s t r i a t i o n s w i t h dimethyl mer-cury over a 2 mm f l u s h cathode. The p i c t u r e i s turned so th a t down i s to the l e f t . (b) Stark e f f e c t s i n H (4 861 A) and He (4922 A) , photo-graphed i n second o r d l r . Other l i n e s v i s i b l e are from OH. (c) S tark e f f e c t s i n molecular hydrogen ( p l a t e 257), photo-graphed i n f i r s t o r d e r , and showing approximately 5915 A to 59 85 A. The p l a t e s were made wit h hydrogen and cyanogen at pressures t o t a l l i n g ~3.5 mm Hg, wit h about 11 kV a p p l i e d v o l t a g e drawing 2 ma. The maximum f i e l d i s about 165,000 kV/cm. 1 CHAPTER I INTRODUCTION Since the d i a t o m i c molecule r e p r e s e n t s the s i m p l e s t p o s s i b l e chemical compound, i t i s important to be a b l e to compare as wide a range of t h e o r e t i c a l and experimental pro-p e r t i e s as p o s s i b l e . T h i s w i l l allow us to e v a l u a t e the accuracy of the p a r t i c u l a r c a l c u l a t i o n a l model used, and to have the f u l l e s t p o s s i b l e d e s c r i p t i o n of the molecules them-s e l v e s . C o n v e n t i o n a l spectroscopy g i v e s a g r e a t d e a l of i n f o r m a t i o n about the p r o p e r t i e s and s t r u c t u r e of the molecu-l a r energy l e v e l s , and r e c e n t work wit h h i g h - r e s o l u t i o n resonance methods has y i e l d e d even more d e t a i l e d i n f o r m a t i o n i n c l u d i n g d e t a i l s of the c o u p l i n g of the v a r i o u s angular momenta wit h the magnetic moment of the n u c l e i . Much of t h i s data, i n c l u d i n g the g - f a c t o r s given Zeeman e f f e c t s t u d i e s , can be accounted f o r i n terms of the c o u p l i n g schemes by which the d i f f e r e n t angular momenta i n t e r a c t (once the broad s t r u c t u r e of the a v a i l a b l e e l e c t r o n i c s t a t e s has been determined), but the dependence upon the d e t a i l e d s t r u c -t u r e o f the m o l e c u l a r wavefunctions i s not c r i t i c a l . Para-meters which comprise r e l a t i v e l y c r i t i c a l t e s t s of the wave-f u n c t i o n s , and are . r e a d i l y c a l c u l a t e d from them, are the m o l e c u l a r e l e c t r i c d i p o l e and t r a n s i t i o n moments. The s i m p l e s t d i a t o m i c molecules f o r c a l c u l a t i o n a l purposes are the monohydrides» However, most of these are u n s t a b l e , so t h a t c o n v e n t i o n a l methods (such as measurement 2 of d i e l e c t r i c constants) used to measure d i p o l e moments of bulk or d i s s o l v e d substances cannot be used. We have made use i n t h i s work of a method (the LoSurdo discharge) which allows the g e n e r a t i o n of h i g h e l e c t r i c f i e l d s i n the same g e n e r a l r e g i o n of the d i s c h a r g e i n which the molecules are formed. Now, there are two d i s t i n c t k i n d s of i n s t a b i l i t y s u f f e r e d by a d i a t o m i c molecule. I t may be i n t r i n s i c a l l y u n s t a b l e so t h a t the i s o l a t e d molecule spontaneously d i s s o -c i a t e s i n t o separate atoms ( i . e . , the ground e l e c t r o n i c s t a t e possesses no s t a b l e minimum of p o t e n t i a l energy w i t h r e s p e c t to the d i s t a n c e between atoms), the most notable example being the He 2 molecule. However, i n the overwhelm-i n g m a j o r i t y o f known d i a t o m i c molecules, i n s t a b i l i t y i s not i n t r i n s i c , but r a t h e r due to very h i g h chemical r e a c t i -v i t y . These molecules w i l l r e a c t not o n l y w i t h most other types of molecule, but w i t h each o t h e r . Thus the molecules with which we are concerned have l i f e t i m e s which depend upon t h e i r c o n c e n t r a t i o n s and environment, and may be c o n s i d e r e d s t a b l e i f kept s u f f i c i e n t l y i s o l a t e d . Such c o n d i t i o n s o f i s o l a t i o n cannot be e a s i l y r e a l i z e d i n the l a b o r a t o r y , but may be found i n the reaches of i n t e r s t e l l a r and i n t e r g a l a c -t i c space. Since hydrogen i s by f a r the most abundant mate-r i a l i n these r e g i o n s , i t i s c l e a r t h a t the h y d r i d e s of f i r s t - r o w elements must be important t h e r e . Indeed, CH was one of the f i r s t i n t e r s t e l l a r s p e c i e s d i s c o v e r e d , and and the importance of OH i n understanding i n t e r s t e l l a r c o n d i t i o n s i s now w e l l known. Furthermore, a l l molecules 3 s t u d i e d i n t h i s work have been d e t e c t e d i n a b s o r p t i o n i n the s p e c t r a of the sun and oth e r s t a r s . Here agai n , a knowledge of the d i p o l e moment i s of p a r t i c u l a r i n t e r e s t s i n c e the a b s o r p t i o n i n t e n s i t i e s of f a r i n f r a r e d and microwave t r a n s i -t i o n s ( i . e . , w i t h i n the ground e l e c t r o n i c ' a n d v i b r a t i o n a l s t a t e ) ' are p r o p o r t i o n a l t o i t s square. The OH and OD molecules have been widely s t u d i e d i n t h e i r ground s t a t e s , and we have used our hig h f i e l d p l a t e s together w i t h a number of low f i e l d exposures to eva l u a t e t h e i r d i p o l e moments more a c c u r a t e l y than had p r e v i o u s l y been done wi t h t h i s method, and t o study v a r i a t i o n s w i t h v i b r a t i o n a l quantum number. By examining second order e f f e c t s i n E s t a t e s where no f i r s t o r d e r e f f e c t s are a v a i l a b l e , we have made the f i r s t r e p o r t e d determinations of the d i p o l e moment i n the ground s t a t e of NH, and i n the f i r s t e x c i t e d s t a t e s of OH and OD. We were a l s o able t o s u b s t a n t i a l l y improve upon a p r e v i o u s l y r e p o r t e d value f o r the f i r s t e x c i t e d s t a t e o f CH. Chapter I I i s intended to i n t r o d u c e the concept and r o l e of "a d i p o l e moment", and to d e t a i l assumptions l a t e r used i n forming matrix elements. Chapters I I I through V pr o v i d e the d e t a i l e d quantum mechanical groundwork upon which our c a l c u l a t i o n s are based. Chapter VI d i s c u s s e s the l a b o r a t o r y procedures and apparatus i n d e t a i l , supplemented by the d e s c r i p t i o n of p l a t e and f i e l d measurements i n Chapter V I I . The r e s u l t s f o r i n d i v i d u a l molecules are d i s c u s s e d i n Chapters V I I I - XI, and are summarized i n Chapter-XII along w i t h those f o r the other f i r s t - r o w d i a t o m i c monohydride molecules. 4 The u n i t s which w i l l normally be used throughout t h i s work are d e s c r i b e d i n Appendix A, while Appendix B p r o v i d e s a g l o s s a r y o f c e r t a i n standard symbols which may be used without d e f i n i t i o n i n the t e x t . I t i s p a r t i c u l a r l y important t o note the unusual s i g n conventions f o r M, A, and fi. Background i n t r o d u c i n g the reader to r e l e v a n t aspects of the p h y s i c s of glow d i s c h a r g e s i s given i n Appendix C. A summary of some r e l a t e d theory and formulae are pr o v i d e d i n Appendices D and E. 5 CHAPTER I I THE MOLECULAR DIPOLE MOMENT 1. Formulation of. the Stark P e r t u r b a t i o n f o r a Diatomic Molecule The p o t e n t i a l energy gained by a p a r t i c l e l a b e l e d i , of charge +e^, w h i l e moving along an a r c segment ds i n an e l e c t r i c f i e l d E ( x , y , z ) , i s g i v e n by dV i = -e i E • ds (l) (the n e g a t i v e o f the k i n e t i c energy gained i f the p a r t i c l e i s f r e e ) . I f the f i e l d i s uniform i n both s t r e n g t h and d i r e c t i o n and chosen to p o i n t i n the p o s i t i v e z d i r e c t i o n o f the l a b o r a t o r y c o o r d i n a t e s (x,y,z -K then, i n t e g r a t i n g from the o r i g i n , the p o t e n t i a l energy becomes V i = - e i r i E cosGj, (2) wit h r e s p e c t to i t s energy a t the o r i g i n . Here r ^ i s the magnitude of the v e c t o r from the o r i g i n to the p a r t i c l e , and 0. i s the angle between r . and the z a x i s . In l l a p p l y i n g t h i s t o a d i a t o m i c molecule with an a x i s of symmetry ( i n t e r n u c l e a r a x i s ) a t angle 0 from the z a x i s , and whose ce n t e r of n u c l e a r charge has been p l a c e d a t the o r i g i n , we a s s i g n 0^ = 0 f o r a l l p a r t i c l e s . While t h i s i s t r u e 6 by d e f i n i t i o n i n the case of the n u c l e i , for the electrons i t represents the assumption of a x i a l symmetry about the inter; nuclear axis. In the absence of molecular rotation or external f i e l d , the time average of any must be Q. In other words, under the assumption that e l e c t r o n i c motions are s u f f i c i e n t l y faster than the rotation or v i b r a t i o n of the n u c l e i , the electrons can be considered to be located at t h e i r time-averaged positions on the internuclear axis, over the duration of a "snapshot" of the molecule with the nuclei fixed. The molecular pote n t i a l energy can then be formed by summing Eq. (2" 1) over a l l p a r t i c l e s : V = -(£ e i ? i ) E cosO (3) i Here £ i s the coordinate along the internuclear axis i n a cartesian coordinate system (5, n, 5) fixed i n the molecule (rotating coordinates), so that i s the C coordinate of p a r t i c l e i . 2. C l a s s i c a l Formulation Eq. ( 3 ) can be used as a s t a r t i n g point for both c l a s s i c a l and quantum mechanical consideration*. F i r s t , we note that the sum i n Eq. ( 3 ) may be taken over the electrons only. (The o r i g i n has been taken to coincide with the center of nuclear charge, so that the nuclear contribution averages to zero.) This sum i s therefore equal *These arguments do not apply to charged molecules. 7 ->• to the negative of the dipole moment u, since i t represents the net displacement of negative charge from the positive, while the dipole moment is defined as a vector from negative to positive charges (however, note that the e^ are a l l negative). Since 0 is the angle between the z axis (E) and the £ axis (p), Eq. ( 3 ) can be rewritten as v = • 2 ( 4 K which is the usual classical expression. t€ a-, molecule possesses a permanent dipole moment different from zero, which is a constant property of the molecule, then Eq. ( 4 ) says that the Stark effect w i l l contribute an energy that is linear in the electric f i e l d . Whenever a Stark effect is found to be proportional to the square of the f i e l d , however, i t is implied that the dipole moment is not constant, but is proportional to the fi e l d u = aE , (5) a and ua is interpreted as being an induced dipole moment, due to the distortion of molecular symmetry by the external f i e l d . The constant a i s called the polarizability, and usually needs to be written as a tensor, since y and E are not, in general, of the same direction. 8 3. . Quantum Mechanical Formulation In the n o t a t i o n of Landau and L i f s h i t z 1 ( p . 2 9 4 ) , n i s the u n i t v e c t o r along the £ a x i s , so t h a t n = cosQ z and V = "()>.£.) n z E . (6) i T h i s i s a form s u i t a b l e f o r the s e p a r a t i o n of the c o o r d i n a t e s of n u c l e a r r o t a t i o n from those o f the n u c l e a r v i b r a t i o n and e l e c t r o n i c motions. n (cose) i s a f u n c t i o n o n l y of angular ( r o t a t i o n a l ) v a r i a b l e s , w h i l e the £^ do not c o n t a i n these v a r i a b l e s . I f we assume t h a t the m a t r i x elements of Eq.(6) can be taken w i t h r e s p e c t to a separated wavefunction * = ^ r *ev , (7) then we may w r i t e < * 1V| * > = < " ' e v l ^ i ^ i ^ e ' v ' > K ^ r ^ z ^ r ' > E ' ( 8 ) T h i s i s the quantum analogue o f Eq. (4). The f i r s t b r a c k e t of Eq. (8) i s taken to be the quantum mechanical e x p r e s s i o n f o r the d i p o l e moment p , and the second b r a c k e t d e s c r i b e s the formation o f the i n n e r product i n Eq. (4), which e s s e n t i a l l y 9 rep r e s e n t s the t r a n s f o r m a t i o n of y from m o l e c u l e - f i x e d to s p a c e - f i x e d c o o r d i n a t e s . Some important p r o p e r t i e s o f the d i p o l e moment can be understood by examining i t s form i n Eq. (8). I f i s the same as i|) , , ( i . e . , we c o n s i d e r the d i a g o n a l m a t r i x element of V i n the s t a t e ^ 1 , then i t makes sense to regard < \p | £e. r,. \ > as the d i p o l e moment of the molecule i i n the s t a t e ii y e v y = < ii I 7e . c. U > . (9) M e v ev 1 4- 1 ^ 1 1 r e v The value o f y g v should be independent of r o t a t i o n a l quantum numbers. The matr i x eliements o f the Stark e f f e c t can thus be expressed i n cm ^ as F = e < i | ) r | n 2 | * r f > where (10) e = yE/hc I f the molecule i s homonuclear ( i . e . , both n u c l e i the same, as i n H 2 or N ^ ) then i t can be changed i n no p h y s i c a l p r o p e r t y ( i n c l u d i n g y) by an i n v e r s i o n of a l l c o o r d i n a t e s through the o r i g i n , r e p l a c i n g the m o l e c u l e - f i x e d c o o r d i n a t e s % 5.. .,-_||!.' -/• 4;. )••; of'"the i p a r t i c l e by 10 (-5^ , -n^, . T h i s means t h a t i ^ e v can be changed by at most a phase f a c t o r e l q (with q r e a l ) The d i p o l e moment i s then g i v e n by* u = < i|» I Ye . t,. U > e v 1 L I I 1 r e v < * e v e i q I K ( - 5 i ) l * e v e i q > < * He. ( - C O |¥ e i q e _ i q > ( L 1 )  r e v 1 L x l 1 e v < t(» I Te . ( - £ . ) U > e v 1 L l I 1 y e v - < ITe.£ . U > e v 1 L l l ' e v which can be s a t i s f i e d o n l y i f y=0. When the molecule i s not homonuclear, a c o o r d i n a t e i n v e r s i o n cannot be expected t o leave the d i p o l e moment unchanged ( i t w i l l c l e a r l y change s i g n ) , and ]x may i n g e n e r a l be non-zero. 4. P a r i t y and I n v e r s i o n The o p e r a t i o n of i n v e r t i n g a l l e l e c t r o n i c and n u c l e a r c o o r d i n a t e s through the o r i g i n w i l l , i f a p p l i e d to the Stark p o t e n t i a l , r e s u l t i n a s e l e c t i o n r u l e which s t a t e s t h a t o n l y s t a t e s p o s s e s s i n g o p p o s i t e p a r i t y may i n t e r a c t . T h i s i s shown by Hougens 2(p.23f) from a r i g o r o u s c o n s i d e r a t i o n of the p r o p e r t i e s o f the a s s o c i a t e d symmetry o p e r a t i o n s , and by Landau and L i f s h i t z 1 (p.94f) by an a l g e b r a i c argument. What f o l l o w s i s a more .geometrical argument. The E u l e r angle 9 i n Eq. (8) ( n z = cosQ) i s d e f i n e d as the angle from .the p o s i t i v e z a x i s to the p o s i t i v e  *Eq. (11) c o n t a i n s t h r e e d i f f e r e n t meanings f o r the symbol "e", 11 x, a x i s . I n v e r s i o n does not change the d i r e c t i o n o f the space-f i x e d z a x i s , by our d e f i n i t i o n , but f o r each p a r t i c l e i i t r e p l a c e s z^ by ~zj_- I t i s n o t obvious, however, whether or not the 5 a x i s changes d i r e c t i o n . We s h a l l see t h a t i t does not matter, as f a r as the e f f e c t upon Eq. (8) i s concerned. I f the p o s i t i v e j; a x i s changes d i r e c t i o n under i n v e r s i o n , then the c, c o o r d i n a t e s of a l l p a r t i c l e s are unchanged but 0 becomes i r-0. Since n -*• cos(n-0) = -cos© changes s i g n but the 5 ^ are i n v a r i a n t , V changes s i g n as an o p e r a t o r . I f , on the o t h e r hand, the p o s i t i v e ?-axis were to not change s i g n upon i n v e r s i o n ( s i m i l a r to the z a x i s ) , then G remains i n v a r i a n t w h i l e a l l the 5 ^ change t h e i r s i g n s . Once again the s i g n o f V changes, but t h i s time from the I e i * * i f a c t o r ' The q u e s t i o n o f which f a c t o r changes s i g n i s s i m i l a r to the q u e s t i o n p e r t a i n i n g to the i n v e r s e o f a u n i t v e c t o r p o i n t i n g n o r t h : Is i t a v e c t o r of magnitude -1 p o i n t i n g n o r t h , or a u n i t v e c t o r p o i n t i n g south? However, the r e i s a c r i t e r i o n f o r d e c i s i o n i n the case of V, as w i l l be seen below. The geometric e f f e c t of an i n v e r s i o n of c o o r d i n a t e s may be d e s c r i b e d e q u i v a l e n t l y as an o p e r a t i o n i n which p a r t i c l e s are u n a f f e c t e d , but the senses of the s p a c e - f i x e d axes are r e v e r s e d . The preceding arguments may be r e a d i l y adapted to t h i s convention. Since the Stark p e r t u r b a t i o n V becomes -V under i n v e r s i o n , and •» ±}\> (according to the p a r i t y of •, i | > ) , then which should not be confused: the e l e c t r o n i c charge e., the e x p o n e n t i a l (2. 718'. ..) i n e l c*, and the s u b s c r i p t " denoting e l e c t r o n i c wavefunction. 12 the m a t r i x elements < • i|» | V | ' • > change s i g n u n l e s s ij» and I{J 1 are wave-functions of o p p o s i t e p a r i t y . Since matrix elements are p h y s i c a l l y measurable q u a n t i t i e s (Stark energies and i n t e r a c t i o n s ) , they must be u n a f f e c t e d by i n v e r s i o n . Thus we have shown t h a t when p a r i t y i s a good quantum number ( i . e . , when the Hamiltonian i s i n v a r i a n t under i n v e r s i o n ) then the m a t r i x elements of the Stark p e r t u r b a t i o n between unperturbed wavefunctions are zero except between wavefunctions of o p p o s i t e p a r i t y . The f a c t t h a t the d i a g o n a l m a t r i x elements of V are zero ( f o r any wavefunction of d e f i n i t e p a r i t y ) o f t e n leads to c o n f u s i o n by seeming to imply t h a t the d i p o l e moment of any pure r o t a t i o n a l l e v e l (a E - s t a t e l e v e l or a A-doublet s u b l e v e l ) i s zero. To i l l u m i n a t e t h i s problem, we may assume t h a t the s i g n change i n V under i n v e r s i o n i s due to the -second f a c t o r i n Eq. (8) <• < | n | ^  >, r a t h e r than the f i r s t f a c t o r 3T Z 3T* ( d i p o l e moment). By our p r e v i o u s d i s c u s s i o n i n t h i s s e c t i o n , we are then f o r c e d to have the m o l e c u l e - f i x e d (c) a x i s change s i g n , although the s p a c e - f i x e d (z) a x i s does not, under i n v e r s i o n . That the s i g n change be a s s o c i a t e d with the E u l e r angles ( r o t a t i o n a l v a r i a b l e s ) i s reasonable, s i n c e we can i n t e r p r e t our r e s u l t as saying t h a t the d i p o l e moment, as a v e c t o r f i x e d i n the molecule, has no component along any s p a c e - f i x e d a x i s as long as the wave f u n c t i o n s possess i n v e r s i o n symmetry. I n t u i t i v e l y , t h i s means t h a t i n any such s t a t e , the molecule spends as much time (or more p r e c i s e l y , has as much 13 p r o b a b i l i t y of p o i n t i n g ) i n any one d i r e c t i o n as i t s o p p o s i t e , thus l e a v i n g the time-averaged e n e r g i e s unchanged. I t i s not s u r p r i s i n g , t h e r e f o r e , t h a t to have a Stark e f f e c t i n any s i n g l e s t a t e , t h i s i n v e r s i o n symmetry must be dest r o y e d . T h i s can onl y be done by mixing wavefunctions o f o p p o s i t e p a r i t y . The formula i n Chapter IV f o r the f i r s t order c o r r e c t i o n to the wavefunction accomplishes e x a c t l y t h i s . From another p o i n t of view, i t i s r e a d i l y seen t h a t the Stark e f f e c t must d e s t r o y the p a r i t y o f the wavefunctions. The unperturbed s t a t e s have d e f i n i t e p a r i t y because the unperturbed Hamiltonian H q i s i n v a r i a n t under i n v e r s i o n (even though the molecule may not possess i n v e r s i o n symmetry, i . e . , be homenuclear). T h i s i s due to the f a c t t h a t space i s i s o t r o p i c , so t h a t t r a n s f o r m i n g a l l c a r t e s i a n p a r t i c l e c o o r d i n a t e s i n t o t h e i r n e g a t i v e s a f f e c t s no i n t e r - p a r t i c l e s e p a r a t i o n s , and thus no p o t e n t i a l e n e r g i e s . Adding an e l e c t r i c f i e l d g i v e s space a p r e f e r r e d d i r e c t i o n : An i n v e r s i o n as d e s c r i b e d above does not change the d i r e c t i o n o f 2 (z axis) but does change the s i g n o f V. I n v e r s i o n thus transforms H Q + V i n t o H - V, and the p a r i t y o p e r a t o r no longer commutes w i t h the f u l l H a miltonian. 5. The T r a n s i t i o n Moment The o f f - d i a g o n a l matrix elements o f the d i p o l e moment are o f t e n c a l l e d t r a n s i t i o n moments and denoted by 14 {Re,e'h " * *ev' K ? i l*e'V * ' ( 1 2 ) The e l e c t r i c d i p o l e t r a n s i t i o n p r o b a b i l i t y ( i n t e n s i t y ) o f a t r a n s i t i o n o c c u r r i n g between the s t a t e s d e s c r i b e d by i j > e v and i\i , , i s p r o p o r t i o n a l t o |R ,| . (See Benedict 6 6 / 6 and P l y l e r 1 * , p.61f, f o r a d i s c u s s i o n o f v a r i o u s modes of ex p r e s s i n g t r a n s i t i o n p r o b a b i l i t i e s . ) R , need not be 6 / 6 zero f o r homonuclear molecules, u n l e s s fy^ and , are of the same e l e c t r o n i c i n v e r s i o n symmetry (g or u ) . Note t h a t Eq. (12) g i v e s only the ? component of R ,, but t h a t 6 / 6 5 and n components are a l s o p r e s e n t , i n g e n e r a l . A permanent d i p o l e moment p l a y s the r o l e of t r a n s i t i o n moment f o r t r a n s i t i o n s from one l e v e l t o another w i t h i n a g i v e n e l e c t r o n i c s t a t e , so t h a t pure r o t a t i o n bands are not found i n homonuclear molec u l e s . One might suppose from the f o r e g o i n g t h a t v i b r a t i o n - r o t a t i o n bands may be found from a non-zero value o f R £ E ? = < | fe^z.^ | * e v i > (or the corres p o n d i n g R E o r R n). The f a c t t h a t such bands are not found e ,e e ,e (Her z b e r g 3 , p.8 0) i n d i c a t e s t h a t a v i b r a t i o n a l wavefunction i s separable from an e l e c t r o n i c p a r t : A c o o r d i n a t e i n v e r s i o n cannot a f f e c t the v i b r a t i o n a l p a r t ( i n t e r n u c l e a r d i s t a n c e s are unchanged), so by an argument s i m i l a r to t h a t surrounding E q . ( l l ) , the t r a n s i t i o n moment i s zero. "Exceptions" t o these r u l e s ( H e r z b e r g 3 , p.279) are found i n a v i b r a t i o n - r o t a t i o n band o f H 2 which i s a t t r i b u t e d to e l e c t r i c quadrupole i n t e r a c t i o n s , and some microwave l i n e s 15 i n 0 2 and more r e c e n t l y i n NO (see Brown, Cole, and H o n e y 4 3 and r e f e r e n c e s i n Blum, N i l l , and Strauss 1** 3) a t t r i b u t e d t o a magnetic d i p o l e mechanism. The quadrupole and magnetic mechanisms are much weaker than the e l e c t r i c d i p o l e , when the l a t t e r i s pr e s e n t . Moreover, v i b r a t i o n - r o t a t i o n bands have been induced i n hydrogen by s t r o n g e l e c t r i c f i e l d s (Crav/ford and Dagg 5), by what Her z b e r g 3 (p. 280) c a l l s "enforced d i p o l e r a d i a t i o n " . These are due to a f i e l d - i n d u c e d d i p o l e moment caused by the p o l a r i z a b i l i t y o f the molecule, as d e f i n e d i n Eq. (5). S i m i l a r bands, probably due to the same mechanism, but induced through c o l l i s i o n s r a t h e r than e x t e r n a l l y a p p l i e d f i e l d s , have been seen i n 0 2 and N 2 (Crawford, Welsh, and L o c k e 6 ) . A second order Stark e f f e c t i s per m i t t e d i n a l l d i a t o m i c molecules due to the t r a n s i t i o n moments to neighbouring e l e c t r o n i c s t a t e s . I|I , v , p o s s e s s i n g the same valu e s of S and A as The K and n components of R ,, permit t r a n s i t i o n s to s t a t e s w i t h A d i f f e r i n g by one from $ e v r ( H e r z b e r g 3 , p.241f). Now, f o r most molecules, such e l e c t r o n i c s t a t e s are too wid e l y separated to produce much e f f e c t . In _ OH, f o r example, the c l o s e s t such s t a t e to the A 2 E + i s the B 2 E * s t a t e , which l i e s some 36,000 cm ^ h i g h e r . A s s i g n i n g a r a t h e r h i g h value of 1 debye t o R „ (the value i s not known) , a maximum va l u e of % to < i|» | n | \> , > , and an extremely h i g h f i e l d of 1000 kV/cm to E, then the Stark e f f e c t (see Chapter IV) i s g i v e n by the square of Eq. (8) d i v i d e d by 36 ,000, or about x 1-0-00/59.5) 2/36,000 = 9 x l O - 4 cm , which i s about two orders of magnitude below our l e v e l o f r e s o l u t i o n . In a few molecules there e x i s t c l o s e l y spaced e x c i t e d s t a t e s among which s i z a b l e second order Stark e f f e c t s are p o s s i b l e . The hydrogen molecule, i n f a c t , shows the l a r g e s t of a l l molecular Stark e f f e c t s and these were observed (see r e f e r e n c e s g i v e n by McDonald 7) some 50 years before e f f e c t s a r i s i n g from permanent d i p o l e moments (Phelps and D a l b y 8 ) . 17 CHAPTER I I I THE MATRIX ELEMENTS OF n — —z Ac c o r d i n g t o Eq.(10) of the pre c e d i n g chapter, i t i s necessary t o eva l u a t e m a t r i x elements o f the form < * r l n z | * r , > i n order to c a l c u l a t e Stark e f f e c t s due to a permanent d i p o l e moment i n a d i a t o m i c molecule. T h i s m a t r i x element can be f a c t o r e d i n t o two or t h r e e segments, depending on the angular momenta presen t and on the c o u p l i n g schemes which predominate. These f a c t o r s are w e l l known, and have here been taken from Landau and L i f s h i t z 1 (pp. 93, 104, and 295). The formulae and t h e ' d e t a i l s of t h e i r a p p l i c a t i o n are to be found i n Appendix E. In t h i s c hapter, we s h a l l b r i e f l y summarize the r e s u l t s i n mat r i x form f o r r e f e r e n c e purposes and as an a i d t o l a t e r c h a p t e r s . 1. The 1E St a t e The simple r o t a t i o n a l s t r u c t u r e o f a l l + s t a t e i s shown to s c a l e i n F i g . 1. The r o t a t i o n a l l e v e l s are l a -b e l l e d by J , and have e n e r g i e s g i v e n by BJ(J+1) cm ^. The p a r i t y s i g n s shown must be r e v e r s e d f o r a lZ s t a t e . 18 12B. f o J B > 2 2B !B F i g . 1 Energy Levels of a, £ State B T Cl D l .+ O P J + 4 -J/2 i 3 3 + 5 / 2 3/2 + 2 i i i i :i. i b • I • I Fig. 2 I • Energy Levels pf a 2 £ + State, with! 3*= B/2 19 I t f o l l o w s t r i v i a l l y from the d i s c u s s i o n of 1E s t a t e s i n Appendix E t h a t the Stark e f f e c t displacements i n a given energy l e v e l , d u e to the two i n t e r a c t i o n s markeded B and D i n F i g . l , are F D = -e / J 2 - ^ ~ M 2 / J 4 J 2 _ 1 } ( 1 3 ) and F D = -e /(J+l) 2 _ r ~ M 2 / ( 4 ( J + l ) 2 _ i ) , (14) where e = uE/hc. We show i n Eq.(15) the squares of the matrix e l e -ments g i v e n i n Eqs. (13) and (14), f o r the f i r s t few values of J i n the ll s t a t e : # - 1 2 3 4 5 6 7 8 9 10 + J 0 1 1 2 2 2 3 3 3 3 4- M -»-+ 0 0 1 0 1 2 0 1 2 3 1 0 0 0 v 3 0 0 0 0 0 0 0 0 2 1 0 v 3 0 0 V i s 0 0 0 0 0 0 3 , 1 1 0 0 0 0 V s 0 0 0 0 0 4 2 0 0 V i s 0 0 0 0 V 3 5 0 0 0 5 2 1 0 0 v 5 0 0 0 0 V 3 5 0 0 6 2 2 0 0 0 0 0 0 0 0 V 7 0 7 3 0 0 0 0 V 3 5 0 0 0 0 0 0 8 3 1 0 0 0 0 V 3 5 0 0 0 0 0 9 3 2 0 0 0 0 0 l / 7 0 0 0 0 10 3 3 0 0 0 0 0 0 0 0 0 0 (15) The x n S t a t e From t h i s p o i n t on, we s h a l l do l i t t l e more than summarize the formulae of Appendix E i n mat r i x form. The squares o f the mat r i x elements o f ,n are g i v e n i n Eq.(16). # + 1 2 3 4 5 6 7 8 9 4- J + 1 1 2 2 2 2 2 2 2 + M -> 0 1 0 1 2 0 1 2 3 4-1 1 0 0 0 X A 0 0 0 0 0 0 2 1 1 0 V - 0 3/ 20 0 0 0 0 0 3 2 0 V , 0 0 0 0 V 35 0 0 0 4 2 1 0 V 2 0 0 V 3 6 0 0 3150 0 5 2 2 0 0 0 0 v 9 0 0 8 / 6 3 0 6 3 0 0 0 V 3 5 0 0 0 0 0 0 7 3 1 0 0 0 6 V 3 1 5 0 0 V 144° 0 8 3 2 0 0 0 0 8 / 6 3 0 0 V 3 6 0 9 3 3 0 0 0 0 0 0 0 0 V i e (16) I t i s t o be understood t h a t each row and column i s doubly degenerate w i t h zero m a t r i x elements between p a r i t i e s of l i k e s i g n . By i l l u s t r a t i o n , we reproduce the top corner of Eq.(16) i n f u l l d e t a i l (with p i n d i c a t i n g p a r i t y ) : i n Eq. (17). H e r e a f t e r , m a t r i c e s w i l l be g i v e n i n the condensed form of Eq. (16) f o r a l l A-doubled e l e c t r o n i c s t a t e s . The order of the p a r i t y s i g n s i n Eq. (17) i s not the same as they might appear,should degeneracy be removed by e f f e c t s not c o n s i d e r e d . 2 1 J 1 1 1 1 2 2 2 2 2 2 4-M -> 0 0 1 1 0 0 1 1 2 2 4-+ - + - + - + - + -4-1 0 + 0 0 0 0 0 0 0 0 0 1 0 - 0 0 0 0 0 0 0 0 0 1 1 + 0 0 0 V , 0 0 0 V20 0 0 1 1 - 0 0 V - 0 0 0 V 2 0 0 0 0 (17) 3. The 2 E S t a t e The f i r s t . s e v e r a l - " (squared) matrix elements of n are shown i n Eq. (18): # ->• 1 2 3 4 5 6 7 8 9 4- N -»- 0 1 1 1 2 2 2 2 2 4- J ->- i i f f f t f f f 4- Mr* 4- i i 1 i f i 1 f i f f 1 0 i i 0 1 1 0 1 0 0 0 0 0 2 1 0 0 0 0 0 0 0 3 1 i" § 0 0 0 0 0 0 4 1 0 0 0 0 0 0 !b 0 5 2 i ' 0 § 0 0 0 0 0 0 6 2 \ 0 0 0 h 0 0 0 0 0 7 2 \ i 0 0 0 0 0 0 0 0 8 2 0 0 0 h 0 0 0 0 0 9 2 1 0 0 0 0 0 0 0 0 0 (18) 22 4 . The 2n S t a t e 2 Eqs. (19) and (20) show the n m a t r i c e s of the 2 I T i and the 2 I I a s t a t e s , r e s p e c t i v e l y : # ->• I 2 3 4 5 6 7 8 9 10 11 12 4- J -»• i 1 4-f -* + 2 " I - i + 2 +* -s -3 - 2 +2 +2 1 2 - i 0 0 0 0 0 0 0 0 0 0 2 + 2 0 0 % 0 0 0 0 0 0 0 0 3 0 0 0 0 0 23 0 0 0 0 23 0 4 -4 0 1 0 0 7 k 0 0 0 0 23 0 0 5 +2 1 0 0 2"23 0 0 0 0 23 0 0 0 6 0 0 h 0 0 0 0 0 0 0 0 7 -2- 0 0 0 0 0 0 0 0 0 0 0 k 8 -1 0 0 0 0 0 23 0 0 0 0 1 f 2 5 0 9 -2 0 0 0 0 23 0 0 0 0 1 22 B 0 0 10 +2 0 0 0 0 0 0 0 1 £ 2 5 0 0 0 11 +3 0 0 23 0 0 0 0 T273 0 0 0 0 12 +1 0 0 0 0 0 0 0 0 0 0 0 # ->• 1 2 3 4 5 6 7 8 9 10 4- j ->- 1 f f ^ - ! - i +2 +i -2 - f - i +i +2 1 -3 0 0 0 0 0 0 0 73 0 2 - i 0 0 k 0 0 0 0 23 0 0 2 3 + 2 0 k 0 0 0 0 73 0 0 0 4 +1 0 0 0 0 h 0 0 0 0 5 -5 0 0 0 '•0 0 0 0 0 0 6 -* 0 0 0 73 0 0 0 0 r § 4 3 0 7 - i 0 0 !h> 0 0 0 0 'i 2 2 a 0 0 8 + 2 0 ihr 0 0 0 0 I 2* 2 5' 0 0 0 9 +2 H 0 0 -0 0 0 0 0 10 +f 0 0 0 0 0 0 0 0 0 (20) 2 3 600— c _ 400-C M 200 - l 6 '~ c *+ 0— c _ d ~ c + d + C _ Al + f BI =1 D l 1 f 2 3_-2 7-2 -4 5 2 3 1 5-r i ' 2 + c !d T 62 D2 1 4 c - c J ^ 2 _ Y.= • Fig. 3 — c 2 + d Energy Levels of the 21T State of OH (v = 0 ) , j to Scale Except the A-doubling. i N 3 i 2z: Hi 0-Bi C l D l • E l — 1 _ T :D3-J L . 1 B3 J - 4.-1^2 £ 3 C3 J L Si A 2 hi ' f 3 0 :Fig. 4 J _ Energy Levels of a Z State, Shovn Schematically 24 5. The 3E S t a t e The squared m a t r i x elements o f n are shown i n Eqs. (21a) and (21b): i * 0 i i O 1 ij 2. 2. Z 0 . 0 . i 0 1. 2 i p | /. ' " ! J ' 1 1 2 2 * 3 3; 3 3 o 1 . .o i 2 . 0 l L _ Z 3 i i 0 1 O i 0 0 o -J- o 4- 0 o o 1 :'i i 2. 0 0 -i 0 i T ! " 2 2. 9 0 I | 0 o : f ) 1 0 2 1 0 9 i 0 k 0 O 6 O O 0 0 0 o 3 I I 0 0 0 0 0 0 2 2 5 ° 0 p ° Is 6 0 0  0 3 6 0 0 ° ° # ° 0 (21a) o o o p £ o q £ o 4.11 2L 3 •3 -3 0 1 0 1 2 0 _ i . 2 3 I. 2 I 3 o o 0 0 .o_ -0 i i f o i 300 O 0 ST j 0 i 0 0 0 0 0 1 3 2 0 o 60 0 0 o 0 0 0 X o 0 f 0 \ 0 0 -0 \ o._ 16 75T D 0 0 \ 1 4 1 0 % IS 0 0 o 0 0 0 25 (3Z state, continued) i if M -i o 1 l £ i 2, 2 3 0 3 i 3 2 3 3 4 „ _ i ! j2 2 2 0 1 2 3 3| 3 3 3 3 1 2 3 25" 0 o o i d o o o o o 0 b -2- o i 50 : 'o 4 4-4 O 1 2 o o; o 90 3ir ° A 0 0 0 0_ 0 o o\ o 0 !l2 --1-4-9 0 i o D 0 _0.. 0 o <5 o ..p.. 0 o o 0 b - J - o o! -4- o o  o ^ £ I 1102S i 2.SZ u \ u 196 P 0 0 0 49 ;o 0 o ol o o 4s 0 0 0 i . 28; 4 4 3 4-0 I t :D I _b_ i i i P t b p i o 3 28 0 0 -0 0 t? o 0 6 O (21b) 6. The 'n State i a . 2 3 .3 _ 3 .4 _4 .. 4 4 0 l 1 • 2 2 •3 3 3 3 1 ir b 0 4 i 2 0 • 1 2 3 loo p i 3 o o; 0 D 0 0 0 I I 0 i 3 6 0 O 0~ 0 _ 0_ -o-~0 ;2 i 4-J b i 0 0 0! 1 5 0 0 0 0 0 1 3 2 0 i 3 2 1 p —o. 4. IS 0 JL -5 i 0 i .OL l 0 o 9 35" 0 JL 3!T 0 _0_ o 0. 3 £ 2 '© 1 o O o ! 1 0 o 0 0 1 7 0 4 3 0 b i 0 0 j£ 35- 0 0 0 0 0 0 i 4 3 1 P 0 0 i 0\ i 8 3S- 0 o 0 4-3 2 b i 0 0 t 0 JL 7 (9 0 0 14- 3 3 b 0 0 *! 0 0 0 0 0 0 3n, (22) 26 Eq. (22) shows the squared m a t r i x elements of n i n the 3 n 0 m u l t i p l e t , w h i l e Eq. (23) shows the and Eq. (24) the 3 n 2 m u l t i p l e t s . These are the pure case (a) m a t r i x elements, and are shown i n the condensed form where p a r i t y i s suppressed (as i n Eq.16). Note t h a t i n Eqs. (19) and (20) f o r the 2n s t a t e , p a r i t y i s shown e x p l i c i t l y and must not be mistaken f o r the s i g n o f M which has been suppressed as per Appendix B. i M ^ i 1 1 1 1 0 J 2 - 2 2 2 2 2 Q 1 2. 1 3 3 ; 3 3 I 4 ; 4 - 4 4 4 3 3 ! 3 3 14 4 4 4 4 o Ii I 3 j 0 i 1 2 3 4 i l l : i 1 0 1 1 i o o 0 4 - i 0 0 r0 tQ ° O " 0 j ~0 0 o ol 0 0 0 | 0 0 0 0 Z Z o t a i . 2 2 2 0 h 0 0 O 0 0 0 5 ? 0 o o ^ 0: 0 o\o\0 0 0 0 0- 0 1-0.1 0—0—Q-. 0-o a o j o : o o.o 0 3 3 0 3 3 1 3 3 1 '3 3 3 0 0 0 0 o o; o o .°__jk.°._.?.. 0 ° \-h 0 0 ° 0 A 0 nk 0 0 0 0 0 ^ 0 0 0 0 0 0 0 ? If 0 o 0 4 4 - 0 4 4 1 " 4-4-2. 4 4 3 -4 4 4 i 0 o -0—0-D 0 o 0 .0-0. 0 0 0 o~-o 0 o o I 0 0 0 0 b .o & o o o o- M 0 0 0 ° 0 4% .o_.ol....o._.o.. 0 0 0 0 0 0 4 5 0 0 0 ° ' 0 ° ife 0 0 0 0 0 440° 0 J . C [ Q . _ . . 0 ^ _ 27 .1_ _ i . 2 2 0 I 2 .2 3 2 ;o 3 3 i 2 .3_3__3 ._J_3 3_ 4- 4 4 i 4 4 0 1 2 3 4 1 2 0 O 0 0 T 0 0 °i 0 O O 0 o i £ 1 I 2. 2. o 4 - o o o 4 0 -h o 6 O O J L o j o o o j o o o o 0 \ 0 0 2 3 0 2 3 1 2 3 2 Z 3 3 o f t * ° ° <§ 0 O 0 0 0 0 Q D _ L ~ 0 o-o o 4 o; 0 0 0 £ o o\o 0 o o 4f o o o o o \ -fa o 3 4 o 3 4 1 3 4 ? 3 4 3 0 -O _0 0 0 0 0 0 0 0 O 0 ^ o o o!. 0 28 0 0 o o 4 0 o 0 0 ^ o„... o _.o..o e>__ ° lk °\° 0 0 0 is,0 0 0 0 o ^ o 3 4 4 o o o b o o o 0 0 o o ^ 4- 5* 0 4 5"1 4 "5 "2." 4 5 3 4 5 4 4- 5 5 0 o 0 o o 0 ~o~ o ~ 0 o . 0 o O O 0 O _ 0 _ 0_ P 0 0 0 0 0 0 0 ; i ; o ~ - o—o~or' i P 0 0 0: pood,, 0 . 0_. 0. . Q £ 6 0,0 0 oOfeo.o 0 o~o §f o-o o o o g o 0 0 0 \0 D 0 0_Lo : 0 3 n . (24) i 28 CHAPTER IV PERTURBATION THEORY 1. Nondegenerate P e r t u r b a t i o n Theory The p e r t u r b a t i o n theory used i n t h i s work f o l l o w s q u i t e c l o s e l y the time-independent theory formulated i n Landau and L i f s h i t z 1 (pp.133-39). T h i s method and the r e s u l t s f o r non-degenerate l e v e l s are summarized i n t h i s s e c t i o n . H Q i s the unperturbed Hamiltonian, f o r which the eig e n v a l u e s W^  and the corresponding wavefunctions are known: H i|>0 = W< V • (25) o n n n The exact Hamiltonian i s H = H + V, and has exact e i g e n -o v a l u e s W and wavef u n c t i o n s il> . We then expand ill i n n r n ^ r n terms o f the complete orthonormal s e t o f wavef u n c t i o n s w i t h expansion c o e f f i c i e n t s c : c nm ib = y c ip r n L nmr: m 0 nmTm (26) The exact Schroedinger equation H ^ n = W n ^n m a ^ ^ e r e " w r i t t e n w i t h Eqs.(25) and (2 6) to g i v e (W - W°)c, = J V, c , n k kn L km nm ' m (27) where = f4^* V ^ m d cZ/ wit h the i n t e g r a l being taken the f u l l range o f a l l c o o r d i n a t e s q. Eq.(27) i s used as the s t a r t i n g p o i n t i n each o r d e r o f approximation by sub-s t i t u t i n g p r e v i o u s l y e v a l u a t e d o r d e r s of approximation f o r W , c. , and c . One f i r s t s e t s k = n to f i n d the n' kn nm next approximation i n the energy, and then one s t i p u l a t e s t h a t k ^ n to f i n d the next order o f approximation to the c , ' s . The va l u e o f c i s e s t a b l i s h e d by r e q u i r i n g nk nn 1 ^ r t h a t J>*i|>ndq = 1 to the r e q u i r e d degree of accuracy. T h i s procedure g i v e s the f a m i l i a r p e r t u r b a t i o n forms: c° = 1 (28) nn C n k = 0 ( k ^ n ) W1 = V (29) n nn c 1 = 0 nn cAk - V n k / U k ( k * n ) (30)-m r . i 1 9 , W = Y * V 2/U , n L 1 nm1 ' m' (31) m where U. E W° - W?, and the prime (') on the sum-5 means I n I ^ t h a t the term with m=n i s omitted. E x t e n s i o n o f t h i s procedure t o t h i r d and f o u r t h o r d e r s g i v e s : 30 , f 2 l = 'nn 'nk ^ ^  1 ran m1 m , Vmn Vkm m k V V nn nk y , rnn k: <• U U,_ U? (32) w m n V V .V. Y 1 V i " " i ni] II n L 4 U .U m j j m V |V | 2 r , nn 1 nm1 i —W m m (33) V V 2 V V. V . _ r , nn mn y , y , mn j n mp nn U 3 4 U^lT m m m j m j (34) V. V V + i l v | 2V, r 3 _ v 2 v /F J3 _ y, km mn nn 2 1 mn1 kn nk " n n v k n / u k 4 U^TT m m k V, V V + V 2 V, V. V . v . y, km mn nn nm kn r , r , km mj j n m m k m j j m k V V . V. V.. V 2 V 2 w r»+ "\ = y • Y i _ J H yi i n i ] _ y • mn y, pn L u 4 u . 4 u . >. z. t p - z. u . m m j . j i l m m j j (35) n V 2 V V. V . + v 2 y - r ^ - 2 V y, nm y, 3 n mj nn ^ LP nn L. U. m m m m j j Morse and Feshbach 9 (p.1005) g i v e a more e l e g a n t i t e r a t i v e f o r m u l a t i o n which, however, does not e x p l i c i t l y separate d i f f e r e n t o r d e r s of approximation. We may t h i n k o f a f i r s t o rder e f f e c t as being the d i r e c t e f f e c t of V on l e v e l n. A second order e f f e c t i s 31 the e f f e c t of. d i r e c t i n t e r a c t i o n s w i t h n e i g h b o r i n g l e v e l s through V. T h i r d o r d e r can then be i n t e r p r e t e d as the e f f e c t on those l e v e l s which cause second order e f f e c t s through the i n t e r a c t i o n s o f s t i l l o t h e r l e v e l s , and a l s o from those changes i n the f i r s t order e f f e c t caused by the f i r s t o r d e r c o r r e c t i o n s to the wavefunction. We s h a l l see below, however, t h a t degeneracy can r a i s e the e f f e c t i v e o r -der o f magnitude of an i n t e r a c t i o n . We s h a l l a l s o make use o f an e x p r e s s i o n c o n t a i n i n g terms to s i x t h order g i v e n by Kusch and Hughes 1 0 (p.139), which i s e v a l u a t e d e x p l i c i t l y f o r the N = 0 l e v e l o f a r i g i d r o t a t o r ( y i e l d i n g nondegenerate energy l e v e l s s u i t a b l e f o r the ground l e v e l of a E s t a t e ) : W° * £ § 2 + T H T T § 3 " 74-5- § 5 (cm" 1) ( 3 g ) . where B = h / ( 8 i r 2 c l ) i s the r o t a t i o n a l constant (I i s the moment o f i n e r t i a o f the r o t a t i n g molecule; e, h, and c are as d e f i n e d i n App. A.,).. 2. Degenerate L e v e l s When energy l e v e l s are spaced s u f f i c i e n t l y c l o s e l y t h a t V /U i s no longer s m a l l , then the r e s u l t s o f the nnr m ' ' prec e d i n g s e c t i o n do not apply, and the m a t r i x d e s c r i b i n g the i n t e r a c t i o n must be d i a g o n a l i z e d d i r e c t l y . I f n 1 and n" r e f e r to d i f f e r e n t l e v e l s among the s e t of c l o s e l y ,.; spaced l e v e l s which i n c l u d e s n, then the a p p r o p r i a t e z e r o -order wavefunction f o r the p e r t u r b a t i o n V . i s some l i n e a r combination f n , c o r r nn' y n ' nn y n " l-*') of whatever s e t o f wavefunctions tyQ ,, ^°„ were chosen f o r r n ' r n the purpose o f t a k i n g the ma t r i x elements of V. I f these energy l e v e l s were not p e r f e c t l y degenerate t o begin w i t h , then the o r i g i n a l wavefunctions ty^,, w i l l n a t u r a l l y be those o f the unperturbed Hamiltonian (which i n c l u d e s whatever terms s p o i l e d the degeneracy). In t h i s case, the matrix of the unperturbed Hamiltonian i s n a t u r a l l y d i a g o n a l and should be added to the matrix v n n i before d i a g o n a l i -z a t i o n . T h i s procedure w i l l i n c l u d e the i n t e r a c t i o n o f V wit h whatever p a r t s o f H Q were r e s p o n s i b l e f o r the un-perturbed energy s e p a r a t i o n s . I n t r o d u c t i o n o f the p e r t u r b b a t i o n w i l l mix these o r i g i n a l wavefunctions, as i s de-s c r i b e d t o f i r s t order by Eq. (30 ) . Note t h a t Eq. 00 ) i s a p p l i c a b l e as long as the p e r t u r b a t i o n i s small^compared t o the (small) energy l e v e l s e p a r a t i o n s . T h i s procedure w i l l now be a p p l i e d to the t w o - f o l d "quasi-degeneracy" which o c c u r s i n ^ t h e r o t a t i o n a l l e v e l s of e l e c t r o n i c s t a t e s having A>0 (A-doubling). 33 3. A-doublets A l l d i a t o m i c molecules are symmetric with r e s p e c t to any plane through the n u c l e i , i n the time average over e l e c t r o n i c motions. A r e f l e c t i o n o f the e l e c t r o n i c c o o r d i -nates i n such a plane can t h e r e f o r e change the wavefunc-t i o n by at most a phase f a c t o r ( s e e Chap.II, Sec.3), which must be ±1 (s i n c e two r e f l e c t i o n s are j u s t an i d e n t i t y o p e r a t o r ) . For A=0 the energy l e v e l s are those of a r i g i d r o t a t o r and are not degenerate. A r e f l e c t i o n w i l l t r a n s f o r m the e l e c t r o n i c wavefunction \l> i n t o +ib or r e r e + — , producing what i s c a l l e d a E or a E s t a t e , r e s p e c t i v e l y . I f A>0, however, there i s angular momen-tum along the i n t e r n u c l e a r a x i s , and the symmetric top model must be used. The symmetric top wavefunctions con-t a i n A and the azimuthal angle <j> o n l y through a phase f a c t o r e ± l A < ^ (Herzberg? p.118), the s i g n depending on the d i r e c t i o n of A along the i n t e r n u c l e a r a x i s . S ince a phase f a c t o r i n the wavefunction cannot a f f e c t the ener-g i e s , every such energy i s doubly degenerate. T h i s degene-racy i s o f t e n p a r t i a l l y removed by what can be viewed as a s l i g h t s h i f t i n the c o u p l i n g of L (see, e.g., H i n k l e y , e t alH or Veseth 1 2-) f rom the i n t e r n u c l e a r a x i s to t h a t of n u c l e a r r o t a t i o n ( t r a n s i t i o n from Hund's case (a) or (b) to case ( d ) ) . A l t e r n a t i v e l y , i t may be viewed as due to an i n t e r a c t i o n w i t h a nearby e l e c t r o n i c s t a t e through the " C o r i o l i s " terms (J,L + J L + L . S + L S t , see Chap.V ) + — — + + — — + of the r o t a t i o n a l H amiltonian. In e i t h e r case, the e f f e c t 34 i s t o s l i g h t l y " s p o i l " A as a good quantum number. These e f f e c t s are u s u a l l y a p p r e c i a b l e o n l y f o r n s t a t e s per-turbed by E s t a t e s , and i n c r e a s e r a p i d l y w i t h r o t a t i o n . I t i s i n t e r e s t i n g to note t h a t A-doubling does occur i n 3 n 0 m u l t i p l e t s wherein ft=0, and may show s i z a b l e s p l i t t i n g s even without r o t a t i o n . Now the assumption t h a t A, S, and E are good quantum numbers i s based on s m a l l v a l u e s of s p i n - o r b i t c o u p l i n g . As the c o u p l i n g i n c r e a s e s , S begins to f o l l o w L i t s e l f r a t h e r than the time-averaged a x i a l component- of L, and Hund's case (c) i s approached making m u l t i p l e t s hard to i d e n t i f y . As t h i s happens, there i s a g r a d u a l widening of the gap between the c and d A-doublet components o f a 3 n 0 s t a t e ; the 3 n 0 c l e v e l s become a 0+ case (c) s t a t e and the 3 n 0 ( j l e v e l s become a 0 s t a t e , having the p r o p e r t i e s of 1 E + and *E r a t h e r than 3n s t a t e s . Each of the A-doublets has an a d d i t i o n a l (2J+1)-f o l d degeneracy, each of the s u b l e v e l s having a d i f f e r e n t component M w i t h r e s p e c t to (say) the s p a c e - f i x e d z a x i s . T h i s degeneracy i s o n l y removed by an e x t e r n a l f i e l d and need not concern us here s i n c e the Stark p e r t u r b a t i o n V i s d i a g o n a l i n M. When a wavefunction belongs to a degenerate e i g e n -v a l u e , a c o o r d i n a t e r e f l e c t i o n may do more than change a phase f a c t o r : I t can t r a n s f o r m the wavefunction i n t o some othe r wavefunction i n the space spanned by the s e t of wave-f u n c t i o n s of t h a t eigenvalue ( i . e . , \p , -> £c i|» ) . A r e f l e c -n' t i o n i n a symmetry plane w i l l change the s i g n of <j> (Hougen, p. 16) and thus change the form of e + l A < ^ i n t o e - 1 ^ (and v i c e v e r s a ) , but not i n t o i e 1 ^ . When i n t e r a c t i o n s remov-in g ? degeneracy -. are i n c l u d e d i n H q , however, each non-degenerate energy l e v e l must have d e f i n i t e p a r i t y so t h a t the r e f l e c t i o n changes ^ e i n t o ± ^ e « T h i s means t h a t the symmetric top wavefunctions ^ e = X e ± l A <' > (where x does not c o n t a i n A o r <j> ) are not the c o r r e c t zero-order wave-f u n c t i o n s f o r the i n t e r a c t i o n s removing degeneracy. Rather, l i n e a r combinations must be used as per Eq.(37). In t h i s case, i t i s easy to see t h a t the simple l i n e a r combinations .+ / iA<f> . „~iA4>\ / / o • = x ' )/>/2 (38) * e = x ( e e ) / / 2 are d e s i r e d f u n c t i o n s , s i n c e ifi*-*- ±ty±. These then are the ' r e r e wavefunctions a p p r o p r i a t e to A-doublets w i t h no e x t e r n a l f i e l d s . The s u b l e v e l s d e s c r i b e d by are c a l l e d "c" l e v e l s , and those d e s c r i b e d by ip are c a l l e d "d" l e v e l s . The p a r i t y o f the s u b l e v e l i s the product o f the p a r i t y of \ii times t h a t of the r o t a t i o n a l wavefunction \b ( t h i s r e r r l a t t e r i s (-1) N f o r case (b) E s t a t e s , and (-1) J f o r case (c) 0 s t a t e s ) . 4 . Stark E f f e c t s i n a A-doublet + — Wavef u n c t i o n s i|» and ij»e (Eq.3 8) are not the zero-order f u n c t i o n s a p p r o p r i a t e to A-doublets i n the p r e -sence of an e l e c t r i c f i e l d . These are found by d i a g o n a l i z -ing the the p e r t u r b a t i o n V, g i v e n by Eq.(7). T h i s i s e q u i v a l e n t t o s o l v i n g Y (V . - W*5 .)c° . = 0 , (39) 4 nj n nj nj • ' v ' where j i s summed over n', n",...,and 6 . i s one i f ' ' ' nj n=j and zero i f n ^ j . Eq.(39) i s d e r i v e d (Landau and L i f -s h i t z 1 , p.137) from Eq.(27) u s i n g a f i r s t order approxima-t i o n , t o the energy, and zero-order approximations t o the wavefunction: c = c° , c ,= c° ,, ..... and c = 0 nn nn' nn 1 nn' ' nm (m^n,n',n",...). (We continue to use n f o r the quantum numbers of the l e v e l i n which we are i n t e r e s t e d , to use n, n', n",... f o r l e v e l s degenerate with or c l o s e to n, and m f o r other l e v e l s which are s u f f i c i e n t l y d i s t a n t so t h a t V /U i s small.) The system of l i n e a r equations nm m J ^ i n Eq.(39) has a s o l u t i o n f o r the c ^ o n l y i f the matrix o p e r a t o r i s s i n g u l a r and thus has a zero determinant: V . - W 6 . = 0 (40) 1 nj n nj 1 v ' The d i a g o n a l matrix elements V are zero, but ^ nn ' t h e r e are o f f - d i a g o n a l elements between the two A-doublet l e v e l s of o p p o s i t e p a r i t y . As noted in.App. E, these m a t r i x elements are d i a g o n a l i n a l l quantum numbers except p a r i t y , and are g i v e n by nn' yE < t J j +|n Id) > r z r (41) yEMfl J(J+1) where the s u p e r s c r i p t s on i n d i c a t e p a r i t y . Hund's case (a) c o u p l i n g was assumed. Igno r i n g other r o t a t i o n a l l e v e l s , we have V = H H L V, HL V. HL 0 (42) where V T „ = -yEMfi/J(J+l). The "H" and "L" i n d i c e s L U i n d i c a t e t h a t these matrix elements were taken with r e s p e c t to two wavef u n c t i o n s a r b i t r a r i l y (at present) l a b e l l e d \ji„ ti and , e i t h e r one o f which can be the s u b l e v e l of p o s i -t i v e p a r i t y (and the other n e g a t i v e ) . Account must now be taken of im p e r f e c t degeneracy: There i s another i n t e r a c t i o n i n the f i e l d - f r e e Hamiltonian which separates the s u b l e v e l s by e n e r g i e s which may be comparable to V „ T . In p r i n c i p l e , these terms causing A-doubling should be s p l i t o f f the unperturbed Hamiltonian and added to the Stark p e r t u r b a t i o n , and the matrix sum (taken with r e s p e c t to the wavefunctions o f p e r f e c t l y degenerate energy l e v e l s ) should be d i a g o n a l -38 i z e d . In fa c t . o n e does not need t o evaluate the o p e r a t o r s r e s p o n s i b l e f o r the s p l i t t i n g i f one knows e m p i r i c a l l y t h e i r e f f e c t upon the e n e r g i e s . They have no o f f - d i a g o n a l m a t r i x elements s i n c e the ^° are t h e i r e i g e n f u n c t i o n s ( i . e . , \\>„ and $ ) . Taking zero energy to l i e a t the ce n t e r o f the unperturbed A-doublet whose s p l i t t i n g i s 26 erg s , we add the doublet e n e r g i e s to Eq.(42), and get H H V. HL V HL -6 (43) b e f o r e d i a g o n a l i z a t i o n . T h i s means t h a t tCH i s the wave-f u n c t i o n o f high e r unperturbed energy, and lower energy, r e g a r d l e s s o f p a r i t y . The perturbed e n e r g i e s W1 are found by p u t t i n g the matrix of Eq.(43) i n t o Eq.(4 0): 6 - W1  VHL VHL •6 - W1 (44) T h i s equation has s o l u t i o n s f o r W1: _/S~2+ v2 HL (4?<) where (again f o r the moment) "S" and "A" are a r b i t r a r y l a b e l s . Eq. (45 ) says t h a t f o r smal l p e r t u r b a t i o n s ( V H L < < | 5 ) ' 39 the e n e r g i e s are ±6 (1 + = ±6 ( 1 + V ^ / 2 6 2 ) which d e v i a t e q u a d r a t i c a l l y i n V" H L (and t h e r e f o r e E) from the unperturbed l e v e l s ±6. For l a r g e p e r t u r b a t i o n s ( v H L > > l S ) the e n e r g i e s are ±|V | ( 1 + 6 2 / V * L ) ^ ~ ± I V H L ' W N I C H A R E l i n e a r i n V T I T . Thus Eq.(45) e x h i b i t s a t r a n s i t i o n from HL ^ second to f i r s t order e f f e c t s with i n c r e a s i n g f i e l d s t r e n g t h as shown i n F i g . 5 . A f i r s t order e f f e c t i s not i n c o n t r a -d i c t i o n w i t h the c o n c l u s i o n s reached i n Chap.II', Sec. 4, s i n c e the wavefunctions a s s o c i a t e d with and Wg (see Eq.(4 9) below) are mixtures of wavefunctions of d i f f e r e n t p a r i t y (ty„ and tyT):; P a r i t y i s no longer a good quantum number, so the r e s u l t i n g wavefunctions no longer possess i n v e r s i o n symmetry and may indeed have d i a g o n a l m a t r i x e l e -ments . • '- •>. v o / v r Two f u r t h e r p o i n t s are important. F i r s t l y , the higher and lower A-doublets move away from each o t h e r , always " r e p e l l i n g " . Secondly, the Stark s h i f t depends on l y on the a b s o l u t e magnitude o f V so t h a t the s i g n s of \i, E, M, and Q do not matter. f i c i e n t s . Eqs.(3 9) may now be s o l v e d f o r the In the case o f Wg, we have coe f -H H V V. HL HL - 6 " w s , 0 'HS , 0 'LS 0 , (46) w i t h a s i m i l a r p a i r f o r W . T h i s system i s s o l v e d by 4 0 F i g . 5." ! i ^ P e r t u r b a t i o n s i n I n t e r a c t i n g Doublets' 54 SINGLET ! DOUBLET ( T = 2.0) TRIPLET (^=1.5, CL3 = 2.0) -.1 0 F i g . 6 Stark E f f e c t s v i n S State M u l t i p l e t s with N = 1 2 1 : V a V a 2 Z 0 0 41 H e r z b e r g 3 (p.282f.) i n somewhat more g e n e r a l form, and the s o l u t i o n s can be w r i t t e n i n m a t r i x form as: A H 'HA HS 'LA LS A -Co S c 2 (47) where the two d i s t i n c t m a t r i x elements Cj and c 2 are g i v e n by C! = +{ \ + 6 / 2 ( 6 2 + V 2 L ) * }* c 2 = +{ £ - 6 / 2 ( 6 2 + V | L ) ^ (48) Eq.(47) says t h a t the new zero-order wavefunctions a p p r o p r i -ate to the Stark p e r t u b a t i o n V „ T are o f the form *A = C l*H " ° 2 * * s = C l ^ n + (49) In d e r i v i n g Ci and c 2 (Eq.48), the c o n d i t i o n cf + c\ =1 was imposed, which ensures t h a t /^*^ gdq = dq = 1, and t h a t fij)*\l>pd.q = /ij/*ij>sdq = 0 , so t h a t i|»s and are orthonormal. Examination of Eq.(48) shows t h a t i f V U T= 0, HL then Ci = 1 and c 2 = 0, so t h a t 42 *A = *H ( VHL + °>' (50) whereas f o r extremely h i g h f i e l d s , Ci = c 2 = l/ * / 2 , and *A = {*H " * L ) / / 2 ( 6 * 0 ) , (51) " j u s t i f y i n g " the l a b e l s \> and i|>c, i n t h i s case, as antisymmetric and symmetric wavefunctions. Remembering t h a t ip H and correspond i n some order t o and *~ (Eq.38), we see t h a t ^ = ±e~ l A < f > and ^ = +e l A* so t h a t v e r y h i g h f i e l d s can be thought of as s e p a r a t i n g the molecular s t a t e s i n t o ones c o n t a i n i n g +A and -A as s e p a r a t e l y v a l i d quantum numbers. T h i s resembles a t r a n s i t i o n to Hund's case (d), although the analogy i s ver y crude (e.g., removal o f M-degeneracy). I t should be noted t h a t C (Eq.47) i s j u s t the u n i t a r y m a t r i x which must d i a g o n a l i z e Eq.(43): V. HL V HL - 6 C = A c|)6 - 2c 1c 2V. HL 2 0 ^ 2 6 + (cf -c^)V HL 2CJC26 - (cf - cjj)V. HL - ( c f - c | ) 6 + 2C!C 2V HL (52) Upon s u b s t i t u t i o n of C i and c 2 from Eq.(48), the o f f -d i a g o n a l elements o f Eq;(52) are found to be « V + V ' HL' HL which i s zero o n l y i f V" H L i s n e g a t i v e . Eq.(52) then becomes /6 2 + V 2 r i l j •/62 + V 2 rlLi (53) i n agreement with Eg.(35). However, the i n e q u a l i t i e s V I T„ < 0 < ( c f - c f ) 6 < 6 < /6 2 + V 2 HL HL imply t h a t the s i g n s shown i n Eq.(53) can o n l y r e s u l t i f c l c 2 > 0/ e n s u r i n g t h a t c j and c 2 are o f l i k e s i g n . Were V H L > 0, however, i t could-be"shown t h a t c i c 2 < 0. We have y e t to d e a l with the o f f - d i a g o n a l m a t r i x elements between s t a t e s o f d i f f e r e n t J . For t h i s purpose we d i s t i n g u i s h the t r a n s f o r m a t i o n s a p p l y i n g t o the two i n t e r a c t i n g p a i r s of A-doublets by marking one o f them with a say the p a i r o f higher energy. I t i s then conven-i e n t to d e f i n e two c o e f f i c i e n t s b and d as f o l l o w s : b = Ci^i + c,c d = c i C 2 — c-j c (54) I t i s r e a d i l y shown t h a t b 2 + d 2 = 1. We then o b t a i n 44 VAA = V S S = b VHH V S A = " VAS = d VHH ' (55) w i t h the remaining elements f o l l o w i n g from matrix symmetry. We have used the f a c t t h a t the order o f p a r i t y i n A-doub-l e t s u s u a l l y a l t e r n a t e s from one r o t a t i o n a l l e v e l t o the next (see F i g . 3) so t h a t the wavefunctions ^ L and ty^ have the same p a r i t y (as do and ^ ) . Thus, we have V H J J = V L £ and = Vj- H = 0 f o r the o l d o f f - d i a g o n a l elements. As an example, V S A = < C2*H + C1*L | V | £1*H " '6lH * C2 S1 VHH C i e 2 V L i : (56) d VHH Using Eqs. (52) and (53) f o r the ma t r i x elements o f n z , we apply these t r a n s f o r m a t i o n s to the 2n s t a t e m a t r i x , which i s shown i n the f i r s t t h ree r o t a t i o n a l l e v e l s o f the 2n^ s t a t e i n l q . ( 5 7 ) and the f i r s t two l e v e l s of the 2II^ s t a t e i n Eq. (58). In these m a t r i c e s , 6^ i s the t h i A-doubling i n t h a t m u l t i p l e t , and p i s the s i g n of the p a r i t y . A f t e r a p p l y i n g the t r a n s f o r m a t i o n s of Eqs. (47)-(49), these m a t r i c e s become those shown i n Eqs. (59) and (60). In Eq.(59)'., ft = \t b i and d^ are combinations o f the c's p e r t a i n i n g to J = \ and \ , w h i l e b 2 and d 2 45 fl = £ J,N 2,1 1,2 2,3 .pM-*- -2 +2 -1 - i +2 +z "z - ! " i +2 +z 4- 4-2,1 "2 0 0 0 0 0 0 0 0 0 +2 0 4 * 0 0 0 0 0 0 0 0 - ! 0 b 0 0 -u 0 0 0 0 0 - i 0 0 - 6 2 0 0 0 0 0 0 4 e +i 0 0 f i e 6 2 0 0 0 4 e 0 0 0 +1 0 0 0 0 6 2 0 - § e 0 0 0 0 - f 0 0 0 0 0 0 s3 0 0 0 0 -u - ! 0 0 0 0 0 0 s3 0 0 H e 0 - i 0 0 0 0 0 0 0 0 0 +i 0 0 0 4 . 0 0 0 0 0 0 0 0 -u 0 0 0 0 & 0 0 - 6 3 0 +i 0 0 0 0 0 0 0 0 0 0 a J,N z,2 4- 1 \ z z z z z z z 2 - l 0 0 0 0 0 0- 0 - i 0 -«1 0 0 0 0 - l e 0 0 -4e - i e +? 0 0 0 0 0 0 0 +1 -h 0 0 0-/Tfe 0 0 0 0 - f 0 0 0 0 + 6 2 0 0 0 0 -1 0 0 0- 0 + 6 2 0 0 - s i * 0 - i 0 0 -h 0 0 0 + 6 2 0 0 § ,2 +z 0 0 0 0 0 - 6 2 0 0 +1 0 0 0 0 0 0 - 6 2 0 +1 0 0 0 0 -h 0 0 0 0 - 6 2 (57) (58) 46 J,N -> i , l 1,2 f ,3 A i A| A i s i s i Af Af A i s i s f s f + + i , l ? : T i 0 0 - f b i -|d! 0 0 0 0 0 0 0 Sz 0 " T i 0 +|dj -ibx o 0 0 0 0 0 0 Af 0 0 T 2 0 0 0 0 -§b 2 0 0 • 0 A Z -§b x +§di 0 T 3 0 0 0 0 - § b 2 -§d 2 0 0 Sz +|d 2 -§d x -^b! 0 0 -T 3 0 0 0 -§b 2 0 0 Sz 0 0 0 0 0 - T 2 0 • ^d 2 0 0 --§b 2 0 Az 0 0 0 0 0 0 Tit 0 0 0 0 0 A| 0 0 - § b 2 0 0 +§d 2 0 T 5 0 0 0 0 z,3 A Z 0 0 -0 -|b 2 +|d 2 0 0 0 T 6 0 0 0 S 2 0 - | d 2 -|b 2 0 0 0 0 0 0 "T G 0 0 Sz 0 0 - f d 2 0 0 -lb2 0 0 0 0 -T 5 0 sf 0 0 0 0 0 0 0 0 0 0 0 -T, fi = 1 . J,N -> 1,1 z,2 4- A Z A i s i s f Af Af A i s i s f s f A Z T l 0 0 0 0 - / ^ 1 0 0- 1 o A 2 0 T 2 0 0 0 0 -4 b i - | d x 0 0 Sz *b 0 0 -T 2 0 0 0 +§d x 1 o 0 s i 0 0 0 - T i o+/^|di 0 0- l o Af 0 0 0 0 T 3 0 0 0 0 0 * e Af o-Z^fd! 0 T,* 0 0 0 0 f ,2 0 -irbl 0 0 0 T 5 0 0 0 +1*1 -§b! 0 s i 0 0 0 0 -T 5 0 0 Sf O-Z^fb! 0 o 0 0 -T 4 0 sf 0 0 0 0 0 0 0 0 0 -T 3 ( 5 9 ) (60) combine the J = £ a n ^ J = f l e v e l s . The T's are d e f i n e d as f o l l o w s : T l (4 + « i 2 A 2 ) * T 2 = (V25 t 6 2 2 / e 2 ) " * T 3 = (T4TT + 6 2 2 A 2 ) * TH = (Vi»9 + 6 3 2 A 2 ) * T 5 = ( V 1 2 2 5 + 6 3 2 A 2 ) * T 6 = ( V l 2 2 5 + 6 3 2 A 2 ) * ft = \ i we have d e f i n e d T l = ( 9/25 + 6 ! 2 / e 2 ) * T 2 = ( V 2 5 + 6 i 2 / e 2 ) ^ T 3 = ( 9 A 9 + 6 2 2 A 2 ) * Tf = ( 8 1 / l 2 2 5 + 6 2 2 / e 2 ) T 5 = ( V l 2 2 5 + <5 2 2 A 2 ) * Note t h a t a f a c t o r e has been e x t r a c t e d from the t r a n s -formed m a t r i c e s . With the f i r s t order e n e r g i e s and zero o r d e r wavefunctions g i v e n by Eqs.(28) and (29), the second order e n e r g i e s can be shown (Landau and L i f s h i t z 1 , p.139) to s t i l l be given by V 2 w m = J . m Um (61) but where the m a t r i x elements are taken w i t h r e s p e c t to the proper zero order wavefunctions (Eq.49) so t h a t the sub-m a t r i c e s between n e a r l y degenerate l e v e l s v n n i a r e d i a g o -n a l and make no c o n t r i b u t i o n - ( E q s . 5 9 and 60). A p p l y i n g Eq.(61) to the transformed m a t r i c e s (and remembering t h a t 48 b? + d? = 1 ) , we see t h a t second order e n e r g i e s are i d e n -t i c a l to what we would have gotten by n a i v e l y a p p l y i n g Eq. (31) to the unperturbed m a t r i c e s o f V (Eqs.5.7 and 58) . 5 . The S e c u l a r Equation f o r l_ S t a t e Mul t i p l e t s ; Hund's case (b) c o u p l i n g d e s c r i b e s most I s t a t e s q u i t e w e l l , s i n c e the i n t e r n u c l e a r a x i s c a r r i e s no component of o r b i t a l angular momentum to which the s p i n may be coupled. In "pure" case (b), the 2S+1 m u l t i p l e t s of energy l e v e l s w i t h N >.0 are degenerate. T h i s degeneracy i s u s u a l l y p a r t l y removed by s p i n - s p i n , s p i n -r o t a t i o n , and s p i n - o r b i t a l c o u p l i n g ( H e r z b e r g 3 , p.223), but these s u b l e v e l s remain c l o s e together compared to the r o t a t i o n a l s p a c i n g , so t h a t case (b) i s s t i l l a f a i r l y good d e s c r i p t i o n . U n l i k e the A-doublets t r e a t e d above, the m a t r i x elements between these nearby s t a t e s are a l l zero, s i n c e a l l l e v e l s i n these m u l t i p l e t s have the same p a r i t y . The m a t r i x o f Eq.(42) and i t s eigenvalues are a l l zero so t h a t the c o r r e c t c h o i c e of zero=:order wavefunctions remains u n s p e c i f i e d . I n t r o d u c i n g the z e r o - f i e l d s e p a r a t i o n e n e r g i e s change nothing s i n c e these are a l r e a d y d i a g o n a l . (The m a t r i c e s o f Eq. (4 2) onwards are r e a d i l y e n l a r g e d to n x n, to d e s c r i b e the problem of Z s t a t e s o f m u l t i p l i -c i t y n.) N e v e r t h e l e s s , i f second order S t a r k e f f e c t s become comparable to the s p i n s p l i t t i n g s , then the i n t e r -a c t i o n w i t h whatever terms are c a u s i n g the s p i n s p l i t t i n g s must be i n c l u d e d , and the r e s u l t i n g d i a g o n a l i z a t i o n must 49 determine the a p p r o p r i a t e zero-order wavefunctions. Landau and L i f s h i t z 1 (p.138) show, by the same methods o u t l i n e d i n S e c . l , t h a t the a p p r o p r i a t e equation i s s i m i l a r to Eq. T39) : 4 nj n nj nj (62) but where V V y nm mj ni f- U  L U (63) J m m v ' r e p l a c e s V .. The m a t r i x X . has the standard second v np n3 order terms along i t s d i a g o n a l , and i t s o f f - d i a g o n a l elements c l e a r l y connect the n e a r l y degenerate but non-i n t e r a c t i n g l e v e l s n and j to each o t h e r through a t h i r d (and p o s s i b l y d i s t a n t ) l e v e l m. T h i s degree o f i n d i r e c t n e s s i s u s u a l l y a s s o c i a t e d with t h i r d order e f f e c t s . However, as wit h A-doubling, degeneracy has i n c r e a s e d the e f f e c t by a f u l l o rder o f approximation. Here, an i n t e r -a c t i o n with a r e l a t i v e l y d i s t a n t l e v e l has g i v e n the wavefunction a smal l component of o p p o s i t e p a r i t y , and has. .weakened the r u l e , p r e v e n t i n g i n t e r a c t i o n s between the s p i n m u l t i p l e t s . I f these l e v e l s have on l y s m a l l components of o p p o s i t e p a r i t y , they make up f o r i t i n p r o x i m i t y t o one another, and may have second order e f f e c t s c o n s i d e r a b l y l a r g e r than Tv 2 /U . 1 ^ L nm/ m 50 6. 2 £ S t a t e s , I n t e r a c t i o n of L e v e l s w i t h N=l The matrix elements of n i n a 2 E s t a t e are z the square r o o t s of those i n the m a t r i x o f Eq.(18), which d e f i n e s the row and column numbers used i n t h i s s e c t i o n . I g n o r i n g s p i n - s p l i t t i n g i n the denominators of Eq.(63), the -1 r o t a t i o n a l l e v e l s are gi v e n by BN(N-fl) cm Only two v a l u e s o f U M are needed: U ( N = 0 ) = 2 B ' A N D U ( N = 2 ) = ~^ B* We may now r e a d i l y e v a l u a t e x n j ' f ° r example, X 23 = ( - y E ) 2Bhc /2 ( y E ) 2 20 hcB + /_ u E) 2 2 5 _ ! L S + * » h ) (-4B)hc ergs, (6^4) or, c o n v e r t i n g t o cm \ X 23 5 X 2 3 / h c khc'20B /2e 2 20B cm -1 (65) S i m i l a r l y , X 44 = e 2 ( v / T 7 7 7 ) 2 + C ? ( / V 7 T ) '-4B £ -4B 20B (66) Thus we o b t a i n Xv 2 OB 2 3 0 /2" /2 1 0 0 it 0 0 -1 2 3 (67) 51 I f we i n c l u d e the s p i n s p l i t t i n g through a simple Hamiltonian o f the.form H r = hcBN 2 + hey(N'£) ,, (68) then i t can be r e a d i l y shown t h a t f 2 l e v e l s are s h i f t e d by -£Y (N+l) , and l e v e l s by +2vyN. Th i s g i v e s the matrix 2 -Y 3 0 0 h 0 0 0 h] (6'9) to be added t o Eq.(67). To s i m p l i f y n o t a t i o n , we d e f i n e 2 OB -T2-Y -(70) The s e c u l a r e quation from Eq.(62) i s now 20B - r - F ' /2 /2 1+2-T-F' 0 = 0 0 - 1 + j T - P f (71) where f = 2 " O B F ' • *-s t n e energy i n cm . The J=M=2i ^ 4 4 ) corner i s a 1 x 1 subdeterminant with immediate s o l u t i o n 20B 52 and the upper l e f t 2 x 2 subdeterminant with M=£ g i v e s (F* + r) ( F 1 - l - £r) 2 = or F' + (£r - 1)F' - (2 + £r2 + r) = (73) where (20B/e 2)F . T h i s equation has s o l u t i o n s F' = £ ( l 4 r ) ± U ( i-£r) 2 + 2 + i r 2 + r ) ^ , which are w r i t t e n s e p a r a t e l y as (74) £ 4 0 B 40B Y 4" ,3e: '2 OB + + 2 Y 2 1 + i r 3 P 2 v 2 <IUB + 2> + 2^2 * , ( J = M = £ ) (75) £ , ( j= | f M=4) . Eqs. (7 2) and (75), which appear i n a paper by Swartz and T r i s c h k a 1 3 ( p . 1 0 8 9 ) , are seen to y i e l d the o r i g i n a l s p i n s p l i t t i n g s i n the l i m i t of e ->• 0. In the hig h f i e l d l i m i t o f y •* 0, F 2 and F^ approach -e 2/20B, w h i l e approaches +e 2/10B. These l i m i t s are j u s t the Stark e f f e c t s i n the M=l and M=0 s u b l e v e l s , r e s p e c t i v e l y , o f the J=N=1 l e v e l o f a 1Z s t a t e (as i s r e a d i l y c a l c u l a t e d from Eqs.(15) and (31) above). We see t h a t the degeneracy added by s p i n s p l i t t i n g does not a l t e r the Stark e f f e c t i n the l i m i t o f p e r f e c t degeneracy, u n l i k e the d i r e c t l y i n t e r a c t i n g A-doublets, From t h i s one would expect t h a t the non-degenerate l e v e l N=0 should have a Stark e f f e c t which i s independent of the m u l t i -53 p l i c i t y of the E s t a t e : T h i s i s r e a d i l y v e r i f i e d by exami-n a t i o n o f the *E, 2 Z , and 3E m a t r i c e s given i n Chap..lil where the sum over J ' o f the matrix elements squared i s i n each case (we use the n o t a t i o n ) . Thus Eq. (36 ) may be a p p l i e d to 2 E s t a t e s even though i t was d e r i v e d f o r 1 E . 7. 3E S t a t e s , I n t e r a c t i o n of N=l L e v e l s T h i s c a l c u l a t i o n i s very c l o s e to t h a t of the l a s t s e c t i o n . In a 3E s t a t e t h r e e l e v e l s a r i s e from N=l, w i t h J= 0, 1, and 2. We use the e m p i r i c a l constants f o r the s p i n s p l i t t i n g : Using the f 2 (J=l) l e v e l , which has h i g h -e s t energy, as a r e f e r e n c e , we take a^ to be the energy d i f f e r e n c e (cm: 1) between the J=2 and J = l l e v e l s , and a^ to be the d i f f e r e n c e between J=0 and J = l . E v a l u a t i n g of Eq.(63) we have f o r the s e c u l a r e q u a t i o n : J M # - 3 i+ 5 6 7 8 4- 4- 4-0 0 3 -a'-F' 0 0 /2 0 0 1 0 0 -•1-F 0 0 0 0 1 1 5 0 0 i - F ' 0 \ 0 2 0 6 /2 0 0 1-aj-F' 0 2 1 7 0 0 z 0 J - a i - -F' 0 2 2 8 0 0 0 0 0 - l - a ' - F 2 OB 0 (76) whe re a£ = ( 2 0 B / E 2 ) a 1 , and i'3 = ( 2 0 B / e 2 ) a 3 (with F' as 54 i n Eq.73). I f we assume t h a t a^ > a^ > 0, so t h a t F J = 1 > F j _ 2 > F j=o a t z e r o f i e l d , then Eq.(7 6) g i v e s the f o l l o w i n g r e s u l t s : J=M=0: f 2 .2, 2 2 2 2 OB ( a 1 + a 3 ) - [(a 3-a 1+2og) ^(j^) 1 J=l,M=0: J=M=1 2 0 B (77) 40B " K + H a ± + ( f ^ ) V 3=2,M=0: J=2,M=1: r e 2 .2 2 .2 2 i l 20B " ( a l + a 3 } + I ( a 3 " a l + 2 0 B ) + 8 ( 2 0 B ) ] j 4 0 B " K " H a 2 + ( § ^ ) V J=M=2: 2 OB " a l These s o l u t i o n s are i l l u s t r a t e d along w i t h those f o r the 2Z (Eqs.72 and 75) and 1l cases, i n F i g . ,6'* 55 CHAPTER V . INTERMEDIATE COUPLING 1. I n t r o d u c t i o n In t h i s chapter we w i l l c o n s i d e r the m o d i f i c a t i o n o f the wavefunctions and m a t r i x elements of the Stark per-t u r b a t i o n i n m u l t i p l e t n and A s t a t e s - due to the f a c t t h a t these s t a t e s may not be p r o p e r l y d e s c r i b e d by Hund's case ( a ) . The matrix elements o f n f o r these s t a t e s were d e r i v e d f o r Chap. I l l u s i n g pure case (a) wavefunctions, o r , more p r e c i s e l y , u s i n g the assumption t h a t case (a) quantum numbers and t h e i r corresponding s e l e c t i o n r u l e s are v a l i d ) . I t i s found t h a t the energy l e v e l s o f a l l s t a t e s c o n s i d e r e d i n t h i s work can be d e s c r i b e d q u i t e c l o s e l y by c o n s i d e r i n g them to l i e along a continuum between pure case (a) and case (b) c o u p l i n g , d e s c r i b e d by a s i n g l e parameter Y. T h i s p a r a -meter i s a measure of the i n t e r a c t i o n energy between the -e l e c t r o n i c s p i n S and the a x i a l l y averaged magnetic f i e l d from the o r b i t a l e l e c t r o n i c angular momentum L. We may d e f i n e Y to be the r a t i o , o f the c o e f f i c i e n t o f t h i s c o u p l i n g (A) to the r o t a t i o n a l s pacing constant (B). Since case (a) wavefunctions form a complete s e t , the wavefunctions f o r case (b) or any i n t e r m e d i a t e c o u p l i n g can be expressed as a l i n e a r combination of them. The problem o f f i n d i n g the a p p r o p r i a t e l i n e a r combination i s e s s e n t i a l l y the problem of d i a g o n a l i z i n g the Hamiltonian. 56 2. The 2n S t a t e  2.1. Hamiltonian The 2n s t a t e i s t r e a t e d i n some d e t a i l by H o u g e n 2 ( p . l O f f ) . There are four p o s s i b l e case (a) wave-f u n c t i o n s , l a b e l e d as e i t h e r |Q > or | f i j >, and c h a r a c t e r -i z e d by the f o l l o w i n g quantum numbers: Wavefunction A S I n j M | + 2 > +1 1 +1 +1 j M 1+1 > +1 1 -1 +1 j M I - I > -1 1 -1 -1 j M 1-1 > -1 1 +2 -1 j M When the 4 x 4 matrix of the Hamiltonian i s e v a l u a t e d w i t h r e s p e c t to these wavefunctions, i n accordance w i t h the d i s c u s s i o n of Appendix D, i t t u r n s out t h a t a l l matrix elements are zero between wavefunctions d i f f e r i n g i n the s i g n o f fi. The m a t r i x i s thus f a c t o r e d i n t o a 2 x 2 block w i t h p o s i t i v e fi and another w i t h n e g a t i v e J J . Moreover the two sub-blocks are the same; f o r example H l l = H33 a n c ^ H12 = H34" T n u s t n e s e c u l a r equation | H - WI | = 0 f a c t o r s immediately i n t o two' i d e n t i c a l q u a d r a t i c equations i n W. We i n t e r p r e t t h i s t o mean t h a t each of the energy l e v e l s o l u t i o n s W i s doubly degenerate, g i v i n g r i s e to the A d o u b l e t s . When t h i s happens, however, they are not separated i n t o ( f o r example) s t a t e s with wavef u n c t i o n | f- > and wavefunction | - f > , but r a t h e r i n t o symmetric and antisymmetric l i n e a r combinations of these which possess d e f i n i t e v a l u e s o f p a r i t y , as d i s c u s s e d i n Chapter IV, Sec. 3. The sub-block o f the wavefunctions 11 > and 12" > i s II > I i > |f > f J(J+1) - I + Y/2 - /(J-i)(J+ 2 ) I i /(J-i)(J+4) J ( J+1) + i - Y/2 (78) where we have a l r e a d y dropped the terms H + B ( L 2 - L 2 ) j ev 5 from the d i a g o n a l and d i v i d e d by hcB. We have d e f i n e d Y = A/B. 2.2. J=} The energy l e v e l J=£ i s a s p e c i a l case; s i n c e cannot be l e s s than J^=n, the. wavefunction | $ > does not e x i s t . Eq. (78) i s a 1 x 1 matrix y i e l d i n g t r i v i a l l y the eigenvalue = B [ J ( J+1) + i - Y/2 ] = B - A/2 (79) E q u a l l y t r i v i a l l y , the wavefunction i s simply |£ >, o r I 1 i i J M > i n the f u l l |A S Z; SI J M > n o t a t i o n , and i s pure case (a). 2.3. J>£ For a l l v a l u e s of J>£/ the f u l l m a trix o f Eq. (78) 58 must be. d i a g o n a l i z e d . The s e c u l a r equation y i e l d s the w e l l known formulae o f H i l l and Van V l e c k 1 8 ( p . 2 6 1 ) : F 1 = Bt (J+z-) - 1 - z-ACJ+i) + Y(Y-4) ] (80) F = B[ ( J + i ) 2 - 1 + J / 4 ( J + ^ ) 2 + Y(Y-4) ] . We s h a l l not bother to add back the e x p e c t a t i o n value of ( H e v + : (L2 - L 2 ) ) / h c i . The matrix of Eq. (78) can be d i a g o n a l i z e d more e a s i l y by f i r s t s u b t r a c t i n g h a l f the t r a c e ( ( J - ^) (J + z ) ) and n o t i n g the s i m i l a r i t y t o Eq. 43) , us i n g the s o l u t i o n of Eq. (45) p l u s (J - ^) (J + \ ) . The wavefunctions are ob t a i n e d immediately from Eq. (49): *1 = C l ' ^ > + > (81) <J>2 = c j f > - c 2 | i > Eq. (48) reduces t o c 1 = /(X + Y - 2)/2X (82) c 2 = /(X - Y + 2)/2X 59 where X = / 4 ( j + £ ) 2 + Y(Y-4) . 2.4. Hund's Cases as L i m i t s When Y >> 2 then 1, c^ ->• 0 so t h a t *1 •* I i > 2 n£ (83) f 2 -»• I | > . 2 H | Which corresponds to a r e g u l a r case (a). When Y=2 (as i s the case f o r CH/ to w i t h i n experimental e r r o r ) then so t h a t c 1 = c 2 = 1//2 * = (|i > + If >)//2 1 (84) * 2 = (II > " Ii >)/'/2 which corresponds t o case (b). When Y<2, then Y/2-1 i s ne g a t i v e , and c 2 becomes l a r g e r than c^. For Y<<2 we have c^ -* 0, c 2 -*• 1 , so t h a t *! * II > ^ 2 n | (85) $2 """Ii > 2 j I^ 60 T h i s corresponds to the i n v e r t e d case ( a ) , whose energy l e v e l s are i l l u s t r a t e d i n F i g . 3, Chap. V. The n e g a t i v e s i g n i n Eq. ( 8 5 ) i s the r e s u l t of our c h o i c e of phases f o r the wavefunction, and causes no d i f f i c u l t i e s . Y 2 i n a l l cases corresponds to the h i g h e r energy ( F 2 i n Eq. 80).. F i g . 7 i l l u s t r a t e s the behavior of the energy l e v e l s o f lowest J as a f u n c t i o n of Y. I t i s seen t h a t the J=i l e v e l f o l l o w s the m u l t i p l e t i n both r e g u l a r and i n v e r t e d s t a t e s , and t h a t the p a i r of l e v e l s of same J shows a mutual r e p u l s i o n which i n c r e a s e s w i t h |Y|. Looking at the s p e c i a l case when Y=0 or 4, then from Eq. ( 8 0 ) we have F, = J 2 + 2J - £ = (J + £)((J + £) +1) - 1 ( 8 6 ) F, = J * - § = (J - i ) ((J - i ) + D - 1 which suggests t h a t , i f we d e f i n e and N = J - £ ( F x l e v e l s ) N = J + £ (F~ l e v e l s ) ( 8 7 ) r then we may w r i t e , f o r both m u l t i p l e t s : F = N(N + 1) - 1 ( 8 8 ) T h i s i s the f a m i l i a r N p r e v i o u s l y d e f i n e d as the t o t a l angular momentum e x c l u d i n g s p i n . The e i g e n f u n c t i o n s d e f i n e d by Eq. (81) can be shown to be e i g e n f u n c t i o n s o f (5-S*) 2 when Y=0. F i g . 7 shows t h a t l e v e l s of equal N do indeed l i e c l o s e together i n the case (b) r e g i o n (small | Y | ) , and are indeed degenerate at Y=0 and 4. The e x c e p t i o n i s J=£ when Y>4, which f o l l o w s the F^ m u l t i p l e t energy scheme but not the F 1 r u l e t h a t N = J ~ i . I t must be concluded t h a t N i s p o o r l y d e f i n e d i n t h i s l e v e l f o r a l l Y>2. I t should be noted t h a t c^ and c 2 are f u n c t i o n s of J (Eq. 82) , and t h a t an i n c r e a s e i n J (and t h e r e f o r e X) has the same e f f e c t on them as a decrease i n Y. Thus case (b) i s approached wi t h i n c r e a s i n g J i n a l l cases, although t h i s e f f e c t i s more obvious when |Y| i s s m a l l e r . 2.5. M a t r i x Elements of Hzj. Example Eq. (81) i s a c t u a l l y a l l we need to c a l c u l a t e the exact m a t r i x elements of n from the pure case (a) forms z 1 g i v e n i n Chap.III. The d i a g o n a l m a t r i x element of the J= 2r. M=f, l e v e l , f o r example, i s now * *ilnzl*l > = ( c i < ?l + c 2 < zH^fcJi > + ' c 2 l z > } <V i|n |i > + <:||n || > x z z (89) 1 2 ^ 3 2 ' ? c l + ? c2 u s i n g Eqs. (19) and (20) . We depend here on the s e l e c t i o n r u l e AE=0, which i m p l i e s A£2=0 f o r e l e c t r i c d i p o l e 63 t r a n s i t i o n s . T h i s assures us, f o r example, t h a t < I |nz|£ > = 0 . (90) S i m i l a r l y , the o f f - d i a g o n a l m a t r i x element between t h i s same l e v e l and the J=|, M=§, F 2 l e v e l ( f o r b i d d e n i n pure case (a) i s g i v e n by < * 1|n 2|?[. 2 > = (c L< £|- + c 2< l | ) n z ( c 1 | | > - c 2|£ >) (91) = 2 C l c 2 / 5 . 3. The 3n S t a t e  3.1. Hamiltonian We take as a b a s i s s e t the wavefunctions c h a r a c t e r -i z e d as f o l l o w s L a b e l A S E u J M 12+ > +1 1 +1 +2 J M |1+ > +1 1 0 +1 J M |0+ > +1 1 -1 0 J M lo- > -1 1 +1 0 J M i i - > -1 1 0 -1 J M |2- > -1 1 -1 -2 J M Here |0 > i s a pure case (a) wavefunction, which may be r e f e r r e d to as a 3 n Q + s t a t e , e t c . The 6 x 6 matrix of the Hamiltonian (Appendix D) taken w i t h r e s p e c t to these f u n c t i o n s f a c t o r s i n t o two 3 x 3 b l o c k s w i t h zero matrix 64 elements between them. As i n the 2 n s t a t e , the two b l o c k s are i d e n t i c a l , producing A-doublets ( p a i r s of l e v e l s of o p p o s i t e p a r i t y which are degenerate to the approximation t h a t we have used the c o r r e c t H a m i l t o n i a n ) . We s h a l l c o n s i d e r o n l y the 3 x 3 matrix correspond-i n g to the s t a t e s | 0 + J >, | 1 + J >, and | 2 + J > and w i l l h e r e a f t e r omit the s u p e r s c r i p t s i g n . The r e s u l t i n g m atrix o f H/hcB i s g i v e n by 0 J > 0 J > - Y + x + l 1 J > 2 J > /2 x 11 J > - /2x x + 1 - /2x - 4 • j 2 J > 0 - /2x - 4 Y + x - 3 (92) where x E J(J+1) and terms i n H „ and L 2 - L 2 ev/hcB t, have been s u b t r a c t e d from the d i a g o n a l . 3.2. J=0 The J=0 energy l e v e l i s a s p e c i a l case s i n c e the |1 > and |2 > s t a t e s are not i n v o l v e d (Jsfi) . We t h e r e f o r e have (93) F Q J * F o o = B t J ( J + 1 ) - Y + 1] = B ( l - Y ) , and a pure case (a) wavefunction * o o = |0 0 > . (94) We m a i n t a i n the n o t a t i o n ^ _ f o r the " t r u e " ( i n t e r m e d i a t e case) wavefunctions, w h i l e c o n t i n u i n g t o use |QJ > f o r the pure case (a) wavefunctions. 3.3. J = l When J = l , we have |OJ > and |1J >, but s t i l l no |2J > i n our b a s i s s e t . Our m a t r i x to be d i a g o n a l i z e d i s now 0 1 > 1 1 > 10 1 > -y + x + l - /2x u 1 > - /2x x + 1 (95) which may be transformed to | 0 1 > 1 1 > 0 1 > 1 1 > Y/2 - 2 - 2 Y/2 (96) by s u b s t i t u t i n g x=2 and s u b t r a c t i n g 3 - Y/2 from the d i a g o n a l . T h i s i s of the same form as Eq. (43) p u t t i n g 6 = -Y/2 and V R L = -2. Adding (3 - Y/2) back to the 66 eigenvalues g i v e n by Eq. (45), we have F Q 1 = B(3 - Y/2 + £/Y 2 + 16) (97) F l l = B ( 3 - Y / 2 - £ / Y 2 + 1 6 ) ( 9 8 ) Here the h i g h e r e i g e n v a l u e has been assig n e d t o the |0 1 > s t a t e , corresponding to an i n v e r t e d 3n s t a t e (Y<0). For a r e g u l a r s t a t e (Y>0), the s i g n s b e f o r e the r a d i c a l must be r e v e r s e d . The wavefun c t i o n s t ifrom Eqs. (48) and (49); are ^01 = C 1 I 0 1 > - C 2 I 1 1 > * n = c 2|0 1 > + c j l 1 > ', ( 9 9 ) w i t h c± = /(X - Y)/2X (100) and c 2 = /(X + Y)/2X (101) In NH, where Y = 2.1344 ( V e s e t h 1 9 , p.235f), we have c1 = 0.857545 c 2 = 0.514410 . (102) These wavefunctions are f o r an i n v e r t e d s t a t e , corresponding 67 to the eigenvalues o f Eqs. (97) and (99), so t h a t > c 2 . For r e g u l a r s t a t e s , the "0" and "1" va l u e s o f the "n" l a b e l s must be everywhere interchanged; equations (101) and (102) are unchanged except t h a t c 2>c^. M a t r i x elements o f n are c a l c u l a t e d e x a c t l y as i n the 2n case z 1 (Eqs. 89-91) . 3.4. J=2 When J i s l a r g e r than one, the f u l l m a trix of Eq. (97) must be used. In these cases, i t was found convenient to c a l c u l a t e the e i g e n v e c t o r s and eig e n v a l u e s n u m e r i c a l l y , u s i n g a U.B.C. computer l i b r a r y program c a l l e d DYSMAL. With a va l u e of Y = -2.1344 a p p r o p r i a t e to the NH molecule, the Hamiltonian m a t r i x f o r J=2 (x=6) i s | 0 2 > 12 2 > 0 2 > 9.1344 3.46410 0 1 2 > -3.46410 7.00000 -2.82843 (103) 2 2 > 0 2.82843 0.8656 i T h i s y i e l d s e i g e n v a l u e s F Q 2 / B = 11.96309 F 1 2 / B = 5.46678 F 2 2 / B = -0.42987 (104) the l a s t o f which i s the ground l e v e l of the 3n s t a t e . The v a l u e s of e i g e n v e c t o r components from the computer 68 p r i n t o u t are the c o e f f i c i e n t s i n the expansion of the proper wavefunctions: ^ Q 2 = 0.764692l|0 2 > - 0.6244334|l 2 > + 0.1591504|2 2 > (105)  * 1 2 = 0.6268996|0 2 > + 0.6637233|l 2 > - 0.408005l|2 2 > i p 2 2 = 0.1491402|0 2 > + 0.4117697|1 2 > + 0.8990011J2 2 > 3.5. M a t r i x Element of n_ z j L Example As an example, we w i l l e v a l u a t e the matrix element of n between the J=l and J=2 l e v e l s of the 3n» z 1 m u l t i p l e t f o r M=0. L a b e l i n g the numerical c o e f f i c i e n t s i n the expansion of i | ) ^ 2 (above) as d^, d^, and d 2 f o r b r e v i t y , we have K- ^12 I nz I ^ 11 > 0 = ( d 0 < 0 2 l + d i < 1 2|+d2< 2 2|)nz x ( c 9 | 0 1 > + c,|1 1 >) X (106) = d Q c 2 < 0 2 j n z j 0 1 > + d 1 c 1 < 1 2|n z 11 1 > = d0c2/47T5 + d 1c 1/i75~" « 0.421 (M = 0) where the c's come from Eqs. (100)-(102), and the pure case (a) m a t r i x elements from Eqs. (22) and (23). S i m i l a r l y , the matri x element f o r M = 1 between the same two l e v e l s i s < ^ J n U n i >•• = d c^/175 + d,c,/3720 12' z' 11 i 0 2 i i (107) « 0.366. (M = 1) . F i n a l l y , we w i l l quote the r e s u l t s f o r NH i n the case of J=3, f o r which the Hamiltonian m a t r i x (Eq. 92) becomes | 0 3 > j1 3 > | 2 3 > 0 3 > 11 3 > |2 3 15.1344 •4. 8990 •4. 8990 13.0000 -4.4721 0 -4.4721 6.8656 (108) The eigenvalues are F Q 3 / B = 19.75088 F 1 3 / B = 11.26576 F 2 3 / B = 3.96254 , (109) and the i n t e r m e d i a t e case wavefunctions: (110) '03 '13 23 0 . 7 0 7 9 9 5 8 | 0 3 > - 0 . 6 6 7 1 7 4 9 | l 3 > + 0 . 2 3 1 5 5 9 l | 2 3 > 0 1 6 6 6 9 3 9 4 | 0 3 > + 0 . 5 2 3 8 3 2 0 | l 3 > - 0 . 5 2 9 8 9 8 0 | 2 3 > 0 . 2 3 2 2 3 6 6 | 0 3 > + 0 . 5 2 9 6 0 1 4 J 1 3 > + 0 . 8 1 5 8 3 6 l | 2 3 > 3.7. Hund's Case (b) Although we w i l l not demonstrate i t a l g e b r a i c a l l y , we can d e f i n e a quantum number N f o r small values of | Y | such t h a t l e v e l s of d i f f e r i n g J but equal N l i e c l o s e t o g e t h e r . F i r s t we must form the standard c l a s s i f i c a t i o n s 7 0 of l e v e l s i n t o F^, F 2 , and F^ groups: When there are t h r e e l e v e l s of equal J ( J > 1 ) then the l e v e l of h i g h e s t energy i s c a l l e d F^, the lowest l e v e l c a l l e d F.^, and the i n t e r m e d i a t e l e v e l F 2 . S p e c i a l d e f i n i t i o n s are needed when J < 2 . For r e g u l a r ( Y > 0 ) s t a t e s , the l e v e l with J = 0 i s c a l l e d F^ as i s the lower of the two l e v e l s w i t h J = l ; The upper J=l l e v e l i s F 2 . For i n v e r t e d s t a t e s ( Y < 0 ) , the l e v e l w i t h J = 0 i s c a l l e d F 3 as i s the J = l l e v e l of h igher energy; the J = l l e v e l of lower energy i s now c a l l e d F 2 > We are now able to d e f i n e N by N N N = J - l f o r F^ l e v e l s = J f o r F 2 l e v e l s = J+l f o r F-, l e v e l s . Our p r e v i o u s d i s c u s s i o n s were based on the assumption o f i n v e r t e d l e v e l s (as e x i s t i n NH). Thus i n Eqs ( 9 3 ) , ( 9 7 ) , ( 1 0 4 ) and ( 1 0 9 ) , the l e v e l s w i t h n= 0 ( F Q 0 , F Q 1 , F Q 2 , and a r e a H F 3 l e v e l s (with N=l, 2 , 3 , and 4 r e s p e c t i v e l y ) ; the l e v e l s with n=l are F 2 , and those For the i n v e r t e d 3IT s t a t e : fi=2 are F l ' F l ~ 3 N 2 F 2 ^ 3 n 1 F 3 ~ 3 n Q 71 For a r e g u l a r s t a t e , the correspondence becomes F 3 ~ H 2 , and the e i g e n v a l u e s must be r e d i s t r i b u t e d among the a v a i l a b l e v a l u e s of fi i n accordance w i t h the above correspondence and the d e f i n i t i o n s o f the p r e c e d i n g paragraph. To c l a r i f y these r e l a t i o n s h i p s as much as p o s s i b l e , F i g . 8 i l l u s t r a t e s the behavior o f the lowest l e v e l s as a f u n c t i o n o f Y. 4^ The *_A S t a t e We s h a l l encounter a 2 A s t a t e i n the CH molecule. I t i s to be t r e a t e d i n a way t h a t i s e n t i r e l y analogous to the h a n d l i n g of 2 I I s t a t e s . Our Hund's case (a) b a s i s s e t i s L a b e l A S i n J M | + f > +2 i + i + f J M | + f > +2 1 - £ + f J M | - f > "2 1 - § J M I-* > - 2 i +i dt - M The Hamiltonian matrix taken w i t h r e s p e c t to these elements does again f a c t o r i n t o two degenerate s e t s of l e v e l s (A-doublets) of o p p o s i t e p a r i t y , and o n l y the f i r s t p a i r of wavefunctions need be c o n s i d e r e d . With r e s p e c t to these, 72 the matrix of the Hamiltonian i s Iz J 13 J If J Y + x - 23/4 II J > - /x - 15/4 - /x - 15/4 - Y + x - 7/4 where x = J ( J +1). We may s o l v e f o r the eigenvalues as bef o r e and a s s i g n them as F 1 / B = ( J + l ) 2 - 4 - / I J + I T + Y ( Y-4) (111) F 2 / B = ( J + i ) - 4 - / ( J + * ) + Y ( Y - 4 ) (112) F J = £ / B = 2 - Y . Note the absence o f a f a c t o r of £ i n f r o n t of the r a d i c a l , and o f a 4 i n s i d e i t as compared to Eq. (80) and i n agreement with the formula o f H i l l and Van V l e c k 1 8 . The wavefunctions are g i v e n by Eq. (81), r e p l a c i n g |£ > by | z >, and 12 > by |f >: * ! = c ! ' l > + c 2 l z > *2 = C l ' ^ > " c 2 ^ > ' where c and c 2 are s t i l l g i v e n by Eq. (82) , i n t e r e s t i n g l y enough, i f X i s now d e f i n e d as X = / ( J + £ ) 2 + Y ( Y - 4 ) (113) I f Y=0 or 4, then i t i s again e a s i l y v e r i f i e d t h a t Eq. (112) may be w r i t t e n i n the form F 2/B N(N+1) - 4 , N E J - £ N(N+1) - 4 , N = J + £ (114) so t h a t F-^  and F 2 l e v e l s of equal N are n e a r l y degenerate when |Y| i s s m a l l , and we have Hund's case (b). The J = Z l e v e l changes from an F^ l e v e l f o r Y>2 (r e g u l a r state) t o an F 2 l e v e l when Y<2 ( i n v e r t e d state) as d i d the J=\ l e v e l i n the 2n case ( F i g . 7). The F^ l e v e l s form the 2A^ m u l t i p l e t and F 2 the 2A^ m u l t i p l e t i n r e g u l a r s t a t e s , and v i c e v e r s a i n i n v e r t e d s t a t e s . e n t i r e l y analogous to the case of the 2n s t a t e and w i l l not be giv e n here. The c a l c u l a t i o n o f mat r i x elements o f n i s z 74 CHAPTER VI APPARATUS 1. E a r l y Work LoSurdo' • f i r s t used the cathode f a l l r e g i o n of a glow d i s c h a r g e to generate Stark e f f e c t s i n atomic s p e c t r a . ...The;'ph.ysi.cs_.of the .glow discharge-and., r e l a t e d l i t e r -ature. 2 L"1^- are..reviewed !in;App.;C I t may seem a b i t un-f a i r , s i n c e the e f f e c t s of e l e c t r i c f i e l d s upon the spec-trum of hydrogen were d i s c o v e r e d independently by LoSurdo 1 + 5 and Stark. 1* 6 ' , t h a t Stark was g i v e the e f f e c t and LoSurdo the tube. E x t e n s i v e work has been done u s i n g t h i s method of g e n e r a t i n g h i g h f i e l d s 7 / ** 7 - 6 3 r e p o r t i n g f i e l d s as high as 1.2 MV/cmlt7. Most o f t h i s work has been on atomic s p e c t r a . Although a number of other elements have been s t u d i e d 5 7 - 6 3 , most a t t e n t i o n has gone to atomic hydrogen and h e l i u m 7 7 - 5 5 , f o r which t h e o r e t i c a l treatments were a v a i l a b l e ! 0 , 6 t* ' 6 5 Stark e f f e c t s were a l s o observed and p a r t i a l l y analysed i n m o l e c u l a r h y d r o g e n 5 6 ' 6 6 - 6 8 , but i n no other molecule u n t i l the work of Phelps and D a l b y 8 i n t h i s l a b o r a t o r y . The l a r g e s t body of p u b l i s h e d work i s t h a t o f I s h i d a and c o - w o r k e r s ' 1 7 - 5 3 ' 5 7 - 6 0 ' 6 2 d u r i n g the p e r i o d 1928-1940. T h i s group seemed able to more or l e s s r o u t i n e l y produce f i e l d s i n excess of 550 kV/cm. The o reasons why they have been able to achieve f i e l d s higher than those produced i n t h i s l a b o r a t o r y are not c l e a r from examination of t h e i r p u b l i s h e d m a t e r i a l . In d i f f e r e n t papers they a t t r i b u t e t h e i r success to such d i v e r s e f a c t o r s as the use o f b e r y l l i u m 5 9 or t a n t a l u m 5 8 cathodes, a d i s -charge v e s s e l of l a r g e volume) 7 a good f i t between cathode wire and surrounding q u a r t z 6 , 0 and the removal o r a d j u s t -ment of the "vanes" of t h e i r synchronous motor r e c t i f i e r ' * 7 ' ' 1 and removal of the inductance 1* 7 (apparently i n t r o d u c i n g s t r o n g ac components to the supply v o l t a g e ) . S c a r c e l y any one of these f a c t o r s has been mentioned more than once, however, making i t seem u n l i k e l y t h a t they were c r u c i a l . I l l u s t r a t i o n s 6 0 / 6 1 make i t appear t h a t the quartz" e n c l o s i n g the cathode wires was s p e c i a l l y formed as p a r t of a l a r g e segment of the d i s c h a r g e tube, a procedure which would have been most awkward i n our experiments which used the order o f 10 3 cathodes! The other method used e x t e n s i v e l y to achieve f i e l d s o f comparable s t r e n g t h i s the "canal r a y " approach. A c a n a l ray d i s c h a r g e i s an o b s t r u c t e d d i s c h a r g e (where the anode i s moved i n s i d e the cathode dark space) with h i g h v o l t a g e (30-60 kV) and low pressure (10~ 2 mm Hg). (See F r a n c i s 2 6 , p . l 6 1 f . ) The p a r t i c l e beam can be e x t r a c t e d through an a p e r t u r e i n the cathode and passed between charged p a r a l l e l p l a t e s to produce the f i e l d (see the i l l u s t r a t i o n and r e f e r e n c e s i n Bethe and S a l p e t e r 6 9 , p.234). 2. Mechanism The d i s t i n g u i s h i n g f e a t u r e of the LoSurdo d i s c h a r g e i s a c i r c u l a r cathode area, u s u a l l y the end of a wire whose s i d e s are i n s u l a t e d by a quartz c a p i l l a r y tube, the sm a l l s i z e of which serves t o i n c r e a s e g r e a t l y the magnitude o f the maximum e l e c t r i c f i e l d i n the cathode f a l l . V a r i a t i o n s are t o be found i n the l i t e r a t u r e to meet the requirements of s p e c i a l m a t e r i a l s , such as I s h i d a and F u k i s h i m a 1 s 6 2 c a n a l ray to v a p o r i s e l i t h i u m , and L i d e n ' s 6 3 metal s h i e l d f o r use i n s u l f u r h e x a f l o u r i d e . However, the p e r s i s t e n t c h a r a c t e r i s t i c of LoSurdo d i s c h a r g e o p e r a t i o n i s t h a t the s m a l l e r the cathode diameter, the l a r g e r i s the e l e c t r i c f i e l d o b t a i n e d f o r a g i v e n m a t e r i a l , v o l t a g e , and p r e s s u r e . The f i r s t r e s u l t o f r e d u c i n g the cathode s i z e must be to i n c r e a s e the c u r r e n t d e n s i t y . To be sure, i t i s observed i n our work t h a t s m a l l e r cathodes draw l e s s c u r r e n t , but as a rough impression ( f o r example), a two m i l l i m e t e r cathode draws about twice as much c u r r e n t as a one m i l l i m e t e r cathode. Since the area has changed f o u r f o l d , the c u r r e n t d e n s i t y i s about twice as l a r g e i n the one m i l l i m e t e r v e r s i o n . I t should be noted t h a t the s i z e o f the cathode does not ent e r d i r e c t l y i n t o any of the c a l c u l a t i o n s mentioned i n s e c t i o n 1 (Ward 3 0, W a r r e n 2 4 , and V o l k o v 3 7 ) , but o n l y through the r e l a t i o n between c u r r e n t and c u r r e n t d e n s i t y . A l l of these c a l c u l a t i o n s , as w e l l as Wa r r e n ' s 2 8 experimental r e s u l t s , c o n f i r m t h a t the f i e l d i s p r o p o r t i o n a l t o 77 f o r most gases, although c o n d i t i o n s i n high v o l t a g e LoSurdo d i s c h a r g e s are s u f f i c i e n t l y d i f f e r e n t as to make comparison suspect. T h i s c o n c l u s i o n can a l s o be r e l a t e d to Eq. (C.9) or ( C l } or other e a r l y c a l c u l a t i o n s ( C o b i n e 2 2 , pp. 223ff, Fig.8.8) s i m i l a r t o Warren's which show the l e n g t h o f the cathode f a l l to be a de c r e a s i n g f u n c t i o n o f c u r r e n t d e n s i t y . Eq.(C.15) then shows (assuming a l i n e a r f i e l d ) t h a t the maximum f i e l d i n c r e a s e s with J . On a more i n t u i t i v e l e v e l , i t can be argued t h a t an i n c r e a s e i n e i t h e r f a c t o r of J=pv must be connected w i t h an i n c r e a s e i n E (see Eqs.C. 17 and C.18 ) . Since Kamke1*2 found t h a t the width at h a l f c u r r e n t , on cathodes i n abnormal glow d i s c h a r g e s , was down around 1 mm a t 20,000 v o l t s , one might expect t h a t i n c r e a s i n g the cathode diameter p a s t 2 mm should make l e s s d i f f e r e n c e i n the f i e l d s o b t a i n e d a t hig h v o l t a g e s . 3. The F l u s h Cathode and Older Tube Design The o r i g i n a l form o f the d i s c h a r g e v e s s e l , as used by P h e l p s 1 ; \ Phelps and D a l b y 8 ' 7 0 , I r w i n 7 1 , Irwin and D a l b y 7 2 , and Wong 7 3 i s shown i n F i g . 9.a . The cathode was an aluminum wire p o l i s h e d f l a t and matched evenly w i t h the ground end of the quartz t u b i n g to which i t i s f i t t e d . T h i s design was used i n c e r t a i n p o r t i o n s o f t h i s work and w i l l be r e f e r r e d to as the f l u s h cathode. Wires o f 0.5, 1.0, and 2.0 mm were used, corresponding t o the bore s i z e s o f quartz t u b i n g which are r e a d i l y a v a i l a b l e . The cathodes were J * -25-cm (b)quartz capillary stopper-^ TfJ^  ground glass joints d * deKhotinsky cement i l F i g . 9 The Discharge Tube CO 79 u s u a l l y mounted i n a h o l e through a rubber stopper ( s i z e 6 £ ) . T h i s was e s s e n t i a l l y the form o f tube used f o r most of our e a r l y work w i t h OH and OD. T h i s tube was pyrex, about an i n c h i n diameter and 10 inches l o n g . The window was s e t on an e x t e n s i o n o f the tube to minimize s p u t t e r i n g o r o t h e r contamination of the o p t i c a l s u r f a c e by the d i s c h a r g e . The anode was aluminum, about £ i n c h i n i t s l a r g e s t diameter, and clamped by a s e t screw to a tungsten wire s e a l e d i n t o the pyrex a t the time o f manufacture. The gases o r vapors were u s u a l l y pumped i n the i n d i c a t e d d i r e c t i o n , attempting to minimize contamination. For a p o r t i o n o f the experiments w i t h OH, the connecting glassware allowed t h i s flow to be r e v e r s e d . No e f f e c t o f any k i n d was noted due to the flow d i r e c t i o n , which i s not s u r p r i s i n g c o n s i d e r i n g the s m a l l flow r a t e s used i n most experiments (1 to 50 cm 3/hr a t standard temperature and p r e s s u r e ) . 4. The Sunken Cathode Fig.9b" shows the p r i n c i p a l i n n o v a t i o n i n cathode d e s i g n , due to F.W. Dalby, which d i f f e r e n t i a t e s the p r e v i o u s l y mentioned experiments from the l a t e r ones of Thomson and Dalby 7**' 7' 5 \ F i s h e r 7 6 , S c a r l and D a l b y 7 7 , and t h i s work. The quartz c a p i l l a r y tube i s simply extended above the s u r f a c e of the cathode, thus conforming more c l o s e l y w i t h the e a r l i e r Japanese d e s i g n s 6 0 ' 6 1 . T h i s change allowed f a r higher v o l t a g e s t o be a p p l i e d to the tube (over 20 k i l o v o l t s , as 80 opposed to around 7 k i l o v o l t s ) , as may be r e a d i l y understood from the d i s c u s s i o n o f Appendix"C, Sec. 8: The presence of c o n s t r i c t i n g w a l l s allow a f a r more e f f e c t i v e f o c u s i n g > of i o n i c c u r r e n t , through p o s i t i v e charge accumulated on the w a l l s over most of the cathode dark space. T h i s not o n l y r e s u l t s i n a h i g h e r c u r r e n t d e n s i t y a t cathode c e n t e r and thus a h i g h e r e l e c t r i c f i e l d per a p p l i e d v o l t , but keeps the d i s c h a r g e away from the cathode edges where a r c i n g i s f a r more probable. T h i s type of cathode w i l l be r e f e r r e d t o as the sunken cathode. I t s o n l y c l e a r disadvantage i s l o s s o f v i s i b i l i t y due to s p u t t e r i n g , chemical accumulation, o r other o b s c u r a t i o n o f the i n n e r s u r f a c e near the cathode. For many m a t e r i a l s , these d i f f i c u l t i e s need not be s e r i o u s , however. The l e n g t h of the c a p i l l a r y above the cathode s u r f a c e d i d not appear to be v e r y important i n t h i s work (perhaps e x c e p t i n g t h a t w i t h methanol), but i t s upper p o r t i o n s ( p o s i t i v e column) would heat to an orange glow under c o n d i t i o n s o f h i g h c u r r e n t i f the c a p i l l a r y were s u f f i c i e n t l y l o n g . There was i n c o n c l u s i v e evidence f o r b e t t e r s t a b i l i t y o f a 1 mm cathode i n cyanogen i f the c a p i l l a r y was 2.2 to 2.8 mm above i t s s u r f a c e . F i s h e r 7 6 found t h a t the CO spectrum generated i n C O 2 was b r i g h t e r , the s h o r t e r the c a p i l l a r y e x t e n s i o n . In g e n e r a l , the optimum d i s t a n c e above the cathode seems to be two or t h r e e m i l l i -meters, o r roughly the l e n g t h of the cathode dark space p l u s the n e g a t i v e glow and p a r t o f the Faraday dark space. 81 5• D e t a i l s o f Tube Assembly  5-rl. Ground G l a s s v s . Rubber, and Other F a i l u r e s A l s o shown i n Fig.9B i s the ground g l a s s j o i n t and cathode h o l d e r i n t r o d u c e d by R. Thomson to p r o v i d e more c o n s i s t e n t cathode p o s i t i o n i n g . The narrowness of the a p e r t u r e , s e n s i t i v i t y o f the j o i n t l u b r i c a n t to heat, and g e n e r a l i n f l e x i b i l i t y o f t h i s d e s i g n l e d to i t s abandon-ment by the author i n f a v o r of a r e t u r n to the rubber stopper. Crude as i t may seem, the use of rubber stoppers gave q u i t e dependable vacuums to the l i m i t o f our pumping range _2 (3 X 10 mm Hg), w i t h no need f o r the e x t r a s e a l e r mentioned by P h e l p s 1 4 ( p . 3 6 ) . P r i o r t o the development of the s t a n d a r d i z e d p o l i s h i n g and r u n n i n g - i n procedures d e s c r i b e d below ("Sees. 7.2. , 7.3.), a number of v a r i a t i o n s i n cathode d e s i g n were not c o n s i d e r e d s u c c e s s f u l , but are herewith d e s c r i b e d f o r completeness: On the theory t h a t cathode f a i l u r e was caused by improper f i t between the cathode and i t s e n c l o s i n g c a p i l l a r y , a number o f v a r i a t i o n s were t r i e d which c o n s t r i c t e d the c a p i l l a r y , a t the, l e v e l o f the cathode s u r f a c e , forming a b a r r i e r r i m a g a i n s t which the cathode was t i g h t l y wedged. T h i s made no p a r t i c u l a r improvement, and i t i s now b e l i e v e d t h a t t h i s f i t i s o f s m a l l importance ( c o n t r a r y to the a s s e r t i o n of I s h i d a and H i y a m a 6 0 ) * On the theory t h a t o v e r h e a t i n g was a cause of f a i l u r e , some designs were t r i e d which i n c o r p o r a t e d an u l t r a - l o n g c a p i l l a r y e x t e n s i o n above the cathode s u r f a c e , w i t h the vacuum s e a l being a t the top i n s t e a d o f the bottom. With the e s s e n t i a l p a r t of the c a p i l l a r y hung i n open a i r , a i r and water c o o l i n g were t r i e d . The added c o o l i n g d i d not n o t i c e a b l y improve performance, and the long c a p i l l a r y seemed d e t r i m e n t a l (Sec.4., above). When a water j a c k e t was added, furthermore, the c r i t i c a l r e g i o n c o u l d not be c l e a r e d o f bubbles generated on the q u a r t z . T i n f o i l wrapped about the c a p i l l a r y , to r e f l e c t l i g h t l e a v i n g i n o t h e r d i r e c t i o n s , s e r i o u s l y i n t e r f e r e d w i t h the d i s c h a r g e . A hole d r i l l e d i n the s i d e o f the c a p i l l a r y a t cathode l e v e l (to prevent o b s c u r a t i o n by s p u t t e r i n g and other d e p o s i t s ) i n t e r f e r e d too much wit h the d i s c h a r g e symmetry, as compared wi t h the much more d i s t a n t s h i e l d used l a t e r (ivSec, 5.-4. ") . I n t e r e s t i n g l y , the top o f the h o l e became white-hot a t times, i n d i c a t i n g e l e c t r o n r a t h e r than i o n bombardment. T h i n k i n g t h a t the presence o f an a l t e r n a t e e l e c t r i c a l path was r e s p o n s i b l e f o r the problem, the c a p i l l a r y top was bent outward and j o i n e d to a l a r g e r s e c t i o n which then e n c l o s e d the cathode r e g i o n completely. T h i s a l s o f a i l e d to normalize the d i s c h a r g e . From what i s w r i t t e n above about the r o l e of the w a l l s i n f o c u s i n g the d i s c h a r g e , t h i s i s not s u r p r i s i n g . 83 5.2. The Cleanable Tube The subsequent d e s i g n i s shown i n F i g . 9c. A l l work here r e p o r t e d was done wi t h t h i s tube except t h a t w i t h OH and OD. I t i s of c o n s i d e r a b l y l a r g e r volume than the o l d e r model, w i t h a l e n g t h o f 13 inches and diameter of n e a r l y two inches (45 mm I.D.). I t was designed f o r ready c l e a n i n g , f i r s t and foremost: Both cathode and anode are r e a d i l y withdrawn and r e p l a c e d and the tube as a whole may be e a s i l y removed (ground g l a s s j o i n t s ) and disassembled (de Khotinsky)cement) f o r c l e a n i n g . 5.3. The Window The window was made l a r g e r to m a i n t a i n the same s o l i d angle of a p e r t u r e as i n the o l d e r v e r s i o n . I t was mounted on an e x t e n s i o n of o r d i n a r y t u b i n g , which i s cemented wit h deKhotinsky as shown. T h i s was intended to a l l o w f l e x i b i l i t y i n window placement and to a l l o w the use of a window h e l d w i t h ceramic cement to withstand h e a t i n g (intended f o r use i n e v a p o r a t i n g a l k a l i metals from the window i n an experiment which has not been performed). In p r a c t i c e , t h i s arrangement makes i t much e a s i e r t o change windows without removing the whole tube, s i n c e they may be mounted on the e x t e n s i o n w h i l e h o r i z o n t a l and then the assembly mounted w i t h the c o o l e d window as a handle. No t r o u b l e was encountered from m e l t i n g o f the deKhotinsky wax except when microwave e x c i t a t i o n was a p p l i e d i n the r e g i o n marked X i n F i g . 9c ( t h i s was p r e v e n t a b l e by c o o l i n g w i t h an a i r f a n ) . 84 The windows were u s u a l l y quartz 2.5 cm (O.D.) diameter and about 12.5 cm from the cathode. 5.4. The S h i e l d The s t r a i g h t c y l i n d r i c a l tube and rubber stopper a l l o w the use o f an i n n e r g l a s s s h i e l d , as shown, to minimize the frequency o f c l e a n i n g the main tube. T h i s s h i e l d simply s i t s on the rubber stopper, and i s used r o u t i n e l y , even w i t h sunken cathodes and c l e a n - r u n n i n g m a t e r i a l s , s i n c e i t s use o f f e r s no problem. The a p e r t u r e i n the s h i e l d , between cathode and window, was e v e n t u a l l y reduced to about 5 mm wide by 11 mm h i g h . With cyanogen d i s c h a r g e s , heavy orange and brown d e p o s i t s were formed w i t h each run. These seemed r e a d i l y removable w i t h a weak s o l u t i o n o f HF, but a p e r s i s t e n t d i s c o l o u r a t i o n was b u i l t up over many runs. The departmental g l a s s b l o w e r s were able to remove t h i s f i l m w i t h concentrated HF. 5.5. The Anode The s i z e and shape o f the anode are u s u a l l y c o n s i d e r e d to make ve r y l i t t l e d i f f e r e n c e t o the d i s c h a r g e 1 ! 2 However, i t has been found t h a t u s i n g a l a r g e anode, which f i t s the i n n e r pyrex s h i e l d q u i t e c l o s e l y , serves to c o n f i n e p o s i t i v e column anode glow a c t i v i t y t o w i t h i n tube and prevent i t from going up and p a s t the s i d e s of the anode. T h i s i s a d i s t i n c t advantage i n gases l i k e cyanogen where b r i g h t areas 85 of the d i s c h a r g e cause a severe c l e a n i n g problem. The anode i s cleaned as needed by e i t h e r t u r n i n g on a l a t h e or f i n e sanding, depending on the amount of the d e p o s i t . In d i r t y gases, the anode w i t h i t s matched g l a s s s h i e l d were u s u a l l y changed along w i t h the cathode a f t e r each run. 6. D e t a i l s of Other Apparatus Fig.10 g i v e s a schematic diagram o f the p h y s i c a l l a y o u t of the vacuum system. The vacuum, e l e c t r i c a l , and o p t i c a l systems are d i s c u s s e d i n some d e t a i l i n the s u b s e c t i o n s below. Some p a r t s of these d i s c u s s i o n s w i l l be of i n t e r e s t o n l y t o persons w i s h i n g to use the same apparatus themselves. 6.1. Sample Input C o n t r o l Adequate c o n t r o l o f sample i n p u t flow was u s u a l l y o b t a i n e d w i t h the Edward's High. Vacuum, L t d . , type LB2B needle v a l v e s . T h i s v a l v e has a brass bellows a c t i v a t e d by screw th r e a d d i a l through a l e v e r system with enough mechanical advantage to compensate f o r the r a t h e r wide i n c l u d e d angle of the needle i t s e l f (around 25°) . While the needle i s s t a i n l e s s s t e e l , the seat i s brass and had some tendency to g a l l and s e i z e shut i f a d j u s t e d to c l o s e under the f u l l p r e s s u r e of i t s r e t e n t i o n s p r i n g , p r o v i d i n g at l e a s t one dangerous moment when used w i t h s i l a n e . The v a l v e should t h e r e f o r e be a d j u s t e d with hydrogen or helium (which have high flow r a t e s ) to j u s t s l i g h t l y p a s t the p o i n t where no p r e s s u r e can be d e t e c t e d i n the system 3 6 Reservoir (ISO ml) discharge tube .^McLeod gauge air bleed 5 - shunt line((not used during runs) rT^D^ *—Input *-trap' or f i l t e r constricting waive rotary pump exhaust line Glass stopcock f£s[ Nupro needle valve Edwards iieedle valve Matheson miniature valve Nupro check valve(pressure activated) • Ground glass joint F i g . 1 0 P h y s i c a l Schematic of Vacuum System 87 under f u l l pumping. The Edwards v a l v e s have been used s a t i s f a c t o r i l y w i t h n o n - c o r r o s i v e gases (He, H 2, N 2, Ar, 0 2 , C0 2) i n c y l i n d e r s w i t h e x t e r n a l p r e s s u r e r e g u l a t o r s . They are a l s o used w i t h n o n - c o r r o s i v e l i q u i d s (H 20, CH3OH) with vapor p r e s s u r e s l e s s than one atmosphere, i n g l a s s c o n t a i n e r s d i r e c t l y con-nected to. a v a l v e ' s " C o r i n g e n t r y p o r t . I t was assumed t h a t the melange o f m a t e r i a l i n the Edward's v a l v e s was i n a p p r o p r i a t e t o c o r r o s i v e substances (NH^, C 2 N 2 , BCl-j) , so one i n p u t l i n e has been f i t t e d w i t h v a l v e s from Nupro (Crawford I n d u s t r i e s ) which are made of e i t h e r s t a i n l e s s s t e e l o r monel, with t e f l o n o r "Viton-A" gaskets and v a l v e s e a t s . In s p i t e of a much sm a l l e r angle o f i n c l u s i o n (1° or 3°) o f the needle, the d i r e c t c onnection o f these v a l v e needles to the o u t s i d e d i a l meant t h a t c o n t r o l was l e s s s a t i s f a c t o r y than with the Edwards v a l v e s . I t must be added, however, t h a t flow r a t e s r e q u i r e d o f these c o r r o s i v e substances were g e n e r a l l y lower (1 cm 3 per hour at room temperature and roughly a i r p r e s s u r e ) . N e v e r t h e l e s s i t was found necessary t o m a i n t a i n a s i z a b l e r e s e r v o i r o f gas at l e s s than atmospheric p r e s s u r e , behind an " M " (metering v a l v e , SS-4M) v a l v e i n s e r i e s w i t h an "S" ( f i n e metering M-2SA) v a l v e , t o get usable c o n t r o l . A f u r t h e r disadvantage o f these v a l v e s i n t h a t they cannot be used f o r o f f - o n c o n t r o l (as can the Edward's v a l v e s ) , and need other v a l v e s i n s e r i e s with them f o r s h u t o f f . A sequence of volumes separated by s h u t o f f v a l v e s 88 was used i n p l a c e of a r e g u l a t o r f o r cyanogen. (Regulators were used with s i l a n e and ammonia.) D i a l gauges capable o f re a d i n g both p r e s s u r e and vacuum were connected to the l a r g e volume preceding the needle v a l v e s , and the s m a l l e r volume preceding i t . A l l components i n t h i s r e g i o n were of s t a i n l e s s s t e e l o r monel, and connected with Swagelok (Crawford) or N a t i o n a l Pipe Thread f i t t i n g s . The s h u t o f f v a l v e s were monel o r s t a i n l e s s s t e e l m i n i a t u r e forged body v a l v e s from Matheson; some d i f f i c u l t y was had w i t h these v a l v e s with ammonia, a p p a r e n t l y due to a hard polymer formed with a l u b r i c a n t (most of these were usable a g a i n a f t e r thorough c l e a n i n g ) . F u r t h e r d e t a i l s are g i v e n below under the v a r i o u s chemical headings. 6.2. Pumping Rate Reduction and Pressure Measurement A r e d u c t i o n i n pumping flow r a t e was sometimes used to reduce consumption of the hazardous samples, or samples a v a i l a b l e i n s m a l l amounts. A t f i r s t the rubber hose t o the vacuum pump was clamped o f f with a l a r g e "C" clamp and a wire i n s i d e (to prevent s e a l i n g s h u t ) . L a t e r a needle v a l v e (Nupro SS-4MA) was p l a c e d between the vacuum gauge and c o l d t r a p s , w i t h s h u t o f f ( s e r i e s ) and bypass ( p a r a l l e l ) stopcocks. T h i s v a l v e c o u l d then be p r e s e t to a d e s i r e d c o n s t r i c t i o n , and then put i n or out of the system w i t h bypasses. The l i q u i d n i t r o g e n t r a p i s an e f f e c t i v e pump f o r substances of h i g h e r b o i l i n g p o i n t than n i t r o g e n (meaning 89 a l l gases used except hydrogen and helium). Thus having a c o n s t r i c t i o n between the c o l d t r a p and vacuum pump was u n s a t i s f a c t o r y , and i t had to be p l a c e d upstream of the c o l d t r a p s t o be e f f e c t i v e f o r a l l substances. Pressure was measured w i t h a McLeod mercury gauge which c o u l d be read to 0.01 mm Hg, and estimated t o 0.002 mm Hg p r e s s u r e . T h i s i s p l a c e d immediately downstream from the d i s c h a r g e tube and was moved t o hang below the l i n e t o which i t i s attached so as to prevent contamination o f the system from mercury s p i l l s . The McLeod gauge operates by compressing the gas, and thus g i v e s i n a c c u r a t e l y low readings f o r substances which as the vapors of l i q u i d s a t room temper-a t u r e (water, methanol, d i m e t h y l mercury). 6 . 3 . C o l d Traps and Pump A l i q u i d n i t r o g e n c o l d t r a p was o f t e n used t o keep samples from f o u l i n g the vacuum pump o i l o r p o l l u t i n g the o u t s i d e a i r . I t was a l s o used to keep mercury from the McLeod gauge out o f the pump, but was e x p r e s s l y not used w i t h s i l a n e t o prevent dangerous accumulations o f the substance i n glassware. The t r a p was loaded with g l a s s beads and NaOH p e l l e t s t o c h e m i c a l l y t r a p cyanogen. The vacuum pump was a smal l Sargen-Welch r o t a r y o i l pump which was capable o f ev a c u a t i n g the system t o the l i m i t s o f the McLeod gauge. The exhaust of the pump was l e d out of doors by some f o r t y f i v e f e e t o f aluminum t u b i n g and s u r g i c a l rubber hose (except d u r i n g the work with OH 90 and OD, where v e n t i n g to room a i r was a c c e p t a b l e ) . When cyanogen was used, the pump exhaust was bubbled through a stro n g s o l u t i o n of NaOH and b l e a c h i n g powder ( a l s o known as c h l o r i n a t e d lime o r c a l c i u m c h l o r i d e h y p o c h l o r i t e ) b e f o r e p a s s i n g outdoors. When s i l a n e was used, the exhaust was bub-b l e d through an outdoors water c o n t a i n e r to p r o v i d e c o n t r o l o f i t s spontaneous combustion. Pressure w i t h i n the system (greater than atmospheric) i s p o t e n t i a l l y dangerous and c o u l d a r i s e i n a d v e r t a n t l y a t u n p r e d i c t a b l e times from a needle v a l v e l e f t open, o r from the m e l t i n g o f l i q u i d oxygen o r other gases caught i n a s e a l e d - o f f c o l d t r a p . To prevent t h i s , a check v a l v e was i n s t a l l e d as shown i n Fi g . 1 0 , upstream from the stopcocks which u s u a l l y c ut o f f pumping, and near the c o l d t r a p . I t i s made o f monel and opens when the pre s s u r e i n the system reaches ^ l b / i n 2 above atmospheric (pump exhaust) p r e s s u r e . I t forms an e f f e c t i v e vacuum s e a l when c l o s e d , j u d ging by McLeod gauge r e a d i n g s . 6.4. The Power Supply The supply used almost e x c l u s i v e l y was made up from what appeared to be heavy-duty war s u r p l u s components. F o r the convenience those who may need i t o r some s i m i l a r d e s i g n , i t i s d e s c r i b e d here b r i e f l y , and diagrammed s c h e m a t i c a l l y i n Fig.! 11. Th i s power supply i s e s s e n t i a l l y a hi g h v o l t a g e transformer c o n t r o l l e d by a v a r i a c i n i t s primary c i r c u i t , and '.I'Af.K H / -VD F i g . 11 Power Supply 92 r e c t i f i e d by a p a i r of diodes arranged w i t h two l a r g e c a p a c i t o r s to form a v o l t a g e doubler: A l t e r n a t e h a l f - c y c l e s o f the AC i n p u t charge a l t e r n a t e c a p a c i t o r s , and the output v o l t a g e i s taken across them as the sum. Between the v a r i a c and high v o l t a g e transformer primary, an o v e r l o a d cutout c i r c u i t has been designed. The s o l e n o i d o f the KRP 14AG r e l a y i s i n s e r i e s w i t h the t r a n s f o r m e r primary, but t h i s c u r r e n t i s p a r t i a l l y shunted pas t the r e l a y by the r e s i s t o r network (top c e n t e r i n Fig.11) which a d j u s t s the t r i p p i n g c u r r e n t from l e s s than 1 mA to over 30 mA. When t h i s r e l a y , i s t r i p p e d by s u f f i c i e n t c u r r e n t i n the primary c i r c u i t , i t i s h e l d c l o s e d by the c u r r e n t f l o w i n g through the 750 ohm r e s i s t o r . To g i v e s u f f i c i e n t time delay f o r t h i s r e s i s t o r to e s t a b l i s h a c u r r e n t path through i t s a s s o c i a t e d r e l a y c o n t a c t s , the primary c i r c u i t o f the power supply i s not i n t e r r u p t e d by the KRP 14AG r e l a y i t s e l f , but by another r e l a y operated by a t h i r d r e l a y which i s operated by the KRP 14AG. The 60 kQ s e r i e s r e s i s t o r i n the h i g h v o l t a g e c i r c u i t i s not c r i t i c a l i n v a l u e , but must be present to l i m i t current; or the cathode i s l i k e l y to a r c immediately. The o v e r l o a d r e l a y c o n t r o l s the power a p p l i e d to a 110 VAC o u t l e t on the s i d e o f the power supply, i n to which the spectrograph s h u t t e r and a t i m i n g c l o c k are u s u a l l y plugged, so t h a t the c l o c k stops and s h u t t e r c l o s e s when o v e r l o a d c o n d i t i o n s occur. Another o u t l e t i s i n s e r i e s w i t h the primary c i r c u i t , s e r v i n g as a p o i n t where an 93 e x t e r n a l timer may break the c i r c u i t through a r e l a y , as shown i n F i g . H . Another power supply was purchased (Peschel RPS 50-5R) w i t h v o l t a g e r e g u l a t i o n , 50 kV c a p a c i t y , and o v e r l o a d t r i p , but was not o f t e n used due to problems i n the r e g u l a t i o n c i r c u i t s and the r e l a t i v e i n f l e x i b i l i t y of i t s c u r r e n t o v e r t r i p . Higher a p p l i e d v o l t a g e s were sometimes o b t a i n e d by e l e v a t i n g the anode p o t e n t i a l w i t h a second power supply. Voltages i n excess of 5 o r 6 kV are dangerous s i n c e the plasma c o u l d conduct the v o l t a g e to a l l p a r t s of the system, i n c l u d i n g t a b l e and pump. (Negative v o l t a g e s a p p l i e d to the cathode can not do t h i s s i n c e they are c o n f i n e d to the cathode r e g i o n . Most of the plasma i s n e a r l y a t anode p o t e n t i a l . ) 6.5. The Spectrograph The u s e f u l p o r t i o n of t h i s work was done e n t i r e l y w i t h a 3.4 meter E b e r t spectrograph ( J a r r e l - A s h 70-000) using a g r a t i n g r u l e d with 30,000 l i n e s per i n c h and o b l a z e d a t an angle corresponding t o 10,000 A. T h i s type of spectrograph has a s i n g l e m i r r o r (masked i n t o two segments) at one end, i n c l i n e d at a s l i g h t angle. At the o t h e r end, corresponding to the f o c a l l e n g t h of the m i r r o r , i s s e t the entrance s l i t and photographic p l a t e s . The g r a t i n g i s mounted near the c e n t e r , and r o t a t e d to s e t the s p e c t r a l range observed. Thus/ l i g h t from the s l i t i s 94 c o l l i m a t e d and r e f l e c t e d toward the g r a t i n g ; the d e s i r e d segment o f the spectrum (dispersed) i s r e f l e c t e d back to the m i r r o r which focuses i t upon the p l a t e s . 6.6. P l a t e Handling i n the Spectrograph o The p l a t e s c o n t a i n a range o f about 1000 A o i n f i r s t order (500 A i n second o r d e r , etc.) over both p l a t e s (50 cm), d e c r e a s i n g s l i g h t l y at the l o n g e r wavelength o s e t t i n g s . Thus we get d i s p e r s i o n s of roughly 2 A/mm o i n f i r s t o r d e r , 1 A/mm i n second o r d e r , e t c . The c e n t e r o o of t h i s range can be s e t from 6000 A to 14 000 A by r o t a t i n g the g r a t i n g , but there are s i g n i f i c a n t i n t e n s i t y o l o s s e s when s e t f a r from 10 000 A. As the p l a t e s s i t i n the h o l d e r , h i g h f r e q u e n c i e s tend toward the h o l d e r handle, and the image i s i n v e r t e d . Most p l a t e s numbered 150 o r g r e a t e r are coded by a s c r a t c h i n the corner o f the emulsion a t the low wavelength (high frequency) end and the upper edge (as mounted, thus towards the bottom of the d i s c h a r g e ) . These s c r a t c h e s are made with a p e n - k n i f e a t the time of placement i n the p l a t e h o l d e r . P l a t e s are numbered i n d i v i d u a l l y , although used i n p a i r s . Thus, p l a t e s 47 and 48 were exposed s i m u l t a n -e o u s l y , and the odd numbered p l a t e (47) i s the low wavelength member o f the p a i r . The p l a t e h o l d e r may be moved v e r t i c a l l y to g i v e more than one exposure per p l a t e . For example, i f the s l i t 95 mask i s s e t to r e v e a l a s l i t 10 mm h i g h , and the p l a t e h o l d e r i s moved 12 mm between exposures, then the exposed r e g i o n s w i l l be separated by 2 mm. Since the LoSurdo d i s c h a r g e leaves the p l a t e v i r t u a l l y unexposed below (the image of) the top of the cathode, exposures may be s e t more c l o s e l y t o g e t h e r , depending on the p o s i t i o n i n g o f the cathode image on the s l i t . Only the c e n t r a l 4 0 mm of the p l a t e s ' 50 mm h e i g h t i s dependably usable. D i f f e r e n t exposures on m u l t i p l y exposed p l a t e s are r e f e r r e d to by lower case Greek l e t t e r s i n c h r o n o l o g i c a l o r d e r o f exposure. The b e s t p l a t e s f o r long exposures are Kodak type I a - o , o r Ia-E i n the y e l l o w and red. When a l i t t l e more i n t e n s i t y i s a v a i l a b l e , I l a (-0, -D, or -F) p l a t e s o f f e r f i n e r g r a i n and lower background fog a t r e l a t i v e l y small c o s t i n s e n s i t i v i t y . 103a-0 p l a t e s are comparable i n speed, but t h e i r advantage over IIa-0 i s q u e s t i o n a b l e . 6.7. Lens An achromatic l e n s composed of two quartz l e n s e s (plano-convex with curved s i d e s inwards) with d i s t i l l e d water between was used to focus the image of the d i s c h a r g e onto the s l i t . T h i s l e n s w i t h about i f " diameter, and o 5" f o c a l l e n g t h t r a n s m i t s l i g h t down to around 2200 A s a t i s f a c t o r i l y . A l e n s of 2\" diameter (5" F.L.) made of S u p r a s i l I ( A m e r s i l , Inc.) i s a l s o a v a i l a b l e , and i s o a l l e g e d t o t r a n s m i t s a t i s f a c t o r i l y to below 1800 A. I t 96 i s not achromatic, however, and must be focused through c a l c u l a t i o n s based on a v a i l a b l e r e f r a c t i v e index data. 6.8. Alignment The equipment i s r e a d i l y a l i g n e d o p t i c a l l y by the use of a double-ended l a s e r . Set on two l a b j a c k s w i t h i n the b a r r e l of the spectrograph, one end i s aimed at the g r a t i n g c e n t e r as determined by a mask s u p p l i e d with the spectrograph. The o t h e r end r e t r a c e s the o p t i c a l a x i s to the m i r r o r and out through the s l i t (set wide). F i r s t the apparatus t a b l e h e i g h t i s s e t so t h a t the l a s e r beam h i t s the window a t mid h e i g h t (lens removed), wi t h the t a b l e l e v e l e d by bubble l e v e l . Then a stopper with a c a r e f u l l y c entered cathode, w i r e , o r wooden rod i s p l a c e d i n the d i s c h a r g e tube, and the t a b l e moved sideways t i l l the beam i s c e n tered upon i t . F i n a l l y , the clamp h o l d i n g the d i s c h a r g e tube i n p l a c e i s loosened and r e t i g h t e n e d w i t h the tube turned so t h a t the beam h i t s the window a t c e n t r e . At runtime (see Subsec. 7.3. below),the l e n s i s focused upon a p i e c e of white cardboard s l i p p e d i n t o the s l i t mask's s l o t , a f t e r b r i g h t e n i n g the d i s c h a r g e w i t h helium. Although the l e n s was a d j u s t e d h o r i z o n t a l l y t o h i t the s l i t , i n most of t h i s work,by aiming at a t h i n i n k l i n e drawn p r e v i o u s l y on the cardboard, a r e c e n t innova-t i o n allows t h i s c r u c i a l adjustment to be made more d i r e c t l y : A heavy duty speedometer c a b l e has been f i t t e d w i t h prongs to f i t h o l e s i n the h o r i z o n t a l p o s i t i o n i n g knob of the l e n s . 97 T h i s may be p u l l e d f r e e of t h a t knob, as w e l l as turned, from the o t h e r end. By t h i s means the l e n s may be h o r i z o n t a l l y p o s i t i o n e d to g i v e the most l i g h t through the s l i t , by l o o k i n g d i r e c t l y over the g r a t i n g and out the s l i t w ith one's head w i t h i n the b a r r e l of the spectrograph. 7. Cathode Manufacture I t i s f e l t t h a t a s i g n i f i c a n t b e n e f i t from t h i s r e s e a r c h has been the development o f a method f o r making and r u n n i n g - i n sunken cathodes which g i v e s a t l e a s t 80% success i n b r i n g i n g cathodes to 20 000 a p p l i e d v o l t s w i t h s a t i s f a c t o r i l y constant o p e r a t i o n . 7.1. M a t e r i a l s Most cathodes used i n t h i s work have been sunken aluminum cathodes o f one m i l l i m e t e r diameter. However, carbon cathodes o f one and two m i l l i m e t e r diameters were used e x t e n s i v e l y d u r i n g the low v o l t a g e work on OH and OD. B e r y l l i u m was a l s o used o c c a s i o n a l l y ; no p a r t i c u l a r improvement was noted, although o p e r a t i o n was s a t i s f a c t o r y . The d i f f i c u l t y of machining b e r y l l i u m s a f e l y ( i t s dust i s h i g h l y t o x i c ) discouraged us from t u r n i n g i t down to a proper f i t w i t h a v a i l a b l e quartz c a p i l l a r y t u b i n g . Attempts at diameter r e d u c t i o n by heated drawing through a drawplate were u n s u c c e s s f u l . Since the a v a i l a b l e wires (0.050" and 0.100" diameters) would have t o be turned down by o n l y 98 about 20% (to 0.038" and 0.078", r e s p e c t i v e l y ) , however, t h i s remains a p o s s i b i l i t y f o r f u t u r e e x perimentation. I r o n , copper,and molybdenum were a l s o t r i e d and found u n s a t i s f a c t o r y due to heavy s p u t t e r i n g or chemical d e p o s i t , although Wong 7 3 has used molybdenum wi t h HC1. Carbon used f o r cathode manufacture was made from s p e c t r o s c o p i c a l l y pure carbon or g r a p h i t e r od ( N a t i o n a l Carbon) which i s a v a i l a b l e i n an e s p e c i a l l y dense form. These rods were u s u a l l y turned down from stock by mounting i n a d r i l l p r e s s and sanding w i t h hand-held emery paper (220 g r i t ) . 2 mm cathodes c o u l d be made e a s i l y by t h i s method, but 1 mm lengths were v e r y d i f f i c u l t to make t h i s way due to t h e i r extreme f r a g i l i t y . I t was d i s c o v e r e d t h a t 1 mm carbon or g r a p h i t e rods can be cut from h e a v i e r stock on a l a t h e , i f the cut i s from the end i n one pass. N e v e r t h e l e s s , these f r a g i l e rods broke so e a s i l y t h a t a system was d e v i s e d wherein a s h o r t l e n g t h of r o d was wedged a g a i n s t the quartz with a p i e c e : o f t h i n wire from below. The advantages of carbon, even f o r low v o l t a g e work, seem d o u b t f u l i n r e t r o s p e c t . Much of the i n s p i r a t i o n f o r working with carbon came from one p a r t i c u l a r cathode (2mm, p l a t e s 67/68) which r a n a t over 10 kV f o r a t o t a l o f more than 15 hours (a r e c o r d f o r any cathode), but n e i t h e r v o l t a g e or d u r a b i l i t y were d u p l i c a t e d i n l a t e r e f f o r t s . The aluminum wire used i n most experiments came from a l o t of £" wire from W i l k i n s o n Co. L t d . , and was drawn down to a v a r i e t y o f s i z e s to f i t 0.5, 1, and 2 mm 99 I.D. t u b i n g . The wire u s u a l l y used f o r 1 ram cathodes was a c t u a l l y 0.033" i n diameter. T h i s wire was not s o l d as q u a l i t y m a t e r i a l , and i t s exact composition i s not known, but i t i s c a l l e d 9 9% pure with i r o n and copper l i k e l y the dominant i m p u r i t i e s . A f a r purer aluminum wire was o b t a i n e d from E l e c t r o n i c Space Products Inc. (Los A n g e l e s ) , r a t e d 99.999% pure, but r e s u l t s w i t h i t were g e n e r a l l y poor. Very pure aluminum i s s o f t and much harder to work by normal methods, but even a f t e r wire had been o b t a i n e d to f i t a v a i l a b l e q u a r t z , no cathodes made therefrom ever achieved h i g h v o l t a g e s . T h i s work was done b e f o r e the p o l i s h i n g and r u n n i n g - i n procedures ( d e s c r i b e d below) were developed, so the r e j e c t i o n of pure m a t e r i a l s may not be j u s t i f i e d . Quartz was General E l e c t r i c type 2 04 c l e a r f u s e d quartz c a p i l l a r y t u b i n g . I t s dimensions are not dependable from sample to sample; 10% v a r i a t i o n s were observed, although manufacturer's t o l e r a n c e i s some 23%! Since attempts a t r e - u s i n g quartz lengths (by c l e a n i n g or moving the wire) were never s u c c e s s f u l , and the quartz i s r e l a t i v e l y i n e x p e n s i v e (ten o r f i f t e e n cents per cathode), our u s u a l p o l i c y was to f i t cathode wires to the quartz r a t h e r than attempting to draw the q u a r t z . Rubber stoppers and deKhotinsky cement are the standard types a v a i l a b l e from C e n t r a l S c i e n t i f i c Co. The l a t t e r has been r e c e n t l y improved and c a l l e d S e a l s t i x cement; i t s r e s i s t a n c e t o p o l y m e r i z a t i o n upon r e h e a t i n g 100 g i v e s a d e c i s i v e advantage over p r e v i o u s l y a v a i l a b l e products. 7.2. P o l i s h i n g Although t h i s work can c l a i m a u t h o r i t y o n l y i n p o l i s h i n g the 1 mm aluminum sunken cathodes which comprised the bulk o f our experiments, the p o l i s h i n g methods used with other types of cathode s h a l l be d e s c r i b e d b r i e f l y . The procedure d e s c r i b e d f o r the standard aluminum 1 mm cathode i s s u f f i c i e n t t o produce dependable r e s u l t s ; s i n c e the e n t i r e process can be c a r r i e d out i n about 15 minutes, no attempt was made to determine whether or not a l l steps d e s c r i b e d are a c t u a l l y necessary. To g r i n d an aluminum cathode from a p r e v i o u s l y used w i r e , about - j ^ " should be removed each time to ensure tha^ c e n t e r hole from the l a s t run (see Subsec. 7.3.) has been e l i m i n a t e d . The wire i s p l a c e d i n a p i e c e o f quartz t u b i n g s i m i l a r t o (but s h o r t e r than) the t u b i n g i n which i t i s t o be used, and h e l d i n p l a c e by a spot of deKhotinsky cement w i t h the wire end to be removed p r o j e c t i n g from a f l a t end. I t i s convenient to keep u s i n g the same p i e c e of quartz f o r t h i s purpose, s i n c e the end w i l l have been p o l i s h e d f l a t by i t s f i r s t use. The bulk of the aluminum stub i s removed w i t h a f a i r l y coarse grade of emery paper (220 g r i t or " f i n e " g r a d e ) . An i n t e r m e d i a t e paper (320 g r i t ) may be used to b r i n g the aluminum n e a r l y down to the quartz end. From t h i s p o i n t 600 g r i t paper ( i . e . , coated w i t h p a r t i c l e s which have been f i l t e r e d through a mesh of 101 600 w i r e s / i n c h ) i s used: The wire i n h o l d e r i s h e l d v e r t i c a l l y by hand and ground w i t h f i r s t moderate (hand-weight) p r e s s u r e , then more l i g h t l y i n a c i r c u l a r or " f i g u r e 8" motion. A drop o r two of water i s p l a c e d on the emery paper and the process continued wet f o r another minute with s o f t backing beneath (e.g., paper t o w e l s ) . Wiping the end c l e a n w i t h t i s s u e , i t i s then i n s p e c t e d w i t h a magnifying l e n s . There w i l l always be r e s i d u a l d e f e c t s v i s i b l e , but the o b j e c t i s to keep them smal l and of uniform t e x t u r e . When the s u r f a c e i s s a t i s f a c t o r y , the wire i s removed from i t s h o l d e r and cleaned o f the deKhotinsky. DeKhotinsky i s b e s t removed by s c r a p i n g with f i n g e r n a i l s , a f t e r a s l i g h t h e a t i n g to s o f t e n i t s g r i p . The wire i s then mounted i n a d r i l l p r e s s chuck where i t may be turned a t slow o r medium speed. Here the corners are rounded with 600 grit? emery, b r a c i n g the t u r n i n g wire l i g h t l y with the f i n g e r s and always moving the g r i t away from the p o l i s h e d f a c e to a v o i d b u i l d i n g up r i d g e s of s o f t metal. The rounded p o r t i o n o f the wire end should be some 10% t o 15% o f i t s diameter 7 as determined by examination under l e n s , a f t e r c l e a n i n g with t i s s u e . A p i e c e o f q u a r t z i s c u t to l e n g t h ; i f t h i s i s to be a f l u s h cathode, one end should be p o l i s h e d f l a t w i t h 600 g r i t paper. The l e n g t h should be j u s t long enough to extend the d e s i r e d d i s t a n c e above the cathode ( u s u a l l y two or t h r e e mm i n a.sunken cathode) and o n l y a few mm below the stopper so as to leave maximum a i r c o o l i n g f o r 102 the w i r e . Both wire and c a p i l l a r y are now washed: f i r s t under a f o r c e d stream of toluene from a wash b o t t l e , next e t h a n o l to remove the t o l u e n e , then a hard j e t of tap water f o l l o w e d by a d i s t i l l e d water r i n s e . A f i n a l r i n s e with e t h a n o l and a d i r e c t e d a i r s t r e a m from a f a n w i l l speed d r y i n g , although there appears to be no harm i n l e t t i n g the cathode dry under vacuum i f - a c o l d t r a p = i s p r e s e n t . The wire i s i n s e r t e d f l a t end (top) i n t o the q u a r t z , then s e t (bottom f i r s t ) i n t o a rubber stopper which has been wiped down w i t h t o l u e n e . DeKhotinsky i s d r i p p e d and then melted onto the bottom quartz-rubber j o i n and then the quartz-metal j o i n . The assembled cathode i s i n s t a l l e d i n t o the d i s c h a r g e tube a l i t t l e a t a time w i t h a s s i s t a n c e from the vacuum, checking c o n s t a n t l y the v e r t i c a l p o s i t i o n i n g . A v e r y s l i g h t amount of vacuum grease ( j u s t enough to produce a f a i n t g l o s s over the c o n t a c t area) on the rubber stopper i s found to improve r e p r o d u c i b i l i t y of p o s i t i o n . I t i s probably unwise to expose a f i n i s h e d cathode to a i r for.jmore than a day o r so p r i o r to r u n n i n g - i n . I t seems l i k e l y t h a t the presence of aluminum oxide may cause a "spray" d i s c h a r g e (Druyveseyn and Penning 2 7,p.139f) i n which p o s i t i v e i ons are trapped on a t h i n i n s u l a t i n g l a y e r , y i e l d i n g high enough f i e l d s t o cause f i e l d e m i ssion ca-pable- of damaging the cathode s u r f a c e . Such processes t a k i n g p l a c e i n s i d e an already-formed hole would be u n l i k e l y to d i s t u r b the d i s c h a r g e , however, and may p r o l o n g cathode 103 l i f e by reducing s p u t t e r i n g (see Sec. 7.4., below). Thus i t may be p r e d i c t e d t h a t o x i d i z i n g gases ( 0 2 , C0 2, CO, H 20) would be poor c h o i c e s f o r i n i t i a t i n g a cathode, but harmless a f t e r running i n . Among a number of a p p a r e n t l y u n s u c c e s s f u l p o l i s h i n g techniques t r i e d were: the use of h y d r o f l u o r i c a c i d o r c h r o m i c - s u l f u r i c c l e a n i n g s o l u t i o n on the aluminum, the use o f Brasso, p o l i s h i n g paper, crocus c l o t h s , and g r i n d s t o n e s , the achievement of a m i r r o r - l i k e p o l i s h e d s u r f a c e w i t h a commercial p o l i s h i n g t a b l e , and the use of a h i g h l y p u r i f i e d aluminum cathode m a t e r i a l . Carbon cathodes can be p o l i s h e d to a shiny f i n i s h merely by the a p p l i c a t i o n o f 600 g r i t emery. Since carbon i s porous, i t cannot be put i n c o n t a c t with o u t s i d e a i r ; u s u a l l y a f i n e wire i s wrapped about the end o f the rod and ' a l l exposed carbon covered w i t h deKhotinsky. (Needless to say, carbon rods are not washed). B e r y l l i u m rods were p o l i s h e d merely by h o l d i n g w i t h gloved hand and rubbing on emery c l o t h under f l o w i n g water. As f u r t h e r p r o t e c t i o n a g a i n s t t o x i c d u s t , a . recommended i n h a l a t o r was worn. 7.3. Running-In The f o l l o w i n g method f o r b r e a k i n g i n sunken 1 mm cathodes has proven s a t i s f a c t o r y . A f t e r "washing" the system once o r twice with enough hydrogen to g i v e c l e a r l y a u d i b l e pumping n o i s e , the system i s evacuated to 104 what should be markedly l e s s than 10 microns (0.010 mm Hg) p r e s s u r e . A f t e r warmup, the power supply i s s e t to 3 kV. Hydrogen p r e s s u r e i s g r a d u a l l y i n c r e a s e d u n t i l the d i s c h a r g e becomes v i s i b l e through blue f l u o r e s c e n c e i n the quartz above sunken cathodes, or a glow becomes v i s i b l e i n the cathode c e n t e r . T h i s must be done somewhat c a u t i o u s l y because bf the^negative" r e s i s t a n c e . r e g i o n ' ( s e e App .C) which can l e a d one to draw e x c e s s i v e l y heavy i n i t i a l c u r r e n t . About 0.3 mA i s proper a t t h i s p o i n t , corresponding to 1.6 mm Hg p r e s s u r e o f hydrogen. As a guess, a l l c u r r e n t s should be doubled f o r 2 mm and halved f o r 0.5 mm cathodes. A f t e r l e a v i n g the cathode running i n t h i s c o n d i t i o n f o r about f i v e minutes to i n i t i a t e the process of forming the c e n t r a l h o l e , the v o l t a g e i s i n c r e a s e d . The p h y s i c s of running i n a cathode s u r f a c e i s probably much as suggested by Loeb 2 3(p.472) f o r the case of a wire l e n g t h (as opposed to end): As the f i e l d i s i n c r e a s e d at the s u r f a c e , a t some p o i n t w i l l be an upward p r o j e c t i n g i r r e g u l a r i t y o f sharper c u r v a t u r e than elsewhere. Here the f i e l d w i l l be h i g h e s t , and e v e n t u a l l y f i e l d e m ission * (Appendix C,Sec.7) w i l l become s i g n i f i c a n t . F u r t h e r f i e l d i n c r e a s e w i l l c r e a t e such high c u r r e n t d e n s i t y t h a t the metal w i l l v a p o r i s e and a sparking process take p l a c e which i s l i k e l y t o d e s t r o y the i r r e g u l a r i t y which gave r i s e t o i t . A c t u a l l y the spark may c r e a t e p o i n t s o f even higher c u r v a t u r e which should cause a sequence of sparks (Fig.12a and 12b) b e f o r e the i r r e g u l a r i t y i s e f f e c t i v e l y removed. P o i n t s o f «9 7 7 ^ 7 7 7 ^ o 777777' '/77T777 -> 7777, O ^7777^77? F i g . 12 E l i m i n a t i o n of Cathodic I r r e g u l a r i t i e s o f P o s i t i v e Curvature, 106 sharp n e g a t i v e c u r v a t u r e ( p i t s ) would not be a f f e c t e d . However, i t i s p o s s i b l e to v i s u a l i z e a number of geometries which, g i v e n s u f f i c i e n t l y d e s t r u c t i v e sparks, c o u l d l e a d to the propaga t i o n o r e x a c e r b a t i o n of s u r f a c e d e f e c t s and the d e s t r u c t i o n of the cathode by continuous s p a r k i n g . One such p o s s i b i l i t y i s shown i n Fig.12c. To bes t h e l p a cathode to achieve a g i v e n v o l t a g e , then,one should r a i s e the v o l t a g e r e l a t i v e l y s l o w l y , or i n smal l jumps, so t h a t the energy per spark i s minimized. In p r a c t i c e , i t i s found t h a t a good r a t e of v o l t a g e i n c r e a s e i s about 0.4 kV/minute. T h i s allows the v o l t a g e t o reach 20 kV i n l e s s than 45 minutes. Shorter r u n n i n g - i n times are d e s i r a b l e simply on the grounds t h a t they use l e s s o f the t o t a l cathode l i f e t i m e . I f f l a s h i n g ( s p a r k i n g a t the cathode) p e r s i s t s more than a few f l a s h e s at any p o i n t , one o r two minutes spent a t t h a t v o l t a g e may s t a b i l i z e i t . I f longer times seem necessary, o r f l a s h i n g p e r s i s t s a t each adjustment, going t o high e r v o l t a g e i s probably u s e l e s s . Hydrogen i s found t o be an e x c e l l e n t gas f o r r u n n i n g - i n . I t tends t o s t a b i l i z e d e f e c t s which would become worse i f the medium were helium, f o r example. The best c u r r e n t f o r 1 mm cathodes i s 1.2 to 1.4 mA, although 0.6 to 1.0 mA may be -• used below 10 kV. The lower c u r r e n t s are d e s i r a b l e simply on the grounds t h a t they cause l e s s s p u t t e r i n g . The c u r r e n t s d e s c r i b e d u s u a l l y correspond to p r e s s u r e s o f about 2.0 mm Hg o f hydrogen a t 107 low v o l t a g e s and 1.0 mm Hg a t hig h v o l t a g e s . I t may be suggested t h a t the s t a b i l i z i n g e f f e c t s o f hydrogen a t pr e s s u r e s g r e a t e r than 1 mm Hg may be r e l a t e d to the o b s e r v a t i o n of Davis and V a n d e r s l i c e 3 6 ( p . 2 2 6 ) t h a t t h i s i s where H + + + and H^ ions begin to predominate over H 2 i o n s . When the cathode reaches 10 kV, enough helium i s i n t r o d u c e d to b r i n g up the c u r r e n t t o 1.8 or 2.0 mA, f o r j u s t l o n g enough to all o w the bri g h t e n e d d i s c h a r g e image to be focused upon white cardboard p l a c e d i n f r o n t o f the spectrograph s l i t (Sec. 6.8.). The helium i s shut o f f be f o r e a l i g n i n g the image h o r i z o n t a l l y onto the s l i t . Running-in by p l a c i n g a hig h v o l t a g e a c r o s s the e l e c t r o d e s w i t h no gas sample was u n s u c c e s s f u l ; s i n c e a high f i e l d g r a d i e n t cannot be formed without the a s s i s t a n c e of space-charge, i t i s u n s u r p r i s i n g t h a t t h i s made l i t t l e d i f f e r e n c e . Once the d e s i r e d running v o l t a g e has been reached the problem remains of i n t r o d u c i n g the m a t e r i a l s under i n v e s t i g a t i o n . I t seems t h a t t h i s i n g e n e r a l i s a unique problem f o r each m a t e r i a l and must be r e s o l v e d by t r i a l and e r r o r . 7.4. Cathode F a i l u r e and S p u t t e r i n g I t i s presumed t h a t the u s e f u l l i f e o f an o p e r a t i n g cathode i s determined by one more of the f o l l o w i n g l i m i t a t i o n s : (1) M a t e r i a l o r i g i n a t i n g from the cathode d e p o s i t e d on the i n t e r n a l w a l l s o f a sunken cathode c a p i l l a r y causes 108 the l i g h t from the d i s c h a r g e to be absorbed or s c a t t e r e d . (2) Products of chemical r e a c t i o n s i n the gas ( p o s s i b l y i n v o l v i n g m a t e r i a l from the cathode) obscure the d i s c h a r g e as i n (1). (3) Products of chemical r e a c t i o n s , as i n (2), d e p o s i t on the cathode so as to d e s t r o y the cathode f a l l r e g i o n and/or r a d i a l f o c u s i n g mechanism (e.g., c r e a t i n g "spray" d i s c h a r g e ; see Sec. 7.2. above). (4) M a t e r i a l l e a v i n g the cathode d e s t r o y s the f i e l d - h a n d l i n g c a p a b i l i t y e s t a b l i s h e d d u r i n g r u n n i n g - i n , by c r e a t i n g s u r f a c e d e f e c t s which are not capable of s e l f -r e p a i r (Sec. 7.3. above). (5) Reaction products or cathode m a t e r i a l s are d e p o s i t e d on the i n t e r n a l w a l l of a sunken cathode, or the q u a r t z end of a f l u s h cathode, so as to form a conducting s u r f a c e which i n c r e a s e s the e f f e c t i v e cathode area and c u r r e n t beyond a l l o w a b l e l i m i t s . Undoubtedly, (4) was the cause o f the v a s t major-i t y of o p e r a t i o n f a i l u r e s encountered i n t h i s work (with H 2 0 ' D 2 0 ' N H 3 ' a n < ^ o t h e r s ) . Evidence f o r t h i s comes from the f a c t t h a t cathodes r a r e l y break down a l l a t once a f t e r f l a w l e s s o p e r a t i o n ; u s u a l l y an i s o l a t e d f l a s h o r a sequence of f l a s h i n g a c t i v i t y (seen by m a g n i f i e r to be s p a r k i n g a t the cathode) w i l l occur d u r i n g a r u n , f o l l o w e d by a r e t u r n to q u i e t , thus i n d i c a t i n g t h a t s e l f - r e p a r a b l e flaws have been c r e a t e d . Runs are r a r e l y terminated due to reason (1) ( s p u t t e r i n g o b s c u r a t i o n ) . A f t e r having stopped f o r ot h e r reasons, however, impaired v i s i b i l i t y (and even damage to the quartz) i s o f t e n noted. Chemical o b s c u r a t i o n (2) o f t e n terminated runs made wit h s i l a n e o r cyanogen. The l a t t e r l e f t d e p o s i t s appearing to be carbon; s i l a n e d e p o s i t s are of unknown composition. F a i l u r e due to chemical d e p o s i t on the cathode s u r f a c e (3) was a l s o u s u a l l y c o n f i n e d to these m a t e r i a l s . I t seems to be t y p i c a l o f such substances t h a t t h e i r i n t r o d u c -t i o n i n t o the d i s c h a r g e caused a decrease i n c u r r e n t . T h i s l a s t e f f e c t was o f t e n used to judge c o n c e n t r a t i o n . F a i l u r e due to conductive d e p o s i t s near the cathode (5) were r a r e i n these experiments, although some such s p e c t a c u l a r f a i l u r e s d i d occur i f a l a r g e excess o f trimethylaluminum, s i l a n e or cyanogen was i n a d v e r t a n t l y admitted. I f these d e p o s i t s were not too heavy, they c o u l d sometimes be burned away i n purer hydrogen. P h e l p s 1 ** (p. 42) noted f a i l u r e s from t h i s cause w i t h v o l a t i l e hydrocarbons. Both P h e l p s 1 4 ( p . 3 9 ) and Wong 7 3(p. 12f) d e s c r i b e an a c t u a l lowering of the cathode s u r f a c e by "burning away" as a cause of t e r m i n a t i o n , but t h i s was not found i n experiments r e p o r t e d here, e x c e p t i n g c e r t a i n carbon cathodes i n which the c e n t r a l h o l e became so enla r g e d t h a t o n l y a t h i n s h e l l remained which was d i f f i c u l t to d i s t i n g u i s h from a w a l l d e p o s i t . The removal of m a t e r i a l from the cathode, d e s c r i b e d i n (1) and (4) above, i s r e f e r r e d to as s p u t t e r i n g . I t i s 110 b e l i e v e d t o be due to the impact of p o s i t i v e i o n s knocking n e u t r a l atoms (or even negative ions) from the cathode. T h i s i s a complex phenomenon dependent upon the nature of the gas and cathode, d i f f u s i o n (pressure) o f atoms i n the gas, and the i o n i c energy and d e n s i t y , and has s i z a b l e l i t e r a t u r e of i t s own (see F r a n c i s 2 6 , p . 154ff, f o r example). E a r l y work found t h a t the r a t e o f mass l o s s by s p u t t e r i n g was p r o p o r t i o n a l . t o the cathode f a l l v o l t a g e , w i t h or without an a d d i t i v e (negative) constant. More r e c e n t work emphasizes a power dependence upon the c u r r e n t . At very low p r e s s u r e s (long f r e e p a t h ) , s p u t t e r i n g i s independent of p r e s s u r e . At the p r e s s u r e s encountered i n our experiments, however, the s p u t t e r i n g r a t e decreases w i t h i n c r e a s i n g p r e s s u r e , supposedly due to d i f f u s i o n of s p u t t e r e d p a r t i c l e s back to the cathode, or perhaps through an i o n i z a t i o n p r o c e s s . Takatsu and T o d a 7 8 , f o r example, have attempted to e v a l u a t e ci ID the s p u t t e r i n g as being p r o p o r t i o n a l to I /p where a and b are p o s i t i v e c o n s t a n t s . S p u t t e r i n g seems to i n c r e a s e with i o n mass; i t has been found t h a t aluminum s p u t t e r s much more h e a v i l y under argon than hydrogen or helium, but mercury compounds d i d not s p u t t e r badly. Loeb 2 3(p.601) g i v e s a l i s t o f s p u t t e r i n g r a t e s f o r a number of metals under hydrogen, which P l e s s e 7 9 has noted corresponds f a i r l y w e l l to t h e i r heats o f s u b l i m a t i o n . T h i s l i s t shows aluminum as having one of the lowest s p u t t e r i n g r a t e s , w i t h o n l y magnesium and tantalum s u b s t a n t i a l l y lower I l l ( b e r y l l i u m i s not i n c l u d e d ) . A c c o r d i n g to Kueck and B r e w e r 8 0 , however, t h i s i s due to the presence of a s t a b l e l a y e r of o x i d e . A c c o r d i n g to t h i s presumption, a cathode should be r u n - i n a f t e r p o l i s h i n g (as has a l r e a d y been noted) bef o r e exposure to a i r so t h a t the c e n t r a l p i t may be s p u t t e r e d out e a s i l y , and then allowed to o x i d i z e b e f o r e a c t u a l use (running i n a i r , perhaps). W i t h i n the p i t , a s p u t t e r e d oxide might l i k e l y r e - a t t a c h i t s e l f and not be l o s t . D e l i b e r a t e o x i d a t i o n has not been t r i e d , although i t seems t h a t a c c i d e n t a l o x i d a t i o n a f t e r r u n - i n has not been d e l e t e r -ious . 8. Experiments In Table I (and the notes t h e r e a f t e r ) i s presented a l i s t o f experimental c o n d i t i o n s p e r t a i n i n g to those photographs which c o n t r i b u t e d to the r e s u l t s of t h i s work. A g e n e r a l d e s c r i p t i o n now f o l l o w s of v a r i o u s substances run, and the r e s u l t s or l a c k t h e r e o f . 8.1. Hydrogen Hydrogen has been d i s c u s s e d i n d e t a i l i n Sec. 7.3. above. I t runs very w e l l and i s n o t i c e a b l y s t a b i l i z i n g a t h i g h e r p r e s s u r e s (over 1.0 mm Hg). H^ i s v e r y b r i g h t , and the s h o r t exposures needed to photograph i t i n e i t h e r f i r s t ( 3 - 7 min) o r d e r o r second order ( 8 - 1 5 min) make i t p l a u s i b l e to photograph H^ b e f o r e and a f t e r an exposure i n regions where a hydrogen l i n e i s not r e a d i l y 112 . T A B L E I E x p e r i m e n t a l C o n d i t i o n s (see notes a t end) Photo V k V . I ma. P mm. Hg t hr. Grating Gases, Vapors Use Spect. Order Field Max,Source 2.6 1.5 2.5 96500 Ho0,He OH 3 95.1 Hp 1.0 1.4 0.5 ti 11 ," leal.He 2 122.4 11 0.7 1.5 3.0 n D?0,» OD 3 179.8 D§ ? 1.6 3.0 11 n<- it • t OD 3 243.0 it 1.8 1.7 it H20," OH 3 276.7 Hp' 1.4 1.5 6.0 96500 H„0,He OH 3 176.7 Hp 2.2 1.3 ? 11 it ^- it . y OH 3 226.9 i t ' 1.3 1.5 1.7 11 tt 11 f OH 3 319.3 it 1.3 1.3 4.7 tt tt 11 » OH 3 285.1 tt 0.8 1.3 1.6 it D20," OD 3 118.9 Dp 0.6 0.8 4.3 96500 D„0,He OD 3 68.4 Dp 0.7 0.6 1.9 it tt 11 t OD 3 54.5 11 1.6 0.5 4.6 ti Dp0 OD 3 25.4 tt 0.7 1.2 0.6 it " ,He OD 3 87.0 11 0.6 0.3 2.2 96710 D20 OD 3 39.3 Dy 0.7 1.8 0.8 96710 D~0fHe OD 3 96.0 D^ 0.7 1.8 0.8 11 n <- ti t OD 3 62.8 M. 2.0 ? 0.4 96596 D?0 OD 3 77.4 Dp 2.1 0.7 1.5 96600 . OH 3. 45.4 Hp 0.6 ? 1.3 n • " OH 3 69.2 Hp. 0.7 0.7 3.2 96600 Ho0 OH 3 54.9 Hp 1.0 0.4 0.9 n OH 3 69.1 HP 0.7 ? 1.8 ti " ,He . OH 59.1 1.0 ? 0.2 it tt it OH 3 46.2 0.6 0^7 2.8 ti . ti it OH 3 56.9 4 1.9 *? 0.6 96600 H?0,He OH 3 47.2 Hp 1.0 0.1 1.7 ti It OH 3 37.4 1.1 0.8 1.0 ti ft tt OH 3 64.4 4 1.6 0.5 ? 96490 11 it OH 3 50.1 •Hp 2.0 2.6 o i l 94990 K2! T C2 W2 t Hoc 2 230.3 H* He n 11 3.1 96825 it it 11 CH, CN 2 it 264/ 2.0 2.5? 0.1 94990 it tt tt Hex 2 208.3 • Hoc n n 5.5 96825 it 11 it CH, CN 2 n 264e 2.4 1.5 2.4 96325 NH N H 3 197.9 He 1.5 2.2 0.2 97575 H 2?C 2N 2 Hoc 1 247.3 Hex 11 n 3.7 96825 it 11 CH, CN 2 ti 27i« 1.9 2.0 0.5 96325 NH ,He NH 3 213.2 He 1.6 2.1 3.7 96340 11 -2 ti » NH 3 216.7 Hp 2.3 2.6 2.8 "n - it 11 NH 3 230.9 He 1.9 1.6 1.3 it ti 11 * NH 3 287.8 He 3.6 3.5 3.4 97575 NH,,He Ho<, cal .(using NH 1.7 ? 0.6 it H2?He 1 372.0 Hoc . 1/2 13/14 35/36 39/40 45/46 12.0 16.0 15.1 19.5 19.0 53/54/3 12.0 55/56 21.6 59/60o< 18.4 59/60p 18.4 125/26* 6.2 125/268 c8.1 127/28* 6.8 127/28p 4.0 127/28* 6.8 131/32* 5.0 -133/34{5 7.6 133/342T 5.3 135/36r 7.1 137/38o< 6.5 137/38^ 6.0 137/38jr 5.3 139/40* 8.5 139/40^ 7.4 139/40s- 6.0 141/42* 7.3 141/42,3 6.6 141/423* 5.3 143/44* 8.2 145/46a 6.1 263/64? 14.4 263/64^ " 263/64€ 12.9 2.63/64^  ** 269/70S 11.6 271/72* 14.9 27l/72p " 273/74* 11.8 273/74A 13.3 275/76* 14.3 275/76JT 17.5 283/84f3 13.2 289/90(5 21.8 113 TABLE I (con't) Notes t o Table I Photo: P l a t e and photograph d e s i g n a t i o n s are e x p l a i n e d i n Sec. 6.6, above. V: T h i s i s the a p p l i e d v o l t a g e i n kV, as read from the power supply meter. The breakdown of a f u n c t i o n a l r e l a t i o n s h i p between V and the r e s u l t i n g e l e c t r i c f i e l d s , c l e a r l y e v i d e n t i n the t a b l e , i s p a r t l y due to v a r i a t i o n i n c u r r e n t or cathode s i z e , and p a r t l y due .(.invruns numbered l e s s than 100) to r e c a l i b r a t i o n o f the v o l t m e t e r , but i s n e v e r t h e l e s s o f t e n unexplained. I: T h i s i s the t o t a l tube c u r r e n t , read a t the power supply. p: The p r e s s u r e i s read on the McLeod gauge downstream from the d i s c h a r g e tube. t : T h i s i s the t o t a l exposure time i n hours. G r a t i n g : T h i s s e t t i n g determines the s p e c t r a l ranges to reach the p l a t e . Use: T h i s i s the use to which the photograph has been put i n t h i s work, not n e c e s s a r i l y the use o r i g i n a l l y intended. F i e l d : T h i s g i v e s the maximum measured f i e l d i n kV/cm, and an i n d i c a t i o n of the source of t h i s measurement. S l i t : The spectrograph s l i t i s u s u a l l y s e t to 40y or 45y width, and o c c a s i o n a l l y 50y. The s l i t was normally masked to a h e i g h t o f more than 7 mm and l e s s than 12 mm, but a p o r t i o n of t h i s image was below the cathode s u r f a c e so t h a t the appar-ent h e i g h t was s m a l l e r . P l a t e type: Kodak type Ia-O, 2" x 10" unbacked s p e c t r o s c o p i c p l a t e s were used except t h a t p l a t e s l/2nand 13/14 were on 103a-0 p l a t e s , and p l a t e 36, p l a t e s 263 through 272, and p l a t e s 283 through 290 used type Ia-E. Cathode: Cathodes were aluminum of diameter 1 mm ex c e p t i n g as f o l l o w s : Runs 1/2' and 13/14 used-2 mml aluminum;u i iruns 125/26$, 127/28 a, 127/28g, 131/32 a, 137/38a, and a l l of plates'139 through 146 used 2. mm carbon cathodes; f i n a l l y runs 127/28 Y, 133/34a, 133/34y, 135/36y, 137/383, and 137/38y used 1 mm carbon. F u r t h e r d e t a i l s : More d e t a i l e d d e s c r i p t i o n i s contained i n a s e r i e s of s i x — l a b o r a t o r y notebooks-; a p l a t e index of a l l p l a t e s made i s to be found a t the back of Book IV. 114 a v a i l a b l e f o r f i e l d c a l i b r a t i o n (Thomson and D a l b y 7 5 ) . and are c o n s i d e r a b l y weaker and may r e q u i r e hours of exposure. The secondary (molecular) spectrum i s e a s i l y o b t a i n e d w i t h l a r g e Stark e f f e c t s , and can be a nuisance o above 5000 A. Standard p u r i t y H 2 gas was s a t i s f a c t o r y . An i n - l i n e (7p) f i l t e r was used i n lat e r ^ w o r k , but i s thought to have been unnecessary. 8.2. Helium Helium runs f a i r l y w e l l and very c l e a n l y (low s p u t t e r i n g ) , but i s r a t h e r u n s t a b l e a t higher v o l t a g e s when used alone. Helium was used f o r r u n n i n g - i n a l l of the cathodes used w i t h H 20 and D 20 (below), but g e n e r a l l y needed those substances f o r s t a b i l i t y above 10 kV. Alone, a t some g i v e n v o l t a g e , helium might r e q u i r e two o r th r e e times as much pr e s s u r e as hydrogen to draw the same c u r r e n t . The n e u t r a l helium l i n e s can be photographed e a s i l y , and p l a t e s have been o b t a i n e d showing h i g h d i s p e r s i o n Stark e f f e c t s on a l l i d e n t i f i a b l e (and some forbidden) l i n e s o o i n r e g i o n s covered. The 5015 A and 4922 A l i n e s have been used f o r f i e l d c a l i b r a t i o n w i t h NH. I s h i d a and Hiyama 5 0 remark t h a t helium b r i g h t e n s the atomic spectrum o f hydrogen, at the expense of the mol e c u l a r . T h i s e f f e c t was not noted here. The hydrogenic l i n e s o f helium I I have o been observed (see, e.g., 4685 A on p l a t e s 53/54), but much more weakly. Standard p u r i t y gas from Canadian L i q u i d A i r , L t d . was s a t i s f a c t o r y . 115 8 . 3 . Argon Argon was found to cause heavy s p u t t e r i n g with aluminum. A carbon cathode l a s t e d a b i t l o n g e r , and was a l s o t r i e d by F i s h e r 7 6 ( p . 3 8 ) , but argon seems u n s a t i s f a c t o r y as a " c a r r i e r " gas. 8.4. N i t r o g e n , Oxygen, A i r , N i t r i c Oxide A l l o f these gases operate without d i f f i c u l t y a t moderate v o l t a g e s . Oxygen shows atomic l i n e s i n the v i s i b l e and i o n i z e d oxygen l i n e s have been observed w i t h c o n s i d e r a b l Stark broadening. The other l i s t e d gases g i v e the f i r s t and second p o s i t i v e bands of N 2 and the f i r s t n e g a t i v e bands o f N* very s t r o n g l y , along w i t h v a r i o u s atomic l i n e s . The spectrum of NO was not observed w i t h these gases i n any combination. Other oxides o f n i t r o g e n (N 20, N 20 3/ N0 2^ a r e a v a i l a k l e but have not been t r i e d as of t h i s w r i t i n g . The mentioned bands o f N 2 and N 2 are common i m p u r i t y s p e c t r a . The (1,0) and (0,0) bands of the N 2 second p o s i t i v e system have shown weak Stark e f f e c t s on s e l e c t e d l i n e s . The yellow-green c o l o r of " a c t i v e n i t r o g e n " has been observed a t the tube edges, but not i n the d i s c h a r g e i t s e l f . 8 . 5 . Aluminum, Trimethylaluminum Aluminum atomic l i n e s appear r e a d i l y with cathode made of the metal, showing Stark broadening and s h i f t s . Attempts to observe the well-known lU - 1 z + band o f AIH a t 4241 A, by running aluminum cathodes under H 2 or H 2 and A l ( C H 3 ) 3 , were u n s u c c e s s f u l . The CH band at 4314 was weakly observed with the l a t t e r combination. A l ( C H 3 > 3 (obtained from A l f a I n o r g a n i c s , Inc.) was not t r i e d e x t e n s i v e l y s i n c e i t i s spontaneously flammable i n a i r . I t was handled much as was s i l a n e , except t h a t no r e g u l a t o r was used (due to i t s low vapor pressure) and the m a t e r i a l was trapped i n a c o l d t r a p c o n t a i n i n g NaOH. ( A f t e r removal, the t r a p was simply allowed to thaw i n p r i v a c y ) . A l ( C H 3 ) 3 was a n o t i c e a b l e c u r r e n t suppressor, o c c a s i o n a l l y l e f t a m e t a l l i c d e p o s i t , and was q u i t e d e s t r u c -t i v e o f cathodes. 8.6. Water and Heavy Water These substances had been found to operate w e l l by Phelps and D a l b y 8 . They were run w i t h helium i n t h i s work, p r i n c i p a l l y because the advantages of hydrogen had not then been r e c o g n i z e d . (Phelps 1 1*, p. 41, r e j e c t e d hydrogen, a p p a r e n t l y because of the c o m p l i c a t i o n of the molecular spectrum.) Sample was u s u a l l y i n t r o d u c e d i n t o the d i s c h a r g e a f t e r having been brought up to s e v e r a l kV. w i t h helium and s i g n s of i n s t a b i l i t y appeared. The (0,0) band of the A 2 i - X 2n system, a t 3064 A, i s very s t r o n g , and appears commonly as an i m p u r i t y . Other v i b r a t i o n a l bands weaken r a p i d l y w i t h i n c r e a s i n g v 1 , v", o r Av, and t r a n s i t i o n s from hi g h e r e l e c t r o n i c s t a t e s have not been observed i n a LoSurdo d i s c h a r g e . The sample was h e l d i n a v e s s e l d i r e c t l y 117 att a c h e d t o a needle v a l v e i n p u t , and was caught i n a l i q u i d n i t r o g e n t r a p . The water's needle v a l v e s e t t i n g was not very c r i t i c a l , sometimes being run wide open. The H 20 was d i s t i l l e d tap water, w h i l e the D 20 was obt a i n e d from Merck, Sharp, and Dohme. 8.7. Methanol o CH^OH was used to generate the 4314 A band o f CH, f o l l o w i n g the method of P h e l p s 1 4 ( p . 4 1 ) and Phelps and D a l b y 7 0 . Oddly, w h i l e the substance seemed t o operate q u i t e w e l l , and l e f t no d i s c e r n a b l e chemical d e p o s i t s , the s p e c t r a o b t a i n e d i n t h i s way were e x c e e d i n g l y weak. Phelps had no d i f f i c u l t y o b t a i n i n g ample i n t e n s i t y . T h i s may be due to the use of the sunken cathode, s i n c e no f l u s h cathodes were t r i e d w i t h t h i s m a t e r i a l . (See a l s o Sec. 8.14., below.) The h a n d l i n g o f methanol i s s i m i l a r to t h a t o f water. 8.8. Dimethyl Mercury Here we attempted, and f a i l e d , to o b t a i n the 4017 A band o f the A 2n^ - X 2Z system o f HgH. S u f f i c i e n t Hg(CH.j) 2 was run i n the d i s c h a r g e (with hydrogen) so t h a t mercury d r o p l e t s coated the s h i e l d w a l l of the d i s c h a r g e tube, but the d e s i r e d spectrum d i d not appear. I t has been noted by McCurdy 8 1 and Compton, Turner, and McCurdy 8 2 t h a t HgH can be seen i n the p o s i t i v e columns of glow d i s c h a r g e s . We cannot v e r i f y t h i s , but onl y note t h a t the spectrum does 118 not appear i n our cathode glow. Larger c o n c e n t r a t i o n s of H g f C H ^ ^ were e f f e c t i v e i n producing b r i g h t s t r i a t i o n s i n the upper tube. Our sample was ob t a i n e d from A l f a I n o r g a n i c s , and was handled much l i k e water, but i t s d i s p o s i t i o n was t r e a t e d w i t h care because of i t s well-known t o x i c i t y i n water. A bulb with a b u i l t - i n s h u t o f f stopcock had to be abandoned when the sample proved too r e a d i l y s o l u b l e i n the stopcock grease. 8.9. D i e t h y l Z i n c T h i s was run w i t h hydrogen i n search o f the o 4300 A band o f ZnH, analogous to HgH, but wit h no b e t t e r l u c k . D i e t h y l z i n c i s a spontaneously combustible substance i n a i r , and l e f t a white powder (undoubtedly ZnO) i n the system when mixed t h i n l y w i t h a i r . Due to the f a i l u r e of t h i s substance, along w i t h methylated mercury and aluminum, to produce the spectrum of the a s s o c i a t e d h y d r i d e , experiments o f t h i s type were c u r t a i l e d even though methylated (or et h y l a t e d ) forms o f t i n , l e a d , cadmium, and germanium were a v a i l a b l e . 8.10. Phosphorous O x y c h l o r i d e T h i s substance, a l s o known as phosphoryl c h l o r i d e (POCl-j) , was used to look f o r the b r i g h t e r bands of the B 2 E + - X 2n (e-band) system o f PO. The f a i l u r e of t h i s search was l i k e l y due to the f a c t t h a t t h i s substance 119 d e s t r o y e d the d i s c h a r g e . The cathode glow was soon r e p l a c e d by a c u r i o u s l y s t a b l e system of t u r q u o i s e sparks, which were photographed but d i d not y i e l d r e p r o d u c i b l e band s p e c t r a i n the r e g i o n o f o b s e r v a t i o n . Due to i t s h i g h l y c o r r o s i v e n ature, the POCl^ was f e d through the monel and s t a i n l e s s s t e e l i n p u t system from a s t a i n l e s s s t e e l sample c y l i n d e r ; otherwise i t was handled as water. 8.11. S i l a n e SiH^ was capable of running f a i r l y w e l l over f l u s h cathodes, but would too q u i c k l y obscure i n n e r quartz o w a l l s to be used w i t h a sunken cathode. The 414 2 A band of the A 2 A -> X 2n t r a n s i t i o n (analogous to the s i m i l a r system of CH) has been photographed, but the f i e l d s a v a i l a b l e with the f l u s h cathode appeared to be too low to g i v e s p l i t t i n g s ( c o n firming t h e o r e t i c a l c a l c u l a t i o n s which i n d i c a t e a r e l a t i v e l y low d i p o l e moment). SiH^ i s c u r r e n t - d e p r e s s i n g , i n d i c a t i n g t h a t a chemical l a y e r may be d e p o s i t e d q u i c k l y over the cathode s u r f a c e .-and burn-i n e q u i l i b r i u m f o r continued o p e r a t i o n . Another p e c u l i a r i t y noted was t h a t e l e c t r i c f i e l d s exceeding some 60 000 V/cm c o u l d not be o b t a i n e d , even though the a p p l i e d v o l t a g e might range from 5.5 t o over 9 kV. The sample, o b t a i n e d from Matheson, Inc., was f e d to the system through copper t u b i n g and a (brass) helium r e g u l a t o r . In s p i t e of Matheson's recommendation to use only c o r r o s i o n - r e s i s t a n t m a t e r i a l s , no d i f f i c u l t y was encountered. No c o l d t r a p was used, to a v o i d 120 accumulations o f the substance i n glassware, but the pump exhaust was fed out of doors and bubbled through a water t r a p t o prevent backup. No problems were encountered, although the ex p l o d i n g bubbles of s i l a n e and hydrogen co u l d be s u r p r i s i n g l y powerful. In l a t e r work d u r i n g a c o l d s p e l l , the pumping e f f i c i e n c y was a d v e r s e l y a f f e c t e d when the o u t s i d e water f r o z e . 8.12. B e r y l l i u m w i t h Hydrogen, Oxygen, or Water o I t was hoped t h a t e i t h e r the 498 8 A band of the A 2 n -»• X 2 Z + t r a n s i t i o n o f BeH, or the b r i g h t e r bands (e.g., 4708 A) of the C1Z+ + X 1! 4" t r a n s i t i o n o f BeO might be seen while running hydrogen or oxygen, r e s p e c t i v e l y , over a b e r y l l i u m cathode. Although the BeO band i s between two E s t a t e s , second o r d e r S t a r k e f f e c t s should be l a r g e s i n c e a l a r g e d i p o l e moment i s expected ( Y o s h i m i n e 8 3 ) . Hydrogen over b e r y l l i u m d e f i n i t e l y f a i l e d to gi v e BeH, and oxygen f a i l e d to g i v e the BeO spectrum. A p e c u l i a r and very f u z z y spectrum d i d appear under oxygen, however, j u s t where the BeH bands were expected! The broadness of the l i n e s makes i d e n t i f i c a t i o n v ery d i f f i c u l t , but the form a t i o n seems to be roughly j u s t i f i a b l e i n terms o f BeH, which has a very confused spectrum with a l l Av=0 bands superimposed. BeH i s expected to have q u i t e a s m a l l d i p o l e moment (around 0.25 debye as compared to about 7.54 D f o r BeO; see Cade and Huo8**, and Bender and D a v i d s o n 8 5 ) , making i t seem u n l i k e l y t h a t the f u z z i n e s s should 121 be due to Stark e f f e c t s . T h i s spectrum was a l s o seen when run under helium and water, but no more s t r o n g l y than under oxygen. The s h o r t d i s t a n c e above the cathode wherein t h i s spectrum i s v i s i b l e g i v e s evidence t h a t b e r y l l i u m i s a c o n s t i t u e n t . T h i s spectrum i s q u i t e weak i n a l l cases, r e q u i r i n g s e v e r a l hours exposure. 8.13. Boron T r i c h l o r i d e T h i s i s a v o l a t i l e c o r r o s i v e l i q u i d which was handled much the same as POCl^. I t seems to be d e s t r u c t i v e of cathodes, and, i n the v e r y b r i e f s e r i e s of t r i a l s here r e p o r t e d , no long exposures were ob t a i n e d . The A 1 n •*• Xl£ o system o f BC1 ( (0,0) band a t 2720 A) was not seen, although i t had been observed by F.W. Dalby and S.Y.- Wong (unpublished) some time e a r l i e r with a much f a s t e r s p e c t r o -graph. T h i s f a i l u r e i s f u r t h e r e x p l a i n e d by our having attempted to see the bands i n f i f t h o r d e r . 8.14. Cyanogen Thomson and D a l b y 7 4 had e a r l i e r found t h a t the v i o l e t bands of CN ( B 2 l + + X 2 Z + , (0,0) band a t 3883 A) c o u l d be e x c i t e d i n a d i s c h a r g e c o n t a i n i n g cyanogen ( C 2 N 2 ) . The main e f f o r t here was d i r e c t e d towards o b s e r v i n g the re d bands (A 2n ->• X 2 Z + , (0,0) band at 11 000 A, but w i t h bands r e p o r t e d down i n t o the b l u e ) . Cyanogen i s a d i r t y m a t e r i a l to run, as w e l l as being h i g h l y t o x i c , and would s p o i l sunken cathodes r e a d i l y by o b s c u r i n g the quartz above 122 the cathode ( p a r t i c u l a r l y a t the negative glow). C e r t a i n of the red bands were observed, although the b e t t e r exposures were taken with f l u s h cathodes. T h i s was accomplished w i t h the a i d o f Ia-E s p e c t r o s c o p i c p l a t e s ; beyond the o range of these p l a t e s (over 6600 A, s a y ) , the hope of r e c o r d i n g r e d band s p e c t r a i s s m a l l . I t has been found t h a t the v i o l e t bands are f a r e a s i e r to o b t a i n , and a number of new Stark phenomena have been observed t h e r e i n . o I t was found, moreover, t h a t the 4314 A band o f CH (Sec. 8.7., above) was obs e r v a b l e w i t h more i n t e n s i t y under a mixture o f cyanogen and hydrogen than with methanol, and i t i s from the work w i t h cyanogen t h a t our i n f o r m a t i o n concerning CH was o b t a i n e d . T h i s i s the o n l y example known to t h i s author wherein elements i n t r o d u c e d without a p r e - e x i s t i n g bond were c h e m i c a l l y j o i n e d by t h i s d i s c h a r g e . Cyanogen was f e d i n t o the system through the sequence o f monel or s t a i n l e s s s t e e l p i p e s and v a l v e s i n d i c a t e d i n F ig.10. T h i s method was made a l l the more f e a s i b l e by the f a c t t h a t cyanogen i n the c y l i n d e r i s a l i q u i d w i t h vapor p r e s s u r e o f 6 0 pounds a t room temperature. I t was caught b e f o r e the pump by a c o l d t r a p loaded w i t h NaOH p e l l e t s and g l a s s beads, and, as a f i n a l p r e c a u t i o n , the pump exhaust was bubbled through a mixture o f NaOH and c a l c i u m c h l o r i d e h y p o c h l o r i t e b e f o r e going out the window. At one p o i n t , s e r i o u s t r o u b l e occured when sunken cathodes were observed to bl a c k e n above the cathode d u r i n g r u n n i n g - i n with hydrogen. T h i s d e p o s i t c o u l d sometimes be 123 burnt o f f a t higher v o l t a g e (12 kV or higher) but more o f t e n caused f l a r e phenomena l e a d i n g to cathode d e s t r u c t i o n . At t h i s p o i n t an i n - l i n e f i l t e r o f 7 micron pore s i z e was p l a c e d i n - l i n e a t the hydrogen i n p u t , but to no a v a i l . I t was f i n a l l y d e c i d e d t h a t cyanogen absorbed by the system's vacuum grease was the c u l p r i t (even though CN s p e c t r a were never observed v i s u a l l y d u r i n g r u n - i n ) , and the problem was s o l v e d by c a r e f u l c l e a n i n g and performing the r u n n i n g - i n sequence at f u l l pumping r a t e . The c o n s t r i c t i n g v a l v e (see Fig.10) was p l a c e d i n l i n e (to reduce sample consumption) onl y a f t e r r u n n i n g - i n and b e f o r e i n t r o d u c i n g cyanogen. B e a u t i f u l s t r i a t i o n s i n the anode r e g i o n c o u l d be o b t a i n e d a t h i g h c o n c e n t r a t i o n s of C 2 N 2 ' P a r t i c u l a r l y w ith f l u s h cathodes. The number of s t r i a t i o n s i s u s e f u l as an i n d i c a t i o n of C 2 N 2 P r e s s u r e . For a sunken cathode, f o r example, an a p p r o p r i a t e c o n c e n t r a t i o n i s found to occur when the f i r s t s t r i a t i o n j u s t appears as a v e l v e t y glow on the anode s u r f a c e . 8.15. Ammonia I t had been found by I r w i n 7 1 and Irwin and D a l b y 7 2 t h a t ammonia would y i e l d the 3359 A A 3n X 3E~ (0,0) band and 3240 A band ( c x n a xA, (0,0)) o f the NH molecule. Ammonia had been found to behave q u i t e n i c e l y i n the d i s c h a r g e a t the lower v o l t a g e s used by the p r e v i o u s workers. At higher p o t e n t i a l s , however, very c u r s o r y work 124 i n d i c a t e d t h a t f a i l u r e was more l i k e l y when r u n n i n g - i n a cathode wi t h ammonia alone . I t thus seemed d e s i r a b l e to r u n - i n our cathodes by the w e l l - e s t a b l i s h e d method u s i n g hydrogen (Sec. 7.3.). A f t e r a number of experiments had been t r i e d , however, i t was found t h a t when ammonia was i n t r o d u c e d i n t o the d i s c h a r g e a t high v o l t a g e , heavy s p a r k i n g i n v a r i a b l y ensued which destroyed the cathode and even s h a t t e r e d the q u a r t z . Upon even more p e r s i s t a n t e f f o r t , the same t h i n g happened w h i l e r u n n i n g - i n hydrogen. The r a t h e r u n l i k e l y c o n c l u s i o n was reached t h a t a s m a l l amount of ammonia contaminating a hydrogen d i s c h a r g e was h i g h l y d e s t r u c t i v e , even though h i g h e r ammonia c o n c e n t r a t i o n s might not be. At f a u l t was a l e a k y s h u t - o f f v a l v e i n the ammonia i n p u t l i n e , and g e n e r a l a b s o r p t i o n by the vacuum greases. The cure i n v o l v e d p l a c i n g a stopcock on the i n p u t l i n e , c l e a n i n g out o l d grease, and abondoning a l l pumping c o n s t r i c t i o n s . A f t e r r u n n i n g - i n the hydrogen, the system was "washed" wit h helium b e f o r e i n t r o d u c i n g the ammonia. In c o n t r a s t to Irwin's work, these experiments showed a p e r s i s t e n t i m p u r i t y spectrum c o n s i s t i n g o f the s t r o n g (0,0) band o f the n i t r o g e n second p o s i t i v e system, which happens to d i r e c t l y o v e r l a p the A -»• X (1,1) and p a r t o f the (0,0) bands. Once r e c o g n i z e d , however, these n i t r o g e n l i n e s proved to be v a l u a b l e as r e f e r e n c e l i n e s . Examination of the nearby (1,0) second p o s i t i v e band l e d to the d i s c o v e r y o f Stark e f f e c t s i n n i t r o g e n . 125 9. Other Means of E x c i t a t i o n A s t r o n g source of e x c i t a t i o n , a l b e i t without e l e c t r i c f i e l d s , i s o f t e n u s e f u l . One such i s a microwave generator. A standard diathermy u n i t ("Radarmed", Deutsche E l e k t r o n i k GMBH) was used which c o u l d d e l i v e r the o r d e r of 100 mW o f power at 2425 MHz. I t was c a p a c i t i v e l y coupled by simply p l a c i n g s h o r t leads across the neck o f the d i s c h a r g e tube a t the p o i n t marked "X" i n F i g . .'9c. Another method i s to merely r e v e r s e the e l e c t r o d e s w i t h an o l d sunken cathode. T h i s produces an i n t e n s e p o s i t i v e column w i t h i n the c a p i l l a r y over (what i s now) the anode. T h i s method i s of b r i g h t n e s s comparable to the microwave e x c i t a t i o n ; i t does not r e q u i r e r e f o c u s i n g o f the l e n s , but should probably not be used w i t h a cathode which i s s t i l l expected to perform a t h i g h f i e l d s . E i t h e r method can produce s t r o n g r e f e r e n c e exposures, f o r a l l s p e c t r a which are normally observable i n the cathode glow r e g i o n (and many which are not) i n a matter o f minutes. 126 CHAPTER VII THE MEASUREMENT OF THE PLATES AND THE CALIBRATION OF ELECTRIC FIELDS 1. P l a t e Measurement A l l measurements were made on a microdensitometer/ comparator k i n d l y made a v a i l a b l e t o us by Dr. A. M. Crooker. T h i s instrument has v e r t i c a l and h o r i z o n t a l p o s i t i o n i n g c o n t r o l s c a l i b r a t e d t o one micron accuracy, a "zoom" len s p r o v i d i n g v a r i a b l e m a g n i f i c a t i o n , and a v i s u a l p r o j e c t i o n as w e l l as p h o t o e l e c t r i c sweep of the c e n t r a l p o r t i o n of the f i e l d , w i t h d i g i t a l readout. Without removing one's gaze from the f i e l d of v i s i o n , one may cause to be punched on an IBM 02 6 keypunch a sequence of data i n c l u d i n g the h o r i z o n t a l s e t t i n g i n microns, an i n t e n s i t y r e a d i n g a t f i e l d c e n t e r , and a c h o i c e of a s i n g l e i d e n t i f i c a t i o n c h a r a c t e r . These data cards were punched i n a format which allowed the i n c l u s i o n o f wavelength (or cm M and l i n e i d e n t i f i c a t i o n d a t a , t o be read d i r e c t l y by a computer program. T h i s program ( c a l l e d WNINT) p i c k s out l i n e s w i t h s u p p l i e d wave-le n g t h s (or wavenumbers) t o be standards, and i n t e r p o l a t e s the remaining h o r i z o n t a l measurements wi t h a f o u r t h o r d e r polynomial f u n c t i o n . I n t e r p o l a t e d wavelengths and/or wavenumbers are p r i n t e d out i n a ch o i c e o f o r d e r s , as are d i s p e r s i o n s formed from the d e r i v a t i v e o f the po l y n o m i a l . 127 The program a l s o sums over (at most two) i n t e r m i x e d sequences of component measurements, p r i n t i n g out separate averages with standard d e v i a t i o n s , and can punch these on cards f o r p r o c e s s i n g by o t h e r programs. Most of the automatic f e a t u r e s o f the comparator c o u l d not be used because of the p e c u l i a r problems a s s o c i a t e d with measuring ( o f t e n v e r y weak) Stark s h i f t e d components. A l l such measurements were done by eye u s i n g t h e . p r o j e c t e d image, although wavelength standards (e.g., OH l i n e s of high angular momentum J) were o f t e n measured u s i n g the o s c i l l o s c o p e d i s p l a y of the photoscan f e a t u r e which g r e a t l y f a c i l i t a t e d the measurement of such " o r d i n a r y " l i n e s . Each measurement was repeated from f o u r to s i x t e e n times (averaging about seven), depending on the p o i n t ' s apparent r e p r o d u c e a b i l i t y . The measurements of molecular S t a r k e f f e c t s and wavelength standards were made a t n e a r l y maximum m a g n i f i c a t i o n (about f i f t y ) . The much broader and more curved helium and Balmer hydrogen l i n e s were measured at c o n s i d e r a b l y lower m a g n i f i c a t i o n without l o s s of accuracy. The p l a t e s were a l i g n e d v e r t i c a l l y r a t h e r than h o r i z o n t a l l y on the comparator p l a t f o r m : They were r o t a t e d u n t i l v e r t i c a l comparator motion produced minimum v a r i a t i o n i n the p o s i t i o n of (high J) l i n e s showing no Stark e f f e c t s , as judged by the o s c i l l o s c o p e photoscan. The v e r t i c a l p o s i t i o n of maximum f i e l d would va r y a c r o s s the p l a t e , i n g e n e r a l , due to i m p e r f e c t p e r p e n d i c u l a r i t y of the spectrum. 128 2. C a l i b r a t i o n of E l e c t r i c F i e l d s E l e c t r i c f i e l d s were measured by e s s e n t i a l l y the same method used by p r e v i o u s workers i n t h i s l a b o r a t o r y : c a l i b r a t i o n by means o f the Balmer l i n e s of hydrogen. E x p r e s s i o n s f o r the Stark e f f e c t through t h i r d order i n these l i n e s have been assembled by I s h i d a and H i y a m a 5 0 , r e p r i n t e d by Condon and S h o r t l e y 6 4 ( p . 3 9 8 f f ) , and found t o agree reasonably w e l l w i t h experiment (see same r e f e r e n c e s ) . Condon and S h o r t l e y express the displacements D' of a g i v e n component as D| = ±aE - b E 2 ± c E 3 , (115) where the upper s i g n s g i v e the displacement o f the component to higher energy, and the lower s i g n s g i v e i t s displacement to lower energy. The c o e f f i c i e n t s a, b, and c are g i v e n f o r the d i f f e r e n t components of , , and by both I s h i d a and Hiyama 5 0 and Condon and S h o r t l e y 6 4 ( p . 4 0 3 ) , r e l a t i n g D 1 i n cm"1 to E i n MV/cm. U n f o r t u n a t e l y , these c o e f f i c i e n t s were c a l c u l a t e d by I s h i d a and Hiyama u s i n g r a t h e r crude v a l u e s f o r Planck's c o n s t a n t , the speed of l i g h t , and the e l e c t r o n i c mass and charge as were then a v a i l a b l e . Using the r e c e n t v a l u e s f o r these c o n s t a n t s 8 6 : h = 6.626196 X 1 0 " 2 7 erg-sec c = 2.997925 X 1 0 1 0 cm/sec 129 e = 4.803250 X 1 0 ~ 1 0 esu m = 9.104599 X 1 0 ~ 2 8 gm (reduced mass of H) the o l d e r v a l u e s o f a are found to be high by about 0.67%, wh i l e b and c were hi g h by about 2% and 3% r e s p e c t i v e l y . Tables I I .arid:.Ill- >give/ improved v a l u e s o f these c o n s t a n t s f o r OH and OD, r e s p e c t i v e l y . T h i s and a l l p r e v i o u s work done i n t h i s l a b o r a t o r y used the o l d c o n s t a n t s 8 ' 1 4 ' 7 0 7 7 . A l l f i e l d s r e p o r t e d i n these researches must be i n c r e a s e d , and d i p o l e moments decreased, by 0.67%. D i p o l e moments w i l l simply be c o r r e c t e d by the 0.67% f a c t o r i n t h i s work, r a t h e r than r e c a l c u l a t i n g f i e l d s and r e p l o t t i n g • t h e graphs. To f a c i l i t a t e comparison w i t h the graphs and o r i g i n a l c a l c u l a t i o n s , the u n c o r r e c t e d f i e l d s are l i s t e d i n the t a b u l a t i o n s . C o r r e c t e d v e r s i o n s o f p r e v i o u s r e s u l t s from t h i s l a b o r a t o r y have been i n c l u d e d i n Chap.- XII f o r r e f e r e n c e purposes. The change i n f i e l d produced by these c o r r e c t i o n s i n higher orders than f i r s t may be g e n e r a l l y i g n o r e d . The l e a s t d e v i a t e d components (e.g., the 2a component of H^) shov; f a r and away the h i g h e s t second and t h i r d o r d e r e f f e c t s ; even i n t h i s worst case a t a h i g h e s t f i e l d o f 32 0 kV/cm, however, the second order term i s 8.3% o f the t o t a l so t h a t a 2% c o r r e c t i o n w i l l change the t o t a l by l e s s than 0.2%; o t h e r components g i v e much l e s s e f f e c t . The d i p o l e moment r a t i o s as r e p o r t e d i n S c a r l and Dalby 7 7(p.2831) are not a f f e c t e d by t h i s c o r r e c t i o n (see 130 ^ TABLE I I Stark E f f e c t C o e f f i c i e n t s f o r the F i r s t Three Balmer L i n e s of Atomic Hydrogen, Derived from Improved Constants L i n e Label a b H« 4 ir 255.85 6.185 0.159 3w 191.89 6.085 0.085 2tf 127.92 6.584 0.003 lo- 63.96 6.035 0.082 O r l 0.00 5.076 0.000 0o-2 0.00 6.574 0.000 Hg 10 ir 639.62 34.41 2.966 8v 511.70 35.27 2.056 6w 383.77 37.61 1.011 6«r 383.77 35.22 2.056 4<rl 255.85 32.87 1.118 4«r2 255.85 37.66 1.008 2«r 127.92 36.80 0.003 H^ 18m 1151.32 127.9 28.35 15 if 959.44 131.6 21.91 13©- 831.51 131.7 21.91 12* 767.55 139.7 14.44 lOo-l 639.62 127.9 15.48 10r2 639.62 139.7 14.45 5TT 319.81 139.4 7.40 3 or 191.89 139.4 7.40 2fr 127.92 143.4 0.003 Oerl 0.00 131.5 0.00 0r2 0.00 143.4 0.00 1 3 1 TABLE III Stark E f f e c t C o e f f i c i e n t s f o r Deuterium, from Improved C o e f f i c i e n t s L a b e l a b c 4TT 255.78 6.181 0.159 3ir 191.84. 6.081 0.085 271 127.88 6.579 0.003 lc r 63.94 6.031 0.082 0<rl 0.00 5.072 0.000 0<r2 0.00 6.569 0.000 10 V 639.44 34.38 2.962 8ir 511.56 35.24 2.053 61T 383.66 37.58 1.010 6<r 383.66 35.19 2.053 4o-l 255.78 32.84 1.116 4cr2 255.78 37.63 1.007 2a- 127.88 36.78 0.003 18 TT 1151.00 127.8 28.31 15 n 959.18 131.4 21.88 13 o- 831.28 131.6 21.88 12 ff 767.34 139.6 14.42 lOo-l 639.44 127.8 15.46 10<r2 639.44 139.6 14.43 5^ 319.72 .139.3 7.39 3(T 191.84 139.3 7.39 277 127.88 143.3 0.003 Oo-l 0.00 131.4 0.00 0(r2 0.00 143.3 0.00 132 Chapter IX) , although y ,^ and u n both are. In computing f i e l d s from observed Balmer d i s p l a c e -ments, i t i s o f t e n convenient to t r e a t the a b s o l u t e v a l u e of Eq. (115) : D+ E I D+ I =" aE ± b E 2 + c E 3 , where the upper and lower s i g n s r e t a i n t h e i r meaning as i n Eq.(115). The formulae f o r the r e v e r s i o n of s e r i e s g i v e , from Eq. (116) : (116) E ± •= D ±/a ± b D 2 / a 2 + (2b 2/a - c j D ^ / a 4 , (117) where E + and E_ a r e , i n p r i n c i p l e , t h e same. An a l t e r n a t i v e f o r m u l a t i o n employs the s p l i t t i n g S, d e f i n e d as the d i f f e r e n c e between the two displacements of Eq. ( 115) , or the sum of those i n Eq. ( 116) : S = 2aE + 2 c E 3 . The r e v e r s i o n o f t h i s form g i v e s E = S/2a - cS 3/8a^ (1\1'81 (119) which i s d i f f e r e n t from the average o f E + and E_ i n Eq.dQ.7). C l e a r l y , Eqs. (1-16) and(10.7 ) are to be used when one s i d e of a component i s u n a v a i l a b l e , w h ile Eqs. (118) and 133 (119) are used when the undeviated p o s i t i o n i s a t a l l u n c e r t a i n . When f u l l i n f o r m a t i o n i s a v a i l a b l e , the two methods add redundancy to the f i e l d measurement, as do the s e v e r a l Balmer components. When only a l i n e a r a p p r o x i -mation i s to be used, however, the l a s t method (S/2a) i s a c c u r a t e to second order but the other i s not. A computer program (FIELD) has been used throughout t h i s work which c a l c u l a t e s a l l p o s s i b l e combinations of a v a i l a b l e components and p r e s e n t s them as two averages, corr e s p o n d i n g to D + and S. A b e s t value i s chosen, p o s s i b l y a f t e r the e l i m i n a t i o n of u n c e r t a i n components, which i s u s u a l l y the one with lowest mean d e v i a t i o n . The c o e f f i c i e n t s of components d i f f e r i n g only i n second (or higher) order terms are averaged t o g e t h e r , s i n c e such components are never r e s o l v e d . In work w i t h OD, the corresponding Balmer l i n e s of deuterium are used. A c c o r d i n g to formulae i n I s h i d a and Hiyama f o r a, b, and c, the c o e f f i c i e n t a i s p r o p o r t i o n a l to 1/m, b to 1/m3, and c to 1/m5, where m i s the reduced mass o f the two p a r t i c l e system. S i n c e the reduced mass o f hydrogen i s s m a l l e r than t h a t of deuterium by a f a c t o r of about 0.99973, the c o e f f i c i e n t s f o r deuterium are s m a l l e r than f o r hydrogen by the correspond-in g powers o f t h i s f a c t o r . The r e s u l t i n g c o e f f i c i e n t s are l i s t e d i n Table I I I . In some of the p l a t e s taken w i t h ammonia (NH), the Balmer l i n e (H Q) intended to be used i n f i e l d determin-134 a t i o n was too weak to be u s e f u l , but helium l i n e s were c l e a r l y v i s i b l e . To use the helium l i n e s f o r the determin-a t i o n of f i e l d s t r e n g t h s (see Chap.VI , Table I ) , i t was decided not to use the c a l c u l a t i o n s o f F o s t e r 6 5 on the Stark e f f e c t i n helium, but to use i n s t e a d an assortment o f p l a t e s (taken d u r i n g . p r e v i o u s work wi t h OH) showing the helium l i n e s (4922 A and 5015 A) w i t h good H Q P l i n e s . The f i e l d s were i n t e r p o l a t e d u s ing the WNINT program (see S e c . l , above) w i t h helium displacements i n cm-"1 used i n p l a c e of p l a t e measurements and e l e c t r i c f i e l d s (times ten) i n p l a c e of wavelengths (the program had a l r e a d y been used to determine the wavenumbers of the helium l i n e s w i t h r e s p e c t to OH or NH standards i n the o r d i n a r y way). F i e l d s thus found were averaged over o the f o u r helium components used (two each from 4 922 A o and 5015 A ) . D e v i a t i o n s quoted were o b t a i n e d from t h i s average. Data used i n c a l i b r a t i n g the helium l i n e s i s l i s t e d i n Table IV. The o n l y o t h e r v a r i a t i o n from the use of Balmer l i n e s i n f i e l d c a l i b r a t i o n was i n the d e t e r m i n a t i o n of the r a t i o v^/v-^ i n OH and OD. In these experiments, the f i e l d s a t each v e r t i c a l l e v e l were ob t a i n e d from the s p l i t t i n g of the ^12^ l i n e a t t h a t l e v e l , as d e s c r i b e d i n Chap. ±ix , below. 135 TABLE IV Helium C a l i b r a t i o n Data Shown are the measured d e f l e c t i o n s i n cm 1 of the two low-frequency allowed components ( l a b e l e d 4 and 6) of the 4922 A l i n e , and the high-frequency compon-ents ( l a b e l e d 1 and 2) of the 5015 A l i n e . OH Run F i e l d Helium Component D e f l e c t i o n number kV/cm 1 2 4 6 13/14 122.3 7.49 4.85 -29.2 -52. 1 53/54 g 176.7 15.5 10.15 -42.9 -83. 8 55/56 226. 9 23.1 15. 5 -56.1 -108. 8 45/46 276.7 32.0 21.8 -70.5 -131. 3 59/60 g 285.1 32.8 22.0 -72. 5 -136. 8 59/60 a 319.3 38.1 27.0 -78.9 -148. 5 1/2 95.1 n. a. n.a. -22. 0 -40. 6 136 CHAPTER V I I I THE 2II STATES OF OH AND OD 1. The L i n e s T h i s work has extended the r e s u l t s o f P h e l p s 1 4 and Phelps and D a l b y 8 by o b t a i n i n g more data, some a t h i g h e r f i e l d s , f o r the (0,0) and (1,1) bands of OH. A number of the l o w - f i e l d p l a t e s were s u f f i c i e n t l y exposed to show d i s c e r n a b l e components o f the a p p r o p r i a t e l i n e s i n the (2,2) band, i n s p i t e of the low v i b r a t i o n a l temperature of t h i s d i s c h a r g e . Furthermore, analogous r e s u l t s were o b t a i n e d f o r the same bands i n OD. See Chap, v i J ( p a r t i c u l a r l y Sec. 8.6.) above f o r a d e s c r i p t i o n of the methods of g e n e r a t i n g these s p e c t r a . There are four l i n e s , d e s i g n a t e d P ^ ( l ) , Q 2 ( l ) , Q - ^ 2 ( l ) , a n d R 2 ( l ) , which correspond to A 2 Z + -»• X2II t r a n s i t i o n s ending a t the J=4 (N=l) l e v e l o f the 2 n ^ s t a t e . A c c o r d i n g to the theory i n Chap. IV above, these l i n e s w i l l be s p l i t by the f i r s t o rder Stark e f f e c t i n t o o n l y two components. In t h i s n o t a t i o n , (see Fig.13). the l e t t e r s 0, P, Q, R, and S s i g n i f y t h a t the quantum number N changes by 2 , 1 , 0 , - 1 , and -2, r e s p e c t i v e l y , d u r i n g the emission p r o c e s s . P - ^ 2 ( l ) , f o r example, i n d i c a t e s a t r a n s i t i o n i n which N i n c r e a s e s by one i n going from an f, to an f~ l e v e l , and Allowed t r a n s i t i o n s and S t a r k E f f e c t s I n v o l v i n g — the 2 H i , J=£ Energy L e v e l of OH. Only second o r d e r e f f e c t s are t o s c a l e . The second o r d e r e f f e c t s shown do not i n c l u d e the i n t e r a c t i o n w i t h s p i n - s p l i t t i n g . 138 i n which the v a l u e of N i n the lower s t a t e i s one. Only the p i 2 ( l ) l i n e i s " c l e a n " , as we s h a l l see, i n the sense t h a t the upper s t a t e a l s o has J=£ which (being non-degenerate w i t h r e s p e c t to d i f f e r e n t M) s h i f t s i n second order (Fig.13) but does not s p l i t i n any order and so cannot a f f e c t the observed s p l i t t i n g . The R 2 ^ l i n e has an upper s t a t e w i t h N=2, J=|/, and i s s p l i t i n second order i n t o and f compon-e n t s . I g n o r i n g any i n t e r a c t i o n s w i t h s p i n s p l i t t i n g , the E s t a t e S t a r k e f f e c t s f o r these l e v e l s are g i v e n by e 2/60B (M=2-) and -e 2/60B (M=|) (using Eq. ( 31 ) and Eq. ( is ) , where e=pE/hc. The upper s t a t e thus s p l i t s by e 2/30B. S e t t i n g e=10, corresponding t o p=2 D and E = 300 kV/cm (see App.. A);,, and p u t t i n g B = 17 cm"1 , we get a s p l i t t i n g of about 0.2 0 cm 1 which i s about 0.028 mm on our p l a t e s (at 7 cm Vmm i n t h i r d o r d e r ) , o r l e s s than a l i n e w i d t h . The s p i n s p l i t t i n g i n the A 2 E + s t a t e of OH (5y/2 = 0.56 cm"1) i s s u f f i c i e n t l y l a r g e r than the St a r k s p l i t t i n g t h a t the i n t e r a c t i o n w i t h s p i n (Chap. IV., Sec. 6) may be n e g l e c t e d . (In OD, the Sta r k e f f e c t s are l a r g e r and s p i n s p l i t t i n g s m a l l e r , but we do not use the R 2 ( l ) l i n e a t f i e l d l a r g e r than 70 kV/cm, so t h a t t h i s approximation s t i l l holds.) In any event, both components o f the f i r s t o r d e r s p l i t t i n g i n the 2 n s t a t e are e q u a l l y a f f e c t e d , so t h a t any skewness i n the l i n e s h a p e due to upper s t a t e e f f e c t s should not a f f e c t the f i r s t - o r d e r s p l i t t i n g . In f a c t , the r e s u l t s from R 9 ( l ) 139 and p i 2 ^ t r a n s i t i o n s are c o n s i s t e n t even a t h i g h e s t f i e l d s . The Q^d) l i n e i s complicated by the presence o f i t s s a t e l l i t e , the Q 1 2 ( l ) l i n e . While the Q 2(D i s o n l y s h i f t e d by second o rder e f f e c t s i n i t s upper s t a t e (and then o n l y through i n t e r a c t i o n with the s p i n s p l i t t i n g ) , the Q ^ 2 ( l ) upper s t a t e e x h i b i t s s p l i t t i n g i n second order ( F i g . 1 3 ) . However, the Q-^ 2(l) has but h a l f the i n t e n s i t y o f the Q 2 ( l ) , a c c o r d i n g to the formulae of E a r l s 8 7 , and the Q^ 2(1) i s f u r t h e r weakened by the s p l i t t i n g i n the upper s t a t e . In OH the two l i n e s can o f t e n be r e s o l v e d , w h i l e i n OD (at our r e s o l u t i o n ) the Q ^ 2 ( l ) can o n l y skew the l i n e p r o f i l e of the Q 2 ( l ) . I t i s f e l t t h a t these e f f e c t s can be taken i n t o account v i s u a l l y d u r i n g measurement. Th i s l i n e has g i v e n r e s u l t s c o n s i s t e n t w i t h o t h e r l i n e s i n OH w i t h v=2, and i n OD w i t h v=0 and 2. F i g . 2 i l l u s t r a t e s the expected appearance of these l i n e s i n h i g h f i e l d s , w h i l e Figs."3.and . 4 show sample photographs o f each band.. . 2. The Methods The measurements were converted i n t o wavenumbers u s i n g the i n t e r p o l a t i o n program WNINT as d e s c r i b e d i n Chap.VII. Wavelength standards were ob t a i n e d by measuring a number of the h i g h J l i n e s of OH and OD with n e g l i g i b l e S t a r k e f f e c t . The wavelengths/wavenumbers of the OH l i n e s were taken from Dieke and C r o s s w h i t e 8 8 f o r a l l bands. 140 F i g . 14 Schematic D e t a i l of the T r a n s i t i o n s of F i g . 13 i n a f i e l d of v a r y i n g i n t e n s i t y . T h i s i s drawn as i t would appear on a h i g h f i e l d p l a t e , w i t h the cathode s u r -face i n d i c a t e d by the h o r i z o n t a l l i n e . 141 P | 2 ( l ) o 2 ( i ) • ( 1 , 1 ) P 1 2 ( l ) Q 2 ( « ) ( 2 , 2 ) Fife. 15 The A •* X Bands of OH at low f i e l d . Plate 140a. The electric f i e l d is 69.1 kV/cm. 142 (0,0) Fig. 16 The A * X Bands of OD at Low Field. Plate 127a The electric f i e l d is 54.5 KV/cm. For OD, i n the (0,0) band standards were taken from ( u s u a l l y ) Oura and N i n o m i y a 8 9 and from I s h a q 9 0 . For the higher v i b r a t i o n a l bands of OD, wavelengths are a v a i l a b l e from I s h a q 9 1 and S a s t r y 9 2 . However, l i n e s o f the o v e r l a p p i n g (0,0) were g e n e r a l l y used as wavelength standards. D i p o l e moments were determined by s e t t i n g S 2 eMfi + 6' z J(J+D. (12v0) = / e V 9 + 6; where S i s the observed s p l i t t i n g (upper minus lower component) i n cm 1 , and 6 i s h a l f the A-doubling. T h i s formula i s d e r i v e d from Eqs. ( 9 ), ( 45), and (E.19(, App.E). The second l i n e i s o n l y t r u e f o r J = M = Q = ^. Eq. (120) may be r e w r i t t e n •'. £ = 8 9 ' 3 2 2 ; / S 2 - ( 2 6 ) 2 ' (121) so t h a t the e l e c t r i c f i e l d (kV/cm) i s i n d i r e c t p r o p o r t i o n to the square r o o t q u a n t i t y , the l a t t e r being r e f e r r e d to as the c o r r e c t e d s p l i t t i n g ( P h e l p s 1 4 ' 8 ) and denoted by S 1, w i t h a constant o f p r o p o r t i o n a l i t y which determines y d i r e c t l y . T h i s constant i s o b t a i n e d from the slope of a p l o t showing the c o r r e c t e d s p l i t t i n g as a f u n c t i o n o f f i e l d . A computer program making use o f a l e a s t squares r o u t i n e (LQF) was used to f i t a s t r a i g h t l i n e through the 144 d a t a . The A-doubling, 26, has been c a l c u l a t e d f o r J=2, v=0 ( i n both OH and OD) u s i n g the formulae o f Dousmanis, Sanders, and Townes 9 3 ( t h e i r Eqs. 21-23). Using t h e i r e x p e r i m e n t a l l y determined constants a p , Bp, and X (which we c a l l Y ) , we o b t a i n 0.158 cm"1 f o r OH and 0.104 cm"1 f o r OD i n t h i s l e v e l . I t should be p o i n t e d out t h a t the v a l u e of Y ( t h e i r X) obtained by Dousmanis, Sanders, and Townes (7.444 f o r OH, v=0) i s now thought to be i n e r r o r (Clough, Curran, and T h r u s h 9 4 ; Lee, Tarn, Larouche, and Woonton 9 5; R a d f o r d 9 6 ; Poynter and B e a u d e t 9 7 ) , with the g r e a t e s t c l a i m f o r accuracy being made by Poynter and Beaudet (Y = 7 . 5009±0.0001). For the e x c i t e d v i b r a t i o n a l s t a t e s , we assumed i n S c a r l and D a l b y 7 7 t h a t the A-doublings were the same as f o r v=0. We made an estimate based on the "pure p r e c e s s i o n h y p o t h e s i s " which assumes t h a t the 21 and 2 n s t a t e s a r i s e from the same e l e c t r o n having L=l (but d i f f e r e n t M^=A) and allows the Dousmanis, Sanders, and Townes' matrix elements a and 8 to be e v a l u a t e d P P simply i n terms of B v and Y v ( S c a r l and D a l b y 7 7 , p. 2827). T h i s estimate i n d i c a t e d t h a t the A-doubling decreases by some 2% per u n i t v, which a f f e c t s the c o r r e c t e d s p l i t t i n g s by 0.1% o r l e s s . Another e s t i m a t e , based on some rough v a l u e s f o r a and 8 i n OH r e c e n t l y p u b l i s h e d i n Lee, P P Tam, Larouche, and Woonton 9 5, g i v e A-doublings of 0.151 cm and 0.143 cm - 1 f o r , r e s p e c t i v e l y , the v=l and 2 l e v e l s 145 w i t h J=£ i n OH. T h i s i s more l i k e a decrease o f 5% per u n i t v, but w i l l s t i l l be expected to make l i t t l e d i f f e r e n c e . T h i s may be v e r i f i e d by examination o f the graphs of v=l and 2 i n F i g s . 18 arid 19. S i n c e - p o i n t s at lower f i e l d s do not d e v i a t e c o n s i s t e n t l y below the l i n e through the o r i g i n ( c o n v e r s e l y , the higher f i e l d p o i n t s do not d e v i a t e above the l i n e ) , we may conclude t h a t the use of A-doublings which are undoubtedly on the hig h s i d e has l i t t l e e f f e c t . f o l d s p l i t t i n g s due to the J=f l e v e l o f the s t a t e ( i . e . , the P ^ l ) , 0^(1), 1^(1), and S 2 1 ( l ) l i n e s and a s s o c i a t e d s a t e l l i t e s ) on a number of p l a t e s . For example, data from the M n = z " components of the Q-^(l) l i n e has been i n c l u d e d i n the data f o r OH, v=2. For the purposes of the p l o t t i n g and s l o p e - f i t t i n g program, the observed s p l i t t i n g s were converted t o the " e f f e c t i v e " s p l i t t i n g o f the e q u i v a l e n t J=£ l e v e l , i n o r d e r to compensate f o r d i f f e r e n c e s i n the matr i x elements and A - d o u b l i n g s . I.e. , i n terms of pre v i o u s n o t a t i o n , we s o l v e f o r an" e f f e c t i v e h a l f - s p l i t t i n g , v S i n terms o f the observed s p l i t t i n g S by s e t t i n g I nformation can be ob t a i n e d a l s o from the f o u r -e = f k [ ( S / 2 ) 2 - 6 2 =^] (L22) , = £ [ ( S e f f / 2 ) 2 - 6 ^ ] where = 0.158 cm -1 and 6 = 0.0556 cm -1 The 146 40 80 120 160 200 240 280 ELECTRIC FIELD ( K V / C M ) FIG. 17, S' vs. E for OH, v = 0. 20 40 60 80 100 ELECTRIC FIELD (KV/CM) FIG. 18, S' vs. E for OH, v = I 147 1 2.8 u w 2.4 o 2.0 1-h -_ l 1.6 Q. CO 1.2 Q TE 0.8 u RE 0.4 CC CO 0 16 32 48 64 ELECTRIC FIELD (KV/CM) FIG. 19, S' vs. E for O H , v » 2 80 20 40 60 80 ELECTRIC FIELD (KV/CM) 100 120 FIG. 20, S' vs. E for OD, v * 0. 148 1.6 1 CM 1.4 1.2 o p 1.0 _ J 0.8 CL-I O 0.6 o L U 1— 0.4 O L U 0.2 ce o o 0. 16 32 48 64 ELECTRIC FIELD (KV/CM) FIG. 21 , S' vs E for 0D, v • I. 16 32 48 64 ELECTRIC FIELD (KV/CM) FIG. 2 2 , S' vs. E for 0D, v= 2. 149 q u a n t i t y k here r e p r e s e n t s a c o r r e c t i o n t o the m a t r i x element | due to the d e v i a t i o n of the c o u p l i n g scheme from pure case ( a ) . T h i s e f f e c t was not c o n s i d e r e d (k=l) i n the r e s u l t s presented below. Since the data from the Q^(l),v=2, f e l l r i g h t inbetween the data from the P]_2 ^  a n < ^ Q2(l) l i n e s , the i n c l u s i o n of these p o i n t s from p l a t e s 140a and 140y had the e f f e c t o f more h e a v i l y weighting the two h i g h e s t f i e l d p o i n t s i n the s l o p e - f i t t i n g ( F i g . 1 9 ) . Taking Y = -8.15 (Lee, Tarn, Larouche, and Woonton 9 5) f o r v=2, and u s i n g Eqs. (82 ) and ( 89), we get k = 0.98, which i s e s s e n t i a l l y the same as the value of Phelps and D a l b y 8 (v=0). Thus i t i s c l e a r t h a t the d i p o l e moment d e r i v e d from the 0^(1) l i n e i s about two p e r c e n t lower than J=£ r e s u l t s . S ince t h i s i s the same o r d e r of magni-tude as the a s s i g n e d t o l e r a n c e s , i t i s d o u b t f u l t h a t t h i s r e p r e s e n t s a s i g n i f i c a n t d i f f e r e n c e . 3. The R e s u l t s Tables I through VI g i v e the data used i n o b t a i n i n g the r e s u l t i n g d i p o l e moments. In the f i r s t column, "R2" means the l i n e , e t c . Those l i n e s marked w i t h an a s t e r i s k (*) f o r v=0 and 1 of OH were taken from the data o f P h e l p s 1 4 (p.80, a l s o Phelps and D a l b y 8 ) , and the a s s o c i a t e d p l a t e d e s i g n a t i o n s are a l s o h i s . S i s the s p l i t -t i n g of the l i n e i n cm ^, and S 1 i s the " c o r r e c t e d s p l i t t i n g " (the square r o o t i n Eq. '121) j i n cm - 1. E i s the e l e c t r i c f i e l d i n kV/cm. AS' and AE are the 150 TABLE V Splittings and Fields Used for Calculating the Dipole moment of the 2n State in OH for v= 0 Line Plate S S« AS' E AE R2* E-27L* 0.675 0.656 0.020 34.90 0.80 P12 141* 0.725 0.708 0.017 37.36 0.75 R2 141 y 0.698 0.680 0.017 37.36 0.75 PI 2 137 * 0.893 0.879 0.020 45.35 2.00 R2 137 x 0.885 0.871 0.043 45.35 2.00 P12 139 S 0.860 0.845 0.020 46.18 0.20 R2 139S 0.869 0.855 0.042 46.18 0.20 Q2 139 6" 0.870 0.856 0.020 46.18 0.20 PI 2 141(3 0.923 0.909 0.031 47.23 0.30 R2 141 p 0.878 0.864 0.016 47.23 0.30 R2* E-22* 1.043 1.031 0.010 49.20 2.40 PI 2 145p 0.961 0.948 • 0.008 50.14 0.50 R2 145e 0.954 0.941 0.022 50.14 0.50 R2* 6/23/63* 1.052 1.040 0.020 52.90 0.50 R2* E-27U* 1.093 1.082 0.010 54.80 0.50 P12 137T . 1.071 1.059 0.026 54.87 0.50 R2 137 f 1.041 1 . 0 2 9 0.016 54.87 0.50 P12 141* 1.067 1.055 0.030 56.87 0.50 R 2 141* 1.066 1.054 0.018 56.87 0.50 PI 2 139 1.125 " 1.114 0.013 59.06 0.60 R2 139/ 1.111 1.100 0.013 59.06 0.60 R2* 6/25/63* 1.156 1.145 0.010 59.90 1.30 R2* F-10* 1.202 1.192 0.010 63.10 1.00 P12 143* 1.179 1 . 1 6 8 0.034 64.45 0.40 R2 143* 1.158 1.147 0 . 0 2 6 64.45 0.40 PI 2 139cx 1.285 1.275 0.021 69.11 0.80 R2 139oc 1.283 1.273 0.021 69.11 0.80 P12 137(3 1.284 1.274 0.019 69.25 0.50 R2 1376 1.302 1.292 0.024 69.25 0.50 R2 1 1.861 1.854 0.O22 95.10 0.40 R2 53 p 3.500 3.496 0.066 176.70 0.70 P12 55 4.494 4.491 0.095 226.90 0.50 R2 55 4.514 4.511 0.125 226.90 0.50 PI 2 45 5.407 5.405 0.066 276.70 2.80 R2 59p 5.750 5.748 0.081 285.10 1.80 •Data from Phelps (see text). !51 TABLE VI Data for Calculating the 2 n " . Dipole Moment in OH v = l Line Plate S S' AS* E AE — __ P 1 2 * E - 2 7 L * 0 . 7 2 9 0 . 7 1 2 0 . 0 2 0 3 4 . 9 0 0 . 8 0 P 1 2 1 4 2 ^ 0 . 7 0 5 0 . 6 8 7 0 . 0 1 9 3 7 . 3 6 0 . 7 0 P12: 1 3 8 o c 0 . 9 0 1 0 . 8 8 8 0 . 0 2 1 4 5 . 3 5 2 . 0 0 P 1 2 142£ 0 . 8 7 0 0 . 8 5 6 0 . 0 2 3 4 7 . 2 3 0 . 3 0 P 1 2 * E - 2 2 * 1 . 0 0 3 0 . 9 9 0 0 . 0 1 0 4 9 . 2 0 2 . 4 0 P 1 2 1 4 5 S 0 . 9 2 8 0 . 9 1 4 0 . 0 1 9 5 0 . 1 4 0 . 5 0 P 1 2 * E - 2 7 U * 1 . 0 4 0 1 . 0 2 8 0 . 0 1 0 5 4 . 8 0 0 . 5 0 P 1 2 1 3 8 J* 1 . 0 3 9 1 . 0 2 7 0 . 0 4 5 5 4 . 8 7 0 . 5 0 P 1 2 142oc 1 . 0 0 3 0 . 9 9 0 0 . 0 1 2 5 6 . 8 7 0 . 5 0 P 1 2 X 4 0 T 1 . 1 0 3 1 . 0 9 2 0 . 0 2 0 5 9 . 0 6 0 . 6 0 P 1 2 * 6 / 2 5 / 6 3 * 1 . 0 9 4 1 . 0 8 3 0 . 0 1 0 5 9 . 9 0 1 . 3 0 P 1 2 * F - 1 0 * 1 . 1 8 1 1 . 1 7 0 0 .0 .10 6 3 . 1 0 1 . 0 0 P 1 2 1 4 3 o c 1 . 2 1 0 1 . 2 0 0 0 . 0 3 0 6 4 . 4 5 0 . 4 0 P 1 2 140o« 1 . 2 5 0 1 . 2 4 0 0 . 0 1 1 6 9 . 1 1 0 . 8 0 P 1 2 1 1 . 8 7 0 1 . 8 6 3 . 0 . 0 4 0 9 5 . 1 0 0 . 4 0 •Data from Whelps (.see text;. .TABLE V I I Data for Calculating the 2 n Dipole Moment in OH v = 2 Line f l a t e S S' AS' E AE PI 2 1421 0.764 0.747 0.023 37.36 0.75 P12 J.42£ 0.868 0.854 0.021 47.23 0.30 P12 142 oc 1.010 0.998 0.027 56.87 0.50 Q2 142o<: 1.111 1.100 0.031 56.87 0.50 PI 2 140^ 1.071 1.061 0.019 59.06 0.60 Q2 140 T 1.088 1.079 0.041 59.06 0.60 Ql 140 It 1.912* 1.062 0.048 59.06 0.60 PI 2 140 of 1.282 1.272 0.029 ' 69.11 0.80 Q2 140 oi 1.276 1.266 0.076 69.11 0.80 Ql 140 CA 2.288* 1.271 0.030 69.11 0.80 •See text with regard to the Q 1 ( l j line. 152 TABLE V I I I p Data for Calculating the IT Dipole Moment in OD v = 0 Line Plate E R2 1278 0.537 0.527 0.034 25.35 0.90 PI 2 132* 0.790 0.783 0.025 39.34 0.80 P12 127<K 1.002 0.997 0.014 - 54.49 0.50 Q2 127 <x 1.021 1.016 0.017 54.49 0.50 R2 127 cx 1.023 1.018 0.022 54.49 0.50 PI 2 134 V 1.216 1.212 0.013 62.84 0.50 Q2 134 t 1.159 1.154 0.033 62.84 0.50 R2 1.218 1.214 0.028 62.84 0.50 PI 2 125p 1.317 1.313 0.012 68.39 0.70 Q2 125^ 1.277 1.273 0.018 68.39 0.70 R2 125£ 1.295 1.291 0.043 68.39 0.70 PI 2 135/ 1.543 1.540 0.038 77.40 0.50 Q2 135/ 1.518 1.514 0.042 77.40 0.50 PI 2 127/ 1.629 1.626 0.020 86.95 0.40 Q2 127 ^  1.631 1.628 0.034 86.95 0.40 Q2 134 § 1.770 1.767 0.050 96.00 2.30 Q2 125* 2.341 2.339 0.081 118.95 0.80 TABLE IX Data for Calculating the 2TT Dipole Moment of OD V 5 l Line Plate s S* . AS' E AE 0.522 0.512 0.037 25.35 0.90 0.758 0.751 0.026 : 39.34 0.80 1.013 1.008 0.019 54.49 0.50 1.152 1.147 0.052 62.84 0.50 1.234 1.236 0.026 68.39 0.70 P12 PI 2 PI 2 PI 2 P12 127fi 132* 127<X 134/ 125 $ L 153 TABLE X Data for 2 Calculating the TT v = 2 State Dipole Moment in OD, Line Plate S S' AS 1 E AE P12 132* P12 127K. 0.729 0.947 0.722 0.941 0.036 39.34 0.80 0.023 54.49 0.50 • TABLE :XI Experimental Values of the Dipole Moment in-the TT State Molecule v u(D; Au(D) Source OH 0 0 o 1.70 1.72? 1.66° 0.02p 0.02^ 0.01 . This work Phelps and Dalby8 Powell and Lid e 9 8 (J-.|;J1=|) 1 • 1 2 1.64* 1.65? 1,62* 0.05. 0.04 0.045 This work Phelps and Dalby8 This work OD 0 1 2 1.68 1.63^ 1155.; 0.03? 0.02^ 0.06£ This work This work This work 154 e r r o r s a s s o c i a t e d w i t h S' and E. We d e r i v e d AS 1 from a standard d e v i a t i o n over s e v e r a l ( u s u a l l y about s i x ) d i f f e r e n t measurements of the same two components, while the AE comes from the standard d e v i a t i o n over the d i f f e r e n t components of the Balmer l i n e which were averaged to o b t a i n E. In the v=l bands o f both OH and OD, o v e r l a p p i n g o f the R ~ ( l ) and Q~(l) l i n e s impaired t h e i r u s e f u l n e s s . In OD with v=2, i t was thought b e s t to e l i m i n a t e a number of d o u b t f u l measurements (these l i n e s were u s u a l l y j u s t b a r e l y v i s i b l e ) , even though t h i s reduced the usable data to two p o i n t s . The l i s t e d v a l u e s of E have not been c o r r e c t e d i n accordance with the t a b l e s g i v e n i n Chap.VII. However, the r e s u l t i n g v a l u e s of p g i v e n i n Table XI". ( i n c l u d i n g those from Phelps and Dalby) have been reduced by the p r e s c r i b e d 0.67% ( a f t e r the c a l c u l a t i o n by the l e a s t squares f i t ) . The r e s u l t s g i v e n i n Table XI are i l l u s t r a t e d i n F i g . 2 3 . I t may be noted t h a t the s e l f - c o n s i s t e n t f i e l d (SCF) c a l c u l a t i o n s o f Cade and Huo 8 4 g i v e a v a l u e of 1.78 D, which i s about as f a r above our r e s u l t f o r OH (v=0) as the measurements o f Powell and L i d e 9 8 (on the Z j I g . ' J = J l e v e l ) and the c o n f i g u r a t i o n i n t e r a c t i o n (CI) c a l c u l a -t i o n s o f Bender and D a v i d s o n 8 5 l i e below. C a d e 9 9 g i v e s a graph ( h i s F i g . 7a) showing c a l c u l a t e d dependence of the. d i p o l e moment upon the i n t e r -n u c l e a r d i s t a n c e . Measurements from t h i s graph g i v e the dependence as 155 L U O U J _J o Q. Q 1.9 r 1.8 U J >-CD UJ £ 1.7 1.6 1.5 1.4 o OH • OD L _ 0 I 2 3 VIBRATIONAL QUANTUM NUMBER F I G . £ 3 Dipole moments of OH and OD as a function of v. The points joined by the solid curve are based on the calculations of Cade (1967). 156 y = 1.7803 + 0.8940(r - r ) - 0.22137(r - r ) 2 6 (123) where y. i s i n debyes and the i n t e r n u c l e a r d i s t a n c e r o (and i t s e q u i l i b r i u m v a l u e r ) are i n Angstroms. Taking the m a t r i x elements of Eq. (123) (see Phelps and D a l b y 8 , S c a r l and D a l b y 7 7 ) f o r the d e s i r e d v a l u e s of v, we o b t a i n the c o r r e s p o n d i n g curve f o r y as a f u n c t i o n o f v, shown as the smooth curve i n F i g . 23. I t i s noted t h a t t h i s f u n c t i o n appears to be i n deci d e d disagreement w i t h our r e s u l t s , both i n magnitude and s l o p e . C a d e 1 0 0 has commented t h a t t h i s disagreement i s "not s u r p r i s i n g " , i n view of c e r t a i n d i f f i c u l t i e s encountered i n h i s c a l c u l a t i o n s . The i m p l i c a t i o n s o f our measured v a l u e s of 3y/3v upon the v i b r a t i o n a l t r a n s i t i o n p r o b a b i l i t y i n the 2 I I s t a t e are d i s c u s s e d i n S c a r l and D a l b y 7 7 . 157 CHAPTER IX THE A  2l + STATES OF OH AND OD 1. I n t r o d u c t i o n The a b i l i t y t o photograph s p e c t r a of OH and OD at high d i s p e r s i o n i n f i e l d s i n excess of 200,000 V/cm has enabled us to measure second order Stark e f f e c t s . A p a r t o f the second order e f f e c t i s due to the 2 E + upper s t a t e of the 3064 A bands, and e v a l u a t i n g t h i s c o n t r i b u t i o n w i l l a llow us t o c a l c u l a t e the d i p o l e moment o f t h i s s t a t e . In d i s c u s s i n g the method by which t h i s i s done, ,the f o l l o w i n g nomenclature i s adopted: C = the u n d j v i a t e d ( z e r o - f i e l d ) l i n e p o s i t i o n ( i n cm ) . U = the p o s i t i o n of the higher frequency component. L = the p o s i t i o n of the lower frequency component. x = U + L - 2C, c a l l e d the asymmetry (note t h a t t h i s i s |U - C| - |L - C|). y = (U - L ) 2 , i n cm , c a l l e d the s p l i t t i n g squared. The P ^ f l ) l i n e has d i s t i n c t advantages over any o t h e r , i n t h a t the upper s t a t e has a l a r g e second order s h i f t without s p l i t t i n g and i s uncomplicated by s p i n i n t e r a c t i o n s . The o n l y other l i n e s coming from t h i s upper s t a t e are the 0^2(1) a n ^ P.^(l), both of which have lower s t a t e s s p l i t i n t o f o u r ( r a t h e r than j u s t two) components. Moreover the 0,9 branch i s q u i t e weak and the P.(1) s u f f e r s from 158 o v e r l a p , so the P 1 2 ( l ) has been used e x c l u s i v e l y f o r t h i s study. For t h i s l i n e , a c c o r d i n g to Eq. ( 45), the f i r s t o rder e f f e c t i n the A-doublets of the 2n s t a t e i s g i v e n by W U ) = ± ( V 2 + 6 2 ) l = ± U 2 / 9 + 6 2)*' . (124) Since second order terms do not a f f e c t the s p l i t t i n g (see F i g . 24 ), we may a l s o w r i t e IW^ 1* | = (U - L)/2 i (125) = V V 2 . g i v e n by For the P ^ 2 ( l ) l i n e , the second order e f f e c t s are 2e 2 w„ = - e and n 27B n (126) e 2 Z 6 B I (127) where the energy l e v e l s have been assumed to be g i v e n by B n J(J+1) and B^ , N(N+1) r e s p e c t i v e l y . C o r r e c t i o n s to Eq. (12 6) due to i n t e r m e d i a t e c o u p l i n g are c o n s i d e r e d i n Sec. 5. 159 Zero f i e l d X = f U-0 First order Second order t J 2 ) FA -r A s JL i i V t r f (1) IT — « - F „ F i g . 24 Schematic diagram of S t a r k e f f e c t s i n a t r a n s i t i o n between s u b l e v e l s of a 2 X -»• 2 n band. F i r s t and second o r d e r e f f e c t s are shown s e p a r a t e l y f o r c l a r i t y , The p a r i t i e s a re shown as they e x i s t i n the n s t a t e o f OH. The s i g n s o f the second order terms are d e f i n e d p o s i t i v e as shown. 160 A l l o w i n g f o r the f a c t t h a t Eq. (124) d e f i n e s the zero of energy t o l i e midway between the A-doublets i n the 12) J n = £ l e v e l , we can express i n terms of the component p o s i t i o n s by W^2) = £(U + L - 2C - 2p£ + 2 W^2*) (2) (128) = x/2 - pa + WR The pr e c e d i n g equations l e a d to a " l i n e a r " r e l a t i o n s h i p between the s p l i t t i n g squared and the asymmetry: y = kx - 2k6 + 45 2 , (129) where k = ^ [ ( y E / y n ) 2 / 3 B i ; - 4 / ( 2 7 B n ) ] " 1 (130) In Eqs. (129) and (130) we have a l i n e a r equation which may be f i t t e d t o measured data t o y i e l d (the a b s o l u t e value of) the r a t i o l i ^ / u ^ , independently - of any e x t e r n a l • measure-ment o f e l e c t r i c f i e l d s . 2. Measurement of the L i n e and Slope F i t t i n g The measurements were done i n the manner d e s c r i b e d i n Chap. V I I , with the p l a t e a l i g n e d so t h a t s p e c t r a l l i n e s not s h i f t e d by the f i e l d (e.g. , h i g h - J l i n e s ) remained centered i n the c r o s s - h a i r s w h i l e the p l a t e was moved v e r t i c a l l y . Then, s t a r t i n g i n a r e g i o n of f a i r l y low f i e l d (at a v e r t i c a l p o s i t i o n where the two Stark components 1 6 1 o f the ^ ^ ( l ) l i n e were j u s t r e s o l v a b l e ) , the p o s i t i o n of the upper and lower components (U and L) were measured a l t e r n a t e l y s e v e r a l times ( u s u a l l y four times each, but i n some cases many more). The p l a t e was then moved v e r t i c a l l y to a higher f i e l d r e g i o n by 0.2 mm, and t h i s process repeated i n t o the r e g i o n of maximum f i e l d . The p o s i t i o n of the l i n e a t zero f i e l d (C) was measured d i r e c t l y at the l i n e top, and the d i s p e r s i o n of the spectrum was c a l i b r a t e d by the measurement of s e v e r a l h i g h - J l i n e s at both z e r o - f i e l d and maximum f i e l d . These measurements were averaged f o r each component and l e v e l by the WNINT computer program (see Chap VII) which punched out cards b e a r i n g the averages and t h e i r standard d e v i a t i o n s . These were then processed (see a l s o Sec. 4 below) by a r a t h e r complex program c a l l e d QUAFF which formed the v a r i a b l e s x and y from the data (see Sec. 3) and f i t t e d them with a s t r a i g h t l i n e by a l e a s t - s q u a r e s computer l i b r a r y r o u t i n e (LQF). T h i s program a s s i g n e d weights to the p o i n t s , d e f i n e d i n v e r s e l y p r o p o r t i o n a l to the standard e r r o r s i n x and/or y, but l i m i t i n g a s i n g l e weight to be no l a r g e r than the mean weight than ( a r b i t r a r i l y ) \ of the standard d e v i a t i o n taken over a l l weights. A subroutine of QUAFF ( c a l l e d PLT) was capable of graphing the data and f i t t e d l i n e s i n a v a r i e t y of ways, drawing the graphs of Figs.28 -30. (A m o d i f i c a t i o n of QUAFF was used to f i t and graph the data o f Chap.VITI f o r the n s t a t e s of OH and OD.) A system of weighting more h e a v i l y p o i n t s i n the higher f i e l d r e g i o n a c c o r d i n g to some power of y was t r i e d , but r e j e c t e d . A number of p o i n t s were r e j e c t e d i n t u i t i v e l y i n the low f i e l d r e g i o n (where the asymmetry i s i n t r i n s i c a l l y i n a c c u r a t e ) . P o i n t s were omitted which seemed to have undue i n f l u e n c e on the s l o p e , w h i l e a t higher f i e l d s they were e l i m i n a t e d where there e x i s t e d reason to b e l i e v e them i n e r r o r based on the appearance of the l i n e a t t h a t l e v e l . F i g s . 25 through 27 show a l l photographs used i n t h i s study except two of those from the (1,1) band of OH. While i t may be seen t h a t o v e r l a p p i n g i s a problem a t some l e v e l o f almost every p i c t u r e , i t i s an advantage of t h i s method t h a t such p o i n t s may be dropped r e a d i l y from the c a l c u l a t i o n . The Z e r o - F i e l d P o s i t i o n There are a number of d i f f e r e n t ways of o b t a i n i n g the p o s i t i o n (C) o f the undeviated l i n e . The measured l i n e top may be r e f e r e n c e d to the standard l i n e s as measured at e i t h e r the top or bottom of the spectrum. A l t e r n a t i v e l y , one may use the value g i v e n i n the same source l i t e r a t u r e as the wave-l e n g t h standards. A l l such d i f f e r e n t v a l u e s are i n agreement to w i t h i n 0.13 cm 1 , but even so the c h o i c e w i l l a f f e c t the r e s u l t s . Since x i s a l i n e a r f u n c t i o n of C, and y does not depend upon C a t a l l , a wrong c h o i c e of C w i l l not a f f e c t the slope of the f i t t e d l i n e (and thus the d e r i v e d y /y ), but w i l l merely s h i f t a l l p o i n t s h o r i z o n t a l l y t o g e t h e r . 163 A L 3092.7 A A L 3082.2 A P I 2 ( I ) Q 2 ( D R 2 ( l ) P L A T E 59 or 319.3 k V / c m Fig. 25 The (0,0) A • X band of OH at high f i e l d . t A L 3092.7 A A L 3082.2 A I P L A T E 3 5 P I 2 ( 2 ) J Q 2 ( l> P | 2 ( l ) R 2 ( l ) Pill HHP Nil 1 7 9 . 8 k V / c m 2 P L A T E 3 9 P.A) I Q2(D 1 2 P « « ' > 2 4 3 . 0 k V / c m Fig. 26 (0,0) A - X band nf nn_. hiah 165 H e 4713 A P L A T E 5 5 2 2 6 . 9 k V / c m P L A T E 5 H e 4 7 1 3 A 2 8 5 . 1 k V / c m Fig. 27 The (1,1) A -*• X band of OH at high f i e l d . Thus, f o r any s i n g l e l i n e , Mj./un can be obtained independently of C as w e l l as E. I f measurements from more than one photograph are to be superimposed on the same graph, however, an i n c o n s i s t e n c y i n the measurement o f C from one p l a t e to another can a f f e c t the r e s u l t a n t b e s t f i t s l o p e . We have chosen to t r e a t C as a f r e e parameter ( f o r each photograph) which may be a d j u s t e d to b r i n g the p o i n t s from d i f f e r e n t photographs t o g e t h e r . These adjustments are l i s t e d i n Table XII"by: the heading " R e l a t i v e Adjustment i n C". They are a l l l e s s than 0.15 cm corresponding to about 2 0 microns on the p l a t e ( l e s s than h a l f a l i n e w i d t h ) , but may a f f e c t r e s u l t s by as much as 2 p e r c e n t . Now, s i n c e the slope k i s of the order of -10 cm i n OD, and the order of -24 cm ^ i n OH, and 6 i s of the order of 0.06 cm , the term 46 2 i n Eq. (129) i s n e g l i g i b l e compared to the 2k6 term. Thus the graph of Eq. (129) should c r o s s the x a x i s near x=2 6. The change i n C r e q u i r e d t o ensure t h a t t h i s a c t u a l l y happens i s g i v e n i n Table XII under the heading "Absolute Adjustment to C". The a c t u a l v a l u e s of C thus obtained are l i s t e d i n the f i r s t d a ta column of Table X I I I . The f a c t t h a t these v a l u e s v' are a l l p o s i t i v e (while s t i l l l e s s than a l i n e w i d t h ) i n d i c a t e s t h a t t h e r e may be a sma l l s y s t e m a t i c e r r o r i n the v e r t i c a l alignment p r o c e s s . I t may a l s o be noted t h a t adding these increments to C tended to cause the asymmetries a s s o c i a t e d w i t h a number of the low f i e l d p o i n t s to become ne g a t i v e , where they had p r e v i o u s l y been p o s i t i v e (by our d e f i n i t i o n s , 167 TABLE XII C o r r e c t i o n s to the Z e r o - F i e l d P o s i t i o n s o f the P12(D P o s i t i o n R e l a t i v e Absolute Band from the Run Adjustment Adjustment L i t e r a t u r e i n C i n C (cm- 1) (cm-1) (cm-1) OH 32314.19 59g 0. 000 0.080 (0,0) (Dieke and C r o s s w h i t e 8 8 ) 45 -0.014 0.066 59a 0.130 0.210 OD 32397.83 39 0.000 0.127 (0,0) ( I s h a q 9 0 ) 35 0.045 0.172 OH 31733.71 59B 0.000 0.021 (1,1) (Dieke and C r o s s w h i t e 8 8) 536 0.145 0.166 59a 0.133 0.154 55 0.050 0.071 168 TABLE XIII Data for p /u : Adjusted Absolute C and Mean Measured Values of U and L, with Errors. Band Run Level Label C Components in AC U -1 cm AU L AL OH 596 3 32314. 132 .025 32314.895 .058 32313.456 .011 (o,o) r 4 ->14.974 .015 '13.308 .061 5 '515.206 .024 13.128 .009 $ 315.276 .006 •13.011 .023 7 15.344 .021 12.805 .039 8 v 15.441 .060 -12.455 .038 9 :15.741 .035 • 12.115 .096 .0 15.997 .094 11.700 .063 B 16.352 .048 10.850 .092 C 16.419 .038 10.647 . .126 "f"\ D 16.518 .020 10.552 .080 4 5 5 32314. 118 .030 32315.202 .003 32313.316 .053 6 15.277 .006 13.197 .039 7 15.315 .006 13.009 .032 8 15.438 .008 12.657 .015 9 15.707 .039 12.299 .022 0 15.939 .050 11.979 .039 A 16.162 .033 11.651 .005 B 16.219 .022 11.390 .030 C 16.334 .027 11.108 .046 D 16.433 .016 10.876 .026 5 9 * 3 32314. 262 .027 32315.148 .040 32313.411 .037 . 4 15.209 .073 13.172 .009 5 15.343 .018 12.984 .054 6 15.397 .046 12.836 .054 7 15.662 .079 12.377 .035 8 15.955 .003 12.111 .021 9 16.380 .017 11.575 .089 0 16.487 .029 11.170 .140 A 16.666 .028 10.641 .075 D 16.626 .054 10.529 .054 1 6 9 TABLE X I I I (cont'd) , n Level Band Run T , . Label C Components in AC U -1 ' cm AU L AL OD 39 3 32397.914 .031 32398.657 .040 32397.033 .033 (0,0) 4 98.730 .056 96.893 .048 5 98.870 .042 96.513 .071 6 98.926 .067 96.357 .110 7 • 98.994 .027 96.026 .062 8 99.042 .0^ 6 95.736 .049 9 99.152 .055 95.466 .087 0 99.180 .070 94.958 .090 D - 99.305 .078 94.700 .046 35 5_ 32397.956 .023 32398.680 .024 32396.750 .062 6 98.794 .014 96.667 .022 ? 98.930 .037 96.453 .016 8 98.969 .006 96.225 .021 9 99.009 .010 95.916 .023 0 99.039 .013 95.748 .020 A 99.114 .024 95.608 .035 D 99.208 .029 95.553 .109 OH 53fi 1 31733.765 .043 31734.235 .040 31733.380 .016 (1,1) P 2 34.310 .040 33.340 .023 3 34.404 .040 33.140 .028 4 34.407 .028 32.915 .017 5 34.536 .045 32.856 .008 6 34.775 .039 32.703 .027 7 34.942 .017 32.465 .044 8 35.004 .023 32.187 .031 9 35.044 .010 31.995 .009 0 35.043 .046 31.884 .027 D 35.162 .037 31.769 .036 55 2 31733.813 .058 31734.399 .022 31733.327 .045 3 34.484 .048 33.218 .029 4 34.455 .036 33.126 .019 5 34.648 .061 33.020 .038 - 6 34.877 .025 32.619 .049 7 34.932 .012 32.300 .075 8 35.140 .048 32.086 .047 59* 5 31733.742 .034 31734.859 .017 31732.377 .003 6 35.029 .007 32.123 .066 7 35:084 .028 31.853 .010 8 35.345 .059 31.558 .097 9 • 35.466 .243 30.916 .045 1 7 0 TABLE X I I I (cont'd) •D J n ~ L e v e l Components i n cm~^ Band Run T , , n » „ T T L a b e l C AC U OH 590 ( 1 , 1 ) v cont* d 1 31733.766 .045 31734.184, 31733.253 2 34.427 33.163 3 34.552 33.073 4 34.635 32.976. 5 34.816 32.754 6 35.017 32.386 7 35.031 32.066 8 35.385 31.802 9 35.732 31.421 #see f o o t n o t e , T a b l e XIV 171 the asymmetry of the P ^ 2 ( l ) l i n e should always be n e g a t i v e ) . T h i s adjustment o f C may be viewed as a "smoothing" of e r r o r i n the s i n g l e measurement of C, much as the s l o p e - f i t t i n g process i s a smoothing of e r r o r s i n the multiply-measured U and L. 4. Higher Order C o r r e c t i o n s The e f f e c t s of higher order terms were i n c l u d e d . E x p r e s s i o n s (33) and (35) were e v a l u a t e d u s i n g the m a t r i c e s o f Eq. (18) and Eq. (59). The r e s u l t s f o r the 2 £ s t a t e were i n agreement w i t h Eq. (36). For the 2 n s t a t e , i n the approximation t h a t ( i n the n o t a t i o n o f Chap. IV) b 2 = l , d 2=0, and ( E < I J J In | u> > / 6 ) 2 i s l a r g e , the r e s u l t i n g e x p r e s s i o n s s i m p l i f y to 2 wn * ±-h n - + T4rT;(en/Bn>+ 2-BTTT ^ \/*\) • ("D No allowance f o r i n t e r m e d i a t e c o u p l i n g has been made i n the p r e c e d i n g formula ( d e s p i t e the erroneous use of " B e f f " i n the denominators o f these e x p r e s s i o n s i n S c a r l and D a l b y 7 7 ) . The d i p o l e moment r a t i o V^/v^ w a s f i r s t c a l c u l a t e d without any high e r order c o r r e c t i o n s , as d e s c r i b e d i n Sec. 2 above. The r e s u l t i n g v a l u e o f y was used i n Eq. (36) and the terms o f hi g h e r than second order were c a l c u l a t e d and s u b t r a c t e d from the observed p o s i t i o n s of U and L. T h i s 172 was i n p r a c t i c e done by a s m a l l computer program c a l l e d HIGH ORDERS which accepted the cards f o r i n p u t to QUAFF and repunched them wi t h c o r r e c t e d U and L v a l u e s . The c o r r e c t e d v a l u e s were r e - r u n by QUAFF. When the new value of was i n t r o d u c e d i n t o HIGH ORDERS, and the o r i g i n a l wavenumbers re p r o c e s s e d , no s i g n i f i c a n t d i f f e r e n c e s were found from the p r e v i o u s run, and so t h i s data was not r e r u n through QUAFF. The accuracy of the data f o r OH, v = l , d i d not appear to j u s t i f y such f i n e c o r r e c t i o n s , and none were made i n t h i s case. 5. C o r r e c t i o n s f o r Intermediate Coupling The v a l u e of W should be c o r r e c t e d from i t s case (a) value of 2/27B n (Eq. 126) to account f o r the f a c t t h a t c o u p l i n g i n the 2 n s t a t e of OH and OD i s intemediate between Hund 1s cases (a) and (b). While the J=£ l e v e l i s pure case (a) and needs no c o r r e c t i o n i n f i r s t o r d e r , the second order term r e q u i r e s c o r r e c t i o n i n two ways: (1) the energy l e v e l s are no longer g i v e n by BJ(J+1) so t h a t the energy denominator HB^) needs c o r r e c t i o n , and (2) s i n c e the two energy l e v e l s w i t h J=f have some mutual mixing i n t h e i r wavefunctions, the m a t r i x elements appearing i n Eq. (59) r e q u i r e c o r r e c t i o n . A c c o r d i n g to Eq. (81), i n an i n v e r t e d s t a t e the wavefunctions ip are g i v e n by 173 *3I = c j i > + c 2 | | > • i f = C l l ^ > ~ C 2 l i > ' U 3 2 ) where the f r a c t i o n s r e f e r to Q, the kets i n d i c a t e case (a) , J=f wavefunctions, and c^ and c 2 are g i v e n by Eq. (82) . By the methods i l l u s t r a t e d i n Chap. V, i t r e a d i l y f o l l o w s t h a t < ' * ^ l n z l * f 3 > = c i < i i | n 2 I H > = /2 C ; L/3 , and (133) < * i j ! n z l * i f > = _ C 2 < H | n 2 I H > = -/2c 2/3 . We t h e r e f o r e have 2 2 2 2 W = [ = ^ — + ± — ] e ; . (134) The v a l u e of Y was taken to be -7.501 i n OH (v=0), -7.800 i n OH (v=l) (from Lee, Tarn, Larouche, and Woonton 9 5), and -13.954 i n OD (Dousmanis, Sanders, and T o w n e s 9 3 ) . From these one o b t a i n s c 1 and c 2 as f o l l o w s : OH (0,0) OD (0,0) OH (1,1) c 2 = 0.03025 0.01138 0.02858 ( 1 3 5 a > 174 OH (0,0) OD (0,0) OH (1,1) c22 = 0.96975 0.98862 0.97142 . ( 1 3 5 b ) The energy denominators i n Eq. (134) were o b t a i n e d d i r e c t l y f o r both OH cases by averaging (over the s p i n s p l i t t i n g and A-doublings) the d i f f e r e n c e s between term v a l u e s g i v e n by Dieke and C r o s s w h i t e 8 8 ( p . 4 9 f ) , g i v i n g v = 0 v = 1 z z ~ r z l = -61.31 cm-'1" r-58.86 cm 1 F z z " F 2 Z = 126.26 cm" 1 126.785 cm - 1. (136) For , OD, term v a l u e s are not d i r e c t l y a v a i l a b l e , and the energy d i f f e r e n c e s were found from the l i n e i d e n t i f i -c a t i o n s of Oura and N i n o m i y a 8 9 and I s h a q 9 0 . For the f i r s t d i f f e r e n c e In Eq. (136), Q 2(2) - R 2 ( 1 ) = _ 3 1 ' 1 6 » 0 1 2 ( 2 ) - P12(D = -30.81, and P 2(2) - Q 2 ( l ) = -31.48, from Oura and Ninomiya, g i v i n g an average of -31.15 cm Ishaq's data was judged l e s s c o n s i s t e n t here. The second d i f f e r e n c e i n Eq. (136) was c a l c u l a t e d from P ^ ( l ) - P 1 2 ( l ) , y i e l d i n g an average o f 130.70 cm 1 from the two sources. In c a l c u l a t i n g f o r OH the 2 Z s t a t e term d i f f e r e n c e s were taken from Dieke and C r o s s w h i t e 8 8 ( p . 4 7 ) (although n e g l i g i b l e e r r o r r e s u l t s from the use o f 2E-J,) , g i v i n g -33.83 cm 1 f o r v=0, and -32.18 cm 1 f o r v = l . In OD, 2Bj, was used w i t h B^ , = 9. 037 cm 1 (Carlone and D a l b y 1 0 1 , p.1955). Spin s p l i t t i n g does not a f f e c t the NJ.=0 l e v e l , s i n c e i t i s s m a l l compared to 2B^. 175 6. R e s u l t s Our experimental data has been summarized i n Tables X I I I and XIV. S o l v i n g Eq. (130) f o r V^/v^'-y E / y n = / f B E ( l / k + l/3B n) (137) Tl$is r e l a t i o n , o p e r a t i n g on the slop e s (k) i n Table XV, does y i e l d the r e s u l t s l i s t e d i n t h a t t a b l e as "y /y , c o r r e c t e d " . 2J ,11 These r e s u l t s d i f f e r from those of S c a r l and D a l b y 7 7 because, i n t h a t paper, the mat r i x elements were not c o r r e c t e d f o r i n t e r m e d i a t e c o u p l i n g . Those v a l u e s l i s t e d as "y /y , from QUAFF" i n c l u d e no c o r r e c t i o n f o r c o u p l i n g , so t h a t U- = -2/27B_, and U = -1/6B . The e r r o r s l a b e l e d " S t a t i s -t i c a l " come from the output of the l e a s t squares f i t r o u t i n e , w h i l e those l a b e l e d "Assigned" were obtained by i n s p e c t i o n o f the range of slope f i t c o n s i s t e n t w i t h the graphs of F i g s . 28 - 30. T h i s data may be compared -with the microwave measurement by Weinstock and Z a r e 1 0 2 , u s i n g Stark e f f e c t l e v e l c r o s s i n g s , which y i e l d e d a c o n s i d e r a b l y lower v a l u e (1. 72±0.10'- D) f o r the d i p o l e moment of OD f o r v=0. SCF o r b i t a l c a l c u l a t i o n s by C a d e 1 0 0 g i v e 2.04 D i n OH (v=0). Weinstock and Z a r e 1 0 2 r e f e r ( t h e i r r e f s . nos. 17 and 25) to c a l c u l a t i o n s by S. Green g i v i n g 1.96 D, and some other unpublished c o n f i g u r a t i o n i n t e r a c t i o n c a l c u l a t i o n s by Mulder and L e s t e r y i e l d i n g 1.9 D, although the i s o t o p e s and v i b r a t i o n a l s t a t e s were not s p e c i f i e d . While these c a l c u l a t i o n s are i n good agreement with r e s u l t s f o r OH 176 TABLE XIV Data f o r v^/v^: U and L as C o r r e c t e d f o r High Order Terms, x and y as Used by QUAFF. Band Run 7 Label D L X Ax ' y Ay OH 5 9 £ 3 32314.895 32313.456 0.247 .077 2.071 .170 (0,0J 4 14 .974 13.308 0.180 .080 2.776' .209 5 15.205 13.128 .231 .056 4.314 .106 6 15.275 13.011 .184 .055 5.126 .108 7 15.342 12.805 .045 .067 6.436 .225 8 15 .437 12.455 -.210 .087 8.892 .424 9 15.734 12.114 -.254 .114 13.104 .740 0 15.985 11.698 -.419 .124 18.378 .970 B 16.327 10.843 -.932 .115 30.074 1.138 C 16.386 10.638 -1.078 .141 33.039 1.513 D 16.481 10.541 -1.080 .096 35.284 .980 45 5 32315.201 32313.316 - .309 .080 3.553 .200. 6 15.276 13.197 • .270 .072 4.322 .164 7 1 5 . 3 1 3 13.009 .119 .068 5.308 .150 8 15 .435 12.657 -.111 .062 7.717 .094 9 15.702 12.299 -.202 .075 11.580 .305 0 15.931 11.978 -.294 .080 15.626 .501 A 16.150 11.649 -.404 .069 20.259 .300 B 16.202 11.386 -.615 .071 23.194 .358 C 16.212 11.102 -.789 .080 27.1^4 .556 D 16.405 10.868 - . 9 3 0 .067 30.658 • 338 59*3 32315.148 32313.411 .162 .077 3.014 .189 4 15.208 13.172 -.011 .091 4.145 .300 5 15.342 12.984 -.065 .078 5.560 .268 6 15.395 12.836 -.160 .089 - 6.548 .363 7 15.658 12.377 -.356 .102 10.765 .567 ' 8 15.948 12.110 -.333 .058 14.730 .163 9 16.364 11.572 -.455 .105 22.963 .868 0 16.460 11.163 -.768 .153 28.058 1.515 A 16.570 10.630 -1.191 .097 35.284 .951 D 16.585 10.596 -1.290 .094 36.833 . .927 Band Run 177 TABLE XIV (cont'd) L e v e l ib e l D L X Ax y Ay 3 3 2 3 9 8 . 6 5 5 3 2 3 9 7 . 0 3 3 . 1 2 0 . 0 8 1 2 . 6 3 1 . 1 6 8 4 9 8 . 7 2 6 9 6 . 8 9 2 . 0 5 6 . 0 9 6 3 . 3 6 4 . 2 7 1 5 9 8 . 8 6 3 9 6 . 5 1 1 — . 1 8 9 . 1 0 5 5 . 5 3 2 . 3 8 8 6 9 8 . 9 1 5 9 6 . 3 5 4 — . 2 9 4 . 1 4 3 6 . 5 5 9 . 6 6 0 7 9 8 . 9 7 6 9 6 . 0 1 9 - . 5 6 7 . 0 9 2 8 . 7 4 4 . 4 0 0 8 9 9 . 0 1 4 9 5 . 7 2 5 — . 8 2 4 . 0 8 7 1 0 . 8 1 8 . 4 0 0 9 9 9 . 1 1 0 9 5 . 4 4 7 - 1 . 0 0 6 . 1 2 0 1 3 . 4 1 8 . . 7 5 4 0 9 9 . 1 1 9 9 4 . 9 2 9 - 1 . 5 1 5 . 1 3 0 1 7 . 5 5 6 . 9 5 5 D 9 9 . 2 1 8 9 4 . 6 5 6 - 1 . 6 8 8 . 1 1 0 2 0 . 8 1 2 . 8 2 6 5 3 2 3 9 8 . 6 7 6 3 2 3 9 6 . 7 4 9 — . 1 5 1 . 0 8 1 3 . 7 1 3 . 2 5 6 6 9 8 . 7 8 8 9 6 . 6 6 6 — . 1 1 6 . 0 5 3 4 . 5 0 3 . 1 1 1 7 9 8 . 9 2 0 9 6 . 4 5 0 — . 2 0 0 . 0 6 1 6 . 1 0 1 . 1 9 9 8 9 8 . 9 5 5 9 6 . 2 2 0 - . 3 9 5 . 0 5 1 7 . 4 8 0 . 1 1 9 9 9 8 . 9 9 0 9 5 . 9 0 9 - . 6 7 1 . 0 5 2 9 . 4 9 3 . 1 5 5 0 9 9 . 0 1 3 9 5 . 7 3 8 — . 8 1 9 . 0 5 2 1 0 . 7 5 3 . 1 5 6 A 9 9 . 0 7 9 9 5 . 5 9 3 — . 8 9 8 . 0 6 3 1 2 . 1 5 2 . 2 9 6 D 9 9 . 1 6 9 9 5 . 5 3 6 — . 8 6 5 . 1 2 2 1 3 . 1 9 9 . 8 2 0 OD 39 (0,0; 35 178 TABLE XIV (cont'd) Band Run ^ , a ^ e ] -Level TJ X Ax y A y .085 .096 .731 .074 .111 .098 .924 .089 .014 .099 1.598 .123 - .208 .092 2.226 .098. .138 .097 2.822 .154 — .052 .098 4.285 .196 — .123 .098 6.126 .233 - .339 .094 7.924 .217 - .513 .087 9.030 .081 .602 .101 9.948 .336 - .599 .100 11.472 .350 .100 .126 1.149 .107 .076 .129 1.603 .142 — .045 .123 1.766 .108 .042 .136 2.650 .234 - .130 .128 5.090 .248 — .394 .139 6.917 .400 - .400 .134 9.303 .410 — .328 .070 6.150 .084 — .412 .095 8.433 .385 — .627 .074 10.414 .192 — .660 .132 14.288 .858 -1.181 .256 20.566 2.241 .181 .090 .867 .090 — .028 .090 1.598 .090 .007 .090 2.187 .090 — .077 .090 2.752 .090 - .048 .090 4.244 .090 _ .215 .090 6.912 .090 .520 .090 8.773 .090 — .430 .090 12.802 .090 — .465 .090 18.516 .090 OH 53'fi 1 (Same as Table II, (1,1) . 2 6 7 8 9 0 55 2 3 4 5 6 7 8 59* 5 6 7 8 ^ n o high order 4 corrections applied) 5 59p* 1 2 3 4 5 6 7 8 9 -*Due t o the low q u a l i t y of the photograph, o n l y one re a d i n g per l e v e l was taken i n Run 59 f i (1,1) , so t h a t s t a t i s t i c a l e r r o r s are meaningless. 180 -0.4 -0.8 -1.2 -1.6 ASYMMETRY FIG. 29, Determination of p /fj,n for OD, v s O . 30 O LU OC < o V ) o I-- J 0_ 0 ) 20 10 V< I i L j L -0.4 -0.8 ASYMMETRY -1.2 -1.6 FIG. 30, Determination of pz/p>n for 0H ,v s l , 181 TABLE XV Di p o l e Moments i n the 2Z S t a t e Band OH (0,0) OD (0,0) OH (1,1) Slope from QUAFF -23.868 1.16, -9.7465 1.28. -15.872 1.3., E r r o r from S l o p e - F i t 0.01, 0.01, 0.05 y E / M n ' C o r r e c t e d 1.13, 1.27. 1.3, Assigned E r r o r ^-^^5 0. 04, 0.2, y (t a k i n g y from Table XI) 1. 94, 2.14. 2.1 E r r o r i n y, 0.08, 0. 08 0 0.4. 182 o b t a i n e d here, the source of the disagreement w i t h the measurements Weinstock and Zare remains u n c l e a r . N e i t h e r i s i t c l e a r t h a t the l a r g e r v a l u e of o b t a i n e d f o r OD (as compared to OH) i s s i g n i f i c a n t . 1L 5 l i -12-^1 L i n e HL 9R As noted i n S c a r l and D a l b y 7 7 , the P]_2^ 2^ l i n e i n OD appears r a t h e r s u r p r i s i n g l y l i k e a t r i p l e t i n high f i e l d s . T h i s may be seen i n the (0,0) band i n F i g . 26, and i n the (1,1) band i n F i g . 31. T h i s l i n e i s a c t u a l l y a composite of twelve components from both the P 2 ^ a n < ^ P ^ 2 ( 2 ) . The Stark e f f e c t i n the upper s t a t e i s comparable t o , and i n t e r a c t s s t r o n g l y w i t h , the s p i n s p l i t t i n g . The r e s u l t i n g Stark e f f e c t s are g i v e n by Eqs. (72) and (75). Due to the i n a c c u r a c y of the o l d e r analyses of the OD spectrum, i t was f e l t a d v i s a b l e to remeasure the s p l i t t i n g constant y from our p l a t e s . Averaging the s p l i t t i n g between main and s a t e l l i t e l i n e s d i v i d e d by (N+£), we get y(OD, V=0) = 0.121 ± 0.005 cm - 1 . (138) T h i s i s i n agreement w i t h a v a l u e of 0.125 cm c a l c u l a t e d from the formula and data of Dousmanis, Sanders, and Townes 9 3 (p.1743) under the assumption t h a t the matrix elements of L^ , are the same i n upper and lower s t a t e s . Using the s p i n - s p l i t t i n g g i v e n by Eq. (133) and a A-doubling i n the 2 I I - , , J=f l e v e l of 0.197 c m - 1 He"*" 4685.7 Fig. 31 The (1,1) A + X band of OD at high f i e l d . 184 ( c a l c u l a t e d from the formulae and data of Dousmanis, Sanders, and T o w n e s 9 3 ) , we o b t a i n the net Stark e f f e c t s shown i n F i g . 32. In t h i s f i g u r e , a l l Stark e f f e c t s are to s c a l e with r e s p e c t to the A-doubling and s p i n s p l i t t i n g . The components a r i s i n g from these l e v e l s are shown i n F i g . 33 separated i n t o ir and a components (AM=0 or ±1, r e s p e c t i v e -l y ) . The p o s i t i o n s of the observed components ( p l a t e 39) are i n d i c a t e d by the l a r g e c i r c l e s ( i n f o r m a t i o n on t h e i r p o l a r i z a t i o n i s not a v a i l a b l e ) . The bar l e n g t h s i n F i g . 33 are p r o p o r t i o n a l t o the component i n t e n s i t i e s , which were c a l c u l a t e d from wavefunctions c o r r e c t e d to f i r s t order i n the Stark e f f e c t , and u s i n g the formulae of E a r l s 8 7 f o r unperturbed i n t e n s i t i e s . In p r i n c i p a l , a p r e c i s e d e t e r m i n a t i o n of the r e l a t i v e i n t e n s i t i e s among the Stark components should y i e l d i n f o r m a t i o n about the s i g n s of the d i p o l e moments (see, f o r example, Thomson and D a l b y 7 h ) . Such an e v a l u a t i o n of i n t e n s i t i e s i s not p o s s i b l e , however, c o n s i d e r i n g the poorness of f i t between observed and c a l c u l a t e d components i n F i g . 33. A comparison between the r e l a t i v e l y b r i g h t l e f t m o s t component (-1.5 cm ^) and the b a r e l y v i s i b l e compon-ent (?) near -0.7 cm ^ ( i n t e r p r e t i n g observed i n t e n s i t i e s as the sum of nearby components) shows t h a t the l e f t m o s t one i s s t r o n g e s t and the next one weakest when the d i p o l e moments i n the upper and lower s t a t e s are o p p o s i t e l y o r i e n t e d , but the p r e d i c t e d i n t e n s i t y of the l e f t m o s t component i s always l e s s than the other ( c o n t r a r y to o b s e r v a t i o n ) , so such i n t e r p r e t a t i o n i s i n s e r i o u s doubt. Note a l s o the absence of t CM ' » x AY -0.0 < * k 1 J I T " • x 2 u 2 1 S£COA/£ OR£)£f> 213 K V / C M F i g . 32 ! f'_«.:. '• S t a r k E f f e c t s i n |the P j ^ t 2 ) L i n e of OD v.y 08SSRVED 7r-. T i c . 33 I n t e n s i t i e s i n the P 1 2 ( 2 ) L i n e of OD 186 the component p r e d i c t e d t o l i e near 2.5 cm \ No simple assumption about observed p o l a r i z a t i o n s makes the e x p l a n a t i o n much c l e a r e r , but the a n a l y s i s does seem to throw some l i g h t on the p e c u l i a r o v e r a l l appearance of the l i n e . 187 CHAPTER X THE A 2 A STATE OF CH 1. The P r o d u c t i o n of the Spectrum o The A 2 A X 2n (0,0) band of CH a t 4314 A has been observed i n second order d u r i n g runs i n which the d i s c h a r g e v e s s e l contained cyanogen ( C 2 N 2 ) and hydrogen. (Unusably weak images of the band had a l s o been obtained while running dimethyl mercury, d i e t h y l z i n c , and methanol. See Chap. VI, Sec. 8.) Since no Balmer l i n e s appear i n t h i s r e g i o n of the spectrum, p r e l i m i n a r y exposures (5 - 10 min.) of H a were taken immediately preceding each photograph. At the s t a r t o f the main exposure, no o p e r a t i n g parameters other than g r a t i n g angle and p l a t e p o s i t i o n were changed. H^ was exposed i n f i r s t o rder f o r run 271 e , and i n second order P f o r runs 264. and 263 . o The 4 314 A band i s c h a r a c t e r i z e d by l a r g e and u b i q u i t o u s Stark e f f e c t s due to the f i r s t o r d e r Stark e f f e c t s i n the ground s t a t e and s t i l l l a r g e r ones i n the 2A s t a t e . However, even the s i m p l e s t l i n e (J = \ J = £•) has e i g h t p r i n c i p a l components and moreover, i s overlapped i n our p l a t e s by components of neighbouring l i n e s . The most i s o l a t e d l i n e s of low J are the Q-^(2) and Q-^(3) , the l a t t e r of 1 k which was a l s o used i n the e a r l i e r work of Phelps and 188 Phelps and D a l b y ^ 0 . Although these l i n e s are composed of fo u r t e e n and twenty p r i n c i p a l components, r e s p e c t i v e l y , they e x h i b i t c l e a r l y v i s i b l e envelopes which have the appearance of d o u b l e t s . Although (being i n second order) we have l o s t the r e s o l u t i o n which allowed Phelps t o make h i s measurement, we nonetheless have high enough f i e l d s t o render the envelopes r e a d i l y measurable. In a d d i t i o n we have e l i m i n a t e d c a l c u l a t i o n a l shortcomings i n Phelps' treatment. The three s p e c t r a used i n c a l c u l a t i n g the d i p o l e moment of the 2 A s t a t e are reproduced i n F i g . 34. These p l a t e s were c a l i b r a t e d u s i n g f r e q u e n c i e s g i v e n by Douglas and R o u t l y 1 0 3 ( p . 3 0 8 ) f o r the 4216 A (0,1) band of CH, as w e l l o o as the 4347.5 A and 4358.35 A l i n e s of mercury which appear on the same p l a t e s . 2. Energy L e v e l S t r u c t u r e The s p e c t r a o f the 2 A •> 2 n t r a n s i t i o n have been analyzed by G e r o 1 0 4 and v i b r a t i o n a l l y extended by K i e s s and B r o i d a 1 0 5 who g i v e improved s p e c t r o s c o p i c c o n s t a n t s . From K i e s s and B r o i d a we take f o r the c o u p l i n g c o n s t a n t s : Y = -0.0659 i n the 2 A s t a t e and Y = 2.00 i n the 2 n s t a t e . (Keiss and Br o i d a a c t u a l l y g i v e Y(Y - 4) = -4.02±0.05 f o r v=0 i n the 2 I I s t a t e , but i t must be noted t h a t v a l u e s o f Y(Y - 4) l e s s than -4 r e s u l t i n complex s o l u -t i o n s f o r Y. K e i s s and B r o i d a a l s o g i v e e m p i r i c a l formulae f o r the 2 n s t a t e A-doubling, but we w i l l p r e f e r the v a l u e s of Douglas and E l l i o t t 1 0 6 . The 2 A s t a t e may be taken t o 189 PLATE 263 £ 208.3 k V / c m •'•j - I -0,(2) 0,(3) PLATE 263 5 230.3 kV/cm I I 0, (2) 0.(3) PLATE 271 /? 247.3 k V / c m 0,(2) 0,(3) R, (I) Fig. 34 The (0,0) A •+ X band of CH have n e g l i g i b l y s m a l l A-doubling (Douglas and E l l i o t t 1 0 6 ) . Using the above v a l u e s of Y, the energy l e v e l s of the two s t a t e s may be found from Eqs. (80) and (112) . These are shown i n F i g . 35, i n which the l e v e l s are drawn to s c a l e w i t h i n each e l e c t r o n i c s t a t e . In the upper s t a t e , the F-^  l e v e l s have dropped s l i g h t l y below the F 2 l e v e l s of same N due to the small negative magnitude of Y. In the ground s t a t e , however, the p o s i t i v e Y i s c o n s i s t e n t with l e v e l s l y i n g lower than II | l e v e l s o f the same N, but the m u l t i p l e t i s r a t h e r c u r i o u s l y c o n s i d e r e d F 2 - The f a c t t h a t the F^ l e v e l s l i e h i g h e r than F 2 i s t y p i c a l of case (b) c o u p l i n g . The Assignment of M In e v a l u a t i n g d i p o l e moments from the 0-^(2) and Q^(3) l i n e s , we are f a c e d w i t h the problem of d e c i d i n g which components form the b r i g h t e s t p a r t s of the envelope. The r e l a t i v e i n t e n s i t i e s of the d i f f e r e n t components are p r o p o r t i o n a l to the squares of the t r a n s i t i o n moments between the two s t a t e s (see Chap. I I ) . Although the t r a n s i t i o n moment i s not e a s i l y e v a l u a t e d , i t s dependence on M i s r e a d i l y e s t a b l i s h e d . L i g h t which i s p o l a r i z e d p a r a l l e l to the e l e c t r i c f i e l d has i n t e n s i t y p r o p o r t i o n a l to the square of the matrix element of n z (where y = |y|n) which i s d i a g o n a l i n M (and a l s o i n J , s i n c e we are d e a l i n g w i t h the Q-branch). Thus, f o r p a r a l l e l p o l a r i z e d l i g h t . , the i n t e n s i t i e s are 1 9 1 F i g . 35 A 2A X 2 n Transitions i n CH 192 p r o p o r t i o n a l to M 2 (see Landau and L i f s h i t z J , p. 93). For l i g h t p o l a r i z e d p e r p e n d i c u l a r l y to the e l e c t r i c f i e l d , the i n t e n s i t y i s p r o p o r t i o n a l to < J M|n x + in^. J J M - 1 > 2, the M-dependence of which i s g i v e n by ^ (J - M + 1) (J + M) (see Condon and S h o r t l e y 6 1 * , p. 63)*. Under assumptions which w i l l be d e t a i l e d i n Sec. 6 below, the p o s i t i o n s of the v a r i o u s components may be c a l c u l a t e d to l i e as shown i n F i g . 36.. T h i s f i g u r e shows the expected components of the 0-^(2) and Q x(3) l i n e s i n a f i e l d of 208 kV/cm ( p l a t e 264 ?) with l e n g t h r e p r e s e n t i n g i n t e n s i t y , h o r i z o n t a l p o s i t i o n r e p r e s e n t i n g displacement i n cm ^ from the l i n e c e n t e r (the average of the c and d components), and the compon-ent being shown above or below the h o r i z o n t a l a x i s a c c o r d i n g as i t s p o l a r i z a t i o n i s p a r a l l e l or p e r p e n d i c u l a r . We see t h a t the averages of the components on each s i d e of c e n t e r f a l l i n roughly the same p o s i t i o n f o r both p a r a l l e l and p e r p e n d i c u l a r components, but t h a t the p a r a l l e l p o l a r i z e d components have the sharper peak ( p a r t i c u l a r l y on the outer s i d e ) . Since our p l a t e s are o f long exposures at low l i g h t i n t e n s i t i e s , we may expect the emulsion's r e c i p r o c i t y f a i l u r e to f u r t h e r emphasize strong components at the expense of weaker ones. Examining the s p e c t r a i n F i g . 34, we observe a sharp d o u b l e t appearance i n both l i n e s , and p a r t i c u l a r l y i n the b e t t e r r e s o l v e d 0^(2). T h i s appearance i s very d i f f i c u l t to e x p l a i n u n l e s s the p e r p e n d i c u l a r components are excluded. We s h a l l t h e r e f o r e assume t h a t we are p r i n c i p a l l y *Note t h a t the f a c t o r of ^ i s m i s s i n g from the e x p r e s s i o n s g i v e n by Landau and L i f s c h i t z l(p.93) . 193 Qi(r2) 208 M d 208 Kv/cw F i g . 36 P r e d i c t e d Components o f the 0^(2) and 0^(3) L i n e s 194 seeing p a r a l l e l p o l a r i z a t i o n f o r i n s t r u m e n t a l reasons. I t i s c l e a r from F i g s . 34 and 36 t h a t the assumption of P h e l p s 1 (p. 87) , t h a t the envelope corresponds to the o u t e r -most p e r p e n d i c u l a r component (M=J -> M=. J-l.L) , i s not warranted here. We i n s t e a d i n t e r p r e t the envelope as represent-i n g the t r a n s i t i o n M=J -*- M=J. 4. The Diagonal M a t r i x Elements of n_z Since both the 2II and 2 A s t a t e s are n e a r l y case (b), we must e v a l u a t e the e f f e c t of i n t e r m e d i a t e c o u p l i n g upon the d i a g o n a l case (a) matrix elements of n z , g i v e n by M f i/J(J+l). We use the n o t a t i o n bf Chap. V wherein | ^ Q J ' > r e p r e s e n t s the a c t u a l ( i n t e r m e d i a t e case) wavefunctions and \QJ > i s the pure case (a) wavefunction, but we should be u n u s u a l l y c a r e f u l s i n c e the n i n | ip A. + > i s a very bad quantum number whose assignment depends on correspondences such as Eqs. (83) and (85). In the case of the 2n s t a t e , Y = 2.00 which i s the s p e c i a l case of pure case (b). Here the wavefunctions | ip^j > and > a r e e x a c t l y g i v e n by Eq. (84) so t h a t c i = c2 = We can s c a r c e l y decide which of and $2 i n t h a t e quation should be c a l l e d | ip^j > , but i t does not matter s i n c e the d i a g o n a l m a t r i x elements are = \ J | n z | ^ J > + < \ J | n z | | J > 195 < * f l J | n z | * n J > = £ {£M / J ( J + 1 ) + f M / J ( j + l ) } = M/J(J+1) , (139) (Cont'd) which are independent of the ch o i c e o f wavefunctions i n the f i r s t l i n e . In the case of the 2A s t a t e , we make the a s s o c i a t i o n a p p r o p r i a t e to i n v e r t e d s t a t e s (analogous t o Eq. 83) and w r i t e *1 = I j > = c l ^ > + c 2 ^ (140) where c^ and c 2 are g i v e n by Eq. (82), wit h Y = -0.0659, as .c± = 0..'40087 c 2 = 0.91613 (J = f) (141) and c± = 0.49386 c 2 = 0.86954 (J = I) (142) With these c o n s t a n t s , the d i a g o n a l m a t r i x elements are r e a d i l y e v a l u a t e d as < * 1 | n z U 1 > = (0.26735)M < < r 2 l n z ^ 2 > = ( ° - 1 4 3 2 4 ) M • (J = I) (J = I) (143) 196 5. The S p l i t t i n g Formulae When a p a i r of A - d o u b l e t s becomes s u b s t a n t i a l l y mixed by the- Stark i n t e r a c t i o n , they are d e s c r i b e d by the wavefunctions i J > A and i f » s of Eq. (51) and i t i s e a s i l y shown t h a t when t r a n s i t i o n s occur between two s t a t e s (each of which c o n t a i n s such A - d o u b l e t s ) , then t r a n s i t i o n s from (fo r example) the i |> A of one s t a t e w i l l go to e i t h e r the i p g or the i |> A of the other s t a t e (but not both) , and the cho i c e w i l l depend on the r e l a t i v e s i g n s of the d i p o l e moments i n the two s t a t e s and not on t h e i r i n i t i a l p a r i t i e s . When the d i p o l e moments have the same s i g n s , then we have i r s -«-->" i p g and i f » A •*•-*• ^ A (see, f o r example, the s i m p l i f i e d d i s c u s s i o n i n P h e l p s 1 4 , p . 2 8 f f * ) . By o b s e r v i n g t h a t t h i s was the case f o r the 2 A •> 2 n bands o f CH, P h e l p s 1 k (p. 76f) showed t h a t the two d i p o l e moments here have l i k e s i g n s . F i g . 37 shows the allowed t r a n s i t i o n s at zero and high f i e l d s , and the r e s u l t i n g appearance of the l i n e . Thus the net s p l i t t i n g o f l i n e i s j u s t twice the d i f f e r e n c e between the Stark s h i f t s of the two components. Denoting the s p l i t t i n g i n the 0-^(2) l i n e by S 1, t h a t of the Q-^(3) l i n e by S 2, and usi n g the mat r i x elements of Eqs. (13 9) and (143) wit h M=J, we have *Note however the e r r o r a t the bottom of p.30 i n Phelps : The l a s t complete sentence on t h a t page should read: "This (changing the r e l a t i v e s i g n of the d i p o l e moments) i s e q u i v a l e n t to r e v e r s i n g the s i g n of the c o e f f i c i e n t . . . ( c 2 ) i n the d e s c r i p t i o n o f one of the s t a t e s and..." (changing the r e l a t i v e s i g n o f the d i p o l e moments). 1 9 ? seccTr.AL LInr. F i g . 37 Appearance of the S t a r k E f f e c t i n a A -*• n L i n e 0S6 0.42. H t | 0.S2 -| i 0,80 A X o A 264S 27i£ a, ( 3 ) F i g . 38 D i p o l e Moment of the A 2 A S t a t e s l S2 where 26^ i s the A-doubling of the 2 n , J=£ l e v e l , and 26 2 i s the A-doubling of the 2 n , J=£ l e v e l . (We have t r e a t e d the A-doubling i n the 2A s t a t e as n e g l i g i b l y small.) These equations may be s e p a r a t e l y s o l v e d f o r e. = y.E/hc, and thus f o r y „ . A A A^ 6. R e s u l t s The Q-^(2) and 0^(3) l i n e s shown i n F i g . 34 were measured on the comparator and t r e a t e d as simple d o u b l e t s . The measured v a l u e s of and S 2, as w e l l as the parameters E and e n = y n E / h c , are shown i n Table XVI along with the c a l c u l a t e d v a l u e s of y^. We have taken y n = 1.46 ± 0.06 from the work of Phelps and D a l b y 7 0 , -1 -1 and A-doublings 26 1 = 0.159 cm , 26 2 = 0.373 cm from Douglas and E l l i o t t 1 0 6 . The observed v a l u e s of y A are i l l u s t r a t e d i n F i g . 3*8. We have adopted ° - 8 9 3 ± ° - 0 4 5 D as a b e s t v a l u e of y . Reducing t h i s by 0.67 % g i v e s our r e s u l t : y A = 0..887 + 0.04 5 . (145) Some a d d i t i o n a l t o l e r a n c e has been i n c l u d e d as an allowance f o r such s y s t e m a t i c e r r o r s as may a r i s e from measuring the e l e c t r i c f i e l d i n non-simultaneous exposures of H (the 198 = 1.337e A - / ( 2 6 ^ ) 2 + ( 4 e n / 7 ) 2 ~ ~ (144) = 1.003e A - • ( 2 6 2 ) 2 + ( 4 E n / 9 ) 2 199 P l a t e 2645 264^ 271 p Phelps (cm" - 1) 1.289 ± .091 1.316 t .056 1.531- .078 — (cm" - 1) 0.945 ±.025 0.811 i.030 0.881 ±.075 0.373 E (kv/cm) 230.3 * 5.0 208.3i0.8 247.3*1.1 66. 5.65 + .26 5.11 i *21 6.06 ± .25 1.62 ^ ( J - 5/2) 0.8741 .038 0.906 +.028 0.901+ .029 — ^ ( J - 7/2) 0.898+ .036 0.887i.028 0.865 i.033 1.06 TABLE XVI Measurements and R e s u l t s f o r the 2 A S t a t e of CH 200 d i s p e r s i o n of which was c a l c u l a t e d from s t i l l other p l a t e s made a t the same g r a t i n g a n g l e ) . That such e r r o r s are reasonably s m a l l , and t h a t our assumption i s a p p r o p r i a t e concerning the i n t e r p r e t a t i o n of the apparent doublet s p l i t t i n g s (Sec. 3) are a t t e s t e d t o by the o v e r a l l c o n s i s t e n c y of the data and, i n p a r t i c u l a r , by the agreement between r e s u l t s from the Q^(2) and Q-^(3) l i n e s . I t should be p o i n t e d out t h a t the v a l u e of Eq. (145) , along w i t h the other data r e f e r r e d to i n t h i s s e c t i o n , was used i n c a l c u l a t i n g the s p l i t t i n g s of F i g . 37. The l a s t column i n Table XVI l i s t s the data used by P h e l p s 1 4 ( p . 8 7 ) i n h i s c a l c u l a t i o n . Phelps a r r i v e d a t a val u e of 1.13 ± .17 f o r p^, but, when c a r r i e d through the c a l c u l a t i o n w i t h c o r r e c t i o n s f o r i n t e r m e d i a t e c o u p l i n g , t h i s data g i v e s p^ = 1.06 ± .15 (maintaining Phelps' e r r o r a t 15%). T h i s i s w i t h i n t o l e r a n c e o f our r e s u l t . 201 CHAPTER XI THE X 3Z~ STATE OF NH 1._ Observed.Spectra The NH (imidogen) molecule has a p r i n c i p l e r e s o -o nance band at 3360 A which was f i r s t analyzed i n d e t a i l by " F u n k e 1 0 7 , and i s known to be the (0,0) band o f an A 3n^ -* X 3 £ t r a n s i t i o n . T h i s band was r e a n a l y z e d by D i x o n 1 0 8 who o b t a i n e d i t i n a b s o r p t i o n and thereby i d e n t i f i e d most of the 27 p o s s i b l e branches. The nearby (1,1) band has been r e a n a l y z e d by Murai and S h i m a u c h i 1 0 9 , who a l s o extended the i d e n t i f i c a t i o n s of s a t e l l i t e s i n the (0,0) band. The (1,1) band i s a l s o v i s i b l e on our p l a t e s , but i t s dense c e n t r a l r e g i o n i s overlapped by the head of the i n t e n s e o (0,0) band (at 3371 A) o f the second p o s i t i v e group of N 2, which always appeared as an i m p u r i t y . Another band t h a t i s o normally o b t a i n e d a t the same time i s the 324 0 A band due to a c 1n ^ a 1A, (0,0) t r a n s i t i o n which has been analyzed by s e v e r a l authors ( P e a r s e 1 1 0 , Dieke and B l u e 1 1 1 , Nakamura and S h i d e i 1 1 2 , and S h i m a u c h i 1 1 3 ) . The A 3 n X 3 Z ~ and c ln -> a 1L bands have been p r e v i o u s l y observed i n t h i s l a b o r a t o r y by I r w i n 7 1 and Irwin and D a l b y 7 2 who were ab l e to determine the d i p o l e moments o f three of these f o u r s t a t e s . Since the f i e l d s a v a i l a b l e t o Irwin and Dalby d i d not exceed 100 kV/cm, the 2 02 second order e f f e c t s due to the X s t a t e were too s m a l l t o permit the d i p o l e moment i n the ground s t a t e t o be e v a l u -o ated. In the experiments r e p o r t e d here, the 3360 A band has been observed with f i e l d s as high as 287 kV/cm, which i s s u f f i c i e n t to allow t h i s dipole-moment-to-be-measured with an e r r o r from i n t e r n a l c o n s i s t e n c y o f about 3%. The f o u r bands r e f e r r e d to above are v i s i b l e i n each exposure o f t e n with.~H g and He l i n e s i n second order An attempt was a l s o made to photograph the c * i i -*• i + 0 b lZ band at 4502 A. While i t i s b e l i e v e d t h a t l i n e s o f t h i s spectrum have been observed, they were much too weak to a l l o w second order e f f e c t s i n the b 1 I + s t a t e t o be observed. Our high-wavelength (even-numbered) p l a t e s c o n t a i n the low-J l i n e s of the A 3 IK ->• X 3 E ~ , (0,0) band, and these have been c a l i b r a t e d u s i n g h i g h - J l i n e s from t h a t same band (wavenumbers from D i x o n 1 0 8 ) , as w e l l as l i n e s from the (1,1) band (using the data of Murai and Shi-v. m a u c h i 1 0 9 ) and from the C 3 I I u -»• B 3 n g , 2nd p o s i t i v e (0,0) band of N 2 (using the i d e n t i f i c a t i o n s of C o s t e r , Brons, and van der Z i e l ^ 1 4 ) . The low wavelength (odd-numbered) o p l a t e s u s u a l l y c o n t a i n H^ as w e l l as the 4 922 A and o 5015 A helium l i n e s . These were c a l i b r a t e d u s i n g the o i d e n t i f i c a t i o n s of P e a r s e 1 1 0 f o r the 3240 A band, and h i g h - J l i n e s o f the t r i p l e t R-branch from D i x o n 1 0 8 . 203 2. Energy L e v e l S t r u c t u r e and Constants The A 3 n . -»• X 3Z bands o f NH have been suc-1 c e s s i v e l y analyzed, with the e x t r a c t i o n of r e v i s e d c o n s t a n t s , by F u n k e 1 0 7 , D i x o n 1 0 8 , Murai and S h i m a u c h i 1 0 9 , Horani, Ros-tas and L e f e b v r e - B r i o n 1 1 5 , and V e s e t h 1 9 . However, we only r e q u i r e t h a t the energy d i f f e r e n c e s between n e i g h b o r i n g l e v e l s be known i n order to e v a l u a t e second order e f f e c t s , and a v a l u e f o r the s p i n - c o u p l i n g parameter Y. We s h a l l use V e s e t h ' s 1 9 c o n s t a n t s : F o r the 3IK s t a t e (v=0) he g i v e s B n = 16.3018 3, and A = -34.79 4; f o r the 3 I + s t a t e , B = 16.374 8-. Thus we use Y = A/B n = -2.134 4 . (146) The magnitude of Y i s s m a l l enough so t h a t H e r z b e r g 3 (p.264) simply r e f e r s to the s t a t e as case ( b ) , although t h i s i s not accurate a t s m a l l N. The near e q u a l i t y o f the upper and lower s t a t e B v a l u e s g i v e s the band no d i s t i n c t head, and the R and P branches proceed f a i r l y u n i f o r m l y toward h i g h e r and lower wavenumbers (although the s l i g h t l y l a r g e r B^ , does cause a head i n the R branch near N=30) . The c e n t r a l p i l i n g up of the Q (and a s s o c i a t e d s a t e l l i t e ) branches forms the most marked f e a t u r e of the spectrum. Rather than r e l y on d e r i v e d constants to p r o v i d e energy d i f f e r e n c e s , we s h a l l d i r e c t l y use the term val u e s r e s u l t i n g from the measurements of D i x o n 1 1 6 . The f i r s t s e v e r a l of these have been l i s t e d i n Table XVII. F i g . 3 9 shows 204 TABLE XVII 1 1 7 Some Term Values of NH, from Dixon State Multiplet XL N J Parity Term (cm - 1) 3r*-5TT X TT: i l t i p l e  (ci 1 0 1 -0.68 3 1 0 — 30.88 1 1 2 — 31.83 2 1 1 - 32.68 3 2 1 — 96.89 1 2 3 - 97.05 2 2 2 - 98.01 1 3 4 — 194.86 3 3 2 - ' 194.95 2 3 3 - 195.90 1 4 5 — 325.12 3 4 3 — 325.39 2 4 4 — 326.23 3 0 1 0 '29826.26 3 0 1 0 - 29828.86 3 0 2 1 — 29878.91 3 0 2 1 - 29880.82 3 0 3 2 — 29970.91 3 0 3 2 - 29972.31 3 0 4 3 — 30097.79 3 0 4 3 — 30098.79 2 1 1 1 29805.90 2 1 1 1 - 29806.58 2 1 2 2 — 29865.40 2 , 1 2 2 - 29866.45 2 1 3 3 29959.99 2 1 3 3 — • 29961.35 1 2 1 2 _ 29769.90 1 2 .1 2 - 29769.90 1 2 2 3 — 29841.52 1 2 2 3 - 29841.62 1 2 3 4 — 29943.05 1 2 3 4 - 29943.28 206 these l e v e l s , drawn to s c a l e e x c e p t i n g the gap between e l e c -t r o n i c s t a t e s . Note the d i f f e r e n c e s i n the o r d e r i n g of the A-doublets and s p i n - s p l i t t i n g s , as compared to Dixon ( F i g . 1, p. 1173). 3. Choice of S p e c t r a l L i n e There are a number of c o n s i d e r a t i o n s i n d e c i d i n g which energy l e v e l s i n the 3 £ + s t a t e at which to look f o r second o r d e r e f f e c t s : Apart from problems of i n t e n s i t y and o v e r l a p p i n g , we must c o n s i d e r the magnitude of the Stark e f f e c t and c o m p l i c a t i o n s due to the presence of more than one v a l u e of M^ .. An i d e a l 3 £ l e v e l f o r study may seem to be the f l e v e l w i t h J=0, which i s s o l e l y comprised o f M=0. How-ever, an examination o f the m a t r i x of n f o r 3E z (column 3 of Eq. 21) shows t h a t , i f e n e r g i e s are g i v e n by BN(N + 1 ) , second order e f f e c t s e x a c t l y c a n c e l to zero, independent of B o r y l Some Stark e f f e c t i s r e - i n t r o d u c e d through i n t e r a c t i o n s w i t h s p i n - s p l i t t i n g (see Fig.39 and the f i r s t l i n e of Eq. 77), but t h i s e f f e c t i s q u i t e s m a l l i n NH a t f i e l d s a v a i l a b l e (due to the l a r g e magnitude o f the s p i n - s p l i t t i n g ) . Thus we must use a l e v e l w i t h J £ = l o r more, so t h a t p o l a r o i d may be needed to separate components of 207 d i f f e r i n g M^ , (note t h a t components of d i f f e r e n t M n are separated by f i r s t o rder e f f e c t s ) . The second order S t a r k e f f e c t s i n the l e v e l N=l, J = l are g i v e n by the second and t h i r d l i n e s (F^ and F^, c o r r e s p o n d i n g t o M £ = 0 and 1, r e s p e c t i v e l y ) of Eq. (77). The N=2, J = l (F 3) l e v e l has Stark e f f e c t s markedly s m a l l e r than the N=l l e v e l (roughly e 2/70B and -e 2/120B, n e g l e c t i n g i n t e r a c t i o n w i t h s p i n s p l i t t i n g , f o r the M=0 and 1 components of N=2, J = l ) . L a s t l y , l o o k i n g a t the l e v e l w i t h N=0, J = l (molecular ground s t a t e ) , t h e r e i s no s p i n s p l i t t i n g to c o n s i d e r , so t h a t a s t r a i g h t -forward a p p l i c a t i o n of the second order energy formula g i v e s : M=0: F ^ 2 ) = - e 2 (^  + §)/2B l. M=l: F ^ } = -e 2(£ + £)/2B We f i n d t h a t the Stark e f f e c t i s the same f o r both values of M, r e n d e r i n g s e p a r a t i o n by p o l a r o i d unnecessary! T h i s Stark e f f e c t i s c o n s i d e r a b l y l a r g e r than f o r the other l e v e l s , furthermore, so t h i s l e v e l i s the l o g i c a l choice f o r study. The S t a r k e f f e c t i n t h i s l e v e l w i l l be examined more c l o s e l y i n s e c t i o n 5, below. There are s e v e r a l p o s s i b l e l i n e s which i n v o l v e T a t r a n s i t i o n to the N=0 l e v e l , namely the R ^ t O ) , S Q 3 1 ( 0 ) , R P 3 1 ( 0 ) , S R 2 1 ( 0 ) , R Q 2 1 ( 0 ) , and R ^ O ) . (The l e f t s u p e r s c r i p t r e f e r s i n the standard way to the change i n N a s s o c i a t e d with the t r a n s i t i o n , and i s omitted when the same = -e 2/6B = -e 2/6B : (147) 208 as the main l e t t e r which d e s c r i b e s the change i n J . The numeric s u b s c r i p t s r e f e r t o m u l t i p l e t cases, as per F i g . 3 9 r and the parentheses e n c l o s e the value of N i n the lower s t a t e . ) T Among these, the R ^ ( 0 ) has not been observed i n t h i s work o r by Dixon, the Q^CO) i s overlapped by the R^(3), and the R^'(0) i s overlapped by the 0^(5) and a number of other l i n e s ( i n c l u d i n g N 2) i n a dense p a r t g of the spectrum. The R 2^(0) seems to appear very f a i n t l y i n some o f our p l a t e s but shows no Stark e f f e c t s as expected and might i n s t e a d be some i m p u r i t y l i n e . T h i s leaves the R P 3 1 ( 0 ) and R Q 2 1 ( 0 ) l i n e s . The P 3 i ^ ^ ^ s t n e b r i g h t e s t (even though i t s i n t e n s i t y should be zero a c c o r d i n g to the formulae of B u d o 1 1 7 ) . U n f o r t u n a t e l y , i t has a h i g h l y skewed p r o f i l e i n most of our p l a t e s , t r a i l i n g o f f vaguely on the low frequency s i d e . T h i s i s probably due to o v e r l a p p i n g by Stark components from the R Q 2 1 ( 4 ) a n d R 2 ^ 4 ) l i n e s i n the (1,1) band (Murai and S h i m a u c h i 1 0 9 , pp. 56 and 58). Our p r i n c i p a l r e s u l t s have t h e r e f o r e come from the R Q 2 ^ ( 0 ) , which i s the same l i n e used by I r w i n 7 1 ' 7 2 i n h i s study of the H s t a t e . We have p r i n c i p a l l y used the center component (Mn=0) which e x h i b i t s no f i r s t o r d e r e f f e c t , and i t has been measured w i t h good accuracy. The ^ 2 ^ ( 0 ) l i n e was a l s o measured, but these r e s u l t s were poor P and c o u l d o n l y c o n f i r m r e s u l t s o b t a i n e d from the Q 2^(0) l i n e . 209 4. Stark E f f e c t s i n the 3 n S t a t e Due to the e f f e c t s of s t r o n g s p i n - u n c o u p l i n g , the m a t r i x elements of the Stark e f f e c t (n z) are not g i v e n by the case (a) m a t r i x elements of Eqs. (22 ) - (24 ) , but r a t h e r by l i n e a r combinations of them. The groundwork f o r t h i s procedure has been l a i d i n Chap. V, where the c o r r e c t l y coupled wavefunctions f o r J= 0 , 1 , 2 , 3 were given by Eqs. (94 ), (99 ) , ( 1 0 5 ) , and (110) r e s p e c t i v e l y . The c o r r e c t l y coupled m a t r i x elements are immediately o b t a i n e d from the above-mentioned r e s o u r c e s . For example, the-matrix element of n z between the 3 n ] _ / J = l / M = 0 ^110^ and the 3 n Q , J=2, M=0 (V q2Q) s t a t e s i s found to be ^ 1 1 0 I n z 1 *020* = c 2 f 1 < 0 1 0 i n z | 0 2 0 > + 2 < 1 1 0 | n z 1120> = ( 0 . 5 1 4 4 ) ( 0 . 7 6 4 7 ) / V T 7 + (0 . 8575) ( - . 6 2 4 4 )/Vs" (148) = - 0 . 0 3 6 3 4 , where the c's are from Eq. ( 1 0 2 ) , the f ' s are from Eq. ( 1 0 5 ) , and the case (a) m a t r i x elements are from Eqs. (22 ) and (23 ). (We r e t a i n the p r e v i o u s n o t a t i o n i n which i ^ J J J M > a r e t * i e c o r r e c t l y coupled wavefunctions, and jfiJM > are the pure case (a) wavefunctions.) We have, of course, a l s o used the f a c t t h a t n z i s d i a g o n a l with r e s p e c t to i n a pure case (a) r e p r e s e n t a t i o n (changes i n the quantum number I are not allowed f o r d i p o l e t r a n s i -210 t i o n s ) . C o l l e c t i n g such m a t r i x elements up to J=3, we have the c o r r e c t l y coupled matrix elements of n z shown i n Eq. (149), wherein the columns (and, i m p l i c i t l y , the rows) are l a b e l e d by n, J p a i r s . 0,0 0,1 0,2 fi, J 1,2 2,2 0,3 1,3 2,3 0 0.49510 .29699 0 0 0 0 0 0 .49510 0 0 0 .23260 .12492 -.03504 0 0 0 0.29699 0 0 0.23260 -0 -0 .03634 .03634 0 .42107 0 .19753 0 0 0.48764 0 .07036 0 -.01898 0 0.12492 .42107 0 0 0 -.02235 .45996 .11607 0 -0.03504 .19753 0 0 0 0.00088 -.02649 .39904 0 0 0 0 .48764 -.02235 .00088 0 0 0 0 0 0 0 .07036 .45996 -.02649 0 0 0 0 0 0 -0 .01898 .11607 .39904 0 0 0 The Stark < e f f e c t p e r t u r b a t i o n s may now be c a l c u l a t e d i n a s t r a i g h t f o r w a r d manner from the u s u a l formulae Eqs.(29 ) - (35 ). We may note t h a t the a r i t h m e t i c of t h i s c a l c u l a t i o n has been s i m p l i f i e d by the f a c t t h a t we are l i m i t i n g o u r s e l v e s t o the case o f M n = 0 . T h i s means t h a t f i r s t o r d e r Stark e f f e c t s are a l l zero, and the A-doublets have no i n t e r a c t i o n w i t h each o t h e r . T h i s permits us to use d i r e c t l y the m a t r i x elements of n with r e s p e c t to the wavefunctions ^ Q J W without combining these i n t o the l i n e a r combinations d e s c r i b e d by Eqs. (48 ) and (49 )• P a r i t y i s conserved to f i r s t o r d e r , and we may t r e a t the r o t a t i o n a l l e v e l s as b eing nondegenerate. Thus i n c o n s i d e r i n g the Q2-|_(0) l i n e , we c o n s i d e r o n l y the higher A-doublet of 3 n 1 , J = l (with p o s i t i v e p a r i t y , see F i g . 39) which i s the upper s t a t e o f the allowed t r a n s i t i o n . For those l e v e l s which i n t e r a c t w i t h t h i s one through the m a t r i x of n z (Eq. 14 9 ) , . we c o n s i d e r o n l y the s u b l e v e l s of n e g a t i v e p a r i t y , and s i m i l a r l y f o r the (°) l i n e . The a c t u a l S t a r k e f f e c t c a l c u l a t i o n s have been accomplished by f e e d i n g the m a t r i x o f Eq. (149) (and the e n e r g i e s o f the a p p r o p r i a t e A-doublet s u b l e v e l s from Table XVII) i n t o a s m a l l computer program c a l l e d PIFIELD, w r i t t e n i n the language ALGOL W. Denoting the r e s u l t i n g n o r d e r term v a l u e s by F_ T.. , we f i n d F i i o = - ° - 0 0 5 8 6 5 e n (150) = -2.838 E 2 F j j f t = -0.0003235 e 2 = -0.1566 E 2 , (151) where E i s i n MV/cm, and we have used the r e s u l t of Irwin and D a l b y 7 2 f o r the d i p o l e moment (1.31 D). Higher order p e r t u r b a t i o n s were a l s o c a l c u l a t e d . T h i r d order c o r r e c t i o n s are n e c e s s a r i l y zero, s i n c e M n = 0 -Fo u r t h order e n e r g i e s are found to be F110 = 5 - 7 6 3 * 1 0 ~ 7 = 0.134 E 4 (152) < 0.0005 (cm - 1) -3.842 x 10~ 7 £ 4 -.0.08998 E 4 -0.0003 (cm - 1) . The l a s t l i n e i s v a l i d f o r E = 0.2 88 MV/cm, the h i g h e s t usable f i e l d o b t a i n e d i n t h i s work. We may thus judge t h a t h i g h e r orde r e f f e c t s i n the 3n s t a t e are s a f e l y n e g l e c t e d . 5- Stark E f f e c t s i n the f j T St a t e (N=0) Examining the second order Stark e f f e c t i n the J = l , N=0 l e v e l o f the (v=0) X 3 E ~ s t a t e , we note t h a t Eq. (147) n e g l e c t e d the e f f e c t s o f s p i n s p l i t t i n g i n the N=l l e v e l s * These may be p a r t i a l l y i n c l u d e d by s u b s t i t u t i n g the a p p r o p r i a t e term d i f f e r e n c e s from Table XVII f o r the energy denominator 2B. Denoting term v a l u e s by F N J M ' t n e second order c o r r e c t i o n s become ,(4) 000 See a l s o the end of Sec. 7, below. 213 M=0 : 32.51 2/9 0.01036 e 2 (154) M=l: 1/6 32.51 0.01012 E 2 (155) 2.8547 y 2 E 2 where the 3E s t a t e d i p o l e moment y i s i n debye and the e l e c t r i c f i e l d E i s i n u n i t s of 10 6 V/cm. Thus a d i f f e r e n c e o f some 2% appears between the components of d i f f e r e n t M. There i s no hope of r e s o l v i n g components so c l o s e , but we should r e s i s t the temptation to average the two, and enquire as to t h e i r r e l a t i v e i n t e n s i t y . In an t r a n s i t i o n , we have J = J so t h a t (Landau and L i f s h i t z 1 , 11 Li p. 93) the i n t e n s i t y of l i g h t p o l a r i z e d p a r a l l e l to the e l e c t r i c f i e l d (M^ = M^ ,) i s p r o p o r t i o n a l to M 2. Since we are d e a l i n g w i t h the c e n t e r component (M n=0), we see t h a t the i n t e n s i t y a r i s i n g from t r a n s i t i o n s to the M £ = 0 s u b l e v e l must be zero. i s somewhat r e l a x e d by the presence of the e l e c t r i c f i e l d . T h i s has been i n v e s t i g a t e d i n the standard manner, assuming t h a t the i n t e n s i t y of the c e n t e r component of the Q 9 1 (0) T h i s " s e l e c t i o n r u l e " a g a i n s t the M =0 component z we have a 214 while n x and n^ y i e l d the i n t e n s i t y ( p e r p e n d i c u l a r component) when M =1. These two wavefunctions are expanded Li to f i r s t o r d e r , by Eq. (30), and terms i n the i n t e n s i t y of higher than second order i n E are dropped. The r e s u l t i n g TT ^ y e x p r e s s i o n f o r < ^ |n|^ > c o n t a i n s s i m i l a r m a t r i x elements, r e l a t e d t o the (square r o o t s of) unperturbed i n t e n s i t i e s of the corresponding components (same Mn and M£) of o t h e r l i n e s . These i n t e n s i t i e s were c a l c u l a t e d from the formulae o f B u d o 1 1 7 , and m u l t i p l i e d by the r a t i o o f the i n t e n s i t y of the component i n q u e s t i o n (M=0 •*• M=M , b e a r i n g i n mind t h a t i f Mj,>0 i t l a b e l s two degenerate s u b l e v e l s ) t o the t o t a l f o r the l i n e (Condon and S h o r t l e y 6 4 , p.63). Assuming t h a t y = 1.3.: and y = 1.5, t h i s procedure g i v e s JL Li I ( p a r a l l e l ) = 0.22 E 2 (156) I (perpendicular) = (0.94 - 0.04 E ) 2 (157) wi t h E i n MV/cm. Thus, even a t the h i g h e s t f i e l d s used i n our experiments (0.3 MV/cm), the r a t i o I ( p a r a l l e l / I ( p e r p e n d i c u l a r ) i s l i t t l e more than two per cent. We s h a l l c o n s i d e r t h a t t h i s r e s u l t j u s t i f i e s t r e a t i n g the ce n t e r component of the Q^CO) l i n e as being s o l e l y the per p e n d i -c u l a r component (Mn = 0 ->• M^=l) . Stark e f f e c t s have been i n v e s t i g a t e d i n t h i r d and f o u r t h o r d e r , u s i n g Eqs. (33) and (35) , m a t r i x elements from Eq. (21) , and energy l e v e l s g i v e n by BJ,N(N+1) . T h i r d 215 order e f f e c t s are zero as expected. With B^ , = 16.375 ( V e s e t h 1 9 ) , the f o u r t h o r d e r e f f e c t i n the ground s t a t e i s found to be M=0: -0.285 v*Eh (158) M=l: -0.327 v*Ek (159) where y^ i s i n debye and E i s i n MV/cm. These formulae w i l l be f u r t h e r e v a l u a t e d a t the end iof :Sec. 7 below. 6. Experimental R e s u l t s moment of the X 31 s t a t e are shown i n F i g . ,40. These were c a l i b r a t e d as d i s c u s s e d i n Sec. 1, o r i g i n a l l y w i t h separate c a l i b r a t i o n s f o r the l i n e tops (zero f i e l d ) and bottoms (maximum f i e l d ) . Although these c a l i b r a t i o n s were used f o r the r e l a t i v e l y i n a c c u r a t e ' P ^ f O ) l i n e , i t was f e l t t h a t improved r e s u l t s c o u l d be obt a i n e d f o r the R Q 2^(0) l i n e by measuring the Stark s h i f t s " d i r e c t l y " by a c y c l i c process o f checking the l i n e top, the l i n e b6ttom, and the v e r t i c a l alignment ( t h i s l a s t u s i n g nearby N 2 l i n e s ) . The r e s u l t i n g measurements are shown i n Table XVIII, together w i t h the e l e c t r i c f i e l d s o b t a i n e d as d e s c r i b e d i n Chap. V I I , Sec. 2. In p l a t e 27 5y, the top of the R P 3 1 ^ l i n e i s not v i s i b l e , so i n t h i s case we have used the average p o s i t i o n o f the l i n e top (29826.822 cm"'1') from The f i v e photographs used i n d e r i v i n g the d i p o l e 216 N2(0,0),2N DPos. Fig. 40 The (0,0) Anr* X band of NH. 217 TABLE XVIII Measured Stark Shifts in the RQ 2 1(0) and ^ ^ ( 0 ) Lines in NH A'—>X (0,0) , Plate RQ 2 1(0) (microns) RQ 2 1(0) (cm"1) R p 3 1 ( 0 ) (cm"1) E (kV/cm) (lO"3X(kV/cm)2) 269^ 17.01 1.0 0.100+ .006 0.27± .01 197.9±1.0 (39.16± .40) 273* 22.0± 5.0 0.120±.020 0.21± .06 213.2± 3.4 (45.45* 1.4) 27 3p 20.3*4.0 0.119 i.024 0.16 ±.03 2l6.7i0.8 (46.96± .34) 275<x 25.6 ±2.8 0.150±.016 0.26± .03 230.9 ±1.2 (53.31^ .55) 2753" 39.9 i3.1 0.235 i.018 0.53(?) 287.8 + 2.3 (82.83* 1.3) Note: the numbers in parentheses under each value of E represent E2 after multiplication by 10~-\ 218 the other four exposures. The data f o r the R Q 2 1 ( 0 ) l i n e from Table XVIII are graphed i n F i g . 41 a g a i n s t the square o f the e l e c t r i c f i e l d . A b e s t f i t s t r a i g h t l i n e was f i t t e d t o t h i s data by means of an IBM l i b r a r y l e a s t squares f i t program c a l l e d DPLQF (Double P r e c i s i o n L e a s t Squares F i t ) . The r e s u l t i n g s l o p e , which we c a l l R, i s A F ( 2 ) -i R = 2^ = 2.74 ± 0.3 0 (cm VMV 2/cm 2) (160) T h i s l i n e i s shown i n F i g . 41, as are the e r r o r l i m i t s of Eq. (160). T h i s t o l e r a n c e i s a c a u t i o u s estimate based on i n s p e c t i o n of F i g . 41. (The DPLQF program gave an e r r o r of only ±0.06.) Now the net St a r k s h i f t i s j u s t the d i f f e r e n c e between t h a t i n the 3n s t a t e and t h a t i n the ground s t a t e (counting s h i f t s to higher frequency as p o s i t i v e ) : A F ( 2 > = A F < 2 > - A F < 2 > 11 1 (161) = 2.8547 y 2 E 2 - 2.8382 E 2 , s u b s t i t u t i n g from Eqs. (150) and (155). S o l v i n g f o r y Li g i v e s y = /(R+2.84 ± 0.13J/2.855 (162) where the e r r o r shown on the term from the 3 n s t a t e i s from Irwin and Dalby's e r r o r i n y n ( = 1.31 ± 0.03). Using the 220 value o f R g i v e n by Eq. (160), we have Z 1.39 8 ± 0.04 3 (uncorrected) (163) and, a f t e r r educing t h i s f i g u r e by 0.67% i n accordance with Chap. V I I , Sec. 2, we a r r i v e at P E = 1.38 9 ± 0.04 3 (164) f o r the d i p o l e moment of the X 3E s t a t e of NH. A wide v a r i e t y o f ab i n i t i o c a l c u l a t i o n s i s a v a i l a b l e f o r comparison, and have been brought t o g e t h e r i n Table XIX. Although no e x p e c t a t i o n s c o u l d be d e r i v e d from such a range of r e s u l t s , the most r e c e n t and s o p h i s t i c a t e d o f these tend toward v a l u e s h i g h e r than our own. The suggestion by Weinstock and Z a r e 1 0 2 t h a t "the o p t i c a l method o f Dalby and coworkers may y i e l d s y s t e m a t i c a l l y high d i p o l e moment v a l u e s " does not seem to apply i n t h i s case u n l e s s these r e c e n t c a l c u l a -t i o n s are s t i l l q u i t e i n a c c u r a t e . 7. Other C o n s i d e r a t i o n s p o s i t i o n found a p p l i c a b l e i n the case o f the Z s t a t e i n OH (see Chap. IX, Sec. 3 ) , the q u e s t i o n might be r a i s e d as t o whether some s i m i l a r c o r r e c t i o n should be a p p l i e d to NH. Since f i r s t o r d e r e f f e c t s are not simultaneously o b s e r v a b l e i n most cases (see Sec. 8 ) , the same method In view of the c o r r e c t i o n s to the z e r o - f i e l d 221 TABLE XIX Comparison wi t h P u b l i s h e d C a l c u l a t i o n s o f the Dipol e Moment of NH (X 3E~) Source D i p o l e Moment (debye) T h i s work l - 3 8 g ± 0.043 Bender and D a v i d s o n 8 5 1.587 Cade and Huo 8 l + 1.627 H a r r i s o n 1 1 8 1.016 Kouba and O h r n 1 1 9 0.36 J o s h i 1 2 0 1.91 Griin 1 2 1 (various b a s i s sets) 1.44-2.17 B o y d 1 2 2 0.908 Krauss and Wehner 1 2 3 0.85 K r a u s s 1 2 1 * 0.90 H i g u c h i 1 2 5 1.24 H u r l e y 1 2 6 ( v a r i o u s methods) 0.83-1.9 4 222 cannot be used here. One can look a t the f i v e data p o i n t s i n F i g . 41 c o l l e c t i v e l y , however. A sm a l l tendency may be noted wherein the p o i n t s at higher f i e l d l i e above the best f i t l i n e , w h i l e p o i n t s a t lower f i e l d s l i e below I i t . Thus, i f the b e s t f i t l i n e were not c o n s t r a i n e d to go through the o r i g i n , we should o b t a i n a h i g h e r s l o p e . A reasonable f i t to the f i v e p o i n t s alone w i l l g i v e R = 3.025 (cm 1/MV 2/cm 2), and a d i p o l e moment of 1.431 D from Eq. (162). Making the 0.67% r e d u c t i o n , would g i v e y^ , = 1.421 D, which i s s t i l l w e l l w i t h i n t o l e r a n c e of Eq. (164) . A c c e p t i n g t h i s note of c a u t i o n t h a t the t r u e v a l u e may l i e to the high s i d e , we s h a l l continue to h o l d the r e s u l t of Eq. (164) as our b e s t v a l u e . While measurements of the ^31^0) l i n e are too i n a c c u r a t e to be used i n e v a l u a t i n g y^., they may be checked f o r c o n s i s t e n c y with the above r e u l t . The d a t a f o r t h i s l i n e i n Table XVIII i s graphed i n F i g . 42. S o l v i n g Eq. (162) f o r R, s u b s t i t u t i n g Eq. (163) f o r y , and s u b s t i t u t i n g 0.157 f o r the 2.838 (Eq. 151), we d e r i v e a slope R = 5.42. T h i s slope i s r e p r e s e n t e d by the s t r a i g h t l i n e i n F i g . 43, which i s as good a f i t as one c o u l d expect f o r such data ( p a r t i c u l a r l y r e c a l l i n g the f a c t t h a t the l i n e c o u l d not be measured a t zero f i e l d i n p l a t e 275V). We can see, however, p t h a t the l a r g e r d e v i a t i o n s of the P3]_(0) l i n e (as compared to the f"Q2^(0)) v i s i b l e i n F i g . 40, i s i n accordance with t h e o r e t i c a l e x p e c t a t i o n s . When we use our d i p o l e moment (Eq. 163) together w i t h our h i g h e s t f i e l d (o.2878 MV/cm) i n the p r e v i o u s ro 224 e x p r e s s i o n s f o r f o u r t h order e f f e c t (Eqs. 158 and 159), we f i n d t h a t these c o r r e c t i o n s are l e s s than 0.01 cm 1 i n the energy l e v e l s under c o n s i d e r a t i o n . Fourth order e f f e c t s w i l l t h e r e f o r e be n e g l e c t e d . The e f f e c t s of the Stark i n t e r a c t i o n w i t h the s p i n s p l i t t i n g can be n e g l e c t e d i n the N =0 l e v e l , as long as the s p i n s p l i t t i n g i n the N =1 l e v e l s i s s m a l l compared to the energy denominators (2B ) of the Stark Li p e r t u r b a t i o n . For M =1 the matrix elements i n Eq. (155) are c o r r e c t e d by f i n d i n g the e i g e n v e c t o r s of the M =1 submatrix (rows and columns l a b e l e d 5 and 7) of Eq. (176). The e i g e n v e c t o r s come from Eq. (48) , a f t e r p u t t i n g the submatrix i n t o the form of Eq. (43) by s u b t r a c t i n g h a l f i t s t r a c e (see App. D, Eq. D.18). At the h i g h e s t f i e l d used (2 88 kV/cm), the matrix elements of Eq.(21a) are m o d i f i e d by over 25%, but the sum i n Eq.(155) i s lowered by l e s s than 0.56%, which r a i s e s the d i p o l e moment by l e s s than 0.3%. E f f e c t s a t lower f i e l d s are much s m a l l e r , and we s h a l l f e e l j u s t i f i e d i n n e g l e c t i n g t h i s i n t e r a c t i o n . 8. The 3n S t a t e Our p l a t e s were examined to see whether the h i g h -er f i e l d s a v a i l a b l e would allow us to improve upon the d i p o l e moment d e r i v e d by Irwin and D a l b y 7 2 f o r the 3 n s t a t e . Components w i t h M J T = 1 were usable i n o n l y two p l a t e s (2693 and 275a) , and then on l y the lower cm - 1 s i d e s i n c e the o t h e r was g e n e r a l l y obscured by Stark components from the R ^ ( l ) l i n e . The observed displacements from the c e n t e r -1 -1 component were 1.88 ± 0.04 cm (2693) and 2.10 ± 0.04 cm (275a) . We take the second order s h i f t o f the M j t = 0 compo-nent t o be as measured i n Table XVIII, and the second order s h i f t i n the 3 E ~ s t a t e t o be gi v e n by Eq. (155) wi t h the d i p o l e moment of Eq.(163). We note t h a t the second order e f f e c t i n the 3 n s t a t e f o r m J T = 1 i s j u s t £ of t h a t f o r Mn=0 (given by Eq. 150). A f t e r s u b t r a c t i n g second o r d er e f f e c t s and the A-doubling ,(0.68 cm - 1, from D i x o n 1 1 6 ) a p p r o p r i a t e l y , the d i p o l e moment i s o b t a i n e d from an equa-t i o n s i m i l a r t o Eq.(120). The r e s u l t i n g d i p o l e moments (0.967 ± 0.023 D f o r 2693, and 0.938 ± 0.023 D f o r 275a) must be c o r r e c t e d f o r i n t e r m e d i a t e c o u p l i n g ( I r w i n 7 1 , p. 23f.) through d i v i s i o n by c\ = 0.7354 (Eq. 148). We o b t a i n y = 1.31 4 ± 0.03, D (2693) il (165) y n = l - 2 7 5 ± D (275a) We t h e r e f o r e have no cause to modify the c o n c l u s i o n s o f I r w i n and Dalby t h a t the d i p o l e moment of the A 3II s t a t e of NH i s 1.31 ± 0.03 D, s i n c e our v a l u e s (no doubt based upon i n f e r i o r data) are e n t i r e l y c o n s i s t e n t w i t h t h e i r s . 226 CHAPTER XII SUMMARY OF RESULTS We summarize the r e s u l t s of t h i s work i n Table XX, along w i t h c e r t a i n other c a l c u l a t e d and experimental r e s u l t s r e l a t i n g t o the d i p o l e moment of f i r s t - r o v ; d iatomic hydride molecules. I t should be p o i n t e d out t h a t the t o t a l body of l i t e r a t u r e , i n the f i e l d of c a l c u l a t i n g wavefunctions and d i p o l e moments, i s q u i t e l a r g e . We have chosen to compare experimental v a l u e s p r i n c i p a l l y w i t h the work of Bender and D a v i d s o n 8 5 and of Cade and Huo 8 l +, s i n c e these r e p r e s e n t good examples o f c o n f i g u r a t i o n i n t e r a c t i o n (CI) and s e l f " c o n s i s t e n t f i e l d (SCF) c a l c u l a t i o n s , r e s p e c t i v e l y . The r i c h n e s s of the l i t e r a t u r e can be surmised from a look a t Table XIX d e a l i n g w i t h NH; a study o f L i H would produce a t a b l e many times t h a t s i z e . A l l v a l u e s r e f e r to the ground v i b r a t i o n a l s t a t e s u n l e s s otherwise noted, and experimental (X) valu e s are d i s t i n g u i s h e d from c a l c u l a t e d (C) by the l e t t e r i n the t h i r d column. TABLE XX Di p o l e Moments of First-Row Diatomic Hydrides Molecule D i p o l e Moment (D) Source and Comments L i H (X 1£ + ) 5.882 X Wharton, e t . a l . 1 2 - 7 5.85 C Bender and D a v i d s o n 8 5 6. 002 C Cade and Huo 8 1 + TABLE XX (Cont'd) Molecule D i p o l e Moment (D) Source and Comments BeH (X 2 Z + ) 0.248 C Bender and D a v i d s o n 8 5 0.282 c Cade and Huo 8 l + BH (X ^ + ) 1.260 + 0.21 X Thomson and D a l b y 7 5 1.470 c Bender and D a v i d s o n 8 5 1.733 c Cade and Huo 8 4 (A xn) 0.58 + 0.04 X Thomson and D a l b y 7 5 CH (X 2 n r ) 1.45 + 0.06 X Phelps and D a l b y 7 0 IT 1.427 c Bender and D a v i d s o n 8 5 1.570 c Cade and Huo81* (A 2A) 1.12 + 0.17 X P h e l p s l h 0.887 + 0. 045 X T h i s Work NH (X 3X") 1.389 + 0.043 X T h i s Work 1.587 c Bender and D a v i d s o n 8 5 1.627 c Cade and Huo 8 4 (A 3n.) 1.30 + 0. 03 X Irwin and D a l b y 7 2 X 1.34 c H U G 1 2 8 (a XA) 1.48 + 0. 06 X Irwin and D a l b y 7 2 1.64 c H u o 1 2 8 (c :n) 1.69 + 0.07 X Irwin and D a l b y 7 2 1. 85 c H u o 1 2 8 OH (X 2n.) 1.720 + 0. 02 X Phelps and Dalby 8 i 1.709 + 0.029 X T h i s Work 1.633 c Bender and D a v i d s o n 8 5 1.780 c Cade and Huo 8 4 1.66 + 0.01 X Powell and L i d e 9 8 1.656 + 0. 04 X Phelps and D a l b y ( v = l ) 8 1.646 + 0.051 X T h i s Work (v=l) 1.626 + 0.045 X T h i s Work (v=2) 1.684 + 0. 032 X T h i s Work (OD,v=0) 1.635 + 0. 023 X T h i s Work (OD,v=l) 1.557 + 0.061 X T h i s Work (od,v=2) (A 1.946 + 0. 085 X T h i s Work 2.142 + 0.080 X T h i s Work (OD) 1.72 + 0.10 X Weinstock and Z a r e 1 0 2 HF (X 1. 820 + 0.003 X W e i s s 1 2 9 1.816 c Bender and D a v i d s o n 8 5 1.942 c Cade and Huo81* For a more d e t a i l e d survey of the l i t e r a t u r e i n v o l v i n g ab i n i t i o c a l c u l a t i o n s o f the d i p o l e moment, see the compendia of K r a u s s 1 3 0 and Ri c h a r d s , Walker, and H i n k l e y 1 3 1 . 228 APPENDIX A UNITS The u n i t s employed i n t h i s work are d e r i v e d from e l e c t r o s t a t i c cgs u n i t s . E n e r g i e s symbolized by V or H (Hamiltonians) are i n e r g s , but the end r e s u l t o f a l l energy c a l c u l a t i o n s w i l l be expressed i n the s p e c t r o s c o p i c u n i t cm 1 (wavelengths/cm, symbolized by F o r o ) , the conver-s i o n f a c t o r being he (erg-cm). The symbols W and 6 ( h a l f the A-doubling) are i n ergs i n Chapter VI, but are i n cm 1 i n l a t e r c h a p t e r s . Other symbols have u n i t s d e f i n e d i n the t e x t . In Chapter VI, a s u p e r s c r i p t "a" i s used t o i n d i c a t e -1 . . . cm u n i t s i n some cases. Thus the equations which i n t r o d u c e the mo l e c u l a r d i p o l e moment (Chapter IV) c o n t a i n the e l e c t r o n i c charge e^ i n statcoulombs, the e l e c t r i c f i e l d $ ?in^statcoulombs/cm 2 (commonly c a l l e d e s u ) , and a l l c o o r d i n a t e s i n cm \ D i p o l e or t r a n s i t i o n moments (y and R, r e s p e c t i v e l y ) are then i n statcoulomb-cm. E- i s u s u a l l y converted t o v o l t s / c m as an a i d to v i s u a l i z a t i o n , and d i p o l e moments converted t o Debye as a numerical convenience. Esu are m u l t i p l i e d by l O ^ / c t o g i v e k i l o v o l t s / c m , and statcoulomb-cm are m u l t i p l i e d by 18 10 to g i v e Debye. P u t t i n g a l l t h i s t o g e t h e r , a term i n the p e r t u b a t i o n expansion u s u a l l y c o n t a i n s some power o f yE e r g s , which becomes yE/hc cm which i s u s u a l l y denoted by e. Using c = 2.99792 x l O 1 ^ cm/sec, and -27 h = 6.6262 x 10 er g - s e c , yE ergs i s expressed conve-n i e n t l y as yE/59.5479 cm 1 where y i s i n Debye and E i s i n kV/cm, or as E/0.0595479 where E i s i n MV/cm. APPENDIX B QUANTUM NUMBERS, HUND'S CASES, AND SELECTION RULES A b r i e f summary of these i s presented here f o r the convenience of the reader, who i s r e f e r r e d to Landau and L i f s c h i t z l (Sees. 2 7 - 3 1 , 8 0 - 8 5 ) , Hougen 2 (pp. I f f . ) , and Her z b e r g 3 (Chap. V), f o r f u l l d i s c u s s i o n . S e l e c t i o n r u l e s c o n s i d e r e d are on l y a p p l i c a b l e t o e l e c t r i c d i p o l e t r a n s i t i o n s Quantum numbers used i n t h i s work are as f o l l o w s : J i s c a l l e d the t o t a l angular momentum o f the molecule, e x c l u d i n g o n l y n u c l e a r s p i n . I.e., t h i s angular momentumjis_." / j ( j+l ) fV>erg-sec. J i s always d e f i n e d (by c o n s e r v a t i o n o f t o t a l angular momentum) i n the abscence o f e l e c t r i c f i e l d s , s i n c e c o u p l i n g w i t h n u c l e a r s p i n i s s m a l l , and obeys A J = 0 , + 1 except t h a t A J = 0 i s f o r b i d d e n i f J = 0 ( a l s o i f ft=0 and A£2=0) . 0. i s the component o f J along the i n t e r n u c l e a r a x i s ; i . e . , n = h*3 where ft i s d e f i n e d as f o r Eq. ( 4 ) below. ft may be d e f i n e d i n a l l Hund's cases (Hougen 2, (p. 2 ) (but i s made e x p l i c i t i n the n o t a t i o n f o r (a) and (c) o n l y ) , and obeys Aft = 0 , ± Except i n case ( c ) , ft i s equal t o A + Z . Many authors d e f i n e ft = ; | „ A + Z , [ ., but we w i l l f i n d i t convenient to r e t a i n the s i g n . M i s the ab s o l u t e v a l u e of the component o f J along a l a b o r a t o r y - f i x e d a x i s , u s u a l l y the z a x i s as d e f i n e d by an e x t e r n a l e l e c t r i c f i e l d . Allowed t r a n s i t i o n s may have A M = 0 , ± 1 but Stark i n t e r -a c t i o n s can o n l y e x i s t between l e v e l s of equal M. I.e., the Stark e f f e c t i s d i a g o n a l i n M. M i s d e f i n e d whenever a z a x i s i s d e f i n e d . Since the Stark e f f e c t does not d i s t i n g u i s h the s i g n o f M, we s h a l l defy quantum mechanical convention by d e f i n i n g M to be the absolute value of J . In i n t e n s i t y c a l c u l a t i o n s , such as the one r e f e r r e d to i n Chap. XIV, Sec. 5 , i t must be remembered t h a t the s e l e c t i o n r u l e on M r e a l l y i n c l u d e s a s i g n . E.g., M=+l does not go to M=-l. 230 L i s the t o t a l angular momentum of the m o l e c u l a r e l e c t r o n s . L i s ortly w e l l d e f i n e d i n Hund's case (d). In the t h r e e other cases c o n s i d e r e d i n t h i s work, L l o s e s d e f i n i t i o n through the c o u p l i n g o f e l e c t r o n i c motion w i t h the i n t e r n u c l e a r a x i s . A i s the component of L along the i n t e r n u c l e a r a x i s ; A=n • t,, d e f i n e d except i n Hund's case (c) . A obeys AA=0,±1. When d e f i n e d , A i s used to name e l e c t r o n i c s t a t e s : S t a t e s with A = 0,1, 2,3,... are c a l l e d £ ,H, A , s t a t e s ( t h i s "I" i s a name, not to be confused with the quantum number d e f i n e d below). Contrary to common usage, we w i l l a l l o w A to have n e g a t i v e v a l u e s ; thus a n s t a t e i s a s s o c i a t e d w i t h A=±l. N i s the m o l e c u l a r angular momentum, but e x c l u d i n g e l e c t r o n i c as w e l l as n u c l e a r s p i n . H e r z b e r g 3 and most e a r l i e r authors use K f o r t h i s q u a n t i t y . N i s d e f i n e d i n case (b) and (d), although i t s d e f i n i t i o n i s o f t e n extended to l e v e l s d e s c r i b e d by c a s e ( a ) , when th e r e i s a g r a d u a l t r a n s i t i o n from case (b) to (a) at low v a l u e s of J . N i s sometimes used i n p l a c e o f J f o r s i n g l e t s t a t e s , where they are the same. N obeys AN=0,±1 except t h a t AN=±1 when A=AA =0. N i s d e f i n e d f o r m a l l y i n Chap. V I I . S i s the t o t a l e l e c t r o n i c s p i n . S i s d e f i n e d except i n case (c) . In other cases,' -< .- s obeys the r u l e AS=0, and i s e x p l i c i t l y i n d i c a t e d by a l e f t super-s c r i p t 2S+1 c a l l e d the m u l t i p l i c i t y . E i s the component of S along the i n t e r n u c l e a r a x i s ; Z =n«S. Z i s d e f i n e d except i n c a s e ( c ) . In case (a) i t obeys AZ=0. R i s the angular momentum of n u c l e a r r o t a t i o n , and i s a v a l i d quantum number o n l y i n case (d). v i s the quantum number o f n u c l e a r v i b r a t i o n along the i n t e r n u c l e a r a x i s . S e l e c t i o n r u l e s are i m p l i e d f o r v o n l y by the o v e r l a p i n t e g r a l R , „ / u ) , u> v„dr whose square i s p r o p o r t i o n a l to the t r a S s i f i o n v p r o b a b i l i t y o f the t r a n s i t i o n from v' t o v". Only f o r i n f r a r e d bands (e'=e"), and o n l y f o r a pure harmonic o s c i l l a t o r p o t e n t i a l does Av=0,±l h o l d . p,± are r o t a t i o n a l energy l e v e l p a r i t y i n d i c a t o r s . "-" means t h a t the t o t a l wavefunction changes s i g n under i n v e r s i o n at the o r i g i n ; "+" means t h a t i t remains unchanged. E l e c t r i c d i p o l e t r a n s i t i o n s may 231 not occur between l e v e l s o f the same p a r i t y . ± are a l s o symmetry i n d i c a t o r s ( s u p e r s c r i p t s ) on E± s t a t e s (or Or s t a t e s , i n case ( c ) ) . Non-degenerate energy l e v e l s i n d i a t o m i c ( l i n e a r ) molecules can s u f f e r no p h y s i c a l change under a r e f l e c t i o n of c o o r d i n a t e s i n any plane c o n t a i n i n g the i n t e r n u c l e a r a x i s . T h e r e f o r e the e l e c t r o n i c wavefunction remains the same, i n a s t a t e d e s i g n a t e d by Only t r a n s i t i o n s between s t a t e s o f l i k e s i g n are allowed. c,d, are r o t a t i o n a l s u b l e v e l d e s i g n a t o r s , i n s t a t e s w i t h near-degeneracy (A>0),.. A s u b l e v e l of a A-doublet i s d e s i g n a t e d "c" i f i t s dependence of p a r i t y upon N i s the same as a ET s t a t e , and "d" i f the same as a 2T s t a t e . a,s, are r o t a t i o n a l energy l e v e l d e s i g n a t o r s i n homonuclear molecules. A l e v e l i s antisymmetric "a" i f the t o t a l wavefunction changes s i g n under interchange o f n u c l e i , and symmetric " s " i f i t remains the same. Only t r a n s i t i o n s between l e v e l s o f the same "a,s" symmetry are allowed. g,u, are e l e c t r o n i c s t a t e s u b s c r i p t s i n homonuclear molecules. T h i s i s a symmetry p r o p e r t y of the e l e c t r o n i c wave-f u n c t i o n I|»T , and i s "g" i f i}> remains unchanged ( i s "even") Snder i n v e r s i o n througS the o r i g i n o f a l l e l e c t r o n c o o r d i n a t e s , and i s l a b e l e d "u" i f if* changes s i g n ( i s "odd") under such an i n v e r s i o n . I t d i f f e r s from p a r i t y ( ±, above) o n l y i n t h a t i t a p p l i e s to i|> i n s t e a d of the t o t a l wavefunction, so t h a t the n u c l e a r c o o r d i n a t e s are not i n v o l v e d , and the "g,u" d e s i g n a t o r a p p l i e s u n i f o r m l y to an e l e c t r o n i c s t a t e ( i n c l u d i n g a l l m u l t i p l e t components) s i n c e i t i s independent of the v i b r a t i o n a l and r o t a t i o n a l wavefunctions (as w e l l as e l e c t r o n i c s p i n ) . T r a n s i t i o n s between e l e c t r o n i c s t a t e s which are a l i k e ( i n t h i s symmetry type) are f o r b i d d e n . '," r e f e r to the upper and lower l e v e l s o f a t r a n s i t i o n , except where otherwise d e f i n e d i n Chap. VI. Hund's cases: Case (a) The e l e c t r i c f i e l d s along the i n t e r n u c l e a r a x i s (which can be v i s u a l i z e d as r e s u l t i n g from the d i s t o r t i o n o f the e l e c t r o n clouds o f the separated atoms through the process o f forming the m olecular bond) are s t r o n g enough to cause complete uncoupling (Paschen-Back e f f e c t ) of the e l e c t r o n i c o r b i t a l and s p i n angular momenta L and S. P r o p e r l y 232 speaking, the f i e l d uncouples L from S, and the r e s u l t i n g component of L along the i n t e r n u c l e a r a x i s (A) produces a s t r o n g magnetic f i e l d t o which S i s coupled. The components o f L and S along the i n t e r n u c l e a r a x i s are good quantum numbers (A and Z, r e s p e c t i v e l y ) . The angular momentum o f n u c l e a r r o t a t i o n R i s added to the a l g e b r a i c sum of A and Z ( t h i s l a t t e r sum being denoted by n) to form J . T h i s s t r o n g c o u p l i n g of the e l e c t r o n s p i n g i v e s l a r g e m u l t i p l e t s p l i t t i n g s . Since Stark e f f e c t s can not remove degeneracy w i t h r e s p e c t to the s i g n of the azimuthal quantum number (A), energy l e v e l s of e l e c t r o n i c s t a t e s w i t h A>0 are doubly degenerate to a f i r s t approximation (A-doubling). S t a t e s w i t h A=0 cannot belong to case ( a ) , i n g e n e r a l , s i n c e L produces no net magnetic f i e l d and the e l e c t r o n s p i n i t s e l f cannot i n t e r a c t with e l e c t r i c f i e l d s . In Chap. VII and l a t e r c h a p t e r s , case (a) wave-f u n c t i o n s are denoted by the ket | ASZ;;fiJM > c o n t a i n i n g quantum numbers c h a r a c t e r i z i n g the s t a t e , and i s o f t e n a b b r e v i a t e d to | QJ- > or even | ft >. Case ( b ) l i s s i m i l a r to case (a) except t h a t the e l e c t r o n i c s p i n i s not coupled to L o r any magnetic f i e l d a long the i n t e r n u c l e a r a x i s . Nuclear r o t a t i o n adds to A to form N, which then adds weakly wi t h S to form J . 2Z s t a t e s always belong to case (b), and Z s t a t e s o f h i g h e r m u l t i p l i c i t y (2S+1) are u s u a l l y c l o s e to case (b). (When more than one e l e c t r o n c o n t r i b u t e s to S, some degree o f s p i n - o r b i t c o u p l i n g o f t e n takes p l a c e which makes a tendency toward case ( c ) . S p i n - s p i n i n t e r a c t i o n s can a l s o augment case (b)'s s m a l l m u l t i p l e t s e p a r a t i o n s . ) Case (c) o c c u r s when S has become more s t r o n g l y coupled t o the magnetic f i e l d of L than L i s coupled to the i n t e r n u c l e a r a x i s . Only the v e c t o r sum o f L and S i s coupled to the i n t e r n u c l e a r a x i s , l e a v i n g a net component n along i t , and t h i s i s the o n l y v a l i d quantum number b e s i d e s J and M (Hougens , p.2). Case (c) s t a t e s are always " s i n g l e t " s t a t e s , t h e r e f o r e , but may have a t w o f o l d degeneracy analogous to A-doubling when n i s g r e a t e r than zero. When fi=0, then 0"*" and 0- s t a t e s are d e f i n e d s i m i l a r l y t o the Z + and Z-~ s t a t e s of cases (a) and (b) . Case (d) occurs when the e l e c t r o n i c angular momenta are completely decoupled from the i n t e r n u c l e a r a x i s , and i n t e r a c t o n l y through n u c l e a r r o t a t i o n . 233 N i s then formed as the v e c t o r sum of R and L which are coupled weakly enough t h a t L remains a v a l i d quantum number. S then couples very weakly with N to form J , so t h a t s p i n s p l i t t i n g can be expected to be n e g l i g i b l e . 234 APPENDIX' C • GLOW DISCHARGES 1. I n t r o d u c t i o n I t i s proposed t h a t the form o f d i s c h a r g e used to produce e l e c t r i c f i e l d s i n t h i s r e s e a r c h i s a s p e c i a l case of a glow d i s c h a r g e . Phenomenological and t h e o r e t i c a l d e s c r i p t i o n s o f t h i s d i s c h a r g e can be found i n a number o f r e f e r e n c e s ? 1 2 7 F i e l d and i o n i z a t i o n measurements have been p u b l i s h e d by many authors over the l a s t 70 years (see e a r l i e r r e f e r e n c e s i n L o e b 2 3 and C o b i n e 2 2 ), those by W a r r e n 2 8 being perhaps the most t r u s t w o r t h y . Numerical s o l u t i o n s f o r f i e l d and space charge c o n d i t i o n s have been achieved by W a r d 3 0 " 3 2 , w i t h some success. T h i s background m a t e r i a l w i l l be summarized i n the f o l l o w i n g s u b s e c t i o n s . The body of l i t e r a t u r e i n t h i s f i e l d i s q u i t e l a r g e , however, and i t should be emphasized t h a t a complete survey of r e c e n t progress has not been attempted. 2. General C h a r a c t e r i s t i c s I f two e l e c t r o d e s of f l a t s u r f a c e are opposed to each o t h e r at some s e p a r a t i o n d, e n c l o s e d i n a tube with moderate gas p r e s s u r e (10~ 2 to l b 2 mm of'Hg)", and-connected to a power supply, then a c h a r a c t e r i s t i c v o l t a g e curve such as i s shown s c h e m a t i c a l l y i n F i g . C l w i l l be o b t a i n e d upon f 235 B 5>. ° 0 F i g . C l C u r r e n t - V o l t a g e C h a r a c t e r i s t i c of a Glow D i s c h a r g e . Cathode E - l e c t f i c F i e l d , ^ L i g h t \ ' I n t e n s i t y \ \ J F Anode F F i g . C2. F i e l d and I n t e n s i t y v s . p o s i t i o n . oA- cathode glow; B- cathode dark space; C- n e g a t i v e glow; D- Faraday dark space; E- p o s i t i v e column; F- s t r i a t i o n s ; G- anode glow. v a r i a t i o n of the c u r r e n t . I t w i l l be n o t i c e d t h a t the p o r t i o n s o f the curve to the r i g h t of p o i n t s B and D have n e g a t i v e slope which r e s u l t s i n i n s t a b i l i t y and r e q u i r e s good c u r r e n t c o n t r o l p r o p e r t i e s i n the power supply. The r e g i o n o f F i g . C l between A and B i s a non-self-s u s t a i n i n g d i s c h a r g e , i n the sense t h a t i t r e q u i r e s some c u r r e n t I of e x t e r n a l o r i g i n ( i n c i d e n t photons, may be s u p p l i e d to the cathode f o r maintenance. T h i s r e g i o n i s c a l l e d the "subnormal" o r Townsend d i s c h a r g e . I f we d e f i n e a (Townsend's f i r s t i o n i z a t i o n c o e f f i c i e n t ) t o be the number of i o n i z i n g c o l l i s i o n s per l e n g t h of path (along the e l e c t r i c f i e l d ) per e l e c t r o n , then dl/dx = a ( x ) I ( x ) , (C.l) which can be i n t e g r a t e d t o g i v e the anode c u r r e n t : T - T cx(x)dx Z a " Toe}° ' (C.2) a may be more p r o p e r l y understood as a f u n c t i o n of the f i e l d which, i n t u r n , v a r i e s w i t h p o s i t i o n . T h i s c u r r e n t w i l l be i n c r e a s e d by any mechanism augmenting the r e g e n e r a t i o n o f e l e c t r o n s a t the cathode. Now, the i o n i z a t i o n s d e s c r i b e d by a w i l l a l s o c r e a t e p o s i t i v e i o n s which w i l l head toward the cathode and may produce e l e c t r o n s when c o l l i d i n g w i t h the cathode m a t e r i a l . I f each i o n h i t t i n g the cathode produces Y e l e c t r o n s , then I = I + T ( i - I ) ° a C (C.3) where I i s the t o t a l c u r r e n t l e a v i n g the cathode. c 3 Y i s c a l l e d the second Townsend c o e f f i c i e n t . Values of Y f o r low energy e l e c t r o n s are g i v e n i n C o b i n e 2 2 (p.15 9). Rep l a c i n g I by I i n Eq. ( C. !2) and combining with Eq.(c.3) g i v e s I = I exp(/adx) a o 1 - Y(exp(/adx) - 1) * (C. 4) At p o i n t B i n F i g . 1, the d i s c h a r g e becomes s e l f - s u s t a i n i n g . I.e., I a becomes independent of I , and l i m i t e d by other c o n s i d e r a t i o n s (space charge). T h i s i s r e f l e c t e d i n Eq.(c.4) by l e t t i n g I be unbounded by the a and y p r o c e s s e s , a s e t t i n g the denominator equal t o zero. That i s , cl 1 = Y(exp(/ oadx) - 1) , ^ which i s o f t e n w r i t t e n 1 + — = ef^adx Y 1 o 1 (C.6) Eq.(C.5.)is r e a d i l y understood: One e l e c t r o n l e a v i n g the cathode r e s u l t s i n exp(/adx) e l e c t r o n s at the anode through exp(/adx) - 1 i o n i z a t i o n s along the way. These exp(/adx) - 1 i o n s produce Y(exp(/adx) - 1) e l e c t r o n s from the cathode s u r f a c e which must be e x a c t l y one, r e p l a c i n g the o r i g i n a l e l e c t r o n . Warren 2 1* (p. 1658) c a l l s t h i s the "economy 238 condition 1; " , i n the r i g h t s i d e of which he i n s e r t s a m u l t i p l i e r (G) to account f o r ions l o s s e s to the tube w a l l s and recombin-a t i o n s . A l t e r n a t i v e mechanisms to y, such as p h o t o e l e c t r i c feedback or i o n i z a t i o n o f the gas by ions o r n e u t r a l molecules, are deemed l e s s probable ( L o e b 2 3 , pp. 383ff) but l e a d t o i n d i s t i n g u i s h a b l y s i m i l a r economy c o n d i t i o n s . F i e l d e m ission processes should a l s o be c o n s i d e r e d at the h i g h f i e l d s encountered i n our experiments, but are not b e l i e v e d by t h i s author t o be important except i n cathode b r e a k - i n and i n breakdown-to-arc processes (see Sec. 6 below). E l e c t r o n e m i s s i o n by impact o f f a s t n e u t r a l p a r t i c l e s would c e r t a i n l y reduce the i o n to e l e c t r o n r a t i o . However, the economy c o n d i t i o n (Eq . c .5 ,C.6) should be of the same form, s i n c e the numbers and e n e r g i e s of f a s t n e u t r a l s should be i n p r o p o r t i o n to those o f ions (see Sec. 7 ) . We need o n l y make y equal t o the number of e l e c t r o n s r e l e a s e d upon the impact of one i o n p l u s a l l the f a s t n e u t r a l s which t h i s i o n generated on i t s way to the cathode. 3. S i m i l a r i t y R e l a t i o n s I t may be shown under reasonably g e n e r a l assumptions t h a t i f a d i s c h a r g e tube i s a l t e r e d such t h a t (1) a l l dimensions o f s i z e i n c l u d i n g mean f r e e paths are m u l t i p l i e d by a f i x e d f a c t o r , ( 2 ) -the gas p r e s s u r e i s d i v i d e d by t h i s same f a c t o r , and 239 (3) the imposed v o l t a g e i s unchanged, then one o b t a i n s a s c a l e d v e r s i o n o f the o r i g i n a l d i s c h a r g e with v o l t a g e d i s t r i b u t i o n i n exact p r o p o r t i o n to the o r i g i n a l . The " s c a l e d " d i s c h a r g e w i l l have d i f f e r e n t v a l u e s of pre s s u r e (p), e l e c t r i c f i e l d ( E ) , c u r r e n t d e n s i t y ( j ) , and i n t e r e l e c t r o d e (or other) spacing (d), but the combinations pd, E/p, and j / p 2 can be shown (see, e.g., C o b i n e 2 2 , pp. 209ff) to be i n v a r i a n t under the change o f s c a l e . Thus E, d••.,„ and j are o f t e n expressed i n these combinations f o r g r e a t e r g e n e r a l i t y . In so f a r as s i m i l a r i t y r e l a t i o n s are obeyed, then j / p 2 and pd (where d i s any c h a r a c t e r i s t i c l e n g t h , such as the cathode dark space o r Aston dark space) are each f u n c t i o n s o f onl y the cathode f a l l v o l t a g e .V '• (Druyvesteyn and P e n n i n g 2 7 , p. 131). 4. Normal: and Abnormal. :Glow. Discharges Once the d i s c h a r g e i s s e l f - s u s t a i n i n g , c u r r e n t w i l l drop with an i n c r e a s e i n v o l t a g e (power supply p e r m i t t i n g ) to the p l a t e a u between B and C i n F i g . 1, c a l l e d the "normal" glow. As the c u r r e n t i n c r e a s e s toward p o i n t C, the glow i s observed t o spread over the cathode i n p r o p o r t i o n t o the t o t a l c u r r e n t , thus m a i n t a i n i n g both a constant v o l t a g e drop V n and a cons t a n t c u r r e n t d e n s i t y 3 n/P 2 which have c h a r a c t e r i s t i c v a l u e s f o r a giv e n gas and cathode m a t e r i a l . For an aluminum cathode under helium these are approximately 140 v o l t s and 20 pA/cm2-mm Hg 2, under hydrogen these are 170 v o l t s and 90 uA/cm2-mm Hg 2. I f the c u r r e n t 240 d e n s i t y j were p l o t t e d along the a b s c i s s a of F i g . C l i n s t e a d of the c u r r e n t I, then between B and C would be a d i p , r a t h e r than a f l a t r e g i o n , with the normal glow a t the bottom p o i n t . W ard 3 2 (p.2791,2794) f i n d s t h a t the c h a r a c t e r i s t i c s o f the Townsend, t r a n s i t i o n a l , and normal glow phenomena (constant V r and j n w i t h changing area) may be accounted f o r by the assumption of a form f o r the i o n i z a t i o n e f f i c i e n c y a/E ( o f t e n denoted by n) as a f u n c t i o n o f E/p which achieves a maximum; he assumes t h a t a/E = (p/E)Aexp(-B(p/E) r) , (C.7) where A and B are constants and r=£ f o r i n e r t gases (He). T h i s has a maximum at E/p = B 2/4. Eq. (c.7) with r = l i s the t h e o r e t i c a l form f o r a d e r i v e d by Townsend and o t h e r e a r l y workers (see L o e b 2 3 , p.358-369 and Sec. 7, below). In t h i s view, when the cathode area has been covered by the d i s c h a r g e glow, ( p o i n t C i n F i g . 1), the c u r r e n t can o n l y r i s e by i n c r e a s i n g the f i e l d and t h e r e f o r e the a p p l i e d v o l t a g e . The i o n i z a t i o n e f f i c i e n c y w i l l decrease r a p i d l y w i t h i n c r e a s i n g E, a c c o r d i n g t o Eq. (c.7)r and even more so f o r m o l e c u l a r gases l i k e H 2 , f o r which Ward 3 2 f i n d s t h a t r = l i s a b e t t e r f i t to experimental data f o r a. Thus an i n c r e a s e i n v o l t a g e i s needed to produce any i n c r e a s e i n c u r r e n t . T h i s r e g i o n i s c a l l e d the "abnormal glow" d i s c h a r g e 241 ( p r i n c i p a l l y t o d i s t i n g u i s h i t from the "normal"). At some p o i n t (D i n F i g . C l depending on the m a t e r i a l s used and the s u r f a c e c o n d i t i o n of the cathode, the abnormal glow w i l l break down i n t o an a r c (E i n F i g . C l . The mechanism of e l e c t r o n r e l e a s e i n the cathode o f an a r c i s q u i t e d i f f e r e n t from t h a t i n the glow (where i n e l a s t i c c o l l i s i o n s between the ions and atoms of cathode metal predominate). In the a r c the dominant mechanism i s t h e r m i o n i c e l e c t r o n emission by Richardson's e q u a t i o n . Breakdown occurs when the cathode becomes s u f f i c i e n t l y hot from the i n c r e a s e d power d i s s i p a t i o n i n the abnormal glow, p o s s i b l y i n i t i a t e d by f i e l d e mission p r o c e s s e s . T h i s power i s mostly d i s s i p a t e d i n the cathode, s i n c e volt-amp l o s s e s are g r e a t e s t i n the cathode f a l l where i o n e n e r g i e s and f a s t n e u t r a l e n e r g i e s are l o s t to the cathode, w h i l e e l e c t r o n and n e g a t i v e i o n e n e r g i e s are d i s s i p a t e d i n the nearby n e g a t i v e glow (see heat l o s s measurements quoted i n Thomson and Thomson 3 3, pp. 3 6 5 f f ) . At some p o i n t thermio-n i c e l e c t r o n e mission from the cathode may become comparable to c o l l i s i o n p r o c e s s e s , with t h i s a d d i t i o n a l c u r r e n t producing s t i l l more heat. The process runs away and r e s u l t s i n the s h a r p l y n e g a t i v e slope ( i n s t a b i l i t y ) shown i n F i g . C l . The normal and abnormal glow d i s c h a r g e s show a s e r i e s o f w e l l known f e a t u r e s , i l l u s t r a t e d i n Fig.C2 which shows the e l e c t r i c f i e l d s t r e n g t h and l i g h t i n t e n s i t y as f u n c t i o n s o f p o s i t i o n between anode and cathode. 242 5. Aston Dark Space and Cathode Glow The weak l i g h t i n t e n s i t y shown r i g h t a t the cathode, marked A i n Fig.c2, i s c a l l e d the cathode glow and i s sometimes observed t o be separated from the cathode by a narrow dark space (not shown i n Fig.C2 nor observed i n our apparatus) c a l l e d the Aston dark space. The cathode glow i s i n the r e g i o n of h i g h e s t f i e l d and thus of g r e a t e s t i n t e r e s t f o r the purposes of t h i s work (along w i t h the r e s t o f the cathode dark space). A d e t a i l e d d i s c u s s i o n o f these f e a t u r e s w i l l be d e f e r r e d u n t i l Sec. 8, a f t e r the cathode dark space as a whole has been d i s c u s s e d . 6. The Cathode Dark Space Most of the p o t e n t i a l a p p l i e d between cathode and anode appears a c r o s s the cathode dark space ( a l s o c a l l e d the cathode f a l l , o r the Crookes or H i t t o r f dark space) which i s l a b e l e d B i n Fig.C2. The cathode glow i s here c o n s i d e r e d as being a p a r t o f the cathode dark space. In the normal glow the cathode dark space has a c h a r a c t e r -i s t i c l e n g t h g i v e n i n c e n t i m e t e r s as 0.72/p f o r H 2 and 1.32/p f o r helium w i t h aluminum cathodes (p i n mm Hg), and i s f u r t h e r c h a r a c t e r i z e d by the cathode f a l l v o l t a g e s and c u r r e n t d e n s i t i e s g i v e n i n Sec. 4. Once the cathode has been covered by the glow, and the d i s c h a r g e has entered the abnormal r e g i o n , then the c u r r e n t d e n s i t y r i s e s w i t h i n c r e a s i n g v o l t a g e , f i r s t v e r y r a p i d l y , then t a p e r i n g o f f . For example, m u l t i p l y i n g 243 the normal f a l l v o l t a g e by seven or e i g h t produces a t h o u s a n d f o l d i n c r e a s e i n c u r r e n t ( C o b i n e 2 2 p.228; Ward 3 2 p.2792). A s t o n 3 4 gave some e m p i r i c a l r e l a t i o n s h i p s among the parameters o f the abnormal d i s c h a r g e ; f o r hydrogen and helium over aluminum cathodes they a r e : He: j = 10"*(V - 2 5 5 ) 2 p 2 (C.8) d = 0.36/p + 0.0049/j^ (C.9) H 2: j = 3 x 10~h(V 1 4 4 ) 2 p 2 (C710) d = 0.265/p + 0.0043/j 2^ (C.'ll) where d i ? i n cm, j i s i n mA/cm2, and p i s i n mm Hg. These formulae do not d e s c r i b e these parameters i n the range o f our experiments, however. The cathode dark space can be c o n s i d e r e d as the p l a c e where most of the a c t i o n i s , with the cathode s i d e of the n e g a t i v e glow s e r v i n g as a s o r t of s u r r o g a t e anode which connects to a conducting (plasma) pathway f o r the r e t u r n o f c u r r e n t t o the p o s i t i v e s i d e of the power supply. The cathode serves as a source o f e l e c t r o n s , not by t h e r m i o n i c emission as i n an a r c , but through the i o n - c o l l i s i o n o r p h o t o e l e c t r i c processes d e s c r i b e d i n Sec. 2. The "anode" at the edge of the n e g a t i v e glow may be a source of p o s i t i v e i o n s , u n l i k e the r e a l anode. Our p i c t u r e of the cathode dark space, then, c o n s i s t s of a c u r r e n t composed o f e l e c t r o n s f l o w i n g one way, 244 and ions f l o w i n g the o t h e r , w i t h i o n i z i n g c o l l i s o n s by e l e c t r o n s throughout the volume which augment both components. The c u r r e n t and space charge i n t h i s r e g i o n are mostly due to the i o n s , however, s i n c e the ion-cathode c o l l i s i o n s are o f low e f f i c i e n c y (y of Sec. 2) i n producing e l e c t r o n s . Moreover, i t i s b e l i e v e d t h a t most o f the i o n i z i n g events w i t h i n the dark space are caused by e l e c t r o n s r a t h e r than the slower-moving i o n s . A f u r t h e r assymetry i s t h a t such i o n s as enter the cathode f a l l r e g i o n do so as a r e s u l t of d i f f u s i o n from the l o w - f i e l d n e g ative glow r e g i o n , and bear no d i r e c t r e l a t i o n s h i p to the a r r i v a l of e l e c t r o n s a t the n e g a t i v e glow "anode". L o e b 2 3 (p. 568, 574) b e l i e v e s t h a t the l e n g t h of the dark space i s determined by the requirement t h a t i t must encompass a s u f f i c i e n t number o f i o n i z a t i o n s to p r o v i d e the e l e c t r o n s needed to c a r r y the c u r r e n t p a s t the dark space through r e g i o n s where ions (or the f i e l d s t o move them) are u n a v a i l a b l e . He hdtes t h a t the l e n g t h of the normal glow dark i s a b i t more con s t a n t ( v a r y i n g by around a f a c t o r of two) f o r d i f f e r e n t cathode-gas combinations when expressed i n terms of m o l e c u l a r f r e e paths than i n terms of d i s t a n c e (where d i f f e r e n c e s by a f a c t o r of f o u r o r f i v e are found): about 100 mean f r e e paths i s average. T h i s argument seems a b i t unreasonable, however, s i n c e the e l e c t r o n s e v i d e n t l y c a r r y enough energy i n t o the n e g a t i v e glow to produce the i n t e n s e e x c i t a t i o n and i o n i z a t i o n found there (see P e n n i n g 3 5 , f o r example), and the e l e c t r i c f i e l d should d i s t r i b u t e i t s e l f 245 a p p r o p r i a t e l y t o draw the needed c u r r e n t . Warren 2 1* (p. 1658) c o n s i d e r s the cathode dark space l e n g t h r a t h e r as t h a t needed f o r an e l e c t r o n to engage i n 1/y i o n i z i n g c o l l i s i o n s , which would ensure the e l e c t r o n ' s replacement, back a t the cathode. He expresses the dark space l e n g t h as about t h i r t y i o n i c mean f r e e paths (Warren 2 1*, p. 1663), although Davis and V a n d e r s l i c e ' s 3 6 work g i v e s lower numbers (and o f t e n much lower, e s p e c i a l l y when the i o n i s not the i o n i z e d form o f an abundant n e u t r a l , see Sec. 7 ) . However,, i t should be the number of e l e c t r o n mean f r e e paths which matters, s i n c e they do the i o n i z i n g . E l e c t r o n f r e e paths are the order o f f i v e o r s i x times longer than m o l e c u l a r f r e e paths by o r d i n a r y k i n e t i c theory ( M i l l m a n 2 1 , p. 284) and may be c o n s i d e r a b l y l a r g e r f o r very f a s t e l e c t r o n s ( C o b i n e 2 2 , P. 30), so r e d u c i n g Loeb's 100 f r e e paths to l e s s than ten or twenty. L l e w e l l y n - J o n e s 2 5 (p.93) says t h e r e are f i v e e l e c t r o n f r e e paths i n the l e n g t h of the dark space. Druyvesteyn and P e n n i n g 2 7 (p.135) say f o u r t o seven a t 200 V drop, and 0.5 a t 1000 V. Warren 2 1* (p. 1660) concludes from a c o n s i d e r a ^ t i o n of a v a i l a b l e energy t h a t t h e r e can be no more than f i v e i o n i z i n g events per e l e c t r o n i n a normal d i s c h a r g e (at hig h e r p o t e n t i a l s the a v a i l a b l e energy goes up but the c o l l i s i o n c r o s s - s e c t i o n and dark space l e n g t h d e c r e a s e ) . He estimates 1 to 10 i o n i z a t i o n s per e l e c t r o n a t low e n e r g i e s and 0.1 to 1 a t h i g h e r e n e r g i e s . On the other hand, the r e q u i r e d number (1/Y) i s a t l e a s t f i f t y . The c o n c l u s i o n i s t h a t a l l necessary i o n i z a t i o n 246 does not take p l a c e i n the dark space, and some ions must be formed i n the n e g a t i v e glow and f a l l i n t o the dark space. V o l k o v 3 7 c a l c u l a t e s t h a t the i o n c u r r e n t d i f f u s i n g down from the n e g a t i v e glow i s comparable i n magnitude to t h a t produced i n the cathode dark space ( h i s F i g . 3 puts the former v e r y n e a r l y equal to the average of the l a t t e r , f o r 1 mm Hg of over aluminum). T h i s author has found no mention i n the l i t e r a t u r e of the p o s s i b i l i t y o f f i e l d e m i ssion o p e r a t i n g i n c o n j u n c t i o n w i t h the y-process under c o n d i t i o n s o f l a r g e cathode f a l l ; the a n a l y s i s of such a s i t u a t i o n does not appear to be t r i v i a l . However, l e t u s - f i r s t assume t h a t the i n t r o d u c t i o n of a s i g n i f i c a n t f i e l d e m i ssion e l e c t r o n c u r r e n t a t the cathode w i l l have no e f f e c t on the Y-process. Then t h i s new c u r r e n t w i l l reduce the net p o s i t i v e space charge i n the cathode f a l l , thereby r e d u c i n g the f i e l d g r a d i e n t . Thus a negative f e e d -back mechanism i s e s t a b l i s h e d which should lengthen the dark space, i n c r e a s e the c u r r e n t , and reduce the g a i n i n f i e l d s t r e n g t h a t the cathode w i t h g i v e n v o l t a g e i n c r e a s e . I f the c u r r e n t i s not r e s t r i c t e d by the power supply, i t would seem t h a t the f i e l d induced c u r r e n t would i n c r e a s e the tube c u r r e n t by a s u b s t a n t i a l amount, i n adding to both the a and Y-processes, assuming t h a t i o n s c o n t r i b u t i n g to these processes are not a l l coming from the negative glow. C l e a r l y , i f more than one i o n i z a t i o n per e l e c t r o n i s o c c u r i n g i n the dark space, then the f i e l d a t the cathode may be i n c r e a s e d r a t h e r than decreased and a runaway f i e l d e mission process would q u i c k l y break down i n t o an a r c . Thus we 247 conclude t h a t a t the c r i t i c a l v a lue of cathode f i e l d , the c u r r e n t should i n c r e a s e and e i t h e r the glow should degener-ate i n t o an a r c o r the cathode f i e l d should prove d i f f i c u l t to i n c r e a s e . As w i l l be seen below, no such phenomena have been noted f o r a p p l i e d v o l t a g e s f a r i n excess of those r e p o r t e d here. 7. I o n i c C o l l i s i o n s and the Shape o f the Cathode F a l l The type of c o l l i s i o n processes a f f e c t i n g ions i n the cathode f a l l has been r a t h e r w e l l e s t a b l i s h e d by the experiments o f Davis and V a n d e r s l i c e 3 6 who measured the energy spectrum by s p e c i e s a t the cathode of moderately abnormal glow d i s c h a r g e s . They r e c e i v e d ions through a c e n t r a l p i n h o l e i n the cathode and sent them through energy and mass spectrometers. T h e i r o b s e r v a t i o n s are w e l l accounted f o r by t h e i r theory which made the f o l l o w i n g assumptions: (1) A l l ions come from the n e g a t i v e glow, o r a t l e a s t s e v e r a l f r e e paths b e f o r e the cathode. (2) The o n l y e f f e c t of a c o l l i s i o n i s a complete t r a n s f e r o f a l l k i n e t i c energy. (3) The c o l l i s i o n c r o s s s e c t i o n , and thus the mean f r e e path f , do not change a p p r e c i a b l y with f i e l d o r k i n e t i c energy. (4) The f i e l d i n c r e a s e s l i n e a r l y from zero a t the n e g a t i v e glow (x=d, v=V ) c to maximum a t the cathode (x = V = 0). A c c o r d i n g to assump-t i o n (2), the t o t a l number o f i o n s N q i s a constant through-out the cathode f a l l , as i s the number dN = N dx/f ' v o ' o f i o n s s t a r t i n g i n the r e g i o n of t h i c k n e s s dx a t p o t e n t i a l 248 V (and thus w i t h p o t e n t i a l energy eV). The f r a c t i o n o f these dN v ions which reach the cathode without f u r t h e r c o l l i s i o n i s a f u n c t i o n of the p o s i t i o n x (at which the p o t e n t i a l - x / f i s V) and i s e ' . Thus the number dN o f ions r e a c h i n g the cathode w i t h k i n e t i c energy eV i s d N c v = N o e " X / f d x / f • (C12) (Compare the d i f f e r e n t meaning o f t h i s formula i n Loeb£ 3 p. 647.) I n t e g r a t i n g a l i n e a r f i e l d dependence 3 V = E = E ( d - X ) 3x m d (C.13) s u b j e c t to the boundary c o n d i t i o n s o f assumption (4), and s o l v i n g f o r x i n terms of V g i v e s x = d(1 - / l - V/V ) , c (C14) and a maximum f i e l d E = -2V /d . m C (C.15) T h i s immediately g i v e s an energy d i s t r i b u t i o n formula d N = cv (C.16) 249 depending on the s i n g l e parameter d/f. For lower values of d/f a number of ions w i l l reach the cathode w i t h f u l l energy eV , having s u f f e r e d no c o l l i s i o n s . The experimental r e s u l t s s t r o n g l y suggested t h a t the dominant c o l l i s i o n type was symmetrical charge t r a n s f e r , where an i o n encountering i t s own s p e c i e s of n e u t r a l imparts very l i t t l e energy to the n e u t r a l but takes an e l e c t r o n or e l e c t r o n s from i t ; a f t e r such a c o l l i s i o n we have a new i o n i d e n t i c a l to the o r i g i n a l but without k i n e t i c energy, and a h i g h energy n e u t r a l moving toward the cathode. For H* i n H 2 , Ne + and N e + + i n neon, A r + and A r + + i n argon, and (with a l a r g e experimental e r r o r ) He + i n helium, good f i t s t o theory were achieved y i e l d i n g s i z a b l e c r o s s s e c t i o n s (small values of f ) . H + i n H2 gave a good f i t to the theory, w i t h a s m a l l c r o s s - s e c t i o n , but w i t h a smaller-than-expected number o f i o n s having f u l l or n e a r l y f u l l energy eV c. A proton exchange r e a c t i o n H + H 2 -»• H 2 + H combined wi t h an abundance o f H near the n e g a t i v e glow may be p o s s i b l e , but E q . ( C l l 6 ) c o u l d no longer apply e x a c t l y s i n c e momentum t r a n s f e r must occur. i n hydrogen seems to f i t theory w i t h a moderately s m a l l apparent c r o s s s e c t i o n and may be due to a proton exchange mechanism and p o s s i b l y to formation from "H-pickup" r e a c t i o n s o f U.^ w i t h J ^ . Based on the apparent c r o s s s e c t i o n , some 40 times too many l i t ions reach the cathode 250 at f u l l energy eV , corresponding to the f a c t t h a t ions o r i g i n a t i n g i n the n e g a t i v e glow cannot undergo symmetric + + + charge exchange. He 2 i n helium, and H and H 2 from hydrogen i m p u r i t i e s i n i n e r t gases, g i v e constant numbers of ions ( i . e . , independent of V"c and d) with f u l l cathode f a l l energy. These c o n s t i t u t e most o f the i o n i c c u r r e n t f o r + + H and H 2 i m p u r i t i e s , as expected s i n c e charge exchanges are r u l e d out. F o r He 2 there i s i n a d d i t i o n a low energy c o n s t i t u e n t which may r e s u l t from the formation of the i o n by the a c t i o n o f low energy e l e c t r o n s near the cathode. These r e s u l t s do not r u l e out charge exchange between d i f f e r e n t s p e c i e s (e.g., H 2 + He H 2 + He f o r hydrogen i m p u r i t y i n helium would l o s e the i m p u r i t y i o n a l t o g e t h e r and g a i n the dominant s p e c i e s i o n i n accordance w i t h i t s own d i s t r i b u t i o n ) . We are l e f t w i t h a c l a r i f i e d p i c t u r e , however, i n which the cathode f a l l i s a r e g i o n wherein e l e c t r o n s (and perhaps protons) are exchanged r a t h e r than momentum. We are a l s o g i v e n reason to expect an abundance of f a s t n e u t r a l s r e a c h i n g the cathode, which c l e a r l y accounts f o r some o f the d i s c r e p e n c y between 1/y and the number of dark space c o l l i s i o n s . The assumption o f a l i n e a r e l e c t r i c f i e l d i n the cathode f a l l (assumption (4) of Davis and V a n d e r s l i c e , above, and Eqs. ( C . l 3 - C t . 15 ) i s based on experimental evidence as much as s i x t y years o l d ( A s t o n 3 8 ; L o e b 2 3 , p. 578, F i g . 271), and shows the importance o f space charge (without which the p o t e n t i a l would be l i n e a r and the f i e l d c o n s t a n t ) . W a r r e n 2 8 251 has made what may be the b e s t measurements to date u s i n g an e l e c t r o n beam as a probe. His curves show a semblance of l i n e a r i t y at h i g h e r p r e s s u r e s but with n o t i c e a b l e departures even then. Warren 2*• proceeded to make some c a l c u l a t i o n s based upon simple assumptions: Maxwell's equation f o r the divergence o f the e l e c t r i c f i e l d i s dE dx = 4ft (p - p ) = 4IT a+ - l v - V (6.17) where x i s d i s t a n c e from the cathode, j i s c u r r e n t d e n s i t y , p i s charge d e n s i t y , v i s v e l o c i t y , and the p o s i t i v e and n e g a t i v e s i g n s make r e f e r e n c e to ions and e l e c t r o n s , r e s p e c t i v e l y , W a r r e n 2 k n e g l e c t e d the r e l a t i v e l y s m a l l e l e c t r o n charge d e n s i t y , c o n s i d e r e d the i o n i c c u r r e n t d e n s i t y as a c o n s t a n t , and f o r i o n i c v e l o c i t y used v + = k ( E / p ) * (C.18) where k i s some cons t a n t ( m o b i l i t y ) . T h i s form was based on the p r e v i o u s work of a number of r e s e a r c h e r s (see Warren 2 1 4, r e f s ; 10 - 13) and i s claimed to h o l d f o r c o l l i s i o n - l i m i t e d motion a t E/p v a l u e s from 100 to a t l e a s t 1000 volts/cm-mm Hg, These premises l e a d to E « ( J x ) * ( c > 1 9 ) where x i s the p o s i t i o n i n the cathode f a l l , as measured from the edge o f the negative glow, determined as f o l l o w s : To e l i m i n a t e c e r t a i n s m a l l , p e c u l i a r , and sharp drops i n the f i e l d near the n e g a t i v e glow, Warren e x t r a p o -l a t e d h i s f i e l d curves to zero (the e f f e c t i v e n e g a t i v e glow "anode" l o c a t i o n ) and r e p l o t t e d them p l a c i n g t h i s p o i n t at a common o r i g i n f o r a l l curves. He found t h a t p l o t t i n g Z E / J 3 a g a i n s t x d i d indeed remove most o f the dependence on J (at constant p) f o r H 2 and He, and, to a somewhat l e s s e r degree, Ar, N 2, and a i r . Furthermore, the r e s u l t a n t Z curves were a c l o s e match to the x ? dependence p r e d i c t e d by Eq.(C.19) a t a p r e s s u r e of 1 mm Hg (and moderately w e l l a t 0.3 mm Hg), but w i t h marked d e v i a t i o n s a t lower Z p r e s s u r e . On the whole, the dependence upon J5 seems Z a p p l i c a b l e over a wider range o f p r e s s u r e than the x 3 dependence. The s y s t e m a t i c inadequacy o f Warren's c a l c u l a t e d curves i s e v i d e n t at low p r e s s u r e s f o r a l l gases t r e a t e d . T h i s i n d i c a t e s t h a t at lower p r e s s u r e s the net p o s i t i v e space charge i s g r e a t e s t near the cathode r a t h e r than near the n e g a tive glow (see Eq. (C.17). T h i s i s r a t h e r reasonable s i n c e at h i g h p r e s s u r e s , c o l l i s i o n s would keep i o n e n e r g i e s too low to cause e x c i t a t i o n . The g r a d u a l i n c r e a s e i n i o n i c v e l o c i t y t o g e t h e r w i t h the h i g h e r space charge from r e l a t i v e l y slow e l e c t r o n s , causes the net space charge to f a l l o f f near the cathode. At lower p r e s s u r e s , i o n i z a t i o n by e l e c t r o n impact becomes r a r e r s i n c e the e l e c t r o n f r e e path becomes longer than the cathode f a l l , but i o n i z a t i o n by i o n s more 253 frequent since the i o n f r e e path becomes long enough to allow i o n s to g a i n the l a r g e e n e r g i e s needed to have a p p r e c i a b l e i o n i z a t i o n c r o s s s e c t i o n . The l a t t e r process would happen i n the v i c i n i t y o f the cathode and y i e l d a s t r o n g e r net p o s i t i v e space charge i n the r e g i o n s i n c e the h i g h f i e l d s would remove e l e c t r o n s more q u i c k l y than i o n s . One might suppose t h a t the i n c l u s i o n of ion-impact i o n i z a t i o n terms should s u b s t a n t i a l l y improve agreement w i t h Warren's d a t a . Examination o f t h a t data r e v e a l s t h a t where E / J 3 t u r n s out to be a f u n c t i o n of J ( c o n t r a r y to Warren's theory) i s at h i g h e r c u r r e n t s (and thus h i g h e r v o l t a g e s ) and lower p r e s s u r e . Warren's l i m i t e d p u b l i c a t i o n of l i g h t i n t e n s i t y measurements a l s o confirms these i d e a s , with l i g h t from near the cathode r i v a l i n g the n e g a t i v e glow i t s e l f under c o n d i t i o n s o f low p r e s s u r e and h i g h c u r r e n t and v o l t a g e i n n i t r o g e n ( W a r r e n 2 8 , F i g . 17). The concave f i e l d v s . d i s t a n c e curves which Warren found a t h i g h c u r r e n t s and low p r e s s u r e s resemble the r e s u l t s o f t h i s work, where much high e r v o l t a g e s (and c u r r e n t d e n s i t i e s ) are p r e s e n t . I s h i d a and Tamura 2 9 (p.173) have found t h a t a f u n c t i o n o f the form E = a i e " b l x - a 2 e ~ b 2 x "(C.20) f i t s t h e i r h i g h - v o l t a g e o b s e r v a t i o n s (a^, b^, a 2 , b 2 are p o s i t i v e c onstants f o r a g i v e n d i s c h a r g e c o n d i t i o n , w i t h a 0 and b~ s m a l l and x measured from the cathode. 254 Ward 3 0 3 2 used a computer to s o l v e n u m e r i c a l l y f o r the parameters o f the glow d i s c h a r g e , u s i n g Eqs. ( C l ) and (C.17)as a p a i r o f coupled d i f f e r e n t i a l equations i n E and J . Eq. (C.7) was used f o r a( E / p ) , with r = l i n di a t o m i c and r=£ i n i n e r t gases. T h i s f u n c t i o n a l form, -1/ r a=e / y , goes from zero to one as y « E/p runs from zero to i n f i n i t y : I t i s a m o n o t o n i c a l l y i n c r e a s i n g but r e a d i l y s a t u r a t e d f u n c t i o n so t h a t (as mentioned e a r l i e r ) the i o n i z i n g e f f i c i e n c y n = a/E passes through a maximum. T h i s was noted by Ward 3 2 (p.27 94) who appears to have been a r g u i n g t h a t t h i s can e x p l a i n the normal d i s c h a r g e c o n d i t i o n : Assuming t h a t E i s some monotonic f u n c t i o n of J a t some pr e s s u r e p, then there i s a value of J a t which n i s a maximum. D e v i a t i o n s from t h i s v a l u e of J i n c r e a s e the e f f e c t i v e r e s i s t i v i t y o f the d i s c h a r g e ; thus i f s u f f i c i e n t c a t h o d i c area i s a v a i l a b l e , the area covered w i l l a d j u s t to g i v e t h i s v a l u e o f J and a path of l e a s t r e s i s t a n c e . The e l e c t r o n m o b i l i t y was found to be not c r i t i c a l and was t r e a t e d as cons t a n t . Ward 3 0 (p.1857) found, i n p a r t i c u l a r , t h a t the e l e c t r o n m o b i l i t y a f f e c t s the Faraday dark space and p o s i t i v e column, w h i l e the cathode r e g i o n i s a f f e c t e d by changes i n i o n i c m o b i l i t y , as one would expect s i n c e these are the r e s p e c t i v e l y dominant c u r r e n t c a r r i e r s . The primary r e s u l t of these c a l c u l a t i o n s was the forma t i o n o f a cathode f a l l r e g i o n w i t h l i n e a r f i e l d connected to the anode by.a n e u t r a l plasma (Ward 3 0, p.1856). The dominant c u r r e n t c a r r i e r s were indeed ions i n the cathode 255 r e g i o n and e l e c t r o n s i n the plasma. C a l c u l a t e d f i e l d s were found to obey Warren's law (Ward 3 0, p.1855), and r a t h e r good agreement was o b t a i n e d with the bulk of Warren's o b s e r v a t i o n s (Ward 3 2, p.2794) f o r the i n e r t gases (except at low p r e s s u r e s ) . Ward's c a l c u l a t i o n s a l s o v e r i f i e d the discharge's.-independence o f the i n i t i a l v a l u e of p h o t o e l e c t r i c s t a r t i n g c u r r e n t ( I Q i n Eqs. (C.2)-(C.4) above), once J 2 exceeds the o r d e r o f 1 mA/cm (c u r r e n t d e n s i t i e s encountered i n experiments r e p o r t e d here were much h i g h e r ) . V o l k o v 3 7 used a s l i g h t l y d i f f e r e n t approach to s o l v i n g Eq. (C.17) and was able to o b t a i n a c l o s e d form s o l u t i o n a p p l i c a b l e o n l y i n the cathode f a l l . (He made what appear to be much si m p l e r assumptions about the form o f v + ( E ) , but i n c l u d e d ions e n t e r i n g from the negative glow). He o b t a i n e d c l o s e agreement with Warren's data f o r H 2 a t 1 mm Hg, accounting f o r the s m a l l sharp drops i n f i e l d mentioned e a r l i e r . Another i n t e r e s t i n g d i s c o v e r y o f Warren's, v e r i f i e d by Ward's and Volkov's r e s u l t s , i s t h a t r e p l o t t i n g the f i e l d -I curves ( a f t e r n o r m a l i z a t i o n by J ? ) with the o r i g i n a t the edge of the n e g a t i v e glow f o r f i x e d p r e s s u r e but d i f f e r e n t cathode m a t e r i a l s , the curves f o l l o w each other q u i t e u n i f o r m l y from the o r i g i n , but extend d i f f e r e n t d i s t a n c e s (corresponding to d i f f e r e n t cathode dark space l e n g t h s ) , depending upon the m a t e r i a l and c o n d i t i o n of the cathode. 256 8. More on the Cathode Glow Thus i t seems t h a t good progress has been made i n understanding the processes c o n t r o l l i n g - the cathode r e g i o n s of glow d i s c h a r g e s , with a s i g n i f i c a n t e x c e p t i o n i n the phenomenon o f the cathode glow. No anomalies are found i n any of the c a l c u l a t e d or observed f i e l d curves near the cathode which c o u l d d i f f e r e n t i a t e cathode glow from a d j o i n i n g dark space, although the glow was d e f i n i t e l y observed by W a r r e n 2 8 (p.1656, F i g . 1 7 ) . The c a l c u l a t i o n s by Ward 3 2(p.2793f., Fig.9) of l i g h t i n t e n s i t y based o n l y on e l e c t r o n c u r r e n t d e n s i t i e s and e a r l y e stimates of e x c i t a t i o n e f f i c i e n c i e s ( P e n n i n g 3 5 ) d i d not produce a cathode glow. A n a t u r a l s u p p o s i t i o n might be t h a t e l e c t r o n s i n the Aston dark space have i n s u f f i e c i e n t energy to e x c i t e the gas and t h a t the cathode glow begins where t h i s energy has been a t t a i n e d . H o i s t and O o s t e r h u i s 3 9 f i r s t observed under c e r t a i n low-pressure c o n d i t i o n s a whole sequence of glows ( s t r i a t i o n s ) extending i n t o the dark space and separated by d i s t a n c e c o r r e s p o n d i n g to the i o n i z a t i o n p o t e n t i a l of the gas (neon). Ward 3 2(p.2790) c a l c u l a t e d v a l u e s f o r the s i z e of the Aston dark space, d e f i n e d as "the d i s t a n c e a t which an e l e c t r o n can f i r s t a t t a i n the lowest e x c i t a t i o n p o t e n t i a l o f the gas", and they agree f a i r l y w e l l w i t h the e a r l i e r measurements on helium. In s p i t e of t h i s evidence l e n d i n g p l a u s i b i l i t y to an e l e c t r o n e x c i t a t i o n p r o c e s s , Loeb 2 3(p.570) f e l t t h a t the e l e c t r o n d e n s i t y would be too low so near the cathode, 257 c o n s i d e r i n g the low v a l u e s of y c i t e d i n h i s book, and t h a t some process i n v o l v i n g ions would be more l i k e l y . W a r r e n 2 8 (pp.1655, 1657) agrees , n o t i n g t h a t i m p u r i t i e s o f low e x c i t a t i o n p o t e n t i a l ( n i t r o g e n , f o r example) are much l e s s dominant i n the s p e c t r a o f cathode glows than i n the other b r i g h t r e g i o n s o f the d i s c h a r g e w h i c h , i n d i c a t e s t h a t high energy p a r t i c l e s are r e s p o n s i b l e . W a r r e n 2 4 (p.1663) estimates t h a t when E/p i s high enough to a l l o w i o n s to g a i n over 100 ev per mean f r e e path (roughly 2000 v/cm f o r H* a c c o r d i n g to Davis and V a n d e r s l i c e 3 6 , p.227), the i o n i z a t i o n p r o b a b i l i t y per c o l l i s i o n i s about one per cent ( V a r n e y 4 0 ) so t h a t each i o n w i l l make an average of one i o n i z i n g c o l l i s i o n . Moreover, e x c i t a t i o n by e n e r g e t i c n e u t r a l s (which should be h i g h l y abundant a c c o r d i n g to Sec. 7) may have even l a r g e r c r o s s s e c t i o n s than f o r ions and may be an even more important cause of gas e x c i t a t i o n a t the cathode.(See the d i s c u s s i o n of Warren's measurements i n Sec. 7. . The glow has been seen to p e n e t r a t e through and beyond small h o l e s i n the cathode, making e l e c t r o n s seem an u n l i k e l y mechanism. F r a n c i s 2 6 (p.96f) and Druyvesteyn and P e n n i n g 2 7 (p.130) r e s o l v e t h i s by suggesting t h a t there are two d i f f e r e n t kinds of cathode glow. Once would occur a t low p r e s s u r e s , e x h i b i t the Aston dark space, and be induced by e l e c t r o n s . The other type i s t h i c k e r , l e s s s h a r p l y d e f i n e d , o f t e n b r i g h t e r ( p o s s i b l y masking the f i r s t t y p e ) , and caused by ions or f a s t n e u t r a l s . The c o r r e l a t i o n of l i g h t i n t e n s i t y w i t h p a r t i c l e 258 e n e r g i e s , as a f u n c t i o n of p o s i t i o n i n the cathode f a l l , would be needed t o c l e a r l y e s t a b l i s h the mechanism of the cathode glow, but t h i s would be a very d i f f i c u l t experiment by the methods of Warren. 9. The R a d i a l Current D i s t r i b u t i o n One f u r t h e r matter o f importance i n the cathode r e g i o n i s the f a c t t h a t , as .the v o l t a g e i s p r o g r e s s i v e l y r a i s e d i n t o the abnormal glow r e g i o n , the c u r r e n t d e n s i t y does not remain uniform over the s u r f a c e o f the cathode. In e a r l y work, Chiplonkar** A measured the c u r r e n t d e n s i t y as a rough f u n c t i o n o f r a d i u s by d i v i d i n g the cathode i n t o three c o n c e n t r i c areas grounded through separate ammeters. Ch i p l o n k a r found t h a t f o r v o l t a g e s between 1000 and 6000 v the c u r r e n t d e n s i t y c o u l d be f i t t e d t o a Gaussian - h r 2 d i s t r i b u t i o n e , where h i s a f u n c t i o n of v o l t a g e . t . (He a c t u a l l y decided t h a t h b e t t e r represented a f u n c t i o n of dark space l e n g t h : a r a t h e r q u e s t i o n a b l e c o n c l u s i o n , e s p e c i a l l y s i n c e p r e s s u r e s were not reported.) In the l a t e r work o f C h i p l o n k a r and J o s h i 4 * 3 '****, the r a d i a l d i s t r i b u t i o n of i o n i c c u r r e n t d e n s i t y was measured by t r a p p i n g the " c a n a l " rays coming through a s m a l l movable hole i n the cathode. C h i p l o n k a r and J o s h i 1 * 3 found t h a t , i n d i s c h a r g e s maintained a t constant c u r r e n t i n hydrogen wi t h an aluminum cathode . low v o l t a g e s , a n d moderate p r e s s u r e s (e.g., 236 v a t 0.5 mm Hg), the c u r r e n t d e n s i t y was f a i r l y u n i f orm over the cathode, dropping o f f s l i g h t l y a t the edges. 259 At h i g h e r v o l t a g e and lower p r e s s u r e (e.g., 1800v at 0.033 mm Hg) the c u r r e n t was h i g h l y c e n t r a l i z e d and dropped to v e r y s m a l l v a l u e s w e l l s h o r t o f the edge. A r e g i o n o f f a i r l y c onstant (maximum) c u r r e n t was always prese n t i n the c e n t e r , but the curves gave the impression t h a t f o r higher v o l t a g e s (lower pressures) they might become t r u l y Gaussian i n appearance. A g l a s s tube supported over the c e n t e r of the d i s c h a r g e would f u r t h e r c e n t r a l i z e a d i s c h a r g e , the g r e a t e r the e f f e c t ( u n s u r p r i s i n g l y ) the c l o s e r the tube to the cathode. Changing the t o t a l c u r r e n t a t constant pressure (by moderate changes i n vol t a g e ) had l e s s e f f e c t on the r a d i a l c u r r e n t d i s t r i b u t i o n than changing v o l t a g e and p r e s s u r e a t constant c u r r e n t . A p p l y i n g an e x t e r n a l p o s i t i v e p o t e n t i a l to the o u t e r tube w a l l s had a s m a l l c e n t r a l i z i n g e f f e c t on the c u r r e n t . V o l t a g e s were a p p l i e d o n l y up to the s u p p l i e d d i s c h a r g e p o t e n t i a l d i f f e r e n c e . Kamke1*1* used much higher v o l t a g e s (20,000 - 30,000 v o l t s ) and obtained r a d i a l d i s t r i b u -t i o n s which c o n s i s t e n t l y showed a c e n t r a l peak f a r too s h a r p l y p o i n t e d to be t r u l y Gaussian. He extended C h i p l o n k a r ' s 4 1 measurements to conclude t h a t a c e n t r a l i z e d d i s c h a r g e with a 4 mm r a d i u s (of h a l f i n t e n s i t y ) a t 1000 v o l t s would drop to around 0.5 mm r a d i u s a t 20,000 v o l t s . Kamke found t h a t the use o f an aluminum cathode caused a n o t i c e a b l y more c e n t r a l i z e d d i s t r i b u t i o n o f c u r r e n t , and reduced by some 25% the p r e s s u r e needed to draw a g i v e n c u r r e n t , as compared to copper, carbon, o r s t e e l cathodes. I t appears t h a t the c e n t r a l i z a t i o n o f d i s c h a r g e a t 260 the cathode i s n e v e r t h e l e s s p r i m a r i l y due to the accumulation o f p o s i t i v e charge a t the w a l l s . In the cathode f a l l r e g i o n there are presumed to be many more ions than e l e c t r o n s c a r r y i n g the c u r r e n t (see p r e c e d i n g s e c t i o n s ) , and t h a t d i f f u s i o n w i l l t h e r e f o r e c a r r y a predominantly p o s i t i v e charge to the w a l l s . I f most i o n i c c o l l i s i o n s are p u r e l y charge exchange then we can expect a number o f ions to be formed wi t h thermal v e l o c i t i e s moving toward the w a l l s so t h a t s u b s t a n t i a l p o t e n t i a l s may develop a t the w a l l s which w i l l serve to r e - f o c u s i o n s toward the c e n t e r . I t would seem t h a t r a d i a l e l e c t r i c f i e l d s a p p l i e d by C h i p l o n k a r and J o s h i 4 * * ( p . l 4 6 f f ) must have been f a r lower than those o c c u r i n g n a t u r a l l y from d i f f u s e d w a l l charge. Only v e r y near the cathode, where the e l e c t r o n s are s t i l l o f low energy (and thus may c o n t r i b u t e s u b s t a n t i a l l y to the net space charge), would t h e i r higher m o b i l i t y a l l o w them to reduce the w a l l p o t e n t i a l . C h i p l o n k a r and J o s h i 1 * 5 (p.l48f) added p l a u s i b i l i t y to r a d i a l charge d i f f u s i o n as a mechanism with t h e i r f i n d i n g t h a t a magnetic f i e l d along the tube a x i s w i l l serve to reduce the c e n t r a l i z a t i o n of c u r r e n t . I t i s hard to v i s u a l i z e an a l t e r n a t i v e mechanism. Ch i p l o n k a r and J o s h i 1 * 3 (p. 1753) found t h a t t h e i r i n t e g r a t e d i o n i c c u r r e n t d e n s i t y accounted f o r o n l y 5-10% o f the t o t a l c u r r e n t , although t h i s d i d not seem to shake t h e i r f a i t h i n the predominance o f p o s i t i v e c u r r e n t c a r r i e r s . Although the r e l e a s e of e l e c t r o n s a t the cathode by f a s t n e u t r a l s would reduce the number of ions needed to m a i n t a i n the d i s c h a r g e , i t i s f e l t t h a t t h i s r e s u l t must be i n s e r i o u s 261 e r r o r . 10. The Negative Glow We s h a l l b r i e f l y d i s c u s s the remaining p o r t i o n s of the glow d i s c h a r g e , h e r e t o f o r e d e s c r i b e d as merely a conductor of n e u t r a l plasma connecting the " e f f e c t i v e anode" a t the end of the cathode f a l l to the r e a l anode; these p o r t i o n s are not e s s e n t i a l i n the sense t h a t they d i s a p p e a r as the anode i s moved c l o s e r to the cathode, without a f f e c t i n g the cathode r e g i o n . Although o f t e n l a r g e r and more s p e c t a c u l a r than the r e g i o n s a l r e a d y c o n s i d e r e d , theyhave l e s s i n t e r e s t here. The negative glow (marked C i n F i g . C 2 ) i s a r e g i o n of h i g h l u m i n o s i t y and low e l e c t r i c f i e l d , which l a t t e r has been measured by Warren 2 8(p.1657) to be ^ to J volt/cm. T h i s can be thought o f as the r e g i o n wherein the excess energy o f e l e c t r o n s a c c e l e r a t e d by the cathode f a l l i s d i s s i p a t e d through i o n i z a t i o n s and e x c i t a t i o n s . Indeed, the l e n g t h o f the negative glow has been c o r r e l a t e d with the range of e l e c t r o n s i n the plasma (Cobine 2 2p.219, F i g . 8.6). I t i s c l e a r t h a t some net e l e c t r i c f i e l d must be maintained i n the n e g a t i v e glow t o m a i n t a i n a n e u t r a l plasma; e l e c t r o n s are s t e a d i l y a r r i v i n g from the cathode and ions l o s t when they stumble i n t o the f a l l . 11. The Faraday Dark Space By the time they reach the Faraday dark space, l a b e l e d D i n Fig.C2,few e l e c t r o n s are s u f f i c i e n t l y e n e r g e t i c 262 to cause e x c i t a t i o n ( l e t alone i o n i z a t i o n ) and the d e n s i t y o f charged p a r t i c l e s d e c l i n e s as our p o i n t o f o b s e r v a t i o n moves towards the anode. The p r i n c i p a l mechanism i s thought to be recombina-t i o n a t the w a l l s . S ince e l e c t r o n s have a much longer mean f r e e path than i o n s , they w i l l reach the w a l l more r a p i d l y u n t i l s u f f i c i e n t n e g a t i v e charge i s e s t a b l i s h e d on the w a l l to prevent f u r t h e r b u i l d u p . The e l e c t r i c f i e l d of t h i s charge enhances the d i f f u s i o n of i o n s t o the w a l l , where they are n e u t r a l i z e d and r e - e n t e r the d i s c h a r g e as gas, but i n the steady s t a t e , a n e g a t i v e w a l l charge i s maintained. (Warren's 2 4 (p.1662) statement t h a t p o s i t i v e charges accumulate on the w a l l s must c e r t a i n l y be an unintended e r r o r . ) T h i s process i s l i k e l y to be p r e s e n t a l l along the d i s c h a r g e except i n the cathode r e g i o n (see Sec.10). W a r r e n 2 8 (p.1654, Fig.16) has measured the w a l l p o t e n t i a l and found i t to be some 20% lower than t h a t c a l c u l a t e d from the average f i e l d a c ross the tube diameter, except near the cathode. With l a r g e k i n e t i c e n e r g i e s p r e s e n t i n the n e g a t i v e glow, t h i s process of charge d e p l e t i o n i s r e l a t i v e l y unimportant. In the Faraday dark space, however, i t i s thought to reduce the a v a i l a b l e space charge to the p o i n t t h a t the tube c u r r e n t causes v o l t a g e d i f f e r e n c e s to r i s e . The r e s u l t i n g f i e l d a c c e l e r a t e s e l e c t r o n s to the i o n i z a t i o n t h r e s h o l d (the s t a r t o f the p o s i t i v e cblumn) whereby the charge c a r r i e r s are r e p l e n i s h e d . W a r ren's 2 8 (p.1657) measurements o f f i e l d s t r e n g t h i n the Faraday dark space are as low as, or lower than those i n the n e g a t i v e 263 glow to £ v o l t / c m ) , so e i t h e r the mechanism or the measurements must be i n doubt. (E.g., the 25 ev i o n i z a t i o n p o t e n t i a l of helium d i v i d e d by £ volt/cm g i v e s 50 cm f o r the l e n g t h of the Faraday dark space, while Warren's e n t i r e tube was o n l y 40 cm long!) F r a n c i s 2 6 ( p . 1 7 9 f ) notes t h a t i n e l e c t r o n e g a t i v e gases, where negative ions r a t h e r than f r e e e l e c t r o n s predominate, most o f the p o t e n t i a l drop of the Faraday dark space e x i s t s across a v e r y narrow l a y e r j u s t p r e c e d i n g the p o s i t i v e column. In H 2 and i n e r t gases, however, the high m o b i l i t y of f r e e e l e c t r o n s makes such a narrow f i e l d r e g i o n seem improbable as an e x p l a n a t i o n f o r Warren's low measurements. In any event i t has been observed t h a t the l a r g e r the tube (and thus the s m a l l e r the w a l l l o s s e s ) the longer the Faraday dark space. Furthermore, i n s p h e r i c a l d i s c h a r g e tubes, where t h e r e are no w a l l l o s s e s , the Faraday dark space reaches the anode. Warren's 2 4(p.1659) o b s e r v a t i o n t h a t the Faraday dark space i s longer f o r higher c u r r e n t s and v o l t a g e s (thus causing g r e a t e r space charge excess) appears to be i n c o n t r a d i c t i o n w i t h an e a r l i e r argument by Thomson and Thomson 3 3(p.358ff) (repeated by C o b i n e 2 2 , p. 2 3 2 f ) , . t h a t t h i s l e n g t h should be i n v e r s e l y p r o p o r t i o n a l to the c u r r e n t (or, a t low p r e s s u r e s , to the square r o o t of the c u r r e n t ) . 12. The P o s i t i v e Column, P l a i n When e l e c t r o n e n e r g i e s are s u f f i c i e n t l y i n c r e a s e d t o cause renewed i o n i z a t i o n (and e x c i t a t i o n ) the p o s i t i v e 264 column beg i n s . Fig.C2 shows the column (marked E) as being s t r i a t e d w i t h peaks of l i g h t i n t e n s i t y marked F, although u n s t r i a t e d columns are more common. In the u n s t r i a t e d column, i t i s imagined t h a t the d i f f u s i o n l o s s e s to recombinations a t the w a l l s , d i s c u s s e d i n the l a s t s e c t i o n , are c o n t i n u i n g and are compensated by p a r t i c l e s from new i o n i z a t i o n s caused by a f i e l d which (through the e f f e c t i v e r e s i s t i v i t y o f the plasma) a d j u s t s i t s e l f to t h i s end. Warren's 2 8(p.1657) f i e l d measurements here g i v e one o r two v o l t s / c m . Ward's 3 0(p.1856, Fig.10) c a l c u l a t i o n s show a f i e l d of over 5 v o l t s / c m along a p o s i t i v e column w i t h pd = 10 cm-mm Hg (d i s the anode-cathode gap), the f i e l d being roughly p r o p o r t i o n a l to p r e s s u r e ; t h i s i s the f i e l d necessary to overcome the i n t r i n s i c r e s i s t i v i t y due to pressure-decreased m o b i l i t y , and does not even i n c l u d e w a l l l o s s e s . Thus i t appears t h a t Warren's measurements may w e l l be i n e r r o r i n these low-f i e l d r e g i o n s , although they have been d e s c r i b e d as a c l a s s i c experiment ( V o l k o v 3 7 ) . I t i s w e l l to p o i n t out, however, t h a t t h e r e i s never any c e r t a i n t y t h a t an a p p a r e n t l y uniform column does not c o n t a i n r a p i d l y moving s t r i a t i o n s ( F r a n c i s 2 6 , p.114). We s h a l l i g n o r e t h i s p o s s i b i l i t y and, i n f a c t , the whole d i f f i c u l t s u b j e c t of moving s t r i a t i o n s , s i n c e they are u s u a l l y not v i s i b l e i n our apparatus. 13. The P o s i t i v e Column, Deluxe With some gases, under the proper c o n d i t i o n s , s t r i a t i o n s of h i g h c o n t r a s t appear. I t i s g e n e r a l l y supposed t h a t the mechanisms u n d e r l y i n g s t r i a t e d and u n s t r i a t e d p o s i t i v e columns are the same, but t h a t c o n d i t i o n s are more u n i f o r m l y d e f i n e d along the d i s c h a r g e a x i s under some c o n d i t i o n s , a l l o w i n g zones of a l t e r n a t e a c c e l e r a t i o n and i o n i z a t i o n / e x c i t a t i o n to become v i s i b l e . Loeb 2 3(p.573) summarizes an argument to the e f f e c t t h a t s t r i a t i o n s may be destroyed through the t r a n s p o r t a t i o n o f e x c i t a t i o n energy by metastable e x c i t e d atoms or molecules, which can t r a v e l independently o f the e l e c t r i c f i e l d s . I t f o l l o w s t h a t mixtures o r i m p u r i t i e s (having a component wi t h i o n i z a t i o n p o t e n t i a l lower than the metastable e x c i t a t i o n energy) can be the v e h i c l e s by which metastable e x c i t e d s t a t e s may be d e a c t i v a t e d ("Penning e f f e c t " ) , d e s t r o y i n g s a i d energy t r a n s p o r t process and a l l o w i n g s t r i a t i o n s to appear. W a r r e n 1 s 2 5 ( p . 1 6 5 7 , Fig.18) f i e l d measurements upon s t r i a t e d p o s i t i v e columns seem eminently more reasonable than those p r e v i o u s l y mentioned f o r the u n s t r i a t e d type. His graph shows f i e l d s averaging around 7 v o l t s / c m and v a r y i n g over the course of each s t r i a t i o n by 1 or l£ v o l t s / c m e i t h e r way. He o b t a i n e d p o t e n t i a l drops between s t r i a t i o n s along the a x i s which correspond., to the i o n i z a -t i o n p o t e n t i a l i n i n e r t gases and to about double the i o n i z a -t i o n p o t e n t i a l i n d i a t o m i c gases. These p o t e n t i a l drops, moreover, appear to depend o n l y on the gas and not on the p r e s s u r e or c u r r e n t , l e n d i n g f u r t h e r p l a u s i b i l i t y to the proposed mechanism. 266 14. The Anode A mechanism i s roughly d e s c r i b e d by L o e b 2 3 ( p . 5 9 6 f ) and C o b i n e 2 2 ( p . 2 4 6 f ) f o r the g e n e r a t i o n of a f a l l o f p o t e n t i a l a t the anode. B a s i c a l l y , i t would seem t h a t the r e p u l s i o n o f the ions i n the plasma by the anode would keep the r e g i o n f a i r l y w e l l c l e a r of i o n s , l e a v i n g a net negative space charge, which, by Maxwell's equation (Eq . c.17 ) , c r e a t e s an e l e c t r i c f i e l d of p o s i t i v e slope ( i n c r e a s i n g toward the anode). Since e l e c t r o n s are f a r more r e a d i l y moved by a f i e l d than are i o n s , the net e l e c t r o n i c space charge near the anode i s f a r s m a l l e r than the i o n i c space charge i n the cathode f a l l , and the e l e c t r i c f i e l d s correspond-i n g l y s m a l l e r . In o t h e r words, where the c a r r i e r s are e l e c t r o n s (at the anode), much s m a l l e r f i e l d s are needed to move the d i s c h a r g e c u r r e n t than where the c a r r i e r s are i o n s (at the cathode). Loeb a l s o argues f o r the n e c e s s i t y of a c c e l e r -a t i n g e l e c t r o n s hard enough to c r e a t e new i o n s which can r e p l a c e i o n l o s s e s to the w a l l s and i n t o the plasma, although there may be some c i r c u l a r i t y i n t h i s . Ward's 2 8(p.1856) computer-generated p o t e n t i a l curves do show a s m a l l anode f a l l , and h i s r a t h e r roughly c a l c u l a t e d l i g h t output curves (Ward 3 2, p.2794) seem to e x h i b i t an anode dark space f o r the h i g h e s t c u r r e n t d e n s i t y shown. 267 APPENDIX D SOME BACKGROUND IN MOLECULAR QUANTUM MECHANICS To review the p r i n c i p l e s i n v o l v e d , when the Hamilton-i a n matrix i s d i a g o n a l , then a l l of the wavefunctions (with r e s p e c t to which the matr i x elements have been taken: H^ .. = / i b * H ib.. dq) are e i g e n f u n c t i o n s of H and thus d e s c r i b e a c t u a l s t a t e s o f the system. For example, c o n s i d e r s t a t e iji^: Operating on i|>^  w i t h the d i a g o n a l Hamiltonian i n a ma t r i x r e p r e s e n t a t i o n : = w * l ' (b.l) E\\>. H l l ° 0 H 22 = H 11 showing t h a t ifi^ i s a p o s s i b l e s t a t e of the system with energy W = H ^ . I f the Hamiltonian i s not d i a g o n a l , then may not be an ei g e n f u n c t i o n of H, but i n g e n e r a l some combination (f>. = 7 c. . ib . (D. 2 w i l l be. The c o e f f i c i e n t s c.. are the elements of a matrix C which transforms each V. (a column v e c t o r which i s a l l 3 zeroes except f o r a 1 i n row j ) i n t o one of the a c t u a l e i g e n v e c t o r s m a t r i c e s , v 268 <K. In matrix n o t a t i o n (C and H are and $ are v e c t o r s , and W i s a s c a l a r ) : H $ i = W (D.3) H (OFj) = W(CVj) = . C(WFj) . ( D * 4 ) Now, i s i s p o s t u l a t e d t h a t C i s a u n i t a r y t r a n s f o r m a t i o n , + + (thereby p r e s e r v i n g the l e n g t h : I f $=CV then $ Q=V ¥ where " t " means the transposed complex c o n j u g a t e ) , -1 t wherefrom i t may be shown t h a t C = C so t h a t Eq. (D.3) becomes (C+HC)17 = m . (D.5) Thus the t r a n s f o r m a t i o n C has two i n t e r p r e t a t i o n s : From Eq. (D.3) i t transforms the v e c t o r of the b a s i s s e t f u n c t i o n s ¥ i n t o the e i g e n v e c t o r s of H. A l t e r n a t i v e l y , by Eq. (D.5) + C transforms H i n t o an o p e r a t o r C HC which i s d i a g o n a l w i t h r e s p e c t to the b a s i s 4 * . Eq. (D.4) may be w r i t t e n out as 7 (H. . - V7 6 . ) c . = 0 . (D. 6) v i j n j n j n I t may be shown t h a t Eq. (D.6) has a s o l u t i o n , i n which not a l l the c. are zero, o n l y i f the determinant of the j n u e x p r e s s i o n i n parentheses v a n i s h e s , and from the r e s u l t i n g equation a l l p o s s i b l e eigenvalues Wn may be found. 2 69 S u b s t i t u t i n g these e i g e n v a l u e s back i n t o Eq. (D.6) g i v e s a system of simultaneous equations which, i n c o n j u n c t i o n with the n o r m a l i z a t i o n c o n d i t i o n (D.7) determines the c. t o w i t h i n a constant phase f a c t o r . In p r a c t i c e , g e n e r a l formulae w i l l be used f o r the case of 2 x 2 m a t r i c e s , while f o r l a r g e r m a t r i c e s i t i s convenient to use numerical s o l u t i o n s . The M o l e c u l a r Hamiltonian The f i r s t step i n f o l l o w i n g the procedure d e s c r i b e d i n the l a s t s e c t i o n i s to s e t up the a p p r o p r i a t e Hamiltonian and to e v a l u a t e i t s m a t r i x elements w i t h r e s p e c t to some convenient b a s i s s e t of wavefunctions. We s h a l l not d e a l w i t h any i n t e r a c t i o n s between e l e c t r o n i c s t a t e s , nor s h a l l we i n c l u d e any of a number of f i n e r i n t e r a c t i o n s such as s p i n - s p i n , s p i n - r o t a t i o n , or r o t a t i o n a l d i s t o r t i o n terms (see M e r e r 1 6 , f o r example). I t w i l l be s u f f i c i e n t f o r our needs t o c o n s i d e r o n l y (1) a pure r o t a t i o n a l Hamiltonian H r along with (2) a s p i n - o r b i t term H s o'- v a n d (3) an u n s p e c i f i e d term H which i s assumed constant f o r the c ev e l e c t r o n i c and v i b r a t i o n a l s t a t e under c o n s i d e r a t i o n : (D.8) 270 H so and H r may be r e a d i l y w r i t t e n as H /he = A i • ? so (D.9) H r/hc = B £ 2 (D.10) where A and B are r e f e r r e d to as the s p i n c o u p l i n g and r o t a t i o n a l c o n s t a n t s , r e s p e c t i v e l y , and have u n i t s of era \ R (nuclear r o t a t i o n ) i s not a good quantum number i n Hund's cases (a) and (b), so we w i l l need to express i t s o p e r a t o r i n terms of o t h e r s which do have v a l i d quantum numbers. In our n o t a t i o n , (£,n , c ) are a s e t of c a r t e s i a n c o o r d i n a t e s f i x e d i n the molecule with o r i g i n a t the center of p o s i t i v e charge and both n u c l e i imbedded i n the t; .axis. R i s n e c e s s a r i l y zero, s i n c e does not i n c l u d e n u c l e a r s p i n or e l e c t r o n i c angular momenta. Now, so we may w r i t e Eqs. (D.9) and (D.10) i n m o l e c u l e - f i x e d components as = R" + £ + S" , ( D . l l ) H s o/hc = A(L 5S 5-+ L n S n + L ^ ) (D.12) and 2 7 1 H /he = B (R2 + R2) R ? N ( D . 1 3 ) = B ( J - L - S R ) + B ( J - L -S ) 2 5 5 S n n n = B ( j 2 + j 2 + L 2 + L 2 + S 2 + S 2 ) - 2 B ( J S + J S + J R L + J _ L ) + 2 B ( L S + L S ) 5 5 n n We d e f i n e the " r a i s i n g " and "lowering" o p e r a t o r s J + and J _ by J A = J r ± i J , (D.1 4 ) ± 5 n and L + and S + a n a l o g o u s l y , i t e a s i l y f o l l o w s t h a t L , S + L S, = 2 ( L R S R + L S ) ( D . 1 5 ) w i t h s i m i l a r formulae o b t a i n e d by s u b s t i t u t i n g J f o r e i t h e r L or S. Using these r e l a t i o n s , and the f a c t t h a t J 2 - J 2 = j 2 + j 2 ( a n d s i m i l a r l y f o r L and S ) , we have (D.16) H/hc = H /he + A L S + B ( J 2 - j 2 ) + B ( L 2 - L 2 ) + B ( S 2 - s 2 ) "V Q L, c, c, - B ( J + S _ + J _ S + ) - B ( J + L _ + J _ L + ) + (B+£A) ( L + S _ + L _ S + ) 272 The Diagonal M a t r i x Elements of the Hamiltonian Since we are assuming case (a) wavefunctions, i n which J , S, ft, A , and E are a l l v a l i d quantum numbers, we may e v a l u a t e most of the f i r s t l i n e of Eq. (D.16) immediately, r e p l a c i n g J2 + . ; J ( J + l)-h2 S 2 + S(S + l ) * 2 j + ah Aft S -> E"fi . I t must be remembered t h a t these terms are a l l zero when eva l u a t e d between wavefunction d i f f e r i n g i n any o f the quantum numbers; i . e . , they are d i a g o n a l with r e s p e c t to a l l o f them. The term i n v o l v i n g L cannot be evaluated s i n c e L i s not a good quantum number f o r case ( a ) . I t would seem t h a t Eq. (D.16) has gained n o t h i n g over Eq. (D.10); however, L ( u n l i k e R) can be taken to be d i a g o n a l with r e s p e c t to A, S, £, and n, and furthermore can be shown to be independent of these quantum numbers. (This r e s u l t i s depen-dent upon the s e p a r a b i l i t y of the s p i n and o r b i t a l wavefunc-t i o n s and t h e r e f o r e the smallness of s p i n - o r b i t c o u p l i n g e n e r g i e s ; see Hougens 2, p.25. Since the magnitude of A i s f i x e d f o r the e l e c t r o n i c s t a t e , i t i s a c t u a l l y the s i g n of A of which < L 2 - L 2 > i s independent.) The term B ( L 2 - L 2 ) i s t h e r e f o r e a constant w i t h i n any g i v e n e l e c t r o n i c 273 s t a t e and w i l l h e r e a f t e r be assumed to be absorbed w i t h i n H e v « The matrix elements of the f i r s t l i n e of Eq. (D.16) are thus r e a d i l y e v a l u a t e d . In summary, they appear o n l y on the d i a g o n a l of the f u l l m a t r i x , and they i n c l u d e a constant term H e v + B ( L 2 - L 2 ) , which w i l l normally be omitted. Omission of the c o n s t a n t terms ( i . e . , s u b t r a c t i n g the i d e n t i t y m a t r i x m u l t i p l i e d by the constant) i s j u s t i f i e d by c o n s i d e r i n g the d i a g o n a l i z a t i o n of H: C +H C = D (D.17) where D i s a d i a g o n a l m a t r i x c o n t a i n i n g the eigenvalues of H ( m u l t i p l i c a t i o n from the r i g h t by a v e c t o r ¥ makes Eq. (D.17) e q u i v a l e n t to Eq. (D.4)). Then i f k r e p r e s e n t s the c o n s t a n t m a t r i x element of H e v/hc + B ( L 2 - L 2 ) , and I i s the i d e n t i t y m a t r i x , C + ( H = D - k l - kI)C = (C +H - C + k I ) C = C +H C - C + k I C = D - kC +C (D.18) which i s a l s o d i a g o n a l . Thus the omission of k makes no change i n the e i g e n v e c t o r s (C) which d i a g o n a l i z e H, but causes the e i g e n v a l u e s to each be low, each by an amount k. 2,74 The O f f - D i a g o n a l M a t r i x Elements of the Hamiltonian The second l i n e of Eq. (D.16) c o n t a i n s o n l y the " r a i s i n g " and "lowering" o p e r a t o r s whose matrix elements are w e l l known; t h e i r e f f e c t s on a wavefunction, c h a r a c t e r i z e d by A, S, I, ft, J , and M and w r i t t e n |ASE; ftJM > , are g i v e n by • : s ± | ASI; ftJM.',>--= -h/S (S+l) - E (E±l) , | AS E±l; ftJM > (D. 19) - L ± |ASE;ftJM > = ft/L(L+l) - A ( A i l ) 1 | A±l SE; ftJM > (D. 20) |ASE;ftJM > = n / j ( J + l ) - ft (ft±l) | ASE ; ft±l JM > . , (D. 21) Thus, i n a t y p i c a l c a l c u l a t i o n , the o n l y non-zero ma t r i x element of L + s _ may be < A'S'E'; ft'J'M'|L+S_|ASE; ftJM > = n 2 / ( S ( S + l ) - E (E-l) ) (L(L+1) - A(A+1) ( D " 2  X < 5A ,A+1' SS ,S < SE'E-l 6ft'ft 6J'J 6M'M The r e v e r s a l of s i g n i n Eq. (D.21), as compared to Eqs. (D.19) and (D.20), r e q u i r e s some e x p l a n a t i o n . I t stems from the f a c t t h a t we are u s i n g m o l e c u l e - f i x e d c o o r d i n a t e s : L and S may be d e f i n a b l e i n t h i s system, but J i n c l u d e s the n u c l e a r r o t a t i o n so t h a t the e f f e c t o f i t s components i n the r o t a t i n g frame i s not obvious. Van V l e c k 1 7 has shown t h a t the ^commutator, of J (e.g., J J - J r J r ) i n molecule-275 f i x e d c o o r d i n a t e s have s i g n s o p p o s i t e to those of the normal commutation r e l a t i o n s ( i . e . , of J i n the s p a c e - f i x e d system or of L and S i n e i t h e r system.), by t a k i n g the (normal) s p a c e - f i x e d r e l a t i o n s through the c o o r d i n a t e t r a n s f o r m a t i o n i n a s t r a i g h t f o r w a r d manner. T h i s i n t u r n produces the s i g n change i n Eq. (D.21). We may n o t i c e however, t h a t t h i s p e c u l i a r s i g n i s necessary to o b t a i n non-zero m a t r i x elements f o r the terms, i n the second l i n e of Eq. (D.16), which are c o n s i s t e n t w i t h the c o n d i t i o n ft = A+E. Whenever ( f o r example) E i n c r e a s e s , Q must i n c r e a s e and the term J _ S + accomplishes j u s t t h i s . T h i s problem stems from the r a t h e r p e c u l i a r r o l e of the quantum- number Q i n the theory of l i n e a r m o l e c u l e s : Because i s always zero, Q, can be c o n s i d e r e d e i t h e r as a s p a c e - f i x e d system quantum number , o r a m o l e c u l e - f i x e d parameter A+E. The requirement t h a t these be equal i m p l i e s t h a t the r a i s i n g and lowering o p e r a t o r s of L,S, and J may occur i n no other combinations (of o r d e r l e s s than q u a r t i c ) than are found i n Eq. (D.16). F i n a l l y , we may note t h a t the l a s t two terms of Eq. (D.16) c o n t a i n L + and L_ and thus connect wavefunc-t i o n s which d i f f e r i n A by one u n i t . I t i s these C o r i o l i s terms which are r e s p o n s i b l e f o r the i n t e r a c t i o n s of n s t a t e s w i t h E and A s t a t e s , causing A-doubling (when the s t a t e s are f a r a p a r t ) . However, a s i n g l e case (a) or (b) e l e c t r o n i c s t a t e c o n s i s t s of wavefunctions with +A and -A o n l y ; s i n c e A may not be h a l f i n t e g r a l these 27,6 never d i f f e r by u n i t y and such terms c o n t r i b u t e n o t h i n g . The o f f - d i a g o n a l m a t r i x elements are thus due o n l y to the term - B ( J + S _ + J _ S + ) . A note of c a u t i o n must a l s o be added i f the Hamiltonian of Eq. (D.16) i s t o be a p p l i e d to i n t e r a c t i o n s between d i f f e r e n t e l e c t r o n i c or v i b r a t i o n a l s t a t e s . The f a c t o r s A and B are i n f a c t f u n c t i o n s o f i n t e r n u c l e a r -2 d i s t a n c e (B i s p r o p o r t i o n a l to r while A v a r i e s as (dV/dr)r ^ ) . Unless i t can be assumed t h a t the two i n t e r a c t i n g s t a t e s have the same v a l u e s of L and S (the "pure p r e c e s s i o n " h y p o t h e s i s ) , A and B cannot be t r e a t e d as constants and ma t r i x elements w i l l have t o be i n t e g r a t e d e x p l i c i t l y over r t o i n c l u d e these dependencies. Furthermore, L and S must then be r e p l a c e d by the sums over i n d i v i d u a l e l e c t r o n s which they normally symbolize. 277 APPENDIX- E •-' . ..CALCULATING. THE MATRIX ELEMENTS OF n - — — —z 1. Summary o f Re l a t e d Formulae This-appendix w i l l summarize the!,formulae and methods of computing the matrix elements of n : / < iii In lili , > = U* n f dq ( E l ) rm' z | rm' ' rm z rm ^ \£>-J-I were m and m' denote the aggregates o f a l l quantum numbers s p e c i f y i n g the wavefunctions, and q the aggregate of a l l c o o r d i n a t e s , the l a t t e r being i n t e g r a t e d over a l l p o s s i b l e v a l u e s . These ma t r i x elements w i l l be s u f f i c i e n t to c a l c u l a t e the Stark e f f e c t , w i t h the h e l p of the p e r t u r -b a t i o n theory .- d i s c u s s e d i n Chap. "IV:: the mat r i x elements of n and n enter i n t o i n t e n s i t y c a l c u l a t i o n s x y J but not the Stark e f f e c t , and w i l l not be d i s c u s s e d here. We w i l l be a b l e to use the v a r i o u s c o u p l i n g schemes of the angular momenta (see App i B ) to s i m p l i f y the c a l c u l a t i o n of E q . ( E . l ) . That e x p r e s s i o n can be r e s o l v e d i n t o a s m a l l number of p a r t i a l f a c t o r s which are commonly a v a i l a b l e i n standard forms, without ever having to d e a l w i t h the wavefunctions themselves. The most complicated g e n e r a l case w i t h which we s h a l l d e a l i s a m a t r i x element of the form 278 < N J M n N' J ' M'> i z 1 (EL 2) which r e p r e s e n t s a case (b) type of c o u p l i n g : the d i a t o m i c molecule has an angular momentum along n, which i s com-bined w i t h the end-over-end (nuclear) r o t a t i o n of the mole-c u l e t o form the angular momentum i\T. Thus, N" = R" + s$«n, and ~ti = S«n. Furthermore, N" i s coupled (weakly) to the s p i n (S) to form 3 which can be q u a n t i z e d w i t h component M along the s p a c e - f i x e d z a x i s . S i m p l i f i c a t i o n s are pos-s i b l e when J=N (S=0) or N=R as i n Z s t a t e s . The f u l l m a t r i x elements of E q . ( E . l ) c a n be f a c t o r e d i n t o n(N) x P(N,J) x X(J,M) (E.3) The f a c t o r n(N) has three u s e f u l forms (Landau and L i f s h i t z } p. 295) : n Q(N) = <N|ix|N> = f2/N(N+l) (E.4) n_(N) = <N|n|N-l> = <N-l|n|N> (E.5) 1 N ' (N-B) (N+fl) H (2N-1)(2N+1)J n (N) = <N|fi|N+l> = <N+l|ft|N> = n_(N+l) 1 (N+l) ' (N-fl+1)(N+n+1)' (2N+1)(2N+3) ( E . 6 ) The f a c t o r p.93) : X(J,M) has a l s o t h r e e forms (Landau and L i f s h i t z } „ / T ... < J Ml ill J x o ( J ' M ) = < J j A i J > 279 = M (E.7) X (J,M) = X +(J,M) < J M n J-1 M > < J n J-1 > < J M n J + l M > < J n J + l > < J-1 M[n|j M > = j 2 _ 2 < J - l | n | J > < J + l M|n|j M > = < J+l n J > = / ( j + l ) z - nz (E.8) (E.9) We w i l l not d e f i n e the q u a n t i t i e s i n the above r a t i o s r i g o r o u s l y (the denominators are simply n ( J ) , of c o u r s e ) , but they help to g i v e a f e e l i n g f o r the k i n d of f a c t o r i z a t i o n t h a t E q . ( E . 3 ) " i n v o l v e s . The f a c t o r P(N,J) we s h a l l w r i t e i n seven forms (Landau and L i f s h i t z 1 , p ,104f): P_(N,J) = < N J n N-1 J-1 > < N-1 J-1 h N J > < N n N-1 > < N-1 n N > (J-S+N)(J+S+N+l)(J+N-S-l)(J+S+N) ' 4 J^(2J-1) (2J+1) (E.10) P (N,J) = < N J r\ N-1 J > < N-1 J fi N J > < N n N-1 > < N-1 n N > /(J-S+N)(J+S+N+l)(S+N-J)(J+S-N+lT 2J(J+1) ( E . l l ) P Q(N,J) < N J-1 n N-1 J > < N-1 J n N J-1 > < N n N-1 > < N--1 n N > = 7 (S+N-J)(J+S-N+l)(S+N-J+l)(J+S-N) 4J 2(2J-1)(2J+1) (E.12) 28 0 . < N J fi N - l J + l > < N n N - l > = P 5(N,J) = P o(N,J-+l) (S+N-J-l) (J+S-N+2) (S+N-J) (J+S-N+l) 4(J+l)^(2J+1)(2J+3) (E..13) Po" < N J ri N+l J - l > < N tt N+l > P Q(N+1,J) P+(N,J) • V (S+N-J+l)(J+S-N)(S+N-J+2)(J+S-N-l) 4J 2 ( 2 J - 1 ) (2J+1) (E.14) < N J fi N+l J > < N n N+l > = P (N+1,J) /(J-S+N+l) (J+S+N+2) (S+N-J+l) (J+S^) ~ 2J(J+1) (E.15) P+(N,J) = < N J n N+l J + l > < N n N+l > = P_(N+l,J+l) - V (J-S+N+2)(J+S+N+3)(J+N-S+l)(J+S+N+2T 4.(J+1) ^  (2J+1) (2J+3) (E. 16) I t may be noted t h a t a l l these e x p r e s s i o n s are r e a l and t h e r e f o r e are elements of symmetric m a t r i c e s , although t h i s i s e x p l i c i t l y i n d i c a t e d o n l y f o r the f i r s t few. Only two each of the formulae f o r n(N) and X(J,M) are d i s t i n c t , as i n d i c a t e d . S i m i l a r l y only t h r e e of the P(N,J) formulae are independent, but i t i s convenient t o l i s t a l l v a r i a t i o n s e x p l i c i t l y . There a l s o e x i s t forms of P(N,J) which are d i a g o n a l i n N, but we need not c o n s i d e r them here, s i n c e we w i l l o n l y use P(N fJ) i n d e a l i n g with E s t a t e s ( i n which H i s e f f e c t i v e l y zero so t h a t m a t r i x elements d i a g o n a l i n N are 281 a l r e a d y m u l t i p l i e d by zero (Eq. E.4). Moreover, f o r N(N) we g i v e o n l y those matrix elements which are d i a g o n a l i n n. Otherwise the formulae g i v e n are complete i n accordance w i t h the s e l e c t i o n r u l e s AN = 0, ±1, AJ = 0, ±1, and (f o r n ) z AM = 0. In the s e c t i o n s t h a t f o l l o w , the v i b r a t i o n of the n u c l e i was not c o n s i d e r e d e x p l i c i t l y . A l l r e s u l t s are d i a g o n a l i n v and apply e q u a l l y w e l l f o r a l l v a l u e s of v. 2. 11 S t a t e s S e t t i n g N=J and S=0 i n Eqs. (E.10) - (E.16), we f i n d t h a t a l l P(N,J) are zero except f o r and P_ which are u n i t y , corresponding r e s p e c t i v e l y to the D and B i n t e r a c t i o n s o f F i g . 1 i n Chap. I I I . The m a t r i x element f o r i n t e r a c t i o n B i s seen to be n_(J) • X_ (J,M) = f4jzT± (E.17) w h i l e t h a t f o r D i s v j> • V J' M> -J(StiuljU) (E>18) _ / ( j + l ) 2 - M * ~ H(J+l)^-1 I t i s to be expected t h a t the i n t e r a c t i o n D should have the same matrix elements as i n t e r a c t i o n B but ev a l u a t e d at one l e v e l h i g h e r J ; t h i s stems from the symmetric nature of a l l m a t r i c e s c o n s i d e r e d : I t i s c l e a r t h a t the "D" i n t e r a c t i o n s 282 f o r one l e v e l are j u s t the "B" i n t e r a c t i o n s f o r the one j u s t above i t . 3. 1 n S t a t e s The r o t a t i o n a l energy l e v e l s o f 1 n s t a t e s are very s i m i l a r t o those o f lI s t a t e s shown i n F i g . l , Chap. I I I . The o n l y d i f f e r e n c e s are t h a t the J=0 l e v e l does not e x i s t (J >^  J^ = A = 1) and a l l l e v e l s are doubly degenerate (A-doubling) w i t h the two s u b l e v e l s having o p p o s i t e p a r i t y s i g n s . The i n t e r a c t i o n s i n d i c a t e d by B and D i n F i g . 1 f o r J=2 must be taken between those s u b l e v e l s w i t h o p p o s i t e p a r i t y . D i f f e r e n c e s i n the St a r k e f f e c t m a t r i x elements come from the f a c t t h a t Q=A=1. We must m u l t i p l y n_(J) i n Eq. (E.17) by / J j - l ) (J+D/J a c c o r d i n g t o Eq. (E.5), and n +(J) by /j(J+2)/(J+l) by Eq. (E.6). Moreover, there now e x i s t matrix elements d i a g o n a l i n J. From Eqs. (E.4) and (E.7) these elements, (which are d i a g o n a l i n a l l quantum numbers except p a r i t y ) are g i v e n by M o ( J ) ' X o ( J ) = jf§+T) ' ( E ' 1 9 ) which i n t h i s case i s j u s t M/J(J+1). Eq. (E.19) i s the ge n e r a l form f o r the r o t a t i o n a l m a t r i x elements of the f i r s t o r d e r Stark e f f e c t i n Hund's case (a). Eq. (E.19) can be i n t e r p r e t e d as the f r a c t i o n of the d i p o l e moment p r o j e c t e d onto the s p a c e - f i x e d z a x i s , v i a i n t e r m e d i a t e p r o j e c t i o n along the d i r e c t i o n of 3 ( t h i s assumes t h a t y precesses .283 very f a s t about j", which i n t u r n precesses much more slo w l y about E ) . Eq. (E.19) i s then the product of cos(y,J) = ft//J(J+l) and cos(J,E) = M//J(J+1) (see P h e l p s 1 4 , p . l O f f ) . The r e s u l t s are shown i n the m a t r i c e s of Eqs. (i6) and '(17), Chap. I I I . 4. 2 I S t a t e s The energy l e v e l s of a 2 E + s t a t e are shown to s c a l e i n F i g . 2, Chap. I l l f o r a very l a r g e s p i n - s p l i t t i n g con-s t a n t y (see Chap. IV, Sec. 6). Comparison with the 1E + s t a t e shows t h a t the p r i n c i p a l d i f f e r e n c e i s t h a t the e l e c t r o n i c s p i n (S=£) s p l i t s a l l l e v e l s with N>0 i n t o two components. A c c o r d i n g to the r u l e s of v e c t o r a d d i t i o n (Landau and L i f s h i t z 1 , p. 100) 5 = N + S* means t h a t J = N - \ o r N + £ as long as N>0. When N = 0, the o n l y p o s s i b i l i t y i s J = S = jr. I t i s convenient to c a t e g o r i z e the r o t a t i o n a l l e v e l s a c c o r d i n g to the d i r e c t i o n of S, or more p r o p e r l y , by i t s component E along the i n t e r n u c l e a r a x i s . Since we are assuming n e g l i g i b l e " i n t e r a c t i o n between N- and S*, we can q u a n t i z e J , N, and S and ^ ? = ^ ? ± £^, or Si ••= A ± E = E = ±£. We denote l e v e l s w i t h fi=+£ as f 1 l e v e l s , i n which J=N+£. The l e v e l s w i t h fi=-£ and J=N-£ are c a l l e d f 2 l e v e l s . The case o f N=0 we c a l l on f ^ l e v e l s i n c e f o r m a l l y J=N+2, even though t h i s l e v e l c o n t a i n s both s i g n s of E, w i t h r e s p e c t to which i t i s degenerate. 284 I n t e r a c t i o n s ( l a b e l e d A) which are d i a g o n a l i n N have zero matrix elements by Eg. (E.4). The ot h e r s are giv e n i n the f o l l o w i n g t a b l e , wherein the order o f the f a c t o r s n(N) • P(N,J) • S(J,M) has been p r e s e r v e d : Table E.I The Squares of the M a t r i x Elements of a 2 E S t a t e . The row and column d e s i g n a t o r s are d e f i n e d by the i n t e r a c t i o n l a b e l s i n F i g . 2, Chap. I I I . (4N 2-l) * (4N 2-1) ' M 2 1 4 2 (2N+1)(2N+3)' (2N+1)(2N+3) ' M W=IT- ( ( N ^ 2 - M 2 ) 1 2N+1 (2N+1) (2N+3) ' 2N+3 ((N+2) M2 ) 1 2N+3 (2N+1)(2N+3) ' 2N+1 ( (N+£) 2 _ M 2 Since we have i n c l u d e d e l e c t r o n i c s p i n through the f a c t o r P(N,J), we omit i t i n e v a l u a t i n g n(N) b y - s e t t i n g ft=A = 0. 285 As noted f o r *E s t a t e s , these matrix elements are not a l l independent; h a l f of them are the same as some other element w i t h N i n c r e a s e d by one. T h i s i s i l l u s -t r a t e d i n the f o l l o w i n g diagram which shows how the p o s i t i o n of a mat r i x element i s s h i f t e d i n Table E.I i f N i s r e p l a c e d by N+l: (E.20) The formulae of Table E.I may be w r i t t e n more c o n c i s e l y i n terms of J as f o l l o w s : Table E . I I Formulae of Table E.I w r i t t e n i n terms of J . 1 2 B J 2-M 2 M 2J(J+l) M ( 2J(J+1) } J 2-M 2 -4J" 2-D ( J + l ) 2 - M 2 4 ( J + l ) 2 ( J + l ) 2 - M 2 4 ( J + l ) * 286 To i n t e r p r e t the e q u a l i t i e s v i s i b l e i n Table E . I I , i t must be remembered t h a t grouping by equal J causes d i f f e r e n t i n t e r a c t i o n s t o be compared than grouping by equal N: s p e c i f i c a l l y , B2, C2, and D2 must be i n t e r p r e t e d as a p p l y i n g to the next h i g h e r value of N than t h a t f o r which they are i l l u s t r a t e d i n F i g . 2. T h i s makes i t obvious t h a t the e n t r i e s f o r C l and B2 should be e q u a l , s i n c e they are the same i n t e r a c t i o n . The r e l a t i o n s i m p l i e d by the v e r t i c a l arrows i n Eq. (E.20) are s t i l l v a l i d , r e p l a c i n g J by J + l . More i n t e r e s t i n g , however, are the e q u a l i t i e s BI = C2 and D l = D2 i n Table E . I I . These imply t h a t i n t e r a c t i o n s which are d i a g o n a l i n Z (but not i n J) are independent of Z. Such c o n s i d e r a t i o n s r e v e a l t h a t there are r e a l l y o n l y two d i s t i n c t m atrix elements: those which are d i a g o n a l i n Z but not J (BI, D l , C2, D2), and those which are d i a g o n a l i n J but not z. (B2, Cl) . 5. 2 ii S t a t e s We s h a l l assume Hund's case (a) c o u p l i n g . The e l e c t r o n i c s p i n i s e n t i r e l y c o n t a i n e d w i t h i n ft, so t h a t the f a c t o r P(N,J) may be dropped, as f o r s i n g l e t s t a t e s . The f a c t o r s X(J,M) are the same as i n 2z s t a t e s , n(N) becomes n ( J ) , s i n c e J i s now the r e s u l t a n t o f a x i a l and r o t a t i o n a l angular momentum. Due to the e l e c t r i c d i p o l e s e l e c t i o n r u l e on z (AE=0), d i f f e r e n t v a l u e s of z (and t h e r e f o r e of ft) can not i n t e r a c t i n pure case ( a). T h i s i s q u i t e convenient, because i t means t h a t each m u l t i p l e t 28? sequence ( 2n^ or 2n^) may be t r e a t e d as a separate 1n s t a t e w i t h h a l f - i n t e g r a l v a l u e s of J and ft. The l e v e l s of a 2n s t a t e are i l l u s t r a t e d i n F i g . 3, Chap. I l l , q u a l i t a t i v e l y as they are found i n the ground e l e c t r o n i c and v i b r a t i o n a l s t a t e of the OH molecule ( a f t e r Moore and R i c h a r d s 1 5 ) . The s u b s c r i p t I i n d i c a t e s an i n v e r t e d s t a t e , meaning t h a t the m u l t i p l e t l i e s e n e r g e t i c a l l y lower than the n^. The r o t a t i o n a l l e v e l s may be c a t e g o r i z e d as f 1 and f 2 , and v a l u e s of N assig n e d a c c o r d i n g to N = J-£ f o r f ^ and N = J+£ f o r f 2 l e v e l s . The i n t e r a c t i o n s shown i n F i g . 3, Chap. I l l are f o r one s u b l e v e l of a A-doublet o n l y ; the other s u b l e v e l enjoys e n t i r e l y e q u i v a l e n t i n t e r a c t i o n s w i t h neighbouring l e v e l s of o p p o s i t e p a r i t y . The same formulae f o r the squares of n z are a p p l i c a b l e to both f ^ and f 2 l e v e l s : B ( J 2 - f t 2 ) ( J 2 - M 2 ) J*(4J*-1). D ( ( J + D 2 - ft2) ( ( J + l ) 2 - M 2 ) ( J + l ) 2 ( 4 ( j + l ) 2 - l ) (E.21) Since a l l non-zero m a t r i x elements are d i a g o n a l i n E ( i . e . , t here are none between f ^ and f 2 l e v e l s ) , t h e r e are o n l y two d i s t i n c t types of terms: those d i a g o n a l (A) and those o f f - d i a g o n a l (B,D) i n J . I t should be 288 remembered t h a t we are e v a l u a t i n g these f o r pure case ( a ) , which ( u n l i k e the s t r u c t u r e shown i n F i g . 3, Chap. I l l ) must be such t h a t the d i s t a n c e between the f.^ and f 2 l e v e l s of same J i s much l a r g e r than the d i f f e r e n c e s between a d j o i n i n g r o t a t i o n a l l e v e l s . 6. 3 I S t a t e s R o t a t i o n a l energy l e v e l s w i t h p a r i t i e s a p p r o p r i a t e f o r a 3E s t a t e are shown s c h e m a t i c a l l y i n F i g . 4, Chap. I I I . I t should be noted t h a t the o r d e r and spacing o f l e v e l s w i t h i n each t r i p l e t i s shown f o r convenience i n l a b e l i n g ; i n a more t y p i c a l case (NH, see F i g . 39, Chap. XI) the f 2 l e v e l i s h i g h e s t , and f ^ lowest. These l e v e l c a t e g o r i e s are d e f i n e d i n Chap. V. In the ground s t a t e , J- equals S, and the degeneracy of l e v e l s of d i f f e r e n t (and thus d i f f e r e n t M) i s not removed. The ma t r i x elements of n z are c a l c u l a t e d i n the same way as f o r the doublet case, s e t t i n g £2=A=0 i n Eqs. (E.4) to (E.6). F i g . 4, Chap. I l l , shows t h a t the f a c t o r s P* and P Q appear i n the i n t e r a c t i o n s l a b e l e d B l and C3, r e s p e c t i v e l y . The r e s u l t s are summarized i n Table E . I I I . 289 Table E . I I I The squares of the matrix elements o f n i n a 3E s t a t e . The row and column d e s i g n a t o r s are d e f i n e d by F i g . 4, Chap. I I I . B (N+l) 2 - M 2  (N+l) z (2N+1)*(2N+3)* M 2  N z(N+l)(2N+1) M 2  N^(N-1)(2N-1) C M 2 (N+l)*(N+2)(2N+3) M 2 N(2N+1) (N+l)* N 2-M 2  N z(2N-1)*(2N+1) z D ((N+2) 2 - M 2)(N+l) (2N+3)*(N+2) ( (N+l) 2-M 2)N(N+2) (N+l) z(2N+1)(2N+3) (N 2-M 2)(N+l) N(2N+1) z E ((N+l) 2-M 2)N (N+l)(2N+1)* (N 2-M 2)(N-1)(N+l) N*(2N-1)(2N+1) ((N-1) 2-M 2)N (N-1)(2N-1) 2 As b e f o r e , h a l f of the formulae i n Table E . I I I can be ob t a i n e d from the remainder by r e p l a c i n g N wi t h N+l. These r e l a t i o n s h i p s are summarized i n the diagram of Eq.(E.22), which i l l u s t r a t e s where ma t r i x elements reappear when the next h i g h e r v a l u e of N i s used: B C D E •1 1> *L (E.22) 290 The e q u i v a l e n t of Table E . I I I expressed i n terms of J may be w r i t t e n out i n a s t r a i g h t f o r w a r d manner; no marked s i m p l i f i c a t i o n r e s u l t s i n the case, however. 7. 3 n S t a t e s The upper h a l f of F i g . 39, Chap. XI, d e p i c t s the energy l e v e l s of a 3IK s t a t e as they l i e i n the e x c i t e d NH molecule. T h i s s t a t e i s t r e a t e d e x a c t l y the same as the 2 n s t a t e , and Eqs. (E.19) and (E.21) are s t i l l v a l i d . 291 REFERENCES 1. L.D. Landau and E.M. L i f s h i t z , "Quantum Mechanics"; (Addison-Wesley P u b l i s h i n g Co.,Inc., Reading, Mass., 1958, 1 s t e d i t . ) . 2. J.T. 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