UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Zero field level crossing in molecular hydrogen 1970

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1970_A1 V35_3.pdf [ 5.03MB ]
Metadata
JSON: 1.0084786.json
JSON-LD: 1.0084786+ld.json
RDF/XML (Pretty): 1.0084786.xml
RDF/JSON: 1.0084786+rdf.json
Turtle: 1.0084786+rdf-turtle.txt
N-Triples: 1.0084786+rdf-ntriples.txt
Citation
1.0084786.ris

Full Text

ZERO FIELD LEVEL CROSSING IN MOLECULAR HYDROGEN by J A C O B V A N D E R L I N D E B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1 9 6 7 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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 to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 7 0 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT The l i f e t i m e s o f t h e 3d 1 Z ( v = 0 ) J = l , 2 and 3 s t a t e s h a v e b e e n m e a s u r e d u s i n g z e r o - f i e l d l e v e l c r o s s i n g t e c h n i q u e s . The t r a n s i t i o n s o b s e r v e d w e re 'the R - b r a n c h members o f t h e 3d JE ->- 2p JE t r a n s i t i o n . The u p p e r s t a t e i s e x c i t e d i n a d i s c h a r g e b e t w e e n two c a p a c i t o r p l a t e s t o w h i c h a r a d i o - f r e q u e n c y v o l t a g e i s a p p l i e d . The m e a s u r e m e n t s were made u s i n g f i r s t a 180 MHz R.F. s o u r c e and l a t e r u s i n g a ^50 MHz s o u r c e . P o l a r i z a t i o n o f e m i t t e d l i g h t was m e a s u r e d by r o t a t i n g a p o l a r o i d i n t h e beam a nd p h a s e s e n s i t i v e d e t e c t i n g t h e r e s u l t i n g m o d u l a t i o n . T h e . d e p o l a r i z a t i o n c u r v e s o b t a i n e d by p l o t t i n g t h e m a g n e t i c f i e l d s t r e n g t h a g a i n s t t h e p o l a r i z a t i o n o f t h e R ( 0 ) , R ( l ) , R(2) l i n e s y i e l d h a l f w i d t h s , when e x t r a p o l a t e d t o z e r o p r e s s u r e , o f 2.37±.12 g a u s s , 2.60±.15 g a u s s • a n d 3.25±.25 g a u s s . The h a l f w i d t h s v a r y l i n e a r l y w i t h p r e s s u r e i n t h e d i s c h a r g e c e l l y i e l d i n g c o l l i s i o n c r o s s - s e c t i o n s o f r o u g h l y 1 . 5 x l 0 - 1 1 * c m 2. U s i n g t h e h i g h f i e l d Lande g v a l u e s o f t h e s e s t a t e s , t h e i r l i f e t i m e s a r e ( 2 . 6 6 ± . 1 2 ) x i o ~ 8 s e c . , ( 3 . 8 3 ± • 2 ) x i o " 8 s e c . , a n d (3-93±.25)xl0 _ 8sec. f o r J = l , 2 and 3 r e s p e c t i v e l y . The d i s c r e p a n c y b e t w e e n t h e f i r s t a n d t h e l a t t e r two l i f e - t i m e s i s d i s c u s s e d . TABLE OF CONTENTS CHAPTER PAGE I INTRODUCTION . . . . . 1 I I THEORY 8 §2.1 I n t r o d u c t i o n . . . . . 8 §2.2 The Magnetic F i e l d Dependence of P o l a r i z a t i o n i n C l a s s i c a l Terms, . 8 §2.3 Quantum-Mechanical D e s c r i p t i o n of Le v e l Crossing . . . . . 14 §2.4 The 3d xE Levels of H 2 . . . . . . 18 §2.5 The E x c i t a t i o n M a t r i x Elements Q .. 22 I I I EXPERIMENTAL DETAILS . . 28 §3.1 Experimental Arrangement . . . . . . . 28 §3.2 The Discharge . . . 30 §3-3 The O p t i c a l System . . . 34 §3-4 The Vacuum System . 37 §3.5 R.F. Supplies and Coupling . . . . 40 §3.6 Helmholtz C o i l s 42 §3.7. Current Supplies f o r Helmholtz C o i l s 46 §3.8 Lo c k - i n A m p l i f i e r 49 §3.9 R o t a t i n g P o l a r o i d 50 §3.10 The V a r i a b l e Quarter-Wave P l a t e . 50 §3.11 P h o t o m u l t i p l i e r 54 §3.12 X-Y Recorder . 54 §3.13 L i q u i d Notrogen Bath 55 §3.14 Data Proce s s i n g . . . . . . . . r . . 56 - i v - CHAPTER PAGE • I V EXPERIMENTAL RESULTS . . ' 57 §4.1 L i f e t i m e s 57 §4.2 C o l l i s i o n C r o s s - s e c t i o n s 64 §4.3 P o l a r i z a t i o n 68 §4.4 U p p e r S t a t e P o p u l a t i o n s . . . . . . 69 §4.5 E x p e r i m e n t a l E r r o r s . . . . . . . 71 a) D i s c h a r g e S t a b i l i t y . . . . 71 b) M a g n e t i c F i e l d 72 c ) P r e s s u r e I n t h e D i s c h a r g e . 73 d) T e m p e r a t u r e i n t h e D i s c h a r g e 73 e) C a s c a d i n g 74 f ) C o h e r e n c e N a r r o w i n g . . . . 74 g) R.F. B r o a d e n i n g 75 ' §4.6 H e l i u m 4 XD L i f e t i m e Compared W i t h T h a t O b t a i n e d F rom O t h e r E x p e r i m e n t s 76 V DISCUSSION OF RESULTS AND CONCLUSION . . . 80 §5.1 I n t r o d u c t i o n 80 §5.2 H y p e r f i n e E f f e c t s . . 80 §5.3 E l e c t r o n i c Wave F u n c t i o n V a r i a t i o n w i t h J 85 §5.4 C o n c l u s i o n and S u g g e s t i o n s f o r F u r t h e r Work 8 6 APPENDIX I THE TRANSITION MATRIX ELEMENTS . . . 87 APPENDIX I I THE 3d 1 STATES OF H 2 . . . . . . . . 91 §A2.1 E n e r g y L e v e l s a n d E i g e n s t a t e s . . 91 §A2.2 The Zeeman E f f e c t 97 - v - PAGE APPENDIX I I I OTHER STATES 102 §A3.1 The 3 d 1 n ~ ( v = 0 ) J=2 State 102 §A3.2 The 3*K State .104 REFERENCES AND FOOTNOTES 105 LIST OF TABLES TABLE • PAGE I R e l a t i v e Populations of the F i r s t Few R o t a t i o n a l Levels of Hydrogen at Thermal E q u i l i b r i u m 2 1 . I I P r o p e r t i e s of E l e c t r o n Motion i n an R.F. E l e c t r i c and D.C. Magnetic F i e l d . . . . . 34 I I I L e v e l Crossing Curve Halfwidths C3d 1 IT and 3 X K ) 1 0 3 IV E x t r a p o l a t e d Halfwidths and L i f e t i m e s . . . 64 V P o l a r i z a t i o n Curves' Halfwidths 6 5 VI C o l l i s i o n C r o s s - s e c t i o n s 6 8 V I I Experimental Upper State Populations . . . 7 0 V I I I L i f e t i m e of the 4*D State of Helium . . . . 7 7 IX Energies of the 3 d 1 Complex of H 2 9 4 X Expansion C o e f f i c i e n t s f o r the 3 d 1 + States 9 5 XI R e l a t i v e I n t e n s i t i e s of P and R T r a n s i t i o n s i n the 3d*E ->• 2 p J E ( 0 , 0 ) Band. . 9 7 X I I g-values of the 3 d 1 ! State . 9 9 ILLUSTRATIONS AND FIGURES FIGURE PAGE 1 The S i n g l e t States of Hydrogen (H 2) .3 2 The T r a n s i t i o n s Observed . . . . 4 3 Hanle E f f e c t Co-ordinate System . 9 4 P o l a r i z a t i o n f o r (J>=0 •". • • 11 5 P o l a r i z a t i o n f o r (J>=TT/4 . 11 6 "Three" L e v e l System ' 14 7 The S c a t t e r i n g Angle 26 8 The Apparatus 29 9 Cross-Section of Discharge C e l l and Capacitor P l a t e s 3 1 10 The Vacuum System 39 11 180 MHz R.F. O s c i l l a t o r 4l 12 T-Matched Resonant C i r c u i t . . . ; 43 . 13 Helmholtz C o i l Power Supply 45 14 D.C. Power A m p l i f i e r 47 15 -10V to +10V Voltage Sweep Mechanism . . . . 48 16 P o l a r o i d Rotator 51 17 Quarter-Wave P l a t e 53 18 P h o t o m u l t i p l i e r Wiring Schematic . . . . . . 54 19 . Experimental L e v e l - C r o s s i n g Curve f o r the R(0) Line Using 450 MHz E x c i t a t i o n 58 20 Least Squares F i t t e d Curve f o r the R(0) Line Using 450 MHz E x c i t a t i o n 59 - v i i i - FIGURE . PAGE 2 1 Experimental L e v e l - C r o s s i n g Curve f o r the R ( 0 ) Line Using 1 8 0 MHz E x c i t a t i o n . . . . . 6 0 2 2 Least Squares F i t t e d Curve f o r the R(o) Line Using 1 8 0 MHz E x c i t a t i o n 6 1 2 3 L e v e l Crossing Curve H a l f w i d t h as a Function of Pressure 62 24 Le v e l .Crossing Curve Halfwidth-as a Function of Pressure 6 3 2 5 Experimental L e v e l - C r o s s i n g Curve f o r the Helium 4 XD -> 21? T r a n s i t i o n . . . . . . . . . 7 8 .26 4*D Curve H a l f w i d t h as a Function of Pressure 7 9 . 2 7 Zeeman E f f e c t i n the Presence of Small Hyperfine S p l i t t i n g 84 2 8 The F i r s t Few Lines of the R Branch of the 3 d 1 Z - ^ 2 p 1 E ( 0 , 0 ) band of H 2 . . . . . . . . . . 1 0 1 ACKNOWLEDGEMENTS I wish to express my g r a t i t u d e to Prof e s s o r F.W. Dalby for' suggesting the problem. His s u p e r v i s i o n of the resear c h , advice and encouragement were i n v a l u a b l e . I a l s o wish to thank my wife f o r her help i n the pr e p a r a t i o n of t h i s t h e s i s . Other c o n t r i b u t i o n s , both t a n g i b l e and i n t a n g i b l e , have been made by my f e l l o w graduate students.. This work was supported by the N a t i o n a l Research C o u n c i l of Canada. - 1 CHAPTER .1 • INTRODUCTION L i f e t i m e s or o s c i l l a t o r strengths of e x c i t e d s t a t e s of atoms and molecules have t r a d i t i o n a l l y been measured through a determination of t r a n s i t i o n p r o b a b i l i t i e s . The most r e l i a b l e of these techniques i s probably the Hook 1 method which y i e l d s , under favourable c o n d i t i o n s , o s c i l l a t o r strengths to 1 0 - 2 0 % accuracy. Common to these methods i s the requirement that the p a r t i a l pressure of the absorber must be a c c u r a t e l y known. I f the absorption occurs from the ground s t a t e of a s t a b l e gaseous atom t h i s may present no problem. I f , however, the absorption i s produced by atoms or molecules of low vapor pressure or by an unstable molecule i t i s very d i f f i c u l t not to make larg e e r r o r s i n determining the number of absorbers.present. More recent techniques such as the p h a s e - s h i f t and delayed coincidence methods 2 do not s u f f e r from the above shortcomings but do r e q u i r e f a s t "gates" w i t h r i s e t i m e s s h o r t e r than the e x c i t e d s t a t e l i f e t i m e ' u n d e r c o n s i d e r a t i o n or other s o p h i s t i c a t e d e l e c t r o n i c s . Since i n many cases l i f e t i m e s are of order 10~ 8seconds or s h o r t e r , i t i s again easy to make e r r o r s of considerable magnitude-. The use of z e r o - f i e l d l e v e l c r o s s i n g to determine l i f e t i m e s s u f f e r s from none of these l i m i t a t i o n s and has been used to measure l i f e t i m e s of many atomic s t a t e s and more r e c e n t l y to measure e x c i t e d s t a t e l i f e t i m e s of NO1*, OH5, and CS 6 . - 2 - An e n e r g y l e v e l d i a g r a m f o r t h e o b s e r v e d s i n g l e t s t a t e s o f m o l e c u l a r h y d r o g e n i s shown i n P I G . l . The l e v e l s shown a r e t h e l o w e s t v i b r a t i o n a l , r o t a t i o n a l l e v e l s o f e a c h e l e c t r o n i c s t a t e . . T h i s t h e s i s c o n c e r n s i t s e l f p r i m a r i l y w i t h t h e 3d*E (v=0) J = l , 2 , a n d 3 s t a t e l i f e t i m e s . The l i g h t o b s e r v e d a r i s e s f r o m t h e t r a n s i t i o n s R ( 0 ) , R ( l ) , a n d R ( 2 ) o f t h e ( 0 , 0 ) b a n d o f t h e 3 d lE -> 2 p lE s y s t e m c o r r e s p o n d i n g t o t h e t r a n s i t i o n s 3d" 12 v=0 J = l -»• 2 p lE v=0 J=0 J=2 -* J = l J=3 J=2 r e s p e c t i v e l y . A more d e t a i l e d e n e r g y l e v e l d i a g r a m f o r t h e s e s t a t e s i s shown i n F I G . 2 . The t r a n s i t i o n s o b s e r v e d i n t h i s t h e s i s a r e m a r k e d w i t h a r r o w s . The e l e c t r o n i c p o r t i o n o f t h e wave f u n c t i o n o f t h e s e s t a t e s o u g h t t o r e s e m b l e t h o s e o f t h e a t o m i c H e l i u m 3D s t a t e s h e n c e we e x p e c t l i f e t i m e s o f t h e same o r d e r o f m a g n i t u d e , o r r o u g h l y 1 0 - 8 s e c o n d s 7 . A l s o o b s e r v e d were s e v e r a l 3 K X E -»• 2p*E t r a n s i t i o n s a n d a 3d*n -> 2p"*E t r a n s i t i o n . I n f o r m a t i o n o b t a i n e d on t h e s e s t a t e s i s c o n t a i n e d i n A p p e n d i x I I I . The H a n l e E f f e c t o I n 1922 R a y l e i g h 8 d i s c o v e r e d t h a t t h e 2537 A l i n e o f m e r c u r y e x c i t e d by r e s o n a n c e r a d i a t i o n , was p o l a r i z e d i f - 3 - ENERGY ( 1 0 3 cm - 1) ns*E np*E np1!! nd*E 20 10 0 0 9 0 8 0 - l e v e l s w i t h The energies of these s t a t e s were obtained from D i e k e 1 8 0 J -1 Figure 1 - The s i n g l e t s t a t e s of hydrogen (H 2) 3d1]! 3d 1 A - 5 - viewed at r i g h t angles to the e x c i t i n g beam. Wood9 and E l l e t 1 0 i n v e s t i g a t e d t h i s e f f e c t f u r t h e r and found that at low pressures, i n the absence of a magnetic f i e l d the emitted r a d i a t i o n was almost completely p o l a r i z e d w i t h i t s e l e c t r i c v e c tor p a r a l l e l to that of the e x c i t i n g l i g h t . Small magnetic f i e l d s i n c e r t a i n d i r e c t i o n s , were found to decrease the degree of p o l a r i z a t i o n . The a d d i t i o n of f o r e i g n gases was a l s o found to decrease the p o l a r i z a t i o n . H a n l e 1 1 , performing a more thorough i n v e s t i g a t i o n found that the a p p l i c a t i o n of a magnetic f i e l d p e r p e n d i c u l a r to the e x c i t i n g l i g h t along the d i r e c t i o n of observation not only decreased but a l s o r o t a t e d the plane of p o l a r i z a t i o n of emitted l i g h t . B r e i t 1 2 explained the e f f e c t i n c l a s s i c a l terms and showed that the degree of p o l a r i z a t i o n , P, i s given by the expression P(H) = 1 / 1 \ PT6T a , rgeHx^ ' K X ) ^ mc / where P = I„ -Ix , Li i s the i n t e n s i t y of l i g h t w i t h e l e c t r i c v e c tor along the e l e c t r i c v e ctor of the e x c i t i n g beam and Ij_ t h a t w i t h p e r p e n d i c u l a r p o l a r i z a t i o n , H i s the a p p l i e d magnetic f i e l d , T i s the mean r a d i a t i v e l i f e t i m e , ge_ i s 2mc the magnetic moment of the atom. A condensed v e r s i o n of the c l a s s i c a l theory i s given i n § 2.2. By p l o t t i n g P(H) we may then e a s i l y o b t a i n the product gx } from.eq. ( 1 ), and an independent measurement of g y i e l d s the r a d i a t i v e l i f e t i m e T.• - 6 - A more complete account of the. e a r l y work on p o l a r i z a t i o n of resonance r a d i a t i o n i s given by M i t c h e l l and Zemansky 1 3. The Hanle e f f e c t i s a s p e c i a l example of " l e v e l - c r o s s i n g " , the appearance of i n t e r f e r e n c e e f f e c t s when two st a t e s are degenerate to w i t h i n t h e i r n a t u r a l l i n e w i d t h . E l e c t r o n Impact P o l a r i z a t i o n L i g h t emitted from atoms e x c i t e d by low energy (-20 e.V.) e l e c t r o n s w i l l i n ' g e n e r a l a l s o e x h i b i t p o l a r i z - a t i o n r e l a t i v e .to a d i r e c t i o n along the e l e c t r o n beam. Measurements of p o l a r i z a t i o n were made f o r a number of atoms by s e v e r a l workers from 1925 to 1935 1 1*. The p o l a r - i z a t i o n i n general depends on the i n c i d e n t e l e c t r o n energy i n a complicated manner, and no adequate theory has yet been devised. The d e p o l a r i z a t i o n of l i g h t emitted by helium e x c i t e d by slow e l e c t r o n s , i n response to a magnetic f i e l d has been observed by Pebay-Peyroula et a l 1 5 . The s i g n a l s have the same magnetic f i e l d and l i f e t i m e dependence, given by e q . ( l ) , as those produced by o p t i c a l l y e x c i t e d atoms. Descourbes 1 6 a l s o r e p o r t s non-zero f i e l d l e v e l - c r o s s i n g i n the 3P s t a t e s of He using e l e c t r o n e x c i t a t i o n . P o l a r i z a t i o n of l i g h t emitted from H 2 t r i p l e t s t a t e s e x c i t e d by e l e c t r o n impact has been reported by P a t r i c k C a h i l l et a l 1 7 . In t h i s work, the r a d i a t i v e l i f e t i m e s of the - 7 - 3d 1! v=0 J = l , 2, a n d 3 r o t a t i o n a l . s t a t e s o f m o l e c u l a r h y d r o g e n h a v e b e e n m e a s u r e d by z e r o f i e l d l e v e l - c r o s s i n g u s i n g e l e c t r o n i m p a c t t o e x c i t e t h e s t a t e s i n a manner s i m i l a r t o t h a t P e b a y - P e y r o u l a e t a l 1 5 . T h e s e l i f e t i m e s h a v e n o t b e e n p r e v i o u s l y m e a s u r e d and s h o u l d p r o v i d e a c h e c k on wave f u n c t i o n s c a l c u l a t e d f o r t h e s e l e v e l s . A b r i e f , v e r y r e a d a b l e r e v i e w o f l i f e t i m e m e a s u r e m e n t s i s g i v e n by S t r o k e 9 . N o t a t i o n and S y m b o l s The o n l y m a j o r d e p a r t u r e f r o m t h e c o n v e n t i o n a l s y m b o l s a r e t h o s e u s e d i n d e n o t i n g t h e v a r i o u s s t a t e s . C o n v e n t i o n a l l y , " r e f e r s t o t h e l o w e r s t a t e a n d ' r e f e r s t o t h e u p p e r s t a t e i n a t r a n s i t i o n . I n t h i s t h e s i s we a d d t o t h i s c o n v e n t i o n t h a t g r o u n d s t a t e quantum numbers a r e u n - p r i m e d . J i s t h e t o t a l a n g u l a r momentum e x c l u s i v e o f n u c l e a r s p i n and v i s t h e v i b r a t i o n a l q u antum number. • Some a m b i g u i t y may a l s o be e n c o u n t e r e d b e t w e e n g, t h e p o l a r i z a t i o n v e c t o r o f a n e m i t t e d p h o t o n and g = g j j t h e Lande g - f a c t o r f o r a s t a t e ; t h e l a t t e r w i l l n e a r l y a l w a y s be a c c o m p a n i e d by u Q o r i t s e q u i v a l e n t , - 5 — . - 8 - CHAPTER I I THEORY . § 2 . 1 I n t r o d u c t i o n I n t h i s c h a p t e r we w i l l d i s c u s s f i r s t t h e H a n l e e f f e c t a n a l o g u e i n c l a s s i c a l t e r m s . F o l l o w i n g t h i s t h e qua n t u m m e c h a n i c a l d e s c r i p t i o n o f t h e e f f e c t u s u a l l y r e f e r r e d t o as l e v e l - c r o s s i n g , w i l l be p r e s e n t e d . The t h e o r y g i v e n w i l l f o l l o w t h e t r e a t m e n t o f F r a n k e n 1 9 q u i t e c l o s e l y . N e x t we w i l l c o n s i d e r t h e s t r u c t u r e o f t h e g r o u n d s t a t e s a n d t h e 3 d 1 ! l e v e l s o f t h e h y d r o g e n m o l e c u l e . The t r a n s i t i o n s I s 1 ! -> 3&1Z w i l l be d i s c u s s e d i n t e r m s o f g r o u n d s t a t e p o p u l a t i o n s a n d a l l o w e d e l e c t r i c m u l t i p o l e t r a n s i t i o n moments. F i n a l l y we w i l l a t t e m p t t o d e s c r i b e t h e r e l e v a n t p o r t i o n s o f t h e e x c i t a t i o n m e c h a n i s m . § 2 . 2 The M a g n e t i c F i e l d D e p e n d e n c e o f P o l a r i z a t i o n i n " C l a s s i c a l Terms I n o r d e r t o d i s c u s s t h e H a n l e e f f e c t c l a s s i c a l l y we r e p l a c e o u r m o l e c u l e by an e l e c t r i c d i p o l e w h i c h p o s - s e s s e s a n a n g u l a r momentum L p e r p e n d i c u l a r t o t h e d i p o l e a x i s and a m a g n e t i c moment y_ = uL . - 9 - Figure 3 - Hanle e f f e c t co-ordinate system F u r t h e r , we suppose that at some time t i the d i p o l e i s set i n t o o s c i l l a t i o n w i t h angular frequency v. The d i p o l e then emits r a d i a t i o n w i t h i t s e l e c t r i c v e ctor p a r a l l e l to the d i p o l e a x i s which at time t i s at some angle 0 ( t ) to the x - a x i s . In the absence of any torques on the d i p o l e , 8(t)=6(0)=0. I f however a magnetic f i e l d H=Hzk i s placed along the z - a x i s , the d i p o l e precesses about the' z-axis w i t h angular v e l o c i t y to=uH and 0( t ) = = y H ( t - t i ) . The amplitude of r a d i a t i o n i s r a d i a t i o n damped wi t h a time constant 2T. Thus an observer l o o k i n g along the z - a x i s , having a reference system £,ri,z where £ and n are i n c l i n e d at angle f t o x and y r e s p e c t i v e l y , w i l l observe a time dependent e l e c t r i c f i e l d w i t h components: E ? ( t ) = A 0 c o s [ a 3 ( t - t i ) - c f ) ] e - 1 ( v t + 6 ) e 2 x t - t i E n ( t ) = A 0 s i n [ a ) ( t - t 1 ) - ( ! ) ] e - 1 ( v t + 6 ) e 2 T or the i n t e n s i t i e s of r a d i a t i o n with, e l e c t r i c vector along the K and n axes: t - t i I ^ ( t ) = I 0 c o s 2 [w(t-ti.)-<f>]e T I ( t ) = I 0 s i n - 2 [ u ( t - t i )-4>]e t - t i ( 2 ) Now, i f our observer does not d i f f e r e n t i a t e l i g h t emitted i n the i n t e r v a l ( t i , t i + T ) where T>>T, the i n t e n s i t i e s observed are 11 +T V Ur V t ) d t 85 / t x V t ) d t ti+T I = /, I ( t ) d t = / " I ( t ) d t n ; t i n • ' t i n The p o l a r i z a t i o n P(<j),H) i s then ^ g ~ ^ r i , or ? ' n t - t i /"{cos 2 [ w ( t - t i )-<|)]-sin2 [ u ( t - t i )-<J>] }e T dt P(cj),H)=—^ t - t / oo f ,e dt 11 performing the i n t e g r a t i o n , P(cJ>,H)= 1 + ( 2COT ) a Ccos2d> -2wTsin2<j>] (3) and s u b s t i t u t i n g OJ=UH P((j),H)= 1 + ( 2 u H T ) 2 [ c o s 2 (t>-2yHTSin2(j)] - 11 - For' the s p e c i a l case of <J> = 0 t h i s reduces to P ( H ) = 1+(2UHT) 2 a 'sketch of t h i s shape i s shown i n FIG. 4. iP H I t should be noted that when Figure 4 - P o l a r i z a t i o n f o r <}>=0 P(H) . p l o y • x of the o s c i l l a t o r i s given by T : For the case tt = TT/4 F { R ) - l+(2yHx) 2 y i e l d i n g the curve shown In FIG 5- 1/2 the " l i f e t i m e " 1_ 2yHi / 2 1 \ 1 \ I I I ' Hl / 2 A 1 1 1 -H 1 / 2 Figure 5 - P o l a r i z a t i o n f o r tt = TT/4 - 12 - T h i s c u r v e n o t o n l y y i e l d s t h e l i f e t i m e b u t a l s o t h e s i g n o f t h e m a g n e t i c moment o f t h e o s c i l l a t o r . F r o m eq.(3) we may a l s o s e e t h a t t h e d i r e c t i o n of. p o l a r i z a t i o n r o t a t e s w i t h i n c r e a s i n g m a g n e t i c f i e l d , f o r P t a k e s on i t s maximum v a l u e a t a g i v e n m a g n e t i c f i e l d dP f o r an a n g l e <j> s u c h t h a t ^ = 0 , i . e . -2sin2<J) - 4UHTCOS2<}) = 0 , o r • • <{> •= t a n - 1 ( 2 y H x ) Up t o t h i s p o i n t we h a v e t a c i t l y a s s u m e d t h a t o u r r a d i a t i n g s y s t e m was composed o f o n l y a s i n g l e o s c i l l a t o r s e t i n t o m o t i o n a l o n g t h e x - a x i s a t t i m e t i . A c t u a l l y we h a v e N o s c i l l a t o r s e x c i t e d a t random t i m e s t . a n d h a v i n g t) t h e i r i n i t i a l m o t i o n s on a x e s d i s t r i b u t e d a b o u t t h e x - a x i s . The i n t e n s i t i e s s e e n by t h e o b s e r v e r due t o n o s c i l l a t o r s i n c o h e r e n t l y e x c i t e d a l o n g t h e x - a x i s a n d d e c a y i n g d u r i n g t i m e T » T i s j u s t n T I ? ( t ) = I I I ? ( t j S t ) d t * n / ~ I c ( t j 5 t ) d t n T w h i c h a g a i n l e a d s t o e q u a t i o n ( 3 ) f o r t h e p o l a r i z a t i o n , V 1 0*.^ I V V t ) d t 88 n ' t . W t ) d t We now c o n s i d e r a n o s c i l l a t o r e x c i t e d i n t h e x-y p l a n e w i t h i t s a x i s a t a n a n g l e 6 0 t o t h e x - a x i s . The ' p o l a r i z a t i o n seen by the o b s e r v e r i s t h e n P ( 4 > j e o , H ) = 1 + ^ y H T ) 2 [ c o s 2 ( ( i ) - 6 Q ) - 2 y H T s i n 2 ( # - 6 0 ) ] i Summing now o v e r n o s c i l l a t o r s w i t h i n i t i a l axes synuretric,= d i s t r i b u t e d about th e x - a x i s i n the x-y p l a n e n P((J) >H)= I P((J>,e H) = 3=1 3 P = l + ( 2 y H T ) ^ c o s 2 ( ^ " 2 l j H T s l n 2 ( H where P D= ± I cos26 , | P 0 l l l j = l J Thus a l t h o u g h the p o l a r i z a t i o n has been d e c r e a s e d , tne p o l a r i z a t i o n shows the same f i e l d dependence. S i m i l a r l y i t can be shown, a l t h o u g h more t e d i o u s t h a t f o r a symmetric d i s t r i b u t i o n o f o s c i l l a t o r s i n the x-y p l a n e the maximum p o l a r i z a t i o n i s a g a i n o n l y decreased: but no change i n i t s m a g n e t i c f i e l d depencence o c c u r s . I t s h o u l d be c l e a r t h a t the p e r t u r b a t i o n which e x c i t e d the o s c i l l a t o r s h o u l d be s h o r t i n d u r a t i o n compare t o the r a d i a t i v e l i f e t i m e f o r the above d e r i v a t i o n t o appl. The time i n v o l v e d i n the c o l l i s i o n i s o f o r d e r 10 1 seconds thus t h i s s h o u l d be w e l l s a t i s f i e d . The o s c i l l a t o r s h o u l d a l s o not be s u b j e c t to r e - o r i e n t a t i o n o r i n t e r r u p t i o n by c o l l i s i o n w i t h i t s neighbor, o r a change i n the p o l a r i z a t i o n w i l l a g a i n r e s u l t . To a p p l y the p r e c e e d i n g t h e o r y t o an a t o m i c o r molecular- - 14 - system we need merely evaluate the magnetic moment y=gy 0L e where y Q i s the Bohr magneton 2mc Thus f i n a l l y : P(<t>,H) = ^ T J r[cos2cj> - £^siri2<|)] K r > ^ |-geHx̂  Y mc Y §2.3 Quantum-Mechanical D e s c r i p t i o n of Lev e l Crossing In 1933 B r e i t 2 0 d e r i v e d a quantum mechanical •expression, the " B r e i t formula", f o r the r a d i a t i o n r a t e from coherently e x c i t e d , n e a r l y degenerate s t a t e s of a system. This was l a r g e l y ignored u n t i l 1958 when Pranken et a l 2 1 reported the measurement of the f i n e s t r u c t u r e i n some Helium l e v e l s u s i n g l e v e l - c r o s s i n g . They r e - d e r i v e d the " B r e i t formula", under c o n d i t i o n s of pulse e x c i t a t i o n s as w e l l as e x c i t a t i o n by white' l i g h t . Consider a "three" l e v e l system w i t h ground s t a t e |a> and e x c i t e d s t a t e s |b> and |c> wit h the energy of Ia> taken f o r convenience to be zero. v Figure 6 - "Three" l e v e l system - 15 - They have m a g n e t i c s u b l e v e l s m 3 y, and v, r e s p e c t i v e l y , i . e . We have t h e u n p e r t u r b e d s t a t e s : |a,J,m;t> = |a,J,m> m=- J , - J + l , . . . , J - ( i a + r v / 2 ) t | b , J ' , y ; t > = |b,J' 5y>e v D y = - J ' , - J ' + 1 , . . . , J ' - ( i u , , + r „ / 2 ) t ,v ; t > = |c,J",v>e E c , J " , v ; t 5  v c v = - J " , - J . " + l , . . . , J " Ja 1 where u =4::—, E i s t h e e n e r g y o f s t a t e a, and r = — a In ' a D J a x e x p r e s s e s t h e r a d i a t i o n damping o f t h e s t a t e . At t i m e t = 0 , t h e atom i s assumed t o be i n one o f i t s g r o u n d s t a t e s |a,Jjm> when i t i s s u b j e c t e d t o a p u l s e p e r t u r b a t i o n Q w h i c h may e x c i t e i t t o some o f t h e s t a t e s | b j j ' 5 y > . The s t a t e v e c t o r o f t h e s y s t e m u s i n g f i r s t o r d e r p e r t u r b a t i o n t h e o r y i s t h e n a t s u b s e q u e n t t i m e s : • . - ( i w + r / 2 ) t |X ;t> = |a,J,m> + I |b,J«,y><b,J',y|Q|a,J,m>e y y Now t h e s t a t e s |b,J',y> a r e assumed c a p a b l e o f d e c a y i n g t o t h e s t a t e s |c,J",v> by e m i s s i o n o f a p h o t o n o f p o l a r i z a t i o n g. The r a t e a t w h i c h t h i s o c c u r s i s g i v e n by R m , b , c ( Q ' g ' t ) = I l < X m ; t | g - r | c , J " , v > | 2 Where g * r i s t h e . d i p o l e moment o p e r a t o r . S i n c e any one o f t h e g r o u n d s t a t e s m c o u l d have b e e n e x c i t e d we sum o v e r m as w e l l t o o b t a i n t h e t o t a l i n s t a n t a n e o u s e m i s s i o n r a t e o f p h o t o n s o f p o l a r i z a t i o n g a t t i m e t . R b c ( Q > E > t ) = I I I <xm; t.| g - r j q, J" ,v> j 2 mv - 1 6 . - • S u b s t i t u t i n g f o r ly ,t> we o b t a i n : ( i w - r / 2 ) t R b c(Q,g,t) = H I ! e <b,J' 5y|Q|a,J,m>- mv y • < b J J ' i y | g - r | c 3 J " , v > | 2 and e x p a n d i n g t h e s q u a r e , t h i s becomes [ i ( w -co , ) - r b ] t R b c ( Q 3 g , t ) = e v » V V V ^ v y -where Q a^ = <a | Q | 3> S a g = < a | g * r I B > Now i f we have N s y s t e m s e a c h s u b j e c t e d t o an i m p u l s e a t random t i m e s w i t h i n t h e i n t e r v a l ( 0 , T ) and o b s e r v e t h e s y s t e m f o r a t i m e T>>^- , t h e r a d i a t i o n r a t e o b s e r v e d becomes: T • R b c ( Q > ^ = N/o R b c ( Q , g _ , t ) d t - N/~ R b c ( Q , g , t ) d t Q Q ,g , g = N HI I ym y y y myy'v r'-i(w - U J ^ , ) Q 0 . g . K = N r I H I y m m y v 1 v yy myy'v 1 - ix(a) -w . ) ( 6 ) T h i s i s t h e " B r e i t f o r m u l a " . Some o f i t s i m p l i c a t i o n s a r e e a s i l y i n v e s t i g a t e d . 1 ) I f t h e v a r i o u s |y> and |y'> a r e w e l l r e s o l v e d ( i . e . |T(U -w , ) | > > 1 ) , t h e sum i n t h e B r e i t f o r m u l a ( e q . ( 6 ) ) r e d u c e s t o j u s t t h o s e t e r m s f o r w h i c h y=y' " b o C a - . e ) - HI l « p m l 2 l g p v l 2 - Ro myv H K t h i s i s j u s t f l u o r e s c e n c e w i t h no i n t e r f e r e n c e t e r m s - 1 7 - 2) I f , h o w e v e r . |x(co -CO , ) ! < 1 we o b t a i n some i n t e r - f e r e n c e t e r m s as w e l l , s e t t i n g A=Q Q ,g , g =A(m,y,y',v) > t . . - ym my y v vp >^»^ > mvy^y' y y' - P , VV V A(m,y ,\i' ,v) A* (m, y , y ' , v ) "° ^ £ , l - i x ( c o -co , ) +. 1+ixCco - 0 ) , ) mvy>y ' y y ' y y' ...... - R D + H I f 1 + T 2 -CO , ) 2 ' mvy>y 1 y y' •{A+A* + 1T(O> -a) , ) ( A - A * ) } (7) I t w i l l be shown i n A p p e n d i x I t h a t A=0 u n l e s s y-y'=2, i t . . w i l l a l s o be shown t h a t i n t h i s e x p e r i m e n t A h a s t h e f o r m A = A 0 e 2 i ^ 3 w h e r e cj> i s t h e a n g l e g makes w i t h t h e x - a x i s . T h e n i f t h e l e v e l s e x h i b i t a l i n e a r Zeeman e f f e c t i n a se x m a g n e t i c f i e l d M = cob + f ^ - H y ge a n d co -co , = e — H y y' mc . Hence 2Ao R h,(Q,g,<iO = Ho + n I mvy>y' 1+ ] •{cos2cJ) - £^sin2<j>} (8) mc The p o l a r i z a t i o n P, w i t h r e s p e c t t o t h e a x i s E, i s t h e n _ \ C ( Q , & A ) - R b c ( Q 5 g , < ! ) + T T / 2 ) R b c ( Q , g 3 ( ( ) ) + R b c(Q,g,cj)+^/2) - 18 - S u b s t i t u t i n g t h e e x p r e s s i o n s f o r f r o m eq.(8), p = ,Lu T, 2 * I I I I A Q(m,y,y',v) , , fgeHx^i 2 R D " . 1+1- 1 ° mvy>y' mc ' p K •{cos2cj> - £^^sin2(f>} mc Po {cos2cf> - ^ ^ s i n 2 ( } ) } (9) e H i i 2 y mc ^ mc J 2 p o = 5- I I I A 0 ( m , y , y ' ,v) ° mvy>y' C o m p a r i n g eq.(9) w i t h t h e e x p r e s s i o n f o r t h e p o l a r i z a t i o n d e r i v e d c l a s s i c a l l y ( e q . ( 5 ) , §2.2) we s e e t h a t , e x c e p t f o r a d i f f e r e n c e i n t h e d e f i n i t i o n o f P Q , t h e two e x p r e s s i o n s a r e t h e same. §2.4 The 3d JE L e v e l s o f H 2 I n t h e h i g h l y e x c i t e d s t a t e s o f l i g h t m o l e c u l e s s u c h as H 2 and H e 2 , t h e r o t a t i o n o f t h e n u c l e i may be s u f f i c i e n t l y r a p i d a t e v e n s m a l l r o t a t i o n a l q u a n t u m numbers t h a t t h e B o r n - O p p e n h e i m e r 2 2 a p p r o x i m a t i o n , t h e r e s o l u t i o n -of t h e t o t a l e i g e n f u n c t i o n i n t o a p r o d u c t o f e l e c t r o n i c , v i b r a t i o n a l a n d r o t a t i o n a l e i g e n f u n c t i o n s , i s no l o n g e r v a l i d . We must d e s c r i b e t h e s t a t e s i n t e r m s o f t h e c o u p l i n g b e t w e e n t h e i n t e r n u c l e a r a x i s a n d t h e e l e c t r o n i c m o t i o n , and t h e c o u p l i n g o f e l e c t r o n i c m o t i o n w i t h t h e a x i s o f r o t a t i o n , The v a r i o u s t y p e s o f c o u p l i n g i n a m o l e c u l e a r e d e s c r i b e d by H e r z b e r g 2 3 and u s u a l l y r e f e r r e d t o as Hund's c a s e a , b , c, d, and e i n t h e i r e x t r e m e l i m i t s . The 3d xE - . 1 9 - s t a t e s o f H 2 c a n be w e l l d e s c r i b e d as h a v i n g a c o u p l i n g b e t w e e n Hund's c a s e b and d. .A t r e a t m e n t ' o f t h i s c o u p l i n g was g i v e n by D a v i d s o n 2 "* when he a n a l y z e d t h e s e s t a t e s . A more r e c e n t d i s c u s s i o n o f c o u p l i n g i n t e r m e d i a t e b e t w e e n c a s e b a n d d h a s b e e n g i v e n by Von I . K o v a c s and A. B u d o 2 5 who a l s o d e r i v e d t h e Zeeman s p l i t t i n g s e x p e c t e d i n a m a g n e t i c f i e l d . • ' The a p p l i c a t i o n o f t h i s t h e o r y t o t h e 3d 1 l e v e l s o f H 2 i s f o u n d i n A p p e n d i x I I . I n t h e n e x t s e c t i o n we w i l l s e e t h a t t h e e x c i t a t i o n p e r t u r b a t i o n may be e x p a n d e d i n e l e c t r i c m u l t i p o l e moments. I t i s c o n v e n i e n t h e r e t o e s t a b l i s h w h i c h o f t h e s e moments w i l l h a v e z e r o , a n d w h i c h w i l l h a v e n o n - z e r o e x p e c t a t i o n v a l u e s b e t w e e n t h e g r o u n d s t a t e l ' s 1 ^ a n d t h e 3d JE s t a t e . As i s w e l l known a m o l e c u l e composed o f atoms w i t h n u c l e a r s p i n s I i a n d I 2 w i l l h a v e a r e s u l t a n t n u c l e a r s p i n I = I i + I 2 , I i + I 2 - 1 , 111 - I 2 | . F o r H 2 t h i s means t h a t I = 1 o r 0. The r o t a t i o n a l l e v e l s a l t e r n a t e l y h a v e t h e s e v a l u e s o f I ; i n p a r t i c u l a r f o r t h e s t a t e s , e v e n J s t a t e s h a v e 1 = 0 a n d odd J s t a t e s h a v e 1 = 1 . The t r a n s i t i o n f r o m a s t a t e w i t h I = 0 t o one w i t h 1 = 1 i s an e x c e e d i n g l y i m p r o b a b l e e v e n t h a v i n g i n t h e p u r e l i q u i d t r a n s i t i o n p r o b a b i l i t y o f o r d e r 1 0 - 7 s e c . 1 . R u l i n g o u t t h i s t r a n s i t i o n we h a v e t h e n t h e s e l e c t i o n r u l e J ' = J , J ± 2 , J ± 4 , - 20 - The p a r i t y o f t h e s t a t e s s p e c i f i e s w h i c h o f t h e m u l t i p o l e moments a r e z e r o between s t a t e J and J ' . We n o t e t h a t £ s t a t e s have p a r i t y (-1) and t h a t t h e e l e c t r i c ( £ ) i 2&-pole moment Q has p a r i t y (-1) . I t i s t h e n e a s y t o see t h a t <a, J | ^ | 3, J±2n> where n i s an i n t e g e r , i s z e r o u n l e s s H i s an e v e n i n t e g e r ; f u r t h e r m o r e , f r o m t h e p r o p e r t i e s o f s p h e r i c a l h a r m o n i c s one c a n show t h a t : <a,J I 3,J±2n> = 0 u n l e s s 2n<&, and i n t h e c a s e n=0, £>2J, so t h a t f o r t h e f i r s t n o n v a n i s h i n g e l e c t r i c m u l t i p o l e moment between o (2) s t a t e s we have t h e e l e c t r i c q u a d r u p o l e moment Q a . 04 p I n t h e d i s c u s s i o n o f l e v e l c r o s s i n g we always" assumed t h a t t h e g r o u n d s t a t e |a,J> was s p e c i f i e d and f o r a t o m i c p r o b l e m s t h i s i s u s u a l l y s o . When c o n s i d e r i n g m o l e c u l e s a t t h e r m a l e q u i l i b r i u m we f i n d t h a t a t room t e m p e r a t u r e n o t one, b u t s e v e r a l r o t a t i o n a l s t a t e s a r e p o p u l a t e d w i t h t h e i r p o p u l a t i o n s g i v e n by t h e M a x w e l l - B o l t z m a n n d i s t r i b u t i o n law, i . e . t h e p o p u l a t i o n N j o f t h e s t a t e J i s g i v e n by E N j = N f j e _ J kT where f j i s t h e d e g e n e r a c y o f t h e s t a t e = (2J+1)(2I+1) E j i s t h e e n e r g y o f t h e s t a t e J _ 1 6 k i s B o l t z m a n n c o n s t a n t =.1.38...x10 e r g s / C ° T i s t h e a b s o l u t e t e m p e r a t u r e N i s a n o r m a l i z i n g f a c t o r - 2 1 - The r e l a t i v e g r o u n d s t a t e p o p u l a t i o n s a t room t e m p e r a t u r e a nd t h o s e when l o w e r e d t o 8 0°K r e m e m b e r i n g t h e o r t h o - p a r a c o n v e r s i o n r e s t r i c t i o n a r e t a b u l a t e d i n T a b l e I . J ' a t 2 9 2°K a t 8 0°K I 0 . 1 3 2 . 2 4 9 1 0 1 . 6 6 3 . 7 4 9 2 1 2 . 1 1 5 . 0 0 1 7 0 3 . 0 8 6 7 x l 0 ~ 6 1 4 .004 0 T a b l e I - R e l a t i v e P o p u l a t i o n s o f t h e F i r s t Few R o t a t i o n a l L e v e l s o f H y d r o g e n a t T h e r m a l E q u i l i b r i u m . . I t c a n be s e e n t h a t a t room t e m p e r a t u r e t h e J'=2 s t a t e w i l l , u n d e r t h e p r e v i o u s s e l e c t i o n r u l e s , be e x c i t e d i h a v i n g e i t h e r t h e J = 0 a n d J = 2 s t a t e s as g r o u n d s t a t e . When t h e t e m p e r a t u r e i s l o w e r e d t o 8 0°K h o w e v e r , no s u c h a m b i g u i t y e x i s t s any more and o n l y t h e J = 0 s t a t e w i l l s e r v e as g r o u n d s t a t e . - 22 - §2.5 The E x c i t a t i o n M a t r i x Elements', Q I n o r d e r t o a p p l y t h e B r e i t f o r m u l a we must f i n d an e x p r e s s i o n f o r t h e p e r t u r b a t i o n m a t r i x e l e m e n t s Q ag- E v e n t h o u g h t h e l a w s g o v e r n i n g e l e c t r o n - a t o m c o l l i s i o n s a t non r e l a t i v i s t i c e n e r g i e s a r e c o m p l e t e l y known, t h e c a l c u l - a t i o n o f l o w e n e r g y s c a t t e r i n g c r o s s s e c t i o n s i s e x t r e m e l y . c o m p l e x a n d h a s o n l y b e e n a t t e m p t e d f o r t h e s i m p l e s t a t o m i c • c a s e s 2 6 . R e f e r r i n g t o e q . ( 9 ) , §2.3, i t s h o u l d be n o t e d t h a t s o l o n g as P o ^ 0 and i s i n d e p e n d e n t o f t h e m a g n e t i c f i e l d , i t s n u m e r i c a l v a l u e i s i r r e l e v a n t t o t h e d e t e r m i n a t i o n o f T. T h e r e f o r e , d i g r e s s i n g somewhat, we w i l l c o n s i d e r what c a n be s a i d o f P D i n t h e a b s e n c e o f a m a g n e t i c f i e l d . The l e v e l - c r o s s i n g f o r m u l i s m i s n o t c o n v e n i e n t f o r t h i s ; I n s t e a d , t h e e x c i t a t i o n a n d d e c a y a r e now c o n s i d e r e d as two i n d e p e n d - e n t p r o c e s s e s . We c o n s i d e r f i r s t t h e d e c a y . To be s p e c i f i c l e t us c o n s i d e r a h y p o t h e t i c a l t r a n s i t i o n J = l ->• J=0 o b s e r v e d a l o n g a n a x i s p e r p e n d i c u l a r t o t h e a x i s o f q u a n t i z a t i o n ( z - a x i s ) . The t r a n s i t i o n s t h u s o b s e r v e d a r e Am=±l a nd Am=0 h a v i n g p o l a r i z a t i o n s p e r p e n d i c u l a r and p a r a l l e l t o t h e z - a x i s r e s p e c t i v e l y . I f a l l t h r e e s t a t e s m=0, +1 a n d -1 o f t h e J = l l e v e l a r e e q u a l l y p o p u l a t e d t h e e m i t t e d l i g h t s h o u l d show no p o l a r i z a t i o n s i n c e t h e z- a x i s was a r b i t r a r i l y ' c h o s e n . I f . h o w e v e r , t h e e x c i t a t i o n m e c h a n i s m p o p u l a t e s t h e s t a t e m=0 d i f f e r e n t l y f r o m t h e s t a t e s m=±l, e i t h e r t h e Am=±l o r t h e Am=0 t r a n s i t i o n s - 23 - d o m i n a t e as t h e s o u r c e o f e m i t t e d r a d i a t i o n , and h e n c e t h e l i g h t w i l l be p o l a r i z e d a l o n g o r p e r p e n d i c u l a r t o t h e z - a x i s . When t h e ' p o p u l a t i o n s o f d i f f e r e n t |m| a r e u n e q u a l t h e m o l e c u l e s a r e s a i d t o be a l i g n e d and i n g e n e r a l t h e r a d i a t i o n e m i t t e d w i l l show, some p o l a r i z a t i o n w i t h r e s p e c t t o t h e a x i s o f q u a n t i z a t i o n . Thus t o o b t a i n a P o ^ 0 i t i s o n l y n e c e s s a r y t h a t t h e e x c i t a t i o n a l i g n t h e m o l e c u l e s . I f t h e c o l l i d i n g e l e c t r o n h a s s u f f i c i e n t l y l o w e n e r g y i t i s e a s y t o show t h a t a l i g n m e n t o c c u r s . L e t t h e e l e c t r o n t r a v e l a l o n g t h e z - a x i s w i t h momentum P = P k , and be s c a t t e r e d i n e l a s t i c a l l y by t h e m o l e c u l e a t a d i s t a n c e R. The a n g u l a r momentum o f t h e e l e c t r o n a b o u t t h e s c a t t e r i n g c e n t r e i s t h e n £_ = P X R , i n p a r t i c u l a r £ z = 0 ; a f t e r t h e c o l l i s i o n we assume t h a t t h e e n e r g y o f t h e e l e c t r o n i s s u f f i c i e n t l y s m a l l t h a t |.£'| = |P'xR|<<h, i . e . t h e e n e r g y p. 2 H 2 o f t h e s c a t t e r e d e l e c t r o n E' = << n p 2 > t h e n £ ' = 0 . 2m 2mR 5 z Thus f o r t h e c o l l i s i o n A£ = 0 and c o n s e r v a t i o n o f a n g u l a r z to momentum f o r t h e w h o l e s y s t e m t h e n i m p l i e s t h a t A J z = 0. Thus i f t h e c o l l i s i o n i n c r e a s e s J , a l i g n m e n t o f t h e u p p e r - s t a t e l e v e l s o c c u r s . I f AJ<_0 we n e e d as an e x t r a c o n d i t i o n t h a t t h e s c a t t e r i n g c r o s s s e c t i o n s f o r d i f f e r e n t |m| be n o n - e q u a l . The c o n d i t i o n on t h e s c a t t e r e d e l e c t r o n e n e r g y O u. ( a b o v e ) i s q u i t e s t r i n g e n t , f o r R = 5A and ^ ' ^ J Q f o r i n s t a n c e „ ^ h 2 n , , , , s o t h a t e v e n i n c a s e s 1 2 m R ^ x i o 2 s 0.15e.V. - 24 - where t h e g r o u n d s t a t e a n g u l a r momentum I s z e r o and t h e p o l a r i z a t i o n c a n be u n i q u e l y p r e d i c t e d i t ' i s n o t e x p e c t e d t o be r e a l i z e d e x p e r i m e n t a l l y where much g r e a t e r e n e r g y e x c e s s e s o c c u r . When P 0 i s n o t i n d e p e n d e n t o f t h e m a g n e t i c f i e l d , as w i l l . b e t h e c a s e f o r n o n - z e r o f i e l d l e v e l c r o s s i n g s , t h e m a t r i x e l e m e n t s A c must be computed as a f u n c t i o n o f m a g n e t i c f i e l d . F o r t h i s r e a s o n as w e l l as f o r t h e s a k e o f c o m p l e t e n e s s , t h e r e l a t i v e m a t r i x e l e m e n t s f o r e x c i t a t i o n t o t h e v a r i o u s y , y' w i l l be c o n s i d e r e d . No a d e q u a t e t h e o r y e x i s t s and v a r i o u s a p p r o x i m a t i o n s s u c h as t h e B o r n a p p r o x - i m a t i o n a r e v a l i d o n l y a t e n e r g i e s above a few h u n d r e d e.V. N e v e r t h e l e s s , b e c a u s e o f i t s r e l a t i v e s i m p l i c i t y and f o r l a c k o f a much b e t t e r t h e o r y , t h e B o r n a p p r o x i m a t i o n w i l l be u s e d . I t i s e x p e c t e d t h a t a l t h o u g h a b s o l u t e c r o s s - s e c t i o n s so o b t a i n e d w i l l be h i g h l y e r r o n e o u s , r e l a t i v e m a t r i x e l e m e n t s Q , Q , w i l l be a t l e a s t q u a l i t a t i v e l y my my' • H . c o r r e c t . T h e s e m a t r i x e l e m e n t s a r e u s e d e x p l i c i t l y o n l y i n t h e d i s c u s s i o n i n §5.2 where t h e e f f e c t o f h y p e r f i n e s p l i t t i n g s o f t h e o r d e r o f t h e n a t u r a l l i n e w i d t h , w i l l be c o n s i d e r e d . We l e t p and p' be t h e momenta o f t h e i n c i d e n t e l e c t r o n b e f o r e and a f t e r t h e c o l l i s i o n . Then f o r t r a n s i t i o n p r o b a b i l i t y a m p t i t u d e , Q R , f r o m a s t a t e a t o I , whose p Q* e n e r g i e s d i f f e r by E 0 , we t a k e i n a c c o r d a n c e w i t h t h e B o r n a p p r o x i m a t i o n 2 7 ; - 2 5 . - - q . r p'-p /2mE.- where q = £ , |q| = / £ 2 r g i s t h e i n c i d e n t e l e c t r o n ' s p o s i t i o n v e c t o r U i s t h e i n t e r a c t i o n p o t e n t i a l ; t a k e n t o be t h e e l e c t r o s t a t i c i n t e r a c t i o n 2 2 U ( { R ,R },r ) = I ^= " %— n 5 e 5 e L i r -R i i r -R i e,n| e n| | e e| w i t h R n = t h e p o s i t i o n v e c t o r o f n u c l e u s n {n} = t h e s e t o f n u c l e i R g = t h e p o s i t i o n v e c t o r o f e l e c t r o n e {e} = t h e s e t o f m o l e c u l e ' s e l e c t r o n s m i s t h e mass o f e l e c t r o n C a r r y i n g o u t t h e i n t e g r a t i o n o v e r r g we o b t a i n - i q • R, i e { e 3 n } Q 3 a a <3| .1 .e x | a > We e x p a n d t h e e x p o n e n t i a l i n a p o w e r s e r i e s a n d o b t a i n Q D = Q D 1 ^ + Q D 2^ +• Qo 3^ + Q D ^ + ... ( 1 0 ) ~2~ ~2T Where {I) M 1 ( i q ) i q R. i — i a> o r t h e e l e c t r i c 2-1 p o l e moment i n t h e q_ d i r e c t i o n b e t w e e n t h e s t a t e s 3 a n d a . - 26 - (Qp°^ = <3|l[a> = 5 a g f r o m o r t h o g o n a l i t y o f |a> and |3> ) F o r E•= l4e.V. and R. - 5 x10~ 8cm I q ^ ' R ^ l ^ l —1 ' — — l 1 s o t h a t t h i s s e r i e s e x p a n s i o n , e q . ( 1 0 ) , c o n v e r g e s a p p r o x - i m a t e l y as pj-p . We h a v e s e e n i n §2.4 t h a t f o r o u r s t a t e s Qp"^ = Qo"^ = 0,-hence we may a p p r o x i m a t e Q R b y : b u t u n f o r t u n a t e l y q i s n o t s p e c i f i e d u n l e s s p' = 0 , i . e . = E n , s o t h a t we must make a f u r t h e r a p p r o x i m a t -2m a3 i o n . We e x p e c t t h a t s o l o n g as p' i s s m a l l , q w i l l l i e p p' a p p r o x i m a t e l y a l o n g - p ; s e t t i n g k = — , k' = — - , q = k'-k "K "ft The d i r e c t i o n s q t a k e s a r e e a s i l y s e e n f r o m t h e f o l l o w i n g d i a g r a m ( F I G . 7 ) . F o r a s p e c i f i e d q t h i s c o m p l e t e s o u r p r o b l e m , F i g u r e 7 - The S c a t t e r i n g A n g l e 0, - 27 - By c o n s e r v a t i o n o f e n e r g y k.' i s r e s t r i c t e d i n m a g n i t u d e t o '2mE k ' I - = I k| - ' a p h 2 - i k ' The maximum a n g l e q makes w i t h -k i s t h e n fl=sin ^ . Thus f o r a 16 e.V. I n c i d e n t e l e c t r o n s c a t t e r e d w i t h a n e n e r g y o f 2 e.V., | fi | <_ = 20°; i f t h e e l e c t r o n i s s c a t t e r e d s o t h a t k' i s s p h e r i c a l l y s y m m e t r i c a l l y d i s - t r i b u t e d , <q> = -k i . e . <Q> = 0, and < s i n 2 f i > = — 2TT k ' 2 s i n 2 g d g ' ( k - c o s g ) 2 o 1 f \ ' 2 s i n 2 g d g n / f k S \ 2TT J o h e n c e , u s i n g t h e same k a n d k' as a b o v e , < s i n 2 f t > < .06 l e a d i n g t o - /<fi2> < 14°. A l t h o u g h t h i s i s n o t a n i m p r e s s i v e l y s m a l l d i s p e r s i o n we s h a l l assume i n d e r i v i n g t h e B r e i t f o r m u l a m a t r i x e l e m e n t s q k» „ t h a t — = - — = i a n d h e n c e t h e m a t r i x e l e m e n t s Q R a r e q k a p r o p o r t i o n a l t o t h o s e o f t h e e l e - c t r i c q u a d r u p o l e moment ( 2 ) Q . S p e c i f i c f o r m u l a e and t h e a p p l i c a t i o n o f t h e s e f o r m u l a e t o t h e t r a n s i t i o n s o b s e r v e d a r e f o u n d i n A p p e n d i x - 28 - CHAPTER I I I EXPERIMENTAL DETAILS §3.1 E x p e r i m e n t a l A r r a n g e m e n t I n v i e w o f t h e d i s c u s s i o n i n t h e p r e c e e d i n g c h a p t e r t h e e s s e n t i a l s o f t h i s e x p e r i m e n t s h o u l d c o n s i s t o f a d i s - c h a r g e e x c i t e d by 15 - 20 e.V. e l e c t r o n s t r a v e l l i n g a l o n g t h e x - a x i s , a homogeneous v a r i a b l e m a g n e t i c f i e l d i n t h e d i s c h a r g e r e g i o n , a f i l t e r f o r r e s o l v i n g a g i v e n t r a n s i t i o n , a n d a d e v i c e f o r d e t e c t i n g and m e a s u r i n g p o l a r i z a t i o n . The b a s i c a p p a r a t u s i s shown i n F I G . 8 ; no a t t e m p t a t s c a l i n g o r r e a l i s m was made i n t h i s s k e t c h . A d i s c h a r g e c e l l i s p l a c e d b e t w e e n a p a i r o f c a p a c i t o r p l a t e s e a c h p a r a l l e l t o t h e y - z p l a n e . B e t w e e n t h e s e p l a t e s a r a d i o - f r e q u e n c y e l e c t r i c f i e l d o f t h e f o r m E = ( E 0 c o s a ) t ) i c a u s e s f r e e e l e c t r o n s t o o s c i l l a t e i n t h e ±x d i r e c t i o n w i t h t h e a p p r o p r i a t e e n e r g y . C e n t e r e d a b o u t t h e d i s c h a r g e a r e 2 p a i r s o f H e l m h o l t z c o i l s ; one t o c a n c e l t h e v e r t i c a l c o mponent o f t h e e a r t h ' s m a g n e t i c f i e l d , t h e o t h e r t o p r o d u c e a f i e l d H = Hk. L i g h t e m i t t e d i n t h e z d i r e c t i o n i s f o c u s s e d by a p a i r o f l e n s e s o n t o t h e e n t r a n c e s l i t o f a m o n o c h r o m a t o r t o s p e c t r a l l y r e s o l v e t h e t r a n s i t i o n o f i n t e r e s t . L i g h t a p p e a r i n g a t t h e e x i t s l i t o f t h e m o n o c h r o m a t o r f a l l s on a p h o t o m u l t i p l i e r whose o u t p u t i s f e d i n t o t h e s i g n a l c h a n n e l o f a l o c k - i n a m p l i f i e r . CAPACITOR PLATES MONOCHROMATOR Z-AXIS X-Y RECORDER F i g u r e 8 - The A p p a r a t u s - 30 - The l i g h t p a s s i n g b e t w e e n t h e two l e n s e s i s p a s s e d t h r o u g h a p o l a r o i d , w i t h i t s p l a n e p e r p e n d i c u l a r t o t h e beam, w h i c h i s r o t a t e d - cfoout t h e x - a x i s . The p o l a r i z e d component o f l i g h t i s t h u s m o d u l a t e d a t t w i c e r o t a t i o n a l f r e q u e n c y . A s m a l l l i g h t and p h o t o d i o d e p l a c e d a t t h e r i m o f t h e p o l a r o i d , one on e i t h e r s i d e p r o v i d e a r e f e r e n c e s i g n a l f o r t h e l o c k - i n a m p l i f i e r . The o u t p u t o f t h e l o c k - i n a m p l i f i e r i s c o n n e c t e d t o t h e y - c h a n n e l o f t h e x-y r e c o r d e r . The x - c h a n n e l i s c o n t r o l l e d by t h e v o l t a g e a c r o s s H e l m h o l t z c o i l s p r o d u c i n g t h e f i e l d H = Hk. T h e s e c o i l s a r e d r i v e n by a p a i r o f p o w e r s u p p l i e s a n d an a m p l i f i - e r whose o u t p u t i s s l o w l y s w e p t t o p r o d u c e f i e l d s o f t h e f o r m H = H c + H i t w h e r e H o~-10 g a u s s a nd H ~+10 g a u s s . ITlclX A q u a r t e r wave p l a t e i s p l a c e d i n t h e beam w i t h i t s f a s t a x i s a t 45° t o t h e x - a x i s i n t h e x-y p l a n e j u s t b e f o r e t h e e n t r a n c e s l i t o f t h e m o n o c h r o m a t o r t o e l i m i n a t e t h e e f f e c t o f t h e m o n o c h r o m a t o r t r a n s m i t t i n g y p o l a r i z e d l i g h t p r e f e r e n t i a l l y o v e r x p o l a r i z e d l i g h t . To e n s u r e t h a t t h e e f f e c t s o b s e r v e d w e re n o t due t o t h e R.P. f i e l d s , t h e e x p e r i m e n t was p e r f o r m e d u s i n g f r e q u e n c i e s f = l 8 0 MHz and l a t e r w i t h f=450 MHz and c o r r e s - p o n d i n g c h a n g e s i n e l e c t r i c f i e l d s . §3-2 The D i s c h a r g e •The d i s c h a r g e a p p a r a t u s c o n s i s t s o f a c y l i n d r i c a l . p yrex d i s c h a r g e c e l l o f 5 cm d i a m e t e r and 2.5-3 cm l e n g t h - 31 - c o n n e c t e d t o t h e vacuum s y s t e m by two 1 cm O.D. p y r e x t u b e s as sho\\rn i n F I G . 10. The ends o f t h e d i s c h a r g e c e l l a r e s l i g h t l y c o n v e x o u t w a r d s t o p r o v i d e m e c h a n i c a l s t r e n g t h a g a i n s t a i r p r e s s u r e . A c i r c u l a r b r a s s p l a t e o f 7 cm d i a m e t e r i s p l a c e d a t e i t h e r end so t h a t i t e x t e n d s a b o u t 1 cm b e y o n d t h e edges o f t h e c e l l . The p l a n e s o f t h e p l a t e s are. p a r a l l e l t o t h e a x i s o f o b s e r v a t i o n . The o u t p u t o f a r a d i o f r e q u e n c y t r a n s m i t t e r i s c o u p l e d t o t h e s e p l a t e s i n a manner d e s c r i b e d i n §3.5- The c e l l and p l a t e s a r e shown f u l l s c a l e i n F I G . 9 below.. F o r c o n v e n i e n c e t h e c o - o r d i n a t e axes u s e d t h r o u g h o u t t h e d i s c u s s i o n a r e a l s o shown: y » X F i g u r e 9 - C r o s s - S e c t i o n o f D i s c h a r g e C e l l and C a p a c i t o r P l a t e s - 32 - The d i s c h a r g e i s assumed t o c o n s i s t o f a d i l u t e g as o f n e u t r a l m o l e c u l e s a nd a much s m a l l e r number o f f r e e e l e c t r o n s . The m o t i o n o f t h e e l e c t r o n s i n a r a d i o - f r e q u e n c y e l e c t r i c f i e l d , a s s u m i n g a c o m p l e t e a b s e n c e o f c o l l i s i o n s and z e r o m a g n e t i c f i e l d i s g o v e r n e d b y : mx = e E 0 c o s o ) t x . = ^ f - c o s w t muK a n d t h e k i n e t i c e n e r g y K.E.: 1 * ? K.E. = -|mx2 = 1 ( i E p _ ) 2 s i n 2 U t 2m OJ The a p p l i c a t i o n o f a s m a l l m a g n e t i c f i e l d H = Hk y i e l d s , s t i l l a s s u m i n g no c o l l i s i o n s o r d a m p i n g , mx = e E D c o s w t - e H y my = eHx s o l v i n g t h e s e e q u a t i o n s : x(oot) = - H i i f l . , x , c o s ( t o t ) y ( c o t ) = - x ( u t + T T / 2 ) , v = — J a) m i n o t h e r w o r d s , t h e e l e c t r o n moves on an e l l i p s e whose r a t i o o f m a j o r t o m i n o r d i a m e t e r i s I n o r d e r f o r t h e a b o v e t o meet t h e c o n d i t i o n s o f t h e d i s c h a r g e c e l l , 3 c o n d i t i o n s must be s a t i s f i e d : 1) e l a s t i c c o l l i s i o n s o f e l e c t r o n s w i t h n e u t r a l - 33 - m o l e c u l e s o u g h t t o o c c u r s u f f i c i e n t l y i n f r e q u e n t l y . T h i s may be r e s t a t e d a s : t h e mean f r e e p a t h o f t h e e l e c t r o n s h o u l d e x c e e d t h e a m p l i t u d e o f i t s •.motion c o n s i d e r a b l y . 2) c o l l i s i o n s o f e l e c t r o n s w i t h t h e w a l l s o f t h e d i s c h a r g e c e l l must a l s o be r a r e e v e n t s . T h i s m e r e l y means t h a t any d i m e n s i o n o f t h e d i s c h a r g e c e l l o u g h t t o e x c e e d c o n s i d e r a b l y t h e a m p l i t u d e o f e l e c t r o n m o t i o n . 3) e l e c t r o n s must be s u f f i c i e n t l y e n e r g e t i c t o i o n i z e t h e o c c a s i o n a l m o l e c u l e i n o r d e r t o make up f o r e l e c t r o n s l o s t by r e c o m b i n a t i o n a n d s u s t a i n t h e d i s c h a r g e . The t h i r d c o n d i t i o n i m p l i e s t h a t we must g i v e o u r e l e c t r o n s a maximum k i n e t i c e n e r g y o r o r d e r 20 e.V. T a b l e 2 l i s t s some o f t h e p r o p e r t i e s o f t h e m o t i o n o f t h e e l e c t r o n s i n an e l e c t r i c f i e l d o s c i l l a t i n g a t f r e q u e n c y f = 2TTOJ . C o n d i t i o n 2 ( a b o v e ) i s c l e a r l y s a t i s f i e d f o r a l l t h e f r e q u e n c i e s t a b u l a t e d i n T a b l e 2 i n a d i s c h a r g e c e l l o f t h e s i z e u s e d . I n o r d e r t o s e e how w e l l t h e f i r s t c o n - d i t i o n i s s a t i s f i e d we compute t h e mean f r e e p a t h , L b e t w e e n m o l e c u l e s o f c r o s s - s e c t i o n o, d i s t r i b u t e d w i t h a d e n s i t y p: _ f r e q u e n c y ( x i O 6 Hz) E Q r e q u i r e d t o p r o d u c e K.E. =20e.V. max a m p l i t u d e o f e l e c t r o n m o t i o n ( m a j o r d i a m e t e r ) m i n o r a x i s t o m a j o r a x i s r a t i o i n 10 g a u s s m a g n e t i c f i e l d 10.0 200 500 100 V/cm 200 V/cm 500 V/cm 4 mm 2 mm 0 . 8 mm .07 .035 .014 T a b l e I I - P r o p e r t i e s o f E l e c t r o n M o t i o n i n an R.F. E l e c t r i c and D.C. M a g n e t i c F i e l d T h e n a t a d e n s i t y o f 1 . 7 7 x 1 0 1 V e r a 3 ( .05 mm Hg a t o room t e m p e r a t u r e ) a n d a s s u m i n g a c r o s s - s e c t i o n o f ~10 A we f i n d t h e mean f r e e p a t h L - 6 mm. Thus a t t h e s e d e n s i t i e s t h e f o r e g o i n g d e s c r i p t i o n o f t h e e l e c t r o n s ' m o t i o n c a n a t b e s t be e x p e c t e d t o be q u a l i t a t i v e . §3.3 The O p t i c a l 'System As shown i n F I G . 8, a p a i r o f p l a n o - c o n v e x l e n s e s o f f o c a l l e n g t h F = 20 cm e a c h and a p e r t u r e F / 6 , a r e p l a c e d , one a t i t s f o c a l l e n g t h f r o m t h e d i s c h a r g e , t h e o t h e r a t i t s f o c a l l e n g t h f r o m t h e e n t r a n c e s l i t o f t h e m o n o c h r o m a t o r , s o t h a t l i g h t o r i g i n a t i n g a t t h e c e n t r e o f t h e d i s c h a r g e t r a v e r s e s t h e s p a c e b e t w e e n t h e l e n s e s i n a p a r a l l e l beam and i s f o c u s s e d o n t o t h e e n t r a n c e s l i t o f t h e m o n o c h r o m a t o r . B e t w e e n t h e l e n s e s a r o t a t i n g p o l a r o i d i s p l a c e d ; t h e a p e r t u r e o f t h e p o l a r o i d i s l a r g e e n o u g h n o t t o c o n s t i t u t e - 35 - a " s t o p " i n t h e o p t i c a l s y s t e m . The r o t a t i n g p o l a r o i d i s shown i n F I G . 16. A t t h e w a v e l e n g t h s t h i s e x p e r i m e n t was o p e r f o r m e d ' (4930 A) t h e m o n o c h r o m a t o r , an F / 8 S p e x 1 m - i n s t r u m e n t , h a s , when u s e d i n s e c o n d o r d e r , a d i s p e r s i o n o o o f ~6A/mm, and a r e s o l u t i o n o f b e t t e r t h a n 0.1A. S l i t - w i d t h s o f 0.3 mm f o r b o t h e n t r a n c e a nd e x i t s l i t s w e r e u s e d o t h r o u g h o u t , y i e l d i n g a r e s o l u t i o n o f ~2A. The m o n o c h r o m a t o r was f o u n d t o p o l a r i z e i n c i d e n t l i g h t i n a manner t h a t o v a r i e d w i t h w a v e l e n g t h . A t 4930 A i t p o l a r i z e d l i g h t a p p r o x i m a t e l y 80% a l o n g t h e y - a x i s ( p a r a l l e l t o t h e g r a t - i n g l i n e s ) . As t h i s l e a d s t o m o d u l a t i o n o f l i g h t i n t e n s i t y a t t h e e x i t s l i t when u n p o l a r i z e d l i g h t i s i n c i d e n t on t h e r o t a t i n g p o l a r o i d , t h e m o n o c h r o m a t o r must be made i n s e n s i t i v e t o p o l a r i z a t i o n . T h i s was a c c o m p l i s h e d by p l a c i n g a q u a r t e r - wave p l a t e w i t h i t s f a s t ( o r s l o w ) a x i s a t 45° t o t h e x- an d y - a x i s . A v a r i a b l e w a v e l e n g t h q u a r t e r - w a v e p l a t e d e s c r i b e d i n §3.10 was p r o d u c e d f o r t h i s p u r p o s e . B e c a u s e i t i s n o t i m m e d i a t e l y o b v i o u s how t h e q u a r t e r - w a v e p l a t e c o r r e c t s f o r t h e m o n o c h r o m a t o r p o l a r i z a t i o n a b r i e f t h e o r - e t i c a l t r e a t m e n t f o l l o w s . We c o n s i d e r a n e l e c t r o m a g n e t i c wave whose e l e c t r i c v e c t o r makes an a n g l e 6' w i t h t h e x - a x i s , t r a v e l l i n g a l o n g t h e z - a x i s . . I t f a l l s on a p o l a r i z e r r o t a t i n g a t a n g u l a r f r e q u e n c y to. The i n c i d e n t l i g h t h a s t h e n a n e l e c t r i c v e c t o r E w i t h c o m p o n e n t s . E x and E^ r e f e r r e d t o t h e u s u a l x , y, z r e f e r e n c e s y s t e m . - 36 - E„ = A n c o s 0 e i ( k z - V t ) i ( k z - v t ) x E = A 0 s i n 6 e y ° and i t s p o l a r i z a t i o n r e f e r r e d t o t h e s e a x e s i s P = c o s 2 6 - s i n 2 i The e l e c t r i c f i e l d E^ a l o n g t h e p o l a r o i d a x i s ( a t an a n g l e wt t o t h e x - a x i s ) i s E p = A o c o s ( w t - 0 ) e 1 ( k z " V t ) The l i g h t t h e n f a l l s on a A/4 p l a t e w i t h i t s f a s t a x i s a t TT/4 t o t h e x - a x i s a n d i t s p l a n e p e r p e n d i c u l a r t o t h e z - a x i s c a l l i n g E „ t h e e l e c t r i c f i e l d a l o n g i t s f a s t a x i s and E A t h e e l e c t r i c f i e l d a l o n g i t s r e t a r d i n g a x i s : i ( k z - v t ) i ( k z - v t ) E q = EpCos (7T/4-cot) e" E L = E p c o s ( 3 i T / 4 - a ) t ) e " p a s s i n g t h r o u g h t h e A/4 p l a t e E |( s u f f e r s a p h a s e s h i f t . 6 and E^ s u f f e r s a p h a s e s h i f t 6+7T/4. The c o m p o n e n t s o f t h e f i e l d a f t e r p a s s i n g t h r o u g h t h e A/4 p l a t e a r e t h e n : E 1 = E e 1 6 E | = E j e 1 ( 6 + i r / 2 , ) o r i n t e r m s o f t h e x and y c o m p o n e n t s : E x = / I ( E ' v - E V ) = ^ ( E , - i E A ) - 37 - The m o n o c h r o m a t o r p a s s e s aE' + 3E' wh e r e i t s p o l a r i z a t i o n = y x a 2 - 6 2 a 2 + g 2 a l o n g t h e y - a x i s . The i n t e n s i t y o f l i g h t a p p e a r i n g a t t h e e x i t s l i t i s t h e n : I cc ( a E ' ) 2 + ( 3 E ' ) 2 y x .= a 2 ^ 2 ( E 2 + E 2 ) a 2 + 3 2 T 7 2 — ~ Cj 2 p = ^ ! lA lcos 2(a)t -0 ) ( 1 1 ) Thus we o b t a i n a s i g n a l o f e x a c t l y t h e same f o r m we w o u l d h a v e o b t a i n e d i f t h e m o n o c h r o m a t o r w e r e n o t t h e r e , r e d u c e d a 2 + 3 2 o n l y by t h e f a c t o r — . The p h o t o m u l t i p l i e r t h e n p r o d u c e s a s i g n a l V ( t ) = C * c o s 2 ( w t - 6 ) w h e r e C i s a c o n s t a n t d e p e n d i n g on t h e p h o t o m u l t i p l i e r e f f i c i e n c y and g a i n , t h e e f f i c i e n c y o f t h e o p t i c s , a n d t h e i n t e n s i t y o f r a d i a t i o n e m i t t e d f r o m t h e d i s c h a r g e . §3.4 The Vacuum S y s t e m The vacuum s y s t e m i s shown i n F I G . 1 0 . I t i s a c o n v e n t i o n a l g l a s s s y s t e m pumped by a " C e n c o - H y v a c " m e c h a n i c a l pump w h i c h . i s c a p a b l e o f r e d u c i n g t h e p r e s s u r e i n t h e s y s t e m t o l e s s t h a n 5 X 1 0 - 1 * mm Hg when t h e l i q u i d n i t r o g e n c o l d t r a p t o p r e v e n t b a c k s t r e a m i n g o f pump o i l i s i n p l a c e . H y d r o g e n gas i s l e a k e d I n t o t h e s y s t e m by a n e e d l e v a l v e ; t h e l e a k a g e r a t e e s t a b l i s h i n g t h e e q u i l i b r i u m p r e s s u r e o f t h e s y s t e m . When t h e d i s c h a r g e w a s . s u b m e r s e d - 38 - i n l i q u i d n i t r o g e n any i m p u r i t i e s i n t h e h y d r o g e n t e n d t o f r e e z e on t h e w a l l s o f t h e d i s c h a r g e c e l l . The h y d r o g e n o b t a i n e d f r o m a h i g h p r e s s u r e b o t t l e i s s u f f i c i e n t l y i m p u r e t h a t w i t h i n a few h o u r s t h e d i s c h a r g e c e l l w a l l s become q u i t e o p a q u e , t h e r e f o r e two c o l d t r a p s w e re i n s t a l l e d f u r t h e r u p s t r e a m . S t a r t i n g f r o m t h e r e d u c t i o n v a l v e on t h e p r e s s u r e b o t t l e , h y d r o g e n a t ~2 p o u n d s / i n 2 a b o v e a t m o s p h e r i c p r e s s u r e p a s s e s t h r o u g h a f l e x i b l e r u b b e r h o s e t o a c o l d t r a p f i l l e d h a l f w a y w i t h a c t i v a t e d c h a r c o a l . F rom t h e c h a r c o a l t r a p i t p r o c e e d s t h r o u g h ~20 cm o f h o s e t o t h e n e e d l e v a l v e and f r o m t h e r e a t l o w p r e s s u r e t o a s e c o n d c o l d t r a p . P r o c e e d i n g d o w n s t r e a m f r o m t h e s e c o n d c o l d t r a p t o t h e d i s c h a r g e c e l l , a b o u t h a l f w a y a l o n g ^ a P i r a n i and a M c L e o d gauge a r e a t t a c h e d t o t h e s y s t e m a n d c o n n e c t e d v i a g l a s s s t o p c o c k s . P r o c e e d i n g d o w n s t r e a m f r o m t h e d i s c h a r g e c e l l a n o t h e r c o l d t r a p i s e n c o u n t e r e d b e f o r e t h e gas i s pumped o u t by t h e m e c h a n i c a l pump. A l l t h e c o n n e c t i n g g l a s s t u b i n g h a s an i n s i d e d i a m e t e r o f ~7 mm. Some f l e x i b i l i t y i n t h e p o s i t i o n o f t h e d i s c h a r g e c e l l was p r o v i d e d by t h e g r o u n d g l a s s s w i v e l j o i n t s by w h i c h i t was a t t a c h e d t o t h e vacuum s y s t e m . When d a t a was b e i n g t a k e n t h e p r e s s u r e i n t h e s y s t e m was m e a s u r e d a t a b o u t 2 h o u r i n t e r v a l s w i t h t h e M c L e o d gauge a nd c o n t i n u o u s l y m o n i t o r e d w i t h t h e r o u g h l y c a l i b r a t e d P i r a n i g a u g e . MCLEOD GAUGE NEEDLE VALVE F i g u r e 10 - The Vacuum S y s t e m - HO - §3.5 R.F. S u p p l i e s a n d C o u p l i n g The 180 MHz s o u r c e was. a s i m p l e o s c i l l a t o r u s i n g a 928 B t u b e . I t s s c h e m a t i c i s shown i n F I G . 1 1 . The p l a t e s t a k e on a n R.F. v o l t a g e a b o u t g r o u n d p o t e n t i a l w h i l e t h e c a t h o d e i s f e d f r o m t h e n e g a t i v e t e r m i n a l o f a power s u p p l y a t a p o t e n t i a l o f -80V t o -600V. The " t a n k " c i r c u i t f o r t h e p l a t e s c o n s i s t s o f a s t r i p o f t h i n c o p p e r s h e e t a p p r o x i m a t e l y 1 cm w i d e and 30 cm l o n g , g r o u n d e d a t t h e c e n t r e and a t t a c h e d t o one anode a t e i t h e r e n d . F e e d b a c k t o t h e g r i d s i s p r o v i d e d by a 30 cm l e n g t h o f 16 guage i n - s u l a t e d c o p p e r w i r e whose l o o p i s l a i d b e t w e e n t h e p l a t e s t r i p l o o p . A t t h e p l a t e s t h e i m p e d a n c e and R.F. v o l t a g e i s s u f f i c i e n t l y h i g h t o o b v i a t e t r a n s f o r m e r s o r s p e c i a l c o u p l i n g c i r c u i t s . The d i s c h a r g e c a p a c i t o r p l a t e s w e re c o u p l e d t o t h e p l a t e " t a n k " c i r c u i t by a l e n g t h o f 300ft ( c h a r a c t e r i s t i c i m p e d a n c e ) t w i n l e a d w i r e . The w i r e was a t t a c h e d t o t h e c o p p e r s t r i p a t p o i n t s , e q u i d i s t a n t f r o m t h e g r o u n d e d p o i n t , w h e r e maximum power t r a n s f e r was e s - t i m a t e d t o o c c u r . I n p r a c t i c e , t h e maximum power t r a n s f e r o c c u r r e d a t c o n t a c t p o i n t s a b o u t 8 cm f r o m t h e c e n t r e . The p o w e r o u t p u t was e s t i m a t e d t o be a b o u t 5 - 10W a t a c a t h o d e p o t e n t i a l o f 300V. T h i s o s c i l l a t o r h a d s e v e r a l d r a w b a c k s . The powe r o u t p u t was q u i t e u n - s t a b l e o v e r p e r i o d s e x c e e d i n g 10 m i n u t e s , r a t h e r l a r g e f r a c t i o n s o f R.F. p o w e r w e r e r a d i a t e d i n t o t h e s u r r o u n d i n g r o o m , a nd t h e o s c i l l a t o r h a d a t e n d e n c y t o p u l s e i t s o u t p u t i n b u r s t s r e p e a t e d a t i n t e r v a l s o f o r d e r o f 10 6 s e c o n d s . F i g u r e 11 - 180 MHz R.F. O s c i l l a t o r - 42 - L a t e r , t h e s o u r c e was r e p l a c e d by a 450 MHz t r a n s m i t t e r c o n s i s t i n g o f a n R.C.A. MI-17436-1 t r a n s m i t t e r u s e d as " d r i v e r " a n d a C a n a d i a n M a r c o n i M o d e l 163-107 h i g h f r e q u e n c y p o w e r a m p l i f i e r w i t h an o u t p u t i m p e d a n c e o f 50ft a n d o u t p u t p o w e r o f a p p r o x i m a t e l y 50V/. I f we h a v e a 1:1 s t a n d i n g wave r a t i o (S.W.R.) on a 50ft l i n e a n d 50W i s t r a n s m i t t e d , t h e R.P. v o l t a g e a v a i l a b l e i s o n l y 50V. S i n c e we n e e d a f i e l d o f o r d e r 400V/cm b e t w e e n t h e d i s c h a r g e c a p a c i t o r p l a t e s a n d t h e p l a t e s a r e s p a c e d a t 3 cm, 1200V i s r e q u i r e d . V o l t a g e s o f t h i s o r d e r w e re o b t a i n e d by t h e "T m a t c h e d " r e s o n a n t c i r c u i t shown i n F I G . 1 2 . A l e n g t h o f RG 8-U c a b l e c a r r i e d t h e power t o t h i s c i r c u i t . By a d j u s t m e n t o f t h e c o n t a c t p o i n t A ( s e e F I G . 1 2 ) an a p p r o x i m a t e i m p e d a n c e m a t c h b e t w e e n t h e r e s o n a n t c i r c u i t and t r a n s m i s s i o n l i n e may be o b t a i n e d . The c i r c u i t i s t u n e d t o r e s o n a n c e by m o v i n g t h e c r o s s b a r s c l o s e r t o g e t h e r o r f u r t h e r a p a r t , w h i l e k e e p i n g them e q u i d i s t a n t f r o m t h e c a p a c i t o r p l a t e s . §3.6 H e l m h o l t z C o i l s The e a r t h ' s m a g n e t i c f i e l d c a n c e l l i n g c o i l s h a v e a 19.5 cm mean d i a m e t e r , a n d a r e s p a c e d 9.7 cm a p a r t ; e a c h h a s 50 t u r n s o f #24 c o p p e r w i r e . When t h e 2 c o i l s a r e p l a c e d i n s e r i e s t h e f i e l d p r o d u c e d a t t h e c e n t r e i s a p p r o x i m a t e l y 5 g a u s s / A . T h e s e c o i l s a r e e x p e c t e d t o r e d u c e t h e e a r t h ' s f i e l d , t o l e s s t h a n .01 g a u s s ; i . e . by a t l e a s t a f a c t o r o f 50. The i n h o g e n e i t i e s n e a r t h e c e n t r e a r e o f _ i | 3 - ' RG8-U TRANSMISSION LINE 5 CM U.H.F CONNECTOR CAPACLTOR PLATES -SLIDING CROSSBAR F i g u r e 12 - r - M a t c h e d R e s o n a n t C i r c u i t - u n - it o r d e r (̂ -) where r i s t h e d i s p l a c e m e n t f r o m t h e c e n t r e and n • r h 1 R i s t h e c o i l r a d i u s . F o r o u r 5 cm d i s c h a r g e c e l l , (̂ -) ~ (if) o r a b o u t 1 p a r t p e r 2 5 0 . The c o i l s w h i c h p r o v i d e t h e a p p l i e d m a g n e t i c f i e l d must h a v e t h e same a b s o l u t e h o m o g e n e i t y b u t t h i s t i m e i n a t o t a l f i e l d o f 10 g a u s s ; i . e . t h e y must p r o v i d e f o r i n - h o m o g e n e i t i e s < 1 0 - 3 . T h e r e f o r e l a r g e r c o i l s w i t h a d i a m e t e r o f 3 7 cm a n d s p a c i n g o f 1 8 . 5 cm w e r e u s e d . The i n h o m o g e n e - i t i e s i n c u r r e d w i t h t h e s e c o i l s s h o u l d be r o u g h l y one p a r t i n 4><10 3 o v e r t h e d i s c h a r g e r e g i o n . T h e s e c o i l s e a c h h a d 1 0 0 t u r n s o f # 1 8 c o p p e r w i r e . The c o i l s w e r e u s e d i n p a r a l l e l a n d i n t h i s c o n f i g u r a t i o n t h e f i e l d p r o d u c e d a t t h e c e n t r e i s a p p r o x i m a t e l y 2 . 4 g a u s s / a m p e r e . T h e r e was no n e e d t o c a l i b r a t e t h e s m a l l e r c o i l s as i t i s o n l y n e c e s s a r y t o a d j u s t t h e c u r r e n t u n t i l a z e r o f i e l d was r e a c h e d . The v e r t i c a l component o f t h e e a r t h ' s f i e l d w i t h i n t h e H e l m h o l t z c o i l s was z e r o e d u s i n g a r o t a t - i n g c o i l w i t h i t s a x i s o f r o t a t i o n a l o n g t h e h o r i z o n t a l c o mponent o f t h e e a r t h ' s m a g n e t i c f i e l d . F i e l d s down t o a p p r o x i m a t e l y . 0 1 g a u s s c o u l d be d e t e c t e d t h i s way. I t was l a t e r f o u n d t h a t a d i p n e e d l e c a n be u s e d t o t h i s e f f e c t w i t h a b o u t t h e same a c c u r a c y . The l a r g e r H e l m h o l t z c o i l s were c a l i b r a t e d i n t e r m s o f v o l t a g e a c r o s s t h e c o i l s as a f u n c t i o n o f f i e l d w i t h i n . The f i e l d m e a s u r e m e n t s were made f i r s t w i t h a F I L . T I 1 5 AA/V^- 2N1132 r 4 7 0 2 5 0 ^ 2 5 0 2 N 3 7 0 2 2 N 3 7 0 2 : 1 K -o U1 A L L RESISTORS 1 WATT CAPACITORS 10 VOLT F i g u r e 13 - H e l m h o l t z C o i l P ower S u p p l y - 46 - " B e l l 240" H a l l probe gaussmeter. Because of the somewhat e r r a t i c behavior of t h i s gaussmeter the measurements were checked w i t h a "Magnion FFC-4" r o t a t i n g c o i l magnetometer. In order to save c a l i b r a t i n g the x-channel of the x-y r e c o r d e r , the p o s i t i o n of the pen was measured as a f u n c t i o n of f i e l d . With the recorder on the 2V/in s c a l e (which was always used i n i t s c a l i b r a t e d mode) the f i e l d i n the d i s - charge r e g i o n was found to be 3.60±.08 gauss per i n c h of pen movement from the centre. The c o i l s were u s u a l l y run w i t h a 1ft r e s i s t o r i n s e r i e s . The f i e l d produced was then 2.00±.04 gauss/in. §3.7 Current Supplies f o r Helmholtz C o i l s The current f o r the earth's f i e l d c a n c e l l i n g c o i l s i s provided by a s m a l l power supply d e l i v e r i n g up to I V and 150mA i n t o the c o i l s . I t s schematic i s shown i n FIG.13. At 0.65V C-llOmA) the r i p p l e i s approximately lmV and the D.C. d r i f t i s l e s s than lOmV. The current d e l i v e r e d to the l a r g e r Helmholtz c o i l s i s s u p p l i e d by a p a i r of "EICO 1064" po;«;er s u p p l i e s and r e g u l a t e d by a D.C. power a m p l i f i e r 2 8 whose schematic i s shown i n FIG.14. The a m p l i f i e r d e l i v e r s up to 7A of e i t h e r p o l a r i t y i n t o a 1ft load w i t h reasonable l i n e a r i t y ; has a voltage gain of approximately u n i t y and an input impedance of approximately lOKft. The a m p l i f i e r i s c o n t r o l l e d by the voltage s u p p l i e d by a mechanically swept -10V to •27K • 2 N 2 7 8 ^ 2N3792 A l l r e s i s t o r s 1 W a t t ^=-12V 2 2 0 22 LOAD WsAiVWMVsAAA 22 12V 3 : 2N3792 —5 2 2 0 ' 2N278/1 27K< o INPUT o ~ Figure- L'l - D . O . P©w§r A m p l i f i e r I K w w 4^1000/ 4 0 A / W V IK O 0 A M P L I F I E R CONTROL VOLTAGE OUT REVE R S I B L E 10 rpm MOTOR 'O-RING -10V t o +10V V o l t a g e Sweep Mechanism - 1J9 - .+10V s o u r c e shown i n F I G . 1 5 . The sweep t i m e i s a b o u t 7 . m i n u t e s . .- §3.8 L o c k - i n A m p l i f i e r The l o c k - i n a m p l i f i e r c o n s i s t s o f a t u n e d p r e - a m p l i f i e r w i t h a Q o f a b o u t 10 and a p h a s e s e n s i t i v e d e t e c t o r . The l o c k - i n a m p l i f i e r u s e d i n t h i s e x p e r i m e n t , P r i n c e t o n A p p l i e d R e s e a r c h M o d e l 1 2 0 , h a s a l i n e a r i t y o f 1% and a g a i n o f 10 1*. The o u t p u t i s D.C. ±10V a t f u l l s c a l e . I n t h e m o d e . i t was u s e d , i t s u p p l i e s i t s own s i n u s o i d a l r e f e r e n c e s i g n a l t r i g g e r e d by a n e x t e r n a l l y s u p p l i e d r e f e r e n c e s i g n a l . I n t h i s e x p e r i m e n t a 3 s e c o n d t i m e c o n s t a n t was u s e d t h r o u g h - o u t . The f u n c t i o n o f t h e p h a s e s e n s a t i v e d e t e c t o r i s e a s i l y u n d e r s t o o d . E s s e n t i a l l y , i f g i v e n an i n p u t s i g n a l V ( t ) i t p r o d u c e s a s i g n a l „ T/2TTV 2 ( n + l ) 7 r s a ^ I / V ( t ) c o s ( v t - c ) ) ) d ( v t ) n=0 2mr w h e r e cj> i s a s e l e c t e d p h a s e a n g l e , v i s t h e t u n e d f r e q u e n - c y , a n d x i s t h e t i m e c o n s t a n t . So l o n g as T i s s u f f i c i e n t - l y l a r g e , i n c o h e r e n t s i g n a l s w i l l a v e r a g e t o z e r o . We c a n now s e e w hat t h e l o c k - i n a m p l i f i e r d o e s t o o u r s i g n a l d e r i v e d i n e q . ( l l ) § 3-3- 2TT • S « / c o s 2 ( c o t - 6 ) c o s ( v t - ( J ) ) d ( v t ) C h o o s i n g v = 2w and cj> = 0, - 50 - 2TT S <* / c o s 2 ( u ) t - e )cos2cot d ( 2 w t ) o • <* c o s 2 0 = c o s 2 6 - s i n 2 6 = P I n o t h e r w o r d s , t h e l o c k - i n a m p l i f i e r o u t p u t s i g n a l i s j u s t p r o p o r t i o n a l t o t h e p o l a r i z a t i o n o f t h e e m i t t e d l i g h t r e f e r r e d t o the. x - a x i s . • §3-9 The R o t a t i n g P o l a r o i d The p o l a r o i d r o t a t o r i s shown i n P I G . 1 6 . The• p o l a r o i d i s g l u e d t o a 2" I.D. b r a s s p i p e w h i c h I s f i t t e d t i g h t l y i n s i d e a l a r g e b a l l - b e a r i n g . A s e w i n g m a c h i n g b e l t l a i d o v e r t h e p i p e and t h e m o t o r p u l l e y r o t a t e s t h e p o l a r o i d . The m o t o r , m o d e l CA3GRH, U n i v e r s a l E l e c t r i c Co., r u n s a t • 1050 r.p.m. a n d d e l i v e r s TJ-Q H.P. The. m o t o r p u l l e y h a s a 2" d i a m e t e r a nd a b e l t g r o o v e o f l j - ^ - i n c h d i a m e t e r . W i t h t h i s a r r a n g e m e n t t h e p o l a r o i d i s r o t a t e d a t a p p r o x i m a t e l y 14 c y c l e s / s e c o n d . The l i g h t t o be. " c h o p p e d " p a s s e s t h r o u g h t h e • c e n t r e o f t h e p i p e . Two q u a r t e r s e g m e n t s o f t h e r i m o f t h e p o l a r o i d a r e p a i n t e d b l a c k t o i n t e r r u p t l i g h t f r o m a s m a l l lamp b e h i n d t h e p o l a r o i d p e r i o d i c a l l y as t h e p o l a r o i d t u r n s . L i g h t f r o m t h e lamp f a l l s on a p h o t o d i o d e , p l a c e d i n f r o n t o f t h e p o l a r o i d d i s k . §3.10 The V a r i a b l e Q u a r t e r - W a v e P l a t e The q u a r t e r - w a v e p l a t e u s e d i s b a s e d on one d e s c r i b e d by H a p p e r and S a l o m a n 2 9 . F i g u r e 16 - P o l a r o i d R o t a t o r - 52 - - When f u s e d q u a r t z i s s t r e s s e d i t becomes b i r e - f r i n g e n t w i t h i t s o p t i c a l a x i s a l o n g t h e d i r e c t i o n o f s t r a i n . The r e l a t i v e r e t a r d a t i o n between a wave w i t h e l e c t r i c v e c t o r , p a r a l l e l - and one w i t h i t s e l e c t r i c v e c t o r p e r p e n - d i c u l a r t o t h e d i r e c t i o n o f s t r a i n i n c r e a s e s w i t h t h e s t r e s s a p p l i e d . T h i s e f f e c t i s made use o f t o c o n s t r u c t t h e v a r i a b l e w a v e l e n g t h q u a r t e r - w a v e p l a t e shown i n FIG.17. The m a j o r components a r e a p i e c e o f f u s e d q u a r t z o n 1" x l " x -̂g. g r o u n d f l a t and p a r a l l e l on two o p p o s i n g e d g e s , a b r a s s c a s e , and a s t e e l p r e s s u r e p l a t e t o d i s - 1" t r i b u t e t h e f o r c e f r o m a N.F. t h r e a d s c r e w o v e r t h e g r o u n d f a c e o f t h e q u a r t z p l a t e . I n an e f f o r t t o r e d u c e t h e p r e s s u r e i n h o m o g e n e i t i e s due t o i r r e g u l a r i t i e s o f t h e c o n t a c t s u r f a c e s ; t h e s u r f a c e s o f t h e s t e e l p l a t e , t h e q u a r t z p l a t e , and t h e i n s i d e b o t t o m s u r f a c e o f t h e c a s e 1 " 3 " a r e s p a c e d by 2 p i e c e s o f -̂g x x 1" t e f l o n . Eye i n s p e c t i o n w i t h c r o s s e d p o l a r o i d s w h i l e t h e s c r e w i s t i g h t e n e d t o s t r a i n t h e q u a r t z shows t h a t e x c e p t a t t h e c o r n e r s a f a i r l y homogeneous q u a r t e r - w a v e p l a t e i s p r o - d u c e d . The q u a r t e r - w a v e p l a t e i s a d j u s t e d by t i g h t e n i n g t h e s c r e w u n t i l u n p o l a r i z e d l i g h t f r o m t h e d i s c h a r g e p r o d u c e s a z e r o o u t p u t s i g n a l f r o m t h e l o c k - i n a m p l i f i e r . F U L L S C A L E F i g u r e 17 - Quarter-Wave P l a t e - 54 - §3 . 1 1 P h o t o m u l t i p l i e r The p h o t o m u l t i p l i e r . u s e d f o r t h i s e x p e r i m e n t i s an E.M.I. 9558QB. I t h a s an S-20 (NaKSbCs) s u r f a c e . o The q u a n t u m e f f i c i e n c y a t 4 9 0 0 A i s s a i d by t h e m a n u f a c - t u r e r t o be -23%. I t was o p e r a t e d w i t h a c a t h o d e t o anode p o t e n t i a l o f - 1 2 8 0 V . The dyn o d e c h a i n r e s i s t o r s w e r e a l l 3 3 K Q w h i l e t h e c a t h o d e t o f i r s t d y node p o t e n t i a l i s m a i n - t a i n e d a t - 1 5 0 V by a z e n e r d i o d e . The anode i s c o n n e c t e d t o g r o u n d by a lOOKft r e s i s t o r . The a b b r e v i a t e d c i r c u i t i s shown b e l o w i n P I G . 1 8 . ANODE CATHODE D i D 2 D 100K 150V AAAA/V1 100K 6 . V = - 1 2 8 0 V 3 D AAAAÂAA/VXÂV--- 33K 3 3 K 11 100K • - v V A A A M 3 3 K D. i s t h e j ' t h d y n o d e J A l l r e s i s t o r s 1 W a t t F i g u r e 1 8 - P h o t o m u l t i p l i e r W i r i n g S c h e m a t i c §3.12 X-Y R e c o r d e r The x-y r e c o r d e r u s e d was a V a r i a n m o d e l F100 h a v i n g a l i n e a r i t y o f 1% and i n p u t i m p e d a n c e o f lOOKft i n t o e a c h c h a n n e l . - 55 - §3-13 L i q u i d N i t r o g e n B a t h The R ( 0 ) , R ( l ) , and.R(2) l i n e s o f t h e (0,0) b a n d o f t h e 3d xE 2p1TI t r a n s i t i o n a r e s e p a r a t e d by a b o u t o- 3A w h i c h i s e a s i l y r e s o l v e d by t h e m o n o c h r o m a t o r w i t h f a i r l y w i d e s l i t s . The R(4) l i n e o f t h e same e l e c t r o n i c , v i b r a t i o n a l t r a n s i t i o n h o w e v e r f a l l s a l m o s t on t h e R ( l ) l i n e a nd has a b o u t t h e same i n t e n s i t y . I t was f o u n d t h a t c o o l i n g t h e d i s c h a r g e e l i m i n a t e d t h e l i n e a l m o s t c o m p l e t e l y . T h i s i s n o t s u r p r i s i n g s i n c e t h e R(4) t r a n s i t i o n a r i s e s f r o m t h e J ' = 5 l e v e l o f t h e u p p e r s t a t e . U n d e r t h e s e l e c t i o n r u l e d e r i v e d i n t h e t h e o r y s e c t i o n t h i s l e v e l i s p o p u l a t e d f r o m t h e J = 3 l e v e l o f t h e g r o u n d s t a t e . Thus t h e l i n e s t r e n g t h o f R(4) w o u l d be e x p e c t e d t o be p r o p o r t i o n a l t o t h e p o p u l a t i o n o f t h e g r o u n d s t a t e J = 3 l e v e l . R e f e r r i n g t o T a b l e I , §2.4, we s e e t h a t w h e r e a s J = 3 h a s a n a p p r e c i a b l e p o p u l a t i o n a t room t e m p e r a t u r e ( c o m p a r e d w i t h t h e J = 0 and J = 2 f r o m w h i c h R ( l ) a r i s e s ) , a t l i q u i d n i t r o g e n t e m p e r a t u r e s i t s p o p u l a t i o n i s n e g l i g i b l e . To l o w e r t h e t e m p e r a t u r e o f t h e d i s c h a r g e , t h e d i s c h a r g e c e l l i s p l a c e d i n s i d e a l a r g e vacuum Dewar f i t t e d w i t h a 2 i n c h d i a m e t e r f l a t window on t h e f r o n t o u t s i d e s i d e , and t h e Dewar I s f i l l e d w i t h l i q u i d n i t r o g e n . _ The l i q u i d n i t r o g e n l e v e l d e c r e a s e d a t a r a t e o f r o u g h l y ^ i n c h p e r h o u r and was o b s e r v e d t o b o i l a t t h e s u r f a c e o f t h e d i s c h a r g e c e l l o n l y when t h e H 2 p r e s s u r e i n s i d e e x c e e d e d 200u. - 56 - §3-14 D a t a P r o c e s s i n g The g r a p h s p l o t t e d by t h e x-y r e c o r d e r ( s e e FIGs.19 a n d 21) w e r e s u b j e c t e d t o n u m e r i c a l p r o c e s s i n g t o e x t r a c t t h e h a l f w i d t h , H i / 2 . R e l a t i v e v a l u e s o f t h e p o l a r i z a t i o n w e r e r e a d f r o m t h e g r a p h s a t 0.2 i n c h i n t e r - v a l s t o p r o v i d e 39 d a t a p o i n t s . The p o i n t s w e re t h e n u s e d t o f i t a f u n c t i o n o f t h e f o r m : P = C i + = 7 - ^ [ c o s 2 6 - =^(H-H 0 ) s i n 2 6 ] 1 + ^ ( H - H 0 ) 2 n 1/2 w h e r e C i , C 2 , H i / 2 , . H D and 0 a r e p a r a m e t e r s f i t t e d by a c o m p u t e r " l e a s t s q u a r e s " f i t t i n g r o u t i n e (U.B.C. L . Q . F . ) . The f i t t e d c u r v e s a r e shown i n F I G s . 2 0 and 22. A t e a c h o f t h e h y d r o g e n p r e s s u r e s u s e d , 6 t o 8 ( d e p e n d i n g on s i g n a l t o n o i s e ) g r a p h s w e r e p r o d u c e d and i n d e p e n d e n t l y f i t t e d by ( 1 2 ) . The a v e r a g e H i / 2 was t h e n c o m p u t e d f o r t h a t p r e s s u r e , p l o t t e d as a f u n c t i o n o f p r e s s u r e and e x t r a p o l - a t e d t o z e r o p r e s s u r e ( s e e FIGs.23 and 2k). From t h e s e H i / 2 v . s . p r e s s u r e g r a p h s t h e r a d i a t i v e l i f e t i m e a n d c r o s s - s e c t i o n s a r e c o m p u t e d as w i l l be s e e n i n t h e n e x t c h a p t e r . - 57 - CHAPTER I V EXPERIMENTAL RESULTS §4.1 L i f e - t i m e s A t y p i c a l e x p e r i m e n t a l p l o t o f t h e p o l a r i z a t i o n c u r v e o b t a i n e d , u s i n g t h e 450 MHz d i s c h a r g e , i s shown i n FIG.19. A s i m i l a r g r a p h i s shown I n FIG.21 f o r t h e d i s c h a r g e e x c i t e d by t h e 180 MHz R.F. f i e l d . T h e i r r e s p e c t i v e " l e a s t s q u a r e s " f i t t e d c u r v e s a r e shown i n FIG.20 and 22. The p o l a r i z a t i o n s c a l e i s a r b i t r a r y and n o r m a l i z e d t o 0.9 on t h e c o m p u t e r g e n e r a t e d p l o t s . The h a l f w i d t h s o f t h e c u r v e s o b t a i n e d a t t h e v a r i o u s p r e s s u r e s a r e t a b u l a t e d i n T a b l e I I I . I n f i g u r e s 23 and 24 t h e a v e r a g e h a l f w i d t h s , H y 2 , o f t h e c u r v e s a r e shown as a f u n c t i o n o f p r e s s u r e i n t h e vacuum s y s t e m . The e r r o r b a r s i n H]/ 2 r e p r e s e n t s t a t i s t i c a l e r r o r s o n l y , w h i l e t h e e r r o r b a r s i n p r e s s u r e r e p r e s e n t t h e maximum e r r o r i n r e a d i n g t h e McLeod g u a g e . The h a l f w i d t h o f a l i n e a t z e r o p r e s s u r e may be c o n v e r t e d t o t h e u p p e r - s t a t e ' s l i f e t i m e , u s i n g : T T — ~ ^ — F j — , u 0 i s t h e B o h r m a g n e t o n . <J 2 g j U 0 h ] / 2 The e x t r a p o l a t e d h a l f w i d t h s and l i f e t i m e s l e v e l s o f t h e J = l , .2, a n d 3 o f t h e v=0 s t a t e a r e t a b u l a t e d i n t a b l e I V . The g f a c t o r s u s e d a r e t h o s e g i v e n by D i e k e 3 0 . POLARIZATION MAGNETIC F I E L D J | - i 1 : ( : 1 1 I 1 1 -8.0 -6.0 -4.0 -2.0 2.0 4.0;: 6.0 8.0 GAUSS F i g u r e 19 - E x p e r i m e n t a l L e v e l - C r o s s i n g C u r v e f o r t h e R ( 0 ) L i n e U s i n g 450 MHz E x c i t a t i o n - 59 - F i g u r e 20 - L e a s t S q u a r e s F i t t e d C u r v e f o r t h e R(0) L i n e U s i n g 450 MHz E x c i t a t i o n F i g u r e 21 - E x p e r i m e n t a l L e v e l - C r o s s i n g C u r v e f o r t h e R ( 0 ) L i n e U s i n g 180 MHz E x c i t a t i o n - 61 - F i g u r e 2 3 - L e v e l C r o s s i n g C u r v e H a l f w i d t h as a F u n c t i o n o f P r e s s u r e 24 - L e v e l C r o s s i n g C u r v e H a l f w i d t h as a F u n c t i o n o f P r e s s u r e - 64 - J g J Hl/2 3 0 0 ° ( g a u s s ) K 180MHz Hi/2 ( g a u s s ) 8 0°K 450 MHz' x( 10 ' 3 0 0°K 8 s e c ) 180MHz _ 8 T( 10 s e c ) 8 0°K 450MHz 1 . 9 0 1 2 . 4 7±.l 2 . 2 7 ± . l 2 . 55 2 . 7 8 2 . 5 7 1 2 . 6 0 ± . 0 8 3 . 8 3 . 3 .445 3 . 4 l ± . l - 3 . 1 0 + . 2 3 . 74 4.12 T a b l e I V - E x t r a p o l a t e d H a l f w i d t h s a n d L i f e t i m e s The h a l f w i d t h s o b t a i n e d f o r e a c h o f t h e i n d i v i d u a l r u n s i s c o n t a i n e d i n T a b l e V. . . • § 4 . 2 C o l l i s i o n C r o s s - s e c t i o n s F r o m t h e s l o p e o f t h e H1/2 v . s . p r e s s u r e g r a p h s , a r e a s o n a b l e e s t i m a t e o f t h e c o l l i s i o n c r o s s - s e c t i o n s may be o b t a i n e d i f one makes some p l a u s i b l e a s s u m p t i o n s a b o u t c o n d i t i o n s i n t h e d i s c h a r g e . ' : ' We assume t h a t t h e number o f f r e e e l e c t r o n s i n t h e d i s c h a r g e i s s m a l l c o m p a r e d t o t h e number o f n e u t r a l m o l e c u l e s , a n d we assume t h a t t h e number o f e x c i t e d m o l e c u l e s i s s m a l l c o m p a r e d t o t h e number o f g r o u n d s t a t e m o l e c u l e s ; t h e n any c o l l i s i o n a n e x c i t e d m o l e c u l e s u f f e r s w i l l be w i t h one i n t h e g r o u n d s t a t e . The e q u a t i o n f o r t h e u p p e r s t a t e p o p u l a t i o n , N*, f o r m o l e c u l e s e x c i t e d a t t i m e t = 0 , may t h e n be w r i t t e n i n t e r m s o f t h e two c o m p e t i n g d e c a y p r o c e s s e s , ~ * = -r 0N*-avN*N ( 1 3 ) - 65 - R (0) 450 a v e r a g e R ( l ) 450 a v e r a g e R (2) 450 a v e r a g e MHz • 90y 3.43194 3.40270 • 3.43443 3.48059 3.437±.l8 MHz 4.61027 4.91537 4.62872 4.52513 • 4.84251 4.81395 4.723±.07 MHz 6.10883 5.54281 6.97308 5.98210 6.74916 5.86058 6.203±.25 70y ' 3.14845 3..25614-- 3.15160 3.20694 3.'33747 3.23944 3.2.23±.031 4.41363 4.26093 4.38411 4.39620 4.12871 4.317±.o6 5.41178 5.91381 5.31858 5.85825 5.61430 5.9H32 5.676+.13 50y 2.71100 3.17868 2.92867 2.98884 3.02103 2.82810 2.943±•07 3.93656 3.76604 3.74870 3.45032 4 .00709 3.96007 3.811+.09 5.18763 4.72888 4.94178 4.80084 4.915±.11 35y 2.75137 2.69689 2.74707 2.68692 2.74468 2.75787 2.731±.0l4 3.44569 3.41708 3.32601 3.38945 • 3.53151 3.39896 3.4l8±.03 4.43328 4.54597- 4.12689 4.17776 4.26244 4.34724 4.3l6±.07 T a b l e V - P o l a r i z a t i o n C u r v e s ' H a l f w i d t h s - 6 6 - 'R(O) 1 8 0 MHz • I 5 y 2 . 6 0 2 6 0 2 . 8 7 5 0 2 - 2 . 5 1 1 4 6 2.64745 2 . 5 7 6 8 8 2.67594 a v e r a g e 2.648±.056 R ( 2 ) 1 8 0 MHz 3 - 7 4 8 5 2 3 . 8 7 7 7 5 3.64140 • 3 . 5 7 2 0 0 3 . 9 6 4 5 9 a v e r a g e 3 . 7 6 l ±.1 3 3 y 2 . 8 5 0 3 8 2 . 8 8 3 0 0 2 . 7 0 5 7 - 2 2 . 7 9 0 1 4 2 . 8 3 6 6 6 2 . 8 l 3 ± . 0 3 5 4 . 3 3 1 8 5 4 . 0 8 8 1 1 . 4 . 1 2 4 3 1 4 . 2 2 2 2 6 4 . 1 9 2 ± . 0 6 5 5 0 y 2 . 9 6 8 1 8 3 . 0 3 0 9 8 2 . 9 0 8 2 1 2 . 9 5 1 3 6 2 . 9 8 5 1 7 2 . 9 6 9 ± . 0 2 2 4 . 3 3 1 8 3 4 . 3 5 5 6 5 4 . 4 3 8 9 7 4 . 3 1 5 9 5 4 . 6 5 7 7 3 4.420±.07 T a b l e V ( c o n t i n u e d ) - P o l a r i z a t i o n C u r v e s ' H a l f w i d t h s - 67 - w h e r e T0 i s t h e r a d i a t i v e t r a n s i t i o n p r o b a b i l i t y , o i s t h e c o l l i s i o n c r o s s - s e c t i o n , v i s t h e r e l a t i v e v e l o c i t y o f c o l l i d i n g m o l e c u l e s , a n d N i s t h e d e n s i t y o f n e u t r a l m o l e c u l e s . S o l v i n g e q . ( 1 3 ) we o b t a i n •. N * ( t ) = N 0 e - r ^ - 0 v N t = N 0 e - r ( N ) t t h u s t h e i n v e r s e l i f e t i m e ' • ̂  = r(N) = r Q+avN •then dr dr/dN , n , . v - T x r = a v o r , a = — ( 1 4 ) dN 5 . v / J T J f r o m o u r g r a p h s we h a v e ^ 2 , w h i c h may be r e l a t e d t o d f by t h e f o l l o w i n g s u b s t i t u t i o n s r = 2 g ju DH N K N 0 P T 0 T w h e r e N D i s A v o g a d r o ' s n u m b e r / M o l a r v o l u m e P i s t h e p r e s s u r e i n s t a n d a r d a t m o s p h e r e s T 0 = 29.7°K T i s t h e a b s o l u t e t e m p e r a t u r e o f t h e gas i n °K t h e n dT_ = 2 s J y ° T dHi/ 2 dN N 0 T 0 dp o r more c o n v e n i e n t l y = n Q 8 x l Q - 7 c m 3 y 1 E d H ^ 2 dN g a u s s s e c T D & J dp - 6 8 - f i n a l l y , s u b s t i t u t i n g t h i s i n t o e q . ( l 4 ) A = 4 . 9 8 x i O - 7 J 5 l ^ — ^ T D v dp g a u s s s e c wh e r e H i / 2 i s m e a s u r e d i n g a u s s a n d P i s m e a s u r e d i n m i c r o n s ( = 10" 3mm H g ) . The v a l u e s o b t a i n e d f o r t h e c o l l i s i o n c r o s s - s e c t i o n , a s s u m i n g T = 3 0 0°K-in t h e room t e m p e r a t u r e d i s c h a r g e a n d 8 0°K i n t h e l i q u i d n i t r o g e n c o o l e d d i s c h a r g e , a n d a s s u m i n g B o l t z m a n n v e l o c i t i e s , a r e l i s t e d i n T a b l e V I . I t i s a l s o a s s u m e d i n c a l c u l a t i n g t h e s e c r o s s - s e c t i o n s t h a t t h e p r e s s u r e i n t h e d i s c h a r g e c e l l i s t h a t m e a s u r e d by t h e McLeod gauge. J a T = 3 0 0 ° K a T=80°-K 1 1 5 7 A 2 1 0 2 A 2 2 1 1 6 A 2 3 1 6 7 A 2 0 0 1 3 1 A T a b l e V I - C o l l i s i o n C r o s s - s e c t i o n s § 4 . 3 P o l a r i z a t i o n The a b s o l u t e p o l a r i z a t i o n o f t h e l i g h t o b s e r v e d a t z e r o m a g n e t i c f i e l d was m e a s u r e d by p l a c i n g a s t a t i o n a r y p o l a r o i d i n t h e l i g h t beam a nd c o m p a r i n g t h e l o c k - i n a m p l i f i e r o u t p u t , S Q , w i t h t h a t o b t a i n e d w i t h o u t t h e s e c o n d p o l a r o i d , S i . The p o l a r i z a t i o n P o f t h e . l i g h t i s t h e n o b t a i n e d f r o m : - 69 - P = , w h e r e T i s t h e p o l a r o i d t r a n s m i t t a n c e . ..Under t h e b e s t c o n d i t i o n s , t h e p o l a r i z a t i o n s t h u s f o u n d w ere a t 20u p r e s s u r e o f o r d e r 10%, 10%, and 5% f o r t h e R ( 0 ) , R ( l ) , and R ( 2 ) l i n e s r e s p e c t i v e l y . P v a r i e s w i d e l y w i t h t h e c o n d i t i o n s o f t h e d i s c h a r g e . Q u a l i t a l l v e l y , t h e p o l a r i z - a t i o n d e c r e a s e s w i t h i n c r e a s i n g p r e s s u r e , a t lOOu i t i s r o u g h l y h a l f o f t h a t o b s e r v e d a t 50u. I m p u r i t i e s i n t h e d i s c h a r g e a l s o d e c r e a s e t h e p o l a r i z a t i o n c o n s i d e r a b l y . §4.4 U p p e r S t a t e P o p u l a t i o n s I f t h e e l e c t r o n i c - v i b r a t i o n a l wave f u n c t i o n o f a m o l e c u l e d o e s n o t v a r y t o o d r a s t i c a l l y w i t h i n c r e a s i n g r o t a t i o n , t h e r e l a t i v e p o p u l a t i o n s o f t h e r o t a t i o n a l l e v e l s o f t h e e x c i t e d s t a t e may be c o m p u t e d f r o m t h e r e l a t i v e i n t e n s i t y o f t r a n s i t i o n s t o t h e l o w e r s t a t e . The r e l a t i v e p o p u l a t i o n N j , o f t h e s t a t e J ' i s t h e n p r o p o r t i o n a l t o t h e sum o f t h e i n t e n s i t i e s o f t h e R, P, a n d Q t r a n s i t i o n s a r i s i n g f r o m t h e same u p p e r s t a t e . D e n o t i n g t h e i n t e n s i t y o f t h e t r a n s i t i o n A ( J ) (A r e p r e s e n t s R, P, o r Q) by I ^ j ^ , N J ' " ̂ R C J ' - l ) + I Q ( J ' ) + I P ( J ' + 1 ) A s s u m i n g a c o n s t a n t s p e c t r a l r e s p o n s e o f t h e p h o t o m u l t i p l i e r o a n d c o n s t a n t m o n o c h r o m a t o r e f f i c i e n c y o v e r t h e ~100A s c a n n e d , t h e r e l a t i v e i n t e n s i t i e s o f t h e r e l e v a n t t r a n s i t i o n s were m e a s u r e d t o -.10% a c c u r a c y a n d t h e p o p u l a t i o n s N j , c o m p u t e d . Q ( J ) i s c o m p l e t e l y a b s e n t f r o m t h e b a n d s h e n c e t h e sum r e d u c e s t o : - 7 0 - N J ' w I R ( J ' - 1 ) + I P ( J ' + 1 ) The numbers N T , e x p e r i m e n t a l l y o b t a i n e d n o r m a l i z e d s o t h a t t h e i r sum e q u a l s u n i t y a r e l i s t e d i n T a b l e V I I . The numbers i n t h e c o l u m n h e a d e d by 300°K p e r t a i n t o t h e d i s c h a r g e a t room t e m p e r a t u r e w h i l e t h o s e h e a d e d by 80°K p e r t a i n t o t h e d i s - c h a r g e i m m e r s e d i n l i q u i d n i t r o g e n . C o m p a r i n g t h e s e p o p - u l a t i o n s t o t h o s e o f t h e g r o u n d s t a t e g i v e n i n T a b l e I (§2.4) s u g g e s t s t h a t t h e s e l e c t i o n r u l e AJ = 0,±2 d e r i v e d i n c h a p t e r I I i s f a i r l y w e l l o b e y e d . J N J ' N J . 32.5°K 80 °K 0 . 0 1 7 * . 0 0 7 * 1 .241 . 2 8 1 2 . 1 2 8 . 1 9 3 3 . 3 5 8 . 4 0 6 . 4 . 0 7 2 * . 0 3 1 * 5 . 1 2 7 . 0 5 2 T a b l e V I I - E x p e r i m e n t a l U p p e r S t a t e P o p u l a t i o n s P ( l ) a n d R ( 3 ) can. n o t be r e s o l v e d by o u r a p p a r a t u s . The v a l u e s g i v e n assume t h a t J'=4 h a s a t l e a s t h a l f t h e p o p u l a t i o n o f J ' = 5 - P r o b a b l y t h e p o p u l a t i o n o f J ' = 0 i s l e s s t h a n t h a t s t a t e d , w h i l e t h e p o p u l a t i o n o f J'=4 i s g r e a t e r . - 71 - 5 E x p e r i m e n t a l E r r o r s D i s c h a r g e S t a b i l i t y The most s e r i o u s e x p e r i m e n t a l s h o r t c o m i n g i s t h e s e n s i t i v i t y o f t h e d i s c h a r g e t o t h e m a g n e t i c f i e l d a p p l i e d . An up t o 10% d e c r e a s e i n i n t e n s i t y c o u l d be o b s e r v e d a t f i e l d s o f a b o u t ±15 g a u s s . The e f f e c t was n o t r e p o r d u c i b l e e n o u g h t o p e r m i t any g a i n i n t a k i n g a c c o u n t o f i t i n t h e c u r v e f i t t i n g s . As m i g h t be e x p e c t - ed t h e e f f e c t was w o r s t a t t h e l o w e s t p r e a s u r e s , w h e r e t h e d i s c h a r g e was somewhat u n s t a b l e e v e n w i t h o u t any m a g n e t i c f i e l d . A s s u m i n g t h a t t h e i n t e n s i t y o f l i g h t e m i t t e d v a r i e s as I 0 ( 1 - C 2 H 2 ) w i t h C ~ 0(10-2gauss~1 ) , t h e a p p a r e n t p o l a r i z a t i o n c u r v e t h e n p r o d u c e d i s o f t h e f o r m ( I - C 2 H 2 ) 1 + ( ^ H T ) , a n d f o r -10 g a u s s <H<_ +10 g a u s s t h e L o r e n t z i a n f i t t e d h a s a h a l f w i d t h o n l y a f r a c t i o n o f a p e r c e n t s m a l l e r t h a n t h a t w i t h o u t t h e f a c t o r . The o p t i c a l s u r f a c e s , h o w e v e r , i n t r o d u c e d a p o l a r i z a t i o n o f ~3% i n t o t h e u n p o l a r i z e d component o f t h e l i g h t . T h e n , a s s u m i n g t h e same f i e l d d e p e n d e n c e o f t h e i n t e n s i t y , t h e s i g n a l p r o d u c e d by t h e l o c k - i n a m p l i f i e r h a s t h e f o r m P o ( 1 - C 2 H 2 ) " d I ° + i + ( 2 y H T ) * s o t h a t t h e r e s u l t a n t s i g n a l h a s a f i t t e d c u r v e somewhat b r o a d e r t h a n t h e t r u e L o r e n t z i a n . The e f f e c t i s e s t i m a t e d - 72 - t o . p r o d u c e c h a n g e s i n t h e a p p a r e n t h a l f w i d t h n o t e x c e e d - i n g 1% f o r o u r c u r v e s . b) M a g n e t i c F i e l d The m a g n e t i c f i e l d c a l i b r a t i o n e r r o r s may be d e v i d - e d i n t o n o n - l i n e a r i t i e s a n d m i s c a l i b r a t i o n . The m a g n e t i c f i e l d i n t h e d i s c h a r g e r e g i o n w i l l i n g e n e r a l n o t be- q u i t e p r o p o r t i o n a l t o t h e v o l t a g e a c r o s s t h e c o i l s . T h i s i s due p a r t l y t o o h m i c h e a t i n g o f t h e c o i l s , a n d p a r t l y t o h y s t e r e s i s e f f e c t s i n n e a r b y f e r r o - m a g n e t i c m a t e r i a l s . B o t h o f t h e s e e f f e c t s s h o u l d h o w e v e r be q u i t e s m a l l i n t h i s e x p e r i m e n t . I n a d d i t i o n , t h e r e s i d u a l e a r t h ' s f i e l d p e r p e n d i c u l a r t o t h e a p p l i e d f i e l d , c a u s e s a s l i g h t n o n - l i n e a r i t y . More s e r i o u s a r e n o n - l i n e a r i t i e s i n t h e x c h a n n e l o f t h e x-y r e c o r d e r , i . e . p e n d i s p l a c e m e n t s a r e n o t q u i t e p r o p o r t i o n a l t o t h e f i e l d . The n o n - l i n e a r i t i e s i n t r o d u c e a maximum e r r o r o f a b o u t '1% i n . I n a d d i t i o n t o ' n o n - l i n e a r i t i e s t h e l i n e a r e r r o r s s u c h as made i n t h e r e a d i n g o f d a t a f r o m g r a p h s r e p r e s e n t a n o t h e r 1% random e r r o r . The g a u s s m e t e r s a v a i l a b l e w e r e o n l y c a p a b l e o f m e a s u r i n g t h e m a g n e t i c f i e l d t o an a c c u r a c y o f .1 t o .2 g a u s s . Thus a 2% s y s t e m a t i c e r r o r - w h i c h d o e s n o t e f f e c t t h e r e l a t i v e l i f e t i m e s o f t h e s t a t e s , b u t d o e s a f f e c t t h e a b s o l u t e l i f e t i m e s - a r i s e s . ~ > I n h o m o g e n e i t i e s o f t h e m a g n e t i c f i e l d i n t h e - 73 - d i s c h a r g e r e g i o n w e re n o t m e a s u r a b l e w i t h o u r g a u s m e t e r and p r o b a b l y do n o t e x c e e d .1 g a u s s l e a d i n g t o n e g l i g i b l e b r o a d e n i n g o f t h e l i n e s h a p e s . The e r r o r s i n t h e m a g n e t i c f i e l d w i l l t h u s c o n t r i b u t e a b o u t 3.5% ' e r r o r i n t h e a b s o l u t e l i f e t i m e s o r 1.5% e r r o r i n t h e r e l a t i v e l i f e t i m e s . c ) P r e s s u r e i n t h e D i s c h a r g e I t s h o u l d be c l e a r f r o m F I G . 1 0 t h a t t h e p r e s s u r e m e a s u r e d . by t h e M c L e o d gauge w i l l d i f f e r f r o m t h e p r e s s u r e i n t h e d i s c h a r g e c e l l . We s t i l l e x p e c t t h e p r e s s u r e s t o be p r o p o r t i o n a l t o t h o s e m e a s u r e d , s o t h a t we i n t r o d u c e no e r r o r s i n t h e e x t r a p o l a t e d l i f e t i m e s . F o r t h e c r o s s - s e c t i o n s , h o w e v e r , t h e p a r t i c l e d e n s i t i e s a r e n e e d e d . P r e s s u r e m e a s u r e m e n t s i n s i d e t h e d i s c h a r g e c e l l w o u l d be d i f f i c u l t a n d a c c u r a t e c a l c u l a t i o n s e q u a l l y d i f f i c u l t . P r e s s u r e s i n t h e d i s c h a r g e c e l l , s h o u l d , j u d g i n g f r o m t h e p r o x i m i t y o f t h e McLeod g a u g e , be no more t h a n 20 o r 30% l e s s t h a n t h o s e measured-. The p r e s s u r e d i f f e r e n c e i s p r o b a b l y g r e a t e s t f o r t h e l i q u i d n i t r o g e n c o o l e d d i s - c h a r g e . d) T e m p e r a t u r e i n t h e D i s c h a r g e The t e m p e r a t u r e o f t h e g a s i n t h e d i s c h a r g e a g a i n a f f e c t s o n l y t h e c a l c u l a t e d c r o s s - s e c t i o n . The mean f r e e p a t h o f g r o u n d s t a t e H 2 a t 50u p r e s s u r e a n d 100°K t e m p e r a t u r e i s o f t h e o r d e r o f t h e s i z e o f t h e d i s c h a r g e - n - c e l l , a n d t h e t i m e b e t w e e n c o l l i s i o n s i s o f o r d e r 1 0 ~ 5 s e c o n d s , h e n c e we e x p e c t t r a n s l a t i o n a l e q u i l i b r i u m t o be r a p i d l y e s t a b l i s h e d w i t h t h e w a l l s o f t h e d i s c h a r g e c e l l . A s s u m i n g a pow e r i n p u t t o t h e c e l l o f 10W, and a t h e r m a l c o n d u c t i v i t y o f p y r e x o f .01 W a t t s c m - 2 c m - 1 / ° K , t h e i n s i d e w a l l s w i l l be a t a t e m p e r a t u r e a b o u t 2°K h i g h e r t h a n t h e o u t s i d e w a l l . S i n c e t h e o u t s i d e w a l l s e x h i b i t no g r e a t t e m p e r a t u r e r i s e when t h e p r e s s u r e i n s i d e i s l e s s t h a n 200 - 300 m i c r o n s , i t i s i n f e r r e d t h a t t h e gas i n t h e d i s c h a r g e w i l l h a v e a t e m p e r a t u r e n o t e x - c e e d i n g t h e a m b i e n t t e m p e r a t u r e by more t h a n 10°K. As t h e c o m p u t e d c r o s s - s e c t i o n v a r i e s as /T, a 10°K e r r o r i n t h e a s s u m e d t e m p e r a t u r e w i l l a f f e c t t h e r e s u l t s o n l y s l i g h t l y . e ) C a s c a d i n g A l t h o u g h t h e r e i s no d i r e c t e v i d e n c e t h a t t h e 3d 1! l e v e l s a r e n o t p o p u l a t e d by r a d i a t i v e t r a n s i t i o n s f r o m h i g h e r e n e r g y s t a t e s , no s u c h t r a n s i t i o n s h a v e e v e r b e e n o b s e r v e d . I t seems l i k e l y t h e r e f o r e t h a t t h e s e t r a n s i t i o n s o c c u r w i t h v e r y much l o w e r p r o b a b i l i t y t h a n e x c i t a t i o n f r o m t h e g r o u n d s t a t e . f) C o h e r e n c e N a r r o w i n g When l i g h t e m i t t e d by one m o l e c u l e i s a b s o r b e d by a n o t h e r b e f o r e l e a v i n g t h e d i s c h a r g e , t h e H a n l e e f f e c t s i g n a l w i l l be " n a r r o w e r " t h a n t h e l i f e t i m e w o u l d i n d i c a t e b e c a u s e t h e c o m p o s i t s y s t e m h a s a l o n g e r l i f e t i m e t h a n - 75 - t h e i n d i v i d u a l m o l e c u l e s . .This phenomenon does n o t o c c u r i n t h e s e . l e v e l s b e c a u s e t h e r e a r e no 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 t o t h e g r o u n d s t a t e , n o r i s t h e r e a m e t a - s t a b l e s t a t e t o w h i c h i t c a n d e c a y . R.F. B r o a d e n i n g The p r e s e n c e o f R.F. f i e l d s w i l l i n g e n e r a l b r o a d e n t h e p o l a r i z a t i o n c u r v e s . I n t h e p r e s e n c e o f a weak mag- n e t i c f i e l d , i t i s p o s s i b l e t o t a k e a c c o u n t o f t h e S t a r k t e r m , p r o v i d i n g t h e R.F. f r e q u e n c y V > > C O ^ - O J ^ , , t h e L a r m o r f r e q u e n c y . The p o l a r i z a t i o n c u r v e t h e n o b t a i n e d i s o f t h e f o r m 7 : P P = r 2 + c 2E" + Cw -oo . ) 2 y y' ( c o m p a r e w i t h e q . ( 9 ) ) w h e r e C i s a c o n s t a n t d e p e n d i n g on t h e p o l a r i z a b i l i t y o f t h e s t a t e , b u t i n d e p e n d e n t o f v, and r = ^ - . The a p p a r e n t l i f e t i m e y i e l d e d f r o m t h e s e c u r v e s i s r a p p = / p ^ T f ? I n t h i s e x p e r i m e n t an E o f a p p r o x i m a t e l y 100-150V/cm and' 300-400V/cm w e r e u s e d . Thus t h e f a c t o r C2Ek i s c h a n g e d by a f a c t o r o f a t l e a s t 80, i . e . t h e c o r r e c t i o n t h a t s h o u l d be a p p l i e d t o T a p p I s c h a n g e d by a f a c t o r o f 9. S i n c e T a p p d o e s n o t a p p e a r t o d e c r e a s e a t t h e l a r g e r f i e l d , we c o n c l u d e t h a t C 2 E 4 < < r and T=T w i t h i n t h e » app e x p e r i m e n t a l a c c u r a c y . - 76 - §4.6 H e l i u m 4*D L i f e t i m e Compared W i t h T h a t O b t a i n e d From O t h e r E x p e r i m e n t s P e r h a p s t h e most c o n v i n c i n g p r o o f t h a t t h e l i f e t i m e s m e a s u r e d a r e n o t s e r i o u s l y a f f e c t e d by m a g n e t i c f i e l d I n - h o m o g e n e i t i e s j R.F. f i e l d s a n d o t h e r b r o a d e n i n g m e c h a n i s m s w o u l d be t o r e m e a s u r e t h e l i f e t i m e o f a s t a t e whose l i f e t i m e h a s a l r e a d y b e e n d e t e r m i n e d by o t h e r w o r k e r s u s i n g v a r i o u s m e t h o d s . I n p a r t i c u l a r a s t a t e w i t h a n a r r o w e r H a n l e e f f e c t c u r v e t h a n t h o s e m e a s u r e d i n t h i s t h e s i s : s h o u l d be c h o s e n . A c o n v e n i e n t e x a m p l e o f s u c h a s t a t e i s o f f e r d by t h e a t o m i c H e l i u m 4 aD s t a t e whose t r a n s i t i o n s 4*D •> 2*P o o c c u r s a t 4922A. T h i s t r a n s i t i o n i s v e r y b r i g h t c o m p a r e d t o t h o s e o b s e r v e d i n m o l e c u l a r h y d r o g e n and h a s a r e l a t i v e l y h i g h p o l a r i z a t i o n . The He 4 1D s t a t e h a s a l i f e t i m e o f a p p r o x i m a t e l y 4xio 8 s e c . a n d a Lande g f a c t o r o f 1 s o t h a t i t s H a n l e e f f e c t c u r v e s h o u l d be some 30% n a r r o w e r t h a n t h e n a r r o w e s t o f t h e s e o b s e r v e d f o r H 2 i n t h i s t h e s i s . A t y p i c a l z e r o f i e l d l e v e l c r o s s i n g e f f e c t c u r v e f o r t h i s s t a t e i s shown i n F I G . 25. The c u r v e e x t r a p o l a t i n g t h e h a l f w i d t h t o z e r o p r e s s u r e i s shown i n F I G . 26.' The l i f e t i m e t h u s o b t a i n e d i s (3.97±•4 )x10" 8 s e c . I n T a b l e V I I I , l i f e t i m e s o b t a i n e d by v a r i o u s o t h e r w o r k e r s a r e l i s t e d f o r c o m p a r i s o n . The e x p e r i m e n t a l w o r k on t h i s l i n e was p e r f o r m e d by R.E. B a r d s l e y o f t h i s l a b o r a t o r y . . T ( X 1 0 8 s e c ) A u t h o r T e c h n i q u e Date 3 . 8 ± . 3 I . M a r t l s o n e t a l . 3 h Beam f o i l , 1 9 6 9 3 . 9 1 ± . 2 Descomps e t a l . 3 5 M a g n e t i c R esonance I 9 6 0 4 . 1 ± . 5 J . P . D e s c o u b e s 3 6 L e v e l C r o s s i n g 1 9 6 7 4 . 7 ± • 5 P e n d l e t o n and H u g h e s 3 7 D i r e c t o b s e r v a t i o n o f d e c a y 1 9 6 5 3 . 0 ± . 5 K l n d l e m a n and B e n n e t t 3 8 D e l a y e d c o i n c i d e n c e 1 9 6 3 3 . 5 ± . 4 F o w l e r e t a l . 3 9 D i r e c t o b s e r v a t i o n o f d e c a y 1 9 6 4 3 . 9 ± . 5 B r l d g e t t and K i n g 1 * 0 D i r e c t o o b s e r v a t i o n o f d e c a y 1 9 6 7 3 . 8 ± . 5 A l l e n e t a l . k 1 D i r e c t o b s e r v a t i o n o f d e c a y 1 9 6 9 3 . 6 6 Wlese e t a l . u T h e o r e t i c a l 1 9 6 5 3 . 9 7 ± . 4 o u r s •' L e v e l C r o s s i n g T a b l e V I I I - L i f e t i m e o f t h e 4*D S t a t e o f H e l i u m - 78 - F i g u r e 25 E x p e r i m e n t a l L e v e l - C r o s s i n g C u r v e f o r t h e H e l i u m 4 1 D -»- 2 1 P. T r a n s i t i o n HALFWIDTH (GAUSS) F i g u r e 26 - 4 JD C u r v e H a l f w i d t h as a F u n c t i o n o f P r e s s u r e - 80' - CHAPTER V . - DISCUSSION OF RESULTS AND CONCLUSION §5.1 I n t r o d u c t i o n U n d e r t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n we e x p e c t t h e l i f e t i m e o f a s t a t e t o d e p e n d p r i m a r i l y on t h e e l e c t - r o n i c a n d v i b r a t i o n a l p a r t s o f t h e wave f u n c t i o n and o n l y v e r y w e a k l y on t h e r o t a t i o n a l p a r t o f t h e wave f u n c t i o n . F o r t h e s t a t e s m e a s u r e d , h o w e v e r , t h e r e a p p e a r s t o be a l a r g e d i s c r e p a n c y b e t w e e n t h e l i f e t i m e s o f t h e J = l s t a t e [(2.66+.1 ) x l 0 _ 8 s e c ] a n d t h e l i f e t i m e s o f t h e J=2, and 3 s t a t e s [(3•85±•15) x10~ 8sec] . The q u e s t i o n n a t u r a l l y a r i s e s , w h e t h e r t h i s d i s c r e p a n c y i s r e a l , a n d , i f i t i s , how m i g h t we a c c o u n t f o r i t . §5.2 H y p e r f i n e E f f e c t s • We w i l l f i r s t c o n s i d e r w h e t h e r t h i s l i f e t i m e d i s - c r e p a n c y c o u l d be r e a l . I t seems v e r y u n l i k e l y t h a t t h e r e a r e e x p e r i m e n t a l e r r o r s l a r g e e nough t o a c c o u n t f o r t h e 50% d i s - c r e p a n c y i n l i f e t i m e s , we w i l l t h e r e f o r e r e - e x a m i n e t h e t h e o r y . P r i m a r y r e q u i r e m e n t s f o r t h e a p p l i c a t i o n o f t h e t h e o r y t o t h i s e x p e r i m e n t a r e t h a t t h e Zeeman e f f e c t be l i n e a r a n d t h a t t h e Lande g f a c t o r be. known. T h r o u g h o u t t h i s d i s c u s s i o n we h a v e i g n o r e d t h e e f f e c t o f n o n - z e r o n u c l e a r s p i n on t h e Zeeman e f f e c t . As i s w e l l known, i n t h e a b s e n c e o f a n e x t e r n a l f i e l d , t h e n u c l e a r s p i n I c o u p l e s t o J t o f o r m a t o t a l a n g u l a r momentum F = I + J , I + J - l , . . . . , | l - j | . When a l a r g e m a g n e t i c f i e l d i s a p p l i e d t h e y become d e c o u p l e d a n d p r e c e s s - 8 l - s e p e r a t e l y a b o u t t h e f i e l d . The g f a c t o r , gp, a t v e r y l o w f i e l d s i s r e l a t e d t o t h e h i g h f i e l d g f a c t o r g j by cr = F ( F + D + J ( J + D ~ K I + D _ g f 2FCF+1) g J F o r t h e J = l s t a t e t h i s y i e l d s 1 S F 2 g J an d u s i n g gp t o compute t h e l i f e t i m e x = (5.32±.2)xl0 8 s e c . w h i c h i s now much t o o l a r g e c o m p a r e d t o t h e J=2 l i f e t i m e (J=2 h a s 1=0 s o t h a t g p = g j ) . B e s i d e s t h e two l i m i t i n g c a s e s o f t h e m a g n e t i c f i e l d d e p e n d e n c e o f t h e e n e r g y l e v e l s we may c o n s i d e r t h e i n t e r - m e d i a t e c a s e w h e r e t h e Zeeman e f f e c t i s n o n - l i n e a r . VJe c o n s i d e r t h e H a m i l t o n i a n "R o f a s y s t e m i n a m a g n e t i c f i e l d H, w i t h a n g u l a r momenta I a n d J c o u p l e d w i t h a c o u p l i n g p a r a m e t e r a. it -- u 0 J ' H + u l«H + a l - J — n — — w h e r e u n i s t h e n u c l e a r m a g n e t o n and as z e r o o r d e r e i g e n - f u n c t i o n s we t a k e t h o s e a t h i g h f i e l d i . e . |J,I,m J,m I> The n o n - z e r o m a t r i x e l e m e n t s o f t h i s H a m i l t o n i a n a r e < J , I ,rrij jm-j. |$| J, I ,rrij »nij> = y 0 m j H + y n m j H + a ^ j ^ j a n d <J,I,m J,m I|-^| J,I,m J±l,m I + l > = / J ( J + l ) - i r i j ( n i j i 1) • • / I ( I + l ) - m I ( m I + l ) - 82 - The s e c u l a r e q u a t i o n may be r e s o l v e d i n t o s e p a r a t e s y s t e m s o f d i m e n s i o n _<2I+1, whose e i g e n v a l u e s .are t h e e n e r g i e s and whose e i g e n v e c t o r s a r e t h e s t a t e s o f t h e s y s t e m . N e g l e c t i n g t h e s m a l l t e r m ynm-j-H t h e s e c u l a r e q u a t i o n s a r e : f o r J = l E±y TH-a = 0 f o r t h e s t a t e s |-J,I,±1,±1> ( i . e . t h e s e a r e a l r e a d y e i g e n s t a t e s o f t h e s y s t e m ) m J = ± 1 m I= 0 m J = 0 m ]-=±l m j = ± l m I= 0 m ] [=±l ± y j H - E a a -E = 0 an d m-j-= 1 m J = 0 m-r=0 m J = 1 m-j-=-l m j = - l m= 1 mJ=° m I = 0 m J = 1 m-j-=-l - y j J - a - E a a -E a a y j H - a - E - 8 3 - The r o o t s o f t h e s e e q u a t i o n s a r e shown as a f u n c t i o n o f m a g n e t i c f i e l d i n F I G . 2 7 - S t a t e s . t h a t c a n i n t e r f e r e t o p r o d u c e l e v e l c r o s s i n g e f f e c t s a r e c o n n e c t e d by d o u b l e e n d - ed a r r o w s . We s u b s t i t u t e t h e f i e l d d e p e n d e n t wave f u n c t i o n s and e n e r g i e s i n t o t h e B r e i t f o r m u l a , and u s e t h e s e l e c t i o n r u l e Am^=0 ( t h i s m e r e l y s a y s t h a t t h e I_«J c o u p l i n g i s so weak t h a t t h e t i m e i n v o l v e d i n c h a n g i n g m . r > > T ) . I f f o r t h e e x c i t a t i o n we a g a i n u s e t h e e l e c t r i c q u a d r u p o l e moment ( 2 ) Q we f i n d a f t e r some l e n g t h y c a l c u l a t i o n s t h a t t h e x x & J L o r e n t z i a n c a n i n d e e d by b r o a d e n e d by an a o f .5 t o 1 0 MHz w i t h o u t s e r i o u s l y d i s t o r t i n g t h e l i n e s h a p e . The d i s p e r s i v e s h a p e , o b t a i n e d when t h e a n g l e cj> ( s e e (§2.5) i s c h a n g e d t o 4 5 ° , shows m a j o r d i s t o r t i o n s n e a r H=0 f o r s u c h an a. No s u c h d i s t o r t i o n s h a v e b e e n o b s e r v e d e x p e r i m e n t a l l y . I t s h o u l d be e m p h a s i z e d t h a t t h e m a g n i t u d e o f t h e d i s t o r t i o n d e p e n d s on t h e e x c i t a t i o n m a t r i x e l e m e n t s ; s i n c e o u r s a r e o n l y q u a l i t a t i v e , we c a n n o t r u l e o u t e n t i r e l y h y p e r f i n e s p l i t t i n g s c o m p a r a b l e w i t h t h e n a t u r a l l i n e w i d t h . F o r t h e J = 3 s t a t e t h e g f a c t o r s g„ f o r F=2, 3 , and 4 a r e r e s p e c t i v e l y ĵ-gj? and ĵ-gj • I f t h e 3 F s t a t e s a r e w e l l r e s o l v e d , t h e a v e r a g e g f a c t o r o f t h e s e s t a t e s i s • 9 5 g j j and a s s u m i n g t h a t e a c h c o n t r i b u t e s t o t h e s i g n a l e q u a l l y t h e w i d t h o b s e r v e d w o u l d be e x p e c t e d t o be much t h e same. The i n t e r m e d i a t e c o u p l i n g c a s e r e q u i r e s t h e h a n d l i n g F i g u r e 27 - Zeeman E f f e c t i n t h e P r e s e n c e o f S m a l l H y p e r f i n e S p l i t t i n g - - 85 - o f 2 1 s t a t e s a n d t h e l a b o r i n v o l v e d i s n o t w a r r a n t e d by t h e p r e s e n t d a t a . § 5 . 3 E l e c t r o n i c Wave F u n c t i o n V a r i a t i o n w i t h J I f we assume t h a t t h e l i f e t i m e d i s c r e p a n c y i s r e a l , we must a t t r i b u t e i t t o t h e f a i l u r e o f t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n . As m e n t i o n e d i n C h a p t e r I I , t h e 3 d 1 s t a t e s o f H 2 s u f f e r f r o m L - d e c o u p l i n g . The s t a t e v e c t o r s o f t h e 3 d * £ s t a t e s a r e d e r i v e d as l i n e a r ' c o m b i n a t i o n s o f t h e Hund's c a s e b c o u p l e d Z, II and A s t a t e s i n A p p e n d i x I I . i . e . |E'(J)> = A z ( J ) | Z > + B z ( j ) | n > + C z ( J ) | A > w h e r e t h e A^ a r e t h e e x p a n s i o n c o e f f i c i e n t s and | a r e t h e c a s e b c o u p l e d s t a t e s . Now u s i n g • 1 . A Z ( J ) B Z ( M ) C E ( J ) • T Z ( J ) " T z Tn TA we a t t e m p t t o s o l v e f o r x^,, T^, a n d x^. U s i n g t h e x o b t a i n e d a s s u m i n g z e r o h y p e r f i n e s p l i t t i n g , s o l u t i o n s t o any p a i r o f t h e s e e q u a t i o n s e x i s t o n l y i f ' . a t l e a s t one o f t h e x's i s n e g a t i v e . Hence i t i s c l e a r t h a t u n d e r t h e s i m p l e c o u p l i n g scheme c o n s i d e r e d we c a n n o t e x p l a i n t h e l i f e t i m e d i s c r e p a n c y . I f f o r t h e . l i f e t i m e s we u s e t h o s e o b t a i n e d a s s u m i n g a l a r g e h y p e r f i n e s p l i t t i n g c o m p a r e d t o t h e n a t u r a l w i d t h , t h e e q u a t i o n s a r e s o l u b l e and y i e l d x ^ , ~ 8 x l 0 ~ 8 s e c . and x n ~ 4 . 5 x l 0 ~ 8 s e c . B u t now, u s i n g g p = . 4 5 0 f o r t h e J = l s t a t e , t h e c o l l i s i o n c r o s s e c t i o n o f t h e J = l s t a t e becomes o n l y h a l f o f t h a t o f t h e J = 2 and J = 3 s t a t e s w h i c h seems r a t h e r u n l i k e l y . - 86 - I n a d d i t i o n t o t h e s i m p l e c o u p l i n g scheme c o n s i d e r e d i n A p p e n d i x I I , t h e r e may be "mixed"- i n t o t h e s t a t e s a v e r y s m a l l f r a c t i o n o f a s t a t e w i t h a v e r y s h o r t l i f e t i m e . The m i x i n g Vr'ould h a v e t o be g r e a t e r f o r J = l t h a n f o r J=2 and J=3 w h i c h i s r a r e b u t n o t i m p o s s i b l e . §5.4 C o n c l u s i o n and S u g g e s t i o n s f o r F u r t h e r Work T h i s t h e s i s h a s r e p o r t e d t h e f i r s t m e a s u r e m e n t o f t h e m o l e c u l a r h y d r o g e n 3d1!, s t a t e l i f e t i m e s and t h e i r c o l l i s i o n c r o s s - s e c t i o n s . The l i f e t i m e s a r e s i m i l a r t o t h a t o f t h e 3*D s t a t e o f H e l i u m as e x p e c t e d . An a p p a r e n t d i s c r e p a n c y b e t w e e n t h e l i f e t i m e o f t h e J = l s t a t e and t h o s e o f t h e J=2 and J=3 s t a t e s may p o s s i b l y be a c c o u n t e d f o r i f one assumes a h y p e r f i n e s p l i t t - i n g o f a few MHz i n t h e e x c i t e d s t a t e s J = l and J = 3 ; no e v i d e n c e o f t h i s was o b t a i n e d i n t h i s t h e s i s . B e c a u s e t h e J=2 s t a t e l a c k s any h y p e r f i n e s t r u c t u r e , i t s l i f e t i m e i s t h e o n l y one t h a t c a n be a c c e p t e d w i t h c e r t a i n t y . I t i s s u g g e s t e d t h a t some o f t h e a m b i g - u i t y i n t h e i n t e r p r e t a t i o n o f t h e r e s u l t s f o r t h e J = l and J=3 s t a t e s may be r e m o v e d by s e a r c h i n g f o r a n o n - z e r o f i e l d l e v e l c r o s s i n g , p r e f e r a b l y w i t h t h e h y p e r f i n e l e v e l s o f t h e J = l s t a t e . T h i s l e v e l c r o s s i n g w o u l d be o b s e r v a b l e o n l y i f t h e h y p e r f i n e s p l i t t i n g e x c e e d s 20 MHz o r s o . T h i s m e t h o d o f m e a s u r i n g l i f e t i m e s s h o u l d be a p p l i c - a b l e t o a l m o s t any m o l e c u l a r o r a t o m i c l i f e t i m e p r o v i d i n g t h e — 7 —3 0 s t a t e i n v o l v e d h a s a gx p r o d u c t i n t h e r a n g e 10 s e c t o 10 s e c . R e f i n e m e n t s t o t h e e q u i p m e n t c o u l d p r o b a b l y e x t e n d t h i s r a n g e c o n s i d e r a b l y . - 87 - APPENDIX I THE TRANSITION MATRIX ELEMENTS To compute the f a c t o r s A and R c i n the B r e i t f o r m u l a , we s h a l l need f o r the e x c i t a t i o n the m a t r i x elements (2) o f t h e q u a d r u p o l e moment, Q , and f o r the decay t h o s e o f X X the d i p o l e moment, g * r . (2) The element Q o f the quadr u p o l e moment t e n s o r X X may be r e w r i t t e n i n terms o f s p h e r i c a l h a r m o n i c s , Y whose m a t r i x elements a r e w e l l k n o w n 3 1 . I n p a r t i c u l a r . Q(2) v _v + / — » Y Sex "2,2 2,-2 V 3 2,0 a b b r e v i a t i n g <J,m|Y£m|J',m-> J ,m J« ,m» Y 2,0 J,m J,m 2[3m 2-J(J+l)] J v x2,0 J ,m J J+2,m [6(J+m+2)(J+m+1)(J-m+2)(J-m+1)]*/2 Y, J J + 2 2, ±2 J ,m J,m+2 = [6(J±m-l)(Jim)(J+m+1)(J+m+2)]^ J Y I 2,±2J J ,m J+2,m+2 [(J±m+1)(J±m+2)(J±m+3)(J±m+4)] V 2 Y, J J + 2 where the f a c t o r s J J are r e d u c e d m a t r i x elements i n d e p e n d - e n t o f the magnetic quantum mumbers. • Because they e n t e r i n t o - 88 - A a n d R 0 o n l y as common f a c t o r s t h e y w i l l be i g n o r e d h e n c e - f o r t h and s e t e q u a l t o u n i t y . The d i p o l e moment o p e r a t o r i s s i m i l a r l y e x p a n d e d i n t o t h e f a m i l i a r r a i s i n g and l o w e r i n g o p e r a t o r . L e t t i n g 6 be t h e a n g l e g makes w i t h t h e x - a x i s : g = gi«coscj) + gj«sincf> t h e n g * r r x c o s t j ) + r ^ s i n c j ) a n d s e t t i n g R + = r ^ t i r ^ R^ + R , . R,. - R o r + - and + r = ~ r x 2 y 21 The m a t r i x e l e m e n t s o f R + a r e : J,m T J - l , m + l ( R ± ) = [(J±m)(J±m-l)] 1/ 2R^_ 1 = (R+J J-l,m±l J,m A g a i n we w i l l n e g l e c t t h e common f a c t o r Rj_-|_ i n a l l s u b s e q u e n t c a l c u l a t i o n s . R e f e r r i n g t o e q . ( 7 ) ( § 2 . 3 ) we s e e t h a t i n t h e t e r m s A, y>y'; s i n c e b o t h s t a t e s d e c a y t o a s t a t e v w i t h v = y±l = y ' + l we f i n d t h a t v = y ' + l and y = y ' + 2 s o t h a t f r ) y ' = (R ) y ' n ( r l y ' = (iR ) y ' ^ xJ v --'u'+l y ; v A --'y'+l <• xJ y - ^ y ' + 2 y^y <• - ; y ' + 2 Thus Svy = ( O v v icj> - 89 - S p e c i f i c A p p l i c a t i o n s The l i n e R ( o ) a r i s e s f r o m t h e t r a n s i t i o n s e q u e n c e j j t J I I ^ -> ^ -»- Q i . e . v = 0, y = l, y ' = - l , a n d m=±l F o r m=-l • Q y m Qmy' " ~^Y2,2^1^-1 / 3^ Y2 , o ) o ' - l r ff • = f R l 1 ' ~ 1 f R e 2 1 * - - ? e 2 i t t > V v g v y 1 - J0,0 i R - J l 3 l 6 " 2 e t h u s A(l,y ,y',v) = l 6 e 2 1 * F o r m=+l 2±4> 2i<J> g , g = 2e &y'v t ovy and- A(+l,y,y',v) = l6e' t h u s H I A(m,y,y',v) = 32e mvy>y ' 2icJ) 2 + R o - i Q ^ . i l M g . ^ o l ^ l ^ i l M s ^ o l ' + l Q ^ n g . ! ^ + I « l , l l 2 l 6 1 > 0 = 101.3 Hence P D = .632 The l i n e R(2) a r i s e s f r o m t h e s e q u e n c e J J ' J " 0 2 1 i . e . y'=-2 m=0 } => v=-l y = 0J °2> -> -  1 - 90 - I I • A(m,y,y ' , - 1 ) = 9 6 e 2 1 < | , my>y 1 s i m i l a r l y I I A(m,y,y',+1) = 9 6 e 2 1 ( J ) my>y ' U I . A(m,y,y',v) = 1926 2 1* mvy>y ' and R0 = 640 192 hence P 0 = = .6 The t r a n s i t i o n R ( 2 ) a r i s e s from the sequence 1 ^ 3 ^ The non-zero c o m b i n a t i o n s o f m a t r i x elements here are m -> -V V - 1 -y -*- - 2 - 1 -»- 0 1 -V 2 1 ->- -»- 0 0 -»- ci> ->• 1 0 -> - 1 and making the u s u a l s u b s t i t u t i o n s , I A(±l,y,y' ,±2) = l S ^ O / I T e 2 1 * y>y 1 I A(±l,y,y',0) = 5 7 6 e 2 1 4 , y>y 1 J A(0,y,y',±l) = l 4 4 0 e 2 i < ! ) y>y' o r H I A(m,y,y',v) = l S ^ e e 2 1 * R0 = 1 1 2 , 9 9 2 mvy>y' and P G = . 2 7 5 - 91 - APPENDIX I I THE 3 d 1 STATES OF H 2 §A2.1 E n e r g y L e v e l s and E i g e n s t a t e s The H a m i l t o n i a n f o r t h e r o t a t i n g m o l e c u l e may be w r i t t e n I n t h e m o l e c u l e f i x e d f r a m e E T I.I + B ( J -L ) 2 + B ( J -L ) 2 n , L , I A | , v x x y y y = E j i A I + B [ J 2 + J 2 + L 2 + L 2 + 2 J L + 2 J L ] n , L j | A | 3 v x x x y x x y y = E T I.I + B [ J 2 - J 2 + L 2 - L 2 + J + L ~ + J ~ L + ] n,L 3IA| ,v L z z 2 w h e r e ..B = p j- , r i s t h e i n t e r n u c l e a r d i s t a n c e u i s t h e r e d u c e d mass L i s t h e e l e c t r o n i c a n g u l a r momentum J i s t h e t o t a l a n g u l a r momentum i n t h e m o l e c u l e f i x e d f r a m e z i s t h e i n t e r n u c l e a r a x i s J 1 = J ±iJ x y IT = L ±iL x y U s i n g Hund's c a s e b b a s i s f u n c t i o n s a n d a s s u m i n g " p u r e p r e c e s s i o n " , t h e n o n - z e r o m a t r i x e l e m e n t s o f a r e 3 2 : <L 3A,j | -Jt|L,A±l JJ> = B [ L ( L + l ) - A ( A ± l ) ] 1 / 2 - [ ( J + A ) ( J ± A + l ) ] l / 2 <L,A,J|;K|L,A,J> = B [ J ( J + I ) - A 2 + L ( L + I ) - A 2 ] Hence f o r t h e 3 d 1 s y s t e m t h e s t a t e s |J,A> o f i n t e r e s t a r e |J,±2> = |A±> |J,±I> = |n ± > |J,O> = |E> F o r t h e s e s t a t e s t h e m a t r i x e l e m e n t s a r e : - 92 - a = < £ | # | » = B E [ J ( J + 1 ) ] •+ - 3 = <n±|#|n±> = B n[J(J.+ l ) - 2 ] + E n 6 = < A ± | « | A ± > = B A [ J ( J + l ) - 6 ] + E A "e = <Z|Jf|n±> = B E n [ J ( J + l ) L ( L + l ) ] 1 / 2 n = <n±|#|A±> = B n A [ ( J + 2 ) ( J - l ) ( L + 2 ) ( L - l ) ] We t h e n o b t a i n t h e e i g e n v a l u e e q u a t i o n : 1/2 a e e 0 0 e 6 0 n 0 e 0 3 0 n o n o 6 0 0 6 n o 6 'AXE ' B A n + V- = x> c x n - V + T r a n s f o r m i n g t o a s y s t e m . o f s y m m e t r i c a n d a n t i - s y m m e t r i c wave f u n c t i o n s : 1, IT A ± v l ( A + ± A - > The a b o v e e q u a t i o n b e c o m e s , a f t e r some r e a r r a n g e m e n t o f t h e t e r m s : a /2e 0 0 0 /2e 3 n 0 0 0 n 6 0 0 0 0 0 3 Tl 0 0 0 Tl 6 = A- B - H - B " n - whose s o l u t i o n s y i e l d t h e e n e r g i e s a n d s t a t e v e c t o r s o f t h e s y s t e m . The e n e r g i e s o f t h e £ and II s t a t e i n t h e a b s e n c e - 9 3 . - o f any J°L c o u p l i n g a r e u n a m b i g u o u s l y known s i n c e £(J= 0 ) and II ( J = l ) a r e u n p e r t u r b e d . _ 1 Thus E v = 1 1 1 , 8 0 4 . 6 3 cm L _ 1 and E n = 1 1 2 , 0 6 4 . 9 1 cm The o b s e r v e d s p e c t r u m c a n t h e n be u s e d t o f i n d B , B , B , Lt J l Z j B E n , B n A , a n d E A . The b e s t v a l u e s o b t a i n e d by t r i a l - a n d - e r r o r w e r e : I E A = 112,488 cm i B £ = 2 7 . 2 cm i B n = 2 7 . 7 6 cm" 1 B A = 2 8 . 1 1 cm 1 B Z I K = 2 6 . 3 9 cm BrtA = 27.54 cm" The e n e r g i e s t h u s f o u n d f o r t h e s t a t e s a r e l i s t e d i n T a b l e I X . The e n e r g i e s g i v e n by D i e k e 1 8 a r e a l s o l i s t e d f o r c o m p a r i s o n . The f i t o f t h e s e e n e r g i e s c a n p r o b a b l y be s o m e w h a t . i m p r o v e d , p a r t i c u l a r l y a t h i g h J , by i n c l u s i o n o f t h e c e n t r i f u g a l d i s t o r t i o n t e r m - D v J 2 ( J + 1 ) 2 . A v a l u e o f . 0 2 f o r D Q f i t s t h e E s t a t e d a t a w e l l . C o r r e s p o n d i n g t o t h e e n e r g y e i g e n v a l u e s f o u n d a b o v e , t h e e i g e n s t a t e s o f t h e s y s t e m may now be f o u n d i n t e r m s o f t h e ± ± " p u r e " s t a t e s , and A . D e n o t i n g t h e e i g e n s t a t e s + + (II ) ' , ( A ) ' a c c o r d i n g t o w h i c h s t a t e t h e y t e n d as B^^-^-0 and B ^ + 0 : E ' = A E | E > + B E | n + > + C E | A + > ( n + ) ' = A N | E > + B n | n + > + c N | A + > ( A + ) ' = A A | E > + B A | n + > + C A | A + > - 94 - J E n e r g y f r o m a b o v e t h e o r y ( c m - 1 ) O b s e r v e d e n e r g y .(cm r ) E n e r g y f r o m a b o v e t h e o r y (cm 1 ) O b s e r v e d e n e r g y . ( c m ~ Y ) E E 0 1 1 1 8 0 4 . 6 3 1 1 1 8 0 4 . 6 3 . 1 1 7 9 6 . 6 4 1 7 9 7 . 1 1 2 1 8 1 9 . 5 4 1 8 1 9 . 7 8 .3 1 8 8 8 . 9 0 1 8 8 5 . 0 7 - 4 2 0 0 9 . 3 8 1 9 9 7 . 4 9 I T n n n~ 1 1 1 2 1 2 7 . 5 0 1 1 2 1 2 7 . 2 3 1 1 2 0 6 4 . 9 1 1 1 2 0 6 4 . 9 1 2 2 2 7 9 . 2 7 2 2 7 4 .24 2 1 4 0 . 9 9 2 1 3 9 . 6 1 3 2 4 7 2 . 0 1 2 4 6 3 .04 2 2 6 5 . 0 6 2 2 6 4 . 0 9 4 . 2 7 0 9 . 5 8 2 6 9 5 . 7 0 2 4 4 0 . 7 4 2441 . 1 2 5 2 6 6 9 . 8 7 2 6 7 1 . 1 0 6 2 9 5 3 . 3 8 2 9 5 3 . 2 1 A + A + A" A~ 2 1 1 2 5 3 3 1 1 2 5 2 8 . 7 5 1 1 2 5 2 2 . 9 6 1 1 2 5 1 7 . 9 5 3 2 7 6 9 2 7 6 6 . 6 0 2 7 3 4 . 1 2 2 7 3 5 . 5 6 4 3 0 7 5 3 0 7 0 . 1 8 3 0 0 5 . 4 2 3 0 1 0 . 3 9 5 3 3 3 5 . 0 1 3 3 3 8 . 5 7 6 3 7 2 1 . 9 7 3 7 1 6 . 8 5 T a b l e I X - E n e r g i e s o f t h e 3 d 1 Complex o f H 2 The c o e f f i c i e n t s A^, B^, C^, f o r t h e f i r s t s i x r o t a t i o n a l l e v e l s o f t h e s y m m e t r i z e d s t a t e s E' , (II ) ' , and (A ) ' a r e l i s t e d i n T a b l e X. U s i n g t h e s e e x p a n s i o n s , a n o t h e r c h e c k on t h e s e wave f u n c t i o n s i s p r o v i d e d by t h e r a t i o s o f t h e i n t e n s i t i e s o f t h e J C E A n B n c n AA BA CA 1 . 9 0 1 - . 4 3 4 0 . 4 3 4 . 9 0 1 0 0 0 0 2 ' . 8 3 0 - . 5 5 0 . 0 9 0 . 5 3 7 .746 - . 3 9 4 .148 • 3 7 5 . 9 1 5 3 . 7 8 7 - . 6 0 2 . 1 3 7 . 5 6 0 . 6 0 3 - . 5 6 9 . 2 6 0 . 5 2 4 . 8 1 1 4 . 7 5 9 - . 6 2 9 . 1 6 9 . 5 5 7 . 4 9 2 - . 6 6 9 . 3 3 8 . 6 0 2 • . 7 2 4 5 . 7 3 9 -.646 . 1 9 2 . 5 5 0 . 4 1 3 - . 7 2 6 . 3 8 9 .642 • . 6 6 0 6 . 7 2 5 - . 6 5 7 . 2 1 0 . 5 4 3 . 3 5 7 - . 7 6 0 .424 .664 . 6 1 6 T a b l e X - E x p a n s i o n C o e f f i c i e n t s f o r t h e 3 d 1 + S t a t e s - 96 - R a n d P l i n e s a r i s i n g f r o m t h e same u p p e r s t a t e . B e c a u s e t h e B v a l u e s o f t h e s t a t e s a r e a l m o s t i d e n t i c a l t h e F r a n c k - C o n d o n f a c t o r s w i l l be a l m o s t i d e n t i c a l a n d w i l l be s e t t o u n i t y . The t r a n s i t i o n m a t r i x e l e m e n t s , A T t T „ f o r t h e 3d 1 (E) ,->2p1E w i l l t h e n be g i v e n by Aj*->j" = A E(J)<E,J"|P|E,J'> + B E ( J ) < E , J " |p|n,j'> w h e r e P i s t h e d i p o l e moment o p e r a t o r summed o v e r a l l d i r e c t i o n s The i n d i v i d u a l m a t r i x e l e m e n t s a r e g i v e n by J . K . L , M a c D o n a l d 3 3 as : <E,J [p E, J + l > = [4CJ+D /2J+1] 1 / 2 P(J+1) <E,J IP E ,J> = 0 Q ( J ) <E, J IP E ,J-1> = [ 4 J / ( 2 J + 1 ) ] ^ 2 R ( J - l ) <n,j IP E , J + 1> = [3J/C2J+1)] 1 / 2 P (J+D <II,J p E ,J> = / 3 Q ( J ) <n,j [p E ,J-1> = - [ 3 ( J + D / 2 J + l ] 1 / 2 R ( J - l ) The t h e o r e t i c a l r e l a t i v e i n t e n s i t i e s t h u s f o u n d a r e l i s t e d i n T a b l e X I f o r t h e f i r s t s i x r o t a t i o n a l s t a t e s ' t r a n s i t i o n s t o t h e 2p*E s t a t e s . T h o s e m e a s u r e d e x p e r i m e n t a l l y a r e a l s o l i s t e d i n T a b l e X I f o r c o m p a r i s o n . The e n e r g y l e v e l s a n d r e l a t i v e i n t e n s i t i e s o f l i n e s a r e n o t s t r i n g e n t t e s t s o f t h e f o r e g o i n g t h e o r y ; t h e f o r m e r b e c a u s e o f t h e l a r g e number o f p a r a m e t e r s u s e d t o f i t t h e d a t a , t h e l a t t e r b e c a u s e o f t h e e x p e r i m e n t a l i n a c c u r a c y . A more s e n s i t i v e t e s t i s o f f e r e d by t h e Zeeman e f f e c t o f t h e s e l e v e l s . - 9 7 - T r a n s i t i o n s I ( T h e o r . ) I ( E x p t l . ) T h e o r . E x p t l . 80°K 300°K- -"-RCJ'-D I R ( J ' - 1 ) I P ( J ' + 1 ) I P ( J ' + 1 ) . 8 0 ° K 3 0 0°X R ( 0 ) P ( 2 ) 2 . 7 4 1 . 0 7 4 2 1 3 46 1 7 2 . 5 3 . 2 2 . 7 R ( D P ( 3 ) 3 . 20 0 . 4 7 40 5 3 3 5 6 . 8 8 6 . 6 R ( 2 ) P ( 4 ) 3 . 3 1 . 2 6 80 3 9 3 5 1 3 26 1 8 . 6 R ( 3 ) P ( 5 ) 3 . 3 3 . 1 6 20 R ( 4 ) P ( 6 ) 3 . 3 2 . 1 1 2 6 R ( 5 ) P ( 7 ) 3 . 3 1 . 0 9 39 T a b l e X I - R e l a t i v e I n t e n s i t i e s o f P and R T r a n s i t i o n s i n t h e 3d*E -*• 2 p J E (0/0) Band § A 2 . 2 The Zeeman E f f e c t The Zeeman e f f e c t H a m i l t o n i a n , i t , may be w r i t t e n 5 m 5 J m °— — u n d e r Hund's c a s e b c o u p l i n g , t h e t e r m L'H may be r e w r i t t e n as L . H = ( L ' J ) ( J - H ) = -2-LgJz + L + J " +• L " J + J > H - - j 2 : ^ - - + + w h e r e L , J , L-, a n d J a r e a l l r e f e r r e d t o t h e m o l e c u l e f i x e d z z ' f r a m e . The n o n - z e r o m a t r i x e l e m e n t s o f $ m a r e t h e n ( a g a i n a s s u m i n g p u r e p r e c e s s i o n ) : Vo'H A 2 <A,J,m J |R m|A,J,m J> = j ^ j + ^ r r i j = g J ( A , A ) y 0 H m J - 98 - <A,J,m T|$ |A±l,J,m T> = — /L(L+l)-A(A±l)/(J+A)(J±A+1) J m J 2 J ( J + 1 > . • - gj(A,A±l)y 0Hmj To s o l v e now f o r t h e e n e r g i e s o f t h e s t a t e s i n a m a g n e t i c f i e l d , we c a n f o l l o w e i t h e r o f two c o u r s e s , may be a d d e d t o Vi (§A2.1) a n d t h e p e r t u r b a t i o n m a t r i x d i a g o n - a l i z e d a g a i n . T h i s p r o c e d u r e w o u l d be n e c e s s a r y i f t h e e n e r g y s p l i t t i n g s due t o t h e m a g n e t i c f i e l d w e r e c o m p a r a b l e t o t h e z e r o - f i e l d s p l i t t i n g s . A l t h o u g h t h i s i s n o t t h e c a s e f o r o u r s t a t e s , t h e r e i s an i n t e r e s t i n g c o n c l u s i o n t h a t c a n be d r a w n f r o m t h i s p r o c e d u r e . L e t us i n p a r t i c u l a r l o o k a t a c o m p l e t e s e t o f s t a t e s w i t h common J and mj = 1. The t r a c e (sum o f t h e d i a g o n a l e l e m e n t s ) o f t h e n o n - d i a g o n a l i z e d m a t r i x i s t h e n j u s t t h e sum o f t h e z e r o - f i e l d e n e r g i e s and t h e g j ( A , A ) y D H p r o d u c t s . D i a g o n a l i z a t i o n o f t h e m a t r i x l e a v e s t h e t r a c e i n v a r i a n t , h e n c e u n d e r any c o u p l i n g scheme, t h e sum £gj = £gj(A,A) where t h e .summation t a k e s p l a c e o v e r a l l c o u p l e d s t a t e s . T h i s c o n c l u s i o n i s i n d e p e n d e n t o f any c o u p l i n g p a r a m e t e r s and s e r v e s as a t e s t t o d e t e r m i n e w h e t h e r t h e s e t o f s t a t e s c o n s i d e r e d i s a c o m p l e t e s e t . The a r g u m e n t a b o v e d o e s n o t y i e l d i n f o r m a t i o n on t h e i n d i v i d u a l g - f a c t o r s u n l e s s t h e c o m p l e t e d i a g o n a l i z a t i o n i s c a r r i e d o u t . A s i m p l e r p r o c e d u r e f o r f i n d i n g g j u n d e r t h e c o u p l i n g scheme c o n s i d e r e d i n §A2.1 i s t o u s e n o n - d e g e n e r a t e f i r s t o r d e r p e r t u r b a t i o n t h e o r y w i t h t h e p e r t u r b a t i o n H a m i l t o n i a n $ . - 9 9 - T h u s ; f o r t h e Z' s t a t e g J = H S ^ T ^ S • , J , m j |*fm I S • J , M J > •• = A 2 ( J ) g j ( Z , Z ) + B 2 ( J ) g j ( n + 5 n + ) + C 2 ( J ) g j ( A + 3 A + ) + A E ( j ) B E ( j ) [ g j ( z , n + ) + g j ( n + , z ) ] + B E ( j ) c z ( j ) [ g J ( n + J A + ) + g j ( A + , n + ) ] The g f a c t o r s t h u s f o u n d f o r t h e f i r s t s i x r o t a t i o n a l l e v e l s o f t h e Z' s t a t e u s i n g t h e c o e f f i c i e n t s f r o m T a b l e X a r e l i s t e d i n T a b l e X I I . J g - t h e o r y g - e x p e r i m e n t a l 1 • 7 7 1 . 9 0 1 2 • 5 ^ 1 . 5 7 1 3 . 4 0 9 .455 4 . 3 2 0 . 3 8 7 5 .245 . 3 3 1 6 . 2 3 3 . 2 8 7 T a b l e X I I - g - v a l u e s o f t h e 3 d J Z S t a t e The g f a c t o r s d e r i v e d a b o v e , a l t h o u g h s h o w i n g a q u a l i t a t i v e a g r e e m e n t w i t h t h o s e e x p e r i m e n t a l l y o b t a i n e d 3 0 , do n o t f i t t h e d a t a n e a r l y as w e l l as m i g h t be e x p e c t e d . I t i s w o r t h w h i l e t o e x a m i n e w h e t h e r t h e Z,IT, and A s t a t e s c o n s i d e r e d do f o r m a c o m p l e t e b a s i s . The p u r e c a s e b c o u p l e d s t a t e s h a v e A 2 g - f a c t o r s j ( • Hence f o r t h e J = l s t a t e s , t h e sum o f t h e g - f a c t o r s i s 0 . 5 . L o o k i n g a t D i e k e ' s e x p e r i m e n t a l d a t a , - 1 0 0 - gj(£) = ± . 9 0 1 and g x (IT ) = ± . 5 0 0 . T h e r e a p p e a r s t o be no way t h a t t h e s e g - f a c t o r s c a n be a d d e d t o ' g i v e a 0 . 5 sum; we c a n t h e r e f o r e c o n c l u d e t h a t o u r s e t o f b a s i s f u n c t i o n s i s n o t c o m p l e t e . The a d d i t i o n o f o t h e r v i b r a t i o n a l l e v e l s t o t h e s e t w i l l n o t h e l p as t h e o v e r l a p i n t e g r a l o f t h e . v i b r a t i o n a l w a v e f u n c t i o n s B. ., , f o r v=v'±l s h o u l d be q u i t e s m a l l . A , v ; A ' , v ' ^ O t h e r c a n d i d a t e s f o r i n c l u s i o n i n t h e p e r t u r b a t i o n t r e a t m e n t i n c l u d e t h e 3*K l e v e l , a *E s t a t e w i t h b o t h e l e c t r o n s e x c i t e d , w h i c h shows a s m a l l Zeeman s p l i t t i n g . I n o r d e r t o e n s u r e t h a t t h e g - f a c t o r s r e p o r t e d by D i e k e 3 0 w e r e n o t t h e r e s u l t o f a r i t h m e t i c o r m e a s u r i n g e r r o r s , t h e s p e c t r u m o f H 2 i n a 2 4 , 5 0 0 g a u s s - f i e l d was p h o t o g r a p h e d on a J a r r e l - A s h 3m g r a t i n g s p e t r o g r a p h . A r e p r o d u c t i o n o f a" p o r t i o n o f t h e p l a t e s h o w i n g t h e R ( 0 ) , R ( l ) and R ( 2 ) l i n e s w i t h p o l a r o i d p e r p e n d i c u l a r ( a ) , and p a r a l l e l ( b ) t o t h e f i e l d i s shown i n F I G . 2 8 . The g - f a c t o r s , g j o b t a i n e d f r o m t h e R ( 0 ) , R ( l ) , and R ( 2 ) l i n e s a r e : g i = . 9 0 0 g 2 = . 5 9 7 g 3 = . 4 5 2 I t s h o u l d be m e n t i o n e d t h a t as R ( 4 ) o v e r l a p s R ( l ) , t h e s p l i t t i n g m e a s u r e d i s o p e n t o q u e s t i o n . T h e s e v a l u e s a r e i n g o o d a g r e e m e n t w i t h t h o s e o f D i e k e .  - 102 - A P P E N D I X . I l l OTHER STATES I n a d d i t i o n t o . t h e w o r k done on t h e 3 d 1 ! s t a t e s d i s c u s s e d b e f o r e , some p r e l i m i n a r y w o r k was a l s o done on t h e 3 1 K ( v = 2 ) J = l , 2, and 3 l e v e l s and on t h e 3 d 1 n ~ ( v = 0 ) J=2 l e v e l . §A3.1 The 3 d ' n " ( v = 0 ) J=2 S t a t e I t was f e l t t h a t a r o u g h k n o w l e d g e o f t h e l i f e t i m e o f t h e 3 d l r [ and 3d*A w o u l d be u s e f u l i n d e t e r m i n i n g t h e r e a s o n f o r t h e d i s c r e p a n c y i n t h e 3 d 1 ! J = l , and t h e J=2 a n d 3 , l i f e t i m e s . The b r i g h t e s t , most e a s i l y r e s o l v e d l i n e i s t h e Q(2) 3 d 1 Il>2p 1 E (0-*0) . The p o l a r i z a t i o n o f t h i s l i n e i s n e g a t i v e . D a t a was o b t a i n e d f o r t h i s l i n e a t p r e s s u r e s o f 30u and 20u. The e x t r a p o l a t i o n o f t h e h a l f w i d t h i s s u b j e c t t o r a t h e r l a r g e e r r o r s , b u t as o n l y a r o u g h e s t i m a t e (±20%) i s r e q u i r e d t h e y s h o u l d s u f f i c e . The h a l f w i d t h s o b t a i n e d a r e 4.805±-045 g a u s s a t 30u p r e s s u r e , a nd 4 . 3 8 5 ± . 0 2 6 g a u s s a t 20u p r e s s u r e . T h e s e h a l f w i d t h s a r e t h e a v e r a g e o f 5 r u n s e a c h a n d t h e e r r o r s q u o t e d a r e s t a - t i s t i c a l o n l y . The e x t r a p o l a t i o n y i e l d s a z e r o - p r e s s u r e h a l f w i d t h o f 3 . 6 ± . 3 g a u s s . The s l o p e , , s u g g e s t s a o c r o s s - s e c t i o n o f - 1 7 0 A 2 . The l i f e t i m e u s i n g D I e k e ' s g- v a l u e o f .412 I s t h e n 3 . 8 4 x i o ~ 8 s e c . The i n d i v i d u a l h a l f - w i d t h s o f t h e l i n e o b t a i n e d on e a c h r u n a r e l i s t e d i n T a b l e I I I . - 1 0 3 - Q(2) o f 3 d ' 1 n - ^ 2 p 1 E P r e s s u r e 3 0 u 4 . 7 5 ^ 5 8 4 . 8 1 7 3 4 4 . 6 8 2 3 1 4 . 9 0 0 7 0 4 . 8 7 1 7 9 2 0 y 4 . 3 7 1 3 8 4 . 3 7 7 7 5 4 . 4 0 1 8 7 4 . 3 0 7 0 0 4 . 5 2 7 6 6 A v e r a g e P r e s s u r e T r a n s i t i o n 3 0 y A v e r a g e 805±. 045 4.385± .026 < 3 1 R(0) R(D R(2) 3.01 4.23 5.88 3 . 5 1 4 . 2 5 6 . 2 9 3.22 4.13 5.97 3 . 2 6 3-97 5 . 6 1 3.14 4.54 . 5.31 3.14 4.13 5.59 3 . 2 1 ± . 0 8 4 . 2 1 + . 0 8 5.77 R(2) a t 20y h a s H i 2 = 5 - 3 6 30y H 1 2 = 7 . 6 6 T a b l e I I I - L e v e l C r o s s i n g C u r v e H a l f w i d t h s ( 3 d 1 ! ! and 3 X K ) - 104 - § A 3 . 2 The 3*K S t a t e A l t h o u g h t h e i n f o r m a t i o n a v a i l a b l e on t h e 3*K s t a t e h a s a l r e a d y b e e n p u b l i s h e d 1 * 3 , t h e d a t a a r e l i s t e d i n . T a b l e I I I . We a t t e m p t e d t o m e a s u r e t h e g - f a c t o r f o r t h e s e s t a t e s w i t h l i t t l e s u c c e s s . The g - f a c t o r f o r t h e s t a t e v = 2 , J = l i s . 2 8 ± . 0 5 ; t h e l a r g e e r r o r a r i s e s b e c a u s e t h e Zeeman p a t t e r n o b s e r v e d i s o n l y b a r e l y r e s o l v e d a n d b e c a u s e l o n g e x p o s u r e s w ere r e q u i r e d w h i c h gave r i s e t o somewhat d i s t o r t e d l i n e s h a p e s . The s p l i t t i n g on t h e h i g h e r r o t a t i o n a l l i n e s c o u l d n o t be m e a s u r e d . The l i n e s o b s e r v e d a r e t h e R ( 0 ) , R ( l ) and R ( 2 ) 3 1 K ( v = 2 ) - > 2 p 1 E ( v = 5 ) . O n l y t h e R ( 2 ) l i n e h a s h a d i t s h a l f w i d t h e x - t r a p o l a t e d t o z e r o p r e s s u r e ; i t s gx p r o d u c t i s (1.6±.1 ' 5 ) 1 0 - 8 s e c . a t z e r o p r e s s u r e . A s s u m i n g a c r o s s - es ' . s e c t i o n o f 100A , I t s g - f a c t o r i s a p p r o x i m a t e l y . 1 7 f r o m w h i c h we w o u l d compute a l i f e t i m e o f a p p r o x i m a t e l y 8 x i o - 8 s e c . A s s u m i n g t h e same l i f e t i m e a n d g A = . 2 8 , we o b t a i n f o r t h e J = l s t a t e a h a l f w i d t h a t z e r o p r e s s u r e o f 2.46 g a u s s . A s s u m i n g t h e same c r o s s - s e c t i o n f o r t h e J = l s t a t e , t h e h a l f w i d t h e x t r a p o l a t e d f r o m 3 0 u t o z e r o p r e s s u r e , i s 2 . 7 g a u s s . On t h e b a s i s o f o u r somewhat u n w a r r e n t e d a s s u m p t i o n s t h i s i s i n s u r p r i s i n g l y g o od a g r e e m e n t . - 1 0 5 - REFERENCES AND FOOTNOTES A.C.G. M i t c h e l l a n d M.W. Zemansky; R e s o n a n c e R a d i a t i o n a n d E x c i t e d Atoms ( C a m b r i d g e U n i v e r s i t y P r e s s , L o n d o n , 1 9 6 1 ) p p . 9 3 - 1 5 3 . G. M. L a w r e n c e and D.B. S a v a g e ; P h y s . Rev. 1 4 1 , 6 7 ( 1 9 6 6 ) • I . B r e w e r , C.G. J a m e s , R.G. B r e w e r , F.E. S t a f f o r d , R.M. B e r g and G.M. R o s e n b l a t h ; Rev. S c i . I n s t r . 2 3 , 145-0 ( 1 9 6 2 ) R.G. B e n n e t t and P.W. D a l b y ; J . Chem. P h y s . 40, 1414 ( 1 9 6 4 ) . O n l y a few o f t h e numerous p a p e r s a r e c i t e d b e l o w : P.. T h a d d e u s and R. N o v i c k ; P h y s . Rev. 13_6_ A, 87 ( 1 9 6 4 J . P . B a r r a t t ; l e J o u r n a l de P h y s i q u e e t l a R a d i u m 20_, 6 3 3 ( 1 9 5 9 ) . W. D e m t r o d e r ; Z e i t s . f . P h y s i k 1 6 6 , 42 ( 1 9 6 2 ) . D.R. C r o s s l e y a nd R.N. Z a r e ; P h y s . Rev. L e t t e r s 1 8 , 942 ( 1 9 6 7 ) . A. M a r s h a l l , R.L. de Z a f r a , H. M e t c a l f ; P h y s . Rev. L e t t e r s 2 2 , 4 4 5 ( 1 9 6 9 ) K.R. German' and R.N. Z a r e ; P h y s . Rev. L e t t e r s 23_, 1 2 0 7 ( 1 9 6 9 ) . S . J . S i l v e r s , T.H. B e r g e m a n , and W. K l e m p e r e r ; J . Chem. P h y s . . 5 2 , 5 3 8 5 ( 1 9 7 0 ) . J . C . P e b a y - P e y r o u l a ; P h y s i c s o f t h e One-and T w o - E l e c t r o n Atoms ( N o r t h H o l l a n d P u b l i s h i n g Co., A m s t e r d a m , 1 9 6 9 ) p p . 3 4 8 - 3 6 1 . J.W.S. R a y l e i g h ; P r o c . Roy. S o c . 1 0 2 , 1 9 0 0 ( 1 9 2 2 ) . H. H. S t r o k e ; P h y s i c s T o d a y , O c t . 1 9 6 6 , p p . 5 5 - 6 0 . R.W. Wood and A. E l e t ; P r o c . Roy. S o c . 1 0 3 , . 3 9 6 ( 1 9 2 3 ) . W. H a n l e Z e i t s ; f . P h y s i k 3 0 , 93 ( . 1 9 2 4 ) . G. B r e i t ; J . O p t . S o c . Amer. 1 0 , 4 3 9 ( 1 9 2 5 ) . A.C.G. M i t c h e l l a n d M.W. Zemansky; l o c . ' c i t . p p . 2 5 8 - 3 1 8 . H. W.B. S k i n n e r ; P r o c . Roy. S o c . A 1 1 2 , 642 ( . 1 9 2 6 ) . M. L o m b a r d ! and J . C . P e b a y - P e y r o u l a ; Compt. Ren d . 2 6 l , 1 4 8 5 ( 1 9 6 5 ) . - J e a n - P i e r r e D e s c o u b e s ; C R . A c a d , S c . P a r i s 2 5 9 , 3 2 7 ( 1 9 6 4 ) . - 1 0 6 - 1 7 P a t r i c k C a h i l l , R i c h a r d S c h w a r t z , and A. Norman J e t t e ; P h y s . Rev. L e t t e r s 1 9 , 2 8 3 ( 1 9 6 . 7 ) . i s G.H. D i e k e ; J . M o l . S p e t r . 2 , 4 9 4 ( 1 9 5 8 ) . 19 P.A. P r a n k e n , P h y s . Rev. 1 2 1 , 5 0 8 ( 1 9 6 1 ) . 20 G.H. B r e i t ; R e v s . M o d e r n P h y s . 5 , 9 1 ( 1 9 3 3 ) . 21 F.D. C o l e g r o v e , P.A. F r a n k e n , R.R. L e w i s a nd R.H. S a n d s ; P h y s . Rev. L e t t e r s 3, 420 ( 1 9 5 9 ) . 22 M. B o r n a nd R. O p p e n h e i m e r ; Ann. P h y s i k 8_4, 4 5 7 ( 1 9 2 7 ) . 23 G. H e r z b e r g ; S p e c t r a o f D i a t o m i c M o l e c u l e s (D. V a n N o s t r a n d Company I n c . 1 9 6 6 ) s e e e s p e c i a l l y pp. 2 1 8 - 2 2 6 . 24 P.M. D a v i d s o n ; P r o c . Roy. S o c . A 1 3 8 , 5 8 0 ( 1 9 3 2 ) . W. R i c h a r d s o n ; M o l e c u l a r H y d r o g e n and I t s S p e c t r u m . ( Y a l e U n i v e r s i t y P r e s s , 1 9 3 5 ) . 2 5 . Von I . K o v a c s a nd A. Budo; Hung. A c t a P h y s i c a 1 , 1 ( 1 9 4 9 ) . 26 P.G. B u r k e , H.M. S c h e y , K. S m i t h ; P h y s . Rev. 1 2 9 . , 1 2 5 8 ( 1 9 6 3 ) A b r i e f a c c o u n t o f t h e s t a t e o f l o w e n e r g y e l e c t r o n - a t o m s c a t t e r i n g t h e o r y as o f 1 9 6 4 i s g i v e n by E. G e r j u o y i n • P h y s i c s Today 18 . , 24 (May 1 9 6 5 ) . 27 L.D. L a n d a u a n d E.M. L i f s h i t z ; Quantum M e c h a n i c s , ( A d d i s o n - W e s l e y P u b l i s h i n g Co. I n c . R e a d i n g M a s s . 1 9 5 8 , s e c o n d e d i t i o n ) §145. A l t h o u g h t h e a u t h o r s g i v e o n l y i n e l a s t i c c o l l i s i o n c r o s s - s e c t i o n s , f o l l o w i n g t h r o u g h t h e i r d e r i v a t i o n , t h e m a t r i x e l e m e n t s a r e e a s i l y f o u n d u n d e r t h e same c o n d i t - i o n s . 28 R.R. B o c k e m u e h l , G e n e r a l M o t o r s R e s e a r c h L a b o r a t o r i e s , W a r r e n , M i c h i g a n . 29 W. H a p p e r and E.B. S a l o m a n ; P h y s . Rev. l 6 0 , .29 ( 1 9 6 7 ) . The a u t h o r s o f t h i s p a p e r a t t r i b u t e t h e p h a s e s h i f t p l a t e s t o A. L u r i o , R. G a r w i n , and A. P a t l a c h o f I.B.M. W a t s o n L a b o r a t o r y , C o l u m b i a U n i v e r s i t y , New Y o r k . 30 G.H. D i e k e , S.P. C u n n i n g h a m , a n d F.T. B y r n e ; P h y s . Rev. 9 2 , 8 1 ( 1 9 5 3 ) . 31 L.D. L a n d a u a n d E.M. L i f s h i t z ; l o c . c i t . § 1 0 , 7 . 32 J-.H. V a n V l e c k ; P h y s . R e v . 33 . , 4 6 7 ( 1 9 2 9 ) . 33 J . K . L . M a c D o n a l d ; P r o c . Roy. S o c . A 1 3 8 , 1 9 3 ( 1 9 3 2 ) . - 1 0 7 - I . M a r t i s o n , W..S. B i c l c e l , J . B r o m a n d e r , H.G. B e r r y , L. L u n d i n , R. B u c h t a , and I . B e r g s t r o m ; J . O p t . S o c . Amer. 60_, 3 5 2 ( 1 9 7 0 ) . B. Descomps, J . C . P e b a y - P e y r o u l a and J . B r o s s e l ; Compt. Rend. A c a d S c i . 2 5 1 , 9 4 l ( 1 9 6 1 ) . J . P . D e s c o u b e s ; P h y s i c s o f t h e One-and T w o - E l e c t r o n Atom l o c . c i t . p. 3 4 l . W.R. P e n d l e t o n and R.H. H u g h e s ; P h y s . Rev. 1 3 8 A, 6 8 3 ( 1 9 6 5 ) P.T. K i n d l e m a n and W.R. B e n n e t ; B u l l . Amer. P h y s . S o c . 8 , 87 ( 1 9 6 3 ) . R.G. F o w l e r , T.M. H o l z b e r l e i n , C.H. J a c o b s o n and S . J . B . C o r r i g a n ; P r o c . P h y s . S o c . ( L o n d o n ) A84, 5 3 9 ( 1 9 6 4 ) . K.A. B r i d g e t t a nd T.A. K i n g ; P r o c . P h y s . S o c . ( L o n d o n ) A 9 2 , 7 5 ( . 1 9 6 7 ) L. A l l e n , D.G.C. Jones', D.G. S c h o f i e l d ' ; J . O p t . S o c . Am. 5 9 , 842 ( 1 9 6 9 ) . W.L. W i e s e , M.W. S m i t h and B.M. G l e n n o n ; A t o m i c T r a n s i t i o n P r o b a b i l i t i e s , V o l . 1 NSRDS-NBS4 (U.S. G o v t . P r i n t i n g O f f i c e , W a s h i n g t o n , D . C , 1 9 6 6 ) . F.W. D a l b y and J . v a n d e r L i n d e ; . C o l l o q u e Ampere XV, N o r t h - H o l l a n d , A m s t e r d a m , 1 9 6 9 .

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
China 5 24
United States 3 0
France 1 0
City Views Downloads
Beijing 5 0
Unknown 3 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items