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ESR on spin-polarized atomic hydrogen at temperatures below 0.5 K Statt, Bryan Wayne 1984

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ESR ON SPIN-POLARIZED ATOMIC HYDROGEN AT TEMPERATURES BELOW 0 . 5 K by BRYAN WAYNE STATT B . S c , U n i v e r s i t y Of B r i t i s h C o l u m b i a , 1976 M . S c , U n i v e r s i t y Of B r i t i s h C o l u m b i a , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA M a r c h 1984 © BRYAN WAYNE STATT, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis f o r scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date March 30. 1984 E-6 (3/81) A b s t r a c t S p i n - p o l a r i z e d a t o m i c h y d r o g e n (H\J/) gas a t low t e m p e r a t u r e s (<0.5 K) has been s t u d i e d u s i n g E l e c t r o n S p i n R e s onance (ESR) t e c h n i q u e s w h i c h a l l o w d i r e c t measurements of t h e d e n s i t i e s of H atoms i n e a c h of t h e two l o w e r h y p e r f i n e s t a t e s (a and b ) . I t has been d e m o n s t r a t e d t h a t ESR c a n be made t o y i e l d a c c u r a t e and d e t a i l e d i n f o r m a t i o n a b o u t t h e d e c a y o f d e n s i t i e s i n t i m e . C r u c i a l t o t h e a n a l y s i s of t h e d a t a i s t h e a b i l i t y t o c o n f i n e t h e sample o f H^ , gas i n s i d e t h e l i q u i d "He c o a t e d microwave c a v i t y where t h e m a g n e t i c f i e l d and t e m p e r a t u r e a r e w e l l known. Measurements of K a a and K a h , t h e two s u r f a c e r e c o m b i n a t i o n r a t e c o n s t a n t s , and t h e i r r a t i o 7 = K a a / K a h have been made and from t h e s e r e s u l t s t h e b i n d i n g e n e r g y of H on l i q u i d "He has been e x t r a c t e d . A l s o r e p o r t e d a r e t h e r e s u l t s of t h e f i r s t d e t a i l e d s t u d y of t h e one-body s u r f a c e r e l a x a t i o n r a t e w h i c h we f i n d i s due t o m i c r o s c o p i c m a g n e t i c i m p u r i t i e s i n t h e c e l l w a l l s . T h e o r e t i c a l c a l c u l a t i o n s of t h e two-body d i p o l e - d i p o l e r e l a x a t i o n r a t e of H\J, i n t h e gas and on t h e s u r f a c e a r e a l s o p r e s e n t e d . i i i TABLE OF CONTENTS A b s t r a c t i i L I S T OF TABLES v L I S T OF FIGURES v i Acknowledgements v i i i CHAPTER I : I n t r o d u c t i o n 1 CHAPTER I I : B a s i c D e s c r i p t i o n of E x p e r i m e n t 7 2.1 D e t a i l e d I n t r o d u c t i o n t o A t o m i c H ydrogen 7 2.2 ESR T e c h n i q u e 14 2.3 D a t a C o l l e c t i o n 21 CHAPTER I I I : E x p e r i m e n t I 25 3.1 C r y o s t a t D e s i g n 25 3.2 R e s u l t s 31 CHAPTER IV: E x p e r i m e n t II 34 4.1 C r y o s t a t M o d i f i c a t i o n s 34 4.2 R e s u l t s 42 CHAPTER V: F i n a l E x p e r i m e n t 45 5.1 M o d i f i c a t i o n s 45 5.2 H e t e r o d y n e S p e c t r o m e t e r 46 5.3 Method o f D a t a C o l l e c t i o n 50 5.4 Thermometry 54 i v CHAPTER V I : D a t a A n a l y s i s and R e s u l t s 60 6.1 I n i t i a l D a t a Format 60 6.2 D e t e r m i n i n g t h e C a v i t y C o u p l i n g P a r a m e t e r 64 6.3 I n t e g r a t e d I n t e n s i t i e s 67 6.4 D e c a y - C u r v e A n a l y s i s 71 CHAPTER V I I : R e s u l t s 85 CHAPTER V I I I : R e l a x a t i o n C a l c u l a t i o n 96 8.1 I n t r o d u c t i o n 96 8.2 D e n s i t y M a t r i x F o r m u l a t i o n f o r T," 1 98 8.3 B u l k R e l a x a t i o n R a t e 102 8.4 S u r f a c e R e l a x a t i o n R ate 113 8.5 S e m i - c l a s s i c a l D e s c r i p t i o n o f t h e 2-D R e s u l t s 128 CHAPTER IX: C o n c l u s i o n s 133 REFERENCES 139 APPENDIX A: A b s o r p t i o n and D i s p e r s i o n Measurements 142 APPENDIX B: M i c r o w a v e P h a s e - L o c k System 146 APPENDIX C: 1460 MHz S p e c t r o m e t e r 149 APPENDIX D: H y p e r f i n e S p i n S t a t e s 151 APPENDIX E: S p i n - E x c h a n g e R e l a x a t i o n 153 V L I S T OF TABLES TABLE I : S p i n M a t r i x E l e m e n t s 108 TABLE I I : Second O r d e r 3-D M a t r i x E l e m e n t I n t e g r a l s 110 TABLE I I I : M a t r i x E l e m e n t s v s . E n e r g y (3-D C a s e ) 112 TABLE IV: F i r s t O r d e r 2-D M a t r i x E l e m e n t I n t e g r a l s 117 TABLE V: Second O r d e r 2-D M a t r i x E l e m e n t I n t e g r a l s 119 TABLE V I : M a t r i x E l e m e n t s (2-D) v s . E n e r g y 123 TABLE V I I : S p i n - E x c h a n g e D e n s i t y M a t r i x : d / d t 157 v i L I S T OF FIGURES 1 H y p e r f i n e e n e r g y l e v e l d i a g r a m 9 2 S i n g l e t and t r i p l e t p o t e n t i a l s 10 3 Microwave c a v i t y and o p e r a t i n g mode 16 4 A b s o r p t i v e and d i s p e r s i v e measurements 20 5 A b s o r p t i o n and d i s p e r s i o n s i g n a l s 23 6 F i l l i n g c u r v e s 24 7 E x p e r i m e n t a l p a c k a g e , e x p e r i m e n t I 27 8 Mic r o w a v e a p p a r a t u s , e x p e r i m e n t I and II 30 9 I n v e r s e d e n s i t y p l o t 32 10 C e l l and f o u n t a i n pump v a l v e 35 1 1 Low t e m p e r a t u r e s o u r c e 38 12 InSb d e t e c t o r mount 40 13 C a v i t y r e s o n a n c e 41 14 E x p e r i m e n t I I d i s p e r s i o n r e s u l t s 44 15 Mi c r o w a v e s p e c t r o m e t e r , f i n a l e x p e r i m e n t 48 16 A b s o r p t i o n l i n e s h a p e s 62 17 S y n t h e t i c s i g n a l l i n e s h a p e s 69 18 T y p i c a l d e c a y c u r v e 73 19 Log d e n s i t y p l o t t o d e t e r m i n e t h e microwave i n d u c e d r e c o m b i n a t i o n r a t e W 77 20 Decay c u r v e f i t s 80 21 C o r r e l a t i o n f u n c t i o n of f i t s 82 22 x 2 dependence on W 83 2 3 Log p l o t of K Q Q and K Q D v s . T " 1 8 6 2 4 T e m p e r a t u r e dependence of gamma 8 7 2 5 Log p l o t of K Q V v s . T" 1 9 0 2 6 One-body r e l a x a t i o n r a t e 9 1 2 7 Log p l o t of g s 9 2 2 8 M a g n e t i c f i e l d g r a d i e n t p l o t s 9 5 2 9 "He s u r f a c e p r o b a b i l i t y d e n s i t y 1 2 1 3 0 T h e r m a l a v e r a g e s of f i r s t o r d e r m a t r i x e l ement i n t e g r a l s \ 1 2 4 3 1 T e m p e r a t u r e dependence of second o r d e r m a t r i x e lement i n t e g r a l s 1 2 5 3 2 D i p o l a r f i e l d of atoms on a s u r f a c e 1 3 0 3 3 A v e r a g e d i p o l a r f i e l d seen by an atom on t h e s u r f a c e . . . 1 3 1 3 4 M i l l i m e t e r wave p h a s e - l o c k c i r c u i t 1 4 7 3 5 1 2 - 1 8 GHz p h a s e - l o c k c i r c u i t 1 4 8 3 6 1 4 6 0 MHz S p e c t r o m e t e r 1 5 0 v i i i A c k n owledgements I would l i k e t o thank D r . W.N. Hardy f o r h i s s u g g e s t i o n of t h i s p r o j e c t . H i s s u p e r v i s i o n and p a r t i c i p a t i o n i n t h i s work a r e g r a t e f u l l y a c k n o w l e d g e d . I have a l s o b e n e f i t t e d from t h e p a r t i c i p a t i o n of D r . A . J . B e r l i n s k y i n t h i s p r o j e c t , e s p e c i a l l y f o r h i s h e l p i n . a n a l y z i n g t h e d a t a . D r . R. Jochemsen and Dr. E. B a t e l l a were of g r e a t h e l p d u r i n g t h e e a r l y s t a g e s of t h e e x p e r i m e n t . Thanks a r e due t o Dr. E. K l e i n and D r . R.W. C l i n e f o r many h e l p f u l d i s c u s s i o n s , d u r i n g t h e i r v i s i t s t o U . B . C . I am a l s o g r a t e f u l t o D r . E. K l e i n f o r making t h e 6 0 C o s o u r c e s u s e d i n t h e a t o m i c H s o u r c e . I am i n d e b t e d t o I . S h i n k o d a who c o n s t r u c t e d and u s e d t h e 3He m e l t i n g c u r v e thermometer t o c a l i b r a t e t h e t h e r m o m e t e r s u s e d i n t h i s e x p e r i m e n t . I am p a r t i c u l a r l y g r a t e f u l t o D r . W.L.H. S h u t e r and D r . W. McCu t c h e o n f o r t h e l o a n o f most of t h e microwave equipment u s e d i n t h e h e t e r o d y n e s p e c t r o m e t e r . C. Chan i s a l s o t h a n k e d f o r h i s a s s i s t a n c e i n t h e microwave work. W i t h o u t t h e h e l p o f t h e s e members o f t h e R a d i o Astronomy g r o u p a t U.B.C. t h i s e x p e r i m e n t w o u l d n o t have been p o s s i b l e . I w o u l d l i k e t o thank D r . M.C.L. G e r r y f o r t h e l o a n of t h e MOS-5 k l y s t r o n s y n c h r o n i z e r and D r . R.R. H a e r i n g f o r t h e l o a n o f s e v e r a l e l e c t r o n i c i n s t r u m e n t s . I wou l d a l s o l i k e t o i x t h a n k D r . B.G. T u r r e l l f o r t h e l o a n of h i s Magnion s u p e r c o n d u c t i n g magnet u s e d i n t h e f i r s t e x p e r i m e n t . I am g r a t e f u l t o t h e members of t h e P h y s i c s D e p a r t m e n t M a c h i n e shop f o r t h e i r s p e e d y and e x c e l l e n t c o n s t r u c t i o n of some of t h e a p p a r a t u s . S p e c i a l t h a n k s a r e due t o B. Meyer f o r h i s p a t i e n c e i n f a b r i c a t i n g t h e microwave c e l l and f o u n t a i n pump v a l v e as w e l l a s P. A c k e r s f o r m a n u f a c t u r i n g t h e c o n f o c a l r e s o n a t o r f i l t e r . Members of t h e P h y s i c s D e p a r t m e n t E l e c t r o n i c s shop a r e g r a t e f u l l y a c k n o w l e d g e d f o r t h e c o n s t r u c t i o n and m a i n t a i n a n c e of much of t h e e l e c t r o n i c equipment u s e d i n t h i s e x p e r i m e n t . I would a l s o l i k e t o thank W. W a l k e r f o r t h e l o a n of e q u i p m e n t and h i s t i m e and e f f o r t s p e n t m a i n t a i n i n g i t . F i n a l l y , I a c k n o w l e d g e t h e s u p p o r t of t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l i n t h e form o f a p o s t g r a d u a t e s c h o l a r s h i p and 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 f o r a G r a d u a t e F e l l o w s h i p d u r i n g t h e c o u r s e of t h i s t h e s i s . 1 CHAPTER I I n t r o d u c t i o n The p o s s i b i l i t y t h a t gaseous s p i n - p o l a r i z e d a t o m i c hydrogen (H^) m i g h t e x h i b i t i n t e r e s t i n g quantum m e c h a n i c a l p r o p e r t i e s was f i r s t r a i s e d by Hecht ( 1 9 5 9 ) . S e r i o u s e x p e r i m e n t a l e f f o r t s t o s t u d y H\J/ began i n the e a r l y 1970 ' s when i t became t e c h n i c a l l y f e a s i b l e t o t r y t o p rod uce H j , . The main t h r u s t b e h i n d the se and c u r r e n t e f f o r t s i s the p o s s i b i l i t y of o b s e r v i n g Bose E i n s t e i n c o n d e n s a t i o n (BEC) of t h e Hj, g a s , s i n c e H i s a c o m p o s i t e b o s o n . There i s , of c o u r s e , a l r e a d y an example of bose c o n d e n s a t i o n , i . e . l i q u i d " H e , but i t o c c u r s i n t h e l i q u i d s t a t e where the i n t e r - a t o m i c i n t e r a c t i o n s a r e v e r y s t r o n g . T h i s o b s c u r e s the c o n n e c t i o n between BEC and t h e onse t of s u p e r f l u i d i t y . I n c o n t r a s t a t o m i c hydrogen i s p r e d i c t e d t o be a gas even a t T=0 ( E t t e r s e t a l . d 9 7 5 ) and S t w a l l e y and Nosanow (1976) ) because of i t s l a r g e z e r o p o i n t m o t i o n . T h e r e f o r e t h e o p p o r t u n i t y may e x i s t t o s t u d y t h i s p r o b l e m f o r t h e case of a w e a k l y i n t e r a c t i n g g a s . S t a b i l i z i n g a sample of H^ r e q u i r e s t h e s u p p r e s s i o n of r e c o m b i n a t i o n of H atoms i n t o H 2 m o l e c u l e s . One s t a r t s by p o l a r i z i n g t h e e l e c t r o n s p i n s w i t h a l a r g e m a g n e t i c f i e l d . Were i t not f o r t h e h y p e r f i n e i n t e r a c t i o n , t h i s w o u l d cause a l l the H atoms t o i n t e r a c t v i a t h e r e p u l s i v e t r i p l e t p o t e n t i a l t h e r e b y e l i m i n a t i n g r e c o m b i n a t i o n e x c e p t a t v e r y h i g h d e n s i t y . The 2 f i r s t s u c c e s s f u l a t t e m p t a t s t a b i l i z i n g a gas of H^, a t low t e m p e r a t u r e was c a r r i e d out by S i l v e r a and W a l r a v e n (1980) i n Amsterdam. The maximum d e n s i t y a c h i e v e d i n t h a t e x p e r i m e n t was 1 . 8 x 1 0 1 " c m " 3 . F u r t h e r e x p e r i m e n t s c a r r i e d out a t Amsterdam ( S p r i k e t a l . d 982 ) ) , MIT ( C l i n e e t a l . d 9 8 l ) ) and C o r n e l l (Yurke e t a l . ( l 9 8 3 ) ) have r a i s e d t h i s maximum d e n s i t y t o 3 x l 0 1 7 c m " 3 , and r e c e n t c o m p r e s s i o n e x p e r i m e n t s ( S p r i k e t a l . d 983) and Hess e t a l . d 983)) have gone as h i g h as 3 x 1 0 1 8 c m - 3 . In o r d e r t o u n d e r s t a n d what t h e l i m i t i n g f a c t o r s a r e , i t i s n e c e s s a r y t o examine more c l o s e l y how t h e atoms behave i n the p r e s e n c e of r e c o m b i n a t i o n p r o c e s s e s . Samples of HJ> have so f a r been c o n t a i n e d by w a l l s c o a t e d w i t h a l i q u i d He f i l m . 3 He and "He f i l m s have t h e l o w e s t known b i n d i n g energy f o r H , = 1 K f o r H on "He (Morrow e t a l . (1981) and M a t t h e y e t a l . d 9 8 l ) ) and =0.4 K f o r H on 3 He (Jochemsen e t a l . (1981) and M a t t h e y e t a l . ( l 9 8 l ) ) . These v a l u e s may be compared t o the l o w e s t known b i n d i n g energy f o r H on a s o l i d , =40 K f o r H on H 2 (Crampton e t a l . ( 1 9 8 2 ) ) . The e f f e c t of b i n d i n g t o t h e s u r f a c e i s t o i n c r e a s e the s u r f a c e d e n s i t y of a toms , n g , w h i c h i s r e l a t e d t o t h e b u l k d e n s i t y , n ^ , by [1 .1 ] n s = n H A e x p ( E g / k B T ) where A= (27rti 2/mkgT) w 2 i s t h e t h e r m a l de B r o g l i e w a v e l e n g t h and E g t h e b i n d i n g e n e r g y . T h i s e x p r e s s i o n h o l d s i n the 3 n o n - d e g e n e r a t e r e g i o n and f o r t h e s i t u a t i o n n s < < n S Q t where t h e s a t u r a t e d s u r f a c e d e n s i t y f o r H on "He, n s a t = 5 x 1 0 1 3 c m " 2 , (Goldman and S . i l v e r a ( 1 9 8 1 ) ) i s t h e d e n s i t y a t whi c h t h e n e t r e p u l s i v e i n t e r a c t i o n between H atoms i s e q u a l t o t h e b i n d i n g e n e r g y E g . C o n s i d e r f i r s t r e c o m b i n a t i o n i n t h e b u l k ( i . e . i n t h e t h r e e d i m e n s i o n a l g a s ) ; t h e r e a r e two p r o c e s s e s w h i c h can p l a y a r o l e . At h i g h t e m p e r a t u r e s where t h e h e l i u m v a p o u r p r e s s u r e i s l a r g e t h e domi n a n t r e c o m b i n a t i o n p r o c e s s i s [1.2] H+H+He —=• H 2+He where t h e t h i r d body, He, i s n e c e s s a r y t o c o n s e r v e e n e r g y and momentum. A t l o w e r t e m p e r a t u r e s and h i g h e r d e n s i t i e s t h e t h r e e - b o d y H c o l l i s i o n c a n p l a y a r o l e , i . e . [1 . 3 ] H+H+H — > H 2+H As m e n t i o n e d a b o v e , d e n s i t i e s h i g h enough t o see t h i s p r o c e s s , n|_|— 1 0 1 8 c m ~ 3 , have j u s t r e c e n t l y been a c h i e v e d . At t h e l o w e r d e n s i t i e s of i n t e r e s t i n t h i s e x p e r i m e n t , two-body r e c o m b i n a t i o n on t h e w a l l d o m i n a t e s : [ 1 . 4 ] H+H+wall—-> H 2 + w a l l Here,' t h e w a l l p l a y s t h e r o l e o f t h e t h i r d body. In t h i s e x p e r i m e n t t h e s u r f a c e r e c o m b i n a t i o n r a t e i s measured. The t e m p e r a t u r e d e p e n d e n c e o f t h i s r e c o m b i n a t i o n r a t e w i l l a l l o w us 4 t o e x t r a c t E g ( s e e [ 1 . 1 ] ) . A s t h e t e m p e r a t u r e i s l o w e r e d , s u r f a c e r e c o m b i n a t i o n w i l l i n c r e a s e u n t i l n s s a t u r a t e s . H o w e v e r a s a p r a c t i c a l m a t t e r , w e l l b e f o r e s a t u r a t i o n c a n o c c u r t h e r e c o m b i n a t i o n r a t e , w h i c h i s d e t e r m i n e d b y t h r e e - b o d y s u r f a c e r e c o m b i n a t i o n a t s u c h h i g h d e n s i t i e s , b e c o m e s l a r g e e n o u g h t o d e s t r o y t h e s a m p l e . F o r t u n a t e l y , t h e r e i s m o r e t o t h e s t o r y t h a n s i m p l e r e c o m b i n a t i o n . A s w i l l b e s h o w n l a t e r t h e l o w e r t w o h y p e r f i n e s t a t e s c o n s i s t o f o n e p u r e s t a t e , |b>, w i t h e l e c t r o n s p i n a n t i - p a r a l l e l t o t h e m a g n e t i c f i e l d H 0 a n d o n e m i x e d s t a t e , |a>, w i t h a s m a l l a d m i x t u r e o f e l e c t r o n s p i n p a r a l l e l t o H 0 . Two a t o m s i n t h e p u r e s t a t e i n t e r a c t i n g s o l e l y v i a t h e t r i p l e t p o t e n t i a l w i l l n o t r e c o m b i n e , w h e r e a s i f o n e o f t h e a t o m s i s i n t h e m i x e d s t a t e r e c o m b i n a t i o n i s a l l o w e d i f t h e a t o m s i n t e r a c t v i a t h e a t t r a c t i v e s i n g l e t p o t e n t i a l . T h i s l e a d s t o a ' b u r n i n g o f f ' o f t h e m i x e d s t a t e l e a v i n g o n l y a t o m s i n t h e p u r e s t a t e , a t o m s w h i c h c a n n o t r e c o m b i n e . T h e r e s u l t i s a n u c l e a r s p i n p o l a r i z e d a s w e l l a s e l e c t r o n s p i n p o l a r i z e d s a m p l e , w h i c h s e e m s t o b e a n i d e a l s t a t e o f a f f a i r s . T h e a c t u a l d e g r e e o f n u c l e a r p o l a r i z a t i o n i s d e t e r m i n e d b y t h e r a t e s o f r e l a x a t i o n b e t w e e n t h e l o w e s t t w o h y p e r f i n e s t a t e s . P a r t o f t h i s t h e s i s i n v o l v e s t h e c a l c u l a t i o n a n d m e a s u r e m e n t o f t h i s r a t e b o t h i n t h e b u l k a n d o n t h e s u r f a c e . I t t u r n s o u t t h a t t h e r e l a x a t i o n r a t e , T , " 1 , w h i c h i s m u c h s l o w e r t h a n t h e r e c o m b i n a t i o n r a t e , d e t e r m i n e s t h e s a m p l e l i f e t i m e . T h i s 5 s i t u a t i o n i s commonly r e f e r r e d t o as a T , b o t t l e n e c k w h i c h i n h i b i t s r e c o m b i n a t i o n . We were u n a b l e t o measure t h e i n t r i n s i c two-body r e l a x a t i o n r a t e s of because of a l a r g e , e x t r i n s i c , one-body s u r f a c e r e l a x a t i o n p r o c e s s w h i c h overwhelmed the two-body r a t e s f o r the d e n s i t i e s o b t a i n e d i n t h e s e e x p e r i m e n t s . S i m i l a r one-body r e l a x a t i o n r a t e s have been o b s e r v e d i n most e x p e r i m e n t s done t o d a t e . We found t h a t we c o u l d v a r y t h i s r a t e i n a s y s t e m a t i c manner by v a r y i n g t h e t h i c k n e s s of t h e s o l i d H 2 on the w a l l s of the sample chamber . E x p e r i m e n t s done so f a r have a l m o s t e x c l u s i v e l y measured t h e t o t a l sample p r e s s u r e w h i c h i s p r o p o r t i o n a l t o the t o t a l d e n s i t y nj_|(t) = n Q ( t ) +n^ ( t ) v i a t h e i d e a l gas l a w . These e x p e r i m e n t s have t h e o b v i o u s d i f f i c u l t y of not m e a s u r i n g n a and n ^ s e p a r a t e l y . Hence t h e d e p l e t i o n of t h e |a> s t a t e can o n l y be i n f e r e d from t h e decay of t h e t o t a l d e n s i t y w i t h o u t any d i r e c t knowledge of t h e n u c l e a r p o l a r i z a t i o n . More i n f o r m a t i o n c o u l d be o b t a i n e d from the e x p e r i m e n t of Y u r k e e t a l . ( l 9 8 3 ) a t C o r n e l l where the n u c l e a r p o l a r i z a t i o n c o u l d be s e t t o a known v a l u e b e f o r e m e a s u r i n g t h e decay of the t o t a l d e n s i t y . F i n a l l y , van Y p e r e n e t a l . (1983) have c a r r i e d out t h e f i r s t r e p o r t e d ESR e x p e r i m e n t on H ^ . The e x p e r i m e n t a l work r e p o r t e d on i n t h i s t h e s i s i n v o l v e s t h e measurement of n Q ( t ) and n ^ f t ) w i t h e l e c t r o n s p i n r e s o n a n c e (ESR) t e c h n i q u e s . T h i s has a l l o w e d a more r e l i a b l e e x t r a c t i o n of the v a r i o u s r a t e c o n s t a n t s t h a n has been p o s s i b l e t o d a t e . 6 The two s u r f a c e r e c o m b i n a t i o n r a t e c o n s t a n t s have been measured as a f u n c t i o n of t e m p e r a t u r e as w e l l as t h e one-body s u r f a c e r e l a x a t i o n r a t e . A more d e t a i l e d i n t r o d u c t i o n t o t h e e x p e r i m e n t i s g i v e n i n C h a p t e r I I . The n e x t t h r e e c h a p t e r s d i s c u s s t h r e e e x p e r i m e n t s , the l a t t e r two b e i n g s u c c e s s i v e improvements on t h e f i r s t . F e a t u r e s of b o t h of t h e s e e x p e r i m e n t s a r e used i n t h e t h i r d and f i n a l e x p e r i m e n t from w h i c h most of t h e u s e f u l r e s u l t s were o b t a i n e d . C h a p t e r VI o u t l i n e s the d a t a a n a l y s i s and C h a p t e r V I I p r e s e n t s t h e f i n a l r e s u l t s . C a l c u l a t i o n s of the b u l k and s u r f a c e two-body r e l a x a t i o n r a t e s a r e p r e s e n t e d i n C h a p t e r V I I I . F i n a l l y , C h a p t e r IX c o n t a i n s a summary of r e s u l t s and t h e c o n c l u s i o n s . 7 CHAPTER II B a s i c D e s c r i p t i o n of E x p e r i m e n t 2.1 D e t a i l e d I n t r o d u c t i o n t o A t o m i c Hydrogen P r o p e r t i e s of a t o m i c h y d r o g e n r e l e v a n t t o t h i s work a r e d e s c r i b e d i n t h i s s e c t i o n . To b e g i n , t h e s i n g l e atom H a m i l t o n i a n and i t s e i g e n v a l u e s and e i g e n s t a t e s a r e g i v e n . T h i s i s f o l l o w e d by a d i s c u s s i o n of the. p o t e n t i a l s f o r two-body i n t e r a t o m i c i n t e r a c t i o n s . F i n a l l y , t h e r a t e e q u a t i o n s g o v e r n i n g t h e p o p u l a t i o n s of t h e l o w e r two h y p e r f i n e s t a t e s i n t h i s e x p e r i m e n t a r e shown. A s i n g l e h y d r o g e n atom i n a m a g n e t i c f i e l d H 0 has c o n t r i b u t i o n s t o i t s H a m i l t o n i a n [2.1] H =-7 p-hf.H 0 + 7 e * S - H 0 + al-S from t h e p r o t o n and e l e c t r o n Zeeman i n t e r a c t i o n and from t h e h y p e r f i n e i n t e r a c t i o n . H e r e I and S a r e t h e p r o t o n and e l e c t r o n s p i n o p e r a t o r s , 7p and 7 e a r e t h e p r o t o n and e l e c t r o n g y r o m a g n e t i c r a t i o s , and a i s t h e h y p e r f i n e i n t e r a c t i o n c o n s t a n t . The e i g e n s t a t e s and a s s o c i a t e d e n e r g i e s a r e : 8 E = l / 2 / a 2 + t i 2 ( 7 e + 7 p ) 2 H 2 o ' - a / 4 E = - V 2 ( 7 e - 7 p ) H 0 + a / 4 E = t i /2 (7 e -7p)Ho+a /4 E = - l / 2 / a 2 + t ) 2 ( 7 e + 7 p ) 2 H l - a / 4 where t a n 2 0 = a / f i ( 7 e + 7 p ) H o . E l e c t r o n and p r o t o n s p i n p r o j e c t i o n s a l o n g H 0 a r e d e s i g n a t e d by j and ^ r e s p e c t i v e l y . The energy l e v e l d i a g r a m i s p l o t t e d i n F i g . 1 . In t h e l i m i t of* l a r g e m a g n e t i c f i e l d cost?^1 and e = s i n 0 ^ a / 2 ' n ( 7 e + 7 p ) H o . A t 40 kG the a d m i x t u r e c o e f f i c i e n t e=0.0063 i m p l i e s t h a t a f r a c t i o n e 2 = 4 x l 0 ~ 5 of t h e e l e c t r o n s p i n s a r e p o i n t i n g t h e 'wrong way' f o r a sample of s p i n - p o l a r i z e d a t o m i c h y d r o g e n . One can see t h a t r a i s i n g t h e f i e l d o n l y r e d u c e s the a d m i x t u r e as H 0 ~ 2 , not f a s t enough t o make r e c o m b i n a t i o n i n s i g n i f i c a n t , on the t i m e s c a l e of p r e s e n t e x p e r i m e n t s , a t p r a c t i c a l l a b o r a t o r y f i e l d s . Hydrogen atoms i n t e r a c t v i a t h e s i n g l e t or t r i p l e t p a i r p o t e n t i a l s d e p e n d i n g on whether the e l e c t r o n s p i n s a r e a n t i - p a r a l l e l o r p a r a l l e l r e s p e c t i v e l y (see F i g . 2 ) . The s i n g l e t p o t e n t i a l has a w e l l d e p t h of 4 .75 eV w i t h i t s g round s t a t e b e i n g t h e hydrogen m o l e c u l e . The t r i p l e t p o t e n t i a l i s e s s e n t i a l y r e p u l s i v e , due t o t h e P a u l i e x c l u s i o n p r i n c i p l e , w i t h a w e l l d e p t h of o n l y 0 .56 meV. Bound s t a t e s a r e not s u p p o r t e d by t h e t r i p l e t p o t e n t i a l f o r e i t h e r a p a i r of atoms i n f r e e space or on a s u r f a c e . H e n c e , two H atoms i n t e r a c t i n g o n l y v i a 9 F i g u r e 1 H y p e r f i n e energy l e v e l d i a g r a m w i t h : 7 e /27r=2.802714 GHz/kG 7 D /2rr=4. 25808 MHz/kG a /h=1.420405752 GHz WQ (j and Whc a r e t h e two ESR t r a n s i t i o n s used i n t h i s e x p e r i m e n t . V(eV) S i n g l e t and F i g u r e 2 t r i p l e t i n t e r a c t i o n p o t e n t i a l s t h e t r i p l e t p o t e n t i a l w i l l not r e c o m b i n e d i r e c t l y . R e c o m b i n a t i o n p r o c e s s e s ( f o r m a t i o n o f H 2 m o l e c u l e s ) and r e l a x a t i o n p r o c e s s e s ( c o n v e r s i o n o f b i n t o a atoms and v i c e v e r s a ) a r e i n t r i n s i c t o t h e d e c a y of H|. In t h i s e x p e r i m e n t r e c o m b i n a t i o n i n t h e b u l k and on t h e s u r f a c e i s d e s c r i b e d by a two-body r a t e e q u a t i o n . In g e n e r a l t h e r a t e c o n s t a n t s K a Q and K a 5 a r e d i f f e r e n t , where K a a i s t h e r a t e c o n s t a n t f o r two a atoms r e c o m b i n i n g and K a b f o r one a and one b atom r e c o m b i n i n g ( s e e G r e b e n e t a l . ( 1 9 8 1 ) ) . R e c o m b i n a t i o n i n v o l v e s t h e l o s s of two h y d r o g e n atoms, c o n t r i b u t i n g t e r m s of t h e form _ K n ^ t o t h e r a t e e q u a t i o n s . I n t r i n s i c r e l a x a t i o n i s due t o two-body d i p o l e - d i p o l e i n t e r a c t i o n s between c o l l i d i n g H^ J, atoms as d i s c u s s e d i n C h a p t e r V I I I w i t h a r a t e c o n s t a n t G, i n c l u d i n g r e l a x a t i o n i n t h e b u l k and on t h e s u r f a c e . T h e r e a l s o e x i s t s a one-body t e r m , w i t h r a t e c o n s t a n t g. T h i s i s f o u n d t o be due t o d i p o l a r i n t e r a c t i o n s between an atom on t h e He s u r f a c e and m a g n e t i c i m p u r i t i e s i n t h e s u b s t r a t e . I f t h e c e l l , w i t h volume V, i s b e i n g f i l l e d w i t h a f l u x 20, a s s u m i n g (pQ=<p^ = <p, t h e t e r m <p/V must be added t o t h e r a t e e q u a t i o n s . A l l t h e s e t e r m s l e a d t o t h e f o l l o w i n g g e n e r a l r a t e e q u a t i o n s : 12 [2.2a] d j i a = - 2 K a ^ - K a b n a n b - [ ' G ( n a + n b ) + g ] [ ( n a - n b ) - ( n a 0 - n b 0 ) ] + »>/V d t [2.2b] d n b = - K Q b n Q n b + [ G ( n a + n b ) + g ] [ ( n Q - n b ) - ( n a 0 - n b 0 ) ] + c>/V cTt where n a 0 and n b 0 a r e t h e i n s t a n t a n e o u s t h e r m a l e q u i l i b r i u m d e n s i t i e s . The r e l a x a t i o n t e r m s a r e of t h e form ± G ( n b - n b 0 ) n ^ f o r t h e two-body p r o c e s s and ± g ( n h - n b 0 ) f o r t h e one-body p r o c e s s . T h e s e p r o c e s s e s a t t e m p t t o r e s t o r e t h e d e n s i t i e s , n b ( n Q o r n b ) , t o t h e i r t h e r m a l e q u i l i b r i u m v a l u e s , n b 0 . In g e n e r a l t h e r a t e c o n s t a n t s f o r o r t h o and p a r a h y d r o g e n f o r m a t i o n a r e d i f f e r e n t . From G r e b e n e t a l . [2.3 ] K a a = s i n 2 t 9 c o s 2 c 9 K p a r a K a b = 1 / 2 s i n 2 0 K o r t n 0 K Q C = l / 4 ( c o s 2 c 9 - s i n 2 c 9 ) 2 K p a r a + l / 4 K o r t h o K b c = 1 / 2 c o s 2 c 9 K 0 r t h 0 K a d = l / 2 c o s 2 t ? K 0 r t n o K D d = l / 4 K p a r a + l / 4 K o r t h o w h i c h i n t h e h i g h f i e l d l i m i t y i e l d s 1 3 [2 .4] K aa pa ra K G b=e 2 / 2 K 0 r t h o K a c = 1 / 4 K p a r a + 1 / 4 K o r t h o K a d = l / 2 K o r t h o K b c = l / 2 K o r t h o K b d = 1 / 4 K pare + 1 / 4 K 0 r t h 0 . As n o t e d e a r l i e r , r e c o m b i n a t i o n i n the l o w e r two h y p e r f i n e l e v e l s i s s u p p r e s s e d by a f a c t o r e 2 . The v a r i o u s r a t e c o n s t a n t s can be e x t r a c t e d f rom t h e d a t a by c o m p a r i n g t h e measured n Q ( t ) and n h ( t ) w i t h decay c u r v e s g e n e r a t e d by c o m p u t e r . A d e t a i l e d d e s c r i p t i o n of t h e p r o c e d u r e i s g i v e n i n C h a p t e r V I . 1 4 2 .2 ESR T e c h n i q u e E l e c t r o n s p i n r e s o n a n c e t e c h n i q u e s used t o measure n a ( t ) and n b ( t ) a r e d e s c r i b e d i n t h i s s e c t i o n . F i r s t , t h e a l l o w e d t r a n s i t i o n s a r e c a l c u l a t e d . Then t h e microwave c a v i t y modes w h i c h can be used a r e g i v e n . F i n a l l y , t e c h n i q u e s f o r m e a s u r i n g t h e a b s o r p t i o n and d i s p e r s i o n s i g n a l s a r e d i s c u s s e d , as i s the b a s i c e x p e r i m e n t a l s e t u p . M i c r o w a v e t r a n s i t i o n r a t e s between the l o w e r and upper two h y p e r f i n e s t a t e s a r e p r o p o r t i o n a l t o t h e d e n s i t i e s n a and n ^ . The two t r a n s i t i o n s used here a r e t h e t r a n s v e r s e e x c i t a t i o n s W a c | and W(oC . One o t h e r n o n - z e r o t r a n s i t i o n r a t e e x i s t s , the l o n g i t u d i n a l W a c , between t h e l o w e r and upper h y p e r f i n e s t a t e s . A p p l y i n g a t r a n s v e r s e o s c i l l a t i n g m a g n e t i c f i e l d H , =H, i c o s c j t , A 1 A A where k i s a u n i t v e c t o r p a r a l l e l t o H 0 and i x k , y i e l d s a t r a n s i t i o n r a t e [ 2 . 5 ] W h j h f =27r/ft|<h f | ( 7 e ' h S x - 7 p ' f i I x ) H 1 / 2 | h i > | 2 6 ( E h f - E h . - h w ) Compar ing | <h ^ i H ' l - 1 ! > l 2 w i t h -h 2 H 2 7* f f / 1 6 y i e l d s an e f f e c t i v e g y r o m a g n e t i c r a t i o of [ 2 . 6 ] 7eQfdf =7 G +e7 p = ^ " ^ p Hence t h e s e n s i t i v i t i e s f o r m e a s u r i n g n Q and n b a r e f o r a l l p r a c t i c a l p u r p o s e s t h e same. The l o n g i t u d i n a l t r a n s i t i o n i s e x c i t e d w i t h H , =H , kcoswt y i e l d i n g 7 ° ^ = 2 e ( 7 e + 7 p ) . C h o o s i n g a r e s o n a n t c a v i t y and i t s o p e r a t i n g mode i n v o l v e s c o n s i d e r a t i o n of s e v e r a l f a c t o r s . A t 7 e/27rx40 kG=115.8 GHz t h e low moded c a v i t y d i m e n s i o n s a r e of o r d e r s e v e r a l m i l l i m e t e r s . A sample c e l l o f t h i s s i z e i s c o n v e n i e n t f o r k e e p i n g t h e t o t a l h e a t o f r e c o m b i n a t i o n down t o a l e v e l e a s i l y h a n d l e d by t h e d i l u t i o n r e f r i g e r a t o r . A c y l i n d r i c a l c a v i t y was c h o s e n f o r e a s e of c o n s t r u c t i o n ( s e e F i g . 3 a ) . E x c i t i n g t r a n s v e r s e t r a n s i t i o n s e f f i c i e n t l y r e q u i r e s a t r a n s v e r s e m a g n e t i c mode. The most c o n v e n i e n t low o r d e r mode i s t h e TM 0,, mode, whose f i e l d p a t t e r n i s s k e t c h e d i n F i g . 3b. Note t h a t t h i s mode has two r e g i o n s o f h i g h H,. The c a v i t y i s a s s e m b l e d from two p i e c e s j o i n e d a t t h e c e n t e r i n o r d e r t o m i n i m i z e t h e e f f e c t of d i s t u r b e d c u r r e n t f l o w s on t h e r e s o n a n t Q. The c a v i t y d i m e n s i o n s a r e a=0.141 cm and Z o=0.0912 cm where a i s t h e r a d i u s and 2 z 0 t h e h e i g h t . M i c r o w a v e e n e r g y i s c o u p l e d i n a t t h e t o p of t h e c a v i t y t h r o u g h a 0.08 cm d i a m e t e r i r i s o f t h i c k n e s s 0.01 cm. A d j u s t m e n t o f t h e c o u p l i n g Q i s f a c i l i t a t e d by p l a c i n g t h e i r i s o f f c e n t e r w i t h r e s p e c t t o t h e c a v i t y . R o t a t i o n o f t h e r e s o n a t o r w i t h r e s p e c t t o t h e r e c t a n g u l a r w a veguide a l l o w s t h e c o u p l i n g t o be c h a n g e d . I t i s n o r m a l l y a d j u s t e d t o be s u b s t a n t i a l l y u n d e r c o u p l e d a t room t e m p e r a t u r e . A l s o n o t e t h a t t h e i r i s i s c o v e r e d w i t h a p i e c e of 0.0025 cm t h i c k m y l a r t o s e a l i n t h e h y d r o g e n atoms w h i c h e n t e r t h e c a v i t y f r o m t h e b o t t o m t h r o u g h t h e f i l l t u b e . The f i l l t u b e ' s d i a m e t e r i s s m a l l enough (=0.036 cm) t o be w e l l beyond 16 F i g u r e 3 a) M i c r o w a v e c a v i t y b) T M 0 , ! mode: M a g n e t i c f i e l d l i n e s , H , , a r e s o l i d and e l e c t r i c f i e l d l i n e s a r e d a s h e d . 1 7 c u t o f f . There a r e two o t h e r l o w e r f r e q u e n c y modes w h i c h can be u s e d , t h e T M 0 , 0 a t 81 .4 GHz and t h e d o u b l y d e g e n e r a t e T E , , , a t 103.25 G H z . One c o u l d , i n p r i n c i p l e , use d a t a from v a r i o u s modes, hence v a r i o u s v a l u e s of H 0 , t o v e r i f y any f i e l d dependence of t h e d i f f e r e n t r a t e c o n s t a n t s . A t o m i c hydrogen d e n s i t i e s can be measured u s i n g the two t r a n s v e r s e t r a n s i t i o n s d i s c u s s e d a b o v e . One can measure e i t h e r t h e a b s o r p t i o n or d i s p e r s i o n s p e c t r u m . I n t r i n s i c a l y t h e a b s o r p t i o n l i n e w i d t h i s v e r y n a r r o w , b e i n g d e t e r m i n e d by t h e d i p o l a r r e l a x a t i o n r a t e . In p r a c t i c e the l i n e w i d t h w i l l be d e t e r m i n e d by the magnet h o m o g e n e i t y . The q u a n t i t y of i n t e r e s t i s the i n t e g r a t e d a b s o r p t i o n i n t e n s i t y , w h i c h i s p r o p o r t i o n a l t o n ^ . The d i s a d v a n t a g e of u s i n g t h e a b s o r p t i o n s i g n a l i s t h a t atoms e x c i t e d i n t o t h e upper two h y p e r f i n e s t a t e s i m m e d i a t e l y r e c o m b i n e . Hence , measurment of an a b s o r p t i o n s p e c t r u m n e c e s s a r i l y d e s t r o y s some of the s a m p l e , the amount d e p e n d i n g on t h e microwave i n t e n s i t y and t i m e spent sweeping t h r o u g h t h e r e s o n a n c e . T h i s p r o b l e m can be a v o i d e d by u s i n g t h e d i s p e r s i o n s p e c t r u m or a l t e r n a t i v e l y by u s i n g v e r y low microwave power . The complex m a g n e t i c s u s c e p t i b i l i t y i s g i v e n by [ 2 . 7 ] X ( " ) = X / ( o ) ) - i x / / ( ^ ) where x / ( ^ ) i s the d i s p e r s i o n p a r t and x ^ ( ^ ) t h e a b s o r p t i o n p a r t . As suming a d e l t a f u n c t i o n f o r t h e a b s o r p t i o n s p e c t r u m , 18 X (u>) = 6(U>-OJQ ) , t h e K r a m e r s - K r o n i g r e l a t i o n + 00 [ 2 . 8 ] x / ( o ) ) - x / ( ° ° ) = J _ P V r x"(u) cW where PV means p r i n c i p a l v a l u e , y i e l d s x -n~ 1 (CJ0-D) ' 1 . I n r e a l i t y the a b s o r p t i o n c u r v e i s not a d e l t a f u n c t i o n . However f o r a w e l l l o c a l i z e d x 7 / ( " ) the d i s p e r s i o n c u r v e i s v e r y c l o s e t o the i d e a l x / (w) i f | O ) - C J 0 | i s g r e a t e r t h a n a few a b s o r p t i o n l i n e w i d t h s . I t has t h e n o t a b l e f e a t u r e t h a t i t e x t e n d s w e l l beyond t h e r e g i o n of a b s o r p t i o n . Thus t h e d i s p e r s i o n c u r v e can be used t o measure n^ (n^ i s e i t h e r n Q o r n^) w i t h o u t l o s i n g any sample t o microwave i n d u c e d r e c o m b i n a t i o n p r o v i d e d we a v o i d t h e c e n t r a l p o r t i o n s where a b s o r p t i o n t a k e s p l a c e . M e a s u r i n g t h e d i s p e r s i o n s i g n a l e s s e n t i a l l y r e q u i r e s t h e measurement of a s h i f t i n t h e c a v i t y r e s o n a n t f r e q u e n c y vc . S t a r t i n g w i t h vQoc ( e j i ) " 1 / 2 (1+47rx y ) " 1 / 2 y i e l d s [ 2 . 9 ] 6 ^ c / ^ c = - 2 7 r x / where x/=Xo^o(v0-v)''/2 and X o=M e n h / H o i s t h e s t a t i c m a g n e t i c s u s c e p t i b i l i t y . The e l e c t r o n m a g n e t i c moment i s M e and u>Q = 2nv0 i s t h e c e n t e r o f t h e a b s o r p t i o n x" (co) . Knowing uG and H 0 a l l o w s one t o d e t e r m i n e n^ once bvQ/vc i s m e a s u r e d . Note t h a t t h e s e methods do not r e q u i r e knowledge of t h e i n p u t microwave power , u s u a l l y a v e r y d i f f i c u l t q u a n t i t y t o measure a c c u r a t e l y . A b s o r p t i o n s p e c t r a a r e o b t a i n e d by m e a s u r i n g t h e change i n 19 t h e i n v e r s e of t h e r e s o n a n t Q of t h e c a v i t y , 5Q" 1. R e l a t i n g 5Q~ 1 t o t h e change i n t h e i m a g i n a r y p a r t o f t h e r e s o n a n t f r e q u e n c y , , one has [2 .10] SQ" 1= + 47r X / / The i n t e g r a t e d a b s o r p t i o n s i g n a l i s now [2 .11] J6Q- 1dj> = 2ir2Xo*'o a l l o w i n g a c r o s s c h e c k on t h e d e n s i t y d e t e r m i n a t i o n . Note t h a t t h i s e x p r e s s i o n a l s o h o l d s f o r i n h o m o g e n e o u s l y b r o a d e n e d l i n e s . R e f l e c t e d power from t h e c a v i t y i s d e t e c t e d i n one o f two modes t o o b t a i n e i t h e r t h e a b s o r p t i o n o r d i s p e r s i o n s i g n a l . M i c r o w a v e power i s s u p p l i e d by a p h a s e - l o c k e d k l y s t r o n w h i c h i s s e t e i t h e r a t t h e minumum i n r e f l e c t e d power f o r a b s o r p t i o n , o r a t t h e s i d e of t h e c a v i t y r e s o n a n c e f o r d i s p e r s i o n . In t h e f i r s t c a s e , a s t h e m a g n e t i c f i e l d , H 0, i s swept t h r o u g h t h e h y d r o g e n r e s o n a n c e t h e c a v i t y Q d e c r e a s e s a t r e s o n a n c e t h e r e b y i n c r e a s i n g t h e r e f l e c t e d power a s d e p i c t e d i n F i g . 4a. To f i r s t o r d e r i n SQ^QQ t h i s i s a measure of t h e a b s o r p t i o n s p e c t r u m , where Q Q i s t h e u n l o a d e d c a v i t y Q. F i g u r e 4b i l l u s t r a t e s how t h e d i s p e r s i o n measurement i s made. Sweeping t h e m a g n e t i c f i e l d t h r o u g h r e s o n a n c e c a u s e s t h e c a v i t y f r e q u e n c y t o s h i f t t h e r e b y y i e l d i n g , t o f i r s t o r d e r i n bv/bvc , t h e d i s p e r s i o n s p e c t r u m . A more d e t a i l e d a n a l y s i s o f t h i s method i s p r e s e n t e d i n A p p e n d i x A. a) \ \ \ s I / Cavity / Resonance Hi " "/A wave b ) \ \ */xwave H -a tom E S R Resonance A b s o r p t i v e (a) measurements. F i g u r e 4 . and d i s p e r s i v e (b) microwave 21 2 . 3 Data C o l l e c t i o n A b r i e f d e s c r i p t i o n of how t h e d a t a was c o l l e c t e d i s g i v e n i n t h i s s e c t i o n . I n a d d i t i o n , samples of a b s o r p t i o n and d i s p e r s i o n c u r v e s , and f i l l i n g and decay c u r v e s a r e shown. I n i t i a l l y the main m a g n e t i c f i e l d , p r o v i d e d by a p e r s i s t e n t s u p e r c o n d u c t i n g s o l e n o i d , i s s e t midway between H Q ( j and H b c . E x p r e s s i o n s f o r t h e m a g n e t i c f i e l d i n terms of t h e t r a n s i t i o n f r e q u e n c y , d e r i v e d from the h y p e r f i n e s t a t e e n e r g i e s , a r e [ 2 . 1 2 a ] H Q d = [ - ( ^ - a / 2 ) ( 7 e - 7 p ) + [ ( ^ - a / 2 ) 2 ( 7 e + 7 p ) 2 - a 2 7 e 7 p ] 1 / 2 ] / 2 7 e 7 p - ( . / - a / 2 ) 7" 1 - a 2 [ 4 ( v-a/2 ) ( 7 ( ? + 7 p ) ]" 1 [ 2 . 1 2 b ] H b c = [ - ( ^ a / 2 ) ( 7 e - 7 p ) + [ ( ^ + a / 2 ) 2 ( 7 e + 7 p ) 2 - a 2 7 e 7 p ] 1 / 2 ] / 2 7 e 7 p = U + a / 2 ) 7 - 1 - a 2 [ 4 ( v+a/2) ( 7 e + 7 p ) 1 " 1 [ 2 . 1 2 c ] , H a c = / ^ 2 - a 2 ' / ( 7 e + 7 p ) = ' » ' ( 7 e + 7 p ) - 1 - a 2 [ 2 I / ( 7 e + 7p) ] " 1 . A s m a l l s u p e r c o n d u c t i n g sweep c o i l , a t t a c h e d t o t h e t h e r m a l s h i e l d , i s t h e n used t o s e t t h e t o t a l f i e l d t o H a d o r H b c . C o l l e c t i n g d a t a f o r a decay c u r v e i n v o l v e s f i l l i n g the c a v i t y w i t h a toms , t u r n i n g o f f t h e s o u r c e t h e n p e r f o r m i n g a s e r i e s of f i e l d sweeps , a l t e r n a t i n g between the two l i n e s . F i g u r e 5a i s an example of t h e a b s o r p t i o n s i g n a l s measured i n the f i r s t e x p e r i m e n t . The l i n e shape i s g o v e r n e d by t h e m a g n e t i c f i e l d p r o f i l e (main magnet p l u s sweep c o i l ) and mic rowave f i e l d 22 p r o f i l e i n t h e c a v i t y . A sample d i s p e r s i o n c u r v e i s shown i n F i g . 5b. Note t h a t the microwave power i s t u r n e d o f f when sweep ing t h r o u g h the a b s o r p t i o n p a r t of the s i g n a l . M o n i t o r i n g the f i l l i n g of t h e c a v i t y w i t h h y d r o g e n can be a c c o m p l i s h e d by s e t t i n g t h e f i e l d on t h e s i d e of t h e d i s p e r s i o n c u r v e ( a r row i n F i g . 5b) t h e n t u r n i n g on t h e a t o m i c hydrogen s o u r c e ( d i s c u s s e d i n t h e n e x t c h a p t e r ) . Two f i l l c u r v e s a r e shown i n F i g . 6. In b o t h c a s e s t h e s o u r c e i s t u r n e d on a t ( i ) and t u r n e d o f f a t ( i i ) a f t e r t h e d e n s i t y has s a t u r a t e d . C o m p r e s s i o n e f f e c t s f rom h e l i u m gas a r e seen i n t h e f i r s t c u r v e : when t h e s o u r c e power i s h i g h enough a s t e a d y s t r e a m of He gas accompanie s t h e H\| and compres se s i t i n the c a v i t y , y i e l d i n g a h i g h e r d e n s i t y . Once the s o u r c e i s t u r n e d o f f t h i s c o m p r e s s i o n d i s a p p e a r s c a u s i n g a d i s c o n t i n u o u s d r o p i n t h e Hvji d e n s i t y . F o l l o w i n g ( i i ) the h y d r o g e n d e n s i t y decays due t o r e c o m b i n a t i o n . II II , . a line "b" line 5 Gauss RFof f F i g u r e 5 a) A b s o r p t i o n s i g n a l . b) D i s p e r s i o n s i g n a l . A r r o w i n d i c a t e s p a r t of d i s p e r s i o n s i g n a l used t o measure f i l l c u r v e s i n F i g . 6 24 F i g u r e 6 F i l l c u r v e s w i t h h e l i u m gas e f f e c t s (a) and w i t h o u t ( b ) . c o m p r e s s i o n 25 CHAPTER I I I E x p e r i m e n t I 3.1 C r y o s t a t D e s i g n A d e s c r i p t i o n of t h e a p p a r a t u s u s e d i n t h e f i r s t e x p e r i m e n t i s g i v e n i n t h i s s e c t i o n . In p a r t i c u l a r , t h e d i s c h a r g e c e l l , i t s o p e r a t i n g c o n d i t i o n s and t h e microwave s y s t e m a r e d i s c u s s e d . R e s u l t s o b t a i n e d i n t h i s e x p e r i m e n t a r e d i s c u s s e d i n s e c t i o n 3.2. The low t e m p e r a t u r e s r e q u i r e d f o r t h i s e x p e r i m e n t a r e a t t a i n e d w i t h a c o m m e r c i a l 3He/"He d i l u t i o n r e f r i g e r a t o r , t h e SHE model DRI-420. A good d e s c r i p t i o n o f t h e o p e r a t i n g p r i n c i p l e s o f a d i l u t i o n r e f r i g e r a t o r can be f o u n d i n Lounasmaa ( 1 9 7 4 ) . As shown i n F i g . 7 t h e e x p e r i m e n t a l p a c k a g e i s mounted on t h e m i x i n g chamber, t h e c o l d e s t p a r t of t h e r e f r i g e r a t o r . The m icrowave c a v i t y i s c e n t e r e d i n t h e main magnet, a Magnion s u p e r c o n d u c t i n g magnet, and t h e r m a l l y c o n n e c t e d t o t h e m i x i n g chamber w i t h Cu b a r s a t t a c h e d t o t h e w a v e g u i d e . H^ atoms e n t e r th e m i c rowave c a v i t y from below t h r o u g h a 1/8* t h i n w a l l s t a i n l e s s s t e e l tube w h i c h i s c o n n e c t e d t o t h e d i s c h a r g e c e l l . A p y r e x b u l b s e r v e s as t h e d i s c h a r g e c e l l and i s c o n n e c t e d t o a Cu h e a t s i n k i n g p l a t e v i a a Cu H o u s e k e e p e r s e a l . The h e a t s i n k i n g p l a t e i s a t t a c h e d t o t h e t h e r m a l s h i e l d w i t h t h r e e Cu b r a i d s . Thus t h e h e a t from t h e d i s c h a r g e l a r g e l y ends up i n t h e s t i l l 26 and n o t t h e m i x i n g chamber. In a d d i t i o n , t h e d i s c h a r g e c o i l i s mounted on t h e b o t t o m vacuum f l a n g e so t h a t t h e RF power d i s s i p a t e d i n t h e c o i l i s a b s o r b e d by t h e main l i q u i d h e l i u m b a t h . M o l e c u l a r h y d r o g e n and h e l i u m must be added t o t h e c e l l b e f o r e t h e d i s c h a r g e c a n be o p e r a t e d . A f i n e c a p i l l a r y c o n n e c t s t h e c e l l t o a room t e m p e r a t u r e gas h a n d l i n g s y s t e m . To b e g i n an e x p e r i m e n t l i q u i d He i s t r a n s f e r e d i n t o t h e dewar p a r t way up t h e vacuum c a n and He exchange gas added t o t h e vacuum c a n . H e a t e r s on t h e vacuum f l a n g e and t h e r m a l s h i e l d a r e t h e n u s e d t o keep t h e t o p of t h e r e f r i g e r a t o r , e s p e c i a l l y t h e f i l l l i n e , above t h e b o i l i n g p o i n t o f H 2 ( 2 0 . 4 K ) . At t h i s p o i n t H 2 i s a d m i t t e d t o t h e f i l l l i n e and i s cryopumped i n t o t h e d i s c h a r g e c e l l : t h e e x c h a n g e gas e n s u r e s t h a t t h e c e l l i s t h e c o l d e s t p a r t of t h e r e f r i g e r a t o r t h e r e b y c o n d e n s i n g most of t h e H 2 . Once t h e H 2 i s i n , t h e e q u i v a l e n t o f 1/2 an a t m o s p h e r e o f gas i n t h e c e l l a t room t e m p e r a t u r e , t h e t r a n s f e r i s c o n t i n u e d and t h e r e f r i g e r a t o r t u r n e d on. Then "He ga s i s added t o t h e c e l l , an amount e q u a l t o t h e H 2 . RF p u l s e s a t 50 MHz a r e u s e d t o d i s s o c i a t e t h e H 2 . T y p i c a l o p e r a t i n g c o n d i t i o n s p r o d u c i n g t h e h i g h e s t d e n s i t i e s a r e : 1 Watt peak power, 3 Msec p u l s e w i d t h and a r e p e t i t i o n r a t e o f 1 0 0/sec. The d i s c h a r g e i s t u r n e d on f o r a b o u t f i v e m i n u t e s t o f i l l t h e c a v i t y t o n ( - | - 1 0 1 6 c m ~ 3 and t h e n t u r n e d o f f . Now t h e a b s o r p t i o n l i n e i s m e a sured, f i r s t f o r n^ t h e n f o r n a , a l t e r n a t i n g i n t i m e MIXING CHAMBER II VACUUM CAN F i g u r e 7 E x p e r i m e n t a l p a c k a g e , e x p e r i m e n t I . 28 a s t h e sample d e c a y s . T y p i c a l l y t h e s c a n n i n g r a t e was s e t t o sweep t h r o u g h t h e l i n e i n a b o u t one s e c o n d . T h i s m i n i m i z e d t h e amount o f microwave i n d u c e d r e c o m b i n a t i o n . B e f o r e f i l l i n g t h e c a v i t y a g a i n f o r t h e n e x t run- t h e atoms a r e d e s t r o y e d by moving o n t o t h e peak o f t h e a b s o r p t i o n l i n e and w a i t i n g f o r t h e r a p i d m i c rowave i n d u c e d r e c o m b i n a t i o n t o d e s t r o y t h e s a mple. D a t a was c o l l e c t e d i n t h i s manner f o r t e m p e r a t u r e s between 175 mK and 600 mK. The f o l l o w i n g i s a b r i e f d e s c r i p t i o n o f t h e m i crowave a p p a r a t u s u s e d i n t h e f i r s t two e x p e r i m e n t s . A V a r i a n VRT-2125A5 k l y s t r o n s e r v e s as t h e s o u r c e of microwave power a t 115 GHz. I t i s phase l o c k e d w i t h a MOS-5 s y n c h r o n i z e r t o a 12-18 GHz backward wave o s c i l l a t o r w h i c h i n t u r n i s p h a s e l o c k e d t o a Rhode and S c h w a r t z 0.01-500 MHz SMDW computer c o n t r o l l e d f r e q u e n c y s y n t h e s i z e r . T h i s s y s t e m , d e s c r i b e d i n A p p e n d i x B, p r o v i d e s , a s t a b l e , narrow band s o u r c e of m i c r o w a v e s f o r t h e e x p e r i m e n t . As i l l u s t r a t e d i n F i g . 8 t h e microwave power i s m o d u l a t e d a t 4.5 KHz and t h e n f e d i n t o t h e t o p of t h e c r y o s t a t . T h e r m a l i s o l a t i o n r e q u i r e s t h e use o f s t a i n l e s s s t e e l w a v e g u i d e w h i c h a l s o p r o v i d e s a good d e a l of a t t e n u a t i o n . T h i s i s a c c e p t a b l e f o r two r e a s o n s . One, o n l y a few m i c r o w a t t s of power i s r e q u i r e d a t t h e c a v i t y and two, any s t a n d i n g waves w h i c h w o u l d have r e s u l t e d f r o m r e f l e c t i o n between t h e t o p and b o t t o m o f t h e c r y o s t a t - a r e e l i m i n a t e d . The microwave power t h e n e n t e r s a 1OdB d i r e c t i o n a l c o u p l e r , t r a v e l s t h r o u g h a n o t h e r s t a i n l e s s 29 s t e e l waveguide s e c t i o n and f i n a l l y t h r o u g h c o i n s i l v e r waveguide t o t h e c a v i t y . L o s s e s i n t h i s p i e c e of s t a i n l e s s a re u n a c c e p t a b l e , and t h i s r e q u i r e d t h e i n s i d e of t h e 0 .050 xO.100 waveguide t o be copper p l a t e d t o a t h i c k n e s s of -1/ ini . T h i s t h i c k n e s s i s s e v e r a l s k i n d e p t h s a t t h e o p e r a t i n g f r e q u e n c y but t h i n enough t o j u s t match t h e t h e r m a l c o n d u c t i v i t y of t h e s t a i n l e s s s t e e l i t s e l f . Power r e f l e c t e d from t h e c a v i t y pa s se s t h r o u g h t h e d i r e c t i o n a l c o u p l e r u n a t t e n u a t e d t o an InSb hot e l e c t r o n b o l o m e t e r . T h i s d e t e c t o r i s o p e r a t e d a t 1.5 K i n t h e c r y o s t a t . In t h i s e x p e r i m e n t an InSb c r y s t a l was mounted a c r o s s t h e waveguide w i t h a matched l o a d b e h i n d i t . F o r d e t a i l s on InSb b o l o m e t e r s s e e , f o r e x a m p l e , K i n c h and R o l l i n (1963) and P u t l e y ( 1 9 6 5 ) . B i a s i n g f o r t h e d e t e c t o r was s u p p l i e d by a b a t t e r y a t room t e m p e r a t u r e i n t h e same box as t h e low n o i s e p r e - a m p l i f i e r . The 4 . 5 KHz s i g n a l i s t h e n d e m o d u l a t e d w i t h a PAR HR-8 l o c k - i n a m p l i f i e r . T h i s s i g n a l i s made a v a i l a b l e t o t h e l a b computer sy s tem f o r s t o r a g e and f u r t h e r p r o c e s s i n g . PHASE LOCK S Y S T E M CRYSTAL DETECTOR HARMONIC MIXER V _ / V * * \ MODULATOR ¥ PRE-AMP T —1>- LOCK-IN AMP. /Ov l n S b Y DETECTOR T CRYOSTAT KLYSTRON A SAMPLE CAVITY F i g u r e 8 Microwave appara tus used i n exper iment I and 1 1 . 31 3 .2 R e s u l t s R e s u l t s o b t a i n e d i n t h e f i r s t e x p e r i m e n t were o n l y q u a l i t a t i v e , f o r r e a s o n s t o be d i s c u s s e d b e l o w . D e s i g n p r o b l e m s r e s p o n s i b l e f o r t h i s were c o r r e c t e d i n the n e x t e x p e r i m e n t . N u c l e a r p o l a r i z a t i o n was not o b s e r v e d i n t h i s r u n . One p o s s i b l e r e a s o n f o r t h i s i s t h e e x i s t e n c e of m a g n e t i c i m p u r i t i e s i n t h e atom tube c o n n e c t i n g t h e s o u r c e t o t h e microwave c a v i t y c a u s i n g an a b n o r m a l l y l a r g e r e l a x a t i o n r a t e and t h u s k e e p i n g n Q - n b « F i g u r e 9 i s a p l o t of n ^ 1 v e r s u s t i m e a t 350 mK. As e x p e c t e d f o r two-body r e c o m b i n a t i o n t h i s y i e l d s a s t r a i g h t l i n e w i t h the s l o p e e q u a l t o t h e e f f e c t i v e r e c o m b i n a t i o n r a t e . One s h o u l d now be a b l e t o t a k e t h e r e c o m b i n a t i o n r a t e s measured as a f u n c t i o n of t e m p e r a t u r e and e x t r a c t t h e s u r f a c e r e c o m b i n a t i o n r a t e and t h e b i n d i n g e n e r g y . U n f o r t u n a t e l y t h i s was not p o s s i b l e f o r t h e f o l l o w i n g r e a s o n . I n t h i s f i r s t e x p e r i m e n t t h e atoms were f r e e t o move between t h e c a v i t y and t h e d i s c h a r g e c e l l . I f t h e e f f e c t of m a g n e t i c c o m p r e s s i o n was i n c l u d e d and' t h e v a r i o u s vo lumes t a k e n i n t o a c c o u n t , one e s t i m a t e d t h a t about o n e - h a l f of t h e atoms were i n t h e c a v i t y and t h e o t h e r h a l f i n t h e d i s c h a r g e c e l l . T h i s l e d t o a v e r y c o m p l i c a t e d e f f e c t i v e a r e a / v o l u m e r a t i o w h i c h was dependent on t h e t e m p e r a t u r e p r o f i l e as w e l l as t h e m a g n e t i c f i e l d p r o f i l e i n t h e s y s t e m . F o r t h i s r e a s o n r e l i a b l e numbers 2 0 0 4 0 0 6 0 0 8 0 0 t ( sec) F i g u r e 9 I n v e r s e d e n s i t y p l o t . 33 c o u l d not be e x t r a c t e d f rom t h e d a t a . O t h e r p r o b l e m s e n c o u n t e r e d i n t h i s run w e r e : l e s s t h a n i d e a l magnet h o m o g e n e i t y , l a r g e magnet d r i f t (=15 G / h r ) , poor s i g n a l t o n o i s e and a l a r g e s t a n d i n g wave r a t i o . A l l of t h e s e p r o b l e m s were l a t e r c o r r e c t e d , as d e s c r i b e d i n t h e nex t c h a p t e r . 34 CHAPTER IV E x p e r i m e n t I I 4.1 C r y o s t a t M o d i f i c a t i o n s S o l u t i o n s t o t h e p r o b l e m s e n c o u n t e r e d i n t h e f i r s t e x p e r i m e n t a r e o u t l i n e d i n t h i s c h a p t e r . A l i q u i d "He v a l v e was u s e d t o c o n f i n e t h e atoms i n t h e c e l l , a n d t h i s r e q u i r e d a new s o u r c e t o be b u i l t . In a d d i t i o n t o t h e s e m o d i f i c a t i o n s a h i g h e r s e n s i t i v i t y t u n e d InSb d e t e c t o r was i n s t a l l e d . A magnet of h i g h e r h o m o g e n e i t y and l o w e r d r i f t was a l s o u s e d . The c l o s i n g o f f of t h e c e l l f r o m t h e r e s t o f t h e s y s t e m was th e most i m p o r t a n t g o a l of t h i s v e r s i o n of t h e e x p e r i m e n t . O p e r a t i n g a m e c h a n i c a l v a l v e o f any s o r t i n a d i l u t i o n r e f r i g e r a t o r o r any o t h e r low t e m p e r a t u r e a p p a r a t u s i s u s u a l l y r a t h e r d i f f i c u l t . T h e r e a r e t h e p r o b l e m s , f o r example, of g e n e r a t i n g t h e f o r c e s n e c e s s a r y t o m a n i p u l a t e t h e v a l v e i n t h e r e s t r i c t e d g eometry a v a i l a b l e , and o f a v o i d i n g f r i c t i o n a l h e a t i n g d u r i n g a c t u a t i o n o f t h e v a l v e . F o r t h e s e and o t h e r r e a s o n s a s u p e r f l u i d "He f o u n t a i n pump v a l v e was c o n c e i v e d . At a b o u t t h e same t i m e t h e C o r n e l l g r o u p r e p o r t e d t h e s u c c e s s f u l o p e r a t i o n o f t h e i r s u p e r f l u i d "He v a l v e , and a d e s i g n b a s e d on t h e i r g e o m e t r y was a d o p t e d as shown i n F i g . 10. O p e r a t i n g t h e v a l v e p r o c e e d s as f o l l o w s . I n i t i a l l y "He i s c o n d e n s e d i n t o t h e c e l l t h r o u g h t h e s o u r c e f i l l l i n e u n t i l t h e WAVEGUIDE F i g u r e 10 C e l l and f o u n t a i n pump v a l v e . Used i n s e c o n d and f i n a l e x p e r i m e n t . 36 He l e v e l c l o s e s o f f t h e c e l l ( l e v e l A ) . A p p l y i n g power t o the f o u n t a i n pump h e a t e r c a u s e s t h e s u p e r f l u i d t o f l o w t h r o u g h the s u p e r l e a k i n t o the f o u n t a i n pump r e s e v o i r , t h e r e b y r a i s i n g the l i q u i d He l e v e l i n t h e r e s e v o i r and d e c r e a s i n g t h e l e v e l i n the f i l l l i n e ( l e v e l B) e x p o s i n g the c e l l t o t h e atom t u b e . Once t h e H atoms a r e i n t h e c e l l t h e f o u n t a i n pump h e a t e r i s t u r n e d o f f , a f t e r w h i c h t h e He l e v e l s r e t u r n t o t h e i r o r i g i n a l p o s i t i o n s , t h e r e b y s h u t t i n g t h e v a l v e . The p r e s e n c e of a f r e e s u r f a c e of l i q u i d h e l i u m r e q u i r e s t h e a t o m i c hydrogen s o u r c e t o be l o c a t e d above t h e c e l l , as opposed t o be low i t as i n t h e p r e v i o u s e x p e r i m e n t . I n an a t t e m p t t o o b t a i n a h i g h e r atom f l u x a 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 o u r c e based on t h e s o u r c e used a t MIT was t r i e d . A l t h o u g h t h e s o u r c e i t s e l f seemed t o p rodu ce an adequate f l u x , most of the atoms r e c o m b i n e d b e f o r e making i t down t h e atom tube i n t o t h e c e l l . A r e l a t e d p r o b l e m i s t h e h e l i u m r e f l u x i n g i n t h e atom t u b e . T h i s p r o v i d e s an e f f e c t i v e t h e r m a l s h o r t between t h e warmer p a r t s of the atom t u b e (>1 K) and t h e c o l d e r p a r t s of t h e r e f r i g e r a t o r . In some of our atom tube d e s i g n s t h e m i x i n g chamber c o u l d not be c o o l e d below 300 mK. The main r e a s o n our r e f r i g e r a t o r c o u l d not h a n d l e the r e f l u x i n g h e a t l o a d was i t s r e l a t i v e l y low c i r c u l a t i o n r a t e . T h i s s y s t e m was f i n a l l y abandoned i n f a v o u r of t h e p r e s e n t low t e m p e r a t u r e s o u r c e . T e m p e r a t u r e g r a d i e n t s between t h e s o u r c e and m i x i n g chamber were m i n i m i z e d by heat s i n k i n g the s o u r c e t o t h e b a s e p l a t e w h i c h .37 i s t y p i c a l l y no more t h a n 50 mK above t h e m i x i n g chamber t e m p e r a t u r e . F i g u r e 11 i l l u s t r a t e s t h e b a s i c f e a t u r e s of the s o u r c e . E n c a p s u l a t e d i n epoxy i s t h e RF d i s c h a r g e c o i l and c a p a c i t o r ( r e s o n a n t f r e q u e n c y of 45 MHz) w h i c h i s used t o d i s s o c i a t e t h e H 2 m o l e c u l e s w h i c h a r e o r i g i n a l l y condensed on t h e i n n e r s u r f a c e of the e p o x y . T h i s l a y e r of H 2 i s put i n p l a c e by f i r s t h e a t i n g t h e c a p i l l a r y f i l l tube t o =21 K (> t h e b o i l i n g p o i n t of H 2 ) , a t t h e same t i m e k e e p i n g t h e s o u r c e c o l d (2-3 K) w i t h t h e f r i d g e . Hydrogen gas i s t h e n s p r a y e d out of the f i l l tube and i t i m m e d i a t e l y condenses on the i n n e r s u r f a c e of t h e s o u r c e . The h e a t i n g i s t h e n removed and "He gas i s a d m i t t e d t o t h e s o u r c e and c e l l , a f t e r w h i c h the d i s c h a r g e can be a c t i v a t e d . As i n t h e p r e v i o u s d e s i g n a s m a l l 6 0 C o sample ( = 1 0MC ) p r o v i d e s f r e e c h a r g e s t o h e l p i n i t i a t i o n of t h e d i s c h a r g e . Hydrogen atoms p r o d u c e d by t h e d i s c h a r g e t h e r m a l i z e by s c a t t e r i n g o f f t h e b a f f l e s b e f o r e l e a v i n g t h e s o u r c e . Most of t h e hea t g e n e r a t e d i s e x t r a c t e d a t t h e b a s e p l a t e a l t h o u g h some of t h e hea t f i n d s i t s way t o t h e m i x i n g chamber and c e l l . T h i s h e a t i n g i s not a p r o b l e m e x c e p t f o r t h e l o w e s t t e m p e r a t u r e s , s i n c e t h e f r i d g e i s a b l e t o r e c o v e r r e a s o n a b l y q u i c k l y f rom the p e r i o d s o f d i s c h a r g i n g . O p t i m i z i n g the d i s c h a r g e c o n d i t i o n s amounts t o a c h i e v i n g a b a l a n c e between too l i t t l e power w i t h a r e s u l t i n g low H f l u x and t o o much power r e s u l t i n g i n a h i g h "He vapour d e n s i t y . T h i s THERMOMETER HEATER F i g u r e 11 Low t e m p e r a t u r e a t o m i c h y d r o g e n s o u r c e . M o l e c u l a r h y d r o g e n a n d l i q u i d ' H e c o a t t h e i n s i d e w a l l s . RF p u l s e s a r e u s e d t o d i s s o c i a t e t h e H 2 . 39 i n d u c e s r e c o m b i n a t i o n and a c o r r e s p o n d i n g l y low H f l u x . E x p e r i m e n t a l l y i t was found t h a t t h e a v e r a g e power was e s s e n t i a l l y t h e o n l y q u a n t i t y t h a t m a t t e r e d . T y p i c a l l y , RF p u l s e s of 0 .4 w a t t s p e a k . p o w e r , 10 MS d u r a t i o n a t a r e p e t i t i o n r a t e of 100 p u l s e s per second were used w i t h an a v e r a g e power of 0 .4 mW. O t h e r m o d i f i c a t i o n s made f o r t h i s run i n c l u d e d a t u n e d InSb d e t e c t o r . A home made mount was made f o r the InSb c r y s t a l , as i l l u s t r a t e d i n F i g . 12, i n o r d e r t o m i n i m i z e t h e r e f l e c t e d power from t h e d e t e c t o r and m a x i m i z e t h e s e n s i t i v i t y . T u n i n g of t h e d e t e c t o r was p e r f o r m e d a t 4 . 2 K i n a n o t h e r c r y o s t a t , the s e t t i n g s t h e n f i x e d and the c o m p l e t e d a s sembly mounted i n t h e d i l u t i o n r e f r i g e r a t o r . T h i s r e s u l t e d i n a much l o w e r s t a n d i n g wave r a t i o , t h e advantage of w h i c h can be seen i n F i g . 13. The c a v i t y r e s o n a n c e i s o b s e r v e d i n r e f l e c t i o n , on an uneven b a c k g r o u n d of s t a n d i n g waves . R e f l e c t i o n s between the c a v i t y and the d e t e c t o r cause most of the p r o b l e m s . F i g u r e 13a shows t h e c a v i t y r e s o n a n c e o b s e r v e d i n r e f l e c t i o n i n t h e f i r s t e x p e r i m e n t , w i t h o u t a matched d e t e c t o r . M a t c h i n g of t h e d e t e c t o r r e d u c e d t h e s t a n d i n g waves t o t h a t shown i n F i g . 13b. F i n a l l y , a magnet w i t h b e t t e r h o m o g e n e i t y , a p p r o x i m a t e l y one p a r t i n 10 5 o v e r t h e c a v i t y , was used ( A m e r i c a n M a g n e t i c s model n o . A - 9 0 4 ) . The d r i f t r a t e was r e d u c e d t o about 0 .066 G / h o u r . F i g u r e 12 InSb d e t e c t o r mount used i n experiment I. frequency (GHz) F i g u r e 13 C a v i t y r e s o n a n c e from the f i r s t e x p e r i m e n t (a) and from the second e x p e r i m e n t ( b ) . 42 4 . 2 R e s u l t s A l t h o u g h t h e f o u n t a i n pump v a l v e was s u c c e s s f u l l y used t o c l o s e t h e c e l l , s e v e r a l p r o b l e m s were e n c o u n t e r e d w h i c h p r e v e n t e d the c o l l e c t i o n of u s e f u l d a t a . Two of t h e s e prob lems were a s s o c i a t e d w i t h the f o u n t a i n pump, and t h e o t h e r was r e l a t e d t o v e r y f a s t microwave s a t u r a t i o n of t h e h y p e r f i n e t r a n s i t i o n when t h e c e l l was c l o s e d . F i l l i n g of the f o u n t a i n pump c o u l d be m o n i t o r e d by m e a s u r i n g the r e s o n a n t f r e q u e n c y of the c a v i t y : as t h e h e l i u m l e v e l r o s e t h e c a p i l l a r y r a d i u s i n c r e a s e d . T h i s i n c r e a s e d the amount of l i q u i d He i n t h e c e l l w h i c h i n c r e a s e d t h e a v e r a g e d i e l e c t r i c c o n s t a n t i n the c e l l t h e r e b y d e c r e a s i n g the r e s o n a n t f r e q u e n c y . The e a s i e s t way t o measure t h e c a v i t y s h i f t i s t o s e t t h e microwave f r e q u e n c y t o t h e s i d e of t h e r e s o n a n c e as i f f o r a d i s p e r s i o n measurement . Once the "He l e v e l s had r e a c h e d e q u i l i b r i u m t h e microwave s i g n a l was found t o be s t e a d y . U n f o r t u n a t e l y , t h e s l i g h t e s t i m p u l s e t o t h e sy s tem e x c i t e d U - t u b e o s c i l l a t i o n s i n t h e l i q u i d ( i . e . t a p p i n g o f the t o p o f t h e c r y o s t a t o r , as i l l u s t r a t e d i n F i g . 14a, r a p i d H r e c o m b i n a t i o n ) . The " H e , b e i n g s u p e r f l u i d , has no v i s c o s i t y and t h i s r e s u l t e d i n v e r y l o n g damping t i m e s , of o r d e r 10 m i n u t e s . I n a d d i t i o n , t h e s i g n a l - t o - n o i s e r a t i o f o r t h e l o w e r d e n s i t i e s was r a t h e r poor as i s i l l u s t r a t e d i n F i g . 14b. D i s p e r s i o n 43 measurements were t h e r e f o r e r e n d e r e d u s e l e s s . A b s o r p t i o n measurements had now t o be c o n s i d e r e d . These a r e t a k e n w i t h the microwave f r e q u e n c y s e t a t t h e c e n t e r of the c a v i t y r e s o n a n c e . To f i r s t o r d e r , t h e U - t u b e o s c i l l a t i o n s s h o u l d not be e x p e c t e d t o a f f e c t t h e s i g n a l , and t h i s has been v e r i f i e d e x p e r i m e n t a l l y . U n f o r t u n a t e l y , t h e microwave power l e v e l (^0.2 MW) was enough t o s a t u r a t e t h e t r a n s i t i o n w i t h a t i m e c o n s t a n t of about 10 msec , and the sy s tem was no t c a p a b l e of sweep ing t h r o u g h t h e a b s o r p t i o n i n a t i m e of 10 msec . or l e s s . H e n c e , the atoms were q u i c k l y d e s t r o y e d by microwave i n d u c e d r e c o m b i n a t i o n , as shown i n F i g . 14c . L a s t , but not l e a s t , was a p r o b l e m of l a r g e u n c o n t r o l l e d s h i f t s i n t h e "He l e v e l a t t h e c e l l . In p a r t i c u l a r t h e r e seemed t o be two s t a b l e l i q u i d l e v e l s a t the c e l l . Changes between the d i f f e r e n t l e v e l s wou ld o c c u r a f t e r o p e r a t i o n of t h e d i s c h a r g e i n an a p p a r e n t l y random manner . One of t h e s e p o s i t i o n s l e f t the c e l l p e r m a n e n t l y o p e n . A f u r t h e r p r o b l e m o c c u r e d near t h e end of t h e e x p e r i m e n t , when t h e r e was v i r t u a l l y no c o n t r o l of the l e v e l even w i t h t h e f o u n t a i n pump. T h i s l e d t o s h u t t i n g down t h e e x p e r i m e n t i n o r d e r t o implement the improvements d i s c u s s e d i n t h e n e x t c h a p t e r . ^ ^ ^ ^ F i g u r e 14 E x p e r i m e n t I I d i s p e r s i o n r e s u l t s a) c e l l o p e n , e x c i t a t i o n of U - t u b e o s c i l l a t i o n s b) c e l l c l o s e d , m i c r o w a v e s o f f d u r i n g a b s o r p t i o n c ) c e l l c l o s e d , microwave i n d u c e d r e c o m b i n a t i o n d u r i n g a b s o r p t i o n d e s t r o y s s a m p l e . Hence upper p a r t of d i s p e r s i o n s i g n a l i s no t p r e s e n t . 45 CHAPTER V F i n a l E x p e r i m e n t 5.1 Modi f i c a t i o n s I f t h e s i g n a l s were t o be o b s e r v e d i n a b s o r p t i o n r a t h e r t h a n d i s p e r s i o n as o r i g i n a l l y p l a n n e d , t h e m i c r o w a v e power i n c i d e n t on t h e c a v i t y had t o be l o w e r e d a b o u t f i v e o r d e r s of m a g n i t u d e i n o r d e r t o r e d u c e m icrowave i n d u c e d r e c o m b i n a t i o n , t o a t o l e r a b l e l e v e l . I t became c l e a r t h a t a h e t e r o d y n e d e t e c t i o n s y s t e m had t o be u s e d t o o b t a i n a r e a s o n a b l e s i g n a l - t o - n o i s e r a t i o . The f o l l o w i n g s e c t i o n d e a l s w i t h t h e s p e c t r o m e t e r i n d e t a i 1 . To f i r s t o r d e r U-tube o s c i l l a t i o n s s h o u l d , n o t a f f e c t t h e a b s o r p t i o n s i g n a l s s i n c e t h e y a r e t a k e n a t t h e minimum o f t h e c a v i t y r e f l e c t i o n s i g n a l ; however i t seemed w i s e t o a v o i d t h e i r e f f e c t s a s much as p o s s i b l e . To t h i s end l a r g e r r e s e r v o i r s on t h e f o u n t a i n pump and emergency pumping l i n e were i n s t a l l e d . T h e s e s h o u l d have t h e e f f e c t o f r e d u c i n g f l u c t u a t i o n s i n h e i g h t due t o any volume f l u c t u a t i o n s p r e s e n t . 46 5 .2 H e t e r o d y n e S p e c t r o m e t e r H e t e r o d y n e d e t e c t i o n i n v o l v e s t h e m i x i n g of a weak s i g n a l v o l t a g e w i t h a s t r o n g r e f e r e n c e ( l o c a l o s c i l l a t o r ) v o l t a g e . T h i s i s i n h e r e n t l y more s e n s i t i v e t h a n square law d e t e c t i o n , as used i n the p r e v i o u s two e x p e r i m e n t s , because a s t r o n g r e f e r e n c e v o l t a g e has r e p l a c e d a weak s i g n a l v o l t a g e a t the d e t e c t i o n m i x e r . W i t h an o u t p u t t i m e c o n s t a n t of 2 m i l l i s e c o n d s and a sy s tem n o i s e t e m p e r a t u r e of 6500 K t h e t h r e s h o l d n o i s e l e v e l wou ld be 1 0 " 1 7 w a t t s . At an i n c i d e n t power l e v e l of 4x10" 1 2 w a t t s , t h i s t r a n s l a t e s t o a minimum o b s e r v a b l e d e n s i t y of about 2 . 5 X 1 0 1 2 a t o m s / c m 3 . The microwave s p e c t r o m e t e r c o n s i s t s of t h r e e b a s i c p a r t s : a) p h a s e - l o c k sys tem b) microwave sys tem c ) 1460 MHz s p e c t r o m e t e r . Phase l o c k i n g of the k l y s t r o n i s done i n two s t e p s . F i r s t a 12-18 GHz backward wave o s c i l l a t o r i s l o c k e d t o a harmonic of a 0-500 MHz f r e q u e n c y s y n t h e s i z e r and t h e k l y s t r o n l o c k e d t o a h a r m o n i c of t h e 12-18 GHz s i g n a l . A p p e n d i x B c o n t a i n s t h e d e t a i l s of t h i s s y s t e m . The p h a s e - l o c k e d o u t p u t of t h e k l y s t r o n d r i v e s t h e microwave s y s t e m , t o be d e s c r i b e d b e l o w , w h i c h t h e n feeds t h e 1460 MHz s p e c t r o m e t e r . T h i s i s a c t u a l l y the same s p e c t r o m e t e r used f o r t h e z e r o f i e l d a t o m i c h y d r o g e n work done 47 i n t h i s l a b (a t 1420.405 MHz) but m o d i f i e d f o r use i n t h i s e x p e r i m e n t (see A p p e n d i x C ) . As shown i n t h e s c h e m a t i c i n F i g . 15 t h e k l y s t r o n o u t p u t (="112.6 GHz) i s f ed i n t o a power s p l i t t e r , one arm f e e d i n g t h e i n j e c t i o n c a v i t y f i l t e r , t h e o t h e r arm f e e d i n g t h e s i d e b a n d g e n e r a t o r . The s i d e b a n d g e n e r a t o r c o n s i s t s of two i s o l a t o r s , an a t t e n u a t o r , a d i r e c t i o n a l c o u p l e r and a m i x e r . I n t h e m i x e r t h e k l y s t r o n f r e q u e n c y i s m i x e d w i t h t h e 1460.405 MHz r e f e r e n c e s i g n a l s u p p l i e d by t h e 1460 MHz s p e c t r o m e t e r , i n j e c t e d i n t o t h e IF p o r t . I s o l a t i o n of t h e g e n e r a t e d s i d e b a n d s from t h e d e t e c t o r i s p r o v i d e d by t h e two i s o l a t o r s and t h e i n j e c t i o n c a v i t y . S i d e b a n d s a t . 111.2 GHz and 114.0 GHz, as w e l l as t h e c a r r i e r , a r e f e d i n t o t h e c r y o s t a t down the o r i g i n a l w a v e g u i d e . The upper s i d e b a n d i s t u n e d t o t h e c a v i t y r e s o n a n c e t h e r e b y e x c i t i n g t h e s p i n sy s tem w i t h t h e a p p r o p r i a t e m a g n e t i c f i e l d ; t h e c a r r i e r and l o w e r s i d e b a n d a r e c o m p l e t e l y r e f l e c t e d . L e a v i n g t h e c a v i t y , t h e m i c r o w a v e s a r e c a r r i e d up t o room t e m p e r a t u r e t h r o u g h an o v e r s i z e d w a v e g u i d e , 2 6 . 5 - 4 0 . 0 GHz b a n d , i n o r d e r t o m i n i m i z e t h e a t t e n u a t i o n . The s t a i n l e s s s t e e l w a v e g u i d e , needed f o r i t s low t h e r m a l c o n d u c t a n c e , i s c o p p e r p l a t e d on t h e i n s i d e t o reduce microwave a t t e n u a t i o n . Once out of t h e c r y o s t a t the m i c r o w a v e s a r e p a s s e d t h r o u g h a t r a n s m i s s i o n f i l t e r c a v i t y w h i c h i s t u n e d t o t h e upper s i d e b a n d ; t h i s i s needed t o r e j e c t the unwanted c a r r i e r and l o w e r s i d e b a n d . The t r a n s m i s s i o n f i l t e r i s an a d j u s t a b l e , h i g h Q, c o n f o c a l 1460 MHz SPECTROMETER 1460 MHz O U T P U T SIDEBAND GENERATOR INPUT KLYSTRON CRYOSTAT^ SIDEBAND FILTER o-QC UJ UJ 1-_ l O D_ Q. V) 1 HARMONIC MIXER MIXER INJECTION CAVITY L N A MIXER BIAS S A M P L E CAV ITY F i g u r e 15 M i c r o w a v e s p e c t r o m e t e r used i n t h e f i n a l e x p e r i m e n t . 49 r e s o n a t o r . These r e s o n a t o r s have i n h e r e n t l y h i g h Qs and a r e no t t o o d i f f i c u l t t o d e s i g n and f a b r i c a t e . D e s i g n d e t a i l s can be found i n ' I n t r o d u c t i o n t o O p t i c a l E l e c t r o n i c s ' (Amnon Y a r i v , 1 9 7 6 ) . F o r t h e mode used i n t h i s e x p e r i m e n t , t h e f i l t e r has a Q of about 4000 and an i n s e r t i o n l o s s of 3 d B . Upon l e a v i n g the f i l t e r the microwave s i g n a l goes d i r e c t l y t o the d e t e c t o r . S i n c e t h e i n j e c t i o n c a v i t y i s t u n e d t o t h e c a r r i e r f r e q u e n c y i t does not a b s o r b or t r a n s m i t any of t h e s i g n a l p o w e r . The s i g n a l i s mixed w i t h t h e c a r r i e r i n a low n o i s e S c h o t t k y b a r r i e r d i o d e (TRG W-965) whose n o i s e t e m p e r a t u r e was measured t o be T =6500 K w i t h t h e s t a n d a r d h o t / c o l d a b s o r b e r t e s t . N o i s e from the k l y s t r o n a t the s i g n a l f r e q u e n c y i s r e j e c t e d by t h e i n j e c t i o n c a v i t y (Q=1000) t h e r e b y p r o v i d i n g a c l e a n r e f e r e n c e s i g n a l . The IF s i g n a l (1460 .405 MHz) i s t h e n f e d i n t o a low n o i s e GaAs FET p r e - a m p l i f i e r w h i c h i n t u r n i s f e d i n t o t h e 1460 MHz s p e c t r o m e t e r . F i n a l l y , the o u t p u t of t h e s p e c t r o m e t e r i s m o n i t o r e d and c o l l e c t e d by t h e computer s y s t e m . 50 5 . 3 Method of Da ta C o l l e c t i o n O u t l i n e d i n t h i s s e c t i o n i s t h e method used t o c o l l e c t d a t a . I n c l u d e d i s a d i s c u s s i o n o f how the one-body s u r f a c e r e l a x a t i o n r a t e , g , was r e d u c e d t o a l e v e l where n u c l e a r p o l a r i z a t i o n c o u l d e v o l v e . The e f f e c t of microwave i n d u c e d r e c o m b i n a t i o n i s a l s o d i s c u s s e d . S e t t i n g up t h e s p e c t r o m e t e r f o r an a b s o r p t i o n measurement i s as f o l l o w s . The i n j e c t i o n c a v i t y , s i d e b a n d f i l t e r and phase ( a d j u s t a b l e i n t h e 1460 MHz s p e c t r o m e t e r ) a r e a d j u s t e d t o g i v e maximum s i g n a l . T h i s i s done i n a few i t e r a t i o n s . W i t h a t o m i c h y d r o g e n i n t h e c e l l t h e microwave f r e q u e n c y i s a d j u s t e d t o g i v e a d i s p e r s i o n f r e e s i g n a l , i . e . t h e f r e q u e n c y i s a d j u s t e d t o be e x a c t l y a t the bo t tom of the c a v i t y r e s o n a n c e . The phase and s i d e b a n d f i l t e r a r e t h e n r e a d j u s t e d and t h e p r o c e s s r e p e a t e d t o g i v e a d i s p e r s i o n f r e e s i g n a l w i t h a l l f r e q u e n c y s e n s i t i v e e l e m e n t s a t t h e i r maximum t r a n s m i s s i o n . Now the d i s c h a r g e i s t u r n e d on f o r about t e n m i n u t e s , w i t h t h e f o u n t a i n pump o p e n , i n o r d e r t o s a t u r a t e the hydrogen d e n s i t y . (Maximum d e n s i t i e s of s l i g h t l y above I 0 1 5 c m " 3 i n the c l o s e d c e l l c o u l d be a c h i e v e d i n t h i s s e t of r u n s . ) The d i s c h a r g e i s t u r n e d o f f , t h e c e l l c l o s e d and t h e computer c o n t r o l l e d d a t a c o l l e c t i o n s t a r t e d . A l t e r n a t e l y t h e b (Wb c ) and a (WQ ( j) l i n e a r e scanned w h i l e n b and n Q d e c a y . T h i s p r o c e d u r e 51 i s r e p e a t e d a t v a r i o u s t e m p e r a t u r e s i n t h e range 200-600 mK. T e m p e r a t u r e s l o w e r t h a n 200 mK c o u l d not be a t t a i n e d i n t h i s run because of a t h e r m a l s h o r t between one of the r e s e r v o i r s , a t t a c h e d t o the c e l l , and the t h e r m a l s h i e l d , a t t a c h e d t o t h e s t i l l . T h i s was u n d e s i r a b l e f o r two r e a s o n s . One, u s e f u l d a t a c o u l d have been o b t a i n e d below 200 mK and two , t h e absence of a c o n s t a n t heat l o a d on the m i x i n g chamber would have improved the sys tems c o o l i n g power and r e s p o n s e t i m e , w h i c h i s p a r t i c u l a r l y u s e f u l when the d i s c h a r g e i s shut o f f . N u c l e a r p o l a r i z a t i o n was no t o b s e r v e d i n the f i r s t s cans due t o t h e f a c t t h a t g > > ( K a a + K Q b ) n b . A c c i d e n t l y , t h e s o u r c e was o p e r a t e d c o n t i n u o u s l y , i n s t e a d of b e i n g p u l s e d , f o r a s h o r t p e r i o d of- t i m e . Scans a f t e r t h i s showed some n u c l e a r p o l a r i z a t i o n . I t i s s p e c u l a t e d t h a t a l a y e r of H 2 was b u i l t up i n t h e c e l l t h e r e b y i n c r e a s i n g t h e d i s t a n c e between the h y d r o g e n atoms on t h e l i q u i d "He s u r f a c e and t h e s u b s t r a t e ( e l e c t r o l y t i c t o u g h p i t c h c o p p e r , =0.999 p u r e ) . In o r d e r t o t e s t t h i s t h e o r y , and t o d e c r e a s e g f u r t h e r , a method f o r e f f i c i e n t l y c o n d e n s i n g H 2 i n t h e c e l l was d e v i s e d . The d i s c h a r g e was run w i t h the s p e c t r o m e t e r r e t u n e d so t h a t t h e s t r o n g c a r r i e r s i g n a l was a t t h e b l i n e r e s o n a n c e . The r e s u l t i n g microwave i n d u c e d r e c o m b i n a t i o n was l o c a l i z e d i n the c e l l and e n s u r e d t h a t most of t h e H 2 condensed e i t h e r i n the c e l l o r t h e f i n e c o n n e c t i n g t u b e t o t h e s o u r c e . D e p o s i t i o n r a t e s where e s t i m a t e d by m e a s u r i n g t h e t e m p e r a t u r e r i s e of t h e 52 c e l l d u r i n g microwave i r r a d i a t i o n . C o m p a r i s o n of t h i s t o the t e m p e r a t u r e r i s e due t o a known amount of power d i s s i p a t e d i n t h e c e l l h e a t e r e f f e c t i v e l y c a l i b r a t e d t h e r a t e a t w h i c h the h e a t of r e c o m b i n a t i o n was. d e p o s i t e d i n t h e c e l l and hence the microwave p o w e r . From t h e hea t d e p o s i t e d i n t h e c e l l due t o r e c o m b i n a t i o n , 4 . 7 5 eV per H 2 m o l e c u l e , a f l u x i n t o t h e c e l l of 2 . 8 x 1 0 1 2 H atoms per second was d e d u c e d . T h i s assumes t h a t a l l t h e atoms recombine and d e p o s i t t h e i r energy i n t h e c e l l . I n any ca se t h e d e p o s i t i o n r a t e was about 57 A per h o u r . L a y e r s of m o l e c u l a r hydrogen were b u i l t up i n t h e c e l l by f o l l o w i n g t h e above p r o c e d u r e f o r about t e n h o u r s a t a t i m e . A f t e r each p e r i o d of d i s c h a r g i n g t h e n u c l e a r p o l a r i z a t i o n was ' measured and found t o have i n c r e a s e d by a s u b s t a n t i a l f a c t o r (=2.5 f o r t h e f i r s t run and even more on subsequent r u n s ) . Two s e t s of d a t a o v e r the f u l l t e m p e r a t u r e range were t a k e n . One s e t was t a k e n when the f i r s t s u b s t a n t i a l n u c l e a r p o l a r i z a t i o n was o b s e r v e d and the o t h e r s e t a f t e r a few runs of t h e d i s c h a r g e . These r e s u l t s a r e p r e s e n t e d i n t h e n e x t c h a p t e r . The main o b j e c t i v e i n r e d u c i n g t h e microwave power i n c i d e n t on t h e c e l l was t o m i n i m i z e microwave i n d u c e d r e c o m b i n a t i o n . I r r a d i a t i n g atoms i n one of t h e l o w e r two h y p e r f i n e s t a t e s a t t h e ESR t r a n s i t i o n f r e q u e n c y r a i s e s them i n t o t h e a p p r o p r i a t e upper h y p e r f i n e s t a t e . As n o t e d i n C h a p t e r I I , the r e c o m b i n a t i o n r a t e i n v o l v i n g one atom i n t h e l o w e r two h y p e r f i n e s t a t e s i s a b o u t e ~ 2 = 2 . 5 x l 0 " f a s t e r t h a n r e c o m b i n a t i o n i n v o l v i n g 53 two atoms i n t h e l o w e r two s t a t e s . Atoms pumped i n t o one of t h e upper s t a t e s w i l l r ecombine on a t i m e s c a l e much f a s t e r t h a n any of t h e r a t e c o n s t a n t s a s s o c i a t e d w i t h atoms i n t h e l o w e r two s t a t e s . To e s t i m a t e t h e amount of microwave i n d u c e d r e c o m b i n a t i o n p r e s e n t i n t h e decay measurements t h e f o l l o w i n g e x p e r i m e n t was c o n d u c t e d . A p o l a r i z e d s ample , n h / n Q = 5 , was p r e p a r e d a t 400 mK ( w i t h t h e c e l l c l o s e d ) . In a p e r i o d of a few m i n u t e s , where t h e e f f e c t s of K Q a , K a b and g s h o u l d be n e g l i g i b l e , t h e b l i n e was scanned t h r o u g h e i g h t y t i m e s . T h i s r e d u c e d t h e d e n s i t y by about 8 p e r c e n t . A s e m i - l o g p l o t y i e l d s a l o s s i n i n t e n s i t y of 6 . 6 x 1 0 " " per sweep t h r o u g h t h e l i n e . A l t h o u g h t h i s seems s m a l l i t can c e r t a i n l y a f f e c t t h e l o n g e r decay measurements (up t o 80 a , b p a i r s ) . T h i s e f f e c t i s c o r r e c t e d f o r as d i s c u s s e d i n t h e n e x t c h a p t e r . 54 5 .4 Thermometry Low t e m p e r a t u r e thermometry r e q u i r e s a g r e a t d e a l of c a r e . In a d d i t i o n t o b e i n g s u b j e c t t o t h e u s u a l p r o b l e m s of t h e r m a l c o n t a c t and s e l f - h e a t i n g d u r i n g measurement , t h i s e x p e r i m e n t r e q u i r e s the c a l i b r a t i o n of the c e l l thermometer i n a f i e l d of 40 k G . I n t h i s s e c t i o n the t y p e s of thermometer s used i n t h i s e x p e r i m e n t a r e d e s c r i b e d , and t h i s i s f o l l o w e d by a d i s c u s s i o n of t h e methods used t o c a l i b r a t e t h e m . C o m m e r c i a l c a r b o n c o m p o s i t i o n r e s i s t o r s f rom M a t s u s h i t a (ERC-18GJ 100ft, 1/8 W) were c h o s e n f o r use i n t h e s e e x p e r i m e n t s f o r t h e f o l l o w i n g r e a s o n s . T h e i r s m a l l s i z e i s more c o n v e n i e n t t o work w i t h t h a n t h e u s u a l Grade 1002 Speer r e s i s t o r s (1 /2 W) , n o r m a l l y used be low 1 K, and a l s o have a l o w e r hea t c a p a c i t y . In a d d i t i o n , m a g n e t o r e s i s t a n c e e f f e c t s up t o 80 kG had been r e p o r t e d t o be s m a l l (see f o r example S a i t o and Sa to ( 1 9 7 5 ) ) . These r e s u l t s were c o n f i r m e d and w i l l be d i s c u s s e d b e l o w . F i n a l l y , t h e s e r e s i s t o r s change t h e i r v a l u e s v e r y l i t t l e upon t h e r m a l c y c l i n g between 4 .2 K and room t e m p e r a t u r e . The c a r b o n r e s i s t o r s were c a l i b r a t e d a g a i n s t a 3 He m e l t i n g c u r v e thermometer and as a c h e c k were compared w i t h two t r a n s i t i o n s of t h e NBS 768 S u p e r c o n d u c t i v e T h e r m o m e t r i c F i x e d P o i n t D e v i c e . A S c i e n t i f i c I n s t r u m e n t s germanium r e s i s t a n c e t h e r m o m e t e r , model 5-He3A, i s a l s o used as a t r a n s f e r s t a n d a r d 5 5 between r u n s . I . S h i n k o d a b u i l t and o p e r a t e d t h e 3He m e l t i n g c u r v e thermometer ( u s e d f o r t h e f i r s t t i m e i n t h e f i n a l e x p e r i m e n t ) , a d e s c r i p t i o n o f w h i c h can be f o u n d i n h i s M.Sc. t h e s i s ( 1 9 8 3 ) . T h i s thermometer was p a r t i c u l a r l y u s e f u l h e r e b e c a u s e of i t s i n s e n s i t i v i t y t o m a g n e t i c f i e l d s i n t h e t e m p e r a t u r e r a n g e u s e d i n t h i s e x p e r i m e n t . The F i x e d P o i n t D e v i c e , on t h e o t h e r hand, i s e x t r e m e l y s e n s i t i v e t o m a g n e t i c f i e l d s and t h e germanium r e s i s t o r t o a l e s s e r e x t e n t . Two c a r b o n r e s i s t o r s were p r e p a r e d and mounted as f o l l o w s . The r e s i s t o r s were f i r s t g r o u n d down on o p p o s i t e s i d e s , e x p o s i n g t h e c a r b o n c o m p o s i t i o n . The m i x i n g chamber thermometer was mounted i n a s l i t m a c h i n e d i n a c o p p e r r o d , w h i c h was a l s o t h r e a d e d on t h e o t h e r end. C i g a r e t t e p a p e r (=1 m i l t h i c k ) was wrapped a r o u n d t h e r e s i s t o r w h i c h was t h e n f o r c e d i n t o t h e s l i t i n t h e c o p p e r r o d w i t h a t h i n f i l m o f epoxy c o a t i n g e a c h p i e c e . Once t h e epoxy has s e t , t h e c o p p e r l e a d s t o t h e r e s i s t o r were wound a r o u n d t h e c o p p e r r o d and GE7031 v a r n i s h e d i n p l a c e . The f i n a l a s s e m b l y was t h e n b o l t e d o n t o t h e m i x i n g chamber. The c e l l t h ermometer was p r e p a r e d i n a s i m i l a r f a s h i o n . A f t e r b e i n g g r o u n d down, wrapped w i t h c i g a r a t t e p a p e r and c o a t e d w i t h epoxy i t was i n s e r t e d i n t o a s l o t i n t h e r e s o n a t o r body, see F i g . 10. T h r e e 0.002 / / c o p p e r shims were a d d e d t o make i t a t i g h t f i t . In o r d e r t o m i n i m i z e m a g n e t o r e s i s t i v e e f f e c t s on t h e m i x i n g chamber (MC) thermometer, i t was c o v e r e d w i t h a h o l l o w c y l i n d e r , w i t h one end c l o s e d , o f N b T i . As l o n g as t h e main magnet i s 56 e n e r g i z e d s l o w l y , f l u x p i n n i n g i n t h e N b T i s h o u l d p r e v e n t p e n e t r a t i o n o f t h e e x t e r n a l f i e l d , w h i c h i s a p p r o x i m a t e l y 5 kG a t the p o s i t i o n of t h e r e s i s t o r ( s e e , f o r e x a m p l e , S h i m i z u and Inoue ( 1 9 8 1 ) ) . The germanium r e s i s t o r was i n s e r t e d i n t o a copper b l o c k w i t h vacuum g r e a s e . The g r e a s e p r o v i d e s t h e r m a l c o n t a c t a t h i g h e r t e m p e r a t u r e s . A t l o w e r t e m p e r a t u r e s , t h e r m a l c o n t a c t i s a c h i e v e d t h r o u g h t h e l e a d s w h i c h a r e wrapped a r o u n d a copper p o s t on the mount w h i c h i s a l s o b o l t e d o n t o the m i x i n g chamber . T h i s thermometer was not m a g n e t i c a l l y s h i e l d e d and c o u l d o n l y be used a t z e r o f i e l d . The F i x e d P o i n t (FP) D e v i c e was o p e r a t e d a c c o r d i n g t o the i n s t r u c t i o n s p r o v i d e d w i t h i t . Care must be t a k e n t o . r e d u c e the ambient f i e l d a t t h e d e v i c e t o l e s s t h a n 10 mG. A mu-meta l s h i e l d p l a c e d a r o u n d t h e dewar r e d u c e d t h e e a r t h ' s f i e l d t o about 5 mG. I n a d d i t i o n t o r e m o v i n g any s t e e l b o l t s f rom t h e immedia te v i c i n i t y of t h e d e v i c e , i t was found t o be n e c e s s a r y t o c a r e f u l l y d e m a g n e t i z e t h e w e l d s on t h e s t a i n l e s s s t e e l vacuum c a n . These w e l d s were found t o c o n t a i n r e g i o n s w i t h u n u s u a l l y h i g h c o e r c i v e f i e l d s . D e s c r i b e d be low i s t h e p r o c e d u r e used t o c a l i b r a t e t h e c e l l t h e r m o m e t e r . The c a l i b r a t i o n c o u l d not be done i n an i d e a l way because of a p a r t i a l t h e r m a l s h o r t between t h e c e l l and the t h e r m a l s h i e l d . D u r i n g t h e run i n w h i c h t h e d a t a was c o l l e c t e d , t h e MC thermometer and t h e germanium r e s i s t o r (GR) were f i r s t 57 c a l i b r a t e d a g a i n s t the 3 He m e l t i n g curve thermometer (MCT) i n ze ro f i e l d . A l l r e s i s t a n c e measurements were c a r r i e d out w i t h a R V - E l e k t r o n i i k k a Oy AVS-45 A C , f o u r - t e r m i n a l , low power r e s i s t a n c e b r i d g e . C l o s e agreement between the MC thermometer, the GR and the SHE p r o v i d e d thermometer, based on p r e v i o u s c a l i b r a t i o n s , i n d i c a t e tha t t h e r e was no a p p r e c i a b l e thermal g r a d i e n t a c ro s s the body of the MC. There was, however, a l a r g e thermal g r a d i e n t between the c e l l and the MC. In the temperature range 200-400 mK the temperature d i f f e r e n c e between c e l l and MC was about 40 mK, d e c r e a s i n g at h i g h e r t empera ture s . Data from the p r e v i o u s run had then to be used to t r a n s f e r the c a l i b r a t i o n of the MC thermometer to the c e l l thermometer. In the p r e v i o u s run the MC and c e l l thermometers were c a l i b r a t e d a g a i n s t the GR i n z e r o f i e l d . The main magnet was then e n e r g i z e d to the working f i e l d and the c e l l thermometer r e c a l i b r a t e d a g a i n s t the s h i e l d e d MC thermometer. A l t h o u g h the a c t u a l temperature v a l u e s w i l l not be used h e r e , the comparison of the c e l l thermometer at 40 kG to the MC thermometer a l l o w s us to t r a n s f e r our r e c a l i b r a t i o n of the MC thermometer d e s c r i b e d above t o the c e l l thermometer at 40 kG. The f o l l o w i n g comments s h o u l d a l l e v i a t e any doubts about the above p r o c e d u r e . The most important assumption made was tha t the r e s i s t a n c e v a l u e s of the MC and c e l l thermometers d i d not change from the l a s t run to t h i s one. ' T h i s c o u l d i n f a c t be checked by comparing the r e s i s t a n c e va lue s o b t a i n e d i n both runs 58 a g a i n s t t h e GR v a l u e s w h i c h a r e e x p e c t e d t o be e x t r e m e l y r e p r o d u c e a b l e under t h e r m a l c y c l i n g . Between 200 and 600 mK the d i s c r e p e n c y between t h e two runs was no more t h a n 1 mK f o r the MC t h e r m o m e t e r . A l t h o u g h t h e c e l l thermometer c o u l d not be c h e c k e d i n t h i s way, r e s u l t s f rom p r e v i o u s runs i n d i c a t e t h a t a l l t h e M a t s u s h i t a r e s i s t a n c e thermometer s used a r e r e p r o d u c e a b l e under t h e r m a l c y c l i n g a t l e a s t f rom one run t o the n e x t . A check on the MCT and GR was p e r f o r m e d u s i n g the FP D e v i c e . T h i s d e v i c e was used i n a p r e v i o u s r u n t o check t h e GR c a l i b r a t i o n where a t each f i x e d p o i n t a v a i l a b l e , t h r e e i n a l l , t h e GR r e s i s t a n c e was r e c o r d e d . The GR c a l i b r a t i o n done t h i s run a g a i n s t t h e MCT g i v e s a t e m p e r a t u r e of 2 0 4 . 9 mK f o r t h e same r e s i s t a n c e v a l u e found i n t h e p r e v i o u s run f o r t h e FP t r a n s i t i o n t e m p e r a t u r e of 205 .7 mK. T h i s d i s c r e p e n c y i s l e s s t h a n t h e a c c u r a c y of the MCT c a l i b r a t i o n , d e t e r m i n e d t o be about 1 mK. A l t h o u g h t h e n e x t l o w e s t FP t r a n s i t i o n t e m p e r a t u r e i s be low t h e o p e r a t i n g range of t h i s e x p e r i m e n t , - i t was c h e c k e d as w e l l . The' GR t e m p e r a t u r e of 161.1 mK (MCT c a l i b r a t i o n ) i s t o be compared w i t h t h e FP t r a n s i t i o n t e m p e r a t u r e of 162.6 mK. C o n s i d e r a t i o n of the v a r i o u s s o u r c e s of e r r o r i n the c a l i b r a t i o n p r o c e d u r e i n d i c a t e s t h a t t h e f i n a l c e l l thermometer c a l i b r a t i o n i s a c c u r a t e t o ±2 mK. I n t h e i n t e r e s t of c o m p l e t e n e s s i t s h o u l d be n o t e d t h a t a l l o f t h e r e s i s t a n c e thermometer c a l i b r a t i o n s were f i t u s i n g t h e f o l l o w i n g e x p r e s s i o n 59 [ 5 . 1 ] l o g ( R ) = A ( l o g ( T ) ) 2 + B l o g ( T ) + C w h i c h a l w a y s gave a f i t t o b e t t e r t h a n 1 mK, even w i t h t h e GR. T h i s s t a n d a r d f o r m u l a was o b t a i n e d from K o b a y a s i e t a l . ( l 9 7 6 ) . The m a g n e t o r e s i s t a n c e o f t h e c e l l r e s i s t o r was a l s o m e a s u r e d . Between 40 kG and z e r o f i e l d , AR/R=-0 .023 a t 200 mK, c r o s s e s z e r o a t =360 mK and AR/R=+0.013 a t 600 mK. These r e s u l t s a r e s i m i l a r t o t h o s e q u o t e d i n van Y p e r e n ' s t h e s i s (1983) . 60 CHAPTER VI Data A n a l y s i s 6.1 I n i t i a l Data Format The f i r s t few s e c t i o n s of t h i s c h a p t e r d e a l w i t h t h e computer p r o c e s s i n g o f t h e raw d a t a w h i c h e x t r a c t s a d e n s i t y from each l i n e s h a p e ; t h e f i t t i n g of t h e d e n s i t y decay c u r v e s t o the r a t e e q u a t i o n s i s d e a l t w i t h i n t h e f i n a l s e c t i o n . A l t h o u g h t h e r e a r e r a t h e r a l a r g e number of d e t a i l s t o d i s c u s s , i t i s i m p o r t a n t t o e x p l a i n t h e f i t t i n g p r o c e s s i n d e t a i l . As w i l l be shown, i t i s sometimes p o s s i b l e t o change two p a r a m e t e r s i n such a way t h a t t h e q u a l i t y of the f i t i s not a p p r e c i a b l y d e g r a d e d . T h e r e f o r e , i f t h e f i n a l r e s u l t s a r e t o be t r u s t e d , one must have c o n f i d e n c e i n the d a t a a n a l y s i s . T h i s f i r s t s e c t i o n b e g i n s by d e s c r i b i n g t h e form of the raw d a t a as i n i t i a l l y c o l l e c t e d by t h e l a b c o m p u t e r . T h i s d a t a i s l a t e r t r a n s f e r r e d t o t h e u n i v e r s i t y ' s c e n t r a l computer where t h e b u l k of t h e p r o c e s s i n g i s p e r f o r m e d . As d e s c r i b e d e a r l i e r , each s e t of d a t a c o n s i s t s of p a i r s of b and a l i n e s p e c t r a c o l l e c t e d as t h e h y d r o g e n atom sample d e c a y s . E a c h s p e c t r u m must be a n a l y z e d t o e x t r a c t t h e d e n s i t y , n ^ , f rom t h e l i n e s h a p e . The l i n e s h a p e measured i s t h e change due t o t h e r e s o n a n c e of t h e H a toms , o f t h e c a v i t y r e f l e c t i o n c o e f f i c i e n t , T, f rom the u n p e r t u r b e d c a v i t y r e f l e c t i o n 61 c o e f f i c i e n t , T 0 . The s i g n a l i s t a k e n t o be [ 6 . 1 ] 6 C = ( r - r 0 ) / r 0 . D e f i n i n g y" and y/ as yy/=4TTQ^X" and y' =4irQ^xf , where i s the measured Q of t h e c a v i t y , a l l o w s one t o w r i t e [ 6 . 2 ] 6 C = 7? 7//+7«'2 + 7'2 + jy> ( 1 + 7 * ) z + y " T h i s e x p r e s s i o n i s d e r i v e d i n A p p e n d i x A . The c o u p l i n g p a r a m e t e r 7 7 = r 0 ~ 1 — 1 i s a c o n s t a n t t o be d e t e r m i n e d . B o t h the r e a l and i m a g i n a r y p a r t s of 6 C can be measured by an a p p r o p r i a t e a d j u s t m e n t of t h e phase of the d e t e c t e d s i g n a l r e l a t i v e t o the l o c a l o s c i l l a t o r . In t h i s e x p e r i m e n t h o w e v e r , as d i s c u s s e d i n the p r e c e e d i n g c h a p t e r , the phase i s a d j u s t e d so t h a t o n l y t h e r e a l p a r t of 6 C , 6, i s measured . I f the s i g n a l s a r e weak enough , i . e . yU«\, t h e s i g n a l i s p r o p o r t i o n a l t o the a b s o r p t i o n , w h i c h c o u l d t h e n be i n t e g r a t e d d i r e c t l y t o o b t a i n t h e d e n s i t y . T h i s i s no t t h e case h o w e v e r . At the b e g i n n i n g of each r u n , the l i n e s u s u a l l y have peak v a l u e s of y"-} . These s i g n a l s a l s o c o n t a i n a s i g n i f i c a n t c o n t r i b u t i o n from t h e d i s p e r s i o n squared terms i n [ 6 . 2 ] as can be seen i n F i g . 16. In o t h e r w o r d s , t h e hydrogen r e s o n a n c e i s s e v e r e l y p e r t u r b i n g the c a v i t y Q. What i s now r e q u i r e d i s a p r o c e s s f o r e x t r a c t i n g the a b s o r p t i o n l i n e s h a p e , y"', f rom t h e s i g n a l , 6. I n o r d e r t o do F i g u r e 16 A b s o r p t i o n 1 i n e s h a p e , . yff, as d e t e r m i n e d f rom s i g n a l , 6 , of a) a b l i n e and b) an a l i n e . 63 t h i s t h e c o u p l i n g p a r a m e t e r , 77, must a l s o be d e t e r m i n e d . S t a n d i n g waves make i t d i f f i c u l t t o e x t r a c t a r e l i a b l e v a l u e of IT0I 2 from t h e c a v i t y r e f l e c t e d power p l o t . One c a n , however, e x t r a c t rj from t h e measured s i g n a l s by c o m p a r i n g a weak s i g n a l w i t h a s t r o n g s i g n a l . Once t h i s i s done, a more r e l i a b l e v a l u e f o r t h e c a v i t y Q c a n be o b t a i n e d . T h i s p r o c e d u r e i s d e s c r i b e d i n t h e n e x t s e c t i o n . In p r a c t i c e t h e b a s e l i n e s i g n a l , r 0 , i s not s t e a d y o v e r t h e l o n g t e r m . D r i f t s i n t h e k l y s t r o n power, c a v i t y f i l t e r t u n i n g and l i q u i d He l e v e l a t t h e c e l l v a l v e c o n t r i b u t e t o a s l o w l y v a r y i n g b a s e l i n e . F o r t u n a t e l y t h e tim e s p e n t m e a s u r i n g one l i n e s h a p e , 0.5 s e c o n d s , i s much f a s t e r t h a n t h e t i m e s c a l e a s s o c i a t e d w i t h t h e d r i f t i n g b a s e l i n e . T h i s a l l o w s us t o assume a l i n e a r b a s e l i n e . In' t h e c a s e of weak s i g n a l s , where t h e d i s p e r s i o n c o n t r i b u t i o n t o t h e l i n e s h a p e i s n e g l i g i b l e , t h e r e g i o n a few l i n e w i d t h s away f r o m t h e c e n t e r of t h e s i g n a l i s f i t t e d t o a s t r a i g h t l i n e , t h e r e b y p r o v i d i n g a b a s e l i n e . • S t r o n g s i g n a l s , on t h e o t h e r hand, a r e f i t t o a s t r a i g h t l i n e p l u s a d i s p e r s i o n s q u a r e d t e r m , i . e . ( x - x 0 ) ~ 2 , where t h e b a s e l i n e i s t a k e n t o be t h e s t r a i g h t l i n e f i t . T h i s p r o c e d u r e r e l i e s on t h e f a c t t h a t t h e a b s o r p t i o n l i n e s h a p e , due t o t h e magnet i n h o m o g e n e i t y , i s r a t h e r w e l l l o c a l i z e d . T h e r e f o r e , 6 s = r j 7 / 2 a few l i n e w i d t h s away f r o m t h e c e n t e r of t h e l i n e . E a c h s i g n a l i s p r o c e s s e d i n t h i s manner t o d e t e r m i n e i t s b a s e l i n e . 64 6 . 2 D e t e r m i n i n g the C a v i t y C o u p l i n g P a r a m e t e r Compar ing two s i g n a l s , e s p e c i a l l y a s t r o n g , 6 S , and weak, 6 W , p a i r a l l o w one t o o b t a i n 77 and t h e r a t i o B=y^/y^ where 7^ i s t h e s t r o n g a b s o r p t i o n l i n e s h a p e and 7 ^ i s the weak a b s o r p t i o n l i n e s h a p e . T h i s assumes t h a t the b a s i c l i n e s h a p e i s d e n s i t y i n d e p e n d e n t as i s the ca se when magnet i n h o m o g e n e i t y d e t e r m i n e s t h e l i n e s h a p e . B e f o r e e l a b o r a t i n g on t h e d e t a i l s of the p r o c e d u r e , a h e u r i s t i c argument w i l l be g i v e n t o show w h i c h i n f o r m a t i o n i n the l i n e s h a p e s may be used t o e x t r a c t 77 and B and why i t i s d e s i r e a b l e t o use a s t r o n g and a weak l i n e . F i r s t assume 5 W i s s m a l l enough t h a t the a p p r o x i m a t i o n 6W=777// h o l d s ( i . e . 7" 2<<1 and 7 / 2 < < 1 ) . The K r a m e r s - K r o n i g r e l a t i o n can now be used t o c a l c u l a t e 777'. Now l e t us c o n s i d e r t h e wings of the s t r o n g s i g n a l where the a b s o r p t i o n i s e s s e n t i a l l y z e r o , i . e . 65—17/32 -y / 2 . T a k i n g t h e r a t i o between t h i s p a r t of t h e s i g n a l and (T77') 2 y i e l d s B2/r). A n o t h e r r e l a t i o n can be o b t a i n e d by c o n s i d e r i n g t h e c e n t e r r e g i o n of each s i g n a l where the d i s p e r s i o n c o n t r i b u t i o n v a n i s h e s . I n t h i s c a se os^r)By" and 6w=TJ7// whence 0 = 5 S / 6 W . These two q u a n t i t i e s t h e n y i e l d 77. S e v e r a l a p p r o x i m a t i o n s were used i n t h i s a r g u e m e n t ; n e v e r t h e l e s s , i t seems r e a s o n a b l e t h a t t h e two s i g n a l s can be used t o r e l i a b l y e x t r a c t 77 and B. In p r a c t i c e an i t e r a t i v e p r o c e d u r e i s u s e d , whereby c u r r e n t 65 v a l u e s f o r TJ and /3 a r e u p d a t e d u n t i l a s t a b l e s o l u t i o n i s r e a c h e d . An i n i t i a l guess f o r TJ i s o b t a i n e d f rom t h e c a v i t y r e f l e c t e d power p l o t . Now t h a t rj i s s e t , e q n . [ 6 . 2 ] can be r e w r i t t e n as [ 6 . 3 ] 7 / / = 6/(17-6) where y/ has been se t t o z e r o . T h i s a l l o w s an i n i t i a l v a l u e f o r P t o be d e t e r m i n e d by r a t i o i n g t h e peak v a l u e of y" f o r the s t r o n g and weak l i n e . The a b s o r p t i o n l i n e s h a p e i s o b t a i n e d from the weak s i g n a l u s i n g e q n . [ 6 . 3 ] and the c o r r e s p o n d i n g d i s p e r s i o n l i n e s h a p e c a l c u l a t e d u s i n g t h e K r a m e r s - K r o n i g r e l a t i o n . A n o n - l i n e a r f i t t i n g r o u t i n e i s now used t o d e t e r m i n e t h e b e s t v a l u e s of rj and 0 needed t o g e n e r a t e a s y n t h e t i c s t r o n g s i g n a l from t h e weak s i g n a l a b s o r p t i o n , y,f, w h i c h f i t s 6 S . The new v a l u e of TJ i s used t o e x t r a c t a b e t t e r e s t i m a t e of y". T h i s p r o c e d u r e i s r e p e a t e d u n t i l a s t e a d y s o l u t i o n i s r e a c h e d . Ten d i f f e r e n t 6 S , 5 W p a i r s f rom f o u r d i f f e r e n t runs were a n a l y z e d . The c o u p l i n g p a r a m e t e r was d e t e r m i n e d t o be TJ= 1 . 5 3 ± 0 . 0 8 . A q u e s t i o n a b l e a s s u m p t i o n was made i n t h i s a n a l y s i s and t h a t i s t h a t 7 / / = 6 w / ( T J - 6 w ) . T h i s e x p r e s s i o n i g n o r e s t h e 7 2 c o n t r i b u t i o n . An i n d i r e c t t e s t can be p e r f o r m e d t o judge t h e s i g n i f i c a n c e of t h i s a s s u m p t i o n . I f t h e weak s i g n a l i s assumed t o be g i v e n by 5 V l / = r j 7 / / , where t h e 7 / / 2 t e r m has been i g n o r e d , a v a l u e of 77= 1 .59 i s o b t a i n e d . T h i s i n d i c a t e s t h a t TJ 66 i s r e l a t i v e l y i n s e n s i t i v e t o s m a l l c o r r e c t i o n s t o t h e l i n e s h a p e o f t h e t y p e j u s t m e n t i o n e d . Once 77 was d e t e r m i n e d , a v a l u e f o r t h e c a v i t y Q was e s t i m a t e d , Q M = 3 7 0 0 ± 2 0 0 , from t h e dependence of t h e r e f l e c t e d power as a f u n c t i o n of f r e q u e n c y . 67 6.3 I n t e g r a t e d I n t e n s i t i e s Decay r u n s a r e a n a l y z e d i n a manner, s i m i l a r t o t h a t d e s c r i b e d i n t h e p r e v i o u s s e c t i o n . A s u i t a b l e weak s i g n a l i s c h o s e n , w i t h o u t t o o much n o i s e on i t , and i t s a b s o r p t i o n , y1', i s e x t r a c t e d . One r e f e r e n c e s i g n a l i s r e q u i r e d f o r e a c h of t h e a and b l i n e s s i n c e t h e t o t a l e f f e c t o f m a g n e t i c i n h o m o g e n e i t y i s d i f f e r e n t f o r e a c h o f t h e two l i n e s . A s y n t h e t i c s t r o n g s i g n a l i s g e n e r a t e d t o f i t t h e r e a l s i g n a l and t h e s c a l i n g p a r a m e t e r m u l t i p l i e s t h e r e f e r e n c e s i g n a l ' s i n t e g r a t e d a b s o r p t i o n i n t e n s i t y t o o b t a i n t h e s t r o n g s i g n a l ' s d e n s i t y . T h i s p r o c e d u r e i s f o l l o w e d u n t i l t h e r e f e r e n c e s i g n a l i s r e a c h e d , a t w h i c h p o i n t t h e a b s o r p t i o n i s e x t r a c t e d d i r e c t l y from t h e s i g n a l by t h e method t o be d e s c r i b e d below. Weak s i g n a l s have much l e s s o f a 7 / 2 c o n t r i b u t i o n , a l l o w i n g t h e f o l l o w i n g i t e r a t i v e p r o c e s s t o c a l c u l a t e y" from 6. R e w r i t i n g e qn. [6.2] t o o b t a i n y" as a f u n c t i o n of 6 and y' y i e l d s [ 6 . 4 ] y" = ( 2 7 c - 1 ) + / l-4 ( l-7c:) V 2 2 ( 1 - 7 s ) where 7 s = 5s/T?. T h i s e q u a t i o n r e q u i r e s y' t o be known i n o r d e r t o e v a l u a t e y" from 7 S . I n i t i a l l y e qn. [6.3] i s u s e d t o o b t a i n a t r i a l f u n c t i o n f o r y" w h i c h i s t h e n u s e d t o c a l c u l a t e y'. T h i s y' i s s c a l e d by a f a c t o r l e s s t h a n one and u s e d i n [6.4] t o 68 y i e l d a new yf/. The d i s p e r s i o n of t h i s 7" i s t h e n s c a l e d by a s l i g h t l y l a r g e r f a c t o r and i n s e r t e d i n t o [6.4] as b e f o r e . T h i s i s r e p e a t e d u n t i l t h e s c a l i n g f a c t o r e q u a l s one and t h e n f u r t h e r r e p e a t e d u n t i l a s t a b l e v a l u e o f y" has been o b t a i n e d . The y11 o b t a i n e d f r o m t h i s d i s p e r s i o n r e m o v a l t e c h n i q u e i s u s e d as a r e f e r e n c e a b s o r p t i o n t o g e n e r a t e s y n t h e t i c s t r o n g s i g n a l s . A s i m i l a r p r o c e d u r e t o t h a t u s e d t o e x t r a c t TJ i s now a p p l i e d t o e a c h s t r o n g s i g n a l , 6 S . The r e f e r e n c e y" i s s c a l e d by /3 t o g e n e r a t e a s y n t h e t i c s t r o n g s i g n a l w h i c h g i v e s t h e b e s t f i t t o 6 S. The i n t e g r a t e d i n t e n s i t y of t h e r e f e r e n c e 7 / / i s m u l t i p l i e d by j3 w h i c h i n t u r n i s u s e d t o c a l c u l a t e t h e d e n s i t y . Examples of t h e f i t s a r e shown i n F i g . 17. The f i r s t p l o t shows what we c o n s i d e r t o be an e x c e l l e n t f i t . Note t h a t t h e s y n t h e t i c l i n e s h a p e i s c o n s i d e r a b l y more n o i s y i n t h e wings t h a n the r e a l s i g n a l . T h i s i s due t o t h e s c a l e d up n o i s e i n t h e r e f e r e n c e y" as i n d i c a t e d by eqn. [ 6 . 2 ] . F i g u r e 17b shows an a c c e p t a b l e f i t and F i g . 17c shows an u n a c c e p t a b l e f i t . U s u a l l y i n t h e s e c a s e s t h e poor f i t r e s u l t s from a c e r t a i n amount of d i s p e r s i o n i n t h e r e f e r e n c e s i g n a l due t o poor p h a s e a d j u s t m e n t . I t i s n e c e s s a r y i n t h e s e c a s e s t o r e j e c t t h e e n t i r e r u n . S i g n a l s weaker t h a n t h e r e f e r e n c e s i g n a l a r e p r o c e s s e d w i t h t h e d i s p e r s i o n r e m o v a l t e c h n i q u e o n l y . F i n a l l y , when t h e i n t e g r a t e d i n t e n s i t y o b t a i n e d u s i n g t h i s t e c h n i q u e a g r e e s w i t h t h a t c a l c u l a t e d f r o m [ 6 . 3 ] , t o w i t h i n 0.1 p e r c e n t , t h e s t r a i g h t f o w a r d i n t e g r a t e d i n t e n s i t y i s u s e d . F i g u r e 17 S y n t h e t i c s i g n a l l i n e s h a p e s , a) an e x c e l l e n t f i t , b) an a c c e p t a b l e f i t and c ) an- u n a c c e p t a b l e f i t . The s y n t h e t i c l i n e s h a p e s have more n o i s e i n t h e wings t h a n the c o r r e s p o n d i n g measured s i g n a l . 70 E a c h run i s a n a l y z e d t o p r o d u c e a t a b l e of d e n s i t i e s v e r s u s t i m e . The d e n s i t i e s a r e a l t e r n a t e l y t h a t o f b atoms and a atoms a t e q u a l l y s p a c e d t i m e i n t e r v a l s . One can now e x t r a c t t h e r e c o m b i n a t i o n r a t e c o n s t a n t s , K Q Q and K a b , and t h e s u r f a c e r e l a x a t i o n r a t e , g, from t h e d e c a y d a t a . T h i s i s d e a l t w i t h i n t h e n e x t s e c t i o n . 71 6 .4 Decay Curve A n a l y s i s A n a l y s i s of t h e decay c u r v e s i n terms of the r a t e e q u a t i o n s g i v e n i n C h a p t e r I I i s d e s c r i b e d i n t h i s s e c t i o n . " To b e g i n , the r a t e e q u a t i o n s a r e r e s t a t e d , f o l l o w e d by a j u s t i f i c a t i o n f o r the o m i s s i o n of o t h e r known r a t e d e t e r m i n i n g p r o c e s s e s . M i c r o w a v e i n d u c e d r e c o m b i n a t i o n , w h i c h p e r t u r b s the decay c u r v e s , had t o be t a k e n i n t o a c c o u n t and t h i s i s d e s c r i b e d n e x t . T h i s i s f o l l o w e d by a d e s c r i p t i o n of t h e f i t t i n g p r o c e d u r e , i n c l u d i n g the e r r o r a n a l y s i s . F i n a l l y , an o u t l i n e of t h e a n a l y s i s of the two d a t a s e t s i s p r e s e n t e d . A t o m i c h y d r o g e n samples c o n f i n e d t o t h e microwave c a v i t y decay a c c o r d i n g t o the r a t e e q u a t i o n s [ 6 . 5 a ] n Q = - 2 K G Q n 2 - K a b n a n b - [ G (nQ+n b)+g ] [ ( n Q - n b ) - ( n a o - n b o ) ] [ 6 . 5 b ] n b = - K Q b n Q n b + [ G ( n Q + n b ) + g ] [ ( n Q - n b ) - ( n Q 0 - n b 0 ) ] as i n i t i a l l y d i s c u s s e d i n C h a p t e r I I . An example of the measured decay c u r v e s i s shown i n F i g . 18. The r a t e c o n s t a n t s t o be d e t e r m i n e d from the d a t a a r e K Q G , K Q b and g . C o n d i t i o n s a r e s u c h i n t h i s e x p e r i m e n t t h a t Gn b <<g. B u l k and s u r f a c e d i p o l e - d i p o l e r e l a x a t i o n c o n t r i b u t e t o G i n t h e f o l l o w i n g way: 72 2 E B / k B T [ 6 . 6 ] G=Gg + G s ( A / V ) A 2 e where A and V a re t h e c e l l a r e a and volume r e s p e c t i v e l y . The b u l k r e l a x a t i o n r a t e , Ggn^ ,• i s a t most f i v e p e r c e n t of the s u r f a c e r e l a x a t i o n r a t e g . T h i s c o u l d not be r e l i a b l y e x t r a c t e d f rom the d a t a , hence t h e t h e o r e t i c a l v a l u e of G g , d e t e r m i n e d i n C h a p t e r V I I I , was used i n t h e a n a l y s i s as a f i x e d p a r a m e t e r . I t s h o u l d be n o t e d t h a t t h e r e i s good agreement between the t h e o r e t i c a l v a l u e s of Gg and v a l u e s d e t e r m i n e d i n o t h e r exper i m e n t s . On the o t h e r h a n d , t h e p r e v i o u s l y r e p o r t e d e x p e r i m e n t a l v a l u e s of G s were about f i f t y t i m e s t h e t h e o r e t i c a l v a l u e s . The r e s u l t i n g c o n t r o v e r s y has r e c e n t l y i n c r e a s e d w i t h t h e s u g g e s t i o n by Hess e t a l . (1984) t h a t no one has y e t measured G s . They b e l i e v e t h a t t h e p r e v i o u s e x p e r i m e n t s were i n f a c t o n l y m e a s u r i n g t h e t h r e e - b o d y s u r f a c e r e c o m b i n a t i o n r a t e . In any c a s e , t h e t h e o r e t i c a l v a l u e of G s i s much t o o s m a l l t o have an e f f e c t i n t h i s e x p e r i m e n t , and o n l y a t 200 mK (where the l a r g e e x p o n e n t i a l f a c t o r i n [ 6 . 6 ] g i v e s a l a r g e v a l u e f o r G) would i t be p o s s i b l e t o see the e f f e c t of G s , even i f i t were as l a r g e as p r e v i o u s e x p e r i m e n t s had s u g g e s t e d . T h e r e f o r e , G s w i l l be d r o p p e d f rom t h e r a t e e q u a t i o n s . T h r e e - b o d y r e c o m b i n a t i o n i n t h e gas and on t h e s u r f a c e i s 73 00, "i 1 1 1 1 r i r -°o o •O ° o o o o o o o o o o 2 -O O Q <7> N o d ° u J I I L , D D o D p o p u p 0.0 0.3 0.6 0.9 TIME ( 105 S ) 1.2 1.5 F i g u r e 18 T y p i c a l d e c a y c u r v e , f r o m l o w g d a t a s e t , T= 3 0 5 m K . C i r c l e s a n d s q u a r e s r e p r e s e n t b a n d a a t o m s r e s p e c t i v e l y . N o t e t h a t t h e l i q u i d "He v a l v e h a s c l o s e d a t a p p r o x i m a t e l y 150 s e c . 74 a l s o i g n o r e d . The d e c a y r a t e f o r t h e s e p r o c e s s e s i s p r o p o r t i o n a l t o t h e d e n s i t y c u b e d and i s t o t a l l y n e g l i g i b l e i n t h i s e x p e r i m e n t . The f i n a l p r o c e s s t o be n e g l e c t e d i s d i p o l e - d i p o l e r e l a x a t i o n f r o m t h e u p p e r two h y p e r f i n e s t a t e s t o t h e l o w e r two h y p e r f i n e s t a t e s . The r a t e c o n s t a n t i s a t most ( 7 e / ( 7 p + e 7 e ) ) 2 G g w h i c h i s much s m a l l e r t h a n t h e r e l e v a n t r e c o m b i n a t i o n r a t e . As a r e s u l t , atoms e x c i t e d i n t o t h e upper two h y p e r f i n e s t a t e s w i l l r e c o m b i n e i n s t e a d o f r e l a x i n g t o t h e l o w e r s t a t e s . M i c r o w a v e i n d u c e d r e c o m b i n a t i o n o c c u r s d u r i n g e a c h measurement of t h e ESR l i n e s h a p e as d i s c u s s e d i n s e c t i o n 5.3. The f o l l o w i n g i s a d e s c r i p t i o n of t h e method us e d t o t a k e t h i s i n t o a c c o u n t i n t h e d e c a y c u r v e f i t t i n g p r o c e d u r e . D u r i n g t h e p e r i o d o f t h e microwave measurement, a p p r o x i m a t e l y 0.1 s e c o n d s t o sweep t h r o u g h t h e ESR a b s o r p t i o n , t h e c o n t r i b u t i o n s t o t h e d e c a y from K Q a , K a b and g a r e n e g l i g i b l e . T h e s e q u a n t i t i e s c a n t h e n be d r o p p e d from t h e r a t e e q u a t i o n s r e l e v a n t t o microwave i n d u c e d r e c o m b i n a t i o n w h i c h a r e : 75 [ 6 . 7 a ] f»d = + w a n a - K Q d n G n d - K b d n b n d - G x ( n b n d - n Q n c ) [ 6 . 7 b ] n c = + w b n b - K Q c n a n c - K b c n b n c + G x ( n b n d - n a n c ) [ 6 . 7 c ] n b = - w b n b - K b c n b n c - K b d n b n d - G x ( n b n d - n Q n c ) [ 6 . 7 d ] n Q = - w a n Q - K a c n Q n c - K Q d n Q n d + G x ( n b n d - n a n c ) . These r e c o m b i n a t i o n r a t e s a r e r e l a t e d t o K Q Q and K a b i n e q u a t i o n s [ 2 . 3 ] and [ 2 . 4 ] . G x i s the s p i n - e x c h a n g e r a t e c o n s t a n t f o r the t r a n s i t i o n ( b , d ) » (a ,c ) . A l l o t h e r s p i n - e x c h a n g e r a t e s a r e too s low t o be e f f e c t i v e . F o r d e t a i l s on the c a l c u l a t i o n of G x see A p p e n d i x E . O n l y one of the microwave t r a n s i t i o n r a t e s i s a c t i v e a t a t i m e . A g e n e r a l s o l u t i o n t o t h e f o u r n o n - l i n e a r , inhomogeneous , c o u p l e d r a t e e q u a t i o n s [ 6 . 7 ] i s , f o r t u n a t e l y , not r e q u i r e d . One r e q u i r e s o n l y the change i n n Q and n b , A n Q and A n b r e s p e c t i v e l y , a f t e r each microwave measurement , w h i c h i s t h e n used t o d e c r e a s e t h e atom d e n s i t i e s of t h e decay c u r v e s used i n t h e f i t t i n g p r o c e d u r e . I n o r d e r t o s o l v e f o r A n Q and A n b i t i s assumed t h a t n Q and n b a r e c o n s t a n t d u r i n g t h e measurement , c o r r e c t t o f i r s t o r d e r i n t h e e x c i t e d s t a t e d e n s i t i e s n c and n d . I n t e g r a t i n g the r a t e e q u a t i o n s o v e r the p e r i o d of t h e measurement y i e l d s , on the l e f t hand s i d e of [ 6 . 7 ] , z e r o f o r the f i r s t two e q u a t i o n s and A n a and A n b f o r t h e second two e q u a t i o n s . The fo rmer r e s u l t i s a consequence of n c and n d b e i n g e s s e n t i a l l y z e r o b e f o r e and 76 a f t e r t h e measurement . E q u a t i o n s [ 6 . 7 a ] and [ 6 . 7 b ] can now be s o l v e d f o r t h e i n t e g r a l s of n c and n d w h i c h a r e t h e n s u b s t i t u t e d i n t o e q u a t i o n s [ 6 . 7 c ] and [ 6 . 7 d ] . The f i n a l r e s u l t i s [ 6 . 8 ] A n h = -W h n h (1+f ) An_h = - W h r » h ( 1 - f ) where i f h i s a or b t h e n - h i s b o r a r e s p e c t i v e l y . A l s o [ 6 . 9 ] f = [ 2 n h n h + ( 1 + 7 / 2 ) n ^ + 2€ 2 G x Ka D (n^-n_ 2 1 ) ] x[ ( 1+7/2) ( n ^ n ^ ) + ( 7 2 / 8 + 7 / 2 + 5 / 2 ) n h n h + 2e 2 G xKa D (n^+n 2 n +( 1+ 7 /2)n h n_ h ) ]" 1 where i s t h e f r a c t i o n of h atoms e x c i t e d i n t o t h e upper s t a t e s d u r i n g t h e microwave measurement and 7 = K a a / K a t ) . Note t h a t t h e s e e q u a t i o n s have the p r o p e r t y t h a t A n a + A n b = - 2 W n n b , i . e . two atoms a r e l o s t per mic rowave t r a n s i t i o n . M e a s u r i n g W=WA=WB was done as f o l l o w s . A sample of r e a s o n a b l y h i g h p o l a r i z a t i o n was p r e p a r e d a f t e r w h i c h many measurements o f e i t h e r t h e b or a l i n e were made i n r a p i d s u c c e s s i o n . P l o t t i n g t h e d e n s i t y v e r s u s sweep number on a s e m i l o g p l o t , as i n F i g . 19, a l l o w s one t o e x t r a c t W=6 .6X10~" from t h e s l o p e . F i t t i n g of the decay c u r v e s w i l l now be d i s c u s s e d . A n o n - l i n e a r f i t t i n g r o u t i n e (UBC Comput ing C e n t e r r o u t i n e : FLETCH, based on a q u a s i - N e w t o n method) was used t o f i t t h e d a t a F i g u r e 19 M i c r o w a v e ' i n d u c e d r e c o m b i n a t i o n d e c a y . Log d e n s i t y v s . sweep number . 78 to a c a l c u l a t e d decay curve w i t h K Q Q , K Q h , g , n Q ( 0 ) , n b ( 0 ) as the f i t t i n g parameter s . A l t h o u g h the i n i t i a l d e n s i t i e s are a l s o f i t t e d , they are i n f ac t very w e l l de termined by the d a t a . Hence there are e s s e n t i a l l y o n l y three v a r y i n g parameters to be f i t t e d to two c u r v e s . Each data set c o n s i s t s of between 20 and 80 b , a p a i r s p r o v i d i n g many e x t r a degrees of freedom for the f i t . Other f i x e d parameters used to c a l c u l a t e the decay curve are G , W and T , the t e m p e r a t u r e . Use of the f i r s t two parameters has a l r e a d y been d i s c u s s e d . The temperature i s r e q u i r e d to c a l c u l a t e the Boltzmann f a c t o r s neces sary to ensure tha t the r e l a x a t i o n p r o c e s s e s tend to r e s t o r e the h y p e r f i n e p o p u l a t i o n s to t h e i r thermal e q u i l i b r i u m v a l u e s . I t shou ld be noted here that h e a t i n g e f f e c t s due to r e c o m b i n a t i o n are n e g l i g i b l e in t h i s exper iment because of the low d e n s i t i e s u sed . The t h e o r e t i c a l decay curve i s generated by i n t e g r a t i n g the r a t e e q u a t i o n s [6 .5] w i t h a s i x t h order Runge-Kutta method. The r e s i d u a l s between the data and the t h e o r e t i c a l decay curve are c a l c u l a t e d and used by the f i t t i n g r o u t i n e to min imize the sum of the r e s i d u a l s squared , x 2 . In order to have c o n f i d e n c e i n the f i t the f o l l o w i n g checks were implemented. An e s t imate of the s t a t i s t i c a l e r r o r i n each parameter was c a l c u l a t e d , w i t h i n a 90 p e r c e n t c o n f i d e n c e i n t e r v a l , u s i n g the c o v a r i a n c e mat r ix s u p p l i e d by FLETCH. More d e t a i l e d i n f o r m a t i o n can be o b t a i n e d by d i a g o n a l i z i n g the mat r ix of second d e r i v a t i v e s of x 2 . The r e s u l t i n g e i g e n v e c t o r s c o r r e s p o n d to the 79 d i r e c t i o n s i n p a r a m e t e r s p a c e of t h e minimum and maximum a s c e n t from t h e optimum v a l u e o f x 2 . The e i g e n v a l u e s e q u a l t h e c u r v a t u r e o f x 2 a l o n g t h e d i r e c t i o n o f t h e c o r r e s p o n d i n g e i g e n v e c t o r . T h e r e f o r e , i f t h e p a r a m e t e r s a r e c h a n g e d from t h e i r optimum v a l u e s a l o n g an e i g e n v e c t o r w i t h a l a r g e e i g e n v a l u e , x 2 c h a n g e s r a p i d l y , r e s u l t i n g i n a s m a l l e r r o r e s t i m a t e f o r t h e c o r r e s p o n d i n g p a r a m e t e r v e c t o r . C o n v e r s e l y , a l a r g e e r r o r r e s u l t s from moving i n t h e d i r e c t i o n of an e i g e n v e c t o r w i t h a s m a l l e i g e n v a l u e s i n c e a l a r g e r change i n t h e p a r a m e t e r s i s needed t o g i v e a s i m i l a r change i n x 2 . T h i s i n f o r m a t i o n i s c r u c i a l i n d e t e r m i n i n g t h e b e s t way t o e x t r a c t t h e r a t e c o n s t a n t s from t h e d a t a . The d a t a s e t w i t h l o w e r v a l u e s of g ( h i g h e r p o l a r i z a t i o n ) was a n a l y z e d f i r s t . V a l u e s of K a a , K a b and g w i t h r e a s o n a b l e e s t i m a t e d e r r o r s were o b t a i n e d . An example of t h e f i t s o b t a i n e d i s shown i n F i g . 20a. A n a l y z i n g t h e d a t a s e t w i t h h i g h v a l u e s of g (low p o l a r i z a t i o n ) i n t h e same way r e s u l t e d i n p o o r l y d e t e r m i n e d p a r a m e t e r s . The e i g e n v a l u e s o f t h e s e c o n d d e r i v a t i v e m a t r i x o f X 2 i n d i c a t e d t h a t a l i n e a r c o m b i n a t i o n i n v o l v i n g t h e d i f f e r e n c e between K Q a and K a b was e s s e n t i a l l y u n d e t e r m i n e d . P h y s i c a l l y t h i s i s r e a s o n a b l e s i n c e s a m p l e s w i t h low p o l a r i z a t i o n d e c a y a t a r a t e d e t e r m i n e d by t h e a v e r a g e r e c o m b i n a t i o n r a t e c o n s t a n t , K a v = ( K a a + K a b ) / 2 . Thus, t h e two d e c a y c u r v e s do n o t p r o v i d e much i n f o r m a t i o n on t h e d i f f e r e n c e o f K a a and K a b . T h e r e f o r e t h e i i i i i i i ' ' ' ; i i 1 0.0 0.4 0.8 12 16 2.0 2.4 2.8 TIME ( 10* S ) TIME ( 10' S ) F i g u r e 20 a ) Low g d a t a s e t , T= 456 mK b) H i g h g d a t a s e t , T= 501 mK C i r c l e s a n d s q u a r e s r e p r e s e n t b a n d a a t o m s r e s p e c t i v e l y . 81 h i g h g d a t a se t was a n a l y z e d by h o l d i n g t h e v a l u e of K a b f i x e d w h i l e K a a and g were f i t t e d . V a l u e s of K a b were o b t a i n e d from t h e p r e v i o u s d a t a s e t , as d i s c u s s e d i n the n e x t c h a p t e r . T h i s p r o c e d u r e gave r e a s o n a b l e e r r o r e s t i m a t e s f o r the f i t t e d p a r a m e t e r s . F i g u r e 20b shows a f i t o b t a i n e d f rom t h i s d a t a s e t . A check f o r t h e p r e s e n c e of s y s t e m a t i c e r r o r s was a l s o p e r f o r m e d . A c o r r e l a t i o n f u n c t i o n , d e f i n e d t o be the t i m e a v e r a g e d p r o d u c t of a r e s i d u a l a t t i m e t w i t h a r e s i d u a l a t t i m e t + r , was p l o t t e d v e r s u s r f o r each f i t . As i l l u s t r a t e d i n F i g . 21a , t h e c o r r e l a t i o n f u n c t i o n of t h e f i t shown i n F i g . 20a can be d e s c r i b e d as a d e l t a f u n c t i o n a t 7=0 and w h i t e n o i s e f o r T>0. T h i s i m p l i e s t h a t i f a s y s t e m a t i c e r r o r e x i s t s , i t i s below the n o i s e l e v e l . Most of t h e d a t a a n a l y z e d have c o r r e l a t i o n f u n c t i o n s of t h i s t y p e . F i g u r e 21b c o n t a i n s t h e c o r r e l a t i o n f u n c t i o n of the f i t i n F i g . 20b . C l e a r l y a s m a l l s y s t e m a t i c e r r o r i n t h e f i t i s p r e s e n t . T h i s t y p e of c o r r e l a t i o n f u n c t i o n i s p r e s e n t m a i n l y i n the h i g h t e m p e r a t u r e , h i g h g d a t a s e t . As d i s c u s s e d b e l o w , t h i s i s b e l i e v e d t o change t h e p a r a m e t e r s by a t most an amount a p p r o x i m a t e l y e q u a l t o t h e i r e s t i m a t e d s t a t i s t i c a l e r r o r . H e n c e , t h e r e s u l t s a r e not a d v e r s e l y a f f e c t e d by t h e p r e s e n c e of such s m a l l s y s t e m a t i c e r r o r s . P r e l i m i n a r y a n a l y s e s o m i t t e d G g as w e l l as t h e e f f e c t s of microwave i n d u c e d r e c o m b i n a t i o n . I n b o t h c a s e s t h e v a l u e s of x 2 were s i g n i f i c a n t l y improved when t h e s e e f f e c t s where t a k e n i n t o a c c o u n t . T h i s i s i l l u s t r a t e d i n F i g . 22 where x 2 i s p l o t t e d 82 la) 2 * m O Ul o ct: d ct: O o in d i I " o o o • • o i r i r n 1 r ° ° ° e 9 ° o 8 ° ° H ° . ° > n n" au£>0 O n 0 " OQ 0^ • uu „ rr °°o ° o0D00oeeo80tt0g°ooo0 • D I I L J I L 0.0 8.0 16.0 24.0 32.0 40.0 TIME DIFFERENCE (TIME/AB-PAIR) Oft lb) Z d O UJ o Ct: d Cr: O O m d I OO I- • OO q T i i i i i i r D e ° D D D ° D D Oo O o. o o 0 QOQO OQOOO ••• 0D|- | ° o ° o 0 o o O Q o o o J I I I L J I L 0.0 8.0 16.0 24.0 32.0 40.0 TIME DIFFERENCE (TIME/AB-PAIR) F i g u r e 21 C o r r e l a t i o n f u n c t i o n of t h e c o r r e s p o n d i n g f i t s i n F i g u r e 2 0 , (a) and ( b ) . 1.3 1.1 B O w 0 .9 x o o o a 0.7 * A A A 0^5 I I I I I I I I 0 8 W ( 10 3 ) F i g u r e 22 X 2 dependence on W f o r t h r e e d i f f e r e n t decay c u r v e s . W i s t h e f r a c t i o n of atoms e x c i t e d i n t o t h e upper h y p e r f i n e s t a t e s per measurement . 84 v e r s u s W. A s u b s t a n t i a l improvement i n x 2 r e s u l t e d when t h e p r o p e r v a l u e o f W was u s e d . The c o r r e l a t i o n f u n c t i o n s a l s o i m p r o v e d when t h e e f f e c t of mic r o w a v e i n d u c e d r e c o m b i n a t i o n was i n c l u d e d . The f i t t e d p a r a m e t e r s c h a n g e d by an amount e q u a l t o a few s t a n d a r d d e v i a t i o n s . A l t h o u g h t h e c o r r e l a t i o n f u n c t i o n s i m p r o v e d , some of them s t i l l i n d i c a t e t h e p r e s e n c e o f s m a l l s y s t e m a t i c e r r o r s i n t h e f i t . Some o f t h i s e f f e c t c o u l d be due t o t h e model u s e d t o d e s c r i b e t h e microwave i n d u c e d r e c o m b i n a t i o n . F o r example, g o i n g from t h e l i m i t i n w h i c h t h e s p i n - e x c h a n g e r a t e G x i s much l e s s t h a n t h e r e c o m b i n a t i o n r a t e s K Q C e t c . t o one i n w h i c h G x i s much l a r g e r t h a n K Q C e t c . c h a n g e s t h e p a r a m e t e r s by, a t most, one s t a n d a r d d e v i a t i o n . A l t h o u g h i t s t i l l a p p e a r s as i f s o m e t h i n g may be e i t h e r wrong w i t h , o r l e f t o u t o f , t h e model f o r t h e d e c a y c u r v e s , t h e r e m a i n i n g d i s c r e p a n c i e s a r e so s m a l l as t o be o f l i t t l e c o n s e q u e n c e t o t h e r e s u l t s p r e s e n t e d i n t h e n e x t c h a p t e r . 8 5 CHAPTER V I I R e s u l t s R e s u l t s o b t a i n e d from t h e d a t a a n a l y s i s d i s c u s s e d i n the p r e v i o u s c h a p t e r a re p r e s e n t e d h e r e . The t e m p e r a t u r e dependence of t h e v a r i o u s r a t e c o n s t a n t s a r e d i s p l a y e d i n t h i s c h a p t e r , i n c l u d i n g an e x p l a n a t i o n of t h e i r p h y s i c a l s i g n i f i c a n c e . F i n a l l y , t h e r e s u l t s of a l i n e s h a p e s t u d y , measured as a f u n c t i o n of a p p l i e d m a g n e t i c f i e l d g r a d i e n t , a r e g i v e n . The r e c o m b i n a t i o n r a t e c o n s t a n t s w i l l be d i s c u s s e d f i r s t . P l o t t e d i n F i g . 2 3 a r e t h e l o g s of k Q Q / T ( s q u a r e s and t r i a n g l e s ) and K Q h / f ( c i r c l e s ) v e r s u s T~ 1 . D a t a from the low g d a t a s e t a r e r e p r e s e n t e d by t h e s q u a r e s and c i r c l e s and t h e t r i a n g l e s r e p r e s e n t d a t a from t h e h i g h g d a t a s e t . R e c a l l t h a t t h e low and h i g h g d a t a s e t s r e f e r t o t h i c k and t h i n H 2 c e l l c o a t i n g s w h i c h g i v e weak and s t r o n g one-body s u r f a c e r e l a x a t i o n r e s p e c t i v e l y . -It i s e v i d e n t t h a t t h e v a l u e s o f K Q a f rom b o t h d a t a s e t s a r e c o n s i s t e n t w i t h each o t h e r . We t a k e t h i s as e v i d e n c e t h a t t h e p r o c e d u r e used t o a n a l y z e t h e l a r g e g d a t a s e t ( u s i n g K G h v a l u e s f rom a f i t t o the low g d a t a s e t ) i s a c c e p t a b l e . The s o l i d l i n e s a r e a f i t t o t h e d a t a s e t , e x c l u d i n g t h e p o i n t s a t 5 0 0 mK. A l t h o u g h t h e d a t a a t 5 0 0 mK i s p r e s e n t e d , i t i s under s u s p i c i o n . A t 6 0 0 mK the c e l l c o u l d not be made t o c l o s e and i t i s s u s p e c t e d t h a t the runs a t 5 0 0 mK a l s o had t h i s p r o b l e m s i n c e F i g u r e 23 Log p l o t of K a C j and K a b v s . T" 1 ( E r r o r b a r s l e s s t h a t t h e symbol s i z e a r e no t p l o t t e d ) . C i r c l e s and s q u a r e s r e p r e s e n t d a t a f rom the l o w ' g d a t a s e t . T r i a n g l e s r e p r e s e n t d a t a from t h e h i g h g d a t a s e t . F i g u r e 24 T e m p e r a t u r e dependence of 7 = K Q a / K a b . Square s and t r i a n g l e s r e p r e s e n t d a t a f rom t h e low g and h i g h g d a t a s e t s r e s p e c t i v e l y . 88 a l l r a t e s ( K Q a , K a b and g) a t t h i s t e m p e r a t u r e l i e above what one w o u l d e x t r a p o l a t e f rom the l o w e r t e m p e r a t u r e d a t a . A l t h o u g h t h e r e c o m b i n a t i o n r a t e s h o u l d i n d e e d i n c r e a s e a t h i g h e r t e m p e r a t u r e s where t h e "He d e n s i t y i s l a r g e enough t o e n a b l e the H+H+He—> H 2+He p r o c e s s , t h i s s h o u l d not o c c u r u n t i l about 800 mK where the s u r f a c e and b u l k "He r e c o m b i n a t i o n r a t e s a r e e q u a l . F i g u r e 23 shows t h a t not o n l y i s Kaa>KQ|-> but t h a t the two r a t e s have a d i f f e r e n t t e m p e r a t u r e dependence . T h i s i s shown more c l e a r l y i n F i g . 24 where t h e q u a n t i t y 7 = K a a / K a b * s p l o t t e d . The t e m p e r a t u r e dependence of 7 i l l u s t r a t e s t h e p r o b l e m of t r y i n g t o e x t r a c t the b i n d i n g e n e r g y of H on "He from the r e c o m b i n a t i o n r a t e c o n s t a n t s . N o r m a l l y one p r o c e e d s by r e l a t i n g t h e measured K t o the i n t r i n s i c s u r f a c e K s ( i n the same way as G i s r e l a t e d t o G s ( e q u a t i o n [ 6 . 6 ] ) ) 2Ep/k QT [ 7 . 1 ] K = K s ( A / V ) A 2 e I f one assumes t h a t the o n l y t e m p e r a t u r e dependence of K s i s t h e f a c t o r of / T i n t h e average H atom v e l o c i t y t h e n t h e s l o p e i n p l o t s l i k e t h a t of F i g . 23 i s e q u a l t o t w i c e the b i n d i n g e n e r g y . However i t i s c l e a r t h a t one o f , and p r o b a b l y b o t h , K Q Q and K a b have t e m p e r a t u r e dependences o t h e r t h a n / T . Most e x p e r i m e n t s t o da te have measured t h e average r a t e c o n s t a n t K Q V = ( K Q Q + K a h ) / 2 . P l o t t i n g l o g ( K a v / T ) v e r s u s T ~ 1 , as i n 89 F i g . 2 5 , f o r t h e low g d a t a s e t y i e l d s a v a l u e f o r E g of 1 .101 .02 K . The average s u r f a c e r e c o m b i n a t i o n r a t e c o n s t a n t i s then K S T ' 1 / 2 B 2 = 5 .71 .8 c m 2 s " ' K " 1 / 2 G 2 . The one-body r e l a x a t i o n r a t e c o n s t a n t f rom b o t h of the d a t a s e t s i s p l o t t e d i n F i g . 26 . R e c a l l t h a t t h e h i g h g d a t a se t c o r r e s p o n d s t o a t h i n n e r H 2 c o a t i n g i n the c e l l . t h a n the low g d a t a s e t . g i s r e l a t e d t o t h e s u r f a c e r a t e c o n s t a n t , g g , by E B / k B T [ 7 . 2 ] g = g s ( A / V ) A e F i g u r e 27 , w h i c h i s a p l o t of In g s v e r s u s T " 1 , shows t h a t the s u r f a c e r e l a x a t i o n r a t e g s i s s t r o n g l y t e m p e r a t u r e d e p e n d e n t . Q u a l i t a t i v e l y , t h e d a t a can be e x p l a i n e d u s i n g the f o l l o w i n g m o d e l . Assume the c e l l w a l l s c o n t a i n a d i l u t e c o n c e n t r a t i o n of m i c r o s c o p i c m a g n e t i c p a r t i c l e s . Hydrogen atoms t r a v e l i n g on the s u r f a c e w i l l e x p e r i e n c e t h e r e s u l t i n g inhomogeneous m a g n e t i c f i e l d s , t h e r e b y c a u s i n g r e l a x a t i o n . The w i d t h of the s p e c t r a l d e n s i t y f u n c t i o n , ACJ , a s s o c i a t e d w i t h t h e f l u c t u a t i n g f i e l d each H atom sees i s of t h e o r d e r of the t h e r m a l v e l o c i t y of H atoms on the s u r f a c e d i v i d e d by the s e p a r a t i o n of t h e "He s u r f a c e f rom the c e l l w a l l s , d . Compared t o t h e t r a n s i t i o n f r e q u e n c y from b—>a, of W o - 1 0 1 o S ~ 1 , ACJ=1 O ' c m s - 1 / 5 0 0 A = 2 X 1 0 9 S " 1 i n d i c a t i n g t h a t co0 i s i n t h e h i g h f r e q u e n c y t a i l of the s p e c t r a l d i s t r i b u t i o n f u n c t i o n . H e n c e , as t h e t e m p e r a t u r e i s r a i s e d , Ao> i n c r e a s e s , and so does g s . 90 F i g u r e 25 Log p l o t of K Q V = ( K a a + K a b ) / 2 v s . T " 1 f rom t h e low g d a t a s e t . T (K) Figure 26 One-body relaxation rate from both data sets. The squares represent data from the low g data set and are plotted with an expanded scale in the bottom graph. The s o l i d l i n e s are f i t s to the data as described in the text. CN 1 1 A 1 i 1 1 A A A — A A in A A in d — s—v < -c n ^ • Z d _ J 1 • • in • — • i - D • — in CN 1 1 1 1.0 2.0 3.0 4.0 r oo F i g u r e 27 Log p l o t of g s , t h e s u r f a c e r e l a x a t i o n r a t e c o n s t a n t . The t r i a n g l e s and s q u a r e s r e p r e s e n t d a t a f rom the h i g h g and low g d a t a s e t s r e s p e c t i v e l y . 93 A t h e o r y c a l c u l a t i n g g s a s suming f r e e p a r t i c l e m o t i o n of t h e H atoms has been d e v e l o p e d by B e r l i n s k y ( 1 9 8 4 ) . R e s u l t s of t h i s t h e o r y f o r d=300 A and d=550 A a r e p l o t t e d w i t h the h i g h g and low g d a t a r e s p e c t i v e l y i n F i g . 26 . I t i s i m p o r t a n t t o n o t e t h a t t h e o n l y p a r a m e t e r changed t o get t h e two c u r v e s i s t h e t h i c k n e s s d . The upper c u r v e i s f i t t o one d a t a p o i n t and t h e t h i c k n e s s d v a r i e d t o f i t t h e s l o p e . The l o w e r c u r v e i s t h e n f i t by i n c r e a s i n g d . Q u a l i t a t i v e agreement e x i s t s a t the low t e m p e r a t u r e s b u t , t h i s model w i l l not e x p l a i n t h e r i s e i n g s a t t h e h i g h e r t e m p e r a t u r e s . I t s h o u l d be n o t e d t h a t the v a l u e s of d used a r e not i n c o n s i s t e n t w i t h t h e e x p e c t e d t h i c k n e s s of H 2 i n the c e l l d u r i n g each of t h e s e r u n s . F i n a l l y , t h e r e s u l t s of an i n v e s t i g a t i o n of t h e l i n e s h a p e as a f u n c t i o n of a p p l i e d m a g n e t i c f i e l d g r a d i e n t w i l l be p r e s e n t e d . In t h e hopes of s e e i n g s p i n o s c i l l a t i o n s , as has been o b s e r v e d i n the NMR l i n e s h a p e of t h e a-b t r a n s i t i o n i n H\ | by Johnson e t a l . ( 1 9 8 4 ) , t h e ESR l i n e s h a p e s where measured as a f u n c t i o n of a p p l i e d f i e l d g r a d i e n t a l o n g t h e d i r e c t i o n of H 0 . Some of t h e s e r e s u l t s a r e shown i n F i g . 2 8 . There a r e c e r t a i n l y no s h a r p f e a t u r e s i n the l i n e s h a p e of t h e t y p e seen i n t h e NMR l i n e s h a p e s . We b e l i e v e t h e l i n e s h a p e i s d e t e r m i n e d s o l e l y by t h e t o t a l m a g n e t i c f i e l d p r o f i l e , i . e . t h a t of the main magnet , sweep c o i l s and a p p l i e d f i e l d g r a d i e n t t o g e t h e r w i t h t h e shape of the c a v i t y mode H , ( t ) . T h i s s t u d y was 94 c o n d u c t e d a t s e v e r a l t e m p e r a t u r e s and a t many d i f f e r e n t , d e n s i t i e s . I n a l l c a s e s t h e l i n e s h a p e remained t h e same. The above m e n t i o n e d d a t a show no e v i d e n c e of s p i n o s c i l l a t i o n s . A f u r t h e r p o i n t i s t h a t i f s p i n o s c i l l a t i o n s were p r e s e n t , t h e i r n o n l i n e a r d e n s i t y dependence would have n o t i c e a b l y d i s t u r b e d the agreement between o b s e r v e d s t r o n g s i g n a l s and t h o s e s y n t h e s i z e d f rom weak s i g n a l s . No such d e n s i t y dependence was e v e r o b s e r v e d . The r e s u l t s r e p o r t e d i n t h i s c h a p t e r a r e d i s c u s s e d f u r t h e r i n C h a p t e r I X . C h a p t e r V I I I i s c o n c e r n e d w i t h t h e c a l c u l a t i o n of t h e d i p o l e - d i p o l e r e l a x a t i o n r a t e i n the b u l k and on the s u r f a c e . T % 420 mK F i g u r e 28 M a g n e t i c f i e l d g r a d i e n t dependence of t h e b l i n e s h a p e . The h o r i z o n t a l s c a l e i s t h e same f o r each p l o t . 96 CHAPTER V I I I R e l a x a t i o n C a l c u l a t i o n 8.1 I n t r o d u c t i o n R e l a x a t i o n and r e c o m b i n a t i o n p r o c e s s e s d e t e r m i n e the t i m e e v o l u t i o n of the h y p e r f i n e s t a t e p o p u l a t i o n s of s p i n - p o l a r i z e d a t o m i c h y d r o g e n . I n t h i s c h a p t e r t h e r e l a x a t i o n r a t e i s c a l c u l a t e d f o r H\| i n the b u l k as w e l l as f o r H\ | on a l i q u i d "He s u r f a c e . A t o m i c h y d r o g e n i n low magne t i c f i e l d s and low t e m p e r a t u r e s r e l a x v i a s p i n exchange d u r i n g 2-body c o l l i s i o n s ( B e r l i n s k y and S h i z g a l ( 1 9 8 0 ) ) . However t h i s p r o c e s s i s . i n o p e r a t i v e i n h i g h m a g n e t i c f i e l d s where t h e e l e c t r o n and p r o t o n s p i n s a r e e f f e c t i v e l y d e c o u p l e d . The o n l y o t h e r mechanism a v a i l a b l e i s a s s o c i a t e d w i t h t h e d i p o l e - d i p o l e i n t e r a c t i o n between s p i n s on two c o l l i d i n g a t o m s . From a s e m i - c l a s s i c a l v i e w p o i n t , the p r o c e s s i s one where the s p i n on one atom p r e c e s s e s i n the d i p o l a r f i e l d of the o t h e r atom d u r i n g a c o l l i s i o n . . B e f o r e d e s c r i b i n g t h e d e t a i l s of t h i s c a l c u l a t i o n a b r i e f o u t l i n e w i l l be g i v e n . U s i n g t h e d e n s i t y m a t r i x f o r m a l i s m , one can o b t a i n an e x p r e s s i o n f o r the l o n g i t u d i n a l r e l a x a t i o n r a t e , T , " 1 , and t h e t r a n s v e r s e r e l a x a t i o n r a t e , T 2 ~ 1 , between t h e l o w e s t two h y p e r f i n e l e v e l s . Here one assumes t h e t ime dependence t o a r i s e from two-body c o l l i s i o n s . The D i s t o r t e d Wave B o r n A p p r o x i m a t i o n 97 (DWBA) i s then used t o c a l c u l a t e the n e c e s s a r y m a t r i x e l e m e n t s . T h i s i n v o l v e s c a l c u l a t i n g the s p a t i a l wave f u n c t i o n s f o r a 2-body c o l l i s i o n n e g l e c t i n g t h e d i p o l e - d i p o l e i n t e r a c t i o n s w h i c h a r e s u b s e q u e n t l y t r e a t e d p e r t u r b a t i v e l y . The u n p e r t u r b e d w a v e f u n c t i o n s a r e used t o c a l c u l a t e t h e m a t r i x e l e m e n t s w h i c h a r e f i n a l l y put i n t o t h e e x p r e s s i o n f o r T , " 1 . A l t e r n a t i v e l y , a mas te r e q u a t i o n f o r " ' can be c o n s t r u c t e d h e u r i s t i c a l l y from t h e t r a n s i t i o n r a t e s between t h e h y p e r f i n e l e v e l s . These t r a n s i t i o n r a t e s a r e e v a l u a t e d u s i n g the G o l d e n R u l e and g i v e the same r e s u l t . C a l c u l a t i o n s based on t h i s method have been p u b l i s h e d f o r the b u l k r e l a x a t i o n r a t e , S t a t t and B e r l i n s k y ( 1 9 8 0 ) , and f o r t h e s u r f a c e r e l a x a t i o n r a t e , S t a t t ( 1 9 8 2 ) . 98 8 .2 D e n s i t y M a t r i x F o r m u l a t i o n f o r T , ~ 1 The r e l a x a t i o n r a t e i n a low d e n s i t y gas of Hv| i s e s s e n t i a l l y d e t e r m i n e d by a s t a t i s t i c a l a v e r a g e over t h e i n t e r a c t i o n s of the Hvjr a t o m s . Thus t h e d e n s i t y m a t r i x f o r m a l i s m of quantum mechan ic s i s w e l l s u i t e d t o d e r i v e an e x p r e s s i o n f o r t h e l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n r a t e s . An i n t r o d u c t i o n t o the d e n s i t y m a t r i x can be found i n a l m o s t any quantum m e c h a n i c s t e x t . F o r a more d e t a i l e d r e v i e w see ' T h e o r y and A p p l i c a t i o n s of t h e D e n s i t y M a t r i x ' by D. t e r H a a r , ( 1 9 6 1 ) . At low t e m p e r a t u r e s and h i g h m a g n e t i c f i e l d s t h e upper two s t a t e s a r e e s s e n t i a l l y d e p o p u l a t e d l e a v i n g a pseudo two l e v e l sys tem t o work i n . I n t h e s p i r i t o f t h e DWBA one w r i t e s t h e H a m i l t o n i a n of the t w o - p a r t i c l e sy s tem as H 0 + H i ( 2 > where K o c o n t a i n s t h e k i n e t i c e n e r g y , i n t e r - a t o m i c p o t e n t i a l ( s i n g l e t o r t r i p l e t ) and the i n t e r n a l s p i n energy (Zeeman and h y p e r f i n e i n t e r a c t i o n ) . W , ( 2 ) i s t h e p e r t u r b i n g H a m i l t o n i a n , namely t h e d i p o l e - d i p o l e i n t e r a c t i o n between s p i n s on t h e two a toms . The e q u a t i o n of m o t i o n of t h e t w o - p a r t i c l e d e n s i t y m a t r i x o p e r a t o r p< 2 ' i s [ 8 . 1 ] d p < 2 ' = J J p ( 2 , - H o + K i < 2 > ] d t * S w i t c h i n g t o t h e i n t e r a c t i o n r e p r e s e n t a t i o n f o r c o n v e n i e n c e one d e f i n e s 99 iHot/ft - i K o t / f t [ 8 . 2 a ] p ( t ) = e p< 2 » ( t ) e and i H o t/fi - i M o t / f i [ 8 . 2 b ] H , ( t ) = e a f t e r w h i c h [ 8 . 3 ] d p ( t ) = j j p ( t ) , H 1 ( t ) ] d t fi f o l l o w s f rom [ 8 . 1 ] , T h i s i s e q u i v a l e n t t o t r a n s f o r m i n g t o t h e r o t a t i n g f r a m e , as i s commonly done i n NMR t h e o r y , w h i c h r o t a t e s a t t h e Larmor f r e q u e n c y about t h e s t a t i c f i e l d H 0 , t h e r e b y b r i n g i n g t h e p r e c e s s i n g m a g n e t i z a t i o n t o r e s t . A more u s e f u l e x p r e s s i o n , t o second o r d e r , i s E v a l u a t i o n o f t h i s e x p r e s s i o n f o l l o w s i n t h e nex t two s e c t i o n s . A f t e r t a k i n g t h e t r a c e o f one of the p a r t i c l e s t a t e s one o b t a i n s an e x p r e s s i o n i n v o l v i n g o n l y t h e o n e - p a r t i c l e d e n s i t y m a t r i x of t h e form t [ 8 . 4 ] d t ft d p ( t ) = i [ p ( 0 ) , H , ( t ) ] + f i l 2 ; [ [ p ( 0 ) , K 1 ( t / ) ] , H 1 ( t ) ] d t ' Ihi 0 100 where [8 .5a] d t ( T 2 P [8 .5b] P = Pba ^bb These a r e t h e d e n s i t y m a t r i x e q u i v a l e n t of t h e B l o c h e q u a t i o n s f o r r e l a x a t i o n i n s p i n sys tems and d e f i n e t h e r e l a x a t i o n r a t e s T , " 1 and T 2 ~ 1 . L o n g i t u d i n a l r e l a x a t i o n , d e n o t e d T , " 1 , t e n d s t o b r i n g t h e s p i n sys tem i n t o t h e r m a l e q u i l i b r i u m where the d i a g o n a l e l e m e n t s o f the d e n s i t y m a t r i x a r e e q u a l . The d i a g o n a l e l e m e n t s a r e p r o p o r t i o n a l t o the s t a t e p o p u l a t i o n s and P c a _ P b b i s p r o p o r t i o n a l t o the l o n g i t u d i n a l m a g n e t i z a t i o n . T r a n s v e r s e r e l a x a t i o n , d e n o t e d by T 2 _ 1 , t e n d s t o r e l a x any c o r r e l a t i o n s , i n the o f f - d i a g o n a l e l e m e n t s of t h e d e n s i t y m a t r i x : t h e s e e l e m e n t s a r e p r o p o r t i o n a l t o the t r a n s v e r s e m a g n e t i z a t i o n . The r e s u l t t h a t PQQ-Pbb decays t o z e r o i n s t e a d of i t s t h e r m a l e q u i l i b r i u m v a l u e , i s a consequence of a h i g h t e m p e r a t u r e a p p r o x i m a t i o n made i n t h e dynamic s of t h e c o l l i s i o n s , w h i c h a r e assumed t o be e l a s t i c ( i . e . t h e Zeeman e n e r g y d i f f e r e n c e i s i g n o r e d ) . F o r n u c l e a r m a g n e t i c sys tems t h i s i s a l m o s t a l w a y s a good a p p r o x i m a t i o n f o r c a l c u l a t i n g T , and T 2 . H e r e , i n f a c t , the a p p r o x i m a t i o n w i l l s t a r t t o b reak down a t t e m p e r a t u r e s below about 100 mK. 102 8 . 3 B u l k R e l a x a t i o n Rate A t o m i c hydrogen can r e l a x e i t h e r i n t h e b u l k (3-D gas) or on t h e s u r f a c e of i t s c o n t a i n e r (2-D g a s ) . The f o l l o w i n g s e c t i o n d e a l s w i t h t h e s u r f a c e r e l a x a t i o n r a t e . Here a t t e n t i o n i s g i v e n t o t h e b u l k T , ' 1 . One s t a r t s by e v a l u a t i n g the t w o - p a r t i c l e s p i n d e n s i t y m a t r i x [ 8 . 6 ] p s s / ( k f k / ) = <ks |p |k ' s> where |ks> i s the 2-body s y m m e t r i z e d r e l a t i v e wave f u n c t i o n i n s p i n s t a t e |s> and u n s y m m e t r i z e d s p a t i a l s t a t e |k> , k b e i n g the r e l a t i v e momentum wave v e c t o r . When t h e atoms a r e s e p a r a t e d , the s p i n s t a t e a s y m p t o t i c a l l y a p p r o a c h e s | h 1 > | h 2 > where |h[> i s e i t h e r |a> o r |b>. I n the Born a p p r o x i m a t i o n the s y m m e t r i z e d w a v e f u n c t i o n f o r two H atoms ( c o m p o s i t e bosons ) i s 1//2 [ | k> | s>+|-k>(P | s> ] where P i s t h e p e r m u t a t i o n o p e r a t o r . The e i g e n v a l u e s of H o a r e g i v e n by [ 8 . 7 ] Ko | k s > = E k s | k s > where t h e wave f u n c t i o n i s c o n f i n e d t o a volume of L 3 . S u b s t i t u t i n g [ 8 . 2 b ] i n t o [ 8 . 4 ] and l e t t i n g ( 2 ' = V y i e l d s 103 [ 8 . 8 ] d p . J k , k ' ) = i L - 3 Z f p ° ( k , k , ) e , S ' M s s ' ( k 1 f k ' ) d t " S S * k , s , S< 1 i ( E . - E k ) t / f i t + n i - L - 6 / d t / L . i ( E k - E k J t 7 * i ( E k - E k / < / ) t / f i { p ° s ( k , k , ) e K , S ' K ^ M S | S U 1 r k 2 ) e ^ K S M S 2 S ' ( k 2 , k ' ) i ( E u - E k J t ' / t t i ( E k , - E u J t / f i -e 5 , S | M s s ( k f k 1 ) p | ( S ( k 1 , k 2 ) e ^ M s ^ ' ( k 2 , k ' ) i ( E k - E k J t / f t i ( E k < . - E k / < / ) t / / « -e K , S ' M s s ( k , M p ° ( k , , k 2 ) e K ^ M s s ' ( k a , k ' ) i ( E k S - E k ' J ) t / f t i ( E k S - E k s) t'/ft 1 +e M s s ( k , k , ) e 1 1 2 2 M s , ^ ! - ^ ' ^ ^ ^ ' / where <ks|V|ks / >= L " 3M s</( k , k') and <ks | p (0) | k'sS= p ° % , ( k , k ' ) . As suming t h e s p a t i a l s t a t e s |k> a r e i n t h e r m a l e q u i l i b r i u m one can w r i t e [ 8 . 9 ] p | s ' ( k , k / ) = 6 k j £ > ( k ) p s s ' where P ( k ) i s t h e t h e r m a l d i s t r i b u t i o n of momentum s t a t e s and p s s / i s t h e t w o - p a r t i c l e s p i n d e n s i t y m a t r i x . C o n v e r t i n g t h e sum o v e r k i n t o an i n t e g r a l and w r i t i n g E k s = E k + E s where E k = - n 2 k 2 / 2 M and M i s t h e r e d u c e d mass y i e l d s 104 i ( E s - E s ' ) t / r i [8.10] dp s s/=d|Lj_ 3 Z / P ( k ) k 2 d k d k ' p s s M s ^ ( k , k ) e d t W 47T S y I I I i ( E s - E S ( ) t / f i i v.t;s-£s;t. T i - M s s ( k , k ) p S ) S ' e } i ( E s - E s ' ) t / f i + I i\ 2 L - 3M I J P ( k ) k 3 d k 1 /dkdk^fp M c c( k , k')M5 , 4 K7, k ) e fe) 8 7 ^ 5 , 5 , 4^ I S S » S ' S ' ^ . K E s + E s - E g - E ^ t / R - 2 M S s ( k , k ) p S i S M S 2 S ' ( k / , k ) e i ( E s - E g J t / n i + M S S ( ( k , k / ) M s s ( k / , k ) p s 'e ' j . In o r d e r t o t r a c e o v e r one of t h e p a r t i c l e s t a t e s one t a k e s t h e t w o - p a r t i c l e s p i n d e n s i t y m a t r i x t o be t h e d i r e c t p r o d u c t of two o n e - p a r t i c l e s p i n d e n s i t y m a t r i c e s P h jh 1 ) h : ! h, = P h . h ^ h . h j where t h e b a r i n d i c a t e s t h e s e c o n d p a r t i c l e . F o r two p a r t i c l e s ts.n] ! | h l h l - T " s f f M ^ A = I IfMl.h.h but f o r an N - p a r t i c l e s y s t e m , i g n o r i n g c o r r e l a t i o n s , t h e r i g h t hand s i d e o f [8.11] s h o u l d be m u l t i p l i e d by ( N - 1 ) , i . e . t r a c e o v e r a l l o t h e r p a r t i c l e s . To p r o c e e d f u r t h e r r e q u i r e s some knowledge o f t h e m a t r i x e l e m e n t s . E v a l u a t i o n o f t h e d i p o l e - d i p o l e m a t r i x e l e m e n t s i s done u s i n g p a r t i a l wave a n a l y s i s . The s p i n wave f u n c t i o n s |s> a r e l i s t e d i n A p p e n d i x D, and t h e u n s y m m e t r i z e d s p a t i a l wave f u n c t i o n s a r e expanded i n t o s t a t e s of a n g u l a r momentum H i : 105 oo 1 +1 _ * A [ 8 . 1 2 ] - |k>=_l I i (21+1) u , ( k , r ) _4jr_ I Y™(r)Y?(k) k r 1=0 21+1 m=-l 1 where r i s t h e v e c t o r between t h e two p a r t i c l e s r e l a t i v e t o the q u a n t i z a t i o n a x i s . Y ^ r ) a re t h e s p h e r i c a l h a r m o n i c s and U | ( k , r ) i s t h e s o l u t i o n t o the r a d i a l wave e q u a t i o n [ 8 . 1 3 ] 3 1 + k2-_2y_ V t ( r ) - l ( l + 1 ) 3 r 2 ft2 r u , ( k r r ) = 0 The r a d i a l wave f u n c t i o n i s o b t a i n e d by s o l v i n g [ 8 . 1 3 ] n u m e r i c a l l y u s i n g t h e Numerov method ( B l a t t (1967) ) w i t h t h e t r i p l e t p o t e n t i a l of K o l o s and W o l n i e w i c z ( 1 9 7 4 ) . The d i p o l e - d i p o l e i n t e r a c t i o n can be w r i t t e n as [ 8 . 1 4 ] V=fi 2 /24JFZ C( 1 l 2 ; m n ) T m n Y m + n * ( r ) r - 3 5 mn where [ 8 . 1 5 ] T m n = 7 e 7 p [ S m ( l ) l n ( 2 ) + S m ( 2 ) i n ( l ) ] - 7 | [ S m ( l ) S n ( 2 )+Sm( 2 ) S n ( 1 ) ] and where 7 e and 7p a r e t h e e l e c t r o n and p r o t o n g y r o m a g n e t i c r a t i o s , and S m ( i ) and I n ( i ) a r e t h e s p h e r i c a l components of the e l e c t r o n and p r o t o n s p i n o p e r a t o r s f o r p a r t i c l e i r e s p e c t i v e l y . The s p h e r i c a l s p i n o p e r a t o r s a r e r e l a t e d t o t h e r a i s i n g and l o w e r i n g o p e r a t o r s as f o l l o w s : S ± 1 = + 1 / / 2 _ S A and S ° = S Z . The c o n v e n t i o n o f Rose (1957) f o r t h e C l e b s c h - G o r d o n 106 c o e f f i c i e n t s i s a d o p t e d . P r o t o n - p r o t o n i n t e r a c t i o n s a r e d r o p p e d s i n c e o n l y t e r m s t o o r d e r e a r e c o n s i d e r e d and 7 p / 7 e - 0 . 0 0 l 5 . U s i n g t h e e x p r e s s i o n s g i v e n f o r t h e wave f u n c t i o n s and t h e i n t e r a c t i o n H a m i l t o n i a n , t h e m a t r i x e l e m e n t s a r e l ' - l [ 8 . 1 6 ] M s«/(k.k /) = 16i / T 7 r 2 ^ 2 I i ( 2 I + 1 ) 1 / 2 (21+1 ) ^ 2 5 llmnn' C ( l'l2 ; 0 0 )C ( l ' l 2 ;m+n+n', -m) r,,/ C (1 1 2 ; 0 r n+n') x ( - ) m Y , r n (^)Y 1T + n + r i k ' ) <s|T n n 1 s ^ where [ 8 . 1 7 ] r [ ( / = f d r u ^ ( k . r ) l y U . r ) Note t h a t t h e h i g h t e m p e r a t u r e a p p r o x i m a t i o n i s u s e d so t h a t k =* k' . i . e . E b-E a<<E( <. The i n t e g r a l r^/ i s e v a l u a t e d n u m e r i c a l l y and t h e s p i n m a t r i x e l e m e n t s a r e e v a l u a t e d u s i n g [ 8 . 1 5 ] and t h e s p i n wave f u n c t i o n s i n A p p e n d i x D. T a b l e I l i s t s t h e r e s u l t s f o r a l l 2" s p i n m a t r i x e l e m e n t s . A l l terms t o o r d e r e have t o t a l e l e c t r o n s p i n S=1, hence t h e i n t e r - a t o m i c i n t e r a c t i o n i s v i a t h e t r i p l e t p o t e n t i a l . Most o f t h e m a t r i x e l e m e n t s have t o t a l n u c l e a r s p i n 1=1 w h i c h r e q u i r e s t h a t 1 must be e v e n . T h i s i s a r e s u l t of t h e s y m m e t r i c n a t u r e o f t h e 2-atom wave f u n c t i o n under p a r t i c l e p e r m u t a t i o n . However, some o f t h e s p i n m a t r i x e l e m e n t s have c o n t r i b u t i o n s from 1=0 t e r m s w h i c h r e q u i r e 1 t o be odd. 107 One now has a l l t h e n e c e s s a r y i n f o r m a t i o n t o e v a l u a t e t h e a n g u l a r i n t e g r a l s on t h e r i g h t hand s i d e of [ 8 . 1 0 ] . The f i r s t o r d e r t e rm i n v o l v e s [ 8 . 1 8 ] J _ f d k M c S ' ( k , k ) = 4i/6~7rn 2£(2l+1 ) C ( 1 1 2 ; 0 0 ) C ( 1 1 2 ; m , - m ) 4TT 5 lm r ( [ C( 1 12; 00) ( - ) m < s | T 0 0 | sS w h i c h v a n i s h e s s i n c e L C ( 1 1 2 ; m , - m ) ( - ) m = 0 . T h i s l e a v e s o n l y t h e m second o r d e r terms i n M : [ 8 . 1 9 ] J J - d & d k ' M s < / (k ,k ' )M s s ' (k ' k)=- i S S e ^ 1 1 4TT ' ' 5 I ( 2 l / + l ) ( 2 l + 1 ) C 2 ( l / l 2 ; 0 0 ) r 2 « n + r / + n / Q l l n n n , ^ , ; C 2 ( 1 I 2 ; 0 , n + n) ( - ) n + n < s | T n n | s / > < s 1 | T n ' n ' | s , ' > . A l l c r o s s terms between 1=0 and 1=1 s p i n m a t r i x e l e m e n t s a r e z e r o l e a v i n g sums o v e r 1 , l ' b e i n g b o t h even or odd as r e q u i r e d . There a r e a t o t a l of 2 8 =256 d i f f e r e n t s econd o r d e r t e r m s . T a b l e I I l i s t s t h e 8 d i s t i n c t terms l e f t a f t e r u s i n g t h e symmetry p r o p e r t i e s o f t h e s p i n m a t r i x e l e m e n t s and e l i m i n a t i n g v a n i s h i n g t e r m s . An i m p o r t a n t p o i n t t o remember when e v a l u a t i n g t h e second o r d e r m a t r i x e l ement i n t e g r a l s i s t h a t when A i n t e g r a t i n g o v e r dk t h e l i m i t s a r e o v e r 2ir s t e r a d i a n s i n s t e a d of 4TT s t e r a d i a n s . T h i s a v o i d s d o u b l e c o u n t i n g of t h e s y m m e t r i z e d s t a t e s s i n c e | k , h , h 2 > = | - k , h 2 h 1 > . A f t e r some a l g e b r a i c m a n i p u l a t i o n [ 8 . 1 0 ] and [ 8 . 1 1 ] y i e l d t h e r a t e of TABLE I S p i n M a t r i x E l e m e n t s <aa <aa <ab <ba rp n n' rp n r/ rp n rY rpn rY ab> ba> bb> bb> <ab <ba <bb <bb rpnn' rp n tY T n r Y rpnn' aa> aa> ab> ba> - ^ e 7 p ( l + e T e / 7 p 8 n ; 0 <aa T n n ' aa> - l / 2 ( 7 | + 7 e 7 p ) 5 n > 0 5 n;o <bb T n r Y bb> - l / 2 ( 7 | - 7 e 7 p ) 6 n j 0 5n',o <aa <bb rpnn' >pnn' bb> aa> 0 <ab <ba >pn rC rpn rY ab> ba> + b u ) <ab <ba T n n ' rp n n' ba> ab> - 1 / 4 7 e 2 6 n > c , 6 n ; 0 ( 6 I ( | - 5 I,-) a) The f i r s t 12 m a t r i x e l e m e n t s i n v o l v e t o t a l n u c l e a r s p i n 1=1 o n l y , whereas t h e l a s t 4 m a t r i x e l e m e n t s i n v o l v e b o t h 1=0 and 1=1 t e r m s . A l l terms have t o t a l e l e c t r o n i c s p i n S=1. b) The m a t r i x e l e m e n t s a r e e v a l u a t e d up t o o r d e r e. 109 c h a n g e o f t h e s i n g l e - p a r t i c l e s p i n d e n s i t y m a t r i x [ 8 . 2 0 a ] d £ h ) h = - | 4 I n H M a ( 7 e 7 p + e 7 » ) » < ( W M ) S t > T ( p h h - p . h . h ) a n d [ 8 . 2 0 b ] d ^ = - 2 4 ^ n H M M ( 7 e 7 p - e 7 | ) 2 + | 7 ^ 2 ] < ( W M ) s t > T P V h . T h e a n g u l a r b r a c k e t s w i t h s u b s c r i p t T i n d i c a t e s a t h e r m a l a v e r a g e ; n ^ i s t h e a t o m i c h y d r o g e n n u m b e r d e n s i t y ; a l s o , i f h i s a ( b ) , t h e n - h i s b ( a ) . F i n a l l y we d i s c u s s t h e n u m e r i c a l e v a l u a t i o n o f t h e r e l a x a t i o n r a t e s . I n e v a l u a t i n g r ^ ' n o t e t h a t o n l y t h e r 0 2 a n d r 2 2 t e r m s , f e e l t h e t r i p l e t p o t e n t i a l a t t h e l o w e n e r g i e s o f c o n c e r n h e r e , ^ 0 . 6 m e V . T h e s e t w o t e r m s a r e e v a l u a t e d n u m e r i c a l l y a s d i s c u s s e d e a r l i e r . F o r t h e r e s t o f t h e t e r m s t h e a s y m p t o t i c f o r m i s u s e d . S p h e r i c a l B e s s e l f u n c t i o n s o f t h e f i r s t k i n d , j j ( x ) , a r e s o l u t i o n s t o t h e f r e e r a d i a l w a v e e q u a t i o n . H e n c e u j ( k , r ) = k r j j ( k r ) f o r l a r g e 1 w h e r e t h e p h a s e s h i f t i s s m a l l a n d [ 8 . 2 1 a ] r M « 1 1>0 1 > I 2 1 ( 1 + 1 ) [ 8 . 2 1 b ] r M , * 1 1>0 1 , 1 6(1+1 ) (1 + 2) T a b l e I I I c o n t a i n s v a l u e s o f r 0 2 , r 2 2 a n d S* f o r t h e e n e r g y v a l u e s u s e d h e r e . TABLE I I S e c o n d O r d e r (3-D) M a t r i x E l e m e n t I n t e g r a l s M 2 ( 1 ) [ a a , a a ; a a , a a ] = 256/257r 3ft" ( 7 | + 7 e 7 p ) 2S* M 2 ( 2 ) [ a a , a a ; a b , a b ] = 1 2 8 / 2 5 7 r 3 h « 7 | ( 7 | + 7 e 7 p ) S t M 2 ( 3 ) [ a a , a a ; b b , b b ] = 256/25Tr 3fc' t ( 7 e 2 + 7 e 7 p ) ( 7 | " 7 e 7 p ) S t M 2 ( 4 ) [ b b , b b ; b b , b b ] = 256/257r 3-h* ( 7 j " 7 e 7 p ) 2 S f M 2 ( 5 ) [ bb, bb;ab, ab] = 1 2 8 / 2 5 7 r 3 f t 4 7 | ( 7 2 " 7 e 7 p ) S t M 2 (6) [ a b , a b ; a b , a b ] = 64/257r 3ft' t7e ( S l +S S) M 2 ( 7 ) [ a b , a b ; a b , b a ] = 6 4 / 2 5 T r 3 * " 7 * ( S f - S s ) M 2 [ a a , a b ; a b , a a ] ' = 96/257r 3fi 4 {yQy^+e7|) 2 S f where M 2 ( i ) [ a a , ab;ab, aa ] = 1 /4?r T d k ^ M ^ ^ k, k ) M Q b (k', k) and even S l = L (21 + 1 ) (21 + 1 ) C 2 ( l / 1 2 ; 0 0 ) r 2 / 11 = 0 " odd S s= L ( 2 1 + 1 ) ( 2 1 + 1 ) C 2 ( 1 1 2 ; 0 0 ) r f / 11 = 0 1 1 111 E v a l u a t i n g <(ftk/u)s\> n u m e r i c a l l y y i e l d s [ 8 . 2 2 a ] Tr 1 = ( 4 8 7 i / 2 5 ) n H M 2 ( 7 e 7 p + e7|) 2 v ( 0 . 140-0.01 54T) [8.22b] T 2 - 2 = ( 247r/25)n H M 2 [ ( 7 e 7 p + e7| ) 2 + ( 4/3 ) 7 | 7 2 ] v ( 0 . 140-0. 01 54T) where v=/8kgT/wy i s t h e r e l a t i v e t h e r m a l v e l o c i t y o f t h e h y d r o g e n atoms. The t h e r m a l a v e r a g e was p e r f o r m e d f o r e n e r g i e s i n t h e ra n g e 0.001meV<E<0.6meV. T h i s e x p r e s s i o n i s v a l i d up t o t h e p o i n t where M 2 becomes n o n l i n e a r i n E (>2K) and down t o =100 mK where t h e c o l l i s i o n s become n o t i c e a b l y i n e l a s t i c . 1 12 TABLE I I I M a t r i x E l e m e n t s v s . E n e r g y (3-D c a s e ) E(meV) r o 2 r 2 2 S l 0 . 6 0 .03943 0 .08510 0 .08715 0 . 5 0 .04744 0 .08603 0 .09520 0 .4 0 .05580 0 .08619 0 .10405 0 . 3 0 .06416 0 .08564 0 .11340 0 . 2 0.07201 0 .08445 0 . 12265 0.1 0 .07889 0 .08287 0 .13115 0 . 0 5 0 .08194 0 .08215 0 .13520 0 .025 0 .08343 0 .08189 0 .13735 0 .010 0 .08445 0 .08180 0 .13895 0 .005 0 .08470 0 .08175 0 .13935 0 .0025 0 .08502 0 .08178 0 .13990 0 . 0 0 .08333 0 .08333 0 .13889 1 13 8 .4 S u r f a c e R e l a x a t i o n Rate R e l a x a t i o n i n t h e gas d o m i n a t e s u n t i l t h e t e m p e r a t u r e i s low enough t h a t t h e i n c r e a s e i n s u r f a c e d e n s i t y , n s , c au se s s u r f a c e r e l a x a t i o n t o t a k e o v e r . In o r d e r t o c a l c u l a t e the s u r f a c e r e l a x a t i o n r a t e one must c o n s i d e r t h e pseudo two d i m e n s i o n a l p r o b l e m of H-H c o l l i s i o n s on the s u r f a c e . The l a r g e e x t e n t of the s u r f a c e wave f u n c t i o n (Edwards and Mantz ( 1 9 8 0 ) ) , =10 A , compared t o the h a r d - c o r e r a d i u s , =3.7 A , does not a l l o w a p u r e l y 2-D t r e a t m e n t . T h i s l e a v e s one w i t h a s c a t t e r i n g p r o b l e m where the m o t i o n p e r p e n d i c u l a r t o the s u r f a c e c a n n o t be s e p a r a t e d f rom t h e t r a n s v e r s e m o t i o n . As suming a r i g i d s u r f a c e t h e f o l l o w i n g t w o - p a r t i c l e S c h r o d i n g e r e q u a t i o n s h o u l d be s o l v e d [ 8 . 2 3 ] V,2 - £ i V 2 2 +V S (z , ) + V S (z 2 ) +V t ( r , - f*2 ) 2m 2m <i>(r , , r l 2 ) = E $ ( r 1 , r 2 ) where V s ( z ) i s t h e e f f e c t i v e s u r f a c e p o t e n t i a l (z norma l t o t h e s u r f a c e ) and r* t h e p o s i t i o n v e c t o r of atom i . T h i s t r e a t m e n t i g n o r e s H-He c o r r e l a t i o n s on the s u r f a c e , one e f f e c t of w h i c h i s t o r e n o r m a l i z e m. Zimmerman (1982) has c a l c u l a t e d m*/m=1.03 l e a d i n g one t o e x p e c t t h a t the c o r r e l a t i o n s have l i t t l e e f f e c t on H-H c o l l i s i o n s . C h a n g i n g t o c e n t e r - o f - m a s s c o o r d i n a t e s p a r a l l e l t o the- s u r f a c e , s e p a r a t i n g out the t r a n s v e r s e c . o f m. m o t i o n and e x p a n d i n g t h e r e s u l t i n g wave f u n c t i o n i n p a r t i a l 114 waves y i e l d s [ 8 . 2 4 ] - f t 2 f 9 2 + ( l / 4 - m 2 ) r ' 2 - f i 2 3 2 - ft j 3j_ +75 (2 , ) + V s ( z 2 ) 2 ^ 3 r 2 2 m 3 z , 2 2 m 3 z 2 2 + V t ( / r 2 + ( Z l - z 2 ) 2 ) * m ( r f z , , z 2 ) = E r T ^ m ( r , z , , z 2 ) . As suming f o r now t h a t * m ( r , z 1 f z 2 ) can be f o u n d , t h e c a l c u l a t i o n p r o c e e d s as i n t h e p r e c e e d i n g s e c t i o n . The 2-D e q u i v a l e n t of [ 8 . 1 0 ] i s , i ( E s - E < v ) t / f i [ 8 . 2 5 ] dp = i L ' 2 Z J P ( k ) k d k d » L , p s r M g > / ( k > k ) e " d T 5 2 «2TT s , L ' • i ( E s - E s ) t / f i 7 - M s s ( k f k ) p S ( S ' e ' j • . i ( E s - E s ' ) t / f t + ( i ) 2 L ' 2 M I ; p ( k ) k d k J _ / d 0 k d ^ k / p s s M s s ( k , k ' ) M s ^ ( k ' f k) . ' f i ' 4 7 T - h S 1 S 2 2X l I I 2 2 >e 1 " 2 M s s - - . -i ( E « . + E « . - E s - E s / ) t / * ( k , k ' ) p S s M s s ' (k ' ,k )e 2 ' i ( E s - E s ) t / f i -j + M S s ( k , k ' ) M ^ s j k ' , k ) p S 2 S ' e 1 ) where now <ks | V | kV>=L" 2 M S s< k , k ' ) . B e f o r e e v a l u a t i n g the m a t r i x e l e m e n t s t h e s p i n q u a n t i z a t i o n a x i s ( l | H 0 ) must be r o t a t e d i n t o t h e s u r f a c e c o o r d i n a t e s y s t e m . T h i s i s done w i t h the r o t a t i o n o p e r a t o r ( see Rose) [ 8 . 2 6 ] Y n ( £ ) = I d 2 m ( 0 o ) Y T ( f i ) m where r i s measured r e l a t i v e t o the q u a n t i z a t i o n a x i s , 80 i s the a n g l e between H 0 and t h e s u r f a c e n o r m a l and Q i s r e f e r r e d t o t h e 115 s u r f a c e c o o r d i n a t e s y s t e m . The s p i n wave f u n c t i o n s a r e as b e f o r e , and t h e u n s y m m e t r i z e d s p a t i a l wave f u n c t i o n can be w r i t t e n as i m ( 0 - t > k ) [ 8 . 2 7 ] |k>= _1 I * m ( r , z 1 f z 2 ) e / k r m u s i n g t h e f a c t t h a t t h e r e i s o n l y one bound s u r f a c e s t a t e f o r i s o l a t e d atoms (Edwards and Mantz ( 1 9 8 0 ) ) . Now [ 8 . 2 8 ] M s ;7 (k t k / )=47r £ 1 7 fx2 L C (1 1 2 ; nn ) d 2 n + n / , ( 60 ) r ^ m + , / 5 m l n n ' ; ; im<t>, - i(m+l)tf>. , , e K e < s | T n n | s > where [ 8 . 2 9 ] 4,m+fk~ 1 ; d r d z , d z 2 4 ^ r , z , , z 2 ) | Y[2 (z , - z 2 , r ) | ( 1 / | 1 | )' ( r 2 + ( z , - z 2 ) 2 ) - 3 / 2 * m ( r , Z l , z 2 ) . As b e f o r e , a l l t e rms t o o r d e r e have t o t a l e l e c t r o n s p i n S=1, i m p l y i n g i n t e r - a t o m i c i n t e r a c t i o n v i a the t r i p l e t p o t e n t i a l . Symmetry under p a r t i c l e p e r m u t a t i o n r e q u i r e s t h a t s p i n m a t r i x e l e m e n t s w i t h 1=1 must have even m and m+1 i n t h e sum of [ 8 . 2 8 ] whereas i f 1=0 t h e n m and m+1 must be o d d . E v a l u a t i n g t h e f i r s t o r d e r a n g u l a r i n t e g r a l s i n [ 8 . 2 5 ] y i e l d s [ 8 . 3 0 ] _ L / d 0 . M c c / ( k , k ) = 4 7 T / 2 i 7 f t 2 Z C(11 2 ; n n ) d 2 U60)r° m<s | T n n | s> 2TT K S S 7 5 mnn' ' ' These i n t e g r a l s do not v a n i s h as i n the 3-D c a s e . T a b l e IV 1 16 l i s t s t h e f i v e d i s t i n c t n o n - v a n i s h i n g te rms of t h e 2" d i f f e r e n t i n t e g r a l s . The second o r d e r t e rms i n M are . [ 8 . 3 1 ] i ; d 0 u d ^ 1 / M C c ( k , k / ) M^2 ( k / , k ) = 7687:"^'' Z 2 i k. k ss, s * s 4 m l n n ' n , n ' , C ( 1 l 2 ; n n / ) C ( l 1 2 ; n l n ^ ) d n + n / 0 ( c 9 0 ) d j n t _ n / ) ( ( e 0 ) ( - ) n , + n / ( r i r ) m + l ) 2 < s | T n n / | s , x s 2 | T n < n - / | s / > where 1=0 and 1=1 s p i n m a t r i x e l e m e n t s have m,m+l odd and even r e s p e c t i v e l y . A l l 1=0 and 1=1 c r o s s t e rms v a n i s h . Of t h e 2 8 d i f f e r e n t second o r d e r t e rms t h e 12 d i s t i n c t n o n - z e r o terms a r e l i s t e d i n T a b l e V . S u b s t i t u t i o n of e q n . [ 8 . 3 0 ] and [ 8 . 3 1 ] i n t o [ 8 . 2 5 ] y i e l d s ( i f l t - i O t [ 8 . 3 2 a ] d ( P a a - P b b ) = 2 i G i n s ( e pab~e P h a * - 2 6 * " * ^ " p b b } d t int -int - 2 G 2 2 n s ( e p Q b +e p b Q ) and - i f i t ( [ 8 . 3 2 b ] dp b = iG\nse ( P a Q - p b b ) + i G , n s p a b - ( G 2 + G 2 ) n s p Q b d t - i f t t 3 - 2 i f t t - G 2 2 n s e ( P a a " ^ b b ) _ G 2 n s e "ob where fi=(Eb-EQ)/fi and TABLE IV F i r s t O r d e r 2-D M a t r i x E lement I n t e g r a l s M 1 ( 1 ) [ aa , aa ] = - 8 7 r / i 75 " ft2 ( 7 2 . + 7 e 7 p ) d 2 0 ( 60 ) R l M 1 ( 2 ) [ b b , bb] = - 8 7 T / ¥ 7 5 " - f i 2 ( 7 | - 7 e 7 p ) d 2 / O(e 0 ) R t M 1 ( 3 ) [ a a , a b ] = -27r /6¥75 f i 2 ( 7 e 7 p + * 7 § ) d 2 0 ( 0 O )R l M 1 ( 4 ) [ a b , ab 3 = -47r /¥7B~ fc27|d20( 0O ) ( R * + R S ) M 1 ( 5) [ a b , b a ] = -$Tr\/775 - n 2 7 | , d 2 / 0 ( 0 O ) (R l " R S ) where M 1 ( i ) [ a b , b a ] = 1 / 2 7 T f d 0 k M a b ( b d ( k , k) and even . odd R = S r ^ _ R = 2 r « — m,m nn,m m m 118 [ 8 . 3 3 a ] G , = 2TT | 6 F f i ( 7 e 7 p + e 7 | ) d f ) O ( e 0 ) < R t > T [ 8 . 3 3 b ] G\= B*j£ f i 7 e 7 p d 2 ) O ( 0 o ) < R t > T [ 8 . 3 3 c ] [ 8 . 3 3 d ] G , = 12TT: ftM(7e7p+e72e)2Z(d2(|(e0))2<SJ>T G 2 = 1|7L! * M ( 7 0 7 n ) 2 2 ( d 2 , ( e 0 ) ) 2 < S t > . [ 8 . 3 3 e ] G 2 = 4 ^ 6 7 L 3 f t M 7 e 7 ( 7 e 7 p + e 7 | ) 2 : d 2 > 1 ( e o ) d f ) , ( e 0 ) < S t > T 5 1 [ 8 . 3 3 f ] G 2 = 36JL1 f iM ( 7 e 7 p +*72 e) 2 £ d 2 } , ( M d_2 ( ( 0 O ) < S * > T . 5 1 B e f o r e a s e m i - c l a s s i c a l d e s c r i p t i o n of t h e s e r e s u l t s i s g i v e n , i n the n e x t s e c t i o n , the n u m e r i c a l e v a l u a t i o n w i l l be d i s c u s s e d . S o l v i n g [ 8 . 2 4 ] f o r ¥ m ( r , z 1 f z 2 ) n u m e r i c a l l y w o u l d t a k e more e f f o r t and computer d o l l a r s t h a n i t i s w o r t h . As an a p p r o x i m a t i o n one can s e p a r a t e t h e wave f u n c t i o n [ 8 . 3 4 ] * m ( r , z 1 , z 2 ) = u m ( k , r ) 0 ( z 1 ) 0 ( z 2 ) w h i c h h o l d s a s y m p t o t i c a l l y where t h e two H-atoms a r e 2-D f r e e p a r t i c l e s bound t o t h e s u r f a c e . The s o l u t i o n o f t h e 2-D r a d i a l TABLE V Se c o n d O r d e r 2-D M a t r i x E l e m e n t I n t e g r a l s M 2 (1 ) [ a a ,aa;aa ,aa ] = 1 28/57r«ft4 ( 7 | + 7 e 7 p ) 2 2 ( d 2 j ( ( © 0 )) 2 Sj M 2 ( 2 ) [ b b , b b ; b b , b b ] = 1 28/57^',ni, ( 7 | " 7 e 7 p ) 2 2 ( d 2 ^ ( 8 0 ) ) 2SJ M 2 ( 3 ) [ a a , a a ; a a , a b ] = ^ / r ^ T r 4 * ! 4 ( 7 2 + 7 e 7 p ) W^p"^ 2?) i d 2 o > , ( e o ) d 2 ; l ( 0 o ) s ! M 2 (4) [ a a , a a ; a b , a b ] = 6 4 / 5 7 r 4 f t 4 7 2 ? ( 7 | + 7 e 7 p ) £ ( d 2 y t? 0 ) ) 2sj M 2 ( 5 ) [ a a , a a ; b b , b b ] = 1 2 B / 5 i r , f t * (y2e+y^yp) W|"7 e7 p) l ( d 2 j [ ( 0 o ) ) 2 s [ M 2 ( 6 ) [ a a , a b ; a a , a b ] = -48/57r 4-n 4 ( 7 e 7 p + e 7 | ) 2 Id 2 j l(t9 0 )d_^|(t9 0 ) s [ M 2 ( 7 ) [ b b , b b ; a a , a b ] = -32/^/57:^" ( 7 | - 7 e 7 p ) ( 7 e 7 p + e 7 | ) L d 2 O ) [ ( e 0 ) d 2 ) l ( e 0 ) s \ M 2 (8) [ a a , a b ; a b , a b ] = - 1 6 / ^ 5 7 ^ * 4 7 2 ( 7 e 7 p + ey\) I d 2 ) [ ( c 9 0 ) d 2 ; [ ( t 9 0 ) s f M 2 (9) [ a b , a b ; a b , a b ] = 32/57r 4 ? i 47 42:(d 2 3 ; !( 0 O ) ) 2 (s| +sf) M 2 ( 10) [ a b , a b ; a b , b a ] = 3 2 / 5 7 r 4 f i 4 7 4 I ( d 2 , [ ( 60 ) ) 2 (sj -Sp) M 2 ( 1 1 ) [ b b , b b ; a b , a b ] = 6 4 / 5 * * * ' 7 | ( 7 | - 7 e 7 p ) Z ( d 2 ) t ( 0 O )) 2 S M 2 [ a a , a b ; a b , a a ] = 48/57r 4ft 4 ( 7 e 7 p + e 7 | )Z(<M,t (0 O )) 2s[ where M 2 ( i ) [ a a , a b ; a b , a a ] = 1 / 2 7 r J d 0 k d 0 ^ M Q a ) C l b ( k, k')M Q b < Q Q( k', k) and I i s a sum o v e r even 1, even odd S - L ( r » , m + | ) » Sf= L (4m+,)a m m 120 wave e q u a t i o n [ 8 . 3 5 ] -_fii(9f_ + ( l / 4 - m 2 ) r - 2 J + v { ( r ) u m ( k , r ) =E ' m u r p (k , r ) 2M>3T 2 i s u m ( k , r ) and <p(z) i s t h e s u r f a c e w a v e f u n c t i o n g i v e n by [ 8 . 3 6 ] -ft2 9JL+V_(z) <Mz)=-EncMz) 2m 9 z 2 0 where E ^ i s t r a n s l a t i o n a l energy ( E ^ O ) and E g i s t h e b i n d i n g e n e r g y (Eg>0) of H t o t h e "He s u r f a c e . As i n t h e 3-D ca se u m ( k , r ) i s s o l v e d f o r n u m e r i c a l l y whereas 0 2 ( z ) i s t a k e n t o be r e c t a n g u l a r w i t h t h e same a v e r a g e p r o b a b i l i t y d e n s i t y as t h e s u r f a c e wave f u n c t i o n c a l c u l a t e d by Edwards and M a n t z . T h i s a p p r o x i m a t i o n can be shown t o g i v e r e a s o n a b l e r e s u l t s by r e p e a t i n g t h e c a l c u l a t i o n w i t h a t r i a n g u l a r 0 2 ( z ) . P a r a m e t e r s f o r t h e t r i a n g l e a r e chosen t o have t h e same p r o b a b i l i t y d e n s i t y and second moment as f o r the r e c t a n g l e , g i v i n g a r e a s o n a b l e f i t t o t h e s u r f a c e p r o b a b i l i t y d e n s i t y of Edwards and Mantz (see F i g . 2 9 ) . C a l c u l a t e d r e s u l t s a r e i d e n t i c a l t o b e t t e r t h a n 1 p e r c e n t . A r e c t a n g u l a r <p2 (z) o f w i d t h a = l 0 . 5 A y i e l d s [ 8 . 3 7 a ] and 0 5 10 15 Z (A) . 2 0 25 F i g u r e 29 P r o b a b i l i t y d e n s i t y of Edwards and Mantz "He s u r f a c e wave f u n c t i o n . I n c l u d e d a r e t h e r e c t a n g u l a r and t r i a n g u l a r <p2 ( z ) used i n the c a l c u l a t i o n . 122 [ 8 . 3 7 b ] r 2 *• rr nyn+2' r 2 ( a 2 + r 2 ) 1 / 2 2 a 2 + r 2 " 1 u ( k , r ) u m+2 ( k , r ) T h e s o l u t i o n o f t h e f r e e 2 - D r a d i a l w a v e e q u a t i o n i s u m ( k , r ) = i / T c r J m ( k r ) w h e r e J m ( x ) i s a f i r s t o r d e r B e s s e l f u n c t i o n . H e n c e , f o r l o w e n e r g y , T h e f i r s t o r d e r . a n g u l a r i n t e g r a l s o n l y r e q u i r e r ^ ^ h i c h a r e l i s t e d i n T a b l e V I o v e r t h e s ame r a n g e o f e n e r g y v a l u e s a s i n t h e 3 - D c a l c u l a t i o n . F i g u r e 30 d i s p l a y s t h e t h e r m a l a v e r a g e s o f s ome o f t h e r ^ p ^ s w e l l a s <R* > w h i c h i s c o n s t r u c t e d f r o m t h e c o m p u t e r c a l c u l a t e d v a l u e s f o r m<6 a n d t h e a s y m p t o t i c v a l u e s f o r m>6. E v a l u a t i o n o f t h e s e c o n d o r d e r a n g u l a r i n t e g r a l s r e q u i r e s c o n s i d e r a t i o n o f o n l y t h e m = 0 , 2 t e r m s s i n c e ( r ^ m + l ^ 2 ~ m ~ * • H i g h e r o r d e r t e r m s c o n t r i b u t e l e s s t h a n 1 p e r c e n t t o t h e r e l a x a t i o n r a t e . T a b l e V I l i s t s v a l u e s o f r ^ m + z a n d F i g . 31 h a s a m m=7< ( r | ! ^ ' J^ ) 2 > T p l o t t e d v e r s u s t e m p e r a t u r e . F i n a l l y t h e s u r f a c e l o n g i t u d i n a l r e l a x a t i o n r a t e i s g i v e n b y [ 8 . 3 8 a ] m>2 a n d [ 8 . 3 8 b ] m>0 TABLE VI M a t r i x E l e m e n t s (2-D) v s . Energy E(meV) r £ 0 ( A - 1 ) r ? 2 ( A " 1 ) r ? 2 ( A " M r ^ U - 1 0 . 6 0 . 3 7 6 1 x 1 0 " " 0 . 3 6 5 9 x 1 0 - " 0 . 5 4 9 5 x 1 0 " " 0 .3279x10 0 . 5 0 .4112 0 .3755 0 .5510 0 .3186 0 .4 0 .4574 0 .3802 0 .5455 0 .3072 0 . 3 0.5221 0 .3760 0 .5290 0 .2906 0 .2 0 .6215 0 .3556 0 .4945 0 .2630 0.1 0 .8042 0 .3012 0.4241 0 .2098 0 .05 0 .9784 0 .2402 0 .3516 0 .1590 0 .025 1.1200 0 .1838 0 .2830 0 .1168 0 .010 1.2365 . 0 .1236 0 .2038 0 .0757 0.001 1 .2125 0 .0410 0 .0763 0 .0243 0 .0 0 .9873 0 .0 0 .0 0 .0 124 0 O.I 0.2 0.3 0.4 0.5 0.6 T(K) F i g u r e 30 T h e r m a l a v e r a g e s of f i r s t o r d e r m a t r i x e lement i n t e g r a l s . 125 F i g u r e 31 T e m p e r a t u r e dependence of s e c o n d o r d e r m a t r i x e l e m e n t i n t e g r a l s . 126 [ 8 . 3 9 ] Tr 1= 2 G 2 n s = 2J_7r 3n s*iM (l^y+nl) 2H s i n 2 260 ( o00 + 2a2 2) 5 p 8 + s i n 2 0 o ( 1 + c o s 2 0 o ) (a02 + a2tr) ] . One i n t e r e s t i n g f e a t u r e of t h i s r e s u l t i s the a n g u l a r dependence of T , ~ 1 on t h e magne t i c f i e l d d i r e c t i o n . S u r f a c e s w i t h H 0 p e r p e n d i c u l a r t o the He f i l m w i l l have T ,~ -0 and s u r f a c e s w i t h H 0 p a r a l l e l t o t h e He f i l m w i l l have a c o n s i d e r a b l y r e d u c e d T , " ' s i n c e t h e c o e f f i c i e n t of a00 v a n i s h e s . A so c a l l e d ' p a n c a k e ' c e l l , w i t h H 0 p e r p e n d i c u l a r t o t h e p a n c a k e , t a k e s advantage of t h i s f a c t t o reduce t h e r e l a x a t i o n r a t e ( S p r i k e t a l . ( l 9 8 2 ) ) . I n p r a c t i c e t h i s r e q u i r e s o p t i c a l l y smooth s u r f a c e s so t h a t t h e He f i l m (a few h u n d r e d angs t roms t h i c k ) can f i l l i n t h e d e p r e s s i o n s and so l e v e l out the s u r f a c e . Comple te a n i s o t r o p y w i l l not e x i s t i n r e a l i t y . E f f e c t s s u c h as i n t e r a c t i o n s w i t h s u r f a c e e x i t a t i o n s w i l l d e s t r o y t h e c o m p l e t e T , " 1 a n i s o t r o p y . R i p p l o n s a r e the s u r f a c e e x c i t a t i o n s f o r "He and a r e e x p e c t e d t o be w e a k l y c o u p l e d as m e n t i o n e d b e f o r e . T h e r e f o r e one e x p e c t s a l a r g e a n i s o t r o p y i n t h e s u r f a c e r e l a x a t i o n r a t e . S u r f a c e e x c i t a t i o n s of the 3 He l i q u i d a r e l e s s w e l l known, and t h e i r e f f e c t on H-H c o l l i s i o n s i s unknown. S e v e r a l a p p r o x i m a t i o n s have been made i n t h i s c a l c u l a t i o n , two of w h i c h a r e t h e f o l l o w i n g . The h i g h t e m p e r a t u r e l i m i t has been used w h i c h p u t s a l o w e r bound on the t e m p e r a t u r e a t w h i c h 127 t h e r e s u l t s a r e v a l i d . A c c o r d i n g t o Ahn e t a l . (1983) T , " 1 i s i n c r e a s e d by about 50 per c e n t a t 0.1 K from the p r e s e n t r e s u l t s . The o t h e r a p p r o x i m a t i o n i s t h e s e p a r a t i o n of v a r i a b l e s made i n [ 8 . 3 4 ] . A s y m p t o t i c a l l y t h i s s e p a r a t i o n i s c o r r e c t , but a t c l o s e range i t i m p l i e s a h a r d - c o r e r e p u l s i v e c y l i n d e r i n s t e a d of t h e u s u a l h a r d - c o r e s p h e r e . E i j n d e e t a l . (1983) f i n d t h a t c o r r e c t i n g f o r t h i s e f f e c t w i l l i n c r e a s e T , _ 1 by about 50 per c e n t . 128 8 . 5 S e m i - c l a s s i c a l D e s c r i p t i o n of t h e 2-D R e s u l t s A s e m i - c l a s s i c a l a n a l y s i s o f t h e s p i n sy s tem i s d e s c r i b e d i n t h i s s e c t i o n . T h i s a l l o w s a s i m p l e i n t e r p r e t a t i o n of t h e above r e s u l t s and g i v e s u s e f u l i n s i g h t i n t o t h e phenomena. C o n s i d e r f i r s t t h e n o n - i n t e r a c t i n g c a s e . The s i n g l e p a r t i c l e s p i n d e n s i t y m a t r i x o p e r a t o r p < 1 ' e v o l v e s a c c o r d i n g t o [ 8 . 4 0 ] _ d p V ' ( t ) = _ i [ p l 1 > ( t ) , X o ] d t ft S o l v i n g f o r t h e s i n g l e p a r t i c l e s p i n d e n s i t y m a t r i x p i 1 , ' y i e l d s n,n2 c o n s t a n t v a l u e s of p 1^ and p b b > and p Q b 1 > ( t ) = p ^ ' (0 )exp( i n t ) . Note t h a t P ^ b ' ^ ' b c i ' • P h y s i c a l l y , t h e t r a n s v e r s e m a g n e t i z a t i o n w h i c h i s p r o p o r t i o n a l t o ^ a b ' ^ ^ ' p r e c e s s e s about H 0 w i t h a f r e q u e n c y ft, t h e Larrnor f r e q u e n c y . I t i s s t a n d a r d p r a c t i s e i n NMR t h e o r y t o t r a n s f o r m t o t h e r o t a t i n g frame w h i c h i s a c c o m p l i s h e d v i a t h e t r a n s f o r m a t i o n iHot/f t - iWot/ft [ 8 . 4 1 ] p ( t ) = e p l 1 ' ( t ) e T h i s l e a d s t o p Q a ( t ) =Pla(\ ' ( t ) and / 3 5 b ( t ) = p b b ) ( t ) b e i n g c o n s t a n t and P a r j ( t ) = e x p ( - i f t t ) p ^ b 1 1 ( t )= P^JMO). I n the r o t a t i n g frame the p r e c e s s i n g m a g n e t i z a t i o n i s b r o u g h t t o r e s t . F o r s i m p l i c i t y we s h a l l work i n t h e h i g h t e m p e r a t u r e a p p r o x i m a t i o n where the t h e r m a l e q u i l i b r i u m v a l u e s of p and 129 P b b a r e e q u a l . A s p i n sy s tem out of t h e r m a l e q u i l i b r i u m r e l a x e s v i a l o n g i t u d i n a l , T , " 1 , and t r a n v e r s e , T2'\ r e l a x a t i o n . The c o r r e s p o n d i n g terms i n e q u a t i o n [ 8 . 3 2 ] a r e [ 8 . 4 2 a ] d ( p Q Q - p b b ) = - 2 G 2 n s ( p Q Q - p b b ) d t [ 8 . 4 2 b ] d p Q b = - ( G 2 + G 2 ) n s p Q b d t where T 1 " 1 = 2 G 2 n s and T 2 " 1 = ( G 2 + G 2 ) n s . L o n g i t u d i n a l r e l a x a t i o n r e t u r n s t h e d i a g o n a l e l e m e n t s of t h e s p i n sy s tem t o t h e i r t h e r m a l e q u i l i b r i u m v a l u e s as ( p Q Q ( t ) - p b b ( t ) ) = ( p a a ( 0 ) - p | D b ( 0 ) ) e x p ( - t / T 1 ) . T r a n s v e r s e r e l a x a t i o n removes c o h e r e n c e i n t h e t o t a l t r a n s v e r s e m a g n e t i z a t i o n as Pab^^~ p a b ( 0 ) e x p ( - t / T 2 ) by d e p h a s i n g t h e i n d i v i d u a l s p i n s w i t h r e s p e c t t o each o t h e r ( i n t h i s ca se v i a t h e d i p o l e - d i p o l e i n t e r a c t i o n ) . M i c r o s c o p i c a l l y one can p i c t u r e t h e r e l a x a t i o n p r o c e s s , w i t h t h e a i d of F i g . 32a , as one s p i n p r e c e s s i n g i n the d i p o l e f i e l d of a n o t h e r d u r i n g a c o l l i s i o n . Note t h a t i f H 0 i s p e r p e n d i c u l a r t o the s u r f a c e (as i n F i g . 3 2 b ) , a g i v e n s p i n w i l l not e x p e r i e n c e a t r a n s v e r s e component of t h e f i e l d due t o any of t h e o t h e r s p i n s . T h i s i s because t h e f i e l d l i n e s of a d i p o l e on a s u r f a c e , a l i g n e d p e r p e n d i c u l a r t o t h e s u r f a c e , a r e a l s o n o r m a l a t the s u r f a c e . T h i s a c c o u n t s f o r t h e f a c t t h a t i n t h e 2-D ca se T^'^-O when H 0 i s p e r p e n d i c u l a r t o t h e s u r f a c e . The o t h e r terms i n e q u a t i o n [ 8 . 3 2 ] a r e p e c u l i a r t o two d i m e n s i o n s . F o r e x a m p l e , the G * t e r m , 130 F i g u r e 32 D i p o l a r f i e l d o f a t o m s o n a s u r f a c e w i t h a ) 0 O = 3 O a n d b ) 6o=0 F i g u r e 33 A v e r a g e d i p o l a r f i e l d , <By>- and <B Z>, seen by an atom on t h e s u r f a c e due t o t h e o t h e r a toms . 132 [ 8 . 4 3 ] d p Q b = i G t n s P Q b dt a r i s e s from the f a c t t h a t the average f i e l d seen by an atom-, due t o the d i p o l a r f i e l d s of a l l t h e o t h e r atoms on t h e s u r f a c e , i s not z e r o as i t i s i n t h r e e d i m e n s i o n s . T h i s i s i l l u s t r a t e d i n F i g . 33 where the average f i e l d a t a p o i n t on the s u r f a c e i s d e p i c t e d by <By> and- <B Z>. The average f i e l d a t t h i s p o i n t a r i s e s from a random d i s t r i b u t i o n of d i p o l e s , o r i e n t a t e d a t an a n g l e t90 f rom the s u r f a c e norma l ( r o t a t e d about the x - a x i s ) , a t —* . . . —* p o s i t i o n s r w i t h r e s p e c t t o t h i s p o i n t . A v e r a g i n g o v e r r r e s u l t s i n <B ( x , y , z)> o c( 0 , 1 / 2 s i n # 0 , - co s t9 0 ) . P r e c e s s i o n about the z - a x i s component of t h e average f i e l d , d e s c r i b e d by G 1 f c au se s a f r e q u e n c y s h i f t i n t h e a-b t r a n s i t i o n of fi-»ft+G*ns. In t h i s e x p e r i m e n t , the f r e q u e n c y s h i f t G*ns i s t o o s m a l l t o o b s e r v e . A l l the o t h e r terms i n e q u a t i o n [ 8 . 3 2 ] a r e t i m e dependent and average t o z e r o . T h i s i s due t o t h e f a c t o r s e x p ( ± i f t t ) i n each of t h e s e t e r m s . In c o n c l u s i o n , t h e p r o c e s s e s p e c u l i a r t o two d i m e n s i o n s , i . e . the f r e q u e n c y s h i f t t e r m , do not a f f e c t the s p i n sys tem i n an o b s e r v a b l e way. 1 3 3 CHAPTER IX C o n c l u s i o n s E l e c t r o n s p i n re sonance t e c h n i q u e s have been used t o s t u d y t h e decay p r o c e s s e s i n s p i n - p o l a r i z e d a t o m i c hydrogen gas a t t e m p e r a t u r e s be low 0 .5 K . T h i s i s t h e f i r s t e x p e r i m e n t t o p r o v e t h e f e a s i b i l i t y o f u s i n g ESR t o measure t h e d e n s i t i e s of H | i n the a and b s t a t e s s e p a r a t e l y and p rodu c e a c c u r a t e r e s u l t s f o r t h e r a t e c o n s t a n t s . Most e x p e r i m e n t s done t o d a t e have measured o n l y t h e t o t a l d e n s i t y from w h i c h the n u c l e a r p o l a r i z a t i o n had t o be i n f e r r e d u s i n g p h y s i c a l model s of t h e decay p r o c e s s e s . ESR on t h e o t h e r hand a l l o w s n u c l e a r p o l a r i z a t i o n t o be o b s e r v e d d i r e c t l y . The ESR e x p e r i m e n t c a r r i e d out by van Yperen e t a l . ( 1 9 8 3 ) was the f i r s t t o d i r e c t l y show n u c l e a r p o l a r i z a t i o n . However t h e i r v e r s i o n , w h i c h had an open geometry and i n c o h e r e n t d e t e c t i o n , has not y e t y i e l d e d a c c u r a t e v a l u e s of K Q Q and K a b . The d e t a i l e d a n a l y s i s t h a t was c a r r i e d out on our own d a t a has r e v e a l e d how c a r e f u l one must be when f i t t i n g t h e s e t y p e s of decay c u r v e s . F o r e x a m p l e , i t i s sometimes p o s s i b l e t o r educe g w h i l e s i m u l t a n e o u s l y i n c r e a s i n g K a b i n such a way as t o not a p p r e c i a b l y degrade t h e q u a l i t y of t h e f i t . I n t h i s r e g a r d we no te t h a t our v a l u e s of 7 i n c r e a s e a t l o w e r t e m p e r a t u r e s as opposed t o t h e r e s u l t s o b t a i n e d by Y u r k e e t a l . ( 1 9 8 3 ) w h i c h have 7 d e c r e a s i n g a t the l o w e r t e m p e r a t u r e s . A t t h i s p o i n t we have c o n s i d e r a b l e c o n f i d e n c e i n t h e c o r r e c t n e s s of our a n a l y s i s 134 and a r e i n c l i n e d t o b e l i e v e t h a t t h e r e s u l t s o f Y u r k e e t a l . a r e e r r o n e o u s . I n t h e i r e x p e r i m e n t , o n l y one decay c u r v e , t h a t of t h e t o t a l d e n s i t y , was a v a i l a b l e f o r a n a l y s i s . H e n c e , l e s s i n f o r m a t i o n was a v a i l a b l e f o r d e t e r m i n i n g b o t h t h e i n d i v i d u a l r a t e c o n s t a n t s , and the e r r o r s i n v o l v e d . The t e m p e r a t u r e dependence of t h e r e c o m b i n a t i o n r a t e s has y i e l d e d a v a l u e f o r t h e b i n d i n g e n e r g y of H on 4 He of E g = l . i 0 ± . 0 2 K. T h i s s h o u l d be compared t o t h e f o l l o w i n g r e s u l t s , a l l of w h i c h where o b t a i n e d from t h e t e m p e r a t u r e dependence of K Q V . i ) E B - 0 • 9 9 ± . 02 K Hess e t a l . (1984) i i ) E B = 1 . 0 6 ± . 04 K Y u r k e e t a l . (1983) i i i ) E B » 0 .89±. 07 K M a t t h e y e t a l . (1981) i v ) E g = 1 . 0 1 ± . 06 K C l i n e e t a l . (1981) v) E g = 1 . 1 5 ± . 05 K Morrow e t a l . (1981 ) I t s h o u l d be n o t e d t h a t r e s u l t (v) was o b t a i n e d from an e x p e r i m e n t c o n d u c t e d a t z e r o m a g n e t i c f i e l d where K Q V i s a d i f f e r e n t c o m b i n a t i o n of t h e r a t e s Kortho a n < ^ K p a r a * I n a d d i t i o n , the same v a l u e of E g was o b t a i n e d from t h e t e m p e r a t u r e dependence of t h e s h i f t i n t h e h y p e r f i n e f r e q u e n c y , a / h . E x p e r i m e n t a l l y we have shown t h a t K Q Q and K a b have d i f f e r e n t t e m p e r a t u r e dependences w h i c h l e a d s one t o b e l i e v e t h a t K S T " 1 / 2 i s t e m p e r a t u r e dependent as w e l l . I t t h e n f o l l o w s 135 t h a t t h e b i n d i n g energy c a n n o t be a c c u r a t e l y d e t e r m i n e d from t h e t e m p e r a t u r e dependence of t h e r e c o m b i n a t i o n r a t e c o n s t a n t s a t p r e s e n t s i n c e a t h e o r y f o r s u r f a c e r e c o m b i n a t i o n has not been d e v e l o p e d . A r e f i n e d v e r s i o n of t h e z e r o f i e l d e x p e r i m e n t (v) may p r o d u c e a b e t t e r v a l u e f o r Eg from t h e t e m p e r a t u r e dependence of the s h i f t i n a / h . The average s u r f a c e r e c o m b i n a t i o n r a t e c o n s t a n t was d e t e r m i n e d t o be K S T - 1 / 2 B 2 = 5 . 7 ± . 8 c m 2 s " 1 K " 1 / 2 G 2 . V a l u e s r e p o r t e d by o t h e r g roups a r e : i ) K S T - 1 / 2 B 2 = 4.1 c m 2 s " 1 K " 1 ' 2 G 2 Y u r k e e t a l . (1983) i i ) K S T " 1 / 2 B 2 = 28 cm 2 s " 1 K ~ 1 / 2 G 2 M a t t h e y e t a l . (1981) i i i ) K S T " 1 / 2 B 2 = 5 .4 c m 2 s " 1 K " 1 / 2 G 2 C l i n e e t a l . (1981) I t s h o u l d be n o t e d i n c o m p a r i n g t h e s e r e s u l t s t h a t t h e v a l u e s o b t a i n e d by each a u t h o r f o r K s , w h i c h a r e e x t r a c t e d from K Q V , a r e s e n s i t i v e t o the v a l u e s of E g t h e y have u s e d . We have a l s o c a r r i e d out t h e f i r s t d e t a i l e d s t u d y of t h e one-body s u r f a c e r e l a x a t i o n r a t e . The r e d u c t i o n of t h i s r a t e by i n c r e a s i n g t h e t h i c k n e s s of t h e H 2 l a y e r on t h e c e l l w a l l i n d i c a t e s t h a t g i s a r e s u l t of m i c r o s c o p i c m a g n e t i c i m p u r i t i e s i n t h e c e l l w a l l s . C o n s i s t e n t w i t h t h e s e f i n d i n g s i s the f a c t t h a t t h e e x p e r i m e n t s p e r f o r m e d a t MIT showed no n u c l e a r p o l a r i z a t i o n u n t i l t h e H atom s o u r c e had been run f o r a l o n g t i m e , d u r i n g w h i c h t i m e a l a y e r of H 2 p r e s u m a b l y was b u i l t up i n 136 t h e i r c e l l ( C l i n e , p r i v a t e c o m m u n i c a t i o n ) . The t e m p e r a t u r e dependence of t h e one-body s u r f a c e r e l a x a t i o n r a t e , g , has been d e m o n s t r a t e d t o be not j u s t p r o p o r t i o n a l t o e x p ( E B / k g T ) as o r i g i n a l l y t h o u g h t by most g r o u p s . A l t h o u g h a d e t a i l e d t h e o r y e x p l a i n i n g t h e c o m p l e t e t e m p e r a t u r e dependence of g has not y e t been d e v e l o p e d , the q u a l i t a t i v e agreement between g and t h e c a l c u l a t i o n of B e r l i n s k y (1984) s u p p o r t s our model c o n c e r n i n g one-body s u r f a c e r e l a x a t i o n . F u r t h e r work s h o u l d be d o n e , e s p e c i a l l y i n m e a s u r i n g g as a f u n c t i o n of the t h i c k n e s s of H 2 on t h e c e l l w a l l s . We have shown i n t h i s t h e s i s t h a t an e f f e c t i v e way t o r educe g i s t o i n c r e a s e the t h i c k n e s s of H 2 on t h e c e l l w a l l s . T h i s i s n e c e s s a r y i f o t h e r r a t e c o n s t a n t s , f o r example the t h r e e - b o d y s u r f a c e r e l a x a t i o n r a t e , a r e t o be measured a t l o w e r t e m p e r a t u r e s i n t h i s a p p a r a t u s . A more s a t i s f y i n g a p p r o a c h may be t o b u i l d t h e sample c e l l u s i n g a b e t t e r grade of c o p p e r . E l e c t r o l y t i c tough p i t c h ( .999 p u r e ) c o p p e r , w h i c h i s known t o c o n t a i n m i c r o s c o p i c i r o n p a r t i c l e s , was used i n t h i s e x p e r i m e n t . SQUID magnetometery , used t o d e t e c t t h e p r e s e n c e of t h e p a r t i c l e s , has shown t h a t h i g h l y r e f i n e d c o p p e r , .99999 p u r e , i s e s s e n t i a l l y f r e e of m a g n e t i c i m p u r i t i e s ( W a l t o n , p r i v a t e c o m m u n i c a t i o n ) . I t w i l l be i n t e r e s t i n g t o see i f the He f i l m on a s u r f a c e o f h i g h p u r i t y c o p p e r i s enough t o r educe g t o an i m p e r c e p t i b l e l e v e l . 137 S p i n o s c i l l a t i o n s have not been o b s e r v e d i n t h e ESR s p e c t r a p r e s e n t e d i n t h i s t h e s i s . On the o t h e r h a n d , t h e NMR s p e c t r u m does c o n t a i n f e a t u r e s due t o s p i n o s c i l l a t i o n s ( Johnson e t a l . ( 1 9 8 4 ) ) . The t h e o r y o f L e v y and R u c k e n s t e i n (1984) assumes d i f f u s e m o t i o n of t h e H a toms , w h i c h i s c e r t a i n l y t h e ca se i n the NMR e x p e r i m e n t . However i n t h i s e x p e r i m e n t , w i t h the v e r y s m a l l c e l l and low d e n s i t i e s , t h e m o t i o n i s e s s e n t i a l l y f r e e p a r t i c l e l i k e . T h i s d i f f e r e n c e may w e l l l e a d t o q u a l i t a t i v e changes i n the t h e o r e t i c a l r e s u l t s ( R u c k e n s t e i n , p r i v a t e c o m m u n i c a t i o n ) . We see no o t h e r r e a s o n why s p i n o s c i l l a t i o n s s h o u l d not be o b s e r v a b l e i n the ESR s p e c t r u m . An e x p e r i m e n t a l check of t h i s h y p o t h e s i s w o u l d be t o measure t h e ESR s p e c t r u m a t h i g h e r d e n s i t i e s . S u p p l e m e n t i n g t h e e x p e r i m e n t a l work a r e t h e c a l c u l a t i o n s of t h e d i p o l e - d i p o l e r e l a x a t i o n r a t e s f o r two-body H c o l l i s i o n s i n the b u l k and on the s u r f a c e . The v a l u e c a l c u l a t e d f o r t h e b u l k r e l a x a t i o n r a t e i s a p p r o x i m a t e l y GgT~ 1 / 2 (1+ e 7 e / 7 p ) ~ 2 = 6 . 7 2 X 1 0 " 2 2 c m 3 s " 1 K " 1 / 2 . E x p e r i m e n t a l l y d e t e r m i n e d v a l u e s of G g T - 1 / 2 ( 1 + e 7 e / 7 p ) - 2 a r e : i ) 7 . 9 ± . 4 X 1 0 " 2 2 c m 3 s - 1 K " 1 / 2 Hess e t a l . (1984) i i ) 6 . 3 4 ± . 8 8 X 1 0 - 2 2 c m 3 s " 1 K - 1 / 2 Y u r k e e t a l . (1983) i i i ) 5 . 9 ± . 5 X 1 0 - 2 2 c m 3 s " 1 K - 1 / 2 S p r i k e t a l . (1982) i v ) 5 . 5 ± . 5 X 1 0 " 2 2 c m 3 s " 1 K - 1 / 2 C l i n e e t a l . (1981) 138 w h i c h a r e a l l i n good agreement w i t h t h e t h e o r e t i c a l r e s u l t . . I n c o n t r a s t , t h e o r e t i c a l r e s u l t s f o r t h e s u r f a c e r e l a x a t i o n r a t e s a r e about 50 t i m e s l o w e r t h a n the e x p e r i m e n t a l l y d e t e r m i n e d r a t e s . As m e n t i o n e d i n C h a p t e r V I , i t i s p o s s i b l e t h a t no one has y e t measured G s (Hess e t a l . ( 1 9 8 4 ) ) . U n t i l t h e p r e s e n t c o n t r o v e r s y o v e r t h e i n t e r p r e t a t i o n of the e x p e r i m e n t a l r e s u l t s i s r e s o l v e d t h e r e i s no r e l i a b l e e x p e r i m e n t a l check on G s . F u r t h e r improvements t o t h e p r e s e n t a p p a r a t u s s h o u l d y i e l d even b e t t e r r e s u l t s . F i x i n g the t h e r m a l s h o r t between the c e l l and t h e t h e r m a l s h i e l d w i l l a l l o w much l o w e r t e m p e r a t u r e s t o be o b t a i n e d . I f the t e m p e r a t u r e i s l o w e r e d t o =150 mK and g r e d u c e d by a f a c t o r of 50 i t s h o u l d be p o s s i b l e t o o b s e r v e t h e s u r f a c e t h r e e - b o d y r e c o m b i n a t i o n p r o c e s s . A low d e n s i t y e x p e r i m e n t a l measurement of t h i s q u a n t i t y w o u l d be a • u s e f u l check on t h e h i g h d e n s i t y r e s u l t s o f Hess e t a l . (1984) and S p r i k e t a l . (1984) as w e l l as a check of the h y p o t h e s i s of Hess e t a l . t h a t no one has y e t o b s e r v e d two-body s u r f a c e r e l a x a t i o n . F i n a l l y , m o d i f y i n g t h e s p e c t r o m e t e r so t h a t i t can measure t h e a b s o r p t i o n and d i s p e r s i o n s p e c t r a s i m u l t a n e o u s l y w i l l g r e a t l y s i m p l i f y the l i n e s h a p e a n a l y s i s a n d , h o p e f u l l y , w i l l a l s o i n c r e a s e t h e a c c u r a c y t o w h i c h d e n s i t i e s can be d e t e r m i n e d . REFERENCES R . M . C . A h n , J . P . H . W . v . d . E i j n d e , B . J . V e r h a a r , t o be p u b l i s h e d . L . C . B a l l i n g , R . J . Hanson , F . M . P i p k i n , P h y s . R e v . A 1 3 3 , 6 0 7 ( 1 9 6 4 ) A . J . B e r l i n s k y , t o be p u b l i s h e d . A . J . B e r l i n s k y and B. S h i z g a l , C a n . J . P h y s . 58 ,881 (1980) J . M . B l a t t , J . Comp. P h y s . J_,382(1967) R .W. C l i n e , T . J . G r e y t a k , D. K l e p p n e r , P h y s . R e v . L e t t . 4 7 , 1 1 9 5 ( 1 9 8 1 ) R .W. C l i n e , p r i v a t e c o m m u n i c a t i o n . S . B . C r a m p t o n , J . J . K r u p c z a k , S . P . S o u z a , P h y s . R e v . B 2 5 , 4 3 8 3 ( 1 9 8 2 ) D.O. Edwards and I . B . M a n t z , J . P h y s . ( P a r i s ) 4J_,C7-257( 1 980) J . P . H . W . v . d . E i j n d e , C . J . R e u v e r , B . J . V e r h a a r , t o be p u b l i s h e d . R . D . E t t e r s , J . V . Dugan, R .W. P a l m e r , J . Chem. P h y s . 6 2 , 3 1 3 ( 1 9 7 5 ) V . V . Goldman and I . F . S i l v e r a , P h y s i c a 107B,515(1981) J . M . G r e b e n , A . W . Thomas, A . J . B e r l i n s k y , C a n . J . P h y s . 5 9 , 9 4 5 ( 1 9 8 1 ) D. t e r H a a r , R e p o r t s on P r o g r e s s i n P h y s i c s 24 ,304 (1961 ) C . E . H e c h t , P h y s i c a 2 5 , M 5 9 ( 1 9 5 9 ) H . F . H e s s , D . A . B e l l , G . P . K o c h a n s k i , R .W. C l i n e , D . K l e p p n e r , T . J . G r e y t a k , P h y s . R e v . L e t t . 5J_, 483 (1 983) H . F . H e s s , D . A . B e l l , G . P . K o c h a n s k i , D. K l e p p n e r , T . J . G r e y t a k , t o be p u b l i s h e d . R. Jochemsen , M . Morrow, A . J . B e r l i n s k y , W . N . H a r d y , P h y s . 140 Rev. L e t t . 47,852(1981) B.R. J o h n s o n , J . S . Denk e r , N. B i g e l o w , L.P. L e v y , J.H. F r e e d , D.M. Le e , t o be p u b l i s h e d . M.A. K i n c h and B.V. R o l l i n , B r i t . J . A p p l . P h y s . J_4,672(1963) S. K o b a y a s i , M. S h i n o h a r a , K. Ono, C r y o g e n i c s , , 5 9 7 ( 1 9 7 6 ) W. K o l o s and L. W o l n i e w i c z , Chem. P h y s . L e t t . 24^,457(1974) L.P. L e v y and A.E. R u c k e n s t e i n , t o be p u b l i s h e d . 0 . V. Lounasmaa, E x p e r i m e n t a l P r i n c i p l e s and Methods Below _1_ K Academic P r e s s , New York (19747^ A.P.M. M a t t h e y , J.T.M. W a l r a v e n , I . F . S i l v e r a , P h y s . Rev. L e t t . 46,668(1981) M. Morrow, R. Jochemsen, A . J . B e r l i n s k y , W.N. Hardy, P h y s . Rev. L e t t . , 46,195(1981) and 4_7 ,455(E)(1981) M. Morrow and A . J . B e r l i n s k y , Can. J . Phy s . 61,1042(1983) E.H. P u t l e y , A p p l . O p t i c s 4,649(1965) M.E. Rose, E l e m e n t a r y T h e o r y o f A n g u l a r Momentum John W i l e y and Sons, New York (1957) A.E. R u c k e n s t e i n , p r i v a t e c o m m u n i c a t i o n . S. S a i t o and T. S a t o , Rev. S c i . I n s t r u m . 46,1226(1975) A. S h i m i z u and M. Inoue, I E E E T r a n s . on M a g n e t i c s MAG-17,2146(1981) 1. S h i n k o d a , M.Sc. T h e s i s , U.B.C. (1983) I . F . S i l v e r a and J.T.M. W a l r a v e n , P h y s . Rev. L e t t . 44,164(1980) R. S p r i k , J.T.M. W a l r a v e n , G.H. van Y p e r e n , I . F . S i l v e r a , P h y s . Rev. L e t t . 49,153(1982) R. S p r i k , J.T.M. W a l r a v e n , I . F . S i l v e r a , P h y s . Rev. L e t t . 5 1,479(1983) and 5J_, 942(E ) (1 983 ) B. W. S t a t t , M.Sc. T h e s i s , U.B.C. (1979) 141 B.W. S t a t t and A . J . B e r l i n s k y , P h y s . R e v . L e t t . 4 5 , 2 1 0 5 ( 1 9 8 0 ) B .W. S t a t t , P h y s . R e v . B25 ,6035(1982) W . C . S t w a l l e y and L . H . Nosanow, P h y s . R e v . L e t t . 3 6 , 9 1 0 ( 1 9 7 6 ) D. W a l t o n , p r i v a t e c o m m u n i c a t i o n . A . Y a r i v , I n t r o d u c t i o n t o O p t i c a l E l e c t r o n i c s 2 n d . E d i t i o n , H o l t , R i n e h a r t and W i n s t o n , (1976) G . H . van Y p e r e n , I . F . S i l v e r a , J . T . M . W a l r a v e n , J . B e r k o u t , J . G . B r i s s o n , P h y s . R e v . L e t t . 50 ,53 (1983 ) G . H . van Y p e r e n , P h . D . t h e s i s , U n i v e r s i t y of Amsterdam (1983) ( u n p u b l i s h e d ) . B . Y u r k e , J . S . D e n k e r , B . R . J o h n s o n , N . B i g e l o w , L . P . L e v y , D . M . L e e , J . H . F r e e d , P h y s . R e v . L e t t . 50 ,1137 (1983 ) D. Zimmerman, M . S c . T h e s i s , U . B . C . (1982) 142 APPENDIX A A b s o r p t i o n and D i s p e r s i o n M e a s u r e m e n t s M e a s u r i n g t h e a b s o r p t i o n o r d i s p e r s i o n s p e c t r u m can be done by s e t t i n g t h e microwave f r e q u e n c y i n t h e a p p r o p r i a t e p l a c e on t h e c a v i t y r e s o n a n c e . In t h e f i r s t two e x p e r i m e n t s the r e f l e c t e d power was measured w i t h t h e m i c r o w a v e f r e q u e n c y c h o s e n i n s u c h a way t h a t e i t h e r t h e a b s o r p t i o n o r d i s p e r s i o n s p e c t r u m was m e a s u r e d . In t h e f i n a l e x p e r i m e n t t h e r e f l e c t e d v o l t a g e was m e a s u r e d w i t h t h e m i c r o w a v e s t u n e d t o t h e c a v i t y r e s o n a n c e . In o r d e r t o c a l c u l a t e w h i c h m i c r o w a v e f r e q u e n c i e s s h o u l d be used, we s h a l l s t a r t by m o d e l l i n g t h e c a v i t y r e s o n a n c e w i t h the f o l l o w i n g e q u i v a l e n t c i r c u i t . c o u p l i n g |_ n:l • * ' ca v 11 y The r e s o n a n t f r e q u e n c y i s u>Q = 1 //L/C and t h e u n l o a d e d Q i s Q 0 = r _ V L / C . The c o u p l i n g c o e f f i c i e n t i s d e f i n d as /3 = Z 0 / n z r . The v o l t a g e r e f l e c t i o n c o e f f i c i e n t r w i l l now be c a l c u l a t e d . By d e f i n i t i o n 143 [A. 1 ] r = z-z, z+z. whereby [ A . 2 ] r = 1 -g+iQo (CJ/CJO-CJQ/CJ) 1 +/3 + i Q 0 ( a)/cj 0-cj 0 / w ) and [ A . 3 ] I T I 2 ^ ( l - g ) 2 Q n - 2 a ; 2 + 4 ( c u - c u n ) 2 ( l + r 3 ) 2 Q 0 - 2 c j 2 + 4 ( c J - c j 0 ) 2 where t h e a p p r o x i m a t i o n (u>/u0-u)0/u))^2 (CJ-CJ0 )/a>0 near r e s o n a n c e has been u s e d . T h i s l e a d s t o a r e s o n a n t l i n e s h a p e of [ A . 4 ] 1- | T | 2= 4/3 ACJ 2 O + 0 ) 2 2 + ( C J - O J 0 ) 2 where Au> =(1+/3)ACJ and Q 0=CJ 0 / 2AOJ , AOJ b e i n g the h a l f w i d t h a t h a l f maximum. Note t h a t 0=1 f o r c r i t i c a l c o u l p l i n g , /3<1 f o r under c o u p l i n g and /3>1 f o r o v e r c o u p l i n g . A l s o , t h e measured ( l o a d e d ) Q i s g i v e n by [ A . 5 ] Q j ^ - Q - ^ Q c ' where t h e c o u p l i n g Q i s g i v e n by QG = r 3 " 1 Q 0 . A t o m i c h y d r o g e n i n the r e s o n a t o r a f f e c t s t h e i n d u c t a n c e i n such a way t h a t L—>L ( 1 +4 rrx) w h i c h i s - e q u i v a l e n t t o L->L( 1+47rx/ ) and r ->r + 4 7 r c j 0 L x / / • Hence 1 44 [A.6] CJ 0 - » CJ 0( 1+4TTX/ ) * " 2 Q O 1 6 J 0 _ ^ Q 0 1 o ; 0 ( 1 + 4 7 T Q O X 7 7 ) ( 1+4TTX 7 ) " 1 /3(1+4TTQ 0X / /)- 1 o r [ A . 7 ] w 0 — ^ w 0 ( 1 ~2 6CJ 0/CJ 0 )" I Y 2 Q O 1 C J 0 -> Q O 1 C J 0 ( 1 + Q 0 5 Q - 1 ) ( 1 - 2 6 W 0 / C J 0 ) - 1 ]3 -> /3(1+Q 06Q- 1 )" 1where 6CJ0 = -2 77X /CJ 0 and 6Q _ 1 = + 4TTX". F i r s t c o n s i d e r t h e r e f l e c t e d power s p e c t r u m , m e a s u r e d i n the f i r s t two e x p e r i m e n t s . S e t t i n g t h e microwave f r e q u e n c y u=u>0 a l l o w s one t o measure t h e a b s o r p t i o n s p e c t r u m , t o f i r s t o r d e r i n Q 0 5 Q _ 1 . The change i n t h e power r e f l e c t i o n c o e f f i c i e n t t h r o u g h the h y d r o g e n r e s o n a n c e i s g i v e n by 6 | r | 2 = | r | 2 - ( 1 - B ) 2 / ( 1 + B)2 y i e l d i n g , t o s e c o n d o r d e r , [A. 8] 5 | r} 2= 4/3 [ ( 1-/3 2)Q 05Q- ' + (2/3-1 ) (Q 06Cy 1 ) 2 + 1 6/3Q2 ( 6w 0/w 0 ) 2 ] (1+/3)« In o r d e r t o measure t h e d i s p e r s i o n s p e c t r u m , t o f i r s t o r d e r , one would l i k e t o f i n d t h e f r e q u e n c y where 3 | r | 2 / 3 ( S Q " 1 ) = 0 . T h i s o c c u r s a t w=w 0±i/l-B 2 • Aw = w 0 ± / ( 1 -B)/( 1 +B) •Aw . Note t h a t one must be u n d e r c o u p l e d i n o r d e r t o a c h i e v e t h i s c o n d i t i o n . D e f i n i n g i n t h i s c a s e 6| T | 2 = | r | 2 - ( 1-B ) / ( 1+B) y i e l d s 1 4 5 [ A . 9 ] 5 | T | 2 = J [ - 4 i / l - / 3 2 Q 0 6 a ) 0 / c j 0 + 4 ( 2 / 3 2 - 1 ) (Q06co0/u>0)  2 ( 1 + 0 ) 2 + 4 / 1 - / 3 2 Q 0 5 C J 0 / " O Q O 6 Q " 1 + ( Q O 5 Q " 1 ) 2 ] . C o n s i d e r now t h e v o l t a g e r e f l e c t i o n c o e f f i c i e n t measured i n the f i n a l e x p e r i m e n t . At r e s o n a n c e , CJ=CJ0 , a l l o w i n g e q u a t i o n [ A . 2 ] t o be r e w r i t t e n a s [A.10] r = r 0 + ye + iyy 1 + 7// + 1 7 ' u s i n g t h e n o t a t i o n i n C h a p t e r V I . D e f i n i n g t h e c o u p l i n g c o e f f i c i e n t 77 = r 0 " 1 — 1 and t h e s i g n a l 5 = ( r - r 0 ) / r 0 y i e l d s [ A . 1 1 ] 6 = T? y" + y"2 + y' 2 + i y' ( 1 + 7 " ) * + 7 / 2 T h i s i s t h e e x p r e s s i o n u s e d i n C h a p t e r VI t o a n a l y z e t h e measured ESR s i g n a l s . 1 46 APPENDIX B M i c r o w a v e P h a s e - L o c k System The p h a s e - l o c k s y s t e m u s e d t o s t a b i l i z e t h e k l y s t r o n f r e q u e n c y i s b r i e f l y d e s c r i b e d h e r e . Two p h a s e - l o c k i n g s y stems a r e u s e d . The f i r s t p h a s e - l o c k s a 12-18 GHz backward wave o s c i l l a t o r t o t h e r e f e r e n c e f r e q u e n c y s y n t h e s i z e r . The 12-18 GHz s i g n a l i s t h e n u s e d as a r e f e r e n c e s i g n a l f o r t h e k l y s t r o n p h a s e - l o c k i n g c i r c u i t . F i r s t , t h e m i l l i m e t e r wave p h a s e - l o c k i n g s y s t e m w i l l be o u t l i n e d . As i l l u s t r a t e d i n F i g . 34, a V a r i a n VRT-2125A5 k l y s t r o n p r o v i d e s t h e LO s i g n a l f o r t h e m i l l i m e t e r wave s p e c t r o m e t e r . P a r t of t h i s s i g n a l i s mixed w i t h t h e 1 2 - 1 8 GHz r e f e r e n c e s i g n a l i n t h e h a r m o n i c m i x e r . The I F , n e a r 400 MHz, i s a m p l i f i e d and f e d t o t h e d i v i d e by t e n p r e s c a l e r , whose o u t p u t i s now n e a r 40 MHz. T h i s f r e q u e n c y d i v i s i o n i s done i n o r d e r t o i n c r e a s e t h e s y s t e m ' s l o c k i n g r a n g e . The 40 MHz IF s i g n a l i s t h e n u s e d by t h e M i c r o w a v e Systems I n c . MOS-5/VT s y n c h r o n i z e r t o p r o v i d e t h e r e q u i r e d e r r o r s i g n a l t o t h e k l y s t r o n . A s c h e m a t i c of t h e 12-18 GHz p h a s e - l o c k i n g c i r c u i t a p p e a r s i n F i g . 35. T h i s i s e s s e n t i a l l y t h e same s y s t e m a s d e s c r i b e d i n t h e M.Sc. t h e s i s of S t a t t (1979) and w i l l n o t be d i s c u s s e d h e r e . 147 KLYSTRON POWER SUPPLY KLYSTRON CRYSTAL D E T E C T O R 4 9 HARMON MIXER M0S-5 SYNCHRONIZER I FROM TO MILLIMETER WAVE SPECTROMETER 12.4-18.0 GHz SWEEPER AMPLIFIER PRESCALEF AMPLIFIER F i g u r e 34 ' M i l l i m e t e r wave p h a s e - l o c k c i r c u i t ISOLATOR | [—» LEVELLER CRYSTAL DETECTOR TO CELL ERROR SIGNAL FM INPUT FROM BACKWARD WAVE OSCILLATOR COMPUTER SWEEP INPUT 16 BIT D/A SYNCHRONIZER 20 MHz BAND PASS 2-*A GHz ISOLATOR SAMPLER AMPLIFIER 200 MHz FILTER BAND PASS FILTER LIMITER 4 10 PRESCALER -* !»-LO MULTIPLIER CHAIN SYNTHESIZER FR OM COMPUTER F i g u r e 3 5 " 12-18 GHz BWO p h a s e - l o c k c i r c u i t 149 APPENDIX C 1460 MHz S p e c t r o m e t e r A b r i e f d e s c r i p t i o n o f t h e 1460 MHz s p e c t r o m e t e r w i l l now be g i v e n . B a s i c a l l y , i t i s a m o d i f i e d v e r s i o n o f t h e 1420 MHz s p e c t r o m e t e r u s e d t o p e r f o r m m a g n e t i c r e s o n a n c e s t u d i e s on a t o m i c h y d r o g e n a t z e r o m a g n e t i c f i e l d . The o p e r a t i n g f r e q u e n c y was c h a n g e d t o 1460 MHz f o r t h e f o l l o w i n g r e a s o n . When s w e e p i n g t h e m a g n e t i c f i e l d f r o m t h e a t o b l i n e t h e f i e l d c h a n g e s by a p p r o x i m a t e l y a//n e. Hence t h e m i l l i m e t e r wave c a r r i e r , 40-50 dB above t h e s i d e b a n d power, m i g h t l e a k i n t o t h e sample c a v i t y and d i s t u r b t h e H atoms. I n c r e a s i n g t h e s i d e b a n d IF t o 1460 MHz a v o i d s t h i s p r o b l e m . R e f e r r i n g t o F i g . 36, a 10 MHz r e f e r e n c e s i g n a l i s m u l t i p l i e d up t o 1460 MHz and added t o t h e s y n t h e s i z e r f r e q u e n c y of 450 KHz i n t h e s i n g l e s i d e b a n d (SSB) g e n e r a t o r . N o r m a l l y t h e SSB g e n e r a t o r i s o p e r a t e d i n t h e p u l s e mode but h e r e i t i s u s e d i n t h e CW mode. A h i g h Q f i l t e r removes unwanted s i g n a l s f r o m the" 1460 MHz s i g n a l w h i c h i s u s e d t o m o d u l a t e t h e m i l l i m e t e r wave c a r r i e r s i g n a l . The 1460.405 MHz s i g n a l from t h e m i l l i m e t e r wave d e t e c t o r m i x e r i s f e d i n t o t h e GaAs low n o i s e p r e - a m p l i f i e r . The s i g n a l i s t h e n m i x e d down t o g i v e t h e n e a r DC o u t p u t s i g n a l w h i c h i s r e c o r d e d by t h e l a b c o m p u t e r . 150 FREQUENCY SYNTHESIZER GATE VOLTAGE 10 MHz REFERENCE GAT PHA SPLI ED SE TTER > ' \ ( SS GENER B ATOR X 2 X 73 MUTLIPLIER AMP POWER SPLITTER 10db 1460.405 MHz 1460 MHz IMAGE REJECTION MIXER MAIN AMP. \ 1 AM P. 1 i LOW PRE NOISE UMP MODULATOR PHASE SHIFTER MIXER PREAMP 405 kHz AMP. SIGNAL OUT 405 kHz OUTPUT FROM MICROWAVE MIXER 1460 MHz F i g u r e 36 S p e c t r o m e t e r 151 APPENDIX D H y p e r f i n e S p i n S t a t e s The s i n g l e atom h y p e r f i n e s t a t e s i n o r d e r o f d e s c e n d i n g e n e r g y a r e |d> = |H> |c> = |ti> + e |W> |b> = \tf> |a> = |j$>-e |U> These a r e t o o r d e r e and f i n d i c a t e the s component of the e l e c t r o n s p i n and 4 the i component of the p r o t o n s p i n . W i t h the f o l l o w i n g n o t a t i o n f o r two atom s t a t e s |tt> = H1> |H> = 1 / / 2 1 |10>+ |00>] \\l> = |1-1> Ilt> = |10>-|00>] where |tt>=jt>i j t>2 and |Ss > a r e t h e t o t a l s p i n s t a t e s , t h e two atom h y p e r f i n e s t a t e s a r e , t o o r d e r e, |a,a> = |1-1>e 11 1 > p — e C |10>6 | 1 0> p - |00>e |00> p] |b ,a> = 1 / / 7 | 1 - 1 > e [ hO> p -|00> p ]-e / /2~[ | l 0> e - |00> e ] |b ,b> = | l -1> e |1-1>P 1 - 1 > p | c , a> = l / 2 [ |l0> e+ |00> e][ | l 0> p - |00> p ] + e[ 11 - 1 >e | H > p - | n > e 11 — 1 >p 1 | c ,b> = l / / 2 [ |10>6+ J00>e] |l-1> p +e//T|l-1> e [ |10>P+ |00> p] |d ,a> = 1//21 |10>6 + |00>e ] |1 1> p -e / /T| l 1>e [ |l0> p+ |00> p] 152 d , b > = l / 2 [ | l O > e + | 00> e ] [ | l O > p + | 0 0 > p ] c,c> = |1 1> e . | l - 1 > p + e[ | l O > e | l O > p - |00> e | 00> p ] c, d> = l / / 2 ~ | l 1 > e [ | l O > p - J 0 0 > p ] + e / / 2 [ | l O > e - J O O > e l | l 1 > p d , d> = |1 1 > e |l 1 > p 153 APPENDIX E S p i n - E x c h a n g e R e l a x a t i o n S p i n - e x c h a n g e s c a t t e r i n g of H atoms i n low m a g n e t i c f i e l d s i s t h e d ominant r e l a x a t i o n mechanism a t low t e m p e r a t u r e s . In h i g h e r f i e l d s t h e s p i n - e x c h a n g e r e l a x a t i o n r a t e between t h e a and b s t a t e s d e c r e a s e s e x p o n e n t i a l l y as t h e t e m p e r a t u r e i s l o w e r e d . T h i s i s b e c a u s e a l l t h e a l l o w e d p r o c e s s e s r e q u i r e one of t h e atoms t o be e x c i t e d i n t o one o f t h e u p p e r h y p e r f i n e s t a t e s . On t h e o t h e r hand, s p i n - e x c h a n g e i s n o t i n s i g n i f i c a n t f o r r e l a x a t i o n between t h e upper and l o w e r h y p e r f i n e s t a t e s . The t i m e d e r i v a t i v e of t h e s p i n - d e n s i t y m a t r i x d e s c r i b i n g t h e s e r e l a x a t i o n r a t e s i s p r e s e n t e d f o r a r b i t r a r y m a g n e t i c f i e l d f o l l o w i n g t h e t r e a t m e n t i n z e r o f i e l d o f B a l l i n g e t a l . ( 1 9 6 4 ) . U n p u b l i s h e d n o t e s of A . J . B e r l i n s k y a r e u s e d t o o b t a i n t h e g e n e r a l e x p r e s s i o n f o r t h e s p i n - d e n s i t y m a t r i x e q u a t i o n and t o c h e c k t h e f i n a l r e s u l t s , i n s p e c i a l c a s e s . The t i m e e v o l u t i o n o f t h e s p i n d e n s i t y m a t r i x i s d e s c r i b e d by [ E . 1 ] dpr'r = I 2 a ( I S ; I V )A! f (h^'h , ;h',h ,h' 2h a ) p./. p , dt i ' h^h, IS i b h , n ' n 2 n * h' 2h 2 l's' where t h e non z e r o t e r m s a r e g i v e n by 1 54 even [E.2] Z o ( I S ; l ' S')A J J = < (*k/u) n u k k - 2 Z (21 + 1) IS I S H L l 2i(S]-6°) -21(61-6°) x [ ( e 1 1 - 1 ) A l 0 J ) + ( e 1 1 - D A ? ? ] odd 2i(6 1.-6?) -2i(6 1-6°) -i + 7rk- 2 Z ( 2 1 + D [ ( e ( 1 -DA?J,+ ( e 1 1 - 1 ) A ^ ] | > . The n o t a t i o n used here i s the same as that i n Chapter VIII with 6° and 6| being the phase s h i f t s i n p a r t i a l wave a n a l y s i s f o r the s i n g l e t and t r i p l e t p o t e n t i a l . Lambda i s given by [E.3] A j g (h,h, ;tiihiti2h2) = Z B j/g/(h', h 2 , h ' , h 2 ) B j g ( h , h 2 , h , h 2) h 2 where [E.4] B j S ( h , h 2 ,h,h 2) = Z A (h , h 2 ; IM j SM^) A (h , h 2 ; IM j SM<-) M,MS and [E.5] A(F 1M 1F 2M 2;IM ISM S) = 6.. .. S C( 1/2, 1/2 ,S;mf M<--m) xC( 1/2,1/2,1;M1-m,M1-M,+m)aH(1/2,1/2,F,;m,M,-m) x a H ( 1/2, l/2,F2;Ms-m,M2-Ms+m) . The non zero e x p r e s s i o n s f o r a ^ a r e : a u ( 1/2,1/2,0;-1/2,1/2) = cos0 1 55 a H(1/2,1/2,0;1/2,-1/2) = - s i n 0 a H ( 1/2, 1/2,1; ± 1 / 2 , ±1/2) = 1 a H ( 1/2, 1/2, 1 ,--1/2, 1/2) = sin© a H ( 1 / 2 , 1 / 2 , 1 ; 1 / 2 , - 1 / 2 ) = c o s 0 The a ^ r e l a t e t h e a t o m i c h y p e r f i n e s t a t e wave f u n c t i o n , |FM>, to t h e e l e c t r o n and p r o t o n s p i n wave f u n c t i o n s . S i m i l a r l y , A r e l a t e s t h e two atom m o l e c u l a r s t a t e s t o t h e two a t o m i c h y p e r f i n e s t a t e s . The f i n a l r e s u l t s , summerized i n T a b l e V I I , a r e w r i t t e n i n terms of t h e t h e r m a l a v e r a g e s of-even odd [ E . 6 a ] o*-(k) = 27rk" 2 Z ( 21+ 1 ) s i n 2 ( 8 ? - 5 ) ) 1 1 1 even odd [ E . 6 b ] X " ( k ) = 27rk" 2 Z (21+1 ) s i n 2 ( 6 ° - 6 M . 1 Note t h a t a l l t h e e l e m e n t s i n T a b l e V I I must be m u l t i p l i e d by v n H t o g i v e dp / d t . The two q u a n t i t i e s x and $ i n v o l v i n g t h e m a g n e t i c f i e l d a r e g i v e n by [ E . 7 a ] x = s i n 2 2 0 ' ^ [E . 7 b ] $ = c o s 2 0 156 T h e r e a r e two b a s i c r e l a x a t i o n r a t e s , g i v e n by [E . 8 a ] G X = v(o*+a-)/4 [E.8b] G * = xvaV 8 w h i c h d e s c r i b e two f u n d a m e n t a l l y d i f f e r e n t t y p e s of r e l a x a t i o n p r o c e s s e s . F i r s t , t h e r e i s G X w h i c h i n v o l v e s t h e t r a n s i t i o n s (a , c (b,d) . T h i s t e n d s t o e q u a l i z e t h e r a t i o of n^/nQ and nc/n^ b u t does n o t c a u s e atoms i n t h e upper h y p e r f i n e s t a t e s t o r e l a x t o t h e lo w e r two s t a t e s . S e c o n d , t h e r e i s G £ w h i c h does r e l a x atoms i n t h e upper two s t a t e s . Note t h a t , i n h i g h m a g n e t i c f i e l d s , x—4e 2 and p=1, t h e r e b y r e d u c i n g t h i s r a t e c o n s i d e r a b l y . • The X - t e r m s r e s u l t i n a f r e q u e n c y s h i f t i n t h e r e l a v e n t t r a n s i t i o n f r e q u e n c i e s . N u m e r i c a l e v a l u a t i o n of t h e s e q u a n t i t i e s c an be c a r r i e d o u t u s i n g t h e 3-D t h e r m a l a v e r a g e s o f B e r l i n s k y and S h i z g a l (1980) and t h e 2-D t h e r m a l a v e r a g e s of Morrow and B e r l i n s k y ( 1 9 8 3 ) . TABLE V I I • S p i n - E x c h a n g e S p i n D e n s i t y M a t r i x : d / d t P u b l i c a t i o n s 1. B.W. S t a t t , W.N. Hardy and R. Jochemsen 1980. Observation of new l i n e s i n the microwave spectrum of o r t h o - ^ p a i r s i n s o l i d hydrogen, Can. J . Phys. _58, 1326. 2. B.W. Stat t and W.N. Hardy 1980. L ine broadening mechanisms of the ortho-H2 p a i r spectrum, Can. J . Phys. _58_, 1341. 3. R. Jochemsen, B.W. Stat t and W.N. Hardy 1980. Microwave absorption by ortho-H2 p a i r s i n s o l i d hydrogen: temperature and densi ty dependence of the f requencies , Can. J . Phys. 58, 1356. 4. W.N. Hardy, M. Morrow, R. Jochemsen, B.W. S t a t t , P .R. Kubik, R.M. Mar so l a i s , A . J . Ber l in sky and A . Landesman 1980. Magnetic resonance s tudies of atomic hydrogen gas at IK and zero magnetic f i e l d , Phys. Rev. Le t te r s 45,453, 5. W.N. Hardy, M. Morrow, R. Jochemsen, B.W. S t a t t , P .R . Kubik, R.M. Mar so l a i s , A . J . B e r l i n s k y , and A . Landesman 1980. Magnetic resonance s tudies of atomic hydrogen gas at low temperatures, Journal de Physique 41, c7-151. 6. B.W. Statt and A . J . Ber l in sky 1980. Theory of spin r e l a x a t i o n and recombination i n s p i n - p o l a r i z e d atomic hydrogen, ' . Phys. Rev. Le t t e r s 4_5, 2105. 7. B.W. S ta t t 1982. Spin r e l a x a t i o n of sp in-po la r i zed atomic hydrogen adsorbed on a surface , Phys. Rev. B25, 6035. 

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