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Orientational ordering of hydrogen molecules adsorbed on graphite Kubik, Philip Roman 1984

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ORIENTATIONAL ORDERING OF HYDROGEN MOLECULES ADSORBED ON GRAPHITE by P H I L I P ROMAN KUBIK B . S c , 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 , 1974 M . S c , 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 , 1977 A THESIS SUBMITTED IN P A R T I A L FULFILLMENT 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 a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA May, 1984 © P h i l i p Roman K u b i k , 1984 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of rkys/c^  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date July l9S</ i ABSTRACT NMR h a s b e e n e m p l o y e d t o m e a s u r e t h e o r i e n t a t i o n a l b e h a v i o u r o f s u b - m o n o l a y e r s o f H2 and D2 a d s o r b e d on g r a p h i t e f r o m .3 K t o 12 K. F o r t h e t e m p e r a t u r e a n d c o v e r a g e s u s e d , t h e h y d r o g e n f o r m s a l a t t i c e c o m m e n s u r a t e w i t h t h a t o f t h e g r a p h i t e ( /3 x ^3 s t r u c t u r e ) and c a n be m o d e l l e d by a h e x a g o n a l l a t t i c e o f i n t e r a c t i n g quantum q u a d r u p o l e s . E a c h m o l e c u l e a l s o e x p e r i e n c e s a c r y s t a l f i e l d a r i s i n g f r o m Van d e r W a a l s i n t e r -a c t i o n s w i t h t h e s u b s t r a t e and w i t h t h e o t h e r h y d r o g e n m o l e c u l e s . F o r s u c h a s y s t e m , mean f i e l d t h e o r y p r e d i c t s a v a r i e t y o f o r i e n t a t i o n a l l y o r d e r e d p h a s e s , d e p e n d i n g on t h e r e l a t i v e s t r e n g t h o f t h e c r y s t a l f i e l d and t h e m o l e c u l a r f i e l d , w h i c h a r e a l w a y s i n o p p o s i t i o n . A k e y q u e s t i o n t o be a n s w e r e d by e x p e r i m e n t was w h e t h e r t h e s t r o n g f l u c t u a t i o n s a s s o c i a t e d w i t h quantum e f f e c t s and t h e two d i m e n s i o n a l n a t u r e o f t h e s y s t e m w o u l d i n f a c t a l l o w a f i n i t e t r a n s i t i o n t e m p e r a t u r e . We h a v e o b s e r v e d t h a t f o r 90% o r t h o - H 2 ( J = l ) , t h e s p l i t t i n g o f t h e h i g h t e m p e r a t u r e NMR d o u b l e t i n c r e a s e s r a p i d l y w i t h d e c r e a s i n g t e m p e r a t u r e n e a r .6 K and a d d i t i o n a l s t r u c t u r e a p p e a r s . I n a d d i t i o n , t h e .3 K s p e c t r u m a g r e e s w e l l w i t h t h e e x p e c t e d T=0 s p e c t r u m f o r t h e o r i e n t a t i o n a l l y o r d e r e d p i n w h e e l p h a s e o f mean f i e l d t h e o r y . C o n s e q u e n t l y , t h e r a p i d i n c r e a s e i n t h e s p l i t t i n g i s i n t e r p r e t e d a s an o r i e n t a t i o n a l o r d e r i n g t r a n s i t i o n . i i I n c o n t r a s t t o t h e r e s u l t s f o r H 21 t h e NMR s p e c t r u m o f 90 % p a r a - D 2 ( J = l ) shows no r a p i d c h a n g e s , e v o l v i n g s m o o t h l y f r o m a d o u b l e t t o a v e r y b r o a d and weak s t r u c t u r e . T h e r e i s no e v i d e n c e o f o r i e n t a t i o n a l o r d e r i n g , a t l e a s t down t o .3 K. The d i f f e r e n t b e h a v i o u r o f H 2 and D 2 i s p a r t i c u l a r l y p u z z l i n g i n l i g h t o f t h e f a c t t h a t t h e b u l k s o l i d s b o t h o r d e r o r i e n t a t i o n -a l l y i n t o t h e same s t r u c t u r e a t s i m i l a r t e m p e r a t u r e s . From t h e 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 e s p l i t t i n g o f t h e h i g h t e m p e r a t u r e NMR d o u b l e t s , we h a v e e x t r a c t e d t h e v a l u e s o f t h e c r y s t a l f i e l d s and e f f e c t i v e q u a d r u p o l e c o u p l i n g c o n s t a n t s o f H 2 and D 2. The q u a d r u p o l e c o u p l i n g c o n s t a n t o f D 2 i s somewhat l e s s t h a n t h e r i g i d l a t t i c e v a l u e as one w o u l d e x p e c t i n t h e p r e s e n c e o f t r a n s l a t i o n a l z e r o p o i n t m o t i o n . However, t h a t o f H 2 i s much t o o s m a l l f o r t h i s m e c h a n i s m t o be r e s p o n s i b l e . i i i TABLE OF CONTENTS Page ABSTRACT i L I S T OF TABLES v i L I S T OF FIGURES '. . v i i ACKNOWLEDGEMENTS , X CHAPTER I : INTRODUCTION 1 1.1 T e c h n i q u e s A v a i l a b l e f o r t h e S t u d y o f P h y s i s o r b e d H y d r o g e n 1 1.2 P r e v i o u s Work 3 1.3 S c o p e o f t h i s T h e s i s 9 CHAPTER I I : ORIENTATIONAL PROPERTIES OF HYDROGEN 17 2.1 O r i e n t a t i o n a l I n t e r a c t i o n s 17 2.2 H y d r o g e n M o l e c u l e s on t h e S u r f a c e 21 CHAPTER I I I : ORIENTATIONAL ORDERING 28 3.1 I n t r o d u c t i o n 28 3.2 O r i e n t a t i o n a l l y O r d e r e d S t a t e s 33 3.3 The L a n d a u E x p a n s i o n 37 3.4 I m p r o v e m e n t s t o Mean F i e l d T h e o r y 44 CHAPTER I V : NMR AS A PROBE OF ORIENTATIONAL BEHAVIOUR .. 54 4.1 U n b r o a d e n e d NMR S p e c t r a 54 4.2 C r y s t a l l i t e O r i e n t a t i o n a l D i s t r i b u t i o n 63 4.3 I n t e r m o l e c u l a r D i p o l a r B r o a d e n i n g 69 4.4 G r a p h i t e D i a m a g n e t i s m 73 i v Page CHAPTER V: SAMPLE PREPARATION AND CONVERSION MEASUREMENTS 76 5.1 I n t r o d u c t i o n 76 5.2 P r e p a r a t i o n o f E n r i c h e d H y d r o g e n S a m p l e s 79 5.3 M e a s u r e m e n t o f t h e J = l C o n c e n t r a t i o n by Raman S c a t t e r i n g 83 5.4 The S u b s t r a t e 85 5.5 A d s o r p t i o n 88 5.6 C o n v e r s i o n R a t e M e a s u r e m e n t s 98 CHAPTER V I : THE NMR CRYOSTATS 112 6.1 The 4 H e C r y o s t a t 112 6.2 The 3 H e C r y o s t a t 115 6.3 The G r a f o i l C e l l 129 3 6.4 T e m p e r a t u r e M e a s u r e m e n t i n t h e He C r y o s t a t ... 133 CHAPTER V I I : THE NMR SPECTROMETER 143 7.1 I n t r o d u c t i o n 143 7.2 D e s c r i p t i o n o f t h e S p e c t r o m e t e r 150 7.3 C o h e r e n t I n t e r f e r e n c e 159 7.4 S i g n a l I n t e n s i t y '. 163 7.5 1 h B a c k g r o u n d 172 CHAPTER V I I I : NMR SPECTRA OF 0-H 2 179 8.1 E v i d e n c e f o r O r i e n t a t i o n a l O r d e r i n g 179 8.2 The O r d e r e d S t a t e 191 8.3 I s T h e r e A n o t h e r T r a n s i t i o n ? 198 8.4 The D i s o r d e r e d S t a t e 201 8.5 B r o a d e n i n g 206 Page CHAPTER I X : NMR SPECTRA OF P-D 2 212 9.1 T e m p e r a t u r e Dependence o f t h e P-D 2 S p e c t r a ... 212 9.2 The J = l D o u b l e t 214 9.3 B r o a d e n i n g 215 CHAPTER X: CONCLUSIONS 219 10.1 P r e c i s 219 10.2 F u t u r e Work 224 BIBLIOGRAPHY 230 APPENDIX A 235 APPENDIX B 237 v i L I S T OF TABLES Page I S t a t e s o f H y d r o g e n 18 I I N o r m a l Modes 41 I I I H y p e r f i n e C o n s t a n t s f o r and 55 I V O r d e r P a r a m e t e r s f o r t h e H e r r i n g b o n e and P i n w h e e l P h a s e s a t T=0 60 V Raman F r e q u e n c i e s and I n t e n s i t y R a t i o s 85 V I P r o p e r t i e s o f E x f o l i a t e d G r a p h i t e 87 V I I 0 _ H 2 C o n v e r s i ° n R a t e C o n s t a n t s 103 V I I I I n i t i a l J = l C o n c e n t r a t i o n s o f t h e H 2 S a m p l e s 106 IX Power D i s s i p a t i o n a nd T e m p e r a t u r e E r r o r s i n t h e 3 H e P o t R e s i s t o r s 137 X T y p i c a l M a g n e t o r e s i s t a n c e M e a s u r e m e n t s 141 X I C h a r a c t e r i s t i c s o f t h e TIXM301 #13 155 X I I R e s u l t s f o r V Q and r f o r 0-H 2 204 v i i L I S T OF FIGURES Page 1 T r a n s l a t i o n a l p h a s e d i a g r a m f o r H 2 and D 2 a d s o r b e d on G r a f o i l 5 2 The /3 x /3 r e g i s t e r e d p h a s e 6 3 O r i e n t a t i o n a l l y o r d e r e d s t a t e s 6 4 The e f f e c t o f v a c a n c i e s on l a t t i c e s o f q u a d r u p o l e s w i t h and w i t h o u t f r u s t r a t i o n 12 5 ) a | v e r s u s 1/T f o r t h e d i s o r d e r e d s t a t e 23 6 O r i e n t a t i o n a l p h a s e d i a g r a m f o r quantum q u a d r u p o l e s on a h e x a g o n a l l a t t i c e 34 7 The f i r s t B r i l l o u i n z one f o r a h e x a g o n a l l a t t i c e . 39 8 The s t r u c t u r e o f t h e o r d e r e d s t a t e s t h a t a r i s e when t h e e i g e n v a l u e s f o r t h e modes c o r r e s p o n d i n g t o t h e Q A^ go s o f t . .. 39 9 The f r e e e n e r g y a s a f u n c t i o n o f t h e o r d e r p a r a m e t e r n e a r T c 43 10 M a g n e t i c e n e r g y o f a J = l h y d r o g e n m o l e c u l e i n a m a g n e t i c f i e l d 58 11 O r i e n t a t i o n s o f H and t h e i n t e r n u c l e a r a x i s i n t h e c r y s t a l r e f e r e n c e f r a m e 58 12 U n b r o a d e n e d T=0 NMR s p e c t r a f o r a 2D powder 61 13 B r o a d e n i n g o f t h e NMR s p e c t r a due t o t h e o r i e n t -a t i o n a l d i s t r i b u t i o n o f t h e a d s o r b i n g s u r f a c e s ... 64 14 O r i e n t a t i o n o f H w i t h r e s p e c t t o t h e c r y s t a l a x e s and t h e f o i l a x e s 65 15 G r a p h o f t h e e q u i l i b r i u m J = l c o n c e n t r a t i o n c v e r s u s T f o r H 2 and D 2 78 16 H y d r o g e n e n r i c h m e n t a p p a r a t u s 80 17 S k e t c h o f t h e wand c o n t a i n i n g t h e G r a f o i l s a m p l e u s e d i n t h e 4 H e c r y o s t a t 90 18 S k e t c h o f t h e a d s o r p t i o n and d e s o r p t i o n a p p a r a t u s u s e d w i t h t h e 3 H e c r y o s t a t 92 v i i i P age 19 A d s o r p t i o n i s o t h e r m f o r 4 H e on G r a f o i l 95 20 G r a p h o f s T 3 v e r s u s t f o r H 2 102 21 G r a p h o f c/c^ v e r s u s t f o r H 2 105 22 G r a p h o f - l n c v e r s u s t f o r D 2 s a m p l e G18 108 23 G r a p h o f - l n c v e r s u s t f o r D 2 s a m p l e G7 110 24 The t a i l o f t h e 3 H e c r y o s t a t 117 25 S c h e m a t i c d i a g r a m o f t h e 4 H e r e f r i g e r a t o r 122 26 ^He p o t t e m p e r a t u r e v e r s u s h e a t e r power 126 27 C r o s s - s e c t i o n o f t h e He p o t and G r a f o i l c e l l .... 128 28 T e m p e r a t u r e m e a s u r e m e n t and c o n t r o l s y s t e m f o r t h e 3 H e c r y o s t a t 134 29 S t a b i l i t y o f t h e r e s i s t a n c e t h e r m o m e t e r s on t h e He p o t 139 30 Q-meter c i r c u i t 145 31 B l o c k d i a g r a m o f t h e NMR s p e c t r o m e t e r 151 32 The t u n e d c i r c u i t 152 33 E q u i v a l e n t c i r c u i t o f a FET a m p l i f i e r 155 34 The sweep . f i e l d and a s s o c i a t e d m o d u l a t i o n p i c k - u p . 162 35 H e i g h t o f t h e d e r i v a t i v e o f t h e a d s o r p t i o n s Q d v e r s u s 1/T 3 f o r o-D 2 165 36 G r a p h o f s-^/c v e r s u s 1/T 3 f o r H 2 166 37 G r a p h o f s Q / ( l - c ) v e r s u s 1/T 3 f o r D 2 169 38 G r a p h o f s-^/c v e r s u s 1/T 3 f o r D 2 170 39 G r a p h o f s v e r s u s 1/T 3 f o r t h e ^H b a c k g r o u n d s i g n a l 174 40 "'"H b a c k g r o u n d s i g n a l , s h o w i n g d i s t o r t i o n due t o s a t u r a t i o n 175 i x P a g e 41 The e f f e c t i v e g a i n o f t h e NMR s y s t e m and t h e u n s a t u r a t e d a b s o r p t i o n s i g n a l 175 42 D e r i v a t i v e s o f t h e H 2 a b s o r p t i o n s i g n a l w i t h and w i t h o u t t h e b a c k g r o u n d s i g n a l 177 43 A b s o r p t i o n s p e c t r a o f H 2 w i t h B=0° 180 44 A b s o r p t i o n s p e c t r a o f H 2 w i t h 6 = 0° 181 45 A b s o r p t i o n s p e c t r a o f H 2 w i t h g = 90° 182 46 A b s o r p t i o n s p e c t r a o f H 2 w i t h 3=90° 183 47 Av/3d v e r s u s 1/T f o r H 2 185 48 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 e NMR s p e c t r u m and P(|a |) f o r b u l k o-H 2 i n t h e d i s o r d e r e d s t a t e 188 49 6v/3d v e r s u s c f o r H 2 192 50 T c v e r s u s c f o r H 2 193 51 C o m p a r i s o n o f t h e .3 K e x p e r i m e n t a l s p e c t r a and t h e T=0 s y n t h e t i c s p e c t r a 197 4 52 P r o p o s e d p h a s e d i a g r a m o f He on g r a p h i t e 200 53 |o | v e r s u s 1/T f o r H 2 203 54 6/3d v e r s u s 1/T f o r H 2 207 55 A b s o r p t i o n s p e c t r a o f D 2 w i t h g=90° 213 56 |a | v e r s u s 1/T f o r D 2 216 57 6 /3d v e r s u s 1/T f o r D ? 218 X ACKNOWLEDGEMENTS T h i s p r o j e c t was p r o p o s e d by my s u p e r v i s o r W a l t e r H a r d y and h i s i n s i g h t i n t o t h e numerous t r i b u l a t i o n s t h a t h a v e a r i s e n h a s b e e n i n v a l u a b l e , n o t o n l y f o r d e a l i n g w i t h t h e p r o b l e m a t h a n d , b u t a l s o f o r f u r t h e r i n g my own a b i l i t y t o d e a l w i t h s u b s e q u e n t m y s t e r i e s . S i n c e t h e h y d r o g e n s a m p l e s u s e d i n t h e s e e x p e r i m e n t s d e t e r i o r a t e w i t h t i m e , i t was s o m e t i m e s n e c e s s a r y t o make m e a s u r e m e n t s d a y and n i g h t f o r p e r i o d s o f up t o t h r e e w e e k s . I am i n d e b t e d t o Hans G l ' a t t l i , E r w i n B a t a l l a , and W a l t e r f o r t h e i r a s s i s t a n c e i n c o l l e c t i n g d a t a a t v e r y i r r e g u l a r h o u r s d u r i n g t h o s e p e r i o d s . O t h e r members o f t h e M i c r o w a v e S p e c t r o s c o p y g r o u p , i n p a r t i c u l a r M i c h a e l Morrow and B r y a n S t a t t , h ave a l s o p r o v i d e d v a l u a b l e h e l p . A s s i s t a n c e w i t h t h e p r e p a r a t i o n and a n a l y s i s o f my e n r i c h e d h y d r o g e n s a m p l e s h a s b e e n g i v e n by two summer s t u d e n t s , S t e p h e n M o r r i s and L o u i s T a i l l e f e r . I a l s o a c k n o w l e d g e t h e g e n e r o s i t y o f C h a r l i e S c h w e r d t f e g e r i n a l l o w i n g me t o u s e h i s magnet f o r a p e r i o d much l o n g e r t h a n e i t h e r o f us a n t i c i p a t e d . The mean f i e l d t h e o r y a n d h i g h t e m p e r a t u r e e x p a n s i o n s t h a t J o h n B e r l i n s k y and B r o o k s H a r r i s d e v e l o p e d w e re v i t a l t o t h e d e v e l o p m e n t o f my u n d e r s t a n d i n g o f t h e o r i e n t a t i o n a l b e h a v i o u r o f a d s o r b e d h y d r o g e n . They b o t h c o l l a b o r a t e d c l o s e l y w i t h me and p r o v i d e d much s t i m u l a t i o n . A s s i s t a n c e h a s b e e n c h e e r f u l l y p r o v i d e d t o me by v i r t u a l l y e v e r y member o f t h e d e p a r t m e n t a l t e c h n i c a l s t a f f s o a l l o f them h a v e e a r n e d my g r a t i t u d e . However, p a r t i c u l a r m e n t i o n s h o u l d be made o f t h e p r o m p t a s s i s t a n c e g i v e n t o me by B i l l W a l k e r and R o l f W e i s s b a c h when a p i e c e o f e q u i p m e n t r e q u i r e d r e p a i r . F i n a l l y , I w i s h t o t h a n k t h e N a t i o n a l R e s e a r c h C o u n c i l and t h e K i l l a m F o u n d a t i o n f o r p o s t g r a d u a t e f e l l o w s h i p s . CHAPTER I INTRODUCTION 1.1 T e c h n i q u e s A v a i l a b l e f o r t h e S t u d y o f P h y s i s o r b e d H y d r o g e n A l t h o u g h t h e r e have b e e n numerous s t u d i e s o f h y d r o g e n c h e m i s o r b e d on v a r i o u s s u r f a c e s , p a r t i c u l a r l y m e t a l s , p h y s i s o r b e d m o l e c u l a r h y d r o g e n 1 h a s r e c e i v e d r e l a t i v e l y l i t t l e a t t e n t i o n . P h y s i s o r p t i o n s t u d i e s h a v e c o n c e n t r a t e d on t h e i n e r t g a s e s . I n many o f t h e s e s t u d i e s t h e b a s a l p l a n e o f g r a p h i t e h a s b e e n u s e d a s a s u b s t r a t e b e c a u s e g r a p h i t e i s a v a i l a b l e i n f o r m s w i t h b o t h a h i g h s p e c i f i c s u r f a c e a r e a a n d a h i g h p r o p o r t i o n o f u n i f o r m b a s a l p l a n a r a d s o r b i n g s u r f a c e s . H y d r o g e n s h o u l d be a g o o d p r o b e o f g a s - s u r f a c e i n t e r a c t i o n s and t w o - d i m e n s i o n a l (2D) p h a s e s b e c a u s e t h e m i c r o s c o p i c i n t e r a c t i o n s o f h y d r o g e n m o l e c u l e s a r e v e r y w e l l known. However t h e r e a r e s e v e r a l s t u m b l i n g b l o c k s t o t h e s t u d y o f p h y s i s o r b e d h y d r o g e n . S u b - m o n o l a y e r s o f h y d r o g e n on g r a p h i t e do n o t s o l i d f y u n t i l t h e t e m p e r a t u r e i s r e d u c e d t o 20K o r l e s s ( N i e l s e n e t a l , 1 9 7 7 ) . T h i s s e v e r e l y r e s t r i c t s t h e u s e o f many m o d e r n s u r f a c e a n a l y s i s t e c h n i q u e s s u c h as l o w e n e r g y e l e c t r o n d i f f r a c t i o n ( L E E D ) , A u g e r e l e c t r o n s p e c t r o s c o p y , a n d f i e l d e m i s s i o n m i c r o s c o p y , t h e a p p a r a t i f o r w h i c h a r e n o r m a l l y d e s i g n e d f o r o p e r a t i o n a t h i g h e r t e m p e r a t u r e s . A l s o , h y d r o g e n , b e i n g a v e r y l i g h t m o l e c u l e , s c a t t e r s e l e c t r o n s a n d X - r a y s v e r y w e a k l y s o t h e r e a r e l a r g e b a c k g r o u n d s i g n a l s f r o m t h e s u b s t r a t e . I n a d d i t i o n t o s u r f a c e m e t h o d s t h e r e a r e s e v e r a l b u l k t e c h n i q u e s t h a t c a n be a d a p t e d t o s u r f a c e s t u d i e s p r o v i d e d t h a t t h e s u b s t r a t e h a s a l a r g e a r e a . However, t h e y s u f f e r f r o m l o w s i g n a l t o n o i s e (S:N) r a t i o s . N e u t r o n d i f f r a c t i o n f r o m H 2 i s i n h i b i t e d by t h e f a c t t h a t s c a t t e r i n g f r o m p a r a - H 2 ( p - H 2 ) , t h e e q u i l i b r i u m s p e c i e s a t l o w t e m p e r a t u r e , i s a l m o s t e n t i r e l y i n c o h e r e n t . The s i t u a t i o n i s more f a v o u r a b l e f o r D 2 w h i c h s c a t t e r s p r i m a r i l y c o h e r e n t l y . S p e c i f i c h e a t m e a s u r e m e n t s h a v e b e e n u s e d e x t e n s i v e l y and e f f e c t i v e l y t o s t u d y t h e 2D p h a s e s o f h e l i u m . I n t h e c a s e o f h y d r o g e n , h o w e v e r , s u c h m e a s u r e m e n t s w o u l d be c o m p l i c a t e d by t h e p r e s e n c e o f h e a t i n g a r i s i n g f r o m 2 c o n v e r s i o n . The t e c h n i q u e o f n u c l e a r m a g n e t i c r e s o n a n c e (NMR), w h i c h h a s b e e n v a l u a b l e f o r t h e s t u d y o f a d s o r b e d He, w o u l d a l s o be e x p e c t e d t o be u s e f u l f o r t h e s t u d y o f a d s o r b e d h y d r o g e n . P ~ D 2 and o - H 2 b o t h h a v e t o t a l n u c l e a r s p i n 1=1. o-D 2 h a s 1=0 and 2 and p - H 2 h a s 1=0. The f i r s t t h r e e s p e c i e s a l l g i v e an NMR s i g n a l ; o - H 2 i s a p a r t i c u l a r l y g o o d s u b j e c t f o r NMR b e c a u s e i t h a s a l a r g e n u c l e a r m a g n e t i c moment. However, a s we s h a l l s e e l a t e r , t h e NMR s p e c t r a o f a d s o r b e d h y d r o g e n a r e c o n s i d e r a b l y b r o a d e r t h a n t h o s e o f He s o t h e S:N r a t i o s a r e n o t a s g o o d . Many o f t h e t h e o r e t i c a l t e c h n i q u e s u s e d i n s t a t i s t i c a l m e c h a n i c s c o u l d be a p p l i e d t o a d s o r b e d h y d r o g e n b u t t h e r e w o u l d be c o m p l i c a t i o n s due t o t h e quantum n a t u r e o f h y d r o g e n m o l e c u l e s . T h i s r e n d e r s d i f f i c u l t t h e a p p l i c a t i o n o f Monte C a r l o a n d r e n o r m a l i z a t i o n g r o u p (RG) m e t h o d s w h i c h h a v e b e e n u s e d e x t e n s i v e l y f o r o t h e r a d s o r b e d s y s t e m s . Mean f i e l d t h e o r y h a s 3 b e e n u s e d q u i t e s u c c e s s f u l l y t o s t u d y t h e o r i e n t a t i o n a l b e h a v i o u r o f h y d r o g e n m o l e c u l e s a d s o r b e d o n g r a p h i t e ( H a r r i s a nd B e r l i n s k y , 1 9 7 9 ) . 1.2 P r e v i o u s Work The e a r l i e s t m e a s u r e m e n t s on H 2 a d s o r b e d o n g r a p h i t e w e re a d s o r p t i o n i s o t h e r m s ( C o n s t a b a r i s e t a l , 1 9 6 1 ) . S i n c e t h e n t h e r e h a v e b e e n o t h e r p r e s s u r e m e a s u r e m e n t s ( D e r i c b o u r g , 1 9 7 6 ; D a u n t e t a l . , 1981) b u t a l l o f t h i s work h a s b e e n p r i m a r i l y c o n c e r n e d w i t h m u l t i l a y e r a d s o r p t i o n . F o r s e v e r a l i n e r t g a s e s , m o s t n o t a b l y K r , a d s o r b e d on g r a p h i t e , t h e a d s o r p t i o n i s o t h e r m s show s t r u c t u r e a r i s i n g f r o m 2D p h a s e s i n t h e s u b - m o n o l a y e r r e g i o n (Thorny and D u v a l , 1 9 6 9 ) . S i m i l a r b e h a v i o u r h a s n o t b e e n o b s e r v e d f o r H 2 . I n e l a s t i c n e u t r o n s c a t t e r i n g ( S t o c k m e y e r e t a l , 1978) and m o l e c u l a r beam s c a t t e r i n g ( M a t t e r a e t a l , 1980) h a v e b e e n e m p l o y e d t o m e a s u r e t h e e n e r g y l e v e l s o f a d s o r b e d h y d r o g e n m o l e c u l e s . From t h e m o l e c u l a r beam work i t h a s b e e n p o s s i b l e t o d e r i v e a l a t e r a l l y a v e r a g e d m o l e c u l e - s u r f a c e i n t e r a c t i o n p o t e n t i a l . A b i n d i n g e n e r g y o f 483 K f o r H 2 o n g r a p h i t e h a s a l s o b e e n o b t a i n e d . S p e c i f i c h e a t ( B r e t z and Chung, 1 9 7 4 ) , e l a s t i c n e u t r o n s c a t t e r i n g ( N i e l s e n e t a l , 1977) and LEED ( S e g u i n and S u z a n n e , 1982) h a v e b e e n u s e d t o a t t e m p t t o s t u d y t h e 2D t r a n s l a t i o n a l p h a s e s . P e a k s i n t h e s p e c i f i c h e a t were o b s e r v e d b e t w e e n 8 K and 16 K f o r t h r e e c o v e r a g e s o f l e s s t h a n h a l f a m o n o l a y e r . F o r t h e two l o w e s t c o v e r a g e s (.0112 and .0277 m o l e c u l e s / A * ) , t h e 4 7 s p e c i f i c h e a t i n c r e a s e d a s T b e t w e e n I K and 4K and t h e n e x p o n e n t i a l l y w i t h 1/T up t o t h e p e a k . An i n c o m m e n s u r a t e 2D 9 Debye s o l i d w o u l d be e x p e c t e d t o g i v e T b e h a v i o u r a t l o w t e m p e r a t u r e s . N e v e r t h e l e s s t h i s i n t e r p r e t a t i o n was r e j e c t e d on t h e b a s i s o f t h e s i z e o f t h e p r o p o r t i o n a l i t y c o n s t a n t b e t w e e n t h e t e m p e r a t u r e o f t h e s p e c i f i c h e a t p e a k ( t h e m e l t i n g t e m p e r a t u r e i n t h i s p i c t u r e ) and t h e s q u a r e o f t h e Debye t e m p e r a t u r e . The p r o p o r t i o n a l i t y c o n s t a n t d i d n o t a g r e e w i t h t h e p r e d i c t i o n o f t h e K o s t e r l i t z - T h o u l e s s t h e o r y o f d i s l o c a t i o n m e d i a t e d m e l t i n g . The v a l i d i t y o f t h e t h e o r y f o r t h i s s y s t e m i s n o t e s t a b l i s h e d , h o w e v e r , and o t h e r t h e o r i e s g i v e d i f f e r e n t p r o p o r t i o n a l i t y c o n s t a n t s . The s p e c i f i c h e a t w i l l i n c r e a s e e x p o n e n t i a l l y w i t h 1/T i f one h a s two p h a s e c o e x i s t e n c e o n t h e s u r f a c e o r i f t h e r e i s a r e g i s t e r e d s o l i d . U s i n g t h e f o r m e r i n t e r p r e t a t i o n , t h e s p e c i f i c h e a t p e a k s w e r e t a k e n t o r e p r e s e n t a 2D l i q u i d t o 2D v a p o u r t r a n s i t i o n . S u b s e q u e n t l y , a t r a n s l a t i o n a l p h a s e d i a g r a m o f c o v e r a g e p v e r s u s t e m p e r a t u r e T f o r H 2 and D 2 o n g r a p h i t e ( F i g . 1) was o b t a i n e d by e l a s t i c n e u t r o n s c a t t e r i n g . N e a r l y a l l o f t h e m e a s u r e m e n t s were made u s i n g o-D 2 b u t t h o s e p - H 2 d a t a t h a t w e r e o b t a i n e d d i d n o t s u g g e s t t h a t t h e p h a s e d i a g r a m f o r H 2 was d i f f e r e n t f r o m t h a t o f D 2-I n t h e l o w t e m p e r a t u r e and l o w c o v e r a g e r e g i o n , s c a t t e r i n g p e a k s c o r r e s p o n d i n g t o a /3" x /3 r e g i s t e r e d s o l i d w e r e o b s e r v e d . I n t h i s s t r u c t u r e ( F i g . 2 ) , a h y d r o g e n m o l e c u l e i s l o c a t e d a t t h e c e n t r e o f e v e r y t h i r d c a r b o n r i n g . T h i s s t r u c t u r e h a s a l s o 4 - , 3 -0 O A A A A X ® I n commensu r a t e Sol id 10 -i— 20 3 0 T C K) — \ 4 0 F i g . 1 T r a n s l a t i o n a l p h a s e d i a g r a m f o r H 2 and D 2 a d s o r b e d o n G r a f o i l , p l o t t e d a s c o v e r a g e p v e r s u s T. The c o v e r a g e i s n o r m a l i z e d t o f u l l c o v e r a g e by t h e / f x ^ 3 s o l i d . P h a s e b o u n d a r i e s were d e d u c e d by N i e l s e n e t a l (1977) f r o m n e u t r o n s c a t t e r i n g d a t a . The s p e c i f i c h e a t p e a k s o b s e r v e d by B r e t z and Chung ( 1 9 7 4 ) a r e a l s o shown. The d a s h e d l i n e i n d i c a t e s m o n o l a y e r c o v e r a g e . S p e c i f i c h e a t peak O x ^5" s o l i d N I n c o m m e n s u r a t e s o l i d I n e u t r o n F l u i d C s c a t t e r i n g D e f o r m e d s c a t t e r i n g p e a k s / p - H 2 • • • o-D-X o 3.4/4 4.26/4 F i g . 2 The /Tx /T r e g i s t e r e d p h a s e o f H 2 a d s o r b e d on t h e g r a p h i t e b a s a l p l a n e . The d i a m e t e r o f t h e H 2 m o l e c u l e s i s e q u a l t o t h e minimum o f t h e H 2 - H 2 p o t e n t i a l . The H 2 m o l e c u l e s a r e n e a r l y s p h e r i c a l . F i g . 3 O r i e n t a t i o n a l l y o r d e r e d s t a t e s , (a) H e r r i n g b o n e : t h e m o l e c u l a r a x e s a r e p a r a l l e l t o t h e s u r f a c e f o r b o t h s u b l a t t i c e s . (b) P i n w h e e l : t h e m o l e c u l a r a x e s o f s u b l a t t i c e s 2, 3, and 4 a r e p a r a l l e l t o t h e s u r f a c e b u t t h o s e o f s u b l a t t i c e 1 a r e p e r p e n d i c u l a r . / / / / / / / \ \ y\ < \ \ / / / V / / / \ \ v s r - ^ x \ \ / / / / / / / i - / 3 - r -\ • \ 1. \2 . \ • \ • \ • \ • (a) 7 b e e n o b s e r v e d f o r 3 H e , 4 H e , N 2 * K r , and C H 4 a d s o r b e d o n g r a p h i t e . N o t e t h a t i n F i g . 1 p i s n o r m a l i z e d t o c o m p l e t e c o v e r a g e by t h e /~3 x /~3 s o l i d (.0637 X ) . As T was i n c r e a s e d t h e /3 x /3 p e a k s d i s a p p e a r e d o v e r a 3K t e m p e r a t u r e i n t e r v a l i n d i c a t i n g a t r a n s i t i o n t o a f l u i d p h a s e . N o t e t h a t t h e s p e c i f i c h e a t p e a k s f o r t h e two l o w e s t c o v e r a g e s u s e d by B r e t z and Chung ( p =.18 a n d .44) c o i n c i d e a p p r o x i m a t e l y w i t h t h e n e u t r o n s c a t t e r i n g p h a s e b o u n d a r y b u t t h a t t h e p e a k f o r P=.64 i s a t a l o w e r t e m p e r a t u r e . The two l o w c o v e r a g e s p e c i f i c h e a t p e a k s a p p a r e n t l y i n d i c a t e m e l t i n g r a t h e r t h a n b o i l i n g . A t c o v e r a g e s g r e a t e r t h a n p = l and a t l o w t e m p e r a t u r e s a n i n c o m m e n s u r a t e , t r i a n g u l a r s o l i d was o b s e r v e d by n e u t r o n s c a t t e r i n g . The x v§~ s t r u c t u r e h a s a l s o b e e n s e e n i n LEED m e a s u r e -m e n t s o f H 2 o n g r a p h i t e a t 10K ( S e g u i n a n d S u z a n n e , 1 9 8 2 ) . U n f o r t u n a t e l y , i n t h i s t e c h n i q u e , t h e p r e s s u r e o f t h e 3D v a p o u r r a t h e r t h a n t h e c o v e r a g e i s m e a s u r e d ; t h e r e l a t i o n s h i p b e t w e e n t h e two i n t h e s u b - m o n o l a y e r r e g i m e i s n o t known. NMR h a s b e e n u s e d t o m e a s u r e t h e l o n g i t u d i n a l a n d t r a n s v e r s e r e l a x a t i o n t i m e s T^ and T 2 o f o - H 2 a d s o r b e d on 3 g r a p h i t i z e d c a r b o n b l a c k ( S p h e r o n 6) i n t h e t e m p e r a t u r e r a n g e 12 K < T < 28 K ( R i e h l and F i s h e r , 1 9 7 3 ) . T h i s s u b s t r a t e h a s a h i g h s p e c i f i c a r e a o f b a s a l p l a n e s u r f a c e s w h i c h a r e f a i r l y homogeneous ( s t e p s i n t h e K r i s o t h e r m s c a n be s e e n ) a nd f l a t b u t l a c k any p r e f e r r e d o r i e n t a t i o n . C o v e r a g e s r a n g e d up t o .3 mono-l a y e r s ( p = . 5 2 ) . T i and T 2 b o t h i n c r e a s e d l i n e a r l y w i t h p a s 8 e x p e c t e d f o r a 2D gas, i n w h i c h r e l a x a t i o n o c c u r s t h r o u g h m o d u l a t i o n o f t h e i n t r a m o l e c u l a r s p i n - s p i n i n t e r a c t i o n s by c o l l i s i o n s , i n t h e s h o r t c o r r e l a t i o n t i m e l i m i t . T h i s i n t e r p r e -t a t i o n i s c o n s i s t e n t w i t h t h e p h a s e b o u n d a r y o f F i g . 1 e x c e p t p e r h a p s f o r t h e h i g h e s t c o v e r a g e s and l o w e s t t e m p e r a t u r e s : h e r e t h e h i g h e x c i t a t i o n f r e q u e n c y o f 30 MHz may have l e d t o RF h e a t i n g . F o r m o l e c u l a r r e o r i e n t a t i o n , t h e s h o r t c o r r e l a t i o n t i m e l i m i t i s e x p r e s s e d by «0Q T C « 1 (1-1) where to Q i s t h e r e s o n a n c e f r e q u e n c y and T c , t h e c o r r e l a t i o n t i m e , i s t h e t i m e between c o l l i s i o n s . I f 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 o c c u r by t h e same mechanism, t h e n i n t h i s l i m i t T-^  = T 2 i n 3D. R i e h l and F i s h e r assumed i m p l i c i t l y t h a t t h i s w o u l d a l s o a p p l y t o a 2D g a s . S i n c e t h e y f o u n d T 2 t o be c o n s i d e r a b l y l e s s t h a n T^r t h e y f e l t t h a t t h e r e must be an a d d i t i o n a l mechanism f o r t r a n s v e r s e r e l a x a t i o n . They were a b l e t o i d e n t i f y a p o s s i b l e mechanism by a s s u m i n g a model i n w h i c h h i n d e r e d r o t a t i o n a l s t a t e s o f t h e m o l e c u l e s were c o u p l e d t o t h e t r a n s l a t i o n a l s t a t e s . More r e c e n t t h e o r e t i c a l work (Cowan, 1980) h a s shown t h a t i n 2D b o t h T± and T 2 depend s t r o n g l y on t h e a n g l e between t h e a p p l i e d s t a t i c m a g n e t i c f i e l d H Q and t h e s u b s t r a t e and t h a t f o r a powder s u b s t r a t e , t h e e f f e c t i v e T 2 w o u l d be c o n s i d e r a b l y s h o r t e r t h a n T± e v e n i n t h e m o t i o n a l n a r r o w i n g l i m i t io 0 t c <<1. C o n s e q u e n t l y t h e c o n c l u s i o n s o f R i e h l and F i s h e r have been u n d e r m i n e d . 9 1.3 Scope o f t h i s T h e s i s T h i s t h e s i s d e s c r i b e s a s t u d y o f t h e o r i e n t a t i o n a l b e h a v i o u r o f J = l H 2 and D 2 m o l e c u l e s a d s o r b e d o n g r a p h i t e u s i n g t h e t e c h n i q u e o f NMR. The m e a s u r e m e n t s s p a n t h e t e m p e r a -t u r e r a n g e o f ,3K t o 12K a t a c o v e r a g e o f p=.85. T h e r e f o r e , a c c o r d i n g t o F i g . 1, t h e h y d r o g e n s h o u l d be i n t h e /3 x /3 s o l i d p h a s e . I n t h i s p h a s e , t h e h y d r o g e n c a n be m o d e l l e d by a h e x a g o n a l l a t t i c e o f i n t e r a c t i n g q uantum q u a d r u p o l e s . E a c h m o l e c u l e a l s o e x p e r i e n c e s a c r y s t a l f i e l d a r i s i n g f r o m Van d e r W a a l s i n t e r -a c t i o n s w i t h n e i g h b o u r i n g m o l e c u l e s and w i t h t h e s u b s t r a t e . H a r r i s and B e r l i n s k y ( 1 9 7 9) h a v e u s e d mean f i e l d t h e o r y (MFT) t o s t u d y s u c h a s y s t e m and h a v e p r e d i c t e d t h e e x i s t e n c e o f a v a r i e t y o f o r i e n t a t i o n a l l y o r d e r e d p h a s e s w h i c h d e p e n d o n t h e r e l a t i v e s t r e n g t h o f t h e c r y s t a l f i e l d a n d t h e m o l e c u l a r f i e l d a n d on t h e t e m p e r a t u r e . S u b s e q u e n t l y a Monte C a r l o c a l c u l a t i o n (O'Shea a n d K l e i n , 1979) f o r t h e a n a l o g o u s s y s t e m o f c l a s s i c a l q u a d r u p o l e s p r e d i c t e d t h e e x i s t e n c e o f two o f t h e p h a s e s w h i c h a r e shown i n F i g . 3. I f t h e c r y s t a l f i e l d w e r e z e r o , t h e y f o u n d o r d e r i n g i n t o t h e p i n w h e e l p h a s e . I f i t w e re l a r g e e n o u g h t o f o r c e a l l o f t h e m o l e c u l a r a x e s t o l i e i n a p l a n e p a r a l l e l t o t h e s u r f a c e , o r d e r i n g i n t o t h e h e r r i n g b o n e p h a s e was s e e n . 4 A s y s t e m c l o s e l y a n a l o g o u s t o p h y s i s o r b e d H 2 i s p h y s i s o r b e d N 2 , w h i c h a l s o f o r m s a /3 x /3 r e g i s t e r e d l a t t i c e on g r a p h i t e ( K j e m s e t a l , 1 9 7 6 ) ; t h e m a j o r d i f f e r e n c e i s t h a t t h e N 0 m o l e c u l e s c a n be t r e a t e d a s c l a s s i c a l r o t o r s b e c a u s e o f 10 their large moment of inertia. Adsorbed N 2 molecules have been observed to undergo an orientational ordering transition into the herringbone phase at 28 K. Techniques used to study this transi-tion were specific heat (Chung and Dash, 1977; Migone et a l , 1983), neutron scattering (Eckert et a l , 1979), LEED (Diehl et a l , 1982), and NMR (Sullivan and Vassiere, 1983). One of the goals of the present work was to answer the question of whether orientational ordering of J=l molecules w i l l occur as predicted by MFT and the classical Monte Carlo calcula-tions or whether i t would be suppressed by either 2D fluctuations or quantum fluctuations. It has been shown that long range positional order cannot exist for T>0 in a ID or 2D lattice on a smooth substrate. For a 3D l a t t i c e the mean square displacement < |u("r)| 2> of a molecule with an equilibrium position R remains f i n i t e as R-x» . For a 2D l a t t i c e < [ti| >^ increases as log R and in ID as R; in both cases i t diverges for an inf i n i t e l a t t i c e . These arguments apply to such diverse systems as superconductors, superfluids, and magnetic spins. Although long range positional order is precluded in a 2D l a t t i c e , i t can s t i l l have long range directional order. If R" i s an equilibrium l a t t i c e vector and r=^+u define e to be the angle between r and an arbitrary axis. On a triangular l a t -tice directional order can be measured by <exp i6[8($)-8(0)]>. It remains f i n i t e R-*-°°. Much of the recent effort in the study of the translational phase transitions of inert gases adsorbed on graphite has been directed towards the discovery of 11 h e x a t i c p h a s e s w h i c h h a v e s h o r t r a n g e p o s i t i o n a l o r d e r b u t p o w e r l a w d e c a y o f t h e d i r e c t i o n a l o r d e r . A p h a s e o f t h i s t y p e , w h i c h i s c l e a r l y a c o n s e q u e n c e o f a m e l t i n g p r o c e s s r a t h e r t h a n s u b s t r a t e i n t e r a c t i o n s , h a s b e e n s e e n f o r X e o n g r a p h i t e ( R o s e n -b a u m e t a l , 1 9 8 3 ) . A t r i a n g u l a r l a t t i c e o f q u a d r u p o l e s i s a n e x a m p l e o f a s y s t e m e x h i b i t i n g f r u s t r a t i o n , a k e y c o n c e p t i n t h e b e h a v i o u r o f g l a s s y s y s t e m s . A s y s t e m i s f r u s t r a t e d i f i t i s i m p o s s i b l e f o r e v e r y p a i r o f n e i g h b o u r i n g m o l e c u l e s t o a c h i e v e t h e i r l o w e s t e n e r g y c o n f i g u r a t i o n . F o r a p a i r o f q u a d r u p o l e s t h i s i s a T c o n f i g u r a t i o n . A s q u a r e l a t t i c e o f q u a d r u p o l e s w i t h n e a r e s t n e i g h b o u r i n t e r a c t i o n s o n l y i s n o t f r u s t r a t e d a s s h o w n i n F i g . 4 ( a ) . I t i s i m p o s s i b l e f o r a l l n e a r e s t n e i g h b o u r s o n a t r i a n g u l a r l a t t i c e t o b e m u t u a l l y p e r p e n d i c u l a r s o t h e r e i s f r u s t r a t i o n [ F i g . 4 ( b ) ] . F r u s t r a t i o n i s r e d u c e d w h e n t h e s y s t e m i s d i l u t e d . F o r h y d r o g e n t h e r e w i l l a l w a y s b e s o m e d i l u t i o n b e c a u s e c o n v e r s i o n p r o d u c e s J = 0 m o l e c u l e s w h i c h h a v e s p h e r i c a l w a v e f u n c t i o n s . T h e s e h a v e n o E Q Q i n t e r a c t i o n s s o t h e y a c t l i k e v a c a n c i e s . I f t h e r e i s n o f r u s t r a t i o n , t h e n t h e s y s t e m i s s t a b l e a g a i n s t s m a l l a m o u n t s o f r a n d o m d i l u t i o n [ F i g . 4 ( c ) ) . I n t h e p r e s e n c e o f f r u s t r a t i o n , r a n d o m d i l u t i o n w i l l c a u s e l o c a l r e o r i e n t a t i o n o f m o l e c u l e s b u t r e o r i e n t a t i o n o f l a r g e n u m b e r s o f m o l e c u l e s w i l l b e b l o c k e d b y e n e r g y b a r r i e r s a t l o w t e m p e r a t u r e s . T h e s y s t e m m a y n o t b e a b l e t o a c h i e v e t h e l o w e s t e n e r g y c o n f i g -u r a t i o n , b e c o m i n g s t u c k i n a m e t a s t a b l e s t a t e ( a g l a s s ) , w h i c h d e p e n d s u p o n t h e t h e r m a l h i s t o r y . L o w ( J = l ) c o n c e n t r a t i o n 12 F i g . 4 The e f f e c t o f v a c a n c i e s on l a t t i c e s o f q u a d r u p o l e s w i t h and w i t h o u t f r u s t r a t i o n : (a) s q u a r e l a t t i c e ( n o t f r u s t r a t e d ) , (b) t r i a n g u l a r l a t t i c e ( f r u s t r a t e d ) , (c) s q u a r e l a t t i c e w i t h v a c a n c i e s , and (d) t r i a n g u l a r l a t t i c e w i t h v a c a n c i e s . 13 a b s o r b e d h y d r o g e n i s a l i k e l y c a n d i d a t e f o r t h e f o r m a t i o n o f a q u a d r u p o l a r g l a s s . S u c h a s t a t e w o u l d be c h a r a c t e r i z e d by f r e e z i n g o f t h e m o l e c u l e s i n t o random o r i e n t a t i o n s . T h i s h a s i n f a c t b e e n p r o p o s e d f o r b u l k H 2 when t h e J = l c o n c e n t r a t i o n f a l l s b e l o w .55 ( S u l l i v a n , 1976) b u t i s s t i l l a s u b j e c t o f c o n t r o v e r s y ( E s t e v e e t a l , 1 9 8 2 ; W ashburn e t a l , 1 9 8 2 ) . E a r l y m e a s u r e m e n t s o f t h e NMR s p e c t r a o f H 2 and D 2 i n t h e o r i e n t a t i o n a l l y d i s o r d e r e d s t a t e ( t h e p a r a r o t a t i o n a l s t a t e ) a r e d e s c r i b e d i n K u b i k and H a r d y ( 1 9 7 8 ) . The r e s u l t s f o r t h e c r y s t a l f i e l d V c and q u a d r u p o l a r c o u p l i n g c o n s t a n t r were H 2 |v c| = .56 K r = 0 D 2 |VC| = 2.4K r = . 5 K W i t h t h e s e v a l u e s t h e MFT o f H a r r i s and B e r l i n s k y ( 1 9 7 9 ) ( d e s c -r i b e d i n C h a p t e r I I I ) p r e d i c t s t h a t p u r e p-D 2 ( J = l ) s h o u l d u n d e r g o a s e c o n d o r d e r p h a s e t r a n s i t i o n n e a r 3K i n t o e i t h e r t h e p i n w h e e l o r h e r r i n g b o n e p h a s e d e p e n d i n g upon t h e s i g n o f V Q . I t was n o t p o s s i b l e t o p r e d i c t an o r i e n t a t i o n a l o r d e r i n g t e m p e r a t u r e f o r o -H 2 ( J = l ) w i t h o u t a more a c c u r a t e v a l u e o f r . A l t h o u g h MFT c a n be e x p e c t e d t o be q u i t e g o o d a t p r e d i c t i n g t h e s y m m e t r y o f t h e o r d e r e d s t a t e s , i t i s l i k e l y t o p r e d i c t t r a n s i t i o n t e m p e r a t u r e s t h a t a r e t o o h i g h b e c a u s e i t i g n o r e s f l u c t u a t i o n s . I n f a c t i n t h e e a r l y work no o r d e r i n g was s e e n down t o 1.2K. O'Shea a n d K l e i n ( 1 9 7 9) a l s o made an e s t i m a t e o f t h e o r d e r -i n g t e m p e r a t u r e f r o m t h e i r c l a s s i c a l M o nte C a r l o c a l c u l a t i o n s . T h ey r e n o r m a l i z e d r i n o r d e r t o t a k e i n t o a c c o u n t q u a n t u m e f f e c t s by c o m p a r i n g t h e e x p e r i m e n t a l b u l k h y d r o g e n o r d e r i n g t e m p e r a t u r e w i t h 3D c l a s s i c a l Monte C a r l o r e s u l t s . They t h e n a p p l i e d t h e same r e n o r m a l i z a t i o n t o t h e i r 2D Monte C a r l o c a l c u l a t i o n s t o g e t a t r a n s i t i o n t e m p e r a t u r e o f 1.0-1.5K d e p e n d i n g upon t h e s t r e n g t h o f t h e i n t e r a c t i o n w i t h t h e s u b s t r a t e . E n c o u r a g e d , we c o n s t r u c t e d a n 8.5 MHz c o n t i n u o u s wave NMR s p e c t r o m e t e r ( d e s c r i b e d i n C h a p t e r V I I ) i n o r d e r t o e x t e n d o u r m e a s u r e m e n t s down t o .3K. C o n s i d e r a b l e e f f o r t was e x p e n d e d t o a t t a i n a d e q u a t e S:N r a t i o s . P a r t i c u l a r l y t r o u b l e s o m e was c o h e r e n t i n t e r f e r e n c e i n d u c e d by m o d u l a t i o n o f t h e m a g n e t i c f i e l d . I t was f o u n d ( C h a p t e r V I I I ) t h a t 9 0 % J = l H 2 u n d e r g o e s a r a p i d o r i e n t a t i o n a l o r d e r i n g t r a n s i t i o n i n t o t h e p i n w h e e l p h a s e a t .6K. NMR s p e c t r a o f H 2 e a c h r e q u i r e d a t l e a s t two h o u r s o f s i g n a l a v e r a g i n g s o we d i d n o t d e t e r m i n e w h e t h e r o r n o t t h e t r a n s i t i o n was f i r s t o r d e r . I n c o n t r a s t , t h e D 2 NMR s p e c t r u m u n d e r w e n t a smooth e v o l u t i o n down t o .3K, b e c o m i n g v e r y b r o a d and weak ( C h a p t e r I X ) . T h i s b e h a v i o u r i s v e r y p u z z l i n g and i s n o t u n d e r s t o o d . A n a l y s i s o f t h e D 2 s p e c t r a was s e v e r e l y hampered by p o o r S:N r a t i o s : 5-20 h o f s i g n a l a v e r a g i n g was u s e d f o r t h e l o w t e m p e r a t u r e s p e c t r a . Some s u g g e s t i o n s f o r i m p r o v i n g t h e S:N r a t i o s a r e c o n t a i n e d i n C h a p t e r X. We were a b l e t o d e d u c e v a l u e s o f V Q and r f r o m t h e h i g h t e m p e r a t u r e s p e c t r a . The D 2 s p e c t r a were a n a l y z e d i n t e r m s o f MFT a n d t h e r e s u l t s a g r e e d w e l l 4 w i t h t h e v a l u e s o b t a i n e d e a r l i e r u s i n g t h e He c r y o s t a t . F o r H 2 we were a b l e t o e m p l o y a h i g h t e m p e r a t u r e e x p a n s i o n o f t h e o r d e r p a r a m e t e r s . I t r e d u c e s t o MFT i f o n l y t e r m s up t o f i r s t o r d e r i n r / T a r e r e t a i n e d . E x p a n d i n g a s f a r a s ( r / T ) , we o b t a i n e d v a l u e s o f [V c| and r s i m i l a r t o t h o s e f o u n d e a r l i e r b u t we were a l s o a b l e t o a s c e r t a i n t h a t V c was p o s i t i v e . F i n a l l y , NMR i n t e n s i t y m e a s u r e m e n t s showed t h a t c o n v e r s i o n o b e y e d a f i r s t o r d e r r a t e e q u a t i o n , p r e s u m a b l y b e i n g c a t a l y z e d by t h e G r a f o i l . T h e s e r e s u l t s a r e p r e s e n t e d i n C h a p t e r V. A s w e l l a s d e s c r i b i n g t h e r e s u l t s o f o u r e x p e r i m e n t s , t h i s t h e s i s was i n t e n d e d t o s e r v e a s an o p e r a t i o n s m a n u a l f o r t h e 3 NMR s p e c t r o m e t e r a nd He c r y o s t a t . C o n s e q u e n t l y i t c o n t a i n s a c o n s i d e r a b l e amount o f d e t a i l c o n c e r n i n g t h e o p e r a t i o n o f t h e s y s t e m . F o r t h i s r e a s o n t h e f o l l o w i n g s e c t i o n s c o u l d be o m i t t e d o n a f i r s t r e a d i n g : 5.1-5.3, 5.5, 6.1-6.4, 7.4, and 7.5. I n a d d i t i o n , s e c t i o n s 3.3 and 3.4 c o n t a i n d e t a i l s o f t h e t h e o r y n o t e s s e n t i a l t o an u n d e r s t a n d i n g o f t h e r e s u l t s . S e c t i o n 10.1 c o n t a i n s a c o n c i s e summary o f t h e r e s u l t s . The r e a d e r may w i s h t o p e r u s e i t a f t e r r e a d i n g C h a p t e r s I and I I i n o r d e r t o g e t a p r e v i e w o f t h e r e s u l t s . N o t e s t o C h a p t e r I 16 1. T h r o u g h o u t t h i s t h e s i s " h y d r o g e n " r e f e r s t o H 2 and D 2. 2. The r o t a t i o n a l a n g u l a r momentum J o f a h y d r o g e n m o l e c u l e may be e i t h e r e v e n ( p a r a - H 2 and o r t h o - D 2 ) o r o d d ( o r t h o - H 2 and p a r a - D 2 ) . The J=0 and J = I s t a t e s a r e s e p a r a t e d by 171 K (86 K) f o r H 2 ( D 2 ) s o t h a t a t l o w t e m p e r a t u r e s one w o u l d e x p e c t o n l y t h e J=0 s t a t e t o be o c c u p i e d . However o d d J t o e v e n J t r a n s i t i o n s ( c o n v e r s i o n ) a r e v e r y s l o w s o i n g e n e r a l one i s w o r k i n g w i t h a n o n -e q u i l i b r i u m m i x t u r e o f J=0 and J = l m o l e c u l e s . 3. M a n u f a c t u r e d by C a b o t C a r b o n Co. I t i s more h e t e r o g e n e o u s t h a n t h e G r a f o i l s u b s t r a t e ( U n i o n C a r b i d e C o r p . ) u s e d i n t h e p r e s e n t work. 4. T h r o u g h o u t t h i s t h e s i s t h e " o r i e n t a t i o n o f t h e m o l e c u l a r a x e s " w i l l be r e f e r r e d t o . However, a c c o r d i n g t o quantum m e c h a n i c s t h e a x i s o f a m o l e c u l e i n a p a r t i c u l a r r o t a t i o n a l s t a t e i s n o t f i x e d i n s p a c e : r a t h e r i t s o r i e n t a t i o n a l d i s t r i b u t i o n i s d e s c r i b e d by t h e m o l e c u l a r wave f u n c t i o n . F o r J=0 m o l e c u l e s t h e wave f u n c t i o n s a r e s p h e r i c a l . F o r J = l m o l e c u l e s t h e wave f u n c t i n s a r e e l l i p s o i d a l . I n t h a t c a s e t h e " o r i e n t a t i o n o f t h e m o l e c u l a r a x e s " s h o u l d be t a k e n t o mean t h e o r i e n t a t i o n o f t h e symmetry a x i s o f t h e wave f u n c t i o n . 5. T h i s r e s u l t d o e s n o t a p p l y t o r e g i s t e r e d p h a s e s . A g o o d d i s c u s s i o n o f t h i s s u b j e c t i s c o n t a i n e d i n s e c t i o n 7.3 o f Dash ( 1 9 7 5 ) . 17 CH/APTER I I ORIENTATIONAL PROPERTIES OF HYDROGEN 2.1 O r i e n t a t i o n a l I n t e r a c t i o n s E v e n i n t h e b u l k s o l i d o r on t h e s u r f a c e o f g r a p h i t e , h y d r o g e n m o l e c u l e s c a n be m o d e l l e d v e r y w e l l by f r e e r o t o r s . The f r e e r o t o r e n e r g i e s E a r e g i v e n by E = B J ( J + l ) ( I I - l ) w h e r e B = 85.35 K ( 4 3 . 0 4 K) f o r H 2 ( D 2) ( S i l v e r a , 1 9 8 0 ) . The d i f f e r e n c e i n t h e v a l u e s o f B a r i s e s f r o m t h e d i f f e r e n t moments o f i n e r t i a . S i n c e t h e J = l s t a t e o f H 2 ( D 2 ) i s 171 K ( 8 6 . 1 K) a b o v e t h e J=0 s t a t e , one w o u l d e x p e c t o n l y t h e l a t t e r t o h ave any s i g n i f i c a n t p o p u l a t i o n a t l i q u i d h e l i u m t e m p e r a -t u r e s . However f o r h o m o n u c l e a r m o l e c u l e s c o n v e r s i o n b e t w e e n odd and e v e n J s t a t e s may be v e r y s l o w . F o r e x a m p l e , *h n u c l e i a r e f e r m i o n s s o t h e H 2 wave f u n c t i o n m u s t be a n t i s y m m e t r i c w i t h r e s p e c t t o t h e i r i n t e r c h a n g e . R o t a t i o n a l s t a t e s w i t h e v e n p a r i t y m u s t be c o m b i n e d w i t h n u c l e a r s p i n s t a t e s w i t h 1=0, w h i c h h a v e odd p a r i t y . Odd r o t a t i o n a l s t a t e s m u s t have 1=1. S i m i l a r a r g u m e n t s a p p l y t o D 2 e x c e p t t h a t d e u t e r o n s a r e b o s o n s s o t h e r o t a t i o n a l and n u c l e a r s p i n s t a t e s h a v e t h e same p a r i t y . The s t a t e s w i t h t h e g r e a t e s t s p i n d e g e n -e r a c y a r e l a b e l l e d o r t h o ; t h e o t h e r s a r e p a r a . T h e s e r e s u l t s a r e s u m m a r i z e d i n T a b l e I . The f a c t o r g j i s t h e d e g e n e r a c y o f t h e n u c l e a r s p i n s t a t e s . F o r h o m o n u c l e a r m o l e c u l e s , c o n v e r s i o n r e q u i r e s a s i m u l -18 t a n e o u s c h a n g e o f p a r i t y o f t h e s p i n and o r b i t a l s t a t e s . On t h e s u r f a c e o f G r a f o i l c o n v e r s i o n i s q u i t e s l o w : o u r m e a s u r e m e n t s ( C h a p t e r V) g i v e a r a t e o f ,40%/h f o r p u r e o-H 2 and .07 %/h f o r p u r e D 2 o n one o f o u r s u b s t r a t e s . T h i s h a s a l l o w e d us t o s t u d y s a m p l e s c o n t a i n i n g up t o 97% J = l m o l e c u l e s . T a b l e I : S t a t e s o f H y d r o g e n M o l e c u l e J I g j p - H 2 e v e n 0 1 o - H 2 o d d 1 3 o-D 2 e v e n 0,2 6 p-D 2 odd 1 3 P r o v i d e d t h a t J i s a g o o d quantum number, t h e m o s t g e n e r a l f o r m o f t h e i n t e r a c t i o n b e t w e e n two h y d r o g e n m o l e c u l e s w i t h o r i e n t a t i o n s ^ ' and at,' w i t h r e s p e c t t o t h e i n t e r m o l e c u l a r a x i s R^ 2 i s a sum o v e r and J 2 o f t e r m s o f t h e f o r m \j2 = 4 * »12> m am Y j " <ty Y j f ( 1 1 - 2) w h e r e Yj™ i s a s p h e r i c a l h a r m o n i c w i t h t h e p h a s e c o n v e n t i o n o f Rose ( 1 9 5 7 ) . O n l y e v e n v a l u e s o f J-^ and J 2 a r e r e q u i r e d b e c a u s e e a c h m o l e c u l e h a s i n v e r s i o n s y m m e t r y . T r a n s f o r m i n g t o an a r b i t r a r y r e f e r e n c e f r a m e V j ^ j 2 becomes V = 4TT E (IL ) J a Y T m(w.) Y T n ( ^ _ ) Y ^ * ( 3 _ , ) ( I I - 3 ) 2 1 2 ' 1 2 1 2 w h e r e , , and n ^ 2 a r e t h e o r i e n t a t i o n s o f t h e two m o l e c u l e s and t h e i n t e r n u c l e a r a x i s . To the e x t e n t t h a t J i s a good quantum number, then f o r each p a i r of J=l molecules, o n l y matrix elements o f V J 1 J 2 w i t n J ] _ a n d J 2 equal to 0 or 2 w i l l be non-zero. Many body i n t e r a c t i o n s w i l l be i g n o r e d . The a n i s o t r o p i c p a r t o f the Hamiltonian i s then H - i «c + kVeQQ = I ^ i r / 5 ) = Y 2 m ( M i ) U(R..)Y™*$..) + • T c (R. .) Y_ m(io . ) Y_ n (u .) Y. (5J •) ( n - 4 b ) i < 3 m,n m,n xy 2 x l 2 j 4 1 3 K ± ± H U > The one body term i s known as the c r y s t a l f i e l d term; i t a r i s e s from Van der Waals i n t e r a c t i o n s between the hydrogen mole c u l e s . Now c o n s i d e r a system of adsorbed, J=l hydrogen molecules i n the fi x phase. I f we take the z a x i s t o be normal to s u r f a c e , then when the c r y s t a l f i e l d i n t e r a c t i o n s of a molecule are summed over a l l of i t s neighbours o n l y the m=0 term w i l l s u r v i v e because the z a x i s i s a s i x - f o l d a x i s of symmetry. There w i l l a l s o be a c o n t r i b u t i o n t o the c r y s t a l f i e l d V c from the i n t e r a c t i o n of the molecule w i t h the s u r f a c e . The a n i s o t r o p i c H a m i l t o n i a n can be r e w r i t t e n as H= -(2/3) ( 5 t t ) 1 / 2 V ?Y °(u. ) C 1 2. 1 + 1 . E . C ( R . . ) Y 9 m ( : . ) Y 9 n ( : . ) Y , m + n * ( ^ . , ) (H-5a) m,n i < 3 m,nv i j ' 2 1 2 j 4 1 3 The hydrogen c o n t r i b u t i o n t o V i s v£= - ( 3 / 5 ) ( 1 6 , / 5 ) 1 / 2 ^ ( R . . ) Y 2 0 ( \ . ) ( I I _ 5 b ) 20 The m a i n c o n t r i b u t i o n t o t h e two bo d y t e r m i s f r o m t h e e l e c t r o s t a t i c q u a d r u p o l e - q u a d r u p o l e (EQQ) i n t e r a c t i o n ( S i l v e r a , 1 9 8 0 ; Van K r a n e n d o n k , 1 9 8 3 ) . I t i s u s e f u l t o r e w r i t e ( I I - 5 a ) i n t e r m s o f t h e s e c o n d r a n k s p h e r i c a l t e n s o r o p e r a t o r s T 2 m ( J ) d e f i n e d by T_° = (5/16 TT) 1 / 2 ( 3 J 2 - 2 ) ( I I - 6 a ) 2. Z T 2 - 1 = + (15/32 T T) 1 / 2 ( J + J z + J z J + ) ( I I - 6 b ) T 2 i 2 = (15/32 T T) 1 / 2 J + 2 ( I I - 6 c ) I t i s a c o n s e q u e n c e o f t h e W i g n e r - E c k a r t t h e o r e m ( R o s e , 1957) and o u r c h o i c e o f n o r m a l i z a t i o n t h a t w i t h i n t h e J = l m a n i f o l d t h e m a t r i x e l e m e n t s o f t h e s p h e r i c a l h a r m o n i c s and s p h e r i c a l t e n s o r s a r e e q u a l . The H a m i l t o n i a n f o r t h e J = l s y s t e m ( I I - 5 a ) i s t h e n H = ( l / 3 ) (16 T T/5) 1 / 2 V . ET°(J.) + 4 . . z. 1 c (R. .) T _ m ( J . ) c I 2 l 25 i <D m,n m,n 1 3 2 1 * T , n ' ( J . ) YAm+n*(n..) ( I I - 7 ) 2 j 4 1 3 I f one c o n s i d e r s o n l y n e a r e s t n e i g h b o u r i n t e r a c t i o n s and a r i g i d l a t t i c e , t h e n t h e c o e f f i c i e n t ? m n i s C (R. .) = (20 T T/9) (70T T) 1 / 2 r C(224;mn) ( I I - 8 ) M 1 j O w h e r e C ( 2 2 4 ; mn) i s a C l e b s c h - G o r d o n c o e f f i c i e n t and r 0 i s t h e n e a r e s t n e i g h b o u r q u a d r u p o l e c o u p l i n g c o n s t a n t . F o r a p a i r o f m o l e c u l e s s e p a r a t e d by a d i s t a n c e R Q, e a c h w i t h q u a d r u p o l e moment eQ, r Q i s g i v e n by 2 2 r = 6e^Cj o 3 ( I I - 9 ) 25R o 21 F o r t h e x ^3 s t r u c t u r e o n g r a p h i t e , R Q = 4.26 7A S O r Q = . 528 K (.510 K) f o r H 2 ( D 2 ) . I n t h e b u l k s o l i d t h e c o u p l i n g c o n s t a n t d e d u c e d f r o m e x p e r i m e n t h a s b e e n f o u n d t o be s m a l l e r t h a n r by a f a c t o r o f .87 f o r b o t h H 2 and D 2 ( S i l v e r a , 1 9 8 0 ) . O r i g i n a l l y i t was f e l t t h a t t h e r e d u c t i o n was due t o z e r o p o i n t m o t i o n o f t h e m o l e c u l a r c e n t r e s and c o u l d be a c c o u n t e d f o r by a v e r a g i n g Y 4 ° ( ^ ^ j ) / R ^ j 5 w e i g h t e d by t h e p a i r d i s t r i b u t i o n f u n c t i o n : H a r r i s ( 1 9 7 0 a ) showed t h a t t h e a v e r a g i n g p r o c e s s c h a n g e s t h e m a g n i t u d e o f t h e e f f e c t i v e EQQ i n t e r a c t i o n b u t n o t i t s f o r m . The p a i r d i s t r i b u t i o n f u n c t i o n c a n be o b t a i n e d f r o m t h e g r o u n d s t a t e wave f u n c t i o n o f t h e p h o n o n s . r i s r e n o r m a l i z e d downwards b e c a u s e s h o r t r a n g e c o r r e l a t i o n s n e e d e d t o m i n i m i z e h a r d c o r e r e p u l s i o n c a u s e t h e p a i r d i s t r i b u t i o n f u n c t i o n t o f a l l o f f more r a p i d l y f o r s m a l l s e p a r a t i o n s t h a n f o r l a r g e . The m a i n d i f f i c u l t y i n t h e c a l c u l a t i o n i s t h a t t h e p h o n o n wave f u n c t i o n i s n o t known v e r y w e l l . More r e c e n t c a l c u l a t i o n s ( G o l d m a n , 1979) i n d i c a t e t h a t z e r o p o i n t m o t i o n a l o n e i s i n s u f f i c i e n t t o a c c o u n t f o r t h e r e d u c t i o n o f r ; i t i s a l s o n e c e s s a r y t o t a k e i n t o a c c o u n t Van d e r W a a l s c o n t r i b u t i o n s t o t h e two body t e r m o f ( I I - 7 ) . F o r t h e 2D s y s t e m z e r o p o i n t m o t i o n i s l i k e l y t o be e v e n g r e a t e r t h a n i n 3D, l e a d i n g t o g r e a t e r r e n o r m a l i z a t i o n . 2.2 H y d r o g e n M o l e c u l e s on a S u r f a c e The r e m a i n d e r o f t h i s c h a p t e r w i l l be r e s t r i c t e d t o a c o n s i d e r a t i o n o f t h e p a r a r o t a t i o n a l ( d i s o r d e r e d ) s t a t e . I n t h a t 22 s t a t e e a c h m o l e c u l e i s i n a s i t e o f s i x - f o l d s y m m e t r y . F o r J = l m o l e c u l e s t h e W i g n e r - E c k a r t t h e o r e m i m p l i e s t h a t t h i s i s e q u i v a -l e n t t o a x i a l s y m m e t r y . The H a m i l t o n i a n o f ( I I - 7 ) w i l l c a u s e t h e J = l s t a t e t o be s p l i t by an e n e r g y A i n t o a n rrtj=0 and d o u b l y d e g e n e r a t e mj=+l s t a t e s . The mj=0 s t a t e h a s a p r o l a t e s p h e r o i d a l wave f u n c t i o n w i t h i t s z a x i s n o r m a l t o t h e s u b s t r a t e ; t h e irij=+l s t a t e s h a v e o b l a t e s p h e r o i d a l wave f u n c t i o n s . N a i v e l y one c a n t h i n k o f t h e s e a s m o l e c u l e s s t a n d i n g up o r l y i n g down on t h e s u r f a c e . The s y s t e m c a n be d e s c r i b e d by an o r i e n t a t i o n a l o r d e r p a r a m e t e r a d e f i n e d by a = ( 5 T i ) 1 / 2 < Y 2 0 ( w i ) > = - ( 4 T T / 5 ) 1 / 2 < T 2 0 ( j i i ) > =- (1/2) < 3 J ± ^ - 2 > (11-10) The b r a c k e t s < > i n d i c a t e a t h e r m a l a v e r a g e . An e q u i v a l e n t e x p r e s s i o n f o r a c a n be o b t a i n e d i n t e r m s o f t h e f r a c t i o n a l p o p u l a t i o n p Q o f t h e mj=0 s t a t e o= (3/2) ( p Q - l / 3 ) ( H - 1 1 ) As T-*» , a l l t h r e e rtij s t a t e s become e q u a l l y p o p u l a t e d s o o-K). A t T=0, a = l o r -1/2 d e p e n d i n g upon w h e t h e r t h e g r o u n d s t a t e h a s irij=0 o r mj=+l. A p l o t o f |oj v e r s u s T _ 1 i s shown i n F i g . 5. I n t h e h i g h t e m p e r a t u r e l i m i t i t i s e a s y t o o b t a i n an e x p r e s s i o n f o r a s i n c e t h e EQQ i n t e r a c t i o n s a r e a v e r a g e d o u t by m o l e c u l a r r e o r i e n t a t i o n . N e g l e c t i n g t h e s e c o n d t e r m i n ( I I - 7 ) , t h e e n e r g i e s o f t h e irtj s t a t e s a r e E = ( V /3) < Y ° | 3 J 2 - 2 | Y . ° > o c 1 1 z 1 1 = - ( 2 / 3 ) V c ( I I - 1 2 a ) 23 24 E ± 1 = ( V c / 3 ) < Y 1 ± 1 | 3 J 2 2 - 2 | Y 1 ± 1 > = ( l / 3 ) V c ( I I - 1 2 b ) The e n e r g y g a p A b e t w e e n t h e g r o u n d and e x c i t e d s t a t e s i s t e m p e r a t u r e i n d e p e n d e n t and e q u a l t o |v c| . The p o p u l a t i o n s o f t h e n i j s t a t e s a r e r e l a t e d by t h e B o l t z m a n n f a c t o r s o p Q = (1 + 2exp( + A A T ) } " 1 ( I I - 1 3 ) w h e r e t h e u p p e r s i g n i s f o r V >0 and t h e l o w e r f o r V <0. I n t h e h i g h t e m p e r a t u r e l i m i t , p u t t i n g A = | v j , e x p a n d i n g t h e e x p o n e n t i a l and r e t a i n i n g o n l y t h e f i r s t o r d e r t e r m P o = ( 1 / 3 ) [ l + ( 2 / 3 ) V c / k T ] (1 1 - 1 4 ) S u b s t i t u t i o n i n t o ( 1 1 - 1 1 ) g i v e s a = V c / 3 k T (1 1 - 1 5 ) I n C h a p t e r I V i t i s shown t h a t t h e NMR s p e c t r u m g i v e s a d i r e c t d e t e r m i n a t i o n o f |a| . T h e r e f o r e a m e a s u r e m e n t o f t h e 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 e s p e c t r u m g i v e s (v c| . I f t h e EQQ i n t e r a c t i o n i s i n c l u d e d , t h e n w i t h i n t h e mean f i e l d a p p r o x i m a t i o n A becomes t e m p e r a t u r e d e p e n d e n t and i s g i v e n by A = V c - 27Ta/2 ( H a r r i s and B e r l i n s k y , 1979) ( 1 1 - 1 6 ) The m o l e c u l a r f i e l d a l w a y s o p p o s e s t h e c r y s t a l f i e l d s i n c e a and V c h a v e t h e same s i g n . The c r y s t a l f i e l d a t t e m p t s t o a l i g n a l l o f t h e m o l e c u l e s i n t h e same d i r e c t i o n w h e t h e r i t be s t a n d i n g up o r l y i n g down. T h i s a l i g n m e n t c o m p e t e s w i t h t h e EQQ i n t e r a c t i o n w h i c h f a v o u r s a "T" c o n f i g u r a t i o n f o r e a c h p a i r . A s y s t e m o f J = l h y d r o g e n m o l e c u l e s i s a n a l o g o u s t o a n a n t i f e r r o m a g n e t i c s y s t e m w i t h s p i n S = l ( H a r r i s , 1 9 7 1 ) . The J=0 m o l e c u l e s a r e a n a l o g o u s t o n o n - m a g n e t i c i o n s . H y d r o g e n r e s e m b l e s a p a r t i c u l a r l y s i m p l e m a g n e t i c a l l o y i n w h i c h a l l o f t h e c o n s t i t u e n t s have t h e same mass and f o r c e c o n s t a n t s . I n t h e m a g n e t i c c a s e t h e o r d e r p a r a m e t e r a i s t h e s u b l a t t i c e m a g n e t i z a t i o n a = <S Z> (11 - 1 7 ) T h i s d e f i n i t i o n i s n o t s u i t a b l e f o r h y d r o g e n b e c a u s e i f one h a s a n o n - d e g e n e r a t e g r o u n d s t a t e a n d t i m e r e v e r s a l s y m metry i s p r e s e r v e d , t h e g r o u n d s t a t e wave f u n c t i o n i s r e a l and t h e e x p e c t a t i o n v a l u e o f t h e momentum i s z e r o ; i t i s s a i d t o be q u e n c h e d ( V a n V l e c k , 1 9 3 2 ) . E v e n i n t h e p r e s e n c e o f a m a g n e t i c f i e l d t h e a s s u m p t i o n o f t i m e r e v e r s a l s ymmetry i s a g o o d a p p r o x i m a t i o n s i n c e t h e m a g n e t i c i n t e r a c t i o n s o f a h y d r o g e n m o l e c u l e (^lmK) a r e much s m a l l e r t h a n t h e o r i e n t a t i o n a l i n t e r -a c t i o n s ( ^ l K ) . H a v i n g e f f e c t i v e a x i a l s y m m e t r y , t h e c r y s t a l f i e l d l e a v e s t h e m T=+l s t a t e s d e g e n e r a t e b u t a t s u f f i c i e n t l y l o w t e m p e r a t u r e s t h e s y m m e t r y w i l l be b r o k e n by t h e EQQ i n t e r -a c t i o n s o t h e g r o u n d s t a t e w i l l be n o n - d e g e n e r a t e a nd <J>=0. F o r h y d r o g e n t h e o r d e r p a r a m e t e r s a r e d e f i n e d i n t e r m s o f t h e s e c o n d r a n k t e n s o r i a l c o m p o n e n t s o f a n g u l a r momentum, t h e q u a d r u p o l e moments. T h e r e a r e n i n e o f t h e s e o f w h i c h f i v e a r e i n d e p e n d e n t s i n c e t h e t e n s o r i s s y m m e t r i c . One o f t h e o r d e r p a r a m e t e r s a was d e f i n e d i n ( 1 1 - 1 0 ) ; t h e r e m a i n i n g f o u r a r e i n t r o d u c e d i n C h a p t e r I I I . The c r y s t a l f i e l d o f h y d r o g e n i s a n a l o g o u s t o a n a p p l i e d m a g n e t i c f i e l d e x p e r i e n c e d by a n t i -f e r r o m a g n e t s : b o t h t r y t o a l i g n t h e m o l e c u l e s i n t h e same d i r e c t i o n . E x p e r i m e n t a l l y t h e r e i s a c r u c i a l d i f f e r e n c e i n t h a t t h e c r y s t a l f i e l d i s d e t e r m i n e d by t h e s u b s t r a t e and t h e n e i g h -b o u r i n g m o l e c u l e s w h e r e a s an a p p l i e d m a g n e t i c f i e l d c a n e a s i l y be c o n t r o l l e d . F o r h y d r o g e n cr w i l l n o r m a l l y be n o n - z e r o i n t h e p a r a r o t a t i o n a l s t a t e ( a n a l o g o u s t o t h e p a r a m a g n e t i c s t a t e ) b e c a u s e t h e c r y s t a l f i e l d c a n n o t be " t u r n e d o f f " . I n t h e a b s e n c e o f a m a g n e t i c f i e l d , t h e H a m i l t o n i a n o f t h e a n t i f e r r o -m a g n e t s i s i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n S_->- - S _ : z z a b o v e t h e t r a n s i t i o n t e m p e r a t u r e <S Z>=0. The EQQ and a n t i f e r r o m a g n e t i c e x c h a n g e i n t e r a c t i o n s a r e a l s o a n a l o g o u s b u t o n c e a g a i n t h e r e a r e some i m p o r t a n t d i f f e r e n c e s . The f o r m e r f a v o u r s a "T" c o n f i g u r a t i o n o f t h e m o l e c u l a r a x e s b u t t h e l a t t e r a t t e m p t s t o a l i g n them a n t i -p a r a l l e l . A l s o , t h e q u a d r u p o l e c o u p l i n g c o n s t a n t r c a n be c a l c u l a t e d w i t h r e a s o n a b l e a c c u r a c y w h e r e a s t h e e x c h a n g e i n t e g r a l i s n o r m a l l y a p h e n o m e n o l o g i c a l p a r a m e t e r . F i n a l l y , t h e EQQ i n t e r a c t i o n f a l l s o f f a s R -^ b u t t h e e x c h a n g e i n t e r -a c t i o n h a s a v e r y s h o r t r a n g e . A r g u m e n t s b a s e d o n t h e G i n z b u r g C r i t e r i o n ( P f e u t y and T o u l o u s e , 1977) show t h a t t h e l o n g e r t h e r a n g e o f f o r c e s , t h e n a r r o w e r t h e c r i t i c a l r e g i o n w i l l b e : f l u c t u a t i o n s i n t h e o r d e r p a r a m e t e r w i l l be r e d u c e d a s more n e i g h b o u r i n g m o l e c u l e s a r e a b l e t o i n t e r a c t w i t h a g i v e n m o l e c u l e . F o r e x a m p l e , i n a s u p e r c o n d u c t o r t h e r e i s no hope 27 of seeing the c r i t i c a l region because Cooper pairs interact over several thousand angstroms. 28 CHAPTER I I I ORIENTATIONAL ORDERING 3.1 I n t r o d u c t i o n T h i s c h a p t e r c o n t a i n s a d i s c u s s i o n o f t h e a p p l i c a t i o n o f mean f i e l d t h e o r y (MFT) t o t h e s t u d y o f t h e o r i e n t a t i o n a l l y o r d e r e d s t a t e s o f an h e x a g o n a l l a t t i c e o f i n t e r a c t i n g quantum q u a d r u p o l e s ( H a r r i s and B e r l i n s k y , 1 9 7 9 ) . I n MFT t h e i n t e r -a c t i o n s o f a m o l e c u l e w i t h i t s n e i g h b o u r s a r e r e p l a c e d by an i n t e r a c t i o n w i t h an a v e r a g e m o l e c u l a r f i e l d ; i t i g n o r e s t h e e f f e c t s o f s h o r t r a n g e c o r r e l a t i o n s b e t w e e n p a i r s o f m o l e c u l e s . C o n s e q u e n t l y one e x p e c t s t h e mean f i e l d t r a n s i t i o n t e m p e r a t u r e t o be t o o h i g h . I n 3D i t w o u l d t y p i c a l l y be t o o l a r g e by a f a c t o r o f two. F o r b u l k H 2 ( D 2 ) t * i e e x P e r " i m e n t a l and mean f i e l d o r d e r i n g t e m p e r a t u r e s a r e 2.8 K (3.8 K) and 7.3 K (9.0 K) r e s p e c t i v e l y ( S i l v e r a , 1 9 8 0 ) . However, MFT d o e s q u i t e w e l l a t p r e d i c t i n g t h e s y m m e t r y o f t h e o r d e r e d s t a t e s s i n c e o n c e l o n g r a n g e o r d e r h a s b e e n e s t a b l i s h e d t h e m a j o r w e a k n e s s o f t h e t h e o r y i s o f s e c o n d a r y i m p o r t a n c e . F o r b u l k h y d r o g e n , MFT p r e d i c t s t h e o r d e r e d s t r u c t u r e c o r r e c t l y . F o r t h e s e r e a s o n s we w i l l c o n c e n t r a t e on t h e s t r u c t u r e o f t h e o r d e r e d p h a s e s r a t h e r t h a n t h e 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 e o r d e r p a r a m e t e r s . H a r r i s a n d B e r l i n s k y t a c k l e d t h e p r o b l e m i n two ways. The f i r s t was a n u m e r i c a l r e l a x a t i o n m e t h o d b a s e d o n t h e work o f James ( 1 9 6 8 ) , who had a p p l i e d i t t o b u l k h y d r o g e n . The s e c o n d was an a n a l y t i c m e t h o d w h i c h c o u l d be f o r m u l a t e d e x a c t l y i f one 29 assumed a s t r u c t u r e f o r t h e o r d e r e d s t a t e , o r i n t e r m s o f a L a n d a u e x p a n s i o n o f t h e f r e e e n e r g y i f one d i d n o t . The r e l a x a -t i o n m e t hod was u s e d t o e l u c i d a t e t h e g e n e r a l f e a t u r e s o f t h e p h a s e d i a g r a m and t o s t u d y t h e f i r s t o r d e r t r a n s i t i o n s . The a n a l y t i c a p p r o a c h was u s e d t o a n s w e r more d e t a i l e d q u e s t i o n s a b o u t t h e s t r u c t u r e o f t h e p h a s e s . The L a n d a u e x p a n s i o n i s v a l i d i n t h e n e i g h b o u r h o o d o f a s e c o n d o r d e r p h a s e t r a n s i t i o n , a r e g i o n where t h e r e l a x a t i o n m e t h o d c o n v e r g e s s l o w l y . B o t h a p p r o a c h e s i n c l u d e d t h e c r y s t a l f i e l d a n d EQQ i n t e r -a c t i o n s i n t h e H a m i l t o n i a n . A r i g i d l a t t i c e o f J = l m o l e c u l e s w i t h o n l y n e a r e s t n e i g h b o u r i n t e r a c t i o n s was assumed. The mean f i e l d a p p r o x i m a t i o n i s e q u i v a l e n t t o w r i t i n g t h e d e n s i t y m a t r i x P a s a d i r e c t p r o d u c t o f t h e s i n g l e p a r t i c l e d e n s i t y m a t r i c e s , p i , i . e . , P = \ P ± ( I I I - 1 ) The b e s t a p p r o x i m a t i o n f o r p i s t h e n o b t a i n e d by c h o o s i n g t h e P ^ ' s s o t h a t t h e y m i n i m i z e t h e f r e e e n e r g y F, w h i c h i s g i v e n by F ( p ) = T r ( p H ) + T { T r ( p l n p ) } ( I I I - 2 ) w h e r e H i s t h e a n i s o t r o p i c H a m i l t o n i a n g i v e n by ( I I - 7 ) . The d e n s i t y m a t r i c e s P ^ c a n be e x p r e s s e d i n t e r m s o f t h e o r d e r p a r a m e t e r s [ s e e ( I I I - 5 ) b e l o w ] a n d , i n t h e L a n d a u e x p a n s i o n , F i s e x p a n d e d i n p o w e r s o f t h e o r d e r p a r a m e t e r s . A l t e r n a t i v e l y o n e c a n o b t a i n t h e p^ s e l f - c o n s i s t e n t l y by s o l v i n g t h e e q u a t i o n p > e x p ( - 0 H e f f 1 ) / T r [ e x p ( ~ ^ e f f 1 ) ] ( I I I - 3 ) 30 w h e r e g = l / ( k T ) and W 1 e f f s a t i s f i e s H e f f i = t f c i + T r ' ^ H E Q Q i j k j i p k > ( " I " * ) The p r i m e i n d i c a t e s t h a t t h e t r a c e i s o v e r t h e s t a t e s o f a l l m o l e c u l e s k s u c h t h a t k ^ i . I n t h e r e l a x a t i o n m e t h o d one t a k e s an n x n l a t t i c e w i t h p e r i o d i c b o u n d a r y c o n d i t i o n s and a s s i g n s a v a l u e t o e a c h p ^ . The v a l u e o f n m u s t be a t l e a s t l a r g e e n o u g h t o c o n t a i n a l l o f t h e s u b l a t t i c e s o f t h e o r d e r e d s t a t e . The e f f e c t i v e H a m i l t o n i a n ^ g f f i s t h e n f o u n d u s i n g ( I I I - 4 ) and new v a l u e s o f P^ a r e o b t a i n e d f r o m ( I I I - 3 ) . The p r o c e s s i s r e p e a t e d a n d h o p e f u l l y c o n v e r g e s . I n t h e a n a l y t i c m e t h o d a s u b l a t t i c e s t r u c t u r e i s assumed and a l l o f t h e p ^ ' s on e a c h s u b l a t t i c e a r e s e t e q u a l . I f two s u b l a t t i c e s a r e r e l a t e d by a s ymmetry o p e r a t i o n t h e number o f i n d e p e n d e n t p ^ ' s i s f u r t h e r r e d u c e d ; i n f a c t t h e r e a r e a t m o s t two f o r t h i s s y s t e m . ( I I I - 3 ) and ( I I I - 4 ) a r e t h e n s o l v e d s i m u l t a n e o u s l y . The same r e d u c t i o n i n t h e number o f i n d e p e n d e n t p ^ ' s c a n be made f o r t h e L a n d a u e x p a n s i o n i f t h e s u b l a t t i c e s t r u c t u r e i s a s sumed. P r o v i d e d t h a t t h e r e i s t i m e r e v e r s a l i n v a r i a n c e , t h e s e c o n d r a n k s p h e r i c a l t e n s o r o p e r a t o r s T 2 m ( j " ) d e f i n e d by ( I I - 6 ) f o r m a c o m p l e t e s e t o f d y n a m i c a l v a r i a b l e s f o r a s y s t e m o f J = l m o l e c u l e s . The ( r e a l ) o r d e r p a r a m e t e r s a i ,yj_ , m , X^i a n d <J>i a r e d e f i n e d i n t e r m s o f t h e t h e r m a l a v e r a g e s o f t h e T 2 r a ( J i ) ' s by o±=- (4ir/5) 1 / 2 < T 2 ° ( J i ) > ( I I I - 5 a ) p . e x p d i x ^ ^ i - i t S . / l S ) < T 2 ± 1 ( 5 . ) > ( i i i - 5 b ) p i = 31 r ^ e x p (±i2<()i) = T)^1 = - (24 T T/5) < T 2 ± 2 ( c L ) > ( I I I - 5 c ) The s i n g l e p a r t i c l e d e n s i t y m a t r i x i s K e r m i t i a n a n d i t s t r a c e i s o n e . I t s most g e n e r a l f o r m i s ( l - a i ) / 3 - ( y i / / 2 ) e x p ( - i x i ) - (n ±/3) e x p (-i2<|>.iT - ( u i / / 2 ) e x p ( i x i ) ( l + 2 a i ) / 3 (u .//2) e x p ( - i X i ) - ( T l i / 3 ) e x p ( i 2 ^ i ) ( v i / / 2 ) e x p ( i x i ) ( l - c r ^ / 3 ( I I I - 6 ) The r o w s and c o l u m n s a r e l a b e l l e d f r o m l e f t t o r i g h t a n d t o p t o b o t t o m b y m=l,0, a nd -1 r e s p e c t i v e l y . F o r some v a l u e s o f t h e o r d e r p a r a m e t e r s , i t i s e a s y t o p i c t u r e t h e a v e r a g e o r i e n t a t i o n s o f t h e m o l e c u l a r wave f u n c t i o n s . I f y^=n^=0 t h e n t h e Y 1 m ( w ^ ) ' s a r e e i g e n -f u n c t i o n s s o t h e r e i s a x i a l s y m m e t r y a b o u t t h e z a x i s , w h i c h i s n o r m a l t o t h e s u r f a c e . The m o l e c u l e s t e n d t o s t a n d up i f cr^>0 a n d l i e down i f o^<o. I f o n l y y^=0, t h e e i g e n f u n c t i o n s a r e Y^0 and (e 1 Y 1 ± e 1 Y )//2 s o one o f t h e p r i n c i p a l a x e s i s a l o n g z a s b e f o r e and one o f t h e o t h e r s makes an a n g l e <f> ^  w i t h r e s p e c t t o t h e x a x i s . I f u ^ O , Y^ 0 i s no l o n g e r a n e i g e n f u n c t i o n and t h e z a x i s i s n o t a p r i n c i p a l a x i s . I t i s n o t p r a c t i c a l f r o m a c o m p u t a t i o n a l p o i n t o f v i e w t o a l l o w a l l o f t h e p ^ ' s t o be d i f f e r e n t . I n a n y e v e n t , one e x p e c t s t h a t t h e r e w i l l be a s m a l l number o f s u b l a t t i c e s w i t h t h e P ^ ' s t h e same o n e a c h s u b l a t t i c e . I f we assume t h a t t h e g r o u n d s t a t e f o r e a c h s u b l a t t i c e i s n o n - d e g e n e r a t e , t h e n a t T=0 a l l o f t h e m o l e c u l e s o n a s u b l a t t i c e 32 w i l l be a l i g n e d p a r a l l e l t o some a x i s . T h i s c a n be s e e n f r o m t h e f o l l o w i n g a r g u m e n t . A t T=0, a l l o f t h e m o l e c u l e s on s u b l a t t i c e i a r e i n t h e i r g r o u n d s t a t e ^ and = | i ( j ^ > < ^ | . S i n c e t h e H a m i l t o n i a n i s r e a l , we c a n t a k e ^ t o be r e a l . A s e t o f r e a l , o r t h o n o r m a l b a s i s f u n c t i o n s $ a c a n be d e f i n e d by $ 1 ( ^ i ) = - ( Y 1 1 ( " . ) - Y 1 " 1 ( ^ i ) )//2 (III-7a) $ 2 ( w i ) = i ( Y 1 1 ( w i ) + Y 1 " 1 ( w i ) )//2 (III-7b) *-(w.)=Y,°(u.) (III-7C) 3 1 x i T h e s e f u n c t i o n s a l l h a v e t h e same f o r m b u t t h e y a r e o r i e n t e d a l o n g t h e x, y, and z a x e s w h i c h a r e p a r a l l e l t o t h e u n i t v e c t o r s x^, x2' a n d x 3 « The ^ ( u ^ ' s a r e p r o p o r t i o n a l t o t h e d i r e c t i o n c o s i n e s ( to ^ ) o f t h e u n i t v e c t o r to ^  a l o n g t h e s e a x e s : $ e < ( w i ) = (3 / 4 7T) 1 / 2 x ( X ( m i ) ( I I I - 8 ) The g r o u n d s t a t e i s a l i n e a r c o m b i n a t i o n o f t h e b a s i s f u n c t i o n s : = S Y « x 1 $ < x ( w i ) ( I I I - 9 ) I f ^ i s n o r m a l i z e d , t h e y^.1 s a t i s f y £ ( ^ 1 ) 2 = 1 ( i n - i o ) T h i s means t h a t t h e Y ^ ' s c a n be t a k e n t o be t h e d i r e c t i o n c o s i n e s o f some u n i t v e c t o r Z 1 w i t h r e s p e c t t o t h e x, y, and z a x e s . The component Z 1(u^) o f a l o n g Z 1 i s z l(Vi YA (V ( i n - i i ) S u b s t i t u t i n g (111-8) and (111-11) i n t o (111-9) g i v e s $ = ( 3 / 4 T T ) 1 / 2 Z i ( w i ) = Y 1 ° ( e i ) (111-12) 33 w h e r e i s t h e a n g l e b e t w e e n Z and m •. T h e r e f o r e a t T=0, t h e wave f u n c t i o n s o f a l l m o l e c u l e s on s u b l a t t i c e i a r e a l i g n e d p a r a l l e l t o Z 1. T h i s c o r r e s p o n d s t o u ^ n ^ O i n t h e p r i n c i p a l a x i s f r a m e . I n t h e c r y s t a l r e f e r e n c e f r a m e ( x , y, z) , t h e o r d e r p a r a m e t e r s c a n be e x p r e s s e d i n t e r m s o f t h e o r i e n t a t i o n ( o f Zi b v a i = ( 3 c o s 2 6 i - l ) / 2 ( I I I - 1 3 a ) y i = s i n 6 i c o s e i = [ ( 2 / 9 ) ( l - a ± ) ( l + o ^ ) ] 1 / 2 ( H I - 1 3 b ) n i = s i n 2 6 i = l - a i ( I I I - 1 3 C ) X ± = ^ ( I I I - 1 3 d ) O n l y two i n d e p e n d e n t v a r i a b l e s a ^ and ^ a r e n e e d e d . 3.2 O r i e n t a t i o n a l l y O r d e r e d S t a t e s The o r i e n t a t i o n a l p h a s e d i a g r a m o b t a i n e d by H a r r i s and B e r l i n s k y a s a f u n c t i o n o f V c / r and T / r i s shown i n F i g . 6. The h i g h t e m p e r a t u r e p a r a r o t a t i o n a l p h a s e h a s a s i n g l e s u b l a t t i c e c h a r a c t e r i z e d by o.= a ( I I I - 1 4 a ) 1 o y.= r,- = 0 ( I I I - 1 4 b ) l l U n l i k e t h e o t h e r p h a s e s , i t h a s no l o n g r a n g e o r i e n t a t i o n a l o r d e r e x c e p t f o r t h a t i n d u c e d by t h e s u b s t r a t e . I f v_<<-kT, t h e n a l l o f t h e m o l e c u l a r a x e s a r e c o n s t r a i n e d t o l i e p a r a l l e l t o t h e s u r f a c e . The EQQ i n t e r a c t i o n d e t e r m i n e s t h e i r a z i m u t h a l o r i e n t a t i o n s . The r e s u l t i s t h e two s u b l a t t i c e h e r r i n g b o n e p h a s e shown i n F i g . 3 ( a ) w i t h o r d e r 3 4 F i g . 6 O r i e n t a t i o n a l p h a s e d i a g r a m , V c / r v e r s u s T/T , f o r quantum q u a d r u p o l e s on a h e x a g o n a l l a t t i c e , d e d u c e d by MFT. The d a s h e d l i n e s a r e f i r s t o r d e r p h a s e b o u n d a r i e s and t h e s o l i d l i n e s a r e c o n t i n u o u s . The r e g i o n n e a r t h e t r i c r i t i c a l p o i n t i s shown w i t h an e x p a n d e d s c a l e a t t h e u p p e r r i g h t . 35 p a r a m e t e r s CTl=a2=CTo ( I l l - 1 5 a ) y ±=0 ( I I I - 1 5 b ) n ^ ^=112 = n 0 ( I I I - 1 5 C ) * 1=180° -< l> 2=45 0 ( I I I - 1 5 d ) A s V c i n c r e a s e s t h e r e i s a s e c o n d o r d e r p h a s e t r a n s i t i o n t o t h e 2 - o u t p h a s e . T h i s i s a two s u b l a t t i c e p h a s e s i m i l a r t o t h e h e r r i n g b o n e p h a s e e x c e p t t h a t y i s no l o n g e r z e r o s o t h e m o l e c u l a r a x e s t i l t o u t o f t h e p l a n e . The o r d e r p a r a m e t e r s a r e 0=0=0 ( I I I - 1 6 a ) 1 1 o U l = u 2 = y o ( I I I - 1 6 b ) n 1 = n 2 = n 0 ( l l l - 1 6 c ) X l = 1 8 0 O - x 2 = x o ( l l l - 1 6 d ) ' ^ l S O 0 - ^ 2=<},o ( I I I - 1 6 e ) I f V c i n c r e a s e s f u r t h e r , t h e r e i s a f i r s t o r d e r t r a n s i -t i o n t o t h e f o u r s u b l a t t i c e p i n w h e e l p h a s e i n w h i c h a m o l e c u l e o n s u b l a t t i c e 1 s t a n d s up and a l l o f i t s n e i g h b o u r s l i e down [ F i g . 3 ( b ) ] # The o r d e r p a r a m e t e r s a r e CT1>0, a 2 = a 3 = a 4 < 0 ( I I I - 1 7 a ) y ±=0 ( I I I - 1 7 b ) n l = 0 , n 2 = n 3 = n 4 ( I I I - 1 7 c ) * 2=<f>3-60O= ^-120° ( I I I - 1 7 d ) A s V c i n c r e a s e s a g a i n , t h e r e i s a f i r s t o r d e r p h a s e t r a n s i t i o n b a c k i n t o t h e 2 - o u t p h a s e . N e x t t h e r e i s a s e c o n d o r d e r t r a n s i t i o n t o t h e s i n g l e s u b l a t t i c e f e r r o r o t a t i o n a l p h a s e , w h i c h i s c h a r a c t e r i z e d by 36 ° i = a o ( I I I - 1 8 a ) v±=v0 ( I l l - 1 8 b ) n i = n o ( I I I - 1 8 C ) x i = * i = < i b ( I I I - 1 8 d ) A l l o f t h e m o l e c u l e s a r e o r i e n t e d i n t h e same d i r e c t i o n b u t t h e i r a z i m u t h a l a n g l e i s a r b i t r a r y . C l e a r l y t h e e n t r o p y d o e s n o t d e p e n d on <f> and i t c a n be shown t h a t f o r a s i n g l e s u b l a t t i c e p h a s e t h e e n e r g y c a n n o t d e p e n d on i t e i t h e r . T h i s p h a s e i s a n a l o g o u s t o t h e p l a n a r XY m o d e l f o r s p i n s w h i c h h a s a H a m i l t o n i a n o f t h e f o r m H = " i j J i j ^ i ( 1 1 1 - 1 9 ) w h e r e J i j Z = 0 a nd J — * = J — ^ i 0. U n l i k e t h e o t h e r o r d e r e d s t a t e s , t h e l i b r o n 1 s p e c t r u m f o r t h i s s t a t e h a s no g a p b e c a u s e o f t h e c o n t i n u o u s s y m m e t r y . F o r f u r t h e r i n c r e a s e s i n V c , t h e r e i s a s e c o n d o r d e r t r a n s i t i o n t o t h e p a r a r o t a t i o n a l p h a s e . H e r e a^>0 and a l l o f t h e m o l e c u l e s a r e e s s e n t i a l l y s t a n d i n g up. I n t h e p i n w h e e l and h e r r i n g b o n e p h a s e s , a t T=0 t h e o r d e r p a r a m e t e r s a r e i n d e p e n d e n t o f V c / r . T h i s i s n o t t h e c a s e f o r t h e 2 - o u t a nd f e r r o r o t a t i o n a l p h a s e s . I n t h e 2 - o u t p h a s e a t T=0, MFT g i v e s a = ( 1 6 4 V / r - 5 5 5 ) / 2 9 5 5 ( 1 1 1 - 2 0 ) o c c o s ( * 1 - * 2 ) = 2 ( l + 2 a o ) / [ 4 1 ( l - a o ) ] ( I I 1 - 2 1 ) a n d i n t h e f e r r o r o t a t i o n a l p h a s e a Q = l / 7 + 4 V c / ( 1 0 5 D ( 1 1 1 - 2 2 ) 37 3.3 The L a n d a u E x p a n s i o n The o r d e r e d p h a s e s w i l l now be e x p l o r e d i n more d e t a i l by m a k i n g a L a n d a u e x p a n s i o n o f t h e f r e e e n e r g y F. We d e f i n e t h e d e v i a t i o n s u ^ o f t h e o r d e r p a r a m e t e r s f r o m t h e i r e q u i l i b r i u m v a l u e s i n t h e p a r a r o t a t i o n a l s t a t e by u l i = n i c o s 2 f . ( I I I - 2 3 a ) U 2 i = n i s i n 2 * i ( I I I - 2 3 b ) U 3 i = a i " 0 e q ( I I I - 2 3 C ) u 4 i = y i c o s x i ( I I I - 2 3 d ) u 5 . = y i s i n x . ( I I I - 2 3 e ) w h e r e a i - s t n e e q u i l i b r i u m v a l u e o f a ^  i n t h e a b s e n c e o f l o n g r a n g e c o o p e r a t i v e o r d e r i n g . I t i s g i v e n by ( 1 1 - 1 1 ) , ( 1 1 - 1 3 ) , and ( 1 1 - 1 6 ) . To s e c o n d o r d e r i n t h e u ;'s t h e f r e e e n e r g y e x p a n s i o n f o r N m o l e c u l e s i s F / ( N r ) = F e g / ( N D + (1/2) | = 1 j j = 1 ! { x _ 1 ( q ) ^ B u * ( q ) u 3 ( q ) ( 1 1 1 - 2 4 ) where u k (q) =u < K * (-q) = N ~ 1 ? u o < i e x p ( i q - r ± ) ( 1 1 1 - 2 5 ) F S g = e q u i l i b r i u m f r e e e n e r g y i n t h e a b s e n c e o f c o o p e r a t i v e o r d e r i n g = NF{ ( 9 / 2 ) a 2 - ( 2 / 3 ) ( V / r ) o + ( k T / 3 r ) x{3 l n 3 + ( l + 2 o ) l n ( l + 2 o ) - 2 ( l - a ) l n ( l - a ) }} e q e q e q e q (1 1 1 - 2 6 ) a n d X ( q ) i s t h e i n v e r s e s u s c e p t i b i l i t y . I t d o e s n o t c o u p l e a ^ and n . o n one hand and P • o n t h e o t h e r . The n o r m a l modes o f t h e s y s t e m Y k ( q ) ( k = 1,2,...,5) a r e f o u n d by -1 "*" "*• d i a g o n a l i z i n g X (q) f o r e a c h q t o g e t t h e e i g e n v e c t o r s 38 xo< (q) and t n e corresponding eigenvalues A (q) since Yk ( q> = c5 x« (9)u(q) (111-27) In terms of the normal modes the free energy i s 5 ^ F / ( N D = F / ( N T ) + (1/2)5 , SMqJX,(q)| 2 (111-28) eg k™j- q K The entropy terms in (111-24) are a l l p o s i t i v e and c o n t r i -bute only to the diagonal elements. At high temperatures the ->-entropy terms w i l l dominate so the A k (q)'s w i l l be a l l p o s i t i v e . A second order phase t r a n s i t i o n w i l l occur i f one of the A (q)'s goes to zero for some q at a temperature T c. If that eigenvalue i s l a b e l l e d k=l, then the c r i t i c a l order para-meter i s determined by x x 0 < ( q ) . The sub l a t t i c e s of the ordered state are determined by q and the orientations of the molecular 1 -> axes by x K (q). We w i l l only consider q vectors that are points 2 of high symmetry i n the f i r s t B r i l l o u i n zone. This means that the supe r l a t t i c e w i l l be commensurate so the p e r i o d i c i t i e s of i n t e r e s t w i l l be temperature independent. Domany et a l (1978) have found that for a triangular l a t t i c e (space group P6mm) the only temperature independent c| vectors for which /V^(q) can go to zero are 0, QA^ and 0 B ^ » which are shown in Fig. 7 and defined by 3 A l ' q o § * 3 A 2 = < 3 o ( ^ ) / 2 ' 3 A 3 - ( - ^ 3 i - j ) / 2 (111-29) ^ B l = " ^ B 2 = ( 4 i r / 3 a ) i (111-30) where a = l a t t i c e parameter = 4.26 TA and q Q = (2ir/a/3). For modes which couple n^ and a ^ ~ o e g i the eigen-values which go to zero f i r s t as T decreases have q = ~5Ai« The corresponding eigenvectors are 3 9 F i g . 7 T h e f i r s t B r i l l o u i n z o n e f o r a h e x a g o n a l l a t t i c e a n d t h e o n l y t e m p e r a t u r e i n d e p e n d e n t w a v e v e c t o r s t h a t c a n p r o d u c e s o f t m o d e s . \ V - 2 \ •- \ (b ) / / / / \ \ 3 \ 4 / / s / l / \ \ \ (a) / - / - r - 4 / - r -A / - / ^ ( c ) F i g . 8 T h e s t r u c t u r e o f t h e o r d e r e d s t a t e s t h a £ a r i s e w h e n t h e e i g e n v a l u e s f o r t h e m o d e s c o r r e s p o n d i n g t o t h e g o s o f t . ( a ) $ A i « ( b ) cLo* 9a3* T n e Y t r a n s f o r m i n t o e a c h o t h e r b y 1 Z 0 ° r o t a t i o n s . y 40 x 1 ( Q A l ) = ( 0 , l , 0 , 0 , 0 ) ( I I I - 3 1 a ) x 1 ( Q A 2 ) = ( - / 3 , - l , 0 , 0 , 0 ) / 2 ( I I I - 3 1 b ) x 1 ( Q A 3 ) = ( / 3 , - l , 0 , 0 , 0 ) / 2 ( I I I - 3 1 C ) The n o r m a l modes w i l l be d e n o t e d by y^ = Y-^  ( 0 /^ ) * The s u p e r l a t t i c e s c o r r e s p o n d i n g t o t h e Q ^ i ' s a r e shown i n F i g . 8. I n t h e Q A^ mode, x ( 0 A i ) h a s no component a l o n g i x s o ^=+45°. The p h a s e s o f t h e e i g e n v e c t o r s o f s u b l a t t i c e s 2, 3, a n d , 4 r e l a t i v e t o t h a t o f 1 a r e ex p { i ^ A l - ( r 2 - r 1 ) } = l ( I I I - 3 2 a ) e x p { i d A l M r 3 - r 1 ) } = e x p { i Q A 1 - ( r 4 - r 1 ) } =-1 ( i l l - 3 2 b ) M o l e c u l e s on s u b l a t t i c e s 1 and 2 h a v e t h e same o r i e n t a t i o n s a n d t h o s e o n 3 and 4 a r e r o t a t e d 90° w i t h r e s p e c t t o them s i n c e n 1 s i n 2 cf>1=n 2 S l n 2 <* >2 =~ n3 s i n 2 * 3 = ~ n 4 s i n 2*4 ( I H - 3 3 ) -»-The s u p e r l a t t i c e s f o r Q A 2 and Q A 3 a r e i d e n t i c a l t o t h a t -y o f Q A 1 e x c e p t t h a t t h e y a r e r o t a t e d by 120° and -120° r e s p e c t i v e l y . •j -y The e i g e n v a l u e s ^  ( Q A ^ ) ( i = l , 2 , 3 ) a r e a l l d e g e n e r a t e s o a n y l i n e a r c o m b i n a t i o n o f t h e i r e i g e n v e c t o r s w i l l g i v e t h e same f r e e e n e r g y . A p a r t i c u l a r l y i n t e r e s t i n g n o r m a l mode t h a t c a n be f o r m e d i n t h i s manner i s y 4 = - ( y 1 + y 2 + Y 3 ) / ^ " (111-34) T h e r e a r e s e v e n o t h e r e q u i v a l e n t modes c o r r e s p o n d i n g t o a l l com-b i n a t i o n s o f s i g n s i n f r o n t o f y x , y 2 , and y 3 . The o r d e r p a r a m e t e r s , n x and n^, f o r v a r i o u s n o r m a l modes a r e t a b u l a t e d i n T a b l e I I . 41 T a b l e I I . N o r m a l Modes v l v 2 v 3 v 4 0 v^/2 0 1 - 1/2 - 1/2 0 0 ^3/2 -^3/2 0 1 1/2 1/2 -2//3 0 ->/3/2 - / J / 2 1 -1 - 1/2 1/2 1/^3 0 v^/2 /3/2 - 1 -1 1/2 - 1/2 1//3 O r d e r p a r a m e t e r \ x % n2x % "4x F o r t h e y 4 mode, <f>-^  i s a r b i t r a r y and * 2=-45°, * 3 = - 1 6 5 ? * 4 = - 1 0 5 ° . ( I I I _ 3 5 ) T h i s i s t h e p i n w h e e l s t r u c t u r e o f F i g . 3 ( b ) . I n o r d e r t o s e e w h i c h mode i s f a v o u r e d , i t i s n e c e s s a r y t o c a l c u l a t e t h e f o u r t h o r d e r t e r m s o f F. T h e r e a r e no o d d o r d e r t e r m s s i n c e F ( u i l f . . . , - u i o c , . . . , u i 5 ) = F ( u i : L , . . . , u i ( X , . ..,u, 5) ( I I 1 - 3 6 ) f o r a l l To f o u r t h o r d e r F h a s t h e f o r m F / ( N r ) = F e g / ( N D + ( 1 / 2 ) A 1 ( $ A 1 ) ( y 1 2 + y 2 2 + y 3 2 ) + A 4 ( y ^ + y ^ + y ^ ) 2  + ^ ^ i S 2 ^ ! 2 ^ 2 ^ 2 ^ 2 ) ( I H - 3 7 ) A 1 , A 4 / and B 4 a r e a l l f u n c t i o n s o f t e m p e r a t u r e . The A 4 t e r m i s i s o t r o p i c i n t h e s p a c e o f ( y ^ r y 21 y 3 ) . I f B 4>0 f l u c t u a t i o n s a l o n g t h e s i x ( 1 , 0 , 0 ) d i r e c t i o n s ( t h e f a c e s o f a 4 2 c u b e ) a r e f a v o u r e d . T h e s e c o r r e s p o n d t o t h e h e r r i n g b o n e p h a s e . I f B 4 < 0 , f l u c t u a t i o n s a l o n g t h e e i g h t ( 1 , 1 , 1 ) d i r e c t i o n s ( t h e b o d y d i a g o n a l s o f a c u b e ) a r e f a v o u r e d and o r d e r i n g i n t o t h e p i n w h e e l p h a s e o c c u r s . The f r e e e n e r g y g i v e n by ( 1 1 1 - 3 7 ) i s s a i d t o h a v e c u b i c a n i s t r o p y . I f t h e f o u r t h o r d e r t e r m w e r e a l w a y s g r e a t e r t h a n z e r o , i . e . , 3 A 4 + >0, t h e n t h e r e w o u l d be a c o n t i n u o u s p h a s e t r a n s i t i o n a t a t e m p e r a t u r e T^ a t w h i c h A"*"(Q A 1) =0. The s e c o n d o r d e r t e r m w o u l d be z e r o a t T . However i f t h e f o u r t h n o r d e r t e r m i s n e g a t i v e a t T^, t h e t r a n s i t i o n w i l l be f i r s t o r d e r 1 a n d w i l l o c c u r a t a t e m p e r a t u r e T^ > T r | • T h i s i s shown i n F i g . 9 w h e r e , f o r s i m p l i c i t y , we w r i t e F a s a f u n c t i o n o f a s i n g l e o r d e r p a r a m e t e r y F = F Q ( T ) + a ( T ) y 2 + b ( T ) y 4 ( 1 1 1 - 3 8 ) and a ( T n ) = 0 . I f b ( T n ) >0, t h e s e c o n d a n d f o u r t h o r d e r t e r m s a r e b o t h p o s i t i v e f o r T>T n. B e l o w T^, a ( T ) becomes n e g a t i v e . S i n c e t h e s e c o n d o r d e r t e r m i s d o m i n a n t n e a r y=0, t h e v a l u e o f |y| w h i c h m i n i m i z e s F i n c r e a s e s c o n t i n u o u s l y f r o m y=0. I f b ( T )<0, n 1 1 F i s a minimum a t y=0 o n l y down t o T d e f i n e d by b ( T )=0. n n B e l o w T^ , t h e f o u r t h o r d e r t e r m becomes n e g a t i v e and F i s m i n i m i z e d f o r |y| + 0 0. T h i s i s c h a r a c t e r i s t i c o f a f i r s t o r d e r t r a n s i t i o n . I n r e a l i t y , F w o u l d c o n t a i n h i g h e r o r d e r t e r m s w h i c h w o u l d p r o d u c e a minimum a t a f i n i t e v a l u e o f y. I t i s a l s o p o s s i b l e t o h a v e o r d e r i n g when a mode i n v o l v i n g u. r a t h e r t h a n n • and aj-CT g o e s s o f t . The s o f t e s t mode h a s "q=0 and t h e e i g e n v a l u e i s d o u b l y d e g e n e r a t e w i t h 43 F i g . 9 The f r e e e n e r g y F-F as a f u n c t i o n o f t h e o r d e r p a r a m e t e r y g i v e n by (111-387 n e a r t h e t r a n s i t i o n t e m p e r a t u r e T c . Here, T>T C f o r t h e s o l i d l i n e , T=T C f o r t h e d a s h e d l i n e , and T<T C f o r t h e d o t t e d l i n e . At T , a=o. In ( a ) , b ( T n ) > 0 , T =T n, and t h e t r a n s i t i o n i s c o n t i n u o u s . I n ( b ) , b ( T j < 0 and b(T n')=0. In t h i s c a s e , T c=T n' and t h e t r a n s i t i o n i s f i r s t o r d e r . 44 e i g e n v e c t o r s x 1 ( 0 ) = ( 0 , 0 , 0 , 1 , 0 ) ( I I I - 3 9 a ) x 2 ( 0 ) = ( 0 , 0 , 0 , 0 , 1 ) ( I I I - 3 9 b ) The d e g e n e r a c y i m p l i e s t h a t F i s i n d e p e n d e n t o f x« T h i s i s t h e f e r r o r o t a t i o n a l p h a s e . The c o n t i n u o u s s ymmetry i n x i s v a l i d t o any o r d e r i n x a s m e n t i o n e d i n s e c t i o n 3.2. 3.4 I m p r o v e m e n t s t o Mean F i e l d T h e o r y The f a c t t h a t MFT n e g l e c t s f l u c t u a t i o n s , w h i c h become v e r y i m p o r t a n t n e a r t h e c r i t i c a l p o i n t , m i g h t be e x p e c t e d t o i n v a l i d a t e some o f t h e r e s u l t s d i s c u s s e d a b o v e . Some o f t h e p r o b l e m s t h a t may a r i s e w i l l be c o n s i d e r e d i n t h i s s e c t i o n . M o n te C a r l o C a l c u l a t i o n s I t h a s b e e n s t r e s s e d t h a t t h e mean f i e l d t r a n s i t i o n t e m p e r a t u r e i s e x p e c t e d t o be t o o h i g h s o t h e t e m p e r a t u r e s c a l e o f F i g . 6 c a n n o t be t a k e n t o o s e r i o u s l y . I n f a c t t h e t r a n s i t i o n t e m p e r a t u r e may go t o z e r o f o r some o r a l l o f t h e o r d e r e d p h a s e s e i t h e r b e c a u s e o f quantum z e r o p o i n t m o t i o n o r 2D f l u c t u a t i o n s . The o n l y o t h e r p r e d i c t i o n o f t h e t r a n s i t i o n t e m p e r a t u r e i s b a s e d o n t h e c l a s s i c a l M o nte C a r l o c a l c u l a t i o n s o f O'Shea and K l e i n ( 1 9 7 9 ) . F o r a r i g i d c u b i c l a t t i c e o f q u a d r u p o l e s , t h e y f o u n d a t r a n s i t i o n t e m p e r a t u r e o f T c (3D) = 1 1 . 5 r Q ( 3 D ) . They t h e n d e f i n e d a c o n s t a n t Y 3 by Y 3 = T C ( 3 D ) / 1 1 . 5 , u s i n g t h e e x p e r i m e n t a l l y o b s e r v e d t r a n s i t i o n t e m p e r a t u r e o f b u l k s o l i d h y d r o g e n . The c o n s t a n t Y 3 was s c a l e d w i t h R Q t o 4 5 g i v e t h e 2 D c o n s t a n t Y » w h i c h was . 1 3 K ( . 1 4 K) f o r H 2 ( D 2 ) i n t h e /3~ x 71 s o l i d p h a s e on g r a p h i t e . F o r V c / r Q = 0 , O'Shea a n d K l e i n f o u n d o r d e r i n g i n t o t h e p i n w h e e l s t a t e a t T c ( 2 D ) = 8 Y 2 = 1 . 0 K ( 1 . 1 K ) . F o r V c / r Q = - < » , t h e y f o u n d o r d e r i n g i n t o t h e h e r r i n g b o n e s t a t e a t T C ( 2 D ) ~ 1 2 Y 2 = 1 . 5 K ( 1 . 6 K ) . I n c o n t r a s t , MFT p r e d i c t s t h a t f o r V c / r Q = 0 , i n i t i a l o r d e r i n g i s i n t o t h e h e r r i n g b o n e p h a s e a t T Q ( 2 D ) = 6 . 1 r Q ( 2 D ) = 3 . 2 K ( 3 . 1 K ) . As T d e c r e a s e s , t h e r e i s a s e c o n d t r a n s i t i o n i n t o t h e p i n w h e e l s t a t e . F o r V c / r 0 = - ° ° r o r d e r i n g i s a l s o i n t o t h e h e r r i n g b o n e s t a t e a t a somewhat h i g h e r t e m p e r a -t u r e . The Monte C a r l o t r a n s i t i o n t e m p e r a t u r e s a r e a f a c t o r o f t h r e e b e l o w t h e mean f i e l d v a l u e s . H owever, e v e n t h e Monte C a r l o r e s u l t s a r e l i k e l y t o be t o o h i g h s i n c e r e n o r m a l i z a t i o n o f r i s e x p e c t e d t o be e v e n g r e a t e r i n 2 D t h a n i n 3 D . When m a k i n g c o m p a r i s o n s w i t h e x p e r i m e n t one m u s t a l s o b e a r i n m i n d t h a t t h e i n e v i t a b l e p r e s e n c e o f J = 0 i m p u r i t i e s w i l l f u r t h e r r e d u c e T c. 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 e c r i t i c a l o r d e r p a r a -m e t e r s h a v e b e e n o b t a i n e d i n a n o t h e r Monte C a r l o s t u d y ( M o u r i t s e n and B e r l i n s k y , 1 9 8 2 ) . A 2 D h e x a g o n a l l a t t i c e o f c l a s s i c a l q u a d r u p o l e s , w h i c h w e r e c o n s t r a i n e d t o l i e i n t h e p l a n e o f t h e l a t t i c e ( i . e . , V c / r o = - ° ° ) , was f o u n d t o u n d e r g o a f l u c t u a t i o n i n d u c e d f i r s t o r d e r t r a n s i t i o n i n t o t h e h e r r i n g b o n e p h a s e . S e c o n d O r d e r P e r t u r b a t i o n C o r r e c t i o n t o t h e MFT G r o u n d S t a t e The o r d e r e d p h a s e p r e d i c t e d a t T=0 by MFT w i l l n o t be t h e 4 6 t r u e g r o u n d s t a t e s b e c a u s e o f z e r o p o i n t l i b r a t i o n . I n a c c o r d -a n c e w i t h (111-12), w i t h i n t h e mean f i e l d a p p r o x i m a t i o n , t h e g r o u n d s t a t e wave f u n c t i o n * o f a s y s t e m o f N p a r t i c l e s c a n be w r i t t e n a s N v = n Y ° ( 0.) ° i = l 1 (111-40) A l l o f t h e m o l e c u l e s on e a c h s u b l a t t i c e a r e a l i g n e d i n t h e same d i r e c t i o n . By a n a l o g y t o a s y s t e m o f a n t i f e r r o m a g n e t i c s p i n s , t h i s i s known a s t h e N e e l s t a t e . I t c a n n o t be t h e t r u e g r o u n d s t a t e b e c a u s e H h a s n o n - z e r o m a t r i x e l e m e n t s b e t w e e n ^ 0 and s t a t e s ^ i° w h i c h a p a i r o f m o l e c u l e s i s e x c i t e d . M a t r i x e l e m e n t s i n v o l v i n g VQ and s t a t e s ¥ ^  i n w h i c h a s i n g l e m o l e c u l e i s e x c i t e d g i v e z e r o when one d o e s t h e sum o v e r a l l n e i g h b o u r s o f a m o l e c u l e . T h i s sum i s c o n t a i n e d i n ^ Eqq« I n a s y s t e m o f a n t i f e r r o m a g n e t i c s p i n s , t h e g r o u n d s t a t e i s a l s o f o r m e d by m i x i n g s t a t e s c o n t a i n i n g e x c i t e d p a i r s i n t o t h e N e e l s t a t e . The e n e r g y s h i f t A E a r i s i n g f r o m t h e a d m i x t u r e c a n be e s t i m a t e d f r o m s e c o n d o r d e r p e r t u r b a t i o n t h e o r y ( H a r r i s and B e r l i n s k y , 1979): AE= g| <Y0|w |Y >|2/E0 *2 EQQ 2 (111-41) w h e r e E 2 i s t h e e n e r g y o f t h e two e x c i t a t i o n s . The r e s u l t i s AE/E o=.06 w h e r e EQ=-30Nr/7 i s t h e e n e r g y o f t h e p i n w h e e l p h a s e a t V c=T=0. By c o m p a r i s o n AE/E q=.04 f o r b u l k s o l i d H 2 ( V a n K r a n e n d o n k , 1983) and AE/E q=.14 f o r a b c c s p i n 1/2 a n t i f e r r o m a g n e t w here E Q i s t h e N e e l s t a t e e n e r g y . F o r 47 h y d r o g e n t h e e f f e c t i s s m a l l e n o u g h t o be n e g l e c t e d . R e n o r m a l i z a t i o n G r o u p A n a l y s i s I n r e c e n t y e a r s t h e r e n o r m a l i z a t i o n g r o u p (RG) a p p r o a c h h a s a l l o w e d m a j o r a d v a n c e s i n t h e s t u d y o f c r i t i c a l phenomena. As y e t t h e r e i s v e r y l i t t l e RG work t h a t c a n be a p p l i e d t o t h e o r i e n t a t i o n a l o r d e r i n g o f a d s o r b e d h y d r o g e n . Q u a d r u p o l e s on a h e x a g o n a l l a t t i c e c a n be a p p r o x i m a t e d by a p l a n a r H e i s e n b e r g m o d e l w i t h c u b i c a n i s o t r o p y s i n c e b o t h a r e d e s c r i b e d by ( 1 1 1 - 3 7) t o f o u r t h o r d e r . I n t h e p l a n a r H e i s e n b e r g m o d e l t h e s p a t i a l d i m e n s i o n d i s 2 and t h e d i m e n s i o n o f t h e o r d e r p a r a -m e t e r n i s 3. The RG m e t h o d i s i n t e n d e d t o be a p p l i e d t o s y s t e m s n e a r t h e i r c r i t i c a l p o i n t . I t s b a s i c p r e m i s e i s t h a t l o n g r a n g e c o r r e l a t i o n s o f t h e f l u c t u a t i o n s n e a r t h e c r i t i c a l p o i n t a r e s o l e l y r e s p o n s i b l e f o r t h e s i n g u l a r b e h a v i o u r o f c e r t a i n o b s e r -v a b l e s s u c h a s t h e s p e c i f i c h e a t and s u s c e p t i b i l i t y . 4 As t h e c r i t i c a l p o i n t i s a p p r o a c h e d t h e c o r r e l a t i o n l e n g t h . U n d e r a RG t r a n s f o r m a t i o n t h e s y s t e m i s mapped t o one i n w h i c h K i s s h o r t e r i n o r d e r t o make t h e c o r r e l a t i o n s l e s s i m p o r t a n t . The f r e e e n e r g y e q u a t i o n ( 1 1 1 - 3 7 ) c o n t a i n s f o u r p a r a m e t e r s F e g f A 1 , A 4 t and B 4 . The RG t r a n s f o r m a t i o n i s c a r r i e d o u t i n t h e s p a c e o f t h e s e p a r a m e t e r s . S u c c e s s i v e i n f i n i t e s s i m a l RG t r a n s f o r m a t i o n s w i l l c a u s e t h e s y s t e m t o d e s c r i b e a p a t h i n p a r a m e t e r s p a c e . T h e r e a r e c e r t a i n p o i n t s i n p a r a m e t e r s p a c e , f i x e d p o i n t s , s u c h t h a t i f t h e RG t r a n s f o r m a t i o n i s a p p l i e d t o 48 them t h e s y s t e m i s c a r r i e d t o t h e same p o i n t . A t a f i x e d p o i n t , £ m u s t be e i t h e r z e r o o r i n f i n i t y . I f i t i s z e r o , t h e r e i s no l o n g r a n g e o r d e r and t h e s y s t e m i s e q u i v a l e n t t o one a t T= ro. T h e s e p o i n t s w i l l be i g n o r e d i n t h e f o l l o w i n g d i s c u s s i o n . The s e t o f a l l p o i n t s c o r r e s p o n d i n g t o £ = m f o r m t h e c r i t i c a l s u r f a c e . C o n s i d e r a s m a l l r e g i o n a r o u n d a f i x e d p o i n t . I f s u c c e s s -i v e RG t r a n s f o r m a t i o n s s t a r t i n g i n t h i s r e g i o n a l w a y s c a r r y t h e s y s t e m i n t o t h e f i x e d p o i n t , i . e . , t h e " f l o w " i s i n t o t h e f i x e d p o i n t , t h e n t h e f i x e d p o i n t c a n be t h o u g h t o f a s a v a l l e y . I f one i s a l w a y s c a r r i e d away, e x c e p t a t t h e f i x e d p o i n t i t s e l f , i t i s a p e a k . I f one i s c a r r i e d i n a l o n g some d i r e c t i o n s , b u t away a l o n g o t h e r s i t i s a s a d d l e p o i n t . A s t h e t e m p e r a t u r e i s v a r i e d t h e s y s t e m f o l l o w s a p a t h i n p a r a m e t e r s p a c e c a l l e d t h e p h y s i c a l l i n e . I f t h e s y s t e m h a s a c r i t i c a l p o i n t , t h e p h y s i c a l l i n e w i l l i n t e r s e c t t h e c r i t i c a l s u r f a c e . A p p l y i n g s u c c e s s i v e RG t r a n s f o r m a t i o n s t o t h e p o i n t o f i n t e r s e c t i o n , t h e s y s t e m w i l l f l o w d o w n h i l l and e v e n t u a l l y e i t h e r s e t t l e i n t o a v a l l e y o r s a d d l e p o i n t o r go o u t t o i n f i n i t y . Any p o i n t on t h e c r i t i c a l s u r f a c e a l w a y s r e m a i n s on t h a t s u r f a c e when one a p p l i e s a n RG t r a n s f o r m a t i o n b e c a u s e a l l p o i n t s o n t h e s u r f a c e h a v e K = 0 0. The f i x e d p o i n t t o w h i c h t h e s y s t e m f l o w s g o v e r n s i t s c r i t i c a l b e h a v i o u r . N o r m a l l y t h e r e a r e o n l y a few f i x e d p o i n t s t h a t a p p e a r i n t h e p a r a m e t e r s p a c e s o f many d i f f e r e n t t y p e s o f s y s t e m s . T h i s i s t h e b a s i s o f t h e c o n c e p t o f u n i v e r s a l i t y , a c c o r d i n g t o w h i c h 49 the c r i t i c a l exponents of a system depend only upon such general properties as the s p a t i a l dimension and the dimension of the order parameter but not on the detailed nature of the i n t e r -actions. Systems in the same u n i v e r s a l i t y class flow to the same fixed point. Cubic anisotropy i s discussed in Chapter 9 of Pfeuty and Toulouse (1977). There i s only one va l l e y fixed point. If the number of components of the order parameter n, i s less than a c r i t i c a l value n c ( d ) , then the va l l e y fixed point i s the iso t r o p i c fixed point. This i s a fixed point that would also occur i f B4=0 i n (111-37) so that the free energy depended symmetrically on the order parameters y-^ , y 2» and y-j. When n=nc the is o t r o p i c fixed point coincides with the cubic fixed point, which was a saddle point for n<nc. For n>nQ, the two fixed points exchange s t a b i l i t y : the cubic fixed point becomes a va l l e y and the i s o t r o p i c fixed point becomes a saddle. Unfortunately the dependence of n Q on d i s not known except at d=4 where nc=4. However Ketley and Wallace (1973) have expan-ded n_ (d) about d=4. Their expansion suggests that n_(d=2)= 2. For the planar Heisenberg model d=2 and n=3 so one expects n>nc. The cubic fixed point should govern the c r i t i c a l behaviour except for regions of parameter space where there are f i r s t order t r a n s i t i o n s so that (111-37) i s not v a l i d . The r e s u l t s of the RG analysis are that i f diagonal aniso-tropy i s favoured (B^O) there w i l l be a continuous t r a n s i t i o n into the pinwheel phase and the c r i t i c a l exponents w i l l be those 50 o f t h e c u b i c f i x e d p o i n t , w h i c h a r e I s i n g - l i k e i f d=2. I f f a c e a n i s o t r o p y i s f a v o u r e d ( B 4 > 0 ) , t h e r e w i l l be a f i r s t o r d e r t r a n s i t i o n i n t o t h e h e r r i n g b o n e p h a s e . The f i r s t r e s u l t i s i n a g r e e m e n t w i t h MFT b u t t h e l a t t e r i s n o t : t h e t r a n s i t i o n t o t h e h e r r i n g b o n e p h a s e was a l s o p r e d i c t e d t o be c o n t i n u o u s . However, t h e t r a n s i t i o n i n t o t h e h e r r i n g b o n e p h a s e was f o u n d t o be f i r s t o r d e r by c l a s s i c a l Monte C a r l o c a l c u l a t i o n s ( M o u r i t s e n and B e r l i n s k y , 1982) i n a g r e e m e n t w i t h t h e RG r e s u l t . T h i s i s a c a s e o f a t r a n s i t i o n t h a t i s d r i v e n t o be f i r s t o r d e r by f l u c t u a t i o n s . I f B^=0, t h e r e i s a t r i c r i t i c a l p o i n t , a p o i n t a t w h i c h t h e s y s t e m c h a n g e s f r o m a f i r s t o r d e r t o a c o n t i n u o u s t r a n s i t i o n . I n t h i s c a s e t h e c r i t i c a l e x p o n e n t s a r e g o v e r n e d by t h e i s o t r o p i c c r i t i c a l p o i n t , w h i c h i s a s a d d l e p o i n t . H i g h T e m p e r a t u r e E x p a n s i o n s Two m a i n p i e c e s o f i n f o r m a t i o n h a v e b e e n e x t r a c t e d f r o m t h e NMR m e a s u r e m e n t s d e s c r i b e d i n t h i s t h e s i s . The f i r s t i s t h e s t r u c t u r e o f t h e l o w t e m p e r a t u r e o r d e r e d s t a t e o f H 2« T h i s was d e r i v e d f r o m t h e s h a p e o f t h e .3K NMR s p e c t r u m . The s e c o n d i s t h e t e m p e r a t u r e d e p e n d e n c e o f a i n t h e p a r a r o t a t i o n a l s t a t e w h i c h a l l o w s one t o d e t e r m i n e V c and r . T h i s was o b t a i n e d f r o m t h e s p l i t t i n g s o f t h e H 2 and D 2 s p e c t r a f o r T>1 K. The way i n w h i c h t h e s e two p i e c e s o f i n f o r m a t i o n c a n be o b t a i n e d i s d e s c r i b e d i n C h a p t e r I V . W i t h i n MFT t h e t e m p e r a t u r e d e p e n d e n c e o f a i s g i v e n by 51 (11-11), (11-13), and (11-16). As one approaches the phase t r a n s i t i o n , MFT w i l l break down so one might achieve better r e s u l t s using a high temperature expansion of o . I n i t i a l l y Harris and Berlinsky (1979) expanded o to second order in g V c and S T . i n ( n - i r j ) (T) was defined to be proportional to o <3J^ Z -2>T where <>T indicates a thermal average at temperature T. At high temperatures <3J^ Z -2>^ can be expanded as < 3 J i z 2 " 2 > T = T r { ( 3 J i z 2 ~ 2 ) e x p ( " B H ) } (III-42a) Tr exp(-eH) = T r U 3 J . z 2 - 2 ) ( l - 3« + (1/2)3 2H 2 * ...)} ( I „ _ 4 2 b ) T r Q - 6H + (1/2)6 2H 2 + ...} = -B <(3J i z 2-2)rf> c o + (l/2)6 2< ( 3 J i z 2 - 2 ) H 2 >ro+ r.. (III-42c) since Tr (3J H„ 2-2) = Tr^=0. If one assumes a random con-X z centration c of J = l molecules then the EQQ interaction with a single J=l neighbour i s weighted by c. To second order in B with nearest neighbour interactions one finds a=BVc/3+B2{Vc 2/18-(3/2) cV^- (25/6) C r 2 } (111-43) MFT gives the same r e s u l t except that the r term i s omitted. More recently Harris and Berlinsky (1983) have presented a high temperature expansion that i s exact in gV c and t h i r d order in g r . It has the form a=A+Bgr + C e 2 r 2+D6 3 r 3 (IH-44) 5 2 c o e f f i c i e n t s , w h i c h a r e d e f i n e d i n A p p e n d i x A , a r e f u n c t i o n s t h e c o n c e n t r a t i o n c a n d 6 V _ . 53 N o t e s t o C h a p t e r I I I 1. A l i b r o n i s a t o r s i o n a l e x c i t a t i o n o f t h e m o l e c u l a r a x e s . 2. T h i s c o n d i t i o n m e a n s t h a t t h e s y s t e m w i l l s a t i s f y t h e " L a n d a u r u l e s " f o r a 2D s y s t e m . T h e s e r u l e s g i v e t h e c o n d i t i o n s u n d e r w h i c h a p h a s e t r a n s i t i o n c a n b e c o n t i n u o u s . They a r e d i s c u s s e d i n Domany e t a l (1978) a n d Mukamel a n d K r i n s k y (19 76).. 3. N o t e t h a t y_ i s n o t a n e f f e c t i v e q u a d r u p o l e c o u p l i n g c o n s t a n t . F o r a s y s t e m o f i n t e r a c t i n g q u a d r u p o l e s , t h e r a t i o o f t h e o r i e n t a t i o n a l o r d e r i n g t e m p e r a t u r e o f a c l a s s i c a l s y s t e m t o t h a t o f a q u a n t u m ( J = l ) s y s t e m i s 5/2 ( H a r r i s a n d B e r l i n s k y , 1 9 7 9 ) . T h u s , o n e c a n n o t s i m p l y c o m b i n e t h e c l a s s i c a l Monte C a r l o r e s u l t t h a t T ( 3 D ) / r Q ( 3 D ) = 1 1 . 5 w i t h t h e e x p e r i m e n t a l T f o r a q S a n t u m s y s t e m t o g e t a n e f f e c t i v e q u a d r u p S l e c o u p l i n g c o n s t a n t . 4. Good i n t r o d u c t i o n s t o t h e RG m e t h o d a r e g i v e n b y P f e u t y a n d T o u l o u s e (1977) a n d Ma ( 1 9 7 6 ) . 54 CHAPTER I V NMR AS A PROBE OF ORIENTATIONAL BEHAVIOUR 4.1 U n b r o a d e n e d NMR S p e c t r a The m a g n e t i c i n t e r a c t i o n s o f h y d r o g e n m o l e c u l e s w i l l now be d i s c u s s e d , m a i n l y f o l l o w i n g t h e t r e a t m e n t g i v e n i n R e i f and P u r c e l l ( 1 9 5 3 ) , H a r r i s ( 1 9 7 0 b ) , and H a r d y and B e r l i n s k y ( 1 9 7 3 ) . The n u c l e a r s p i n d e g r e e s o f f r e e d o m c o n s t i t u t e a n e g l i g i b l e p e r t u r b a t i o n on t h e m o l e c u l a r o r i e n t a t i o n a l d e g r e e s o f f r e e d o m s i n c e t h e i n t e r a c t i o n e n e r g i e s d i f f e r by a f a c t o r o f 10 . F o r a n i s o l a t e d h y d r o g e n m o l e c u l e c o n t a i n i n g n u c l e i w i t h s p i n s l 1 a n d i £ i n a m a g n e t i c f i e l d H Q a l o n g t h e z a x i s , t h e m a g n e t i c H a m i l t o n i a n i s h _ 1 H M = - a I z - b J z - c i - 3 + ( 5 / 2 i ) d M { ? 1 . ? 2 - 3 ( ? 1 . r ) ( t 2 - r ) > + ( 5 / 2 ) d Q { 3 ( i ^ r ) 2 + 3 ( i ^ r ) 2 - 2 i ( i + l ) } . ( I V - 1 ) w h e r e I = t 1 + i 2 i = I i 1 ! = | i 2 | = 1/2 (1) f o r H 2 ( D 2 ) r = u n i t v e c t o r a l o n g t h e i n t e r n u c l e a r a x i s The f i r s t and s e c o n d t e r m s a r e t h e n u c l e a r and r o t a t i o n a l Zeeman i n t e r a c t i o n s . The t h i r d t e r m i s t h e s p i n - r o t a t i o n i n t e r a c t i o n . The f o u r t h t e r m i s t h e i n t r a m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n . The f i f t h t e r m i s t h e i n t e r a c t i o n b e t w e e n t h e n u c l e a r q u a d r u p o l e moment and t h e l o c a l e l e c t r i c f i e l d g r a d i e n t . I t i s z e r o f o r H 2 b u t n o t D 2. F o r a f r e e m o l e c u l e t h e c o n s t a n t s a r e d e f i n e d and e v a l u a t e d ( w i t h H Q i n t e s l a ) i n T a b l e I I I . 55 T a b l e I I I H y p e r f i n e C o n s t a n t s f o r H 2 and D 2 ( H a r d y and B e r l i n s k y , 1973) C o n s t a n t D e f i n i t i o n V a l u e ( k H z ) H 2 D 2 a ( 2 y i / h ) H 0 42576 H Q 6535.7 H( c b ( U j / h J ) H 0 6717 H Q 3368 H Q ( 2 u i / h ) H * _ 113.8 8.773 d M ( 2 u 2 / 5 i h ) < r " 3 > Q ( f o r J = l ) 57.68 2.74 d Q ( 2 / 5 ) e Q N 0 2V/3 r 2 ) ( f o r J = l ) 0 22. 50 H e r e u ^  i s t h e m a g n e t i c moment o f e a c h n u c l e u s ; p j i s t h e m o l e c u l a r r o t a t i o n a l m a g n e t i c moment; H i s t h e m a g n e t i c f i e l d a t t h e n u c l e u s a r i s i n g f r o m m o l e c u l a r r o t a t i o n ; r i s t h e i n t e r -n u c l e a r d i s t a n c e and <> 0 r e p r e s e n t s an a v e r a g e o v e r t h e v i b r a t i o n a l a n d e l e c t r o n i c g r o u n d s t a t e ; e Q N i s t h e n u c l e a r q u a d r u p o l e moment; V i s t h e e l e c t r i c p o t e n t i a l a t t h e n u c l e u s . I n s e c t i o n 2.2 we a r g u e d t h a t t h e o r b i t a l a n g u l a r momentum w i l l be q u e n c h e d f o r o r i e n t a t i o n a l l y o r d e r e d h y d r o g e n . However a t h i g h t e m p e r a t u r e s t h a t a r g u m e n t may n o t be v a l i d b e c a u s e t h e it i j = +1 s t a t e s a r e d e g e n e r a t e . E v e n i f t h e g r o u n d s t a t e h a s mj=0, t h e r e w i l l be s i g n i f i c a n t p o p u l a t i o n o f t h e m j = j t l s t a t e s i f t h e c r y s t a l f i e l d s p l i t t i n g i s o f t h e o r d e r o f kT. I t i s n o t c l e a r t h a t <J> w o u l d be z e r o a n d i f t h a t i s n o t t h e c a s e t h e s p i n - r o t a t i o n i n t e r a c t i o n w o u l d s p l i t t h e NMR s p e c t r u m . When one c o n s i d e r s t h e d y n a m i c s h o w e v e r , i t i s f o u n d t h a t t h e s p i n - r o t a t i o n t e r m d o e s i n f a c t a v e r a g e t o z e r o . The t r a n s i t i o n 5 6 r a t e f „ , b e t w e e n t h e raT s t a t e s w i l l b e o f t h e o r d e r o f t h e l i b r o n b a n d w i d t h w h i c h i s ' v r / h . T h u s t h e NMR f r e q u e n c y w i l l b e m o d u l a t e d b y a n a m o u n t f D ^ c , d M , a n d d g a t t h e r a t e f m > > f D . O n e c a n s e e , f o r e x a m p l e f r o m t h e t h e o r y o f f r e q u e n c y m o d u l a t i o n , t h a t i f f m > > f D , n o s i d e b a n d s a r e p r o -d u c e d . T h e r e f o r e W M s h o u l d b e a v e r a g e d o v e r t h e i r i j s t a t e s a n d t h e s p i n - r o t a t i o n i n t e r a c t i o n c a n b e n e g l e c t e d . T h e a v e r a g e d H a m i l t o n i a n tfM c a n b e r e w r i t t e n u s i n g a s p h e r i c a l b a s i s a s h _ 1 H M = - a I z + ( 5 / 2 ) ( 4 T T / 5 ) 1 / 2 £ v C (112 ; p V ) Y / + V * ^ ' ) r , . 1 . 2 . , . 1 . 1 . . 2. 2v, x { d . . i I - d „ ( i i , . + i l ) } ( I V - 2 ) -> i i i * w h e r e <o. = (6 , p ) = t h e o r i e n t a t i o n o f r y , v = 0 , + l S i n c e a > > d M a n d d g i n a n NMR e x p e r i m e n t , w e o n l y n e e d t h e s e c u l a r c o m p o n e n t s o f t h e s e c o n d a n d t h i r d t e r m s i . e . , t h o s e 1 2 t h a t c o m m u t e w i t h I z = i Q + i Q ( A b r a g a m , 1 9 6 1 ) . O n l y t h e s e c o m p o n e n t s w i l l a f f e c t t h e NMR s p e c t r u m i n t h e r e g i o n n e a r t h e L a r m o r f r e q u e n c y v Q = a . T h e s e c u l a r p a r t o f t h e H a m i l t o n i a n i s h _ 1 H ^ = - a I z + ( 5 / 2 ) ( 4 T T / 5 ) 1 / 2 < Y 2 ° ( U ' ) > I C ( 1 1 2 ; y , - y ) ^ y ' s ' ^ Q ^ y ^ - y ^ y 2 1 - ^ 1 < I V " 3 > = - a I z + ( 5 / 4 ) < 3 c o s 2 e ' - l > ( - d M ( 3 i z 1 i z , 2 - i 1 . i 2 } + d n { 3 ( i „ 1 ) 2 + 3 ( i ^ 2 ) 2 - 2 i ( i + l ) } ) ( I V - 4 ) v2 Z Z Q u a n t i t i e s o f t h e f o r m 3 i " i 6 - w h e r e cx, 6 = 1 , 2 z z a r e t h e O t h c o m p o n e n t s o f s e c o n d r a n k s p h e r i c a l t e n s o r o p e r a t o r s 57 s o , a c c o r d i n g t o t h e W i g n e r - E c k a r t Theorem ( R o s e , 1 9 5 7 ) , t h e y a r e p r o p o r t i o n a l t o 3 I Z - 1 ( 1 + 1 ) . E v a l u a t i n g t h e m a t r i x e l e m e n t s f o r 1=1, one f i n d s / 2 31 -2 i f o^g 3 i c x i 6 - = < Z 7. 1 (IV-5) , - (31 2 - 2 ) if O<=B Thus f o r 1=1 t h e d i p o l a r and q u a d r u p o l a r t e r m s have t h e same fo r m and t h e s e c u l a r H a m i l t o n i a n becomes h _ 1 H M S = - a I 2 - (5/4)d<3 c o s 2 6 ' - l > { 3 I z 2 - 2 } (IV-6) where d=d M+dg. The r e s u l t i n g s p l i t t i n g o f t h e NMR s p e c t r u m i s d e p i c t e d i n F i g . 10. F o r t h e s u b s e t o f m o l e c u l e s whose i n t e r n u c l e a r a x e s make an a n g l e 6 w i t h r e s p e c t t o H Q , t h e r e w i l l be a p a i r o f NMR f r e q u e n c i e s v = v o 1 ( 1 5 / 4 ) d < 3 c o s 2 6 ' - l > (IV-7) Now, g i v e n t h a t t h e o r d e r p a r a m e t e r s d e s c r i b e t h e o r i e n t a t i o n s o f t h e m o l e c u l e s w i t h r e s p e c t t o t h e s u b s t r a t e and n o t t h e d i r e c t i o n o f H Q, i t i s c o n v e n i e n t t o t r a n s f o r m t o a r e f e r e n c e frame i n w h i c h t h e z a x i s i s n o r m a l t o t h e s u b s t r a t e ( F i g . 1 1 ) . T a k i n g Z and Z = (6 ,6) t o be t h e o r i e n t a t i o n s o f t h e i n t e r -n u c l e a r a x i s and H Q i n t h e s u b s t r a t e f r a m e , t h e S p h e r i c a l H a r monic A d d i t i o n Theorem ( R o s e , 1957) g i v e s < 3 c o s 2 6 ' - l > = ( 8 7 r / 5 ) ZY ™(u.) <Y ™* (Z )> ( I V - 8 ) in £• £• 3, F i r s t c o n s i d e r t h e p a r a r o t a t i o n a l s t a t e . Each m o l e c u l e i s i n a s i t e w h i c h has s i x - f o l d symmetry a b o u t t h e z a x i s . O n l y t e r m s t h a t have t h i s symmetry ( t h o s e w i t h m=0) w i l l be n o n - z e r o / Si 1=1 nnI-0 ld<3cos 2e'-f> V + V -a -Td<3cos20'-1> Zeeman Intramolecular Int eract ion Interact ions F i g . 10 Magnetic energy of a J=l hydrogen molecule i n a magnetic f i e l d . The i n t r a m o l e c u l a r i n t e r a c t i o n s are d i p o l a r f o r H 2 and d i p o l a r p l u s quadrupolar f o r 5 D 2« The NMR f r e q u e n c i e s are v Q+v + and v 0 + v _ -F i g . 11 O r i e n t a t i o n s of 'ft0 and the i n t e r n u c l e a r a x i s with r e s p e c t t o the c r y s t a l r e f e r e n c e frame. z i s normal to the s u r f a c e . 59 i n ( I V - 8 ) . The s h i f t s i n t h e NMR f r e q u e n c i e s r e l a t i v e t o v o a r e t h e n v + = + ( 3 / 2 ) d o ( 3 c o s 2 g - l ) ( I V - 9 ) w h e r e a i s t h e o r i e n t a t i o n a l o r d e r p a r a m e t e r d e f i n e d i n ( 1 1 - 1 0 ) . The NMR s p e c t r u m c o n s i s t s o f a d o u b l e t w h i c h h a s a s p l i t t i n g d i r e c t l y p r o p o r t i o n a l t o |a| . From t h e 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 e s p l i t t i n g one c a n o b t a i n V c and F w i t h i n t h e mean f i e l d a p p r o x i m a t i o n u s i n g ( 1 1 - 1 1 ) , ( 1 1 - 1 3 ) , a n d ( 1 1 - 1 6 ) . I n t h e o r d e r e d s t a t e s t h e s i x - f o l d s y m m e t r y w i l l i n g e n e r a l be b r o k e n by t h e EQQ i n t e r a c t i o n . U s i n g t h e o r d e r p a r a -m e t e r s d e f i n e d i n ( I I I - 5 ) , t h e s h i f t s o f t h e NMR f r e q u e n c i e s n o r m a l i z e d t o t h e i r maximum v a l u e s a r e v + / 3 d = + ( 4 / 5 T r ) 1 / 2 [ a Y 2 ° ( e r 6 ) - ( 3/2) 1 / 2 {y -Y 2 1 ( B , 5 ) - y + Y 2 _ 1 ($ , 6 ) } + ( l / 6 ) 1 / 2 { r f Y 2 2 ( 6 , 6 ) + n + Y 2 " 2 ( B , 6 ) }] ( I V - l O a ) 2 2 =+ (1/2) {a ( 3 c o s e-1) + 6 y s i n 6 c o s 6 c o s (6-x) + n s i n B c o s 2 (5-<|>) } ( I V - l O b ) On a s i n g l e c r y s t a l s u b s t r a t e , e a c h s u b l a t t i c e o f a n o r i e n t a t i o n a l l y o r d e r e d s t a t e w o u l d p r o d u c e a p a i r o f d e l t a f u n c t i o n NMR l i n e s i n t h e a b s e n c e o f b r o a d e n i n g . F o r G r a f o i l t h e a d s o r b i n g s u r f a c e s a r e p r i m a r i l y b a s a l p l a n e s . The c r y s t a l x a x e s a r e o r i e n t e d u n i f o r m l y t h r o u g h t h e r a n g e ( 0 , 2 i r ) . T y p i c a l l y a b o u t 30% o f t h e c r y s t a l s h a v e t h e i r z a x e s i n a n a n g u l a r d i s t r i b u t i o n w i t h a f u l l w i d t h a t h a l f maximum o f 30° w i t h r e s p e c t t o t h e n o r m a l t o t h e f o i l and t h e r e m a i n i n g 7 0 % a r e u n i f o r m l y o r i e n t e d t h r o u g h t h e r a n g e ( 0 , 2^) ( B i r g e n e a u e t . a l , 1 9 8 2 ) . 60 I n C h a p t e r V I I I t h e .3 K e x p e r i m e n t a l s p e c t r a o f H 2 a r e c o m p a r e d w i t h t h e T=0 s y n t h e t i c s p e c t r a e x p e c t e d f o r v a r i o u s o r d e r e d s t a t e s . T h e m i s o r i e n t a t i o n o f t h e G r a f o i l h a s b e e n t a k e n i n t o a c c o u n t . T h e i m p o r t a n t f e a t u r e s o f t h e s p e c t r a c a n b e s e e n b y a s s u m i n g t h a t a l l o f t h e z a x e s a r e p a r a l l e l b u t t h a t t h e x a x e s h a v e a u n i f o r m a n g u l a r d i s t r i b u t i o n o v e r t h e r a n g e ( 0 , 2 i r ) . T h i s d i s t r i b u t i o n o f c r y s t a l l i t e s w i l l b e c a l l e d a "2D p o w d e r . " A t T=0 a l l o f t h e m o l e c u l e s o n e a c h s u b l a t t i c e w i l l b e a l i g n e d a l o n g s o m e a x i s . T h e o r d e r p a r a m e t e r s a r e e x p r e s s e d i n t e r m s o f t h e o r i e n t a t i o n o f t h a t a x i s b y ( 1 1 1 - 1 3 ) . S u b s t i t u t i n g 6 ^ = 0 ° , 90° f o r m o l e c u l e s t h a t a r e s t a n d i n g u p o r l y i n g d o w n g i v e s t h e T=0 o r d e r p a r a m e t e r s a ^, y ^ , a n d n ^ f o r t h e p i n w h e e l a n d h e r r i n g b o n e p h a s e s . T h e s e a r e t a b u l a t e d i n T a b l e I V . F o r t h e 2 - o u t a n d f e r r o r o t a t i o n a l p h a s e s , d e p e n d s o n V c / r b u t a Q , y Q , a n d n Q w i l l a l l b e n o n - z e r o . T h e e x p e c t e d T=0 NMR s p e c t r a f o r t h e 2D p o w d e r a r e s h o w n i n F i g . . o 12 f o r t h e c a s e s w i t h H 0 n o r m a l t o t h e s u b s t r a t e (3 =0 ) a n d p a r a l l e l t o t h e s u b s t r a t e ( 3 = 9 0 ° ) . T a b l e I V O r d e r P a r a m e t e r s f o r t h e H e r r i n g b o n e a n d P i n w h e e l P h a s e s a t T=0  P h a s e S u b l a t t i c e 6 a y n <f> o 0 0 H e r r i n g b o n e 1,2 90° -1/2 0 3/2 +45° P i n w h e e l 1 0° 1 0 0 -P i n w h e e l 2,3,4 90° -1/2 0 3/2 - 4 5 ° , -16 5°, -105° (a) —j | | | I -1 -5 0 .5 1 ( v - v 0 ) / 3 d ( b ) 1 \ ^ 1 /' / 1 1 1 1 1 i I I i r --1 -.5 O .5 1 F i g . 12 U n b r o a d e n e d T=0 NMR s p e c t r a f o r a 2D powder. The s o l i d l i n e s a r e p r o d u c e d by m o l e c u l e s w i t h t h e i r a x e s n o r m a l t h e s u r f a c e and t h e d a s h e d l i n e s by t h o s e p a r a l l e l t o t h e s u r f a c e . The h e r r i n g b o n e p h a s e w o u l d have o n l y t h e d a s h e d s p e c t r a . The p i n w h e e l p h a s e w o u l d h a v e t h e d a s h e d and s o l i d s p e c t r a i n a 3:1 i n t e n s i t y r a t i o . I n ( a ) "H0 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 and i n (b) i t i s p a r a l l e l . For 3 = 0 ° o n l y the f i r s t term o f (IV-lOb) i s non-zero so the up molecules and the down molecules each g i v e a p a i r o f l i n e s with a s p l i t t i n g t h a t i s p r o p o r t i o n a l t o |a| . The e x p r e s s i o n f o r v i s the same as i n the p a r a r o t a t i o n a l s t a t e . Thus f o r 6=0° the d i s t i n c t i o n between the s p e c t r a o f the herringbone and the p i n -wheel phases i s very c l e a r . The former c o n t a i n s o n l y two l i n e s w i t h a s p l i t t i n g o f Av/3d=l. The l a t t e r c o n t a i n s two p a i r s o f l i n e s with s p l i t t i n g s of Av/3d=l and Av/3d=2. The i n n e r l i n e s have three times the i n t e n s i t y o f the out e r ones. Now c o n s i d e r the case of 8=90° ( H Q i s i n the plane of the s u b s t r a t e ) . The molecules s t a n d i n g up have y=n=0 so f o r 6=90° the f i r s t term of (IV-lOb) i s s t i l l the onl y non-zero one. For the molecules l y i n g down, n i s a l s o non-zero so the t h i r d term a l s o c o n t r i b u t e s . I t c o n t a i n s the azimuthal angle 6 which must be averaged over a l l x a x i s o r i e n t a t i o n s . The spectrum i s thus spread over the e n t i r e range of ( v - v Q ) / 3 d from -1 t o 1. The o n l y d i f f e r e n c e between the herringbone and pinwheel s p e c t r a i s t h a t the inn e r peaks would be s t r o n g e r f o r the l a t t e r . In the 2-out and f e r r o r o t a t i o n a l phases u i s non-zero but i t i s m u l t i p l i e d by s i n 3 cos3, which i s zero f o r g = 0 o r 90°. The magnitude of the s p l i t t i n g w i l l depend upon V c / r but otherwise the 3=0 and 90° NMR s p e c t r a w i l l be very s i m i l a r t o those of the herringbone phase. The major source o f broadening o f the NMR s p e c t r a i s the o r i e n t a t i o n a l d i s t r i b u t i o n o f the b a s a l p l a n e s . I f H Q i s para-l l e l o r normal t o the G r a f o i l , o n l y the i n s i d e edges o f the 63 p e a k s a r e b r o a d e n e d b e c a u s e i n t h e s e two c a s e s v + and v _ h a v e e i t h e r t h e i r maximum o r minimum v a l u e s i f t h e r e i s no m i s a l i g n m e n t o f t h e z a x e s . NMR s p e c t r a f o r a s i n g l e c r y s t a l , a 3D powder and G r a f o i l a r e c o m p a r e d i n F i g . 13. N e x t a r e t h e i n t e r m o l e c u l a r 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 c a u s e e a c h m o l e -c u l e t o e x p e r i e n c e a l o c a l m a g n e t i c f i e l d o f t h e o r d e r o f 1 0 ~ 4 T . F i n a l l y , t h e s u b s t r a t e i t s e l f p r o d u c e d l o c a l m a g n e t i c f i e l d s due t o i t s a n i s o t r o p i c d i a m a g n e t i s m . T h e s e m e c h a n i s m s w i l l be d i s c u s s e d i n t h e n e x t t h r e e s e c t i o n s . 4.2 C r y s t a l l i t e O r i e n t a t i o n a l D i s t r i b u t i o n S y n t h e t i c NMR s p e c t r a w h i c h i n c l u d e t h e raisorientation o f t h e g r a p h i t e c r y s t a l s h a v e b e e n g e n e r a t e d . I n ( i V - l O a ) t h e NMR f r e q u e n c i e s a r e e x p r e s s e d i n t e r m s o f t h e s p h e r i c a l h a r m o n i c s Y 2 m ( " ) w h e r e w = ( 3 , < 5 ) i s t h e o r i e n t a t i o n o f H Q i n t h e c r y s t a l ( x y z ) r e f e r e n c e f r a m e . D e f i n e ( a , r , , y ) t o be t h e E u l e r r o t a t i o n w h i c h c a r r i e s t h e XYZ a x e s f i x e d i n t h e f o i l i n t o t h e x y z a x e s . The Z a x i s i s p a r a l l e l t o t h e n o r m a l o f t h e f o i l and t h e o r i e n t a t i o n o f t h e X a x i s i s a r b i t r a r y f o r t h e t i m e b e i n g . U s i n g t h e c o n v e n t i o n o f Rose ( 1 9 5 7 ) , t h e E u l e r r o t a t i o n i s d e f i n e d by t h e f o l l o w i n g o p e r a t i o n s . 1) R o t a t i o n by a a b o u t Z, XYZ -t-X'Y'Z' 2) R o t a t i o n by t a b o u t Y' , X'Y'Z' ->X"Y"Z" 3) R o t a t i o n by y a b o u t Z" , X"Y"Z" ->-x y z The o r i e n t a t i o n s o f ~tiQ and z i n t h e XYZ f r a m e a r e n = (F ,6 ) and n'=(B',6') a s shown i n F i g u r e 14. 64 1 1 1 1 1 \ : \ : \ ! \ : \ j 1 1 1 1 \ : / •: / /= / - - - - - - - - - • - - . - - . - 1 - - - . - - . . . . . . . . 1 1 1 -.5 0 .5 1 ( v - i * ) / 3 d F i g . 13 B r o a d e n i n g o f t h e NMR s p e c t r a due t o t h e o r i e n t a t i o n a l d i s t r i b u t i o n o f t h e a d s o r b i n g s u r f a c e s . The h y d r o g e n m o l e c u l e s a r e t a k e n t o be s t a n d i n g up on t h e s u r f a c e . The s o l i d l i n e i s f o r a s i n g l e c r y s t a l s u b s t r a t e w i t h "ft n o r m a l t o t h e s u r f a c e . The d a s h e d l i n e i s f o r G r a f o i l w i t h ftQ n o r m a l t o t h e f o i l . The d o t t e d l i n e i s f o r a powder s u b s t r a t e . The d a s h e d and d o t t e d c u r v e s a r e n o r m a l i z e d t o t h e same a r e a . 65 F i g . 14 O r i e n t a t i o n o f H Q w i t h r e s p e c t t o t h e c r y s t a l a x e s ( x y z ) and t h e f o i l a x e s ( X Y Z ) . The z and Z a x e s a r e n o r m a l t o t h e c r y s t a l b a s a l p l a n e and t h e G r a f o i l s h e e t r e s p e c t i v e l y . 66 I n o r d e r t o t a k e i n t o a c c o u n t t h e o r i e n t a t i o n a l d i s t r i b u -t i o n o f t h e a d s o r b i n g s u r f a c e s , we w i l l f i r s t t r a n s f o r m t h e e x p r e s s i o n f o r t h e NMR f r e q u e n c i e s f r o m t h e x y z f r a m e t o t h e XYZ f r a m e u s i n g t h e r o t a t i o n a l t r a n s f o r m a t i o n p r o p e r t i e s o f t h e s p h e r i c a l h a r m o n i c s . The p r o b a b i l i t y d i s t r i b u t i o n P' (a , o f t h e E u l e r r o t a t i o n s d e p e n d s upon t h e G r a f o i l o r i e n t a t i o n a l d i s t r i b u t i o n . The n e x t s t e p i s t o o b t a i n t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e NMR f r e q u e n c i e s p ( v ) . T h i s was done n u m e r i -c a l l y u s i n g an Amdahl 470 V/8 c o m p u t e r . The r a n g e s o f t h e E u l e r a n g l e s were d i v i d e d i n t o i n t e r v a l s and f o r e a c h s e t o f i n t e r v a l s P ' ( o t / ? f Y ) and v were d e t e r m i n e d . The r a n g e o f v was d i v i d e d i n t o i n t e r v a l s and a f t e r g o i n g t h r o u g h a l l o f t h e E u l e r a n g l e s p ( v ) was d e t e r m i n e d by w e i g h t i n g e a c h v a l u e v ( a , C , Y ) by P ' ( a f C f Y ) . T h i s i s t h e NMR s p e c t r u m . The s p h e r i c a l h a r m o n i c s i n ( I V - l O a ) t r a n s f o r m u n d e r t h e E u l e r r o t a t i o n s by Y 2 m ( J ) = I Y 2 m ( £ ) D 2 m ( a , c , Y ) ( I V - 1 1 ) o w h e r e D (a , £,Y ) i s a r o t a t i o n m a t r i x ( A p p e n d i x A o f Van K r a n e n d o n k , 1 9 8 3 ) . I n s p e c t i o n o f F i g . 14 shows t h a t t h e t r a n s -f o r m a t i o n c a n be made by m a k i n g a = 6 1 , C=S 1> and y a r b i t r a r y . A p p l y i n g ( I V - 1 1 ) t o ( I V - l O a ) g i v e s Y 2 2 ( J ) = ( l / 2 ) e " i 2 y { 2 c o s 4 ( r / 2 ) e ~ i 2 a Y 2 2 (ft) + s i n ? ( l + c o s c ) e ~ l a Y 2 1 (ft) + ( 3 / 2 ) * / 2 s i n 2 ? Y 2 ° ( n ) + s i n ^ ( l - c o s C ) e l a Y 2 " 1 ( n ) + 2 s i n 4 U / 2 ) e l 2 a Y 2 " 2 ( f t ) } ( I V - 1 2 a ) Y 2 1(w) = (l/2)e" 1 Y {-sin£ (i+coss)e~ l 2 aY 2 2(^) 6 7 + (l+cos c) (2cos?-l)e~ l aY 2 1(^) + 6 1/ 2sin?cosCY 2 0(n) + (1-cos.r.) (2cos?+l)e i aY 2 _ 1(n) + sine(l-aosS)el2aY2~2$)} (IV-12b) Y°ti)={l/2) { ( 3 / 2 ) 1 / 2 s i n 2 ? e " i 2 a Y 2 2 ( ^ ) - 6 1 / 2 s i n ? c o s neT%H21 (ft -. + (3cos e-l)Y2°(fi) + 6 1 / 2sinecosee i aY 2 _ 1(Jt) + ( 3 / 2 ) 1 / 2 s i n 2 ? e i 2 a Y 2 " 2 ( n ) ) (IV-12c) Y 2 _ 1 ( S ) =-Y 2 1*(5) (IV-12'd) Y 2 " 2 ( 5 ) = Y 2 2 * ( S ) (IV-12e) The orientation o f the X axis can be chosen such that the projection o f HQ onto the XY plane is along X ie, 5=0°. With this choice, substitution o f ( IV-12) into ( IV-lOa) gives v+/3d=+(l/2) {(a/2) ((3cos 2B-l) (3cos 2?-l) + 3sin23sin2S c o s c t + 3sin 2Bsin 2?cos2a) - 3M ((1/2) (3cos23-1) sin2C cos ( x + Y ) - sin23 {cos2?cos (X+Y) c o s c t - cosSsin(x+Y)sina } 2— - sin 3sin?{cosC cos ( x+Y) c o s 2 c t - sin (x+Y)sin2a}) + n((l/2) (3cos 23-l)sin 2Ccos2 (<f>+Y) - sin23sin£ x {cos ?cos2 (i>+y) cosa - sin2 (<)>+Y) since } 2~ 2 + sin 3 {(1/2) (cos X, +1) cos2 (<!>+Y) cos 2 a - cos ?sin2 (<J>+Y) x sin2a}) } (IV-13) A l l o f the NMR measurements were made with 3=0° or 3=90°. The results for these cases are given below. 1) 3 = o ° : v+/3d=+ (1/2) { 0 (3cos 2 ?-l) - 3ysin2; cos ( x+Y) + nsin 2? cos2 (<f>+Y) } (IV-14) 2) 3 =90° : v+/3d=+(l/2){(a/2)(-(3cos2C-l) + 3sin\ cos2a ) + 3y (sin2 Scos ( x + Y ) 6 8 2 2 * c o s a - s i n ^ s i n (x+y) s i n 2 a ) + n ( - ( l / 2 ) s i n £COS(<|>+Y) 2 + (1/2) ( c o s c+D c o s 2 (<f)+Y ) c o s 2 a - c o s ? s i n 2 (<}>+Y ) s i n 2 a ) } ( I V - 1 5 ) H a v i n g t r a n s f o r m e d t h e e x p r e s s i o n f o r v + t o t h e f o i l r e f e r e n c e f r a m e , we n o t e t h a t P ' ( C X , ? , Y ) a c t u a l l y d e p e n d s o n l y on S: t h e o r i e n t a t i o n a l d i s t r i b u t i o n o f t h e G r a f o i l a d s o r b i n g p l a n e s i s c h a r a c t e r i z e d by t h e a n g l e 3 =C b e t w e e n t h e z and Z a x e s . D e f i n e P ( S ) d n t o be t h e p r o b a b i l i t y t h a t z l i e s i n t h e s o l i d a n g l e dfi= s i n ^ d c d a . The E u l e r a n g l e y d o e s n o t a f f e c t t h e o r i e n t a t i o n o f t h e z a x i s s o P ( C ) i n v o l v e s an i n t e g r a t i o n o v e r a l l a n g l e s Y , g i v i n g P ( C ) d $ = 6Q / ^ P ' ( a »Y) d Y ( I V - 1 6 ) P' (a»C »Y> = P ( t ) / 2 u ( I V - 1 7 ) I f 3 = 0 ° , t h e n m o s t c r y s t a l s w o u l d have 3-0° and t h e f i r s t t e r m o f (IV=10 b ) w o u l d d o m i n a t e ; t h e p i n w h e e l and h e r r i n g b o n e s p e c t r a w o u l d r e s e m b l e F i g . 1 2 ( a ) s o t h e y w o u l d e a s i l y be d i s t i n g u i s h e d . I n t h e p a r a r o t a t i o n a l s t a t e , t h e f i r s t t e r m o f ( I V - l O b ) i s t h e o n l y n o n - z e r o t e r m r e g a r d l e s s o f t h e o r i e n t a t i o n o f H 0• The s p e c t r u m w o u l d a l w a y s c o n s i s t o f a d o u b l e t b u t i t s s p l i t t i n g w o u l d be t w i c e a s g r e a t f o r 6=0° a s f o r ^ = 9 0 ° . H o w e v e r , i t i s n o t o b v i o u s f r o m ( I V - 1 4 ) a n d ( I V - 1 5 ) w h e t h e r t h e b r o a d e n i n g i n t h e f o r m e r c a s e w o u l d be more o r l e s s t h a n t w i c e a s g r e a t a s i n t h e l a t t e r c a s e . I f 3 = 0 ° , t h e n a n y m i s a l i g n m e n t o f t h e z a x e s w i l l c h a n g e 3 . On t h e o t h e r hand i f 3 = 9 0 ° , r o t a t i o n s o f t h e z a x i s a b o u t H Q , i e . a =+90° a nd c,^0°, w o u l d n o t c h a n g e 3 s o t h e y w o u l d n o t s h i f t t h e NMR f r e q u e n c i e s . T h i s m i g h t l e a d one t o t h i n k t h a t t h e b r o a d e n i n g f o r 3 =90° w o u l d be 69 l e s s t h a n t w i c e t h a t f o r 3=0°. However f o r 3=90°, i t i s a l s o n e c e s s a r y t o a v e r a g e o v e r a l l a n g l e s a a s ( I V - 1 5 ) shows. E x p e r i -m e n t a l l y we h a v e f o u n d t h e s p e c t r a f o r 3=0° t o be b e t t e r r e s o l v e d t h a n t h o s e f o r 3=90° i n t h e p a r a r o t a t i o n a l s t a t e s o f o r o u r G r a f o i l s a m p l e t h e s e c o n d m i s o r i e n t a t i o n a l e f f e c t d o m i n a t e s . ^ 4.3 I n t e r m o l e c u l a r D i p o l a r B r o a d e n i n g The d i p o l e - d i p o l e i n t e r a c t i o n b e t w e e n t h e n u c l e a r s p i n s o f n e i g h b o u r i n g m o l e c u l e s r e s u l t s i n b r o a d e n i n g o f t h e NMR l i n e . I n t h i s s e c t i o n t h e b r o a d e n i n g w i l l be c a l c u l a t e d a s s u m i n g t h a t t h e l a t t i c e i s r i g i d . The i n t e r a c t i o n c a n h a v e two e f f e c t s . I n t h e p r e s e n c e o f a l a r g e m a g n e t i c f i e l d "JH0, an i s o l a t e d m a g n e t i c moment y-^  w i l l p r e c e s s a b o u t H Q a t t h e L a r r a o r f r e q u e n c y t o 0 g i v e n by oi = Y-.H ( I V - 1 8 ) o ' 1 o w h e r e Y^ i s t h e g y r o m a g n e t i c r a t i o . I f we i n t r o d u c e a s e c o n d moment y 2 , t h e n a t t h e s i t e o f y^ i t i n d u c e s a s t a t i c f i e l d h g p a r a l l e l t o H Q and a f i e l d h r t r a n s v e r s e t o H 0 , w h i c h i s r o t a t i n g a t a f r e q u e n c y w 2 d e t e r m i n e d by i t s own l o c a l f i e l d . To f i r s t o r d e r t h e Zeeman l e v e l s o f y-j. a r e s h i f t e d o n l y by h_. T h i s i s t h e f i r s t e f f e c t . I f t h e two moments a r e o f t h e same t y p e , i . e . Y 1=Y2 =Y» t h e n h r w i l l be r o t a t i n g n e a r t h e r e s o n a n c e f r e q u e n c y to ^  o f t h e f i r s t moment. T h i s p r o d u c e s t r a n s i t i o n s b e t w e e n t h e Zeeman l e v e l s w h i c h c a u s e p ^ t o t i p away f r o m H Q , t h e same e f f e c t a s t h e a p p l i e d t r a n s v e r s e RF f i e l d u s e d i n NMR. I n o r d e r f o r t h e s e c o n d m e c h a n i s m t o be e f f e c t i v e i t i s n e c e s s a r y t h a t ( p . 2 1 o f Abragam, 1961) h r ^ l u 2 " u l l ( I V - 1 9 ) The w i d t h o f t h e r e s o n a n c e i s h r w h i c h i s o f t h e same o r d e r o f m a g n i t u d e a s t h e s t a t i c s h i f t h g . N o r m a l l y f o r a 3D s o l i d , t h e NMR l i n e s h a p e p r o d u c e d b y i n t e r m o l e c u l a r d i p o l a r b r o a d e n i n g c a n be a p p r o x i m a t e d f a i r l y w e l l by a G a u s s i a n f u n c t i o n w i t h a s t a n d a r d d e v i a t i o n a G w h i c h i s e q u a l t o t h e s e c o n d moment M 2 i w h i c h i s d e f i n e d by 2 M 9= / ( v - v ) f ( v ) d v ( I V - 2 0 ) w here f ( v) i s t h e n o r m a l i z e d NMR l i n e s h a p e . I n t h e r i g i d l a t t i c e a p p r o x i m a t i o n , t h e c a l c u l a t i o n o f M 2 i s s t r a i g h t -f o r w a r d i f t h e l a t t i c e s t r u c t u r e i s known. I n o r d e r t o f i t t h e e x p e r i m e n t a l l i n e s h a p e we h a v e assumed t h a t t h e i n t e r m o l e c u l a r b r o a d e n i n g p r o d u c e s a G a u s s i a n l i n e s h a p e . T h i s s o u r c e o f b r o a d e n i n g i s s m a l l s o t h i s a s s u m p t i o n i s a c c e p t a b l e . The e x p r e s s i o n f o r M 2 i s o b t a i n e d f o l l o w i n g t h e t r e a t -m ent i n C h a p t e r I V o f Abragam ( 1 9 6 1 ) . The H a m i l t o n i a n tfd f o r t h e i n t e r m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n i s ^ * ( f l 2 Y 2 / r j k 3 ) { ? i - f k " 3 ( f j - V ( t k - V J ( I V " 2 1 ) w h e r e r j k = v e c t o r b e t w e e n t h e j t h and k t h m o l e c u l e s r j k = ? j k / r j k A s i n t h e c a s e o f t h e i n t r a m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n d i s c u s s e d i n s e c t i o n 4.1, we w i l l o n l y k e e p t h e s e c u l a r t e r m s . A s s u m i n g t h a t H Q i s a l o n g t h e z a x i s , t h e s e a r e H d S = A ^ ^ ^ j k ^ ^ - ^ ^ j k ^ ^ ^ z 1 " " d / 4 ) ( I + j I _ k + I _ V ) l ( I V - 2 2 ) 71 = i Z < k ( Y 2 f t 2 / r n k 3 ) ( l - 3 c o s 2 6 . , ) (1/2) [31 j I k - f ] , 3 ^ 3 K 3 k z z 3 k J ( I V - 2 3 ) fi ->-wh e r e D j k = a n g l e b e t w e e n r ^ k and H Q . The f i r s t t e r m o f ( I V - 2 2 ) i n t h e s q u a r e b r a c k e t s a r i s e s f r o m t h e s t a t i c p a r a l l e l f i e l d h s . The s e c o n d t e r m i s due t o t h e r o t a t i n g t r a n s v e r s e f i e l d h r . I f t h e RF f i e l d u s e d t o p r o d u c e t h e NMR t r a n s i t i o n s i s a p p l i e d a l o n g t h e x a x i s , t h e n t h e e x p r e s s i o n f o r t h e s e c o n d moment g i v e n , by Abragam i s M2 =-Tr ( [ H d S ^ x ] 2 ) ( I V - 2 4 ) 2 2 h T r ( I z ) x The e x p r e s s i o n f o r H^s g i v e n i n ( I V - 2 3 ) was o b t a i n e d a s s u m i n g t h a t a l l l a t t i c e s i t e s were o c c u p i e d by s p i n s o f t h e same t y p e w i t h r e s o n a n t f r e q u e n c i e s u ^ s u c h t h a t f o r a n y p a i r o f s p i n s l u . - u . I < Yh ( I V - 2 5 ) 1 I j 1 ^  ' r However f o r h y d r o g e n , t h e i n t r a m o l e c u l a r d i p o l e - d i p o l e i n t e r -a c t i o n c a n be much l a r g e r t h a n t h e i n t e r m o l e c u l a r i n t e r a c t i o n a t l o w t e m p e r a t u r e s s o t h a t ( I V - 2 5 ) w o u l d n o t be s a t i s f i e d f o r a l l p a i r s o f s p i n s . I n t h a t c a s e h r i s n o t r e s o n a n t and t h e s e c o n d t e r m i n t h e s q u a r e b r a c k e t s o f ( I V - 2 2 ) s h o u l d be d e l e t e d . I t h a s be e n f o u n d f o r b u l k s o l i d H 2 t h a t M 2 o f one pe a k o f t h e f u l l y s p l i t d o u b l e t i s 5/6 t h a t o f t h e u n s p l i t l i n e ( I s h i g u r o e t a l , 1 9 5 1 ) . S i n c e i t was n o t p o s s i b l e t o o b t a i n a v e r y p r e c i s e m e a s u r e m e n t o f M 2 i n t h e e x p e r i m e n t s d e s c r i b e d i n t h i s t h e s i s , we w i l l s i m p l y u s e ( I V - 2 3 ) i n ( I V - 2 4 ) g i v i n g M,," (3/4) ( Y / 2 7 r ) 4 h 2 I ( I + l ) £ ( l - 3 c o s 2 6 j k ) 2 r j k " 6 ( I V - 2 6 ) 72 I f t h e J = l c o n c e n t r a t i o n i s d e n o t e d by c and 1=1, t h e n we 2 g e t M 2 = ( 3 / 2 ) c ( Y / 2 7 r ) 4 h 2 £ ( 1 - 3 c o s 2 e j k ) 2 r j k " 6 ( I V - 2 7 ) F o r D 2 i 5/6 o f t h e J=0 s p i n s h a v e 1=2 s o t h e s e c o n t r i b u t e t o t h e b r o a d e n i n g o f t h e J = l s p e c t r a . A t l o w t e m p e r -a t u r e s m o s t p a i r s c o n s i s t i n g o f a n 1=1 and an I'=2 s p i n w i l l n o t s a t i s f y ( I V - 2 5 ) s o i t i s more a p p r o p r i a t e t o d e l e t e t h e s e c o n d t e r m o f ( I V - 2 2 ) . I f t h i s i s d o n e , t h e c o n t r i b u t i o n o f t h e I'=2 s p i n s t o t h e s e c o n d moment o f t h e 1=1 s p i n s i s M 2 - ( l / 3 ) ( Y / 2 7 r ) 4 h V (1+1) ( 1 - c ) (5/6) jE ( l - 3 c o s 2 6 ) 2 r j R ~ 6 ( I V - 2 8 ) where t h e sum i s o v e r a l l m o l e c u l e s . The t o t a l s e c o n d moment f o r t h e 1=1 s p i n s o f D 2 i s M2- (3/2) (c+ (10/9) ( 1 - c ) } ( Y / 2 7 r ) 4 h 2 £ ( l - 3 c o s 2 9 j k ) 2 r j k " 6 ( I V - 2 9 ) I n v i e w o f t h e S:N i t i s s u f f i c i e n t t o a p p r o x i m a t e t h e G r a f o i l by a 2D powder when c a l c u l a t i n g M 2. Two c a s e s w i l l be c o n s i d e r e d . I f H Q i s n o r m a l t o t h e f o i l (3=0°), t h e n 9j k=90° f o r a l l j and k s o ( 1 - 3 c o s 2 6 ^ R ) 2 = i . i f H Q l i e s i n t h e f o i l p l a n e (&=90°) t h e n a l l a n g l e s 9 j k i n t h e r a n g e ( 0 , 2 " ) h a v e e q u a l p r o b a b i l i t y . A v e r a g i n g o v e r t h e s e g i v e s < ( l - 3 c o s 2 e j k ) 2 > = ( l / 2 T T ) / 2 l T ( l - 3 c o s 2 e ) 2 d e ( I V - 3 0 a ) =11/8 ( I V - 3 0 b ) The f a c t t h a t M 2 d e p e n d s upon t h e o r i e n t a t i o n o f H Q i s a w e l l known r e s u l t f o r NMR o f 2D s y s t e m s (Cowan, 1 9 8 0 ) . Now t h e sum o f r j ^ ~ 6 w i l l be c a l c u l a t e d f o r t h e /3 x s o l i d . The n e i g h b o u r s o f t h e j t h m o l e c u l e l i e on a 73 s e r i e s o f c o n c e n t r i c h e x a g o n s . L e t be t h e c o n t r i b u t i o n o f r o m t h e i t h h e x a g o n t o t h e sum and l e t a=1.420A be t h e i n t r a p l a n e r C-C s e p a r a t i o n . Then f o r t h e f i r s t t h r e e h e x a g o n s 51 = 8.231 x 1 0 " 3 a ~ 6 ( I V - 3 1 a ) 5 2 = .434 x 1 0 " 3 a ~ 6 ( I V - 3 1 b ) , - 3 - 6 S 3 = .059x10 J a ( I V - 3 1 c ) The s q u a r e r o o t s o f t h e s e c o n d moments a r e ( IV-32) . 0 2 7 7 c 1 / 2 i f 6=0° H 0 : M 2 1 / 2 / 3 d = V 0 3 2 5 c 1 / 2 i f B=90° ( I V - 3 3 ) .00149 {c+(10/9) ( 1 - c ) } 1 / 2 i f 8=0° D 2 : M 2 1 / 2 / 3 d = S 00173 {c+(10/9) ( 1 - c ) } 1 / 2 i f B=90° ( I V - 3 4) 4.4 G r a p h i t e D i a m a g n e t i s m G r a p h i t e h a s a l a r g e a n i s o t r o p i c d i a m a g n e t i s m . C o n -s e q u e n t l y t h e m a g n e t i c f i e l d a t t h e s u r f a c e o f t h e c r y s t a l i n t h e G r a f o i l w i l l d e p e n d upon t h e d i r e c t i o n o f t h e a p p l i e d f i e l d w i t h r e s p e c t t o t h e n o r m a l t o t h e c r y s t a l . The d i s t r i b u t i o n o f c r y s t a l o r i e n t a t i o n s w i l l l e a d t o b r o a d e n i n g o f t h e NMR s p e c t r a . H i c k e r n e l l e t a l (1974) m e a s u r e d t h e room t e m p e r a t u r e f r e q u e n c y s h i f t o f e t h y l a l c o h o l a d s o r b e d o n G r a f o i l w i t h t h e m a g n e t i c f i e l d p a r a l l e l a n d p e r p e n d i c u l a r t o t h e f o i l . They u s e d a c y l i n d r i c a l s a m p l e i n w h i c h t h e n o r m a l t o t h e G r a f o i l s h e e t s was p e r p e n d i c u l a r t o t h e a x i s o f t h e c y l i n d e r . A t 30 MHz t h e 74 s h i f t was 2.4 kHz. If one assumes that the s h i f t i s the same at l i q u i d helium temperatures, then for the 8 MHz experiments described in thi s thesis one would expect the contribution to M 2 1/ 2/3d to be of the order of 3.7xl0 - 3 (9.3xl0~ 3) for H 2 (E^)* Comparing with (IV-33) and (IV-34) one can see that for H 2 the intermolecular dipolar broadening i s l i k e l y to be much larger than the e f f e c t of the graphite diamagnetism but for D 2 the reverse i s true. 75 Notes to Chapter IV 1. This corrects a misconception in the paper of Kubik and Hardy (1978). At that time we believed that the f i r s t e f f e c t would dominate. 2. In general we w i l l define c to be the odd J concentration. At l i q u i d helium temperatures t h i s i s the same as the J=l concentrat ion. 3. This i s the same sample geometry as used in the work described in thi s thesis. 76 CHAPTER V SAMPLE PREPARATION AND CONVERSION MEASUREMENTS 5.1 Introduction For bulk s o l i d H 2 and D 2 the temperature of the or i e n t a t i o n a l ordering t r a n s i t i o n decreases l i n e a r l y with the J=l concentration c down to c=.6. It then drops rapidly to zero at c=.55. For adsorbed hydrogen the depression of the t r a n s i -t i o n temperature i s l i k e l y to be even more rapid so i t was desireable to l i m i t the J=0 concentration to a few percent. In t h i s chapter we w i l l consider the preparation of enriched samples of J=l hydrogen and measurements of the conversion rate. In thermodynamic equilibrium the odd J concentration ceq l s given by Ce q = g J Z _ 1 o d d J(2J+l)exp(-E J/kT) (V-l) where Ej = energy of the Jth r o t a t i o n a l l e v e l g j = nuclear degeneracy factor which i s d i f f e r e n t for odd J and even J (see Table I) 2 = a l l J 9j(2J+l) exp (-Ej/kT) A graph of c e g versus T i s shown i n Fig. 15. Room tempera-ture i s e s s e n t i a l l y the high temperature l i m i t and c__ = e q .749 (.333) for H 2 (D 2). These are referred to as normal-H 2 and normal-D 2 (n-H 2 and n-D 2). At temperatures below 10 K f c = 0 for H 0 and D0. eq 2 2 Transitions from odd J states to even J states require a simultaneous change of p a r i t y of the o r b i t a l and nuclear s p i n wave f u n c t i o n s . For an H 2 molecule, c o n v e r s i o n n e c e s s i t a t e s the presence o f a magnetic f i e l d g r a d i e n t . For D 2 e i t h e r an e l e c t r i c o r magnetic f i e l d g r a d i e n t i s s u f f i c i e n t because deuterons have a quadrupole moment. I f the magnetic f i e l d g r a d i e n t i s pr o v i d e d by the n u c l e a r and r o t a t i o n a l magnetic moments of neighbouring m o l e c u l e s , the c o n v e r s i o n i s c a l l e d " i n t r i n s i c " . Paramagnetic ions have e l e c t r o n i c magnetic moments that are one thousand times g r e a t e r than n u c l e a r moments so t h e i r presence can s t r o n g l y c a t a l y z e c o n v e r s i o n . As a p r a c t i c a l matter, n o n - e q u i l i b r i u m hydrogen samples can be s t o r e d a t room temperature i n c l e a n g l a s s b u l b s . At 1 atmosphere p r e s s u r e , H 2 w i l l c o n v e r t 1% i n a week whereas f o r D 2 i t may take a year f o r a s i m i l a r change. Conversion o f the gas i s b e l i e v e d to be due' p r i m a r i l y to the presence o f 0 2 i m p u r i t i e s . A glance a t F i g . 15 suggests t h a t the c r e a t i o n o f hydrogen samples i n which the J=l c o n c e n t r a t i o n i s hig h e r than i n normal hydrogen ( e n r i c h e d samples) might be a hopeless task. However at low temperatures, J=l molecules are p r e f e r e n t i a l l y adsorbed on some s u b s t r a t e s such as alumina. S i l v e r a and N i e l s e n (1976) used i n e l a s t i c neutron s c a t t e r i n g t o measure the s e p a r a t i o n c o e f f i c i e n t S which i s d e f i n e d as S= {c (1-c )}/{c (1-c )} (V-2) s s g g where c e and c are the c o n c e n t r a t i o n s o f odd J molecules s g + 2 on the s u r f a c e and i n the gas. They found S=39_g (5.Q+.3) f o r H 2 (D 2) on A l c o a a c t i v a t e d a l u m i n a . 1 79 5.2 P r e p a r a t i o n o f E n r i c h e d H y d r o g e n S a m p l e s E n r i c h e d h y d r o g e n s a m p l e s w ere p r e p a r e d i n an a p p a r a t u s ( F i g . 16) l a r g e l y c o n s t r u c t e d by W.N. H a r d y , b a s e d o n t h e m e thod o f D e p a t i e and M i l l s ( 1 9 6 8 ) . The n-H 2 h a d a p u r i t y o f 9 9 . 9 9 9 % ( M a t h e s o n U l t r a - h i g h P u r i t y ) a nd t h e n-D 2 was 9 9 . 6 5 % ( B i o - R a d L a b o r a t o r i e s ) . The h y d r o g e n was a d s o r b e d o n t o A l c o a a c t i v a t e d a l u m i n a c o n t a i n e d i n a 1.23 m l o n g , 1.27 cm d i a m e t e r s t a i n l e s s s t e e l t u b e h e l d a t 20 K. The a l u m i n a had i n i t i a l l y b e e n washed i n c o n c e n t r a t e d HC1 i n o r d e r t o remove p a r a m a g n e t i c i m p u r i t i e s , 3 + m a i n l y Fe . How e f f e c t i v e l y t h i s r e d u c e d c o n v e r s i o n on t h e s u r f a c e ( b a c k c o n v e r s i o n ) i s n o t known. A f t e r b e i n g p l a c e d i n t h e wand, t h e a l u m i n a was b a k e d a t 120 C u n d e r vacuum f o r a d a y and f l u s h e d s e v e r a l t i m e s w i t h h y d r o g e n . The a l u m i n a c a n w i t h -s t a n d f a r h i g h e r b a k i n g t e m p e r a t u r e s b u t i t h a s b e e n f o u n d by H a r d y t h a t b a k i n g a t a t e m p e r a t u r e o f 200 C c a u s e d a n i n c r e a s e i n b a c k c o n v e r s i o n . T h i s p r o c e s s may h a v e b r o k e n some b o n d s o f t h e s u r f a c e m o l e c u l e s , c r e a t i n g p a r a m a g n e t i c i o n s . B e f o r e e a c h e n r i c h m e n t r u n t h e wand was pumped o u t and f l u s h e d w i t h h y d r o g e n a t room t e m p e r a t u r e . I d e a l l y t h e t e m p e r a t u r e o f t h e wand w o u l d be c o n t r o l l e d by p l a c i n g i t i n l i q u i d H 2 a t i t s b o i l i n g p o i n t o f 20.4 K. A t t h a t t e m p e r a t u r e t h e v a p o u r p r e s s u r e a b o v e a s u b - m o n o l a y e r w o u l d be v e r y l o w and t h e r e w o u l d be no d a n g e r o f p o c k e t s o f b u l k s o l i d h y d r o g e n f o r m i n g s i n c e t h e t r i p l e p o i n t t e m p e r a t u r e i s 13.8 K ( 1 8 . 7 K) f o r H 2 ( D 2 ) . U n f o r t u n a t e l y t h e u s e o f a l i q u i d h y d r o g e n b a t h had t o be r e j e c t e d f o r s a f e t y o T h ermocoxp I * Pressure Cauge D.ff Pump Hydrogen Supply Tank A v^ H ydrogen Collection ent Bulb W a n d (20 K ) O Valv« A Needle Volv* Bourdon Tube Pressure Gauge -<- Conical Glass Joint " l 4 o - o -M e r c u r y -Mo nom«Ur 0", 15 6 v<* Fig. 16 Hydrogen enrichment apparatus. 81 c o n s i d e r a t i o n s . I n s t e a d t h e wand was p l a c e d i n s i d e a c o p p e r c y l i n d e r w h i c h was c o o l e d by pumping l i q u i d 4 H e f r o m a s t o r a g e d e w a r a t a b o u t .5 1 ( l i q u i d ) / h t h r o u g h c o p p e r t u b e s s o l d e r e d o n t o t h e c y l i n d e r . B e t t e r s t a b i l i t y was o b t a i n e d by pumping t h e 4 H e t h r o u g h r a t h e r t h a n p r e s s u r i z i n g t h e s t o r a g e d e w a r . T y p i c a l l y , f o r H 2 t h e b o t t o m o f t h e c y l i n d e r was h e l d a t a b o u t 19 K and t h e t o p a t 23 K. T e m p e r a t u r e s were m e a s u r e d u s i n g an Au (.07% F e ) / C h r o m e l - P t h e r m o c o u p l e (Rosenbaum, 1969) w i t h a l i q u i d N 2 r e f e r e n c e t e m p e r a t u r e . A f t e r c o o l i n g t h e wand t o 20 K, n-H 2 was pumped t h r o u g h i t by a r o t a r y pump a t a r a t e o f .30 l ( S T P ) / m i n a n d an i n p u t p r e s s u r e P^~ 150 t o r r . I n i t i a l l y t h e e f f l u e n t g a s had a l o w J = l c o n c e n t r a t i o n b u t a f t e r a few m i n u t e s i t s t a b i l i z e d a t =.65. S i n c e t h e i n c o m i n g n-H 2 h a s c = . 7 5 , t h e r e m u s t h a v e b e e n some b a c k c o n v e r s i o n on t h e a l u m i n a . T h e s e c o n d i t i o n s were m a i n t a i n -ed f o r h a l f an h o u r t o e n s u r e t h a t t h e a d s o r b e d h y d r o g e n was i n e q u i l i b r i u m w i t h t h e f l o w i n g g a s . The v a l v e v ^ a t t h e i n p u t o f t h e wand was t h e n c l o s e d and t h e wand was r a i s e d 2 cm e v e r y 15 s. E n r i c h e d H 2 d e s o r b e d f r o m t h e t o p end o f t h e wand, f l o w e d down t h r o u g h t h e a l u m i n a , f u r t h e r e n r i c h i n g t h e a d s o r b e d H 2 , and t h e n up t h e c e n t r e . When t h e wand had b e e n r a i s e d .76 m t h e c o n c e n t r a t i o n o f t h e e f f l u e n t g a s had r i s e n t o c=.97. The v a l v e s Vg and V - ^ Q t o t h e r o t a r y pump were c l o s e d , t h e s t o p c o c k V 5 o n t h e 2 1 P y r e x c o l l e c t i o n b u l b was o p e n e d a nd t h e wand was r a p i d l y p u l l e d o u t . I m m e d i a t e l y a f t e r w a r d s v 5 was c l o s e d t o p r e v e n t i m p u r i t i e s on t h e a l u m i n a f r o m d e s o r b i n g 82 i n t o t h e b u l b . I n t h i s manner a b o u t 2 l ( S T P ) o f H 2 w i t h c>.98 c o u l d be c o l l e c t e d . The r e s u l t s f o r D 2 were s i m i l a r t h o u g h t h e r e were m i n o r m o d i f i c a t i o n s t o t h e p r o c e d u r e . I t i s i m p o r t a n t t o be a b l e t o m o n i t o r t h e J = l c o n c e n t r a -t i o n o f t h e e f f l u e n t g a s . T h i s was done u s i n g a m a t c h e d p a i r o f P i r a n i g a u g e s . E a c h P i r a n i g a u g e c o n s i s t s o f a c e l l c o n t a i n i n g 40 t o r r o f h y d r o g e n and a f i l a m e n t . The f i l a m e n t was h e a t e d by p a s s i n g a 650 mA ( 5 0 0 mA f o r D 2) c u r r e n t t h r o u g h i t . The t e m p e r a t u r e o f t h e f i l a m e n t a s m e a s u r e d by i t s r e s i s t a n c e d e p e n d s upon t h e t h e r m a l c o n d u c t i v i t y o f t h e g a s . A t t h i s p r e s s u r e t h e mean f r e e p a t h o f t h e h y d r o g e n was much l e s s t h a n t h e d i m e n s i o n s o f t h e g auge and c o n v e c t i o n was i n s i g n i f i c a n t s o t h e t h e r m a l c o n d u c t i v i t y was i n d e p e n d e n t o f t h e p r e s s u r e . The o r t h o and p a r a s p e c i e s h a v e d i f f e r e n t t h e r m a l c o n d u c t i v i t i e s b e c a u s e t h e r o t a t i o n a l e n e r g i e s a r e d i f f e r e n t ; t h e d i f f e r e n c e i s m a x i m i z e d n e a r t h e b o i l i n g p o i n t o f N 2 s o t h e g a u g e s were i m m e r s e d i n a f l a s k o f l i q u i d N 2 . The f i l a m e n t s o f t h e two g a u g e s were c o n n e c t e d t o a r e s i s t a n c e b r i d g e w h i c h had b e e n b a l a n c e d w i t h n-H 2 i n e a c h o f t h e g a u g e s . The n-H 2 i n t h e l e f t arm was t h e n r e p l a c e d w i t h s l o w l y f l o w i n g e f f l u e n t g a s f r o m t h e wand. V a l v e s V-^ Q and v ^ w e re a d j u s t e d t o m a i n t a i n 40 t o r r p r e s s u r e and t o r e p l a c e t h e g a s i n t h e g a u g e a p p r o x i -m a t e l y e v e r y 15 s. The b r i d g e i m b a l a n c e i s a p p r o x i m a t e l y l i n e a r l y d e p e n d e n t on c and w o u l d be a c c u r a t e t o a b o u t 1% i f i t was c a l i b r a t e d u s i n g s a m p l e s o f known c o n c e n t r a t i o n . I n p r a c t i c e t h e b r i d g e was o n l y u s e d t o m o n i t o r t h e p r o d u c t i o n 83 process; absolute measurements of c were made using Raman scattering as described in Section 5.3. 5.3 Measurement of the J=1 Concentration by Raman Scattering It i s important that c be measured both before and af t e r an NMR experiment. During the course of NMR measurements, the r e l a t i v e concentration was monitored by the inte n s i t y of the NMR sig n a l . An absolute c a l i b r a t i o n of the NMR inte n s i t y would have been d i f f i c u l t . It was found that during the processes of adsorption and desorption conversion was much faster than when the hydrogen was in the /3 x /3 s o l i d . We suspect that when the hydrogen i s f l u i d on the surface many molecules can come close to paramagnetic surface impurities and are converted quickly but in the s o l i d phase only those molecules at s i t e s close to the impurities w i l l be converted rapidly. In order to get a uniform f i l m the hydrogen must be adsorbed at temperatures high enough (>40 K) that there w i l l be a s i g n i f i c a n t 3D vapour pressure (>1 torr) to allow exchange between the f i l m and vapour. At these temperatures the adsorbate i s f l u i d . We could not measure c during adsorption or desorption but by measuring i t before adsorption and afte r desorption the conversion was estimated. About 3 cnr(STP) of hydrogen was used in the NMR experiments. This i s too small a quantity to be analyzed with our P i r a n i gauges. Consequently the sample was transferred to a 84 s q u a r e o p t i c a l c e l l a n d c was d e t e r m i n e d f r o m t h e i n t e n s i t y r a t i o o f a p a i r o f Raman l i n e s . I f J i s a g o o d quantum number, t h e s e l e c t i o n r u l e s f o r Raman s c a t t e r i n g b e t w e e n s t a t e s J and J ' a r e J'-J=0,+_2. We w i l l o n l y be c o n c e r n e d w i t h t h e c a s e J ' - J = 2 . F o r u n p o l a r i z e d l i g h t t h e d i f f e r e n t i a l s c a t t e r i n g c r o s s - s e c t i o n p e r u n i t s o l i d a n g l e and p e r u n i t f r e q u e n c y i s d a J J ( / d f t d t o ' = A w (a) 1 ) 3 P J S J 6 ( E j - E , +fiw-fia>') ( V - 3 ) where A i s i n d e p e n d e n t o f J and J ' b u t d e p e n d s on t h e a n i s o t r o p i c p o l a r i z a b i l i t y a) = f r e q u e n c y o f t h e i n c i d e n t l i g h t i f = f r e q u e n c y o f t h e s c a t t e r e d l i g h t S j = ( J + 2 ) ( J + l ) / ( 2 J + 3 ) f o r j ' > J ( S t o k e s l i n e s ) E j = e n e r g y o f t h e J t h r o t a t i o n a l s t a t e = B J ( J + l ) i n t h e h a r m o n i c a p p r o x i m a t i o n P j = p r o b a b i l i t y t h a t t h e J t n s t a t e i s o c c u p i e d = e x p ( - E j / k T ) s',, ( 2 J " + I ) e x p ( - E J M /kT) The p r i m e on t h e s u m m a t i o n s y m b o l i n d i c a t e s t h a t t h e sum i s o v e r a l l e v e n (odd) J " i f J i s e v e n ( o d d ) . F o r t h e 19,435 c m - 1 A r l a s e r l i n e , t h e i n t e n s i t y r a t i o s do j J+2^<iai 3 a r e g i v e n i n T a b l e V . P j and w-o)1 h a v e b e e n e v a l u a t e d i n t h e h a r m o n i c a p p r o x i m a t i o n . Raman s p e c t r a w e r e o b t a i n e d u s i n g a Spex R a m a l o g s p e c t r o -m e t e r w i t h 400 mW u n p o l a r i z e d l i g h t f r o m a n A r l a s e r . The s l i t w i d t h o f 150 ym c o r r e s p o n d e d t o a b a n d w i d t h o f 2.8 c m - 1 , 85 which i s much greater than the i n t r i n s i c linewidth. For a pure o-H2 NMR sample (3 cm 3(STP)) with a pressure =350 t o r r in the o p t i c a l c e l l , the S:N with a 2 s time constant was =40. Concentrations could be measured with an accuracy of 2%. Table V. Raman Frequencies and Intensity Ratios H 2 D2 ^ \ k , / v / : v y — v 0 356.0 1.025 178.3 .757 1 593.4 1 296.8 1 2 830.8 .434 413.9 .785 3 1068 .099 529.4 .420 4 1306 - 642.2 .160 5. 4 The Substrate The substrate employed in these experiments was GTA grade G r a f o i l , a form of ex f o l i a t e d graphite. It i s prepared by in t e r c a l a t i n g graphite c r y s t a l s with a strong oxidizing agent such as s u l f u r i c acid, r i n s i n g , and rapid heating. Then i t i s r o l l e d into sheets with a density s1 g/cm , about half that of graphite. The main impurities are Fe, S i , A l , Mg, and Mn with l e v e l s ^10-100 ppm. These leve l s can be reduced to 86 <10 ppm by b a k i n g a t 1700 C u n d e r vacuum (Hegde a nd D a u n t , 1978) a l t h o u g h t h i s was n o t done w i t h o u r s a m p l e . The e x f o l i a t i o n p r o c e s s r e s u l t s i n an enormous i n c r e a s e i n t h e a r e a o f t h e e x p o s e d b a s a l p l a n e s b u t l i t t l e c h a n g e i n t h a t o f t h e edge p l a n e s ( D a s h , 1 9 7 8 ) . B r o a d e n i n g o f x - r a y d i f f r a c t i o n p e a k s i n d i c a t e s t h a t t h e c r y s t a l s a r e ^ 1 0 0 p l a n e s t h i c k . The s u r f a c e a r e a i s t y p i c a l l y 20-30 m / g . G r a f o i l h a s q u i t e homogeneous s u r f a c e s . A common c r i t e r i o n f o r d e t e r m i n i n g t h e h o m o g e n e i t y o f an a d s o r b e n t i s t o m e a s u r e t h e s h a r p n e s s o f t h e s t e p s i n t h e 3D v a p o u r p r e s s u r e i s o t h e r m s o f an i n e r t g a s s u c h a s K r . The s t e p s r e s u l t f r o m t h e c o e x i s t e n c e o f two s u r f a c e p h a s e s i n a d d i t i o n t o t h e 3D v a p o u r . F o r s u c h a s y s t e m we c a n t a k e t h e t h e r m o d y n a m i c v a r i a b l e s t o be t h e 3D v a p o u r p r e s s u r e P, t h e 2D s p r e a d i n g p r e s s u r e <j> , T, and N. When t h e r e i s c o e x i s t e n c e o f two s u r f a c e p h a s e s on a u n i f o r m s u r f a c e , t h e c h e m i c a l p o t e n t i a l s o f t h e two p h a s e s p ^ f ^ T ) and p 9 ( i j),T) a nd t h a t o f t h e 3D v a p o u r P„(P,T) m u s t a l l be e q u a l i n d e p e n d e n t o f N. T h e r e f o r e P w i l l be i n d e p e n d e n t o f N a l s o . A t l o w t e m p e r a t u r e s , t h e two s u r f a c e p h a s e s a r e u s u a l l y a s o l i d and a g a s and t h e s t e p s c a n c o v e r a l m o s t t h e e n t i r e r a n g e o f N f o r e a c h l a y e r . F o r K r a d s o r b e d o n G r a f o i l , f i v e s t e p s c a n be o b s e r v e d i n a d d i t i o n t o s u b - s t e p s c a u s e d by p h a s e t r a n s i t i o n s b e t w e e n o t h e r 2D p h a s e s d u r i n g f o r m a t i o n o f t h e f i r s t l a y e r . E x f o l i a t e d g r a p h i t e e x h i b i t s b e t t e r h o m o g e n e i t y t h a n g r a p h i t i z e d c a r b o n b l a c k w h i c h was p o p u l a r i n e a r l i e r s u r f a c e s t u d i e s (Thorny a n d D u v a l , 1 9 6 9 ) . The u n i f o r m i t y a n d s p e c i f i c s u r f a c e a r e a s o f 87 s e v e r a l f o r m s o f e x f o l i a t e d g r a p h i t e a r e c o m p a r e d i n T a b l e V I ( B i r g e n a u e t a l , 1 9 8 1 ; Rosenbaum e t a l , 1 9 8 3 ; R. C l a r k e , P.M. H o r n , S.E. N a g l e r , a n d T.F. Rosenbaum, t o be p u b l i s h e d ) . T a b l e V I . P r o p e r t i e s o f E x f o l i a t e d G r a p h i t e S u r f a c e S p e c i f i c z A x i s D i s t r i b u t i o n C o h e r e n c g a r e a S u b s t r a t e L e n g t h (A) (m /g) G r a f o i l ' P a p y e x t UCAR Foam* UCAR ZYX* E x f o l i a t e d s i n g l e c r y s t a l 100-200 200-300 900 2000 4000 20-30 20 • 27 2.8 .3 7 0 % : i s o t r o p i c 3 0 % : FWHM = 30° 7 0 % : i s o t r o p i c 3 0 % : FWHM = 30° i s o t r o p i c FWHM = 20-30° FWHM = 17° * U n i o n C a r b i d e C o r p . , C a r b o n P r o d u c t s D i v . , 270 P a r k Ave. , New Y o r k , USA t C a r b o n e L o r r a i n e , 37-41 r u e J e a n - J a u r e s , 9 2 2 3 1 , G e n n e v i l l i e r s , F r a n c e A n o t h e r p r o p e r t y o f G r a f o i l t h a t i s i m p o r t a n t f o r NMR m e a s u r e m e n t s i s t h a t t h e n o r m a l s t o t h e a d s o r b i n g p l a n e s , t h e c r y s t a l z a x e s , a r e p r e f e r e n t i a l l y a l i g n e d p a r a l l e l t o t h e f o i l n o r m a l , t h e Z a x i s ( s e e T a b l e V I ) . The d i s t r i b u t i o n was o b t a i n e d f r o m an a n a l y s i s o f t h e s h a p e o f t h e d i f f r a c t i o n p e a k s o f a d s o r b e d g a s e s : one c a n n o t u s e t h e g r a p h i t e p e a k s b e c a u s e one w a n t s o n l y t h e c o n t r i b u t i o n s f r o m t h e s u r f a c e p l a n e s . F i g . 13 shows t h a t t h e i n t e n s i t y o f t h e p e a k s i n t h e NMR s p e c t r u m c a n 8 8 be enhanced considerably i f the adsorbing surfaces are p r e f e r e n t i a l l y aligned. Among the various forms of exfoliated graphite, G r a f o i l was chosen as the substrate because i t makes a good compromise among the properties l i s t e d in Table VI. Paramount among these for t h i s work i s the s p e c i f i c surface area because the NMR S:N r a t i o i s quite poor. Apparently the surface coherence length of G r a f o i l i s s u f f i c i e n t to allow o r i e n t a t i o n a l ordering of H 2» Other substrates w i l l be discussed in Chapter X. 5.5 Adsorption A t y p i c a l procedure for adsorbing a gas onto G r a f o i l begins by pumping out the c e l l at room temperature and then f i l l i n g i t with a measured quantity of gas. The c e l l would then be cooled over a period of several hours to adsorb the gas. Afterwards i t would be slowly heated u n t i l the vapour pressure had risen to about 1 t o r r to anneal the sample. F i n a l l y i t would be slowly cooled again. This procedure was found to be unsuitable for hydrogen because during the adsorption process the conversion rate was very rapid. In order to keep the con-version to <5% i t was necessary to adsorb the sample in less than 15 min. In early experiments the samples were held at temperatures of 30 - 40 K corresponding to vapour pressures of several hundred microns for about 40 min. Pure o-H2 samples converted to about 25% o-H2. Fig. 15 shows that t h i s i s close to c„_ at those temperatures. e Q 89 The s a m p l e c e l l s w e r e q u i t e d i f f e r e n t f o r t h e 4 H e and 3 He c r y o s t a t s s o t h e a d s o r p t i o n p r o c e d u r e s w i l l be d e s c r i b e d 4 s e p a r a t e l y . I n t h e He c r y o s t a t t h e G r a f o i l was a t t h e b o t t o m o f a l o n g g l a s s wand shown s c h e m a t i c a l l y i n F i g . 17. The v o l u m e V Q c o n t a i n e d t h e h y d r o g e n g a s , V 2 was a c a l i b r a t e d v o l u m e , a n d V 3 c o n t a i n e d t h e G r a f o i l . V 3 was a l w a y s e v a c u a t e d e x c e p t when f i l l e d w i t h h y d r o g e n . P r i o r t o a n e x p e r i m e n t V-^, V 2 * and V 3 w o u l d be pumped o u t a t room t e m p e r a t u r e by a d i f f u s i o n pump w i t h a l i q u i d N 2 t r a p f o r a t l e a s t one d a y and f l u s h e d a few t i m e s w i t h h y d r o g e n . O c c a s i o n a l l y t h e b o t t o m o f t h e wand was b a k e d a t 180 C d u r i n g t h e pumping and f l u s h i n g . The t e m p e r a t u r e was l i m i t e d by t h e p r e s e n c e o f T e f l o n s h e e t s i n t h e c e l l a s d e s c r i b e d i n C h a p t e r V I . S t o p c o c k v 2 3 was t h e n c l o s e d and V 2 was f i l l e d t o a p r e s s u r e P 2 m e a s u r e d by a B o u r d o n t u b e p r e s s u r e g a u g e . V a l v e v ^ 2 was c l o s e d , v 2 3 was o p e n e d and t h e wand was l o w e r e d f a i r l y r a p i d l y i n t o t h e l i q u i d 4 H e b a t h . As s o o n a s t h e p r e s s u r e P 3 m e a s u r e d by t h e V e e c o DV-1M t h e r m o c o u p l e p r e s s u r e gauge (1-2000 y) b e g a n t o d r o p b e l o w t h e 1 a t m o s p h e r e mark t h e wand was r a i s e d a c o u p l e o f c e n t i m e t e r s t o b r i n g P 3 b a c k up. The wand was t h e n l o w e r e d o v e r a p e r i o d o f 12 - 15 m i n u n t i l P 3 was down t o 100 p w h e r e u p o n i t was l o w e r e d r a p i d l y i n t o t h e 4 H e b a t h . The o b j e c t o f t h i s p r o c e d u r e was t o a d s o r b m o s t o f t h e g a s f a i r l y q u i c k l y b u t t h e n t o h o l d t h e p r e s s u r e n e a r 1 t o r r f o r s e v e r a l m i n u t e s t o a l l o w i t t o a n n e a l . Once e s s e n t i a l l y a l l o f t h e g a s was a d s o r b e d i t was c o o l e d r a p i d l y t o m i n i m i z e c o n v e r s i o n . 90 Conical joint 500 ml H 2 storage bulb V 23 T h e r m o c o u p l e .^^p^ p r e s su r e gauge —Press u re gauge v . 3a II Quart; -v, - P y r e x Pyrex-quartz graded jo int — Grafoi l Fig. 17 Sketch of the wand containing the G r a f o i l sample used in the 4He cryostat (not to scale ) . 91 3 I n t h e He c r y o s t a t t h e s a m p l e c e l l was m o u n t e d o n t h e 3 He p o t s o t h e h y d r o g e n was i n t r o d u c e d v i a a n i n s u l a t e d , h e a t e d s t a i n l e s s s t e e l t u b e . The s y s t e m i s shown i n F i g . 18. I t was pumped o u t and f l u s h e d a t room t e m p e r a t u r e a s d e s c r i b e d 4 f o r t h e He c r y o s t a t . On t h e o c c a s i o n s t h a t t h e G r a f o i l c e l l was b a k e d , i t was o n l y h e a t e d t o 120 C t o a v o i d d a m a g i n g t h e c o p p e r - q u a r t z j o i n t d e s c r i b e d i n A p p e n d i x B. The l i q u i d 4 H e b a t h was f i l l e d t o a few cm a b o v e t h e 4 K p l a t e s o t h a t by t h e t i m e t h a t t h e h y d r o g e n e n t e r e d t h e c e l l t h e l e v e l was j u s t b e l o w t h e p l a t e . T h i s p r e v e n t e d t h e f o r m a t i o n o f c o l d s p o t s w h e r e t h e h y d r o g e n m i g h t c o n d e n s e . The H 2 f i l l l i n e and 4 H e p o t h e a t e r s w e re u s e d t o h e a t t h e 4 H e p o t t o T 4=52 K and t h e 3 H e p o t t o T 3=40 K. T h r o u g h o u t t h e p r o c e s s o f a d s o r p t i o n T 4 > T 3 s o t n a t t n e G r a f o i l s h o u l d h a v e b e e n i n t h e c o l d e s t r e g i o n o f t h e f i l l l i n e . The h y d r o g e n was g r a d u a l l y a l l o w e d t o e n t e r t h e G r a f o i l c e l l V 4 f r o m t h e 16.96 cm 3 c a l i b r a t e d v o l u m e V 2» A f t e r a b o u t two m i n u t e s , l e s s t h a n 1% o f t h e h y d r o g e n r e m a i n e d i n V 2 a " d t h e p r e s s u r e P 4 a b o v e t h e G r a f o i l was a b o u t 1 t o r r . T h i s p r e s s u r e was m a i n t a i n e d f o r 6 m i n a t w h i c h t i m e 30\i o f 4 H e was a l l o w e d i n t o t h e vacuum c a n . F i v e m i n u t e s l a t e r P 4 had d r o p p e d t o l O O y , T 4=40 K and T-j=30 K. The vacuum c a n p r e s s u r e was i n c r e a s e d t o 200y o f 4 H e a n d 2 m i n l a t e r P 4 < l y , T 4=27 K, and T 3=20 K. The h e a t e r s were t h e n t u r n e d o f f and w i t h i n a m i n u t e T 4 went t o 10 K a n d T 3 t o 4 K. The t o t a l t i m e e l a p s e d s i n c e t h e s t a r t o f a d s o r p t i o n was 16 m i n . Rftmao H y d r o g e n bulb Q va lve ^ \ needle vqlro Thermocouple pressure gauge Heater D e w a r L i q u i d 4 He 4 K plate V a c u u m j a c k e t H y d r o g e n fill t u b e 4 H e pot J He pot Craf oil F i g . 18 3 S k e t c h o f t h e a d s o r p t i o n a n d d e s o r p t i o n a p p a r a t u s u s e d w i t h t h e He c r y o s t a t . to F o r a f e w r u n s a d i f f e r e n t p r o c e d u r e w a s t r i e d . T 3 w a s r a i s e d t o o n l y a b o u t 25 K u s i n g t h e f i l l l i n e a n d 4 H e p o t h e a t e r s . H y d r o g e n w a s t h e n l e a k e d i n t o V 4 t h r o u g h t h e n e e d l e v a l v e V 2 3 a t s u c h a r a t e t h a t P 4 w a s m a i n t a i n e d a t a b o u t 500u. T h i s w a s w e l l a b o v e t h e e q u i l i b r i u m v a p o u r p r e s s u r e b u t w e l l b e l o w t h e b u l k h y d r o g e n v a p o u r p r e s s u r e . A f t e r 10-15 m i n v i r t u a l l y a l l o f t h e h y d r o g e n w a s i n V 4 a n d P 4 b e g a n t o d r o p . T h e h e a t e r s w e r e t u r n e d o f f a n d 50y o f 4 H e w a s a d d e d t o t h e v a c u u m c a n . I n a b o u t 3 m i n T 3 d r o p p e d t o 4 K. T h i s p r o c e d u r e w a s a b a n d o n e d b e c a u s e i t a p p a r e n t l y r e s u l t e d i n n o n -u n i f o r m f i l m s : t h e NMR s i g n a l s w e r e v e r y w e a k a n d b r o a d . S i n c e t h e c o v e r a g e p i s a n i m p o r t a n t e x p e r i m e n t a l p a r a m e t e r , i t w a s n e c e s s a r y t o d e t e r m i n e t h e s u r f a c e a r e a o f t h e G r a f o i l s u b s t r a t e . T h e a m o u n t o f g a s N 3 t h a t w i l l g i v e f u l l c o v e r a g e i n t h e /3 x /3 s o l i d , i e . p = l , c a n b e d e t e r m i n e d f r o m t h e h e a t c a p a c i t y , n e u t r o n s c a t t e r i n g , o r a d s o r p t i o n i s o t h e r m s . T h e l a t t e r w a s b y f a r t h e s i m p l e s t f o r u s . I n p r i n c i p l e , i t s h o u l d b e p o s s i b l e t o d e t e r m i n e N 3 b y i d e n t i f y i n g t h e t r a n s i t i o n f r o m t h e /3 x /3 s o l i d t o t h e i n c o m m e n s u r a t e s o l i d i n t h e l o w t e m p e r a t u r e a d s o r p t i o n i s o -t h e r m s a s h a s b e e n d o n e , f o r e x a m p l e , f o r K r a d s o r b e d o n g r a p h i t e . H o w e v e r , t h i s h a s n o t b e e n d o n e f o r l i g h t g a s e s s u c h a s h y d r o g e n a n d h e l i u m b e c a u s e t h e p r e s s u r e s i n v o l v e d a r e q u i t e l o w , £ l 0 y , a n d t h e r m o m o l e c u l a r p r e s s u r e e f f e c t c o r r e c t i o n s w o u l d b e l a r g e . T h e r e i s c o n t r o v e r s y c o n c e r n i n g t h e e f f e c t o f t h e w a l l s o f t h e c o n t a i n e r o n t h e s e c o r r e c t i o n s ( M c C o n v i l l e , 1969; 94 B e r n a t a n d Coh e n , 1974) ( s e e p. 9 7 ) . The a p p r o a c h t h a t we a d o p t e d was t o u s e t h e a d s o r p t i o n i s o t h e r m s t o d e t e r m i n e t h e m o n o l a y e r c o v e r a g e N m o f 4 H e ; t h i s o n l y i n v o l v e s t h e m e a s u r e m e n t o f p r e s s u r e s g r e a t e r t h a n a few t e n s o f m i c r o n s . The h e a t c a p a c i t y o f 4 H e on G r a f o i l e x h i b i t s a s h a r p peak a t N=N 3 and a minimum i n t h e h e a t c a p a c i t y h a s b e e n i n t e r p r e t e d a s m o n o l a y e r c o m p l e t i o n ( B r e t z e t a l , 1 9 7 3 ; H u f f and D a s h , 1 9 7 6 ) . T h e s e m e a s u r e m e n t s g a v e N 3 / N m=.556 a t 1.0 K. A l s o , t h e l a t t i c e p a r a m e t e r s o f 4 H e on g r a p h i t e a t N=N 3 and N=N m h a v e b e e n m e a s u r e d by e l a s t i c n e u t r o n s c a t t e r i n g ( C a r n e i r o e t a l , 1 9 7 6 ) . A s s u m i n g a u n i f o r m s u r f a c e , t h e s e r e s u l t s i m p l y N 3 / N m = .561 a t 4.2 K i n g o o d a g r e e m e n t w i t h t h e h e a t c a p a c i t y m e a s u r e m e n t s . We d e t e r m i n e d N m f o r 4 H e by t h e " P o i n t B" m e t h o d (Young a n d C r o w e l l , 1 9 6 2 ) . T h i s i s a n e m p i r i c a l p r o c e d u r e t h a t h a s b e e n u s e d e x t e n s i v e l y f o r d e t e r m i n i n g m o n o l a y e r c o v e r a g e s a l t h o u g h i t i s n o t a s p r e c i s e a s t h e h e a t c a p a c i t y m e t h o d . F o r e a c h l a y e r o f a d s o r b e d m o l e c u l e s , i n t e r a c t i o n s b e t w e e n m o l e c u l e s c a n be n e g l e c t e d a t l o w c o v e r a g e s s o t h a t H e n r y ' s Law (Young and C r o w e l l , 1962) s h o u l d be v a l i d . I t s t a t e s t h a t f o r a u n i f o r m s u r f a c e t h e p r e s s u r e i s p r o p o r t i o n a l t o t h e c o v e r a g e . I n p r a c t i c e , t h e r e w i l l a l w a y s be some r o u n d i n g o f t h e i s o t h e r m s due t o s u r f a c e h e t e r o g e n e i t y a n d t h e r m a l p r o m o t i o n t o t h e n e x t l a y e r . The p o i n t a t w h i c h t h e i s o t h e r m becomes l i n e a r i s " P o i n t B"; t h i s h a s b e e n i n t e r p r e t e d a s l a y e r c o m p l e t i o n . A p p l y i n g t h e P o i n t B m e t h o d t o t h e i r 4 H e on G r a f o i l a d s o r p t i o n i s o t h e r m s a t U.O-J , 1 1 1 , ( • 1 • 1 i 1— 0 200 400 600 ^ 800 1000 1200 P + (/A F i g . 19 A d s o r p t i o n i s o t h e r m f o r 4 H e on G r a f o i l : P 4 v e r s u s N/N . M o n o l a y e r c o v e r a g e N m was d e t e r m i n e d by t h e P o i n t B m e t h o d . Dashed l i n e s a r e e x t r a p o l a t i o n s o f t h e H e n r y ' s Law r e g i o n . P o i n t B i s d e n o t e d by an a s t e r i s k . The s o l i d c i r c l e s [ c u r v e ( a ) ] were o b t a i n e d f r o m o u r G r a f o i l s u b s t r a t e i n c e l l B a f t e r c o r r e c t i n g f o r t h e v a p o u r i n t h e d e a d v o l u m e . The e r r o r b a r s a r e p r i m a r i l y due t o t h e e r r o r i n m e a s u r i n g t h e c a l i b r a t e d v o l u m e V*2. The o p e n c i r c l e s [ c u r v e ( b ) ] show t h e t o t a l amount N o f h e l i u m removed f r o m t h e c a l i b r a t e d v o l u m e . I n t h i s c a s e , N/N m was c a l c u l a t e d u s i n g N m f r o m c u r v e ( a ) . The c r o s s e s [ c u r v e ( c ) ] a r e t h e d a t a o f G o e l l n e r e t a l ( 1 9 7 5 ) . 96 4.2 K, G o e l l n e r e t a l (1975) f o u n d a m o n o l a y e r d e n s i t y o f °-2 .115A , w h i c h a g r e e d w i t h t h e h e a t c a p a c i t y r e s u l t . F i g . 19(a) shows o u r 4.23 K i s o t h e r m o f 4 H e o n t h e G r a f o i l s a m p l e i n c e l l B, w h i c h i s d e s c r i b e d i n s e c t i o n 6.3. The number o f a d s o r b e d m o l e c u l e s i s n o r m a l i z e d t o m o n o l a y e r c o v e r a g e N/N m. N o t e t h a t t h e r e i s c l e a r l y a r e g i o n w h e r e H e n r y ' s Law i s v a l i d . The p r e s s u r e a b o v e t h e G r a f o i l , P 4 , was m e a s u r e d by an MKS B a r a t r o n m o d e l 170 c a p a c i t a n c e manometer w i t h a 1 t o r r s e n s o r . C o r r e c t i o n s f o r t h e t h e r m o m o l e c u l a r p r e s s u r e d i f f e r e n c e b e t w e e n 4.2 K and 300 K h a v e b e e n made u s i n g t h e d a t a o f R o b e r t s and S y d o r i a k (1956); t h e y f o u n d g o o d a g r e e m e n t w i t h t h e W e b e r - S c h m i d t e q u a t i o n (Weber and S c h m i d t , 1936). The c o r r e c t i o n s were s i g n i f i c a n t a t t h e l o w e r p r e s s u r e s , b e i n g -14% a t 145 y b u t o n l y -3% a t 520y . The a d s o r p t i o n i s o t h e r m was m e a s u r e d by m e t e r i n g a q u a n t i t y N T o f g a s f r o m t h e c a l i b r a t e d v o l u m e V"2 i n t o V-j a n d V 4 ( s e e F i g . 18). F o r p r e s s u r e s P 4 n e a r lOOy , i t was n e c e s s a r y t o w a i t up t o two h o u r s f o r P 4 t o a p p r o a c h e q u i l i -b r i u m b u t t h i s t i m e was r e d u c e d t o a few m i n u t e s f o r P 4>200y. To d e t e r m i n e t h e q u a n t i t y N o f h y d r o g e n a d s o r b e d o n t h e G r a f o i l , i t was n e c e s s a r y t o c o r r e c t f o r t h e 3D v a p o u r i n a n d V 4 ; t h i s i s c a l l e d t h e d e a d v o l u m e c o r r e c t i o n . Of t h i s v o l u m e , o 3 12 cm was a t room t e m p e r a t u r e , 8 cm was a t 4.2 K, and o 12 cm was i n a vacuum j a c k e t e d t u b e b e t w e e n 4.2 K and room t e m p e r a t u r e . S i n c e t h e t e m p e r a t u r e g r a d i e n t down t h e t u b e was n o t known, i t was d i f f i c u l t t o p r e d i c t t h e amount o f g a s i n i t . 97 H o w e v e r , t h e r e was a s i m i l a r t u b e c o n n e c t e d t o a v a p o u r p r e s s u r e 3 4 c e l l i n t h e He p o t w h i c h c o u l d be f i l l e d w i t h He and s e a l e d o f f . By m o n i t o r i n g t h e p r e s s u r e i n t h a t t u b e a s a f u n c t i o n o f t h e l e v e l o f t h e l i q u i d h e l i u m b a t h , t h e amount o f g a s i n t h e f i l l l i n e f o r t h e G r a f o i l c e l l c o u l d be e s t i m a t e d . I n f a c t t h e p r e s s u r e i n t h e v a p o u r p r e s s u r e l i n e was a l m o s t i n s e n s i t i v e t o t h e l i q u i d h e l i u m l e v e l . A l t h o u g h t h e d e a d v o l u m e c o r r e c t i o n was q u i t e l a r g e f o r P 4 > 1 0 0 y , i t o n l y c h a n g e d t h e p o s i t i o n o f P o i n t B by 2% [ compare F i g . 1 9 ( a ) and ( b ) ] . F o r c o m p a r i s o n p u r p o s e s , t h e 4.24 K 4 H e on G r a f o i l 2 a d s o r p t i o n i s o t h e r m o b t a i n e d by G o e l l n e r e t a l i s a l s o shown i n F i g . 1 9 ( c ) . The d a t a h a s b e e n n o r m a l i z e d u s i n g t h e m o n o l a y e r c o v e r a g e o f 6.56 cm (STP) d e t e r m i n e d by G o e l l n e r e t a l u s i n g t h e P o i n t B m e t h o d . The s l o p e s o f F i g . 1 9 ( a ) and ( c ) a r e s l i g h t l y d i f f e r e n t ; t h i s i s p r o b a b l y due t o a n e r r o r i n o u r d e a d v o l u m e c o r r e c t i o n . The s h i f t i n t h e c u r v e s i s p r o b a b l y t h e r e s u l t o f an e r r o r i n t h e P o i n t B d e t e r m i n a t i o n : G o e l l n e r e t a l c h o s e P o i n t B t o be a t P 4 = 200v , w h e r e a s we h a v e t a k e n i t t o be a t 430M. T h e r e a r e s e v e r a l p o s s i b l e e x p l a n a t i o n s . A c c o r d i n g t o M c C o n v i l l e ( 1 9 6 9) , t h e t h e r m o m o l e c u l a r p r e s s u r e e f f e c t c o r r e c t i o n f o r s t a i n l e s s s t e e l t u b e s ( w h i c h w e re u s e d i n o u r He c r y o s t a t ) i s l a r g e r t h a n t h a t p r e d i c t e d by t h e W e b e r - S c h m i d t e q u a t i o n . T h i s w o u l d mean t h a t o u r v a l u e s o f P 4 a r e t o o h i g h e s p e c i a l l y a t p r e s s u r e s b e l o w a b o u t 200u . H owever, t h i s p o i n t i s c o n t r o v e r s i a l . G o e l l n e r e t a l u s e d a c a p a c i t a n c e manometer o p e r a t i n g a t 4.2 K s o t h e y had no t h e r m o m o l e c u l a r 9 8 p r e s s u r e d i f f e r e n c e . F o r p r e s s u r e s n e a r 100 y, e q u i l i b r a t i o n t i m e s f o r t h e a d s o r p t i o n o f h y d r o g e n were g r e a t e r t h a n an h o u r s o o u r p r e s s u r e s i n t h i s r a n g e c o u l d be t o o h i g h . U s i n g P o i n t B, we h a v e e s t i m a t e d m o n o l a y e r c o v e r a g e t o be 6.63 c m 3 ( S T P ) / g +3% f o r 4 H e . F o r a m o n o l a y e r d e n s i t y o f °-2 .115 A , t h i s c o r r e s p o n d s t o a s p e c i f i c s u r f a c e a r e a o f 2 15.6 m / g f o r t h e G r a f o i l i n c e l l B - somewhat l e s s t h a n t h e 2 u s u a l r a n g e o f 20-30 m^/g f o r G r a f o i l b u t q u i t e c l o s e t o G o e l l n e r e t a l . Upon c o m p l e t i o n o f an NMR e x p e r i m e n t , t h e h y d r o g e n was q u i c k l y d e s o r b e d and t r a n s f e r r e d t o t h e Raman c e l l f o r m e a s u r e -ment o f t h e J = l c o n c e n t r a t i o n . The p r o c e d u r e f o r d o i n g t h i s i s a s f o l l o w s . F i r s t t h e l i q u i d 4 H e was b o i l e d o f f u n t i l i t s l e v e l was b e l o w t h e 4 K p l a t e . The h y d r o g e n f i l l l i n e was t h e n h e a t e d a t a r a t e o f 2.3 W u n t i l T 3 = 10 K. Then t h e 3 H e p o t was a l s o h e a t e d a t 2.3 W. M e a n w h i l e , one arm o f t h e e v a c u a t e d Raman c e l l was e x p o s e d t o a s t r e a m o f c o l d 4 H e g a s f r o m a l i q u i d 4 H e s t o r a g e d e war. A f t e r t h e 3 H e p o t had b e e n h e a t e d f o r 5-8 m i n , t h e p r e s s u r e P 4 a p p r o a c h e d a maximum and t h e h y d r o g e n was a l l o w e d t o e n t e r t h e Raman c e l l . I t l i q u i f i e d i n t h e c e l l s o t h a t w i t h i n a m i n u t e v i r t u a l l y a l l o f i t was remov e d f r o m t h e c r y o s t a t . The c e l l was t h e n s e a l e d a n d t a k e n t o t h e Raman s p e c t r o m e t e r . 5.6 C o n v e r s i o n R a t e M e a s u r e m e n t s H 9 m o l e c u l e s w i t h 1=1 p r o d u c e a d i p o l a r m a g n e t i c f i e l d , 99 which re s u l t s in a f i e l d gradient at neighbouring molecules. This i s the source of the i n t r i n s i c conversion of H 2. It leads to a second order rate equation dc/dt = -kc 2 (V-4) which i s obeyed in bulk s o l i d H 2 with k = 1.90(5)%/h ( S i l v e r a , 1980). The same process occurs in D 2« In addition, 1=1 mole-cules interact with the dipolar f i e l d s produced by the 1=2 molecules and, since deuterons have a nuclear spin i = l , they also interact with the e l e c t r i c f i e l d gradient produced by those D 2 molecules which have an e l e c t r i c quadrupole moment (1=1 or 2). Thus, the rate equation contains both f i r s t and second order terms dc/dt = -kc 2 - k'c(l-c) (V-5) For bulk s o l i d D 2, i t has been found that k=k' = ,060(3)%/h. Consequently, the rate equation i s approximately f i r s t order. For hydrogen adsorbed on graphite, one might expect interactions with the surface to catalyze conversion, p a r t i c u l a r l y i f there were paramagnetic impurities or dangling bonds with associated unpaired electrons. Conversion would then be a f i r s t order process dc/dt = -kc (V-6) We have determined the conversion rate of o-H2 and p-D 2 in the /3 x /3~ s o l i d phase on G r a f o i l by measuring the time dependence of the int e g r a l of the NMR adsorption spectrum S. It i s proportional to the s t a t i c magnetization M produced 100 by the magnetic f i e l d H Q i n the absence of the RF e x c i t a t i o n f i e l d H^. In the high temperature l i m i t , which p e r t a i n s i n our case, M Q obeys the C u r i e Law M Q = C/T (V-7) For N n u c l e i with s p i n I and gyromagnetic r a t i o y, the C u r i e c o n s t a n t C i s g i v e n by (Abragam, 1961) C= N y 2 f t 2 I (I+l)H Q/3k (V-8) For s u f f i c i e n t l y low v a l u e s of S i s p r o p o r t i o n a l to H-^  and the shape of the spectrum i s independent of H^. 2 7 However, i f the s a t u r a t i o n parameter T T ^ i where Tj^ and T 2 are the 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 times, becomes comparable to 1, S i n c r e a s e s more s l o w l y than H-^  and the shape of the spectrum w i l l depend on H-^ . U s u a l l y f o r s o l i d s , i n c r e a s e s as T decreases so s a t u r a t i o n becomes more l i k e l y . In order t h a t we be a b l e t o p l o t data o b t a i n e d under d i f f e r e n t c o n d i t i o n s on the same graph, S has been normal i z e d . The normalized amplitude s i s d e f i n e d by s = a S/dnNGQV^ (V-9) where a = a r b i t r a r y constant m = number of sweeps through the resonance N = number of hydrogen molecules G = t o t a l g a i n of the measuring system Q = q u a l i t y f a c t o r o f the NMR tuned c i r c u i t V^ = RF input v o l t a g e to the tuned c i r c u i t QWi i s the RF v o l t a g e a c r o s s the c o i l which c o n t a i n s the 101 G r a f o i l . In the absence of saturation, s should be proportional to the f r a c t i o n of hydrogen molecules with 1=1 and to T - 1 but independent of a l l other parameters. Our f i n a l H 2 spectra were obtained from three samples l a b e l l e d G19, G20, and G21. A l l of these were contained in c e l l B. The f i r s t two samples were used to study the temperature dependence of the spectra at high J=l concentrations and were removed a f t e r the hydrogen had been adsorbed for a t o t a l time t of less than 80 h. Sample G21 was studied u n t i l c had dropped to .2 aft e r nearly 400 h. Fig. 20 i s a plot of sT 3 versus t for t h i s sample at 1.81, .575, and .333 K. At each temperature the data were f i t t e d to the equation l n (sT 3) = -kt + l n ( s i T 3 ) (V-10) with k and s^ as parameters. (V-10) i s derived from the f i r s t order rate equation (V-6). The 1.81 K data has the le a s t scatter because i t i s near the temperature at which the S:N r a t i o was the highest. At higher temperatures, the absorption signal was a l i t t l e narrower but i t s amplitude had dropped because of the Curie Law. At temperatures below 1 K, the signal broadened considerably so that the amplitude dropped despite the net increase in area due to the Curie Law. In addition, there was coherent noise (discussed in Chapter VII) which became more severe at low temperatures. The scatter in the data increased with time because of the decrease i n J=l concentration c. o ° 1.0 H o CO .6 " . 4 -0 1 0 0 2 0 0 t (h) 3 0 0 400 F i g . 20 G r a p h o f s T 3 v e r s u s t h e t i m e s i n c e a d s o r p t i o n , t , f o r H 2 s a m p l e G21 a t : ( a ) T 3=1.81 K ( c i r c l e s , s o l i d l i n e ) , ( b) T 3=.575 K ( s q u a r e s , d o t - d a s h e d l i n e ) , and ( c ) T 3=.333 K ( d i a m o n d s , d o t t e d l i n e ) . A l l o f t h e d a t a have t h e same RF i n p u t a t t e n u a t i o n , A=13 dB. The l i n e s a r e two p a r a m e t e r l e a s t s q u a r e s f i t s t o t h e d a t a a t e a c h t e m p e r a t u r e . The s l o p e s k and i n t e r c e p t s s ^ T 3 a r e g i v e n i n T a b l e V I I . 103 The r e s u l t s o f t h e two p a r a m e t e r l e a s t s q u a r e s f i t t o ( V - 1 0) a r e shown i n T a b l e V I I . The d a t a w e re w e i g h t e d by t h e s q u a r e o f t h e S:N r a t i o . The v a l u e s o f s^T^ a t 1.81 K and .575 K a r e a b o u t t h e same b u t s^T^ i s d e c i d e d l y l o w e r a t .333 K. The r e a s o n i s d i s c u s s e d i n s e c t i o n 7.3. On t h e o t h e r h a n d , t h e r e i s m o d e r a t e l y g o o d a g r e e m e n t b e t w e e n t h e t h r e e v a l u e s o f k. C o n s e q u e n t l y t h e w e i g h t e d mean o f t h e t h r e e v a l u e s , k = . 4 0 ( 2 ) % / h , was u s e d and one p a r a m e t e r l e a s t s q u a r e s f i t w ere made t o t h e d a t a a t e a c h t e m p e r a t u r e t o f i n d t h e new y - i n t e r c e p t s s^'T-j ( s e e T a b l e V I I ) . A g r a p h o f l / s T 3 was a l s o p l o t t e d ; i t w o u l d be l i n e a r i f t h e r a t e e q u a t i o n were s e c o n d o r d e r . The f i t t o a s t r a i g h t l i n e was much p o o r e r t h a n i n F i g . 20. T a b l e V I I O-H^ C o n v e r s i o n R a t e C o n s t a n t s T 3 ( K ) k ( % / h ) s i T 3 s i ' T 3 1.81 .402 ( 2 7 ) 3.43 (15) 3.40 .575 .355 (28) 3.67 (14) 3.82 .333 .457 (36) 2.40 (10) 2.27 H a v i n g m e a s u r e d t h e c o n v e r s i o n r a t e , i t was n e c e s s a r y t o d e t e r m i n e t h e a b s o l u t e c o n c e n t r a t i o n s o f t h e H 2 s a m p l e s . An a b s o l u t e c a l i b r a t i o n o f t h e NMR s p e c t r o m e t e r w o u l d h a v e b e e n d i f f i c u l t b e c a u s e i t w o u l d h a v e e n t a i l e d a c a l i b r a t i o n o f t h e s p e c t r o m e t e r g a i n . T h i s m i g h t h a v e b e e n done by m e a s u r i n g S f o r a known q u a n t i t y o f o-D 9 o r HD. I t w o u l d h a v e b e e n n e c e s s a r y 104 t o e n s u r e t h a t t h e r e was no s a t u r a t i o n o r RF h e a t i n g by c h e c k i n g t h e s i g n a l a t l o w e r RF l e v e l s , r e d u c i n g t h e a l r e a d y m a r g i n a l S:N r a t i o s . I n s e c t i o n 7.4, we p r e s e n t e v i d e n c e t h a t t h e 0-D2 s i g n a l was s a t u r a t i n g a t t h e n o r m a l RF l e v e l s u s e d f o r 0-H2 and p-D 2 s a m p l e s . We were a b l e t o a v o i d t h e c a l i b r a t i o n by m e a s u r i n g t h e J = l c o n c e n t r a t i o n b e f o r e a d s o r p t i o n a n d a f t e r d e s o r p t i o n u s i n g t h e r o t a t i o n a l Raman s p e c t r a . The amount o f c o n v e r s i o n d u r i n g a d s o r p t i o n and d e s o r p t i o n was unknown b u t i t was p o s s i b l e t o make e s t i m a t e s b a s e d o n t h e t i m e f o r a d s o r p t i o n a n d d e s o r p t i o n , t h e i n i t i a l a n d f i n a l c o n c e n t r a t i o n s , a n d t h e r e l a t i v e c h a n g e o f c o n c e n t r a t i o n o b t a i n e d f r o m t h e NMR m e a s u r e m e n t s . I n a d d i t i o n , t h e NMR s p e c t r a o f H2 s a m p l e G21 p r o v i d e d i n f o r m a t i o n a b o u t t h e s h a p e o f t h e NMR s p e c t r a a t a f i x e d t e m p e r a t u r e a s a f u n c t i o n o f t i m e . F o r e x a m p l e , u s i n g t h a t i n f o r m a t i o n , i t was c l e a r t h a t f o r a G21 s p e c t r u m t a k e n a t t=3.55 h and T 3=.573 K, c must h a v e b e e n l o w e r t h a n f o r two G20 s p e c t r a t a k e n a t t=10.19 h and T3 = .382 K and a t t = 7.88 h and T3 = .563 K. T a b l e V I I I c o n t a i n s t h e d a t a u s e d t o e s t i m a t e t h e c o n c e n -t r a t i o n c^ a t t h e c o m p l e t i o n o f a d s o r p t i o n ( t = 0 ) . The c o n c e n -t r a t i o n s b e f o r e a d s o r p t i o n a n d a f t e r d e s o r p t i o n a s m e a s u r e d b y Raman s p e c t r o s c o p y a r e c a a n d c ^ . The t i m e s e l a p s e d d u r i n g a d s o r p t i o n a n d d e s o r p t i o n a r e t a and t ^ . The t i m e b e t w e e n a d s o r p t i o n and d e s o r p t i o n i s t f . U s i n g t h e s e v a l u e s o f c ^ , a g r a p h o f c / c ^ v e r s u s t h a s b e e n p l o t t e d i n F i g . 2 1 . The d a t a f o r G21 and G19 f i t t h e 105 F i g . 21 G r a p h o f c / c ^ v e r s u s t f o r t h r e e s a m p l e s , a l l i n c e l l B, a t t h r e e t e m p e r a t u r e s T 3 : (a) G21,- 1.81 K ( o p e n c i r c l e s ) ; (b) G21, .575 K ( o p e n s q u a r e s ) ; ( c ) G21, .333 K ( o p e n d i a m o n d s ) ; (d) G21, Raman ( # ) ; ( e ) G19, 1.8 K ( s o l i d c i r c l e s ) ; ( f ) G19, .57 K ( s o l i d s q u a r e s ) ; (g) G19, .31 K ( s o l i d d i a m o n d s ) ; ( h ) G19, Raman ( * ) ; ( i ) G20, 1.8 K ( x ) ; ( j ) G20, .57 K ( o p e n t r i a n g l e s ) ; ( k ) G20, .33 K ( i n v e r t e d o p en t r i a n g l e s ) ; and (1) G20, Raman ( + ) . I n e v e r y c a s e , A=13 dB. The s t r a i g h t l i n e h a s a s l o p e k=.40%/h, w h i c h was o b t a i n e d f r o m t h e G21 d a t a . 106 s t r a i g h t l i n e f a i r l y w e l l . A l l o f t h e d a t a f o r G20 f a l l o n o r b e l o w t h e l i n e . N e v e r t h e l e s s , t h e v a l u e o f t a k e n i n t h a t c a s e i s s u p p o r t e d by t h e c o m p a r i s o n o f t h e s h a p e s o f some o f t h e s p e c t r a a s d e s c r i b e d a b o v e and by t h e Raman r e s u l t s . F o r G19 and G20, c d was a b o v e t h e e q u i l i b r i u m c o n c e n t r a t i o n a t a l l t e m p e r a t u r e s d u r i n g d e s o r p t i o n s o i t i s a l o w e r l i m i t f o r c ( t = t f ) . F o r G21, c d was e q u a l t o c e q a t T=40 K, a t e m p e r a t u r e a t w h i c h t h e h y d r o g e n w o u l d h a v e b e e n p a r t i a l l y d e s o r b e d . T h e r e f o r e c ^ c o u l d be e i t h e r s m a l l e r o r g r e a t e r t h a n c ( t = t f ) . T a b l e V I I I . I n i t i a l J = l C o n c e n t r a t i o n s o f t h e S a m p l e s S a m p l e c a C d t ( m i n ) a t .(min) d t f (h) c . l c . e I + .02 + .03 G21 >. 96 .17 8 13.5 388.10 .93 .20 G20 >. 98 .79 5 14 43.00 .96 .81 G19 >. 98 .56 12 16 76.38 .95 .70 k t f F o r D 2 r c o n v e r s i o n r a t e m e a s u r e m e n t s were l e s s p r e c i s e t h a n f o r H 2 b e c a u s e t h e S:N r a t i o was much p o o r e r and t h e r a t e c o n s t a n t was s m a l l e r . C o h e r e n t n o i s e made t h e d e t e r m i n a t i o n o f t h e b a s e l i n e o f t h e s p e c t r a v e r y d i f f i c u l t i n many c a s e s . D a t a c o n c e r n i n g t h e e f f e c t s o f t h e t e m p e r a t u r e a n d t h e RF l e v e l on t h e m a g n i t u d e o f t h e i n t e g r a l o f t h e a b s o r p t i o n s p e c t r u m w ere m i n i m a l ( s e e s e c t i o n 7 . 4 ) . T h o s e J=0 m o l e c u l e s t h a t h a v e 1=2 g i v e a n a r r o w l i n e a t t h e c e n t r e o f t h e s p e c t r u m . A t h i g h t e m p e r a t u r e s , t h e J = l m o l e c u l e s p r o d u c e a d o u b l e t w i t h a s p l i t t i n g t h a t i n c r e a s e s a s T d e c r e a s e s : t h e same as f o r H 2» N e a r 1 K, t h e b r o a d e n i n g b e g i n s t o i n c r e a s e much more r a p i d l y t h a n t h e s p l i t t i n g s o t h a t a t l o w t e m p r a t u r e s t h e d o u b l e t t r a n s f o r m s i n t o a weak, b r o a d hump. C o n s e q u e n t l y , t h e c o n v e r s i o n r a t e m e a s u r e m e n t s were r e s t r i c t e d t o T-j>. 1 » 6 K« A b o v e 4 K t h e s p l i t t i n g becomes s m a l l and i t i s d i f f i c u l t t o s e p a r a t e t h e J=0 and J = l c o m p o n e n t s : t h o s e s p e c t r a were n o t u s e d e i t h e r . F o r 1.6 K i . T 3 - i 4 • 2 K ' t n e n o r m a l i z e d i n t e g r a l s s Q a n d s^ o f t h e J=0 and J = l c o m p o n e n t s o f t h e a d s o r p t i o n s p e c t r a o b e y e d t h e C u r i e Law ( s e e s e c t i o n 7 . 4 ) . Had i t b e e n n e c e s s a r y t o r e l y on s-^ a l o n e t o d e t e r m i n e t h e c o n v e r s i o n r a t e , i t w o u l d n o t h a v e b e e n p o s s i b l e t o g e t a r e l i a b l e e s t i m a t e b e c a u s e t h e s c a t t e r was s o l a r g e . H o wever, t h e a b s o l u t e J = l c o n c e n t r a t i o n c a n be o b t a i n e d f r o m t h e r a t i o s-,/s^ s i n c e 1 o S l / s o = ^ I 1 ( I 1 + 1 ) / I 2 ( I 2 + 1 ) } c / { ( 5/ 6> ^ - c » ( V - l l a ) = ( 2 / 5 ) c / ( l - c ) ( V - l l b ) w h e r e I i = l a n d I 2 = 2 . T n e f a c t o r o f 5/6 i s t h e f r a c t i o n o f J=0 m o l e c u l e s t h a t h a v e 1=2. The D 2 c o n v e r s i o n r a t e d a t a i s n o t e x t e n s i v e e n o u g h t o a s c e r t a i n w h e t h e r t h e r a t e e q u a t i o n i s f i r s t o r s e c o n d o r d e r . H o w e v e r , f o r H 2 c o n v e r s i o n was f o u n d by e x p e r i m e n t t o f o l l o w a f i r s t o r d e r r a t e e q u a t i o n s o t h a t i n t h i s c a s e c o n v e r s i o n must • 3 - l F i g . 22 G r a p h o f - l n c v e r s u s t f o r D? s a m p l e G18 i n c e l l B. 1.6 K <T 3<4.2 K. The RF i n p u t a t t e n u a t i o n was A=13 dB ( c i r c l e s ) o r A=19 dB ( d i a m o n d ) . The c o n c e n t r a t i o n s b e f o r e a d s o r p t i o n c g and a f t e r d e s o r p t i o n c d , d e t e r m i n e d by Raman s c a t t e r i n g , a r e a l s o shown ( * ) . The l i n e i s a l e a s t s q u a r e s f i t t o t h e NMR d a t a . I t g i v e s c i = . 9 9 ( 2 ) and k = . 0 6 9 ( 1 5 ) % / h . o CO 109 be c a t a l y z e d b y t h e s u r f a c e . P r e s u m a b l y t h e same i s a l s o t r u e f o D 2 g i v e n t h a t i t s i n t r i n s i c c o n b e r s i o n r a t e i s l e s s t h a n t h a t o f H 2 > A g r a p h o f - I n c v e r s u s t f o r D 2 s a m p l e G18 i n c e l l B i s shown i n F i g . 2 2 . The Raman m e a s u r e m e n t s b e f o r e a d s o r p t i o n and a f t e r d e s o r p t i o n a r e a l s o shown. The f o r m e r i s a l o w e r l i m i t on - I n c a n d t h e l a t t e r i s a n u p p e r l i m i t . A l e a s t s q u a r e s f i t t o t h e NMR d a t a g i v e s c i = . 9 9 ( 2 ) and k = . 0 6 9 ( 1 5 ) % / h . Some D 2 c o n v e r s i o n r a t e m e a s u r e m e n t s w e r e a l s o made f o r c e l l A w h i c h c o n t a i n e d a d i f f e r e n t t y p e o f G r a f o ' i l f r o m c e l l B (s e e s e c t i o n 6 . 3 ) . F i g . 23 i s a p l o t o f - I n c v e r s u s t o b t a i n e d f r o m P _ D 2 s a m p l e G7 i n c e l l A. The d a t a c o v e r s a v e r y l i m i t e d c o n c e n t r a t i o n r a n g e f r o m .94 t o .97. A l e a s t s q u a r e s f i t t o a f i r s t o r d e r r a t e e q u a t i o n g i v e s c^=.976(4) a nd k = . 0 2 6 ( 7 ) % / h . The r a t e c o n s t a n t i s o n l y .38 o f t h a t m e a s u r e d f o r c e l l B; i t i s n o t s u r p r i s i n g t h a t d i f f e r e n t t y p e s o f G r a f o i l g i v e d i f f e r e n t c o n v e r s i o n r a t e s . 110 Fig. 23 Graph of -In c versus t for D 2 sample G7 in c e l l A. .3 K £To£1.4 K. A=14 dB. The l i n e i s a least squares f i t . I t gives ci=.976(4) and k=.026(7)%/h. I l l N o t e s t o C h a p t e r V 1. A l c o a C h e m i c a l s , a c t i v a t e d a l u m i n a g r a d e F - l , mesh 8-14. 2. The d a t a were a c t u a l l y t a k e n f r o m Novaco (1975). The d a t a o f G o e l l n e r e t a l f o r P„<388y were n o t t a b u l a t e d . 112 CHAPTER V I THE NMR CRYOSTATS 6.1 The 4 H e C r y o s t a t Our f i r s t NMR m e a s u r e m e n t s f o r h y d r o g e n a d s o r b e d on G r a f o i l were made i n t h e t e m p e r a t u r e r a n g e 1.3-4.2 K u s i n g a c r y o s t a t w h i c h was immersed i n l i q u i d 4 H e . The p r i m a r y p u r p o s e s o f t h e m e a s u r e m e n t s were t o d e t e r m i n e t h e c r y s t a l f i e l d a nd t o s e a r c h f o r e v i d e n c e o f o r i e n t a t i o n a l o r d e r i n g . I n t h e s e e x p e r i m e n t s , t h e G r a f o i l was a t t h e b o t t o m o f a l o n g P y r e x wand w i t h a q u a r t z t i p ( F i g . 1 7 ) , w h i c h c o u l d be l o w e r e d i n t o t h e l i q u i d 4 H e . The t e m p e r a t u r e was c o n t r o l l e d by pumping on t h e b a t h u s i n g e i t h e r a r o t a r y pump t h r o t t l e d by a W a l k e r r e g u l a t o r ( W a l k e r , 1959) o r a d i f f u s i o n e j e c t o r pump. The 4 H e v a p o u r p r e s s u r e was m e a s u r e d by a m e r c u r y m a n o m eter, an o i l m a n o m e t e r , o r a McLeod gauge d e p e n d i n g upon t h e p r e s s u r e . C o r r e c t i o n s f o r t h e t h e r m o m o l e c u l a r p r e s s u r e e f f e c t were made a t t h e l o w e s t p r e s s u r e s . The p r e s s u r e was c o n v e r t e d t o t h e t e m p e r a t u r e o f t h e 4 H e b a t h u s i n g t h e 1958 4 H e t e m p e r a t u r e s c a l e ( v a n D i j k e t a l , 1 9 6 0 ) . The t i p o f t h e 9 mm o u t s i d e d i a m e t e r (OD) wand f i t s n u g l y i n s i d e a c o i l h o l d e r c o n t a i n i n g a 10 mm l o n g , 10 t u r n , 1.1 yH c o p p e r c o i l w h i c h p r o d u c e d t h 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 t o e x c i t e t h e s p i n s i n t h e s a m p l e . The c o i l , composed o f 26 AWG w i r e , was w r a p p e d o n a t h i n , t h r e a d e d T e f l o n c y l i n d e r w h i c h was p l a c e d i n s i d e a more r i g i d o u t e r c y l i n d e r o f T e f l o n . The 113 t h e r m a l c o n t r a c t i o n o f T e f l o n e x c e e d s t h a t o f b o t h c o p p e r and q u a r t z s o t h e w h o l e a s s e m b l y f i t t i g h t l y t o g e t h e r a t l o w t e m p e r a t u r e s , r e d u c i n g t h e p o s s i b i l i t y o f m i c r o p h o n i c s . M o s t o f t h e NMR m e a s u r e m e n t s were made u s i n g a s a m p l e c o n s i s t i n g o f 39 g o f .25 mm t h i c k , r e c t a n g u l a r G r a f o i l 1 3 s h e e t s w i t h a d e n s i t y o f 1.12 g/cm . T h e s e were i n t e r -l e a v e d w i t h .01 mm T e f l o n s h e e t s i n o r d e r t o r e d u c e RF s h i e l d i n g s i n c e G r a f o i l i s an e l e c t r i c a l c o n d u c t o r (Hedge e t a l , 1 9 7 3 ) . C o p p e r p l a t e s were i n s e r t e d a g a i n s t t h e o u t e r G r a f o i l s h e e t s t o s q u e e z e t h e s a m p l e t i g h t l y i n s i d e t h e q u a r t z t u b e . T h i s i m p r o v e d t h e t h e r m a l c o n t a c t b e t w e e n t h e G r a f o i l a nd t h e q u a r t z t u b e and h e l p e d t o p r e v e n t m i c r o p h o n i c s . P r i o r t o a s s e m b l y o f t h e s a m p l e , t h e G r a f o i l s h e e t s had b e e n b a k e d u n d e r vacuum a t 1000 C f o r 12 h and p u r g e d s e v e r a l t i m e s w i t h p u r e H 2 g a s . The G r a f o i l was e x p o s e d t o a i r w h i l e t h e s a n d w i c h was b e i n g a s s e m b l e d and was t h e n b a k e d u n d e r vacuum a t 220 C f o r 15 h and p u r g e d w i t h H 2• We were c o n c e r n e d t h a t T e f l o n v a p o u r m i g h t c o n t a m i n a t e t h e G r a f o i l s o a s e c o n d s a m p l e was c o n s t r u c t e d i n w h i c h t h e T e f l o n was r e p l a c e d by .1 mm t h i c k s a p p h i r e p l a t e s . A f t e r a s s e m b l y t h i s s a m p l e was b a k e d a t 900 C. I t g a v e i d e n t i -c a l NMR s i g n a l s t o t h e G r a f o i l and T e f l o n s a m p l e . We e m p l o y e d an 8.5 MHz c o n t i n u o u s wave, s i n g l e c o i l NMR s p e c t r o m e t e r . The a b s o r p t i o n s i g n a l ( t h e i m a g i n a r y p a r t o f t h e s u s c e p t i b i l i t y ) was o b s e r v e d by s w e e p i n g t h e m a g n e t i c f i e l d t h r o u g h t h e r e s o n a n c e , w h i c h was a t .2 T f o r H 2 and 1.3 T f o r D 9. S i m u l t a n e o u s l y , t h e f i e l d was m o d u l a t e d a t 280 Hz by 114 .17 mT pe a k t o p e a k . The AC component o f t h e a m p l i f i e d v o l t a g e a c r o s s t h e c o i l was t h e n d e m o d u l a t e d u s i n g a l o c k - i n d e t e c t o r and f e d i n t o a s i g n a l a v e r a g e r . T h i s y i e l d e d t h e d e r i v a t i v e o f t h e a b s o r p t i o n , w h i c h c o u l d be i n t e g r a t e d u s i n g t h e " i n t e g r a t e " f a c i l i t y o f t h e s i g n a l a v e r a g e r . I n o r d e r t o p r e v e n t s p u r i o u s NMR s i g n a l s f r o m b e i n g d e t e c t e d when we o b s e r v e d t h e r e s o n a n c e o f t h e H 2 m o l e c u l e s , f a i r l y e x a c t i n g m e a s u r e s were n e c e s s a r y t o k e e p t h e number o f *H n u c l e i n e a r t h e c o i l a t an a c c e p t a b l y l o w l e v e l . The m o d u l a t i o n c o i l s , composed o f e n a m e l l e d c o p p e r w i r e , w e re c o v e r e d by a l u m i n u m f o i l t o s h i e l d them f r o m t h e RF f i e l d . I n i t i a l l y t h e G r a f o i l had b e e n p l a c e d i n s i d e a P y r e x t u b e b u t i t was f o u n d t h a t a t 4 K t h e P y r e x p r o d u c e d an a b s o r p t i o n p e a k w i t h a f u l l w i d t h a t h a l f maximum (FWHM) o f .16 mT. I t s a m p l i t u d e was c o m p a r a b l e t o t h a t o f t h e o - H 2 s i g n a l w h i c h had a FWHM o f .56 mT a t 4 K. No *H s i g n a l c o u l d be d e t e c t e d i n q u a r t z s o t h e P y r e x was r e p l a c e d by q u a r t z . D e s p i t e t h e s e m e a s u r e s , a t r i a n g u l a r "''H s i g n a l w i t h a FWHM o f 1.1 mT p e r s i s t e d . T e f l o n g a v e an ^H s i g n a l o f t h e same s h a p e . I t c o u l d be r e d u c e d by b a k i n g t h e T e f l o n u n d e r vacuum a t 220 C f o r two d a y s s o t h e T e f l o n c o i l h o l d e r and T e f l o n s h e e t s i n t h e wand w e r e a l s o b a k e d u n d e r vacuum. A f t e r w a r d s , t h e b a c k g r o u n d s i g n a l h a d an a m p l i -t u d e o f a b o u t one t h i r d t h a t o f o-H 2 a t 4.2 K; t h e b a c k g r o u n d was s t i l l s i g n i f i c a n t . C o n s e q u e n t l y , i t was n e c e s s a r y t o m e a s u r e t h e b a c k g r o u n d s i g n a l b e f o r e t h e H 2 was a d s o r b e d o n t h e G r a f o i l and l a t e r s u b t r a c t i t f r o m t h e s i g n a l o b t a i n e d w i t h 115 0-H2 a d s o r b e d . P r e s u m a b l y t h e b a c k g r o u n d s i g n a l was f r o m t h e T e f l o n , a l t h o u g h we c o u l d n o t r u l e o u t t h e p o s s i b i l i t y t h a t some o t h e r m a t e r i a l p r o d u c e d an x H s i g n a l o f t h e same sh a p e . In o r d e r t o m a i n t a i n t h e h o m o g e n e i t y o f t h e m a g n e t i c f i e l d , no s u p e r c o n d u c t i n g m e t a l s c o u l d be a l l o w e d n e a r t h e NMR c o i l . F o r t y p e I s u p e r c o n d u c t o r s below t h e i r c r i t i c a l t e m p e r a -t u r e T Q , m a g n e t i c f l u x p e n e t r a t i o n i s z e r o ( a p a r t f r o m a v e r y t h i n r e g i o n a t t h e s u r f a c e ) i n a m a g n e t i c f i e l d H l e s s t h a n a c r i t i c a l v a l u e H c and i t i s c o m p l e t e f o r H > H C « F o r t y p e I I s u p e r c o n d u c t o r s , i t i s z e r o f o r H<H C^ and c o m p l e t e f o r H>H c 2 where H c l H c 2 = H c 2 (VI-1) T y p i c a l l y H c 2 ~ H c l ' V / . 1-. 5 H c. H Q was above H c f o r any t y p e I s u p e r c o n d u c t o r b u t d i d n o t e x c e e d H c 2 o f many t y p e I I s u p e r c o n d u c t o r s , i n p a r t i c u l a r some s o l d e r s . Many o f t h e s e a l s o have T c>4 K. S o l d e r j o i n t s n e a r t h e NMR c o i l were made u s i n g t h e f o l l o w i n g m a t e r i a l s : I n ( t y p e I , T c = 3.4 K, H c = .0293 T ) , Pb ( t y p e I , T c = 7.2 K, H c = .080 T ) , and 60 Sn/40 Pb [ t y p e I I , T c = 7.0 K, H c = .083 T, ( H c 2 ~ H c l ) / H c = . 1 8 ] . 2 6.2 The 3 H e C r y o s t a t The NMR measurements made u s i n g t h e 4 H e c r y o s t a t a l l o w e d us t o d e t e r m i n e (v c| and r f o r H 2 and D 2 b u t d i d n o t show any e v i d e n c e o f c o o p e r a t i v e o r i e n t a t i o n a l o r d e r i n g ( K u b i k and H a r d y , 1 9 7 8 ) . However, t h e y d i d show t h a t EQQ i n t e r a c t i o n s were i m p o r t a n t , a t l e a s t f o r D 2« T h i s m o t i v a t e d us t o b u i l d a 3 He c r y o s t a t t h a t w o u l d a l l o w us t o e x t e n d t h e m e a s u r e m e n t s down t o .3 K. The c r y o s t a t was a l s o b u i l t w i t h a v i e w t o e x t e n d i n g t h e m e a s u r e m e n t s - up t o 25 K i n o r d e r t o s t u d y t h e s o l i d - f l u i d t r a n s i t i o n . Due t o l a c k o f t i m e , no m e a s u r e m e n t s have b e e n made a b o v e 12 K: t h e S:N r a t i o d e t e r i o r a t e s r a p i d l y a b o v e 8 K. We c o u l d h a v e a t t a i n e d e v e n l o w e r t e m p e r a t u r e s u s i n g a d i l u t i o n r e f r i g e r a t o r b u t t h i s w o u l d h a v e b e e n more d i f f i c u l t t o c o n s t r u c t . I t a l s o seemed v e r y l i k e l y t h a t t h e i n t e r e s t i n g o r i e n t a t i o n a l b e h a v i o u r w o u l d o c c u r a b o v e .3 K. 3 The p r i n c i p l e o f He c r y o s t a t s i s s i m p l e and g o o d d i s c u s s i o n s o f t h e i r d e s i g n a r e c o n t a i n e d i n R o s e - I n n e s ( 1 9 7 3 ) and Lounasmaa ( 1 9 7 4 ) . The t a i l o f o u r c r y o s t a t i s d e p i c t e d i n F i g . 24. The c r y o s t a t was imm e r s e d i n a b a t h o f l i q u i d 4 H e . The f i r s t c o o l i n g s t a g e b e l o w 4 K was a c o n t i n u o u s l y o p e r a t i n g 4 H e r e f r i g e r a t o r w h i c h s e r v e d t o c o n d e n s e t h e 3He g a s f r o m a 3 t a n k a t room t e m p e r a t u r e and t o t h e r m a l l y s h i e l d t h e He p o t . L i q u i d 4 H e was pumped f r o m t h e b a t h t h r o u g h a f l o w i m p e d a n c e i n t o t h e 4 H e p o t where i t b o i l e d a t 1.3 K. When t h e v a l v e on 3 3 t h e J H e s u p p l y t a n k was o p e n e d , 1.6 l ( S T P ) o f He g a s was a d s o r b e d by a c h a r c o a l a d s o r p t i o n pump a t s u b - m o n o l a y e r 3 c o v e r a g e . The He v a l v e a t t h e t o p o f t h e c r y o s t a t was c l o s e d a nd t h e pump was h e a t e d a t .5 W t o d e s o r b t h e J H e , w h i c h w o u l d t h e n c o n d e n s e a s i t p a s s e d t h r o u g h t h e 4 H e p o t and d r i b b l e i n t o t h e 3 H e p o t . A f t e r 20 m i n t h e pump t e m p e r a t u r e r e a c h e d 35 K and n e a r l y a l l o f t h e 3 H e was d e s o r b e d . I t was t h e n 117 H 2 fill line 4 H Hea t e r Vacuum j a c k e t e pumping line F i l t e r o o p p e r wire pump) 4 H e f i l l line F l o w i m p e d a n c e ^He p o t S ta in less steel be l l ows ( H 2 ) oppe r 3 He pot R, a n d R 2 Pump valve stem ft^He f i l l line F) Vapour pressure line Vacuum jacke t •j-Vacuum c an pump ing line 4 K p l a t e | v — P u m p K e a t e r A d s o r p t i o n pump C o a x i a l c a b l e s Superconduc t i ng w i r e ^—Copper plate (4K " 3 He pumping line) He pumping line V apou r pressure l ine. Epoxy R, >PP G r a f o i l cell ( q u a r t z ) T e f l o n ho l de r f o r t h e N M R c o i l C apa c i t a n c e t h e r m o m e t e r Coppe r rod (Grafo i l _ J H e po t ) C l a m p F i g . 24 The t a i l of the He c r y o s t a t , drawn approximately to s c a l e . The s e c t i o n below the 4 K p l a t e was enclosed i n a vacuum can, which was s o l d e r e d to the 4 K p l a t e . n e c e s s a r y t o t u r n o f f t h e pump h e a t e r o r t h e He p o t w o u l d be d r a i n e d and warm up t o 4 K. As t h e c h a r c o a l c o o l e d , i t w o u l d b e g i n t o a d s o r b He p r o d u c i n g a minimum t e m p e r a t u r e o f T3 = .295 K a f t e r a b o u t 30 m i n . O p e r a t i o n a t .3 K c o u l d be s u s t a i n e d 3 f o r 18 h b e f o r e t h e He had t o be r e c o n d e n s e d , i m p l y i n g a h e a t 3 l e a k o f 30yW i n t o t h e He p o t . F o r o p e r a t i o n a t h i g h e r 3 t e m p e r a t u r e s , i t was o f t e n n e c e s s a r y t o h e a t t h e He p o t ; a t a 3 t e m p e r a t u r e o f 1.2 K, t h e He o n l y l a s t e d a b o u t 3 h. Up t o .6 K, t h e t e m p e r a t u r e c o u l d be c o n t r o l l e d by means o f a t h r o t t l e v a l v e a t t h e b a s e o f t h e a d s o r p t i o n pump. T h i s c o n s i s t e d o f a n 8 mm l o n g , t a p e r e d b r a s s p l u g i n a 12.7 mm OD, s t a i n l e s s s t e e l t u b e . As t h e c h a r c o a l s u r f a c e became c o v e r e d 3 w i t h He t h e p u m p ing s p e e d d e c r e a s e d . However, t e m p e r a t u r e s t a b i l i t y o f <1 mK c o u l d be m a i n t a i n e d by o p e n i n g t h e t h r o t t l e v a l v e s l i g h t l y e v e r y 15 m i n o r s o . A l t e r n a t i v e l y , t h e t e m p e r a -t u r e c o u l d be c o n t r o l l e d u s i n g a L a k e s h o r e C r y o t r o n i c s CS-400 GR 3 S r T i 0 3 c a p a c i t a n c e s e n s o r on t h e He p o t , a d e t e c t o r c o n s i s t i n g o f a G e n e r a l R a d i o 1615A c a p a c i t a n c e b r i d g e w i t h a P r i n c e t o n A p p l i e d R e s e a r c h 122 l o c k - i n a m p l i f i e r , an A-05 t e m p e r a t u r e c o n t r o l l e r c o n s t r u c t e d by t h e d e p a r t m e n t a l e l e c t r o -3 n i c s s h o p , and a h e a t e r on t h e He p o t . T y p i c a l l y , 5 yW o f h e a t e r power was u s e d . The c a p a c i t a n c e s e n s o r e x h i b i t e d c o n s i d e r a b l y more d r i f t t h a n t h e c a r b o n and g e r m a n i u m r e s i s t o r s u s e d f o r t e m p e r a t u r e m e a s u r e m e n t s . The r a t e o f d r i f t i n c r e a s e d a s T d e c r e a s e d , s i n c e b e l o w 1 K t h e s e n s i t i v i t y o f t h e c a p a c i -t a n c e s e n s o r d r o p p e d . Even a t .3 K, t h i s was n o t a p r o b l e m b e c a u s e t h e d r i f t r a t e was o n l y -.3 p F / h , c o r r e s p o n d i n g t o a b o u t -2 mK/h. A t .6 K t h e d r i f t r a t e was -.05 p F / h , c o r r e s p o n d i n g t o -.3 mK/h. Ab o v e .6 K t h e t h r o t t l e v a l v e was k e p t c l o s e d . I n o r d e r 3 t o a v o i d p u t t i n g l a r g e amounts o f h e a t d i r e c t l y i n t o t h e He 3 3 p o t and b o i l i n g o f f He, t h e He p r e s s u r e i n t h e pump was r e g u l a t e d by h e a t i n g t h e pump i n o r d e r t o l o w e r i t s pumping s p e e d . The p r e s s u r e was c o n t r o l l e d u s i n g an MKS B a r a t r o n m o d e l 170 c a p a c i t a n c e manometer w i t h a 1000 t o r r s e n s o r and a n o t h e r A-05 t e m p e r a t u r e c o n t r o l l e r . I t was f o u n d t o be b e s t t o c o n t r o l 3 t h e He p o t t e m p e r a t u r e d i r e c t l y a s w e l l b e c a u s e t h e pump had a l o n g t h e r m a l t i m e c o n s t a n t . Once t h e p r e s s u r e had s t a b i l i z e d , r e g u l a t i o n by t h e pump p r e s s u r e a l o n e was a s g o o d a s w i t h t h e 3 c a p a c i t a n c e s e n s o r on t h e He p o t . A t T 3 = 1.3 K, t h e pump h e a t e r power was .2 W. I f t h e h e a t i n g was i n c r e a s e d much more t h e 4 H e p o t w o u l d b e g i n t o 3 d r a i n c a u s i n g t h e He t o b o i l o f f r a p i d l y . I t was i m p o s s i b l e t o o p e r a t e i n t h e r a n g e 1.3 K <T 3 <1.6 K f o r more t h a n a few m i n u t e s s o no NMR m e a s u r e m e n t s were made t h e r e . 3 B e t w e e n 1.6 K and 4.2 K, t h e He p o t was empty and i t s t e m p e r a t u r e was m a i n t a i n e d by t h r o t t l i n g t h e 4 H e p o t pump and u s i n g t h e c a p a c i t a n c e s e n s o r t o r e g u l a t e T 3 . The 4 H e p o t 3 c o u l d o p e r a t e a s l o w a s 1.3 K b u t t h e minimum He p o t t e m p e r a -t u r e d u r i n g NMR m e a s u r e m e n t s was 1.6 K due t o e d d y c u r r e n t h e a t i n g by t h e m a g n e t i c f i e l d m o d u l a t i o n and t o a l e s s e r e x t e n t by t h e RF f i e l d . A b o v e 4.2 K o n l y t h e c a p a c i t a n c e s e n s o r was 120 u s e d t o r e g u l a t e t h e t e m p e r a t u r e . •a Some o f t h e c o m p o n e n t s o f t h e He c r y o s t a t w i l l now be d i s c u s s e d i n more d e t a i l . The a d s o r p t i o n pump was a 9.0 cm l o n g , 3.5 cm d i a m e t e r s e m i - c i r c u l a r c o p p e r c y l i n d e r w h i c h c o n t a i n e d 16.3 g o f F i s h e r a c t i v a t e d c o c o n u t c h a r c o a l (6-14 m e s h ) . C o c o n u t c h a r c o a l was c h o s e n o v e r z e o l i t e , a n o t h e r common a d s o r b e n t , b e c a u s e p e r u n i t v o l u m e o f b u l k m a t e r i a l c h a r c o a l h a s a m o n o l a y e r c a p a c i t y o f 170 c m J ( S T P ) / c n r a t 4.2 K ( R o s e -I n n e s , 1973) c o m p a r e d t o 130 c i r r ( S T P ) / c m J f o r z e o l i t e ( D a u n t and R o s e n , 1 9 7 0 ) . Above 4 K t h e m o n o l a y e r c a p a c i t y o f c h a r c o a l f o r h e l i u m d r o p s r a p i d l y f r o m i t s l o w t e m p e r a t u r e v a l u e o f 390 cm ( S T P ) / g s o f o r e f f e c t i v e pump o p e r a t i o n a t l o w t e m p e r a t u r e s , i t i s c r u c i a l t h a t t h e c h a r c o a l be w e l l a n c h o r e d t h e r m a l l y t o t h e 4 H e b a t h . C o n s e q u e n t l y p a r a l l e l c o p p e r p l a t e s were p l a c e d i n s i d e t h e pump 6 mm a p a r t t o a i d i n c o o l i n g t h e c h a r c o a l . The p l a t e s c o n t a i n e d h o l e s t o f a c i l i t a t e g a s f l o w . The pump was c o o l e d v i a a .1 mm d i a m e t e r , 18 mm l e n g t h o f c o p p e r w i r e a t t a c h e d t o t h e 4 K p l a t e . I d e a l l y t h e t h e r m a l l i n k s h o u l d h a v e v e r y h i g h t h e r m a l c o n d u c t a n c e a t 4 K and v e r y l o w t h e r m a l c o n d u c t a n c e b e t w e e n a b o u t 20 K and 40 K. T h u s , when t h e pump i s a d s o r b i n g He, i t w i l l n o t be h e a t e d much a b o v e 4 K b u t when i t i s h e a t e d t o 30-40 K t o d e s o r b t h e J H e , i t w i l l n o t p r o v i d e a l a r g e t h e r m a l l o a d f o r t h e 4 H e b a t h . Some h i g h p u r i t y m e t a l s s a t i s f y t h i s c r i t e r i o n f a i r l y w e l l b e c a u s e t h e y h a v e a peak i n t h e i r t h e r m a l c o n d u c t i v i t y n e a r 10 K. The g r e a t e r i s t h e p u r i t y , t h e h i g h e r t h e peak and t h e l o w e r t h e p e a k t e m p e r a t u r e . We u s e d c o m m e r c i a l c o p p e r w i r e b u t i t s t h e r m a l c o n d u c t i v i t y was i n c r e a s e d by a n n e a l i n g f o r 24 h a t 970 C u n d e r a p r e s s u r e o f 1.0 x 1 0 - 4 t o r r o f d r y a i r ( R o s e n b l u m e t a l , 1 9 7 7 ) . A s s u m i n g a 30 yW h e a t l e a k i n t o t h e 3 He p o t , t h e h e a t o f a d s o r p t i o n w o u l d g e n e r a t e 1.7 mW i n t h e pump. The t e m p e r a t u r e r i s e o f t h e pump was f o u n d t o be <1 K. The pump c o u l d be m a i n t a i n e d a t 45 K by .5 W o f h e a t e r power. 4 The c o n t i n u o u s l y o p e r a t i n g He r e f r i g e r a t o r shown s c h e m a t i c a l l y i n F i g . 25 was b a s e d on t h e d e s i g n o f De Long e t 4 a l ( 1 9 7 1 ) . When pumping o n t h e He p o t commenced, l i q u i d 4 H e was dr a w n f r o m t h e m a i n b a t h a t 4.2 K, t h r o u g h t h e f i l t e r 4 and f l o w i m p e d a n c e , i n t o t h e He p o t . T h e r e was a p r e s s u r e 4 d i f f e r e n t i a l a c r o s s t h e f l o w i m p e d a n c e c a u s i n g t h e He t o be c o o l e d t o 1.3 K. A b o u t h a l f o f t h e l a t e n t h e a t o f e v a p o r a t i o n 4 4 o f t h e He was r e q u i r e d t o c o o l t h e i n c o m i n g He. The r e m a i n d e r was u s e d t o b a l a n c e t h e h e a t f l u x i n t o t h e 4 H e p o t . I f t h e r e w e re no e x t e r n a l l o a d , t h e l i q u i d 4 H e r o s e up t h e pumping t u b e u n t i l t h e h e a t f l u x down t h e t u b e e q u a l l e d t h e r e f r i g e r a t i o n c a p a c i t y . As t h e l o a d i n c r e a s e d , t h e l i q u i d l e v e l d r o p p e d u n t i l , a t t h e c r i t i c a l power Q c, t h e 4 H e p o t i t s e l f b e g a n t o be d e p l e t e d . E v e n t u a l l y i t was e m p t i e d and t h e t e m p e r a t u r e r o s e r a p i d l y . ( ) c i s t h e l o n g t e r m power i n p u t t h a t t h e 4 H e p o t c a n s u s t a i n . D u r i n g t h e 3 H e c o n -d e n s a t i o n , we f o u n d t h a t 6C was e x c e e d e d ; t h e l e n g t h o f t i m e f o r w h i c h TA c o u l d be m a i n t a i n e d a t 1.3 K u n d e r o v e r l o a d Filter Vacuum can To 3He supply 1 2 2 ^ a n ( d f To diffusion pump Devva r L iqu id + He (4 .2 K ) The rmo ink A d sorpt ion pump a rcoo Throt t le valve Copper plate *He pot ' L iqu id 4 H e (1.3 K ) H e pot L i q u i d 3He F i g . 25 S c h e m a t i c d i a g r a m o f t h e He r e f r i g e r a t o r . The G r a f o i l c e l l and h y d r o g e n f i l l l i n e h ave b e e n o m i t t e d . The t h e r m a l l i n k a t t h e t o p o f t h e a d s o r p t i o n pump i s a c o p p e r w i r e . The c o p p e r p l a t e b e t w e e n t h e He pumping l i n e and t h e vacuum c a n p r o v i d e s a t h e r m a l l i n k f r o m 4 K t o t h e b a f f l e s , w h i c h c o o l t h e He g a s d e s o r b e d f r o m t h e pump. 123 c o n d i t i o n s d e p e n d e d on t h e c a p a c i t y o f t h e 4 H e p o t . I n o r d e r t o r e d u c e t h e h e a t l o a d on t h e 4 H e p o t d u r i n g 3 H e c o n d e n -3 s a t i o n a 5 mm s e c t i o n o f t h e s t a i n l e s s s t e e l He pumping l i n e , s t a r t i n g a t a p o i n t 5 mm a b o v e t h e 4 H e p o t , was t h e r m a l l y a n c h o r e d t o 4 K by a c o p p e r p l a t e s o l d e r e d t o t h e vacuum c a n . S e m i - c i r c u l a r c o p p e r b a f f l e s w e r e a l s o p l a c e d i n s i d e t h e pumping 3 l i n e t o c o o l t h e i n c o m i n g He g a s . I t w o u l d h a v e b e e n 3 d e s i r e a b l e t o c o o l t h e i n c o m i n g J H e more e f f e c t i v e l y by i n c r e a s i n g t h e l e n g t h o f t h e h e a t e x c h a n g e r o r t o h a v e i n c r e a s d t h e 5.2 cm 3 i n t e r n a l v o l u m e o f t h e 4 H e p o t . S p a c e l i m i t a t i o n s p r e c l u d e d t h i s . De Long e t a l f o u n d t h a t t h e c r i t i c a l power p e r u n i t f l o w r a t e f o r t h e i r r e f r i g e r a t o r was 4.5 mW/(10~ 4 m o l e s / s ) . U s i n g t h i s r e s u l t , we d e t e r m i n e d t h e r e q u i r e d f l o w r a t e n f r o m t h e e x p e c t e d h e a t l e a k i n t o o u r 4 H e p o t . The f l o w r a t e i s f i x e d by t h e v a l u e o f t h e f l o w i m p e d a n c e z. G i v e n t h e f l o w r a t e , T 4 i s f i x e d by t h e s p e e d S o f t h e 4 H e pump. F o r l a m i n a r , i n c o m p r e s s i b l e f l o w t h r o u g h a p o r o u s medium, t h e v o l u m e f l o w r a t e V i s g i v e n by D a r c y ' s L a w 4 ( C o l l i n s , 1961) V = ( P B - P 4 ) / ( n z ) ( V I - 2 ) w h e r e P B = p r e s s u r e a b o v e t h e 4 H e b a t h = 760 t o r r P 4 = p r e s s u r e i n t h e 4 H e p o t n = f l u i d v i s c o s i t y We e x p e c t P 4 ^ l t o r r s o i t c a n be d r o p p e d . ( V I - 2 ) i m p l i e s t h a t nz s h o u l d be i n d e p e n d e n t o f z. De Long e t a l f o u n d t h a t t h i s i s a c t u a l l y n o t q u i t e t r u e . T h i s i s h a r d l y s u r p r i s i n g b e c a u s e t h e r e w i l l be a t u r b u l e n t m i x t u r e o f l i q u i d and g a s i n t h e f l o w i m p e d a n c e s o t h e a s s u m p t i o n s o f ( V I - 2 ) w o u l d n o t be f u l f i l l e d . N e v e r t h e l e s s , ( V I - 2 ) i s u s e f u l a s a g u i d e . G a u t h i e r and V a r o q u a u x (1973) f o u n d t h a t t h e y c o u l d p r e d i c t t h e f l o w i m p e d a n c e o f a t u b e o f l e n g t h L a n d c r o s s - s e c t i o n a l a r e a A c o n t a i n i n g a powder o f p a r t i c l e s o f d i a m e t e r d u s i n g z = L/AB ( V I - 3 a ) w h e r e B = 8 x 1 0 ~ 4 d 2 ( V I - 3 b ) F o r o u r p o w d e r s , t h i s v a l u e o f B was n o t r e l i a b l e , p r o b a b l y b e c a u s e o f t h e d i s t r i b u t i o n o f g r a i n s i z e s a nd d i f f e r e n c e s i n p a c k i n g , s o z had t o be m e a s u r e d . T h i s was done by f l o w i n g N 2 g a s t h r o u g h t h e f l o w i m p e d a n c e a t room t e m p e r a t u r e . The i n p u t p r e s s u r e was 1500 t o r r and t h e o u t p u t p r e s s u r e was 760 t o r r . A f l o w i m p e d a n c e was c o n s t r u c t e d f r o m i n d u s t r i a l g r a d e 3 ym A l 2 0 3 powder i n a 16 mm l o n g , 1.1 mm i n s i d e d i a m e t e r s t a i n -12 l e s s s t e e l t u b e . The m e a s u r e d v a l u e o f z was 4.1 x 10 —3 12 —3 cm c o m p a r e d t o a v a l u e o f 2.3 x 10 cm o b t a i n e d f r o m ( V I - 3 ) . E x t r a p o l a t i n g t h e v a l u e s o f fiz g i v e n by De • 8 3 L o n g e t a l , we e x p e c t e d n z ~ 1 . 5 x 10 ( m o l e s / s ) / c m s o o u r m e a s u r e d v a l u e o f z i m p l i e s n r 4 . 5 x 1 0 ~ 4 m o l e s / s . T h e r e f o r e we a n t i c i p a t e d a c r i t i c a l p ower Q c o f 20 raw. 4 The e x p e c t e d He p o t t e m p e r a t u r e was d e t e r m i n e d f r o m • 4 n and t h e p u m ping s p e e d S o f t h e He p o t pump, a S a r g e n t - W e l c h m o d e l 1397. A t 1 t o r r , i t had a p u m p i n g s p e e d o f 4 7 1/s w h i c h was n o t l i m i t e d by t h e p u m ping l i n e . The He p o t p r e s s u r e was o b t a i n e d f r o m t h e i d e a l g a s e q u a t i o n P 4 = hRT/S ( V I - 4 ) w h e r e R i s t h e g a s c o n s t a n t . T h i s g a v e P 4 = 1.2 t o r r i m p l y i n g T 4 = 1.30 K. The t h e r m a l b o u n d a r y r e s i s t a n c e b e t w e e n t h e l i q u i d 4 H e and t h e c o p p e r 4 H e p o t ( L o u n a s m a a , 1974) a t 1.3 K 2 2 i s 10 mK-cm /mW. I f t h e 25 cm i n t e r i o r s u r f a c e o f t h e 4 H e p o t had b e e n s m o o t h , a 20 mW h e a t f l u x i n t o t h e 4 H e p o t w o u l d h a v e p r o d u c e d an i n s i g n i f i c a n t 8 mK d i f f e r e n c e b e t w e e n t h e t e m p e r a t u r e s o f t h e 4 H e and t h e c o p p e r . I n f a c t , t h e i n t e r i o r o f t h e 4 H e p o t was s a n d b l a s t e d s o t h e d i f f e r e n c e w o u l d be e v e n l e s s . 4 The o p e r a t i o n o f t h e He p o t was t e s t e d by f i l l i n g i t w i t h l i q u i d 4 H e and a p p l y i n g e n o u g h h e a t e r power & n t o empty i t . T hen 6 n was r e d u c e d i n s t e p s and T 4 was m e a s u r e d when e q u i l i b r i u m was r e a c h e d . The r e s u l t i n g g r a p h o f T 4 v e r s u s Q h i s shown i n F i g . 26. F o r h e a t e r power i n p u t s up t o 9.1 mW, a c o n s t a n t t e m p e r a t u r e o f T 4 = 1.28 K was m a i n t a i n e d . F o r h i g h e r p o w e r s , T 4 r o s e r a p i d l y . S i n c e we e x p e c t 20 mW, t h i s i m p l i e d a h e a t l e a k i n t o t h e 4 H e p o t o f 11 mW. T h i s w o u l d come p r i m a r i l y t h r o u g h t h e c o p p e r p l a t e s o l d e r e d t o t h e vacuum c a n a t 4.2 K and t h e 4 H e pumping l i n e . F i l l i n g t h e 4 H e p o t r e q u i r e d 40 m i n s o i t was d e s i r e a b l e t o e n s u r e t h a t i t was n o t e m p t i e d by an o v e r l o a d s u c h a s c o n d e n -•3 4 s a t i o n o f t h e He. I t was p o s s i b l e t o t e l l t h a t t h e He p o t was d r a i n i n g by m e a s u r i n g T 4 . When t h e p o t was f u l l , T 4 was 1.28 K b u t a s t h e p o t e m p t i e d T 4 d r o p p e d t o 1.23 K b e f o r e 126 Fig. 26 He pot temperature versus heater power. The c r i t i c a l heater power, obtained from the intersection of the two straight l i n e s , i s 9.1 mw. 127 r i s i n g r a p i d l y when t h e p o t was c o m p l e t e l y empty. The J H e p o t , shown i n F i g . 2 7 , was c o n s t r u c t e d f r o m 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 . The l i d was b r a z e d w i t h E a s y - f l o 45 s o l d e r 5 (.45 Ag/.15 Cu/.24 Cd/.16 Z n ) , w h i c h i s n o t s u p e r c o n d u c t i n g a b o v e 64 mK. O r i g i n a l l y a l a y e r o f c o p p e r powder had b e e n s i n t e r e d i n t o t h e b o t t o m o f t h e He p o t i n o r d e r t o i n c r e a s e t h e s u r f a c e a r e a and l o w e r t h e t h e r m a l b o u n d a r y r e s i s t a n c e : f o r a s m o o t h s u r f a c e , we e x p e c t e d a t e m p e r a t u r e g r a d i e n t o f 13 mK f o r a 30 yW h e a t l e a k i n t o t h e p o t . The s i n t e r i n g was done a t 870 C i n an H 2 a t m o s p h e r e . L a t e r , we d i s c o v e r e d t h a t a f t e r t h e r m a l c y c l i n g t o l i q u i d h e l i u m t e m p e r a t u r e s v e r y s m a l l h o l e s w e re p r o d u c e d i n t h e 3 mm t h i c k w a l l s o f t h e J H e p o t . The H 2 had d i f f u s e d i n t o t h e c o p p e r d u r i n g s i n t e r i n g and r e a c t e d e x p l o s i v e l y w i t h o x y g e n i m p u r i t i e s t o f o r m w a t e r . C o n s e q u e n t l y , t h e He p o t had t o be r e m o v e d f r o m t h e c r y o s t a t - a n d r e p l a c e d . The s e c o n d v e r s i o n was s a n d -b l a s t e d i n s i d e t o l o w e r t h e b o u n d a r y r e s i s t a n c e and f i l l e d w i t h c o p p e r foam. The foam n o t o n l y i n c r e a s e d t h e s u r f a c e a r e a o f t h e c o p p e r b u t , more i m p o r t a n t l y , e q u a l i z e d t h e t e m p e r a t u r e o f -3 t h e l i q u i d He, w h i c h i s a p o o r t h e r m a l c o n d u c t o r . "3 W i t h i n t h e He p o t , t h e r e was a s e p a r a t e c e l l e n c l o s i n g 3 ( a f t e r b e i n g f i l l e d w i t h c o p p e r foam) a v o l u m e o f .43 cm . I t c o u l d be u s e d t o c a l i b r a t e t h e t e m p e r a t u r e s e n s o r s a g a i n s t t h e v a p o u r p r e s s u r e o f h e l i u m o r h y d r o g e n . The h y d r o g e n f i l l l i n e 3 w e n t t h r o u g h two 90° b e n d s i n t h e He p o t i n o r d e r t o t r a p r a d i a t i o n . 128 Copper foam Heater Resisiors R F COI He pot a pour pressure eel Copper - qua r t z joint Teflon ( .01 mm) ium o-nncj Copper sleeve Copper rod Qua r t z t u b e C lamp Copper foil (.02 mm) Grafoil (.25 mm) on co i i l hold er Fig. 27 Cross-section of the JHe pot and the G r a f o i l c e l l , drawn to scale. 129 The m a i n h e a t l e a k s i n t o t h e He p o t w e r e due t o 3 c o n d u c t i o n down t h e h y d r o g e n f i l l l i n e a n d t h e He pumping l i n e . The f o r m e r was a 35 mm l o n g , 10 mm d i a m e t e r , .15 mm t h i c k s t a i n l e s s s t e e l b e l l o w s w h i c h p r o d u c e d a h e a t l e a k o f 11 yW b e t w e e n 1.3 K and .3 K. The l a t t e r was a 13 mm l o n g , 13 mm d i a m e t e r , .6 mm t h i c k t u b e c o n s t r u c t e d f r o m f i b r e g l a s s mat and S t y c a s t 2850 F T 6 e p o x y w h i c h was e p o x i e d t o two .25 mm t h i c k b r a s s t u b e s . I t p r o d u c e d a h e a t l e a k o f 12 yW b e t w e e n 1.3 K and .3 K. T h e s e e x p e c t e d h e a t l e a k s compare q u i t e w e l l w i t h t h e t o t a l m e a s u r e d h e a t l e a k o f 30 yW. 6.3 G r a f o i l C e l l The G r a f o i l c e l l shown i n F i g . 27 was a t t a c h e d t o t h e 3 3 He p o t u s i n g an i n d i u m 0 - r m g s e a l b e t w e e n t h e He p o t and t h e b r a s s f l a n g e a t t h e t o p o f t h e c e l l . The s i m p l e s t way t o j o i n t h e q u a r t z c e l l t o t h e b r a s s f l a n g e w o u l d h a v e b e e n t o u s e a q u a r t z - P y r e x g r a d e d j o i n t and a P y r e x - c o p p e r H o u s e k e e p e r s e a l . G i v e n o u r s p a c e l i m i t a t i o n s , t h i s w o u l d h a v e b e e n much t o o l o n g s o a s o l d e r e d q u a r t z - c o p p e r j o i n t was f a b r i c a t e d a s d e s c r i b e d i n A p p e n d i x B. I t was d e s i r e a b l e t o a v o i d u s i n g e p o x y i n t h e j o i n t i n o r d e r t o p r e v e n t p o s s i b l e c o n t a m i n a t i o n o f t h e G r a f o i l by v a p o u r f r o m t h e e p o x y . The j o i n t h a s s u r v i v e d c y c l i n g f r o m 4 K t o 300 K o v e r t w e n t y t i m e s . 4 H e a t i n g o f t h e G r a f o i l h a d b e e n a p r o b l e m i n t h e He 3 c r y o s t a t s o we w i s h e d t o a v o i d s u c h h e a t i n g i n t h e He c r y o s t a t . G r a f o i l i s a v e r y p o o r t h e r m a l c o n d u c t o r a t l o w 130 t e m p e r a t u r e s b e c a u s e t h e t h e r m a l c o n t a c t b e t w e e n t h e c r y s t a l l i t e s i s p o o r . M e a s u r e m e n t s b e t w e e n 1.5 K and 4.2 K h a v e shown t h a t t h e t h e r m a l c o n d u c t i v i t y i n t h e p l a n e o f t h e f o i l <^ i s p r o p o r t i o n a l t o T and b e t w e e n 1.5 K and 20 K t h e o c o n d u c t i v i t y n o r m a l t o t h e f o i l K F C i s p r o p o r t i o n a l t o T (Hedge e t a l , 1 9 7 3 ) . E x t r a p o l a t i n g t h e s e m e a s u r e m e n t s t o .3 K, we e s t i m a t e K - ^ 2 . 0 x 1 0 ~ 5 W/cm/K and K f c = 3 . 2 x 1 0 ~ 9 W/cm/K. U s i n g o u r m e a s u r e d c o n v e r s i o n r a t e s f o r c e l l B, t h e c o n v e r s i o n h e a t i n g f o r p u r e J = l h y d r o g e n w i t h a c o v e r a g e p=.85 w o u l d be 200 (20) nW f o r H 2 ( D 2 ) . The RF h e a t i n g a t o u r n o r m a l power l e v e l ( i n p u t a t t e n u a t i o n A=13dB) w o u l d h a v e b e e n 24 nW c o r r e s p o n d i n g t o a m a g n e t i c f i e l d o f 160 nT RMS. H e a t i n g by 4.2 K r a d i a t i o n w o u l d h a v e b e e n a b o u t 8 nW. The t r a n s v e r s e t h e r m a l c o n d u c t a n c e o f t h e G r a f o i l l i m i t s t h e r a t e a t w h i c h h e a t i s c a r r i e d away. F o r .25 mm t h i c k G r a f o i l s h e e t s c o v e r e d w i t h p u r e o.-H2 and b o t h s i d e s a t .3 K, t h e t e m p e r a t u r e r i s e a t t h e c e n t r e o f t h e e a c h s h e e t w o u l d be o f t h e o r d e r o f .6 mK s o h e a t i n g o f t h e G r a f o i l s h o u l d n o t be a p r o b l e m i f t h e s i d e s o f e a c h s h e e t a r e a d e q u a t e l y t h e r m a l l y a n c h o r e d . A l t h o u g h t h e e x t r a p o l a t i o n o f t h e t h e r m a l c o n d u c t i v i t y d a t a may n o t be v e r y p r e c i s e , i t i s n o t l i k e l y t o be t o o f a r f r o m t h e t r u t h . •a Two G r a f o i l c e l l s w e r e c o n s t r u c t e d f o r t h e He c r y o s t a t . C e l l A c o n t a i n e d 1.15 g o f .13 mm t h i c k , 13 mm l o n g s h e e t s o f o GTA g r a d e G r a f o i l w h i c h w e r e c o v e r e d o n one s i d e by a 200 A e v a p o r a t e d f i l m o f c o p p e r . The G r a f o i l had p r e v i o u s l y b e e n a c t i v a t e d a t 900 C f o r 71 h. P a i r s o f s h e e t s w e re t h e n s i n t e r e d 131 t o .05 mm c o p p e r f o i l a t 900 C i n an H 2 a t m o s p h e r e . A f t e r s i n t e r i n g , t h e s a m p l e was b a k e d u n d e r vacuum a t 850 C f o r 13 h. E a c h p i e c e o f f o i l had a t a b w h i c h was c l a m p e d t o one end o f a 37 mm l o n g p i e c e o f 12 AWG c o p p e r w i r e . The o t h e r e n d o f t h e w i r e was c l a m p e d t o t h e He p o t . E a c h p a i r o f G r a f o i l s h e e t s was w r a p p e d i n .01 mm T e f l o n t o r e d u c e RF s h i e l d i n g . The T e f l o n c o n t a i n e d numerous s l i t s t o a i d g a s f l o w . The t h i n n e s t G r a f o i l a v a i l a b l e , w i t h a t h i c k n e s s o f .13 mm, was c h o s e n s o t h a t g a s f l o w i n t o t h e s a m p l e , RF f i e l d h o m o g e n e i t y , and t h e r m a l c o n d u c t a n c e n o r m a l t o t h e s h e e t s w o u l d be m a x i m i z e d . A f t e r m e a s u r i n g t h e 4 H e a d s o r p t i o n i s o t h e r m , we f o u n d t h i s G r a f o i l had t h e a d d e d a d v a n t a g e o f a s p e c i f i c s u r f a c e a r e a o f 27.7 m^/g c o m p a r e d t o o n l y 15.6 m / g f o r t h e .25 mm G r a f o i l u s e d i n t h e 4 H e c r y o s t a t . T h i s e n h a n c e d t h e S:N r a t i o c o n s i d e r a b l y . A l s o , t h e J = l t o J=0 c o n v e r s i o n was a f a c t o r o f .38 t i m e s s m a l l e r on t h e .13 mm G r a f o i l , a t l e a s t f o r D 2. On t h e o t h e r h a n d , X - r a y m e a s u r e m e n t s had shown t h a t t h e i s o t r o p i c p owder component o f t h e g r a p h i t e c r y s t a l s i n o u r .25 mm G r a f o i l was s m a l l e r t h a n u s u a l ( J . G . D a s h , p r i v a t e c o m m u n i c a t i o n ) . However, X - r a y s a r e s e n s i t i v e t o a l l o f t h e b a s a l p l a n e s , n o t j u s t t h e o n e s on t h e s u r f a c e , s o t h e r e was no i n f o r m a t i o n a b o u t t h e o r i e n t a t i o n a l d i s t r i b u t i o n o f t h e s u r f a c e p l a n e s . I t was n o t c l e a r w h e t h e r t h e b r o a d e n i n g due t o m i s -o r i e n t a t i o n o f t h e s u r f a c e w o u l d be s i g n f i c a n t l y l e s s i n t h e .25 mm G r a f o i l t h a n i n t h e .13 mm G r a f o i l . S p e c t r a o b t a i n e d w i t h c e l l A w e re n o t a s c l e a r l y r e s o l v e d 132 a s t h o s e o b t a i n e d w i t h t h e 4 H e c r y o s t a t . T h i s was p a r t i c u l a r -l y e v i d e n t f o r D 2. I n t h e 4 H e c r y o s t a t , t h e J = l p e a k s o f D 2 h a d b e e n c l e a r l y r e s o l v e d o v e r t h e e n t i r e t e m p e r a t u r e r a n g e o f 1.2 K - 4.2 K b u t i n c e l l A t h e p e a k s w ere b a r e l y r e s o l v e d a t t h e l o w t e m p e r a t u r e e nd o f t h a t r a n g e . F o u r p o s s i b l e e x p l a n a t -i o n s o c c u r r e d t o u s : 1) The G r a f o i l i n c e l l A was l e s s w e l l o r i e n t e d . 2) I t may h a v e b e e n c o n t a m i n a t e d by c o p p e r d u r i n g e v a p o r a t i o n o r s i n t e r i n g . 3) I t may h a v e b e e n c o n t a m i n a t e d by T e f l o n d u r i n g t h e b a k e - o u t o f t h e s a m p l e . 4) The h y d r o g e n may n o t h a v e p l a t e d t h e G r a f o i l u n i f o r m l y . The a d s o r p t i o n p r o c e d u r e was d i f f e r e n t f r o m t h a t u s e d i n t h e 4 H e c r y o s t a t . T e f l o n c o n t a m i n a t i o n seemed u n l i k e l y b e c a u s e t h e same b a k e - o u t p r o c e d u r e was u s e d i n t h e 4 H e c r y o s t a t . C o p p e r c o n t a m i n a t i o n was a l s o d o u b t f u l b e c a u s e s i m i l a r p r o c e d u r e s h a v e b e e n u s e d i n s p e c i f i c h e a t m e a s u r e m e n t s o f 3 H e and 4 H e on G r a f o i l ( B r e t z e t a l , 1973) w i t h no a p p a r e n t d e l e t e r i o u s e f f e c t s . Some c a u t i o n must be e x e r c i s e d w i t h r e g a r d t o t h i s l a s t c o n c l u s i o n b e c a u s e a l t h o u g h a f i l m may be u n i f o r m w i t h r e s p e c t t o some p r o p e r t i e s , i t may n o t be f o r o t h e r s . S i n c e t h e r e was r e a s o n t o s u s p e c t t h a t t h e p o o r s p e c t r a w ere due t o t h e m i s o r i e n t a t i o n o f t h e G r a f o i l , a new s a m p l e , c a l l e d c e l l B, was c o n s t r u c t e d u s i n g .911 g o f t h e o r i g i n a l .25 mm G r a f o i l o b t a i n e d f r o m J.G. Dash. As an a d d i t i o n a l p r e c a u t i o n , c o p p e r was n o t e v a p o r a t e d o r s i n t e r e d o n t o t h e G r a f o i l . B e t w e e n e a c h p a i r o f 13 mm l o n g G r a f o i l s h e e t s was a s h e e t o f .01 mm T e f l o n f i l m , a s h e e t o f .03 mm c o p p e r and a 133 s e c o n d s h e e t o f T e f l o n . T h e T e f l o n s h e e t s c o n t a i n e d s l i t s a s i n c e l l A a n d t h e y h a d b e e n b a k e d f o r o n e w e e k a t 2 0 0 C t o r e d u c e t h e * H b a c k g r o u n d N M R s i g n a l . T h e c o p p e r s h e e t s c o n t a i n e d 1 0 mm l o n g , 1 . 6 mm w i d e s l i t s , s p a c e d 1 . 6 mm a p a r t , t o i m p r o v e g a s f l o w a n d p e n e t r a t i o n o f t h 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 s . T h e y 3 w e r e a n c h o r e d t o t h e H e p o t i n t h e s a m e m a n n e r a s i n c e l l A . T h e w h o l e a s s e m b l y w a s t i g h t l y w r a p p e d w i t h T e f l o n f i l m . T h e i n t e n s i t i e s o f t h e N M R a b s o r p t i o n s p e c t r a o b t a i n e d w i t h c e l l s A a n d B w e r e a b o u t t h e s a m e a s t h e t e m p e r a t u r e a n d R F l e v e l w e r e v a r i e d , s o t h e t h e r m a l c o n t a c t t o t h e G r a f o i l s h e e t s m u s t h a v e b e e n c o m p a r a b l e . S p e c t r a o b t a i n e d w i t h c e l l B w e r e s t i l l p o o r s o t h e a d s o r -p t i o n p r o c e d u r e b e c a m e t h e p r i m e s u s p e c t . A l l o f t h e s a m p l e s u p t o t h i s p o i n t h a d b e e n a d s o r b e d a t t e m p e r a t u r e s o f 3 0 K o r l e s s s o t h a t t h e h y d r o g e n v a p o u r p r e s s u r e a b o v e t h e G r a f o i l w o u l d h a v e b e e n a t m o s t 1 0 0 y . S u b s e q u e n t l y , w e w e r e a b l e t o c o n s i s t e n t l y o b t a i n s p e c t r a o f t h e s a m e q u a l i t y a s i n t h e 4 H e c r y o s t a t b y a d s o r b i n g t h e h y d r o g e n a t 4 0 K , w h e r e t h e v a p o u r p r e s s u r e w a s a b o u t 1 t o r r , a s d e s c r i b e d i n s e c t i o n 5 . 5 . S i n c e t h e u n i f o r m i t y o f a l l o f t h e f i l m s i n c e l l A i s s u s p e c t , i t w a s n e v e r d e t e r m i n e d w h e t h e r t h e b e t t e r a l i g n m e n t o f t h e g r a p h i t e c r y s t a l s i n c e l l B h a d a s i g n i f i c a n t e f f e c t . 3 6 . 4 T e m p e r a t u r e M e a s u r e m e n t i n t h e J H e C r y o s t a t T h e t e m p e r a t u r e m e a s u r e m e n t a n d c o n t r o l s y s t e m o f t h e 3 H e c r y o s t a t i s s h o w n i n F i g . 2 8 . T h e 1 / 8 W O h m i t e c a r b o n UBC A O S Temperqture C ontrollcr Out Oat PAR 122 Lock-in Amplifier 1n R»f«r«nc» Out In PAR 112 Amp CR 1615A Capacitance Br id je I n Meat er MK 5 Barotron f70 C opac'rtance Manometer Out Adsorption P ump ' H e Pot ~Eh '-0--o 5! In UBC A - 0 5 Temperature Controller Out R 3 v Flare 8 0 3 0 A Dtgitol M ulf imeter H P 34-65B Digital M ultimeter LaKesKori C r y o t r o n ' s CCS" BRS Current Source  Fig. 28 Temperature measurement and control system for the He cryostat. r e s i s t o r s R-^ , R 2, R 3, and R 4 were 10ft, 680ft , 220 ft, and 680 ft r e s p e c t i v e l y . M o st o f t h e e p o x y c o a t i n g was r emoved by s a n d b l a s t i n g . The 100 ft g ermanium r e s i s t o r R G e was a L a k e s h o r e C r y o t r o n i c s m o d e l GR-200A, w h i c h was f i l l e d w i t h 3 H e g a s . The S rT^O^ c a p a c i t a n c e s e n s o r C was a L a k e s h o r e C r y o t r o n i c s m o d e l CS-400GR. The c u r r e n t s e n s e r e s i s t o r R-j-c o u l d be s e l e c t e d t o be e i t h e r a 1 kft o r a 100 kft w i r e wound r e s i s t o r e a c h o f w h i c h had a t e m p e r a t u r e c o e f f i c i e n t o f 50 ppm/C. S e n s o r s R-^ , R 2 r R G e , and C w e re a l l c o a t e d w i t h A p i e z o n M g r e a s e and p l a c e d i n h o l e s i n t h e b a s e o f t h e He p o t . R-^  was u s e d b e t w e e n .3 K and 2 K, w h e r e a s R 2 was u s e d a b o v e 2 K. R 3 and R 4 were v a r n i s h e d t o t h e 4 H e p o t and t h e a d s o r p t i o n pump r e s p e c t i v e l y w i t h GE 7031 v a r n i s h . R 3 and R 4 were m e a s u r e d u s i n g t h e r e s i s t a n c e s e t t i n g o f a F l u k e 8030A d i g i t a l m u l t i m e t e r . T h r e e l e a d s w e re u s e d . A s s u m i n g t h a t t h e y were i d e n t i c a l , t h i s a l l o w e d one t o c o m p e n s a t e f o r t h e l e a d r e s i s t a n c e . The He p o t r e s i s t o r s w ere m e a s u r e d u s i n g a f o u r l e a d c o n f i g u r a t i o n . A L a k e s h o r e C r y o t r o n i c s CCS-BRS DC c u r r e n t s o u r c e d r o v e a c u r r e n t t h r o u g h t h e s e l e c t e d r e s i s t o r v i a two o f t h e l e a d s and t h e v o l t a g e a c r o s s t h e r e s i s t o r was m e a s u r e d by a H e w l e t t - P a c k a r d (HP) 3465B d i g i t a l m u l t i m e t e r c o n n e c t e d t o t h e o t h e r two l e a d s . S i n c e t h e HP 3465B h a s a h i g h i n p u t i m p e d a n c e , t h e v o l t a g e d r o p down t h e v o l t a g e l e a d s was i n s i g n i f i c a n t . The c u r r e n t was d e t e r m i n e d by u s i n g t h e HP 3465B t o m e a s u r e t h e v o l t a g e a c r o s s t h e s e n s e r e s i s t o r R T i n s e r i e s w i t h t h e t e m p e r a t u r e s e n s o r s . An 136 a v e r a g e o f t h e a b s o l u t e v a l u e o f t h e v o l t a g e o b t a i n e d w i t h t h e c u r r e n t i n o p p o s i t e d i r e c t i o n s was u s e d i n o r d e r t o c o m p e n s a t e f o r t h e r m o e l e c t r i c v o l t a g e s i n t h e l e a d s . The power d i s s i p a t i o n and t e m p e r a t u r e e r r o r s r e s u l t i n g f r o m t h e m e a s u r e m e n t s y s t e m , a t s e v e r a l t e m p e r a t u r e s , a r e shown i n T a b l e I X . The v o l t a g e s a c r o s s R-^  and R G e were a l w a y s 1-4 mV. i t was n e c e s s a r y t o d i s c o n n e c t t h e HP 3465B f r o m t h e c a r b o n r e s i s t o r s d u r i n g NMR m e a s u r e m e n t s b e c a u s e i t i n c r e a s e d t h e n o i s e s l i g h t l y . A G e n e r a l R a d i o 1615A c a p a c i t a n c e b r i d g e was u s e d t o m e a s u r e t h e two c o m p o n e n t s o f t h e i m p e d a n c e o f t h e c a p a c i t a n c e t h e r m o m e t e r . The o u t p u t o f t h e b r i d g e was d e t e c t e d by a P r i n c e t o n A p p l i e d R e s e a r c h (PAR) m o d e l 122 l o c k - i n a m p l i f i e r w i t h a m o d e l 112 p r e a m p l i f i e r . The 5 kHz, 1 V r e f e r e n c e s i g n a l f r o m t h e l o c k - i n a m p l i f i e r was u s e d t o e x c i t e t h e s e n s o r . I t was n e c e s s a r y t o e n s u r e t h a t t h e b r i d g e was n o t g r e a t l y u n -b a l a n c e d o r t h e v o l t a g e a p p l i e d a c r o s s t h e s e n s o r w o u l d c h a n g e . Changes i n t h e a m p l i t u d e o r f r e q u e n c y o f t h i s v o l t a g e w o u l d c a u s e t h e s e n s o r t o d r i f t . C o a x i a l - l e a d s t o t h e c a p a c i t a n c e s e n s o r were u s e d t o r e d u c e i n t e r l e a d c a p a c i t a n c e . Power d i s s i -p a t i o n was l e s s t h a n 1 nW. R-j and R^ w e r e o n l y u s e d t o m o n i t o r t h e o p e r a t i o n o f t h e c r y o s t a t s o t h e y d i d n o t r e q u i r e a c c u r a t e c a l i b r a t i o n . A two p o i n t c a l i b r a t i o n was made b e t w e e n 4.2 K and 77 K u s i n g t h e e q u a t i o n T= A l o g R ( V I - 5 ) ( l o g R - B) 137 where A and B a r e c o n s t a n t s d e t e r m i n e d by t h e r e s i s t a n c e a t 4.2 K and 77 K. R-^  was a l s o c a l i b r a t e d a g a i n s t t h e 4 H e v a p o u r p r e s s u r e b e t w e e n 1.2 K and 4.2 K. T a b l e IX Power D i s s i p a t i o i n t h e n a nd T e m p e r a t u r e He P o t R e s i s t o r s E r r o r s T ( K ) Power D i s s i p a t i o n (nW) T e m p e r a t u r e E r r o r (mK) R-, R„ T, T T^ 1 2 Ge 1 2 Ge 4 400 14 15 4 1 3 1 3 - 1.5 1 1 .6 1 - .5 .5 .4 .3 .2 _ . 5 .04 .1 The c a p a c i t a n c e s e n s o r was t h e o n l y s e n s o r t h a t had no o b s e r v a b l e m a g n e t i c f i e l d d e p e n d e n c e f o r t h e r a n g e o f f i e l d s t h a t we u s e d (H< 1.3 T ) . H o w e v e r , i t had p o o r r e p r o d u c i b i l i t y o n t h e r m a l c y c l i n g and i t was s u b j e c t t o c o n s i d e r a b l e d r i f t , p a r t i c u l a r l y d u r i n g t h e f i r s t c o u p l e o f h o u r s a f t e r c o o l i n g f r o m 77 K t o 4.2 K. The g e r m a n i u m r e s i s t o r R G e had t h e b e s t r e p r o d u c i b i l i t y a n d s t a b i l i t y b u t e x h i b i t e d t h e g r e a t e s t 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 , e s p e c i a l l y i f t h e y w e r e g r e a t e r t h a n .3 T. I t s r e s i s t a n c e d e p e n d e d on b o t h t h e m a g n i t u d e and o r i e n t a t i o n o f t h e f i e l d . A l t h o u g h t h e c a r b o n r e s i s t o r s w e r e n o t a s s t a b l e a n d r e p r o d u c i b l e a s t h e Ge r e s i s t o r , t h e y were 138 a d e q u a t e f o r t h e s e e x p e r i m e n t s . A f t e r t h e i n i t i a l c a l i b r a t i o n , t h e r e s i s t o r s w e r e c h e c k e d a t 4.2 K i n z e r o m a g n e t i c f i e l d 4 a g a i n s t t h e He v a p o u r p r e s s u r e a t t h e b e g i n n i n g o f e a c h NMR e x p e r i m e n t ( s e e F i g . 2 9 ) . The 1958 4 H e t e m p e r a t u r e s c a l e was u s e d t o d e t e r m i n e T H e « A DC c a l i b r a t i o n f r o m .3 K t o 6 K by L a k e s h o r e C r y o t r o n i c s I n c . g a v e T G e - The c a r b o n r e s i s t o r s w e r e c a l i b r a t e d a g a i n s t T H e a t 318 d a y s and a t o t h e r t i m e s t h e i r t e m p e r a t u r e s were d e t e r m i n e d u s i n g t h e 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 e i r r e s i s t a n c e s . F o r t h e c a r b o n r e s i s t o r s , t h e t h r e e m e a s u r e m e n t s up t o 263 d a y s have much more s c a t t e r t h a n l a t e r m e a s u r e m e n t s . I n t h o s e t h r e e c a s e s R-^ , R 2 1 and R G e were m e a s u r e d u s i n g an SHE m o d e l ARB a u t o m a t i c r e s i s t a n c e b r i d g e o p e r a t i n g a t 27.5 Hz w i t h 20 PV e x c i t a t i o n . A t h r e e t e r m i n a l c o n n e c t i o n was u s e d t o c o m p e n s a t e f o r l e a d r e s i s t a n c e . O t h e r w i s e t h e s y s t e m d e s c r i b e d a b o v e was u s e d . The Ge r e s i s t o r shows a c h a n g e o f a b o u t 10 mK i n i t s c a l i b r a t i o n b e t w e e n 117 and 263 d a y s . T h i s may h a v e b e e n c a u s e d by i n a d v e r t e n t h e a t i n g o f t h e He p o t t o 50 C d u r i n g t h a t p e r i o d . E v en w i t h t h i s s h i f t , t h e c a l i b r a t i o n was s a t i s f a c t o r y f o r o u r w o r k . The m a g n e t o r e s i s t a n c e o f t h e c a r b o n r e s i s t o r s , A R = R ( H ) - R ( 0 ) i s a c o m p l i c a t e d f u n c t i o n o f H and T. A t 1.35 K, AR^/R-^ was f o u n d t o be p o s i t i v e a n d p r o p o r t i o n a l t o H b u t a t .3 K, i t had a p o s i t i v e p e a k o f a few p e r c e n t b e l o w .1 T and t h e n d e c r e a s e d , a p p e a r i n g t o l e v e l o u t a t - 5 . 6 % n e a r 1.3 T ( m e a s u r e m e n t s were o n l y made up t o 1.4 T ) . A t .2 T, t h e f i e l d 139 (a) 20 -• ^ - 2 0 - 1 E 200 400 600 ~1 600 Ti'me (days) .-40 -I -60 - ' •80-( b ) 30 -I— 2 0 -10 i r~ 4 0 0 i r T 1 800 i r 200 6 0 0 Time (days) F i g . 29 S t a b i l i t y o f t h e r e s i s t a n c e t h e r m o m e t e r s on t h e J H e p o t a t 4.2 K i n z e r o m a g n e t i c f i e l d . The t e m p e r a t u r e s o b t a i n e d f r o m t h e c a r b o n and Ge r e s i s t o r s a r e T^, T ? r and T G e ; t h a t o b t a i n e d f r o m t h e v a p o u r p r e s s u r e o f t h e He i n t h e m a i n d e w a r i s T „ e . The vacuum c a n c o n t a i n e d 4 H e g a s a t p r e s s u r e s b e t w e e n .1 and 1 t o r r . I n ( a ) , we show T-, ( c i r c l e s ) a n d T 2 ( d i a m o n d s ) and i n ( b ) , T G e i s shown. N o t e t h e d i f f e r e n c e i n t h e v e r t i c a l s c a l e s . 140 u s e d f o r H 2 NMR, t h e e q u i v a l e n t t e m p e r a t u r e s h i f t AT o f R^ a n d R 2 was n e v e r more t h a n .2% t h r o u g h o u t t h e t e m p e r a t u r e r a n g e i n w h i c h e a c h was u s e d . S i n c e AT was e i t h e r c o m p a r a b l e t o o r l e s s t h a n t h e u n c e r t a i n t y w i t h w h i c h t h e He p o t t e m p e r a -t u r e c o u l d be m e a s u r e d and l e s s t h a n t h e t e m p e r a t u r e d r i f t d u r i n g an NMR m e a s u r e m e n t , i t was i g n o r e d . A t 1.3 T, t h e f i e l d u s e d f o r D 2 NMR, t h e m a g n e t o r e s i s t a n c e was s m a l l b u t s i g n i f i -c a n t s o t h e 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 t t h a t f i e l d . T a b l e X c o n t a i n s a few t y p i c a l r e s u l t s , oT^ and a T 2 a r e t h e e r r o r s i n T^ and T 2 , t a k i n g i n t o a c c o u n t t h e a c c u r a c y o f t h e m e a s u r i n g i n s t r u m e n t s and t h e s t a b i l i t y o f t h e r e s i s t o r s . T a b l e X T y p i c a l M a g n e t o r e s i s t a n c e M e a s u r e m e n t s T ( K ) H(T) A R G e A R 1 A R 2 A T 1 a T l A T 2 a T R G e ( % ) R 1 ( % ) R 2 ( % ) T 1 ( % ) T x ( %) T 2 ( % ) T, 4.22 .19 .05(4) - . 2 3 ( 6 ) - . 0 8 ( 5 ) .3 (1) .3 . 0 4 ( 3 ) .2 1.35 .19 .1(1) .02(3) - - . 0 1 ( 2 ) .2 - -.32 .19 3.0(6) .9 (4) - - .2 (1) .06 - -4.22 1.3 4 0.0 (3) . 2 1 ( 7 ) 0.0 (3) .3 - . 1 0 ( 3 ) .2 2.0 1.3 - .5 (4) 1.0 (7) - .3 (3) .2 - . 2 2 ( 2 ) . 1 1.35 1.3 11 1.4 (4) - - .7 (2) .2 - -.33 1.3 80 -5.6 (3) — 1. 07 (6) .06 — — 2 142 N o t e s t o C h a p t e r V I 1. T h i s G r a f o i l was k i n d l y s u p p l i e d t o us by J.G. D a s h , P h y s i c s D e p t . , U n i v e r s i t y o f W a s h i n g t o n , U.S.A. 2. V a l u e s o f H c a r e f o r T=1.3 K ( W a r r e n and B a d e r , 1 9 6 9 ) . 3. The d r i f t r a t e was m e a s u r e d s e v e r a l h o u r s a f t e r c o o l i n g t o l i q u i d h e l i u m t e m p e r a t u r e s . I m m e d i a t e l y a f t e r c o o l i n g , i t was much g r e a t e r . 4. (VT-2) c a n a l s o be u s e d f o r an i d e a l g a s , i n w h i c h PV r a t h e r t h a n V i s c o n s t a n t , p r o v i d e d t h a t one i n t e r p r e t s V a s t h e a v e r a g e v o l u m e f l o w r a t e . 5. E a s y - f l o 45 i s a p r o d u c t o f Handy and Harman Co. I t i s v e r y s i m i l a r i n c o m p o s i t o n t o C a s t o l i n 1802 ( C a s t o l i n S.A., L a u s a n n e S t - S u l p i c e , S w i t z e r l a n d ) s o t h e t r a n s i t i o n t e m p e r a t u r e o f t h e l a t t e r h a s b e e n q u o t e d ( L a n d a u and Rosenbaum, 1 9 7 2 ) . 6. Emerson and Cuming I n c . , C a n t o n , M a s s . , U.S.A. 143 CHAPTER V I I THE NMR SPECTROMETER 3.1 I n t r o d u c t i o n A t t h e p r e s e n t t i m e , m o s t NMR work i s c o n d u c t e d u s i n g p u l s e d , r a t h e r t h a n c o n t i n u o u s wave, s p e c t r o m e t e r s . I n a c o n t i n u o u s wave s p e c t r o m e t e r , l o w l e v e l RF e x c i t a t i o n i s a p p l i e d i n o r d e r t o t i p t h e n u c l e a r s p i n s by a s m a l l a n g l e . One m e a s u r e s t h e power a b s o r b e d a s one s weeps e i t h e r t h e f i e l d o r t h e f r e q u e n c y t h r o u g h r e s o n a n c e . I n a p u l s e d s p e c t r o m e t e r , one o r more s h o r t , h i g h power RF p u l s e s a r e a p p l i e d a t t h e r e s o n a n c e f r e q u e n c y . T h e s e t i p t h e s p i n s by a l a r g e a n g l e ( u s u a l l y 90° o r 180°) and one o b s e r v e s t h e r e c o v e r y o f t h e m a g n e t i z a t i o n a f t e r t h e p u l s e s . P u l s e t e c h n i q u e s d o m i n a t e t h e s c e n e b e c a u s e i n m o s t c a s e s t h e y p r o d u c e b e t t e r S:N r a t i o s and b e c a u s e s o p h i s t i c a t e d p u l s e s e q u e n c e s w i l l o f t e n a l l o w one t o i s o l a t e t h e e f f e c t s o f s p e c i f i c i n t e r a c t i o n s . I n o u r c a s e , t h e y w e re r e j e c t e d b e c a u s e l a r g e a n g l e p u l s e s w o u l d l e a d t o i n t o l e r a b l e h e a t i n g o f t h e G r a f o i l : n e a r l y a l l o f t h e l o s s e s i n t h e t u n e d c i r c u i t a r e due t o t h e G r a f o i l . S i n c e t h e r m a l t i m e c o n s t a n t s become l o n g a t l o w t e m p e r a t u r e s , t h e t i m e r e q u i r e d f o r t h e t h e r m a l r e c o v e r y o f t h e s a m p l e w o u l d g r e a t l y e x c e e d t h e r e c o v e r y t i m e o f t h e m a g n e t i z a t i o n . We f o u n d i n t h e 4 H e c r y o s t a t t h a t t h e h e a t i n g o f t h e s a m p l e was a b o u t .1 K a t 1 K u s i n g o u r n o r m a l RF f i e l d o f a b o u t 8 x 10 T. I n t h e He c r y o s t a t , t h e r m a l s i n k i n g o f t h e G r a f o i l was much b e t t e r b u t t h e t r a n s v e r s e t h e r m a l 144 c o n d u c t i v i t y < t o f t h e G r a f o i l was i n s u f f i c i e n t t o a l l o w t h e us e o f l a r g e a n g l e p u l s e s . The s i m p l e s t s i n g l e c o i l n u c l e a r r e s o n a n c e c i r c u i t , a Q-meter, i s shown i n F i g . 30. A c u r r e n t g e n e r a t o r d r i v e s a c u r r e n t i i n t o a p a r a l l e l r e s o n a n t c i r c u i t . A t r e s o n a n c e t h e v o l t a g e v a c r o s s t h e c o i l , d e t e c t e d by t h e a m p l i f i e r , i s m a x i m i z e d and i s g i v e n by v = i0u> oL ( V I I - 1 ) w h ere Q = q u a l i t y f a c t o r o f t h e t u n e d c i r c u i t w = r e s o n a n c e f r e q u e n c y o f t h e t u n e d c i r c u i t L = i n d u c t a n c e o f t h e c o i l I f t h e NMR s a m p l e i s p l a c e d i n s i d e t h e c o i l , t h e n , a s one sweeps 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 , power i s a b s o r b e d by t h e s a m p l e , w h i c h l o w e r s t h e Q. T h i s r e s u l t s i n a d r o p i n v. The c h a n g e i n t h e 0 i s p r o p o r t i o n a l t o t h e i m a g i n a r y p a r t o f t h e m a g n e t i c s u s c e p t i b i l i t y , w h i c h i s c a l l e d t h e a b s o r p t i o n ( A b r a g a m , 1 9 6 1 ) . The Q-meter h a s t h e d i s a d v a n t a g e t h a t i t i s f a i r l y s e n s i t i v e t o m i c r o p h o n i c s . V i b r a t i o n s o f t h e c r y o s t a t r e s u l t i n s m a l l c h a n g e s AC i n t h e t u n e d c i r c u i t c a p a c i t a n c e C. T h e s e s h i f t w0, a d d i n g n o i s e t o v. A t r e s o n a n c e , t h e c h a n g e i n 1/Q i s s e c o n d o r d e r i n AC: A (1/0) = Q(AC/C) 2 ( V I I - 2 ) I n p r a c t i c e , t h e c i r c u i t w i l l a c t u a l l y be m i s t u n e d by an amount AC Q SO t h a t A (1/Q) = 0(AC QAC/C 2) ( V I I - 3 ) 145 Ampl i f ier F i g . 30 Q-meter c i r c u i t . 146 T y p i c a l l y , one m i g h t e x p e c t t o m a x i m i z e t h e c i r c u i t r e s p o n s e t o w i t h i n 1%, i n w h i c h c a s e t h e m i s t u n i n g w o u l d be Ac Q/C = .1/Q ( V I I - 4 ) T h e r e f o r e t h e s e n s i t i v i t y t o n o i s e w o u l d be A(1/Q) = .1AC /C ( V I I - 5 ) I n s t e a d o f a Q-meter, we c h o s e t o u s e a R o b i n s o n o s c i l l a t o r c i r c u i t ( R o b i n s o n , 1959) b e c a u s e i t i s l e s s s e n s i t i v e t o m i c r o p h o n i c s . A R o b i n s o n o s c i l l a t o r c o n s i s t s o f a c l o s e d l o o p c o n t a i n i n g a t u n e d c i r c u i t , an a m p l i f i e r , a p h a s e s h i f t e r , a l i m i t e r , and an a t t e n u a t o r . Our l i m i t e r was a h i g h g a i n a m p l i f i e r w i t h d i o d e c l i p p i n g s o t h a t i t s o u t p u t was r e l a t i v e l y i n s e n s i t i v e t o w i d e v a r i a t i o n s i n t h e i n p u t a m p l i t u d e . The l e v e l o f t h e RF e x c i t a t i o n was s e t by t h e a t t e n u a t o r . The o s c i l l a t o r f r e g u e n c y w was d e t e r m i n e d by t h e r e q u i r e m e n t t h a t , i n o r d e r f o r o s c i l l a t i o n s t o o c c u r , t h e t o t a l p h a s e s h i f t a r o u n d t h e l o o p must be a n i n t e g r a l m u l t i p l e o f 360°. The p h a s e s h i f t e r was a d j u s t e d s o t h a t u)=w0. N e a r <x>Q, t h e p h a s e s h i f t p r o d u c e d by t h e t u n e d c i r c u i t i s a s t r o n g f u n c t i o n o f f r e q u e n c y . I f t h e t u n e d c i r c u i t c a p a c i t a n c e c h a n g e s , ut m u s t f o l l o w u Q i n o r d e r t o m a i n t a i n t h e same p h a s e s h i f t . The s e n s i t i v i t y t o s m a l l c h a n g e s i n c a p a c i t a n c e becomes A ( l / Q ) = (1/2Q)(AC/C) ( V I I - 6 ) T h i s i s a f a c t o r o f Q/5 ^ 20 t i m e s l e s s t h a n f o r t h e Q-meter f o r a t y p i c a l NMR c i r c u i t . S i g n i f i c a n t i m p r o v e m e n t i n t h e S:N r a t i o c a n be a c h i e v e d by c o o l i n g t h e t u n e d c i r c u i t t o a l o w t e m p e r a t u r e T ( A l d e r m a n , 1 9 7 0 ) . F o r a s a m p l e a t a t e m p e r a t u r e T s w i t h m a g n e t i z a t i o n M ( T S ) t t h e S:N r a t i o a t t h e o u t p u t o f an a m p l i f i e r w i t h n o i s e f i g u r e F i s p r o p o r t i o n a l t o M ( T S ) ( Q / F T ) 1 / / 2 , a s s u m i n g t h a t t h e n o i s e i s J o h n s o n n o i s e f r o m t h e t u n e d c i r c u i t ( A b r a g a m , 1 9 6 1 ) . As T d e c r e a s e s , Q w i l l u s u a l l y i n c r e a s e t o a l i m i t i n g v a l u e . I n o u r c a s e , i t c h a n g e d f r o m 3 0 a t room t e m p e r a t u r e t o 7 0 a t l i q u i d h e l i u m t e m p e r a t u r e s . ( T h i s a p p l i e s when t h e c o i l c o n t a i n s t h e G r a f o i l ; t h e i n t r i n s i c Q i s much h i g h e r ) . I t i s p o s s i b l e t o r e d u c e F by c o o l i n g t h e a m p l i f i e r a s w e l l . However, e v e n i f F r e m a i n e d u n c h a n g e d , c o o l i n g m i g h t s t i l l be d e s i r e a b l e . S a y t h a t a c o o l e d t u n e d c i r c u i t was c o n n e c t e d t o a room t e m p e r a t u r e a m p l i f i e r v i a a c o a x i a l c a b l e . The c a b l e f o r m s p a r t o f t h e t u n e d c i r c u i t . The c a b l e m u s t h a v e l o w t h e r m a l c o n d u c t a n c e w h i c h g e n e r a l l y means t h a t i t w i l l h a v e p o o r e l e c t r i c a l c o n d u c t a n c e . T h e r e f o r e i t w i l l l o w e r t h e 0 and a d d h i g h t e m p e r a t u r e J o h n s o n n o i s e . I n f a c t a l o n g l o s s y c a b l e may make t h e S:N r a t i o w o r s e t h a n i f t h e e n t i r e t u n e d c i r c u i t was a t room t e m p e r a t u r e . Our s o l u t i o n was t o e m p l o y a l i q u i d 4 H e c o o l e d a m p l i f i e r w h i c h a l l o w e d us t o k e e p t h e c a b l e s h o r t and e n t i r e l y a t l i q u i d h e l i u m t e m p e r a t u r e s . The a m p l i f i e r had a n o i s e t e m p e r a t u r e o f 8 K. T h e r e a r e some d i s a d v a n t a g e s t o u s i n g a c o o l e d a m p l i f i e r . The t h r e e p-Ge j u n c t i o n f i e l d - e f f e c t t r a n s i s t o r s ( J F E T s ) i n o u r a m p l i f i e r d i s s i p a t e d 9 mw e a c h when o p e r a t e d a t 6 V s o t h a t t h e a m p l i f i e r was r e s p o n s i b l e f o r a b o u t h a l f o f t h e t o t a l l i q u i d 4 H e c o n s u m p t i o n o f t h e c r y o s t a t . The e l e c t r o n i c c o m p o n e n t s 148 o f t h e a m p l i f i e r h a d t o be a b l e t o w i t h s t a n d r e p e a t e d t h e r m a l c y c l i n g . A l s o , i t i s c o n v e n i e n t i f a l l o f t h e p a s s i v e c o m p o n e n t s a r e i n d e p e n d e n t o f t e m p e r a t u r e . Our s p e c t r o m e t e r o p e r a t e d a t a f i x e d f r e q u e n c y b u t i f one w i s h e d t o h a v e a t u n a b l e s p e c t r o m e t e r , i t w o u l d be n e c e s s a r y t o t u n e i t f r o m room t e m p e r a t u r e . F i n a l l y , a d j u s t m e n t and r e p a i r o f a c o o l e d a m p l i f i e r c a n be t e d i o u s . I n g e n e r a l , t h e S:N r a t i o c a n be i m p r o v e d by i n c r e a s i n g t h e t o t a l m e a s u r i n g t i m e . T h e r e a r e two m a i n a p p r o a c h e s t h a t c a n be a d o p t e d . The f i r s t i s t o u s e a s h o r t d e t e c t o r t i m e c o n s t a n t , sweep t h r o u g h t h e r e s o n a n c e r e l a t i v e l y r a p i d l y ( t y p i c a l l y ^ 10 m s ) , and a v e r a g e many s p e c t r a . The s e c o n d i s t o sweep s l o w l y ( t y p i c a l l y ^ 100 s) o n l y a few t i m e s , u s i n g a l o n g t i m e c o n s t a n t . I f t h e n o i s e s p e c t r u m i s f l a t and t h e r e s o n a n c e i s n o t s a t u r a t e d , t h e n i n p r i n c i p l e t h e two m e t h o d s a r e e q u a l l y e f f e c t i v e . I n p r a c t i c e , f a s t sweeps o f t e n h a v e s e v e r a l a d v a n t a g e s o v e r s l o w o n e s . N o r m a l l y , i f t h e r e s o n a n c e s i g n a l i s weak, one o p e r a t e s n e a r s a t u r a t i o n i n o r d e r t o g e t a s l a r g e a s i g n a l a s p o s s i b l e w i t h o u t d i s t o r t i n g i t . I f s a t u r a t i o n o c c u r s , a f a s t sweep i s l e s s l i k e l y t o d i s t o r t t h e s i g n a l b e c a u s e t h e s p i n s a r e s a t u r a t e d more u n i f o r m l y t h a n w i t h a s l o w sweep. I n t h e l a t t e r c a s e , t h e r e may be a t r a n s f e r o f p o l a r i z a t i o n among t h e s p i n s d u r i n g t h e sweep. A n o t h e r c o n s i d e r a t i o n i s t h a t t h e n o i s e t e n d s t o i n c r e a s e a t l o w f r e q u e n c i e s w i t h a n a p p r o x i m a t e 1/f s p e c t r u m . F o r e x a m p l e , d r i f t i n t h e s p e c t r o m e t e r g a i n w i l l c a u s e f l u c t u a t i o n o f t h e b a s e l i n e d u r i n g a s l o w sweep. D u r i n g 1 4 9 a f a s t s weep, t h e g a i n w i l l be a p p r o x i m a t e l y c o n s t a n t p r o d u c i n g u n i m p o r t a n t s h i f t s i n t h e b a s e l i n e f r o m one sweep t o t h e n e x t . F i n a l l y , a s t r o n g b u t b r i e f d i s t u r b a n c e w i l l a f f e c t o n l y one s c a n s o i t s i n f l u e n c e w i l l be d i m i n i s h e d i f many f a s t s c a n s a r e a c c u m u l a t e d r a t h e r t h a n a few s l o w o n e s . I f t h e NMR s p e c t r u m i s q u i t e w i d e and t h e s p e c t r o m e t e r o p e r a t e s a t l o w t e m p e r a t u r e s , f a s t sweeps c a n p r o d u c e s e r i o u s s i d e e f f e c t s . A c h a n g i n g m a g n e t i c f i e l d w i l l i n d u c e e d d y c u r r e n t s i n t h e e l e c t r i c a l l y c o n d u c t i n g s e c t i o n s o f t h e c r y o s t a t , c r e a t i n g o s c i l l a t i n g m a g n e t i c moments. I n t h e p r e s e n c e o f a l a r g e s t a t i c f i e l d H Q , t h e r e w i l l be o s c i l l a t i n g t o r q u e s w h i c h c a u s e v i b r a t i o n c o h e r e n t w i t h t h e sweep. The v i b r a t i o n s m o d u l a t e t h e t u n e d c i r c u i t c a p a c i t a n c e , w h i c h i n t u r n m o d u l a t e s t h e r e s o n a n c e f r e q u e n c y . I n a d d i t i o n t h e e d d y c u r r e n t s w i l l c a u s e h e a t i n g . We c h o s e o u r sweep r a t e o f .3 mT/s t o be a s f a s t a s p o s s i b l e w i t h o u t p r o d u c i n g s i g n i f i c a n t e ddy c u r r e n t h e a t i n g o f t h e He p o t . I t i s p o s s i b l e t o e s c a p e f r o m t h e e f f e c t s o f t h e l o w f r e q u e n c y n o i s e i n h e r e n t w i t h s l o w sweeps by s h i f t i n g t h e d e t e c t i o n f r e q u e n c y away f r o m z e r o . We d i d t h i s by m o d u l a t i n g H Q s i n u s o i d a l l y a t 280 Hz u s i n g a s m a l l p a i r o f c o i l s m o u n t e d o n t h e i n s i d e o f t h e vacuum c a n . W i t h t h i s t e c h n i q u e , i n t h e l i m i t o f s m a l l m o d u l a t i o n , t h e d e r i v a t i v e o f t h e a b s o r p t i o n i s o b s e r v e d . The m o d u l a t i o n f i e l d 1^ i n d u c e d c o h e r e n t v i b r a t i o n b u t i t was much l e s s s e v e r e t h a n t h e v i b r a t i o n p r o d u c e d by a f a s t sweep f o r two r e a s o n s . F i r s t , t h e m o d u l a t i o n f i e l d was 150 o n l y .17 mT p e a k t o p e a k c o m p a r e d t o 10 -20 mT f o r t h e sweep f i e l d . S e c o n d l y , s i n c e t h e m o d u l a t i o n f i e l d i s s o s m a l l i t n e e d n o t be v e r y homogeneous s o t h e c o i l s c a n be made q u i t e s m a l l . On t h e o t h e r h a n d , t h e sweep c o i l s had t o be q u i t e l a r g e and homogeneous s o e d d y c u r r e n t s were i n d u c e d i n a much l a r g e r s e c t i o n o f t h e c r y o s t a t . N e v e r t h e l e s s , c o h e r e n t i n t e r f e r e n c e a r i s i n g f r o m t h e m o d u l a t i o n f i e l d was a s e r i o u s p r o b l e m ( s e e s e c t i o n 7 . 3 ) . E v e n w i t h s l o w s w e e p s , we f o u n d i t n e c e s s a r y t o a c c u m u l a t e s p e c t r a w i t h a s i g n a l a v e r a g e r . T y p i c a l l y , a b o u t 100 s c a n s were a d d e d i n t h e s i g n a l a v e r a g e r and t h e n t r a n s f e r r e d t o an Amdahl 470 V/8 c o m p u t e r f o r c o m p a r i s o n w i t h t h e s y n t h e t i c s p e c t r a . The H 2 s p e c t r a r e q u i r e d 2-4 h o f s i g n a l a v e r a g i n g w h e r e a s t h e l o w t e m p e r a t u r e D 2 s p e c t r a n e e d e d up t o 20 h b e c a u s e D 2 h a s a s m a l l e r m a g n e t i c moment and a b r o a d e r s p e c t r u m . 7.2 D e s c r i p t i o n o f t h e S p e c t r o m e t e r F i g . 31 i s a b l o c k d i a g r a m o f t h e s p e c t r o m e t e r , s h o w i n g t h e R o b i n s o n o s c i l l a t o r , t h e d e t e c t i o n s y s t e m , and t h e d r i v i n g s y s t e m f o r t h e m o d u l a t i o n c o i l s . The s u p p o r t s f o r t h e c r y o s t a t a n d d e w a r s were embedded i n s a n d t o dampen v i b r a t i o n s . The p umping l i n e s f o r t h e r o t a r y pumps c o n t a i n e d b a l l a s t v o l u m e s t h a t were a l s o embedded i n s a n d . The t u n e d c i r c u i t i s shown i n F i g . 32. The 1.2 yH i n d u c t a n c e o f t h e NMR c o i l i s d e n o t e d by L and R i s t h e e q u i v a l e n t p a r a l l e l r e s i s t a n c e o f t h e t u n e d c i r c u i t . A l l o f Attenuator Tuned Circu it L iquid 4H« H P 5 3 2 7 A Timer - Counter Li m it« r Pnoje S n i f t e r Tuned Preamplifier Modulotion Coils r Wide Band Amplif ier A . Tekt r o n i J f 54-40 Oscilloscope w i t h SA22N /Amplifier Diode Peak Detector PA R 185 Preamplifier PAR 114 Amplifier J H P 7004B "XY Recorder Nicolet <170 Signal Averager with f72/ + B Plug-in Transformer D vnotrac 3 9 1 A j Lock-in A mplifier Reference I E C F 5 4 A Signal Generator General Radio 1192 Counter F i g . 31 B l o c k d i a g r a m o f t h e NMR s p e c t r o m e t e r . F i g . 32 T h e t u n e d c i r c u i t . to 153 the capacitors were silver-mica and the 50ft r e s i s t o r s were metal glaze. These components show l i t t l e temperature dependence between 300 K and 4 K. A 2 pF coupling capacitor C c made the RF power source look l i k e a current generator. Two capacitors, and C 2 1 were used to match R = 4.3 kf2 to the optimum source resistance of the l i q u i d helium cooled amplifier A^, which was 2 . 9 kft at 4 K. This was necessary in order to minimize the noise figure of the amplifier. The 7 pF input capacitance of A-^  i s included in C 3« The values of C 2 and C 3 were chosen as follows. Define Z^  to be the impedance across the c o i l and Z Q to be the impedance seen by A^ across i t s input. One can e a s i l y show that 2i = Q/[ WL + iQ(uL - 1/uC)] (VII-7) where C = C x + C 2C 3/(C 2+C 3) 2 At resonance u LC = 1 so Z JL = 0/uL (VII-8a) = R (VII-8b) Since the matching impedances are both capacitors, V Q i s in phase with and i s a maximum at resonance. It i s straightforward to show that the output impedance of the matching network i s Z = 1 +iQ((o 2LC =-l) (VII - 9 ) O a • iu)C b[l + iQ(u LC-1)] where C a = C-^C^ and C b = C 2+C 3 Although Zi i s real at resonance, Z i s in general c o m p l e x . T h i s i s u n d e s i r e a b l e b e c a u s e t h e s o u r c e i m p e d a n c e o f A-^  s h o u l d be r e a l i n o r d e r t o o b t a i n t h e b e s t n o i s e f i g u r e . Z Q w i l l be a p p r o x i m a t e l y r e a l i f R > > ( C 2 + C 3 ) / ( o ) C 2 2 ) ( V I I - 1 0 ) We s a t i s f i e d ( V I I - 1 0 ) and o b t a i n e d Z Q = 2.9 kft by m a k i n g C 1 = 300 p F , C 2 = 180 p F , and C 3 = 40 pF. The c o a x i a l c a b l e b e t w e e n t h e t u n e d c i r c u i t and A-^  was 3 4 t h e r m a l l y a n c h o r e d t o t h e He p o t , t h e He p o t , and t h e 4 K p l a t e . I n t h o s e s e c t i o n s , i t was c o n s t r u c t e d f r o m c o p p e r c o n d u c t o r s w i t h S t y c a s t 2850 FT e p o x y ^ a s t h e d i e l e c t r i c . B e t w e e n t h e s e s e c t i o n s , i t was n e c e s s a r y t o h a v e l o w t h e r m a l c o n d u c t a n c e b u t h i g h e l e c t r i c a l c o n d u c t a n c e o r t h e Q w o u l d be d e g r a d e d . T h e s e c o n d i t i o n s were met by u s i n g s h o r t s e c t i o n s o f s u p e r c o n d u c t i n g N b - T i w i r e . A l t h o u g h t h e w i r e h a s n o n - z e r o r e s i s t a n c e a t RF f r e q u e n c i e s , i t i s s m a l l . The l i q u i d h e l i u m c o o l e d a m p l i f i e r A-^  c o n s i s t e d o f a c a s c o d e s t a g e ( d e s i g n e d by W a l t e r H a r d y ) and a s o u r c e f o l l o w e r . The t r a n s i s t o r s i n i t were TIXM12 and TIXM301 p-Ge J F E T s 2 . F o r o u r p u r p o s e s , a FET a m p l i f i e r c a n be t r e a t e d a s a n o i s e l e s s a m p l i f i e r w i t h b o t h a c u r r e n t n o i s e g e n e r a t o r and a v o l t a g e n o i s e g e n e r a t o r a t i t s i n p u t . The v o l t a g e n o i s e g e n e r a t o r c a n be r e p r e s e n t e d by a f i c t i t i o u s r e s i s t o r R n, w h i c h g e n e r a t e s J o h n s o n n o i s e a s shown i n F i g . 33. V s i s t h e s i g n a l v o l t a g e a n d R g i s t h e s o u r c e r e s i s t a n c e . The n o i s e a d d e d by t h e l i q u i d h e l i u m c o o l e d a m p l i f i e r came p r i m a r i l y f r o m t h e i n p u t FET. S i n c e t h e n o i s e p r o p e r t i e s o f 155 F i g . 33 E q u i v a l e n t c i r c u i t o f a FET a m p l i f i e r . The a m p l i f i e r A i s n o i s e l e s s . The c u r r e n t n o i s e i s i n . The n o i s e r e s i s t o r R n a t t h e a m p l i f i e r t e m p e r a t u r e T g e n e r a t e s v o l t a g e n o i s e , v = ( 4 k T R n A f ) 1 / , i n b a n d w i d t h Af. T a b l e X I : C h a r a c t e r i s t i c s o f TIXM301 #13 v D S = - 3 . 4 0 V, I D S = - 4 . 0 mA, V G G = . 6 9 V T ( K ) g m(mmho) R n ( o h m s ) I G g ( n A ) 4.2 21 52 -4 77 16 48 -3 298 8 190 -800 156 t h e TIXM12 and TIXM301 J F E T ' s v a r y c o n s i d e r a b l y f r o m one d e v i c e t o t h e n e x t , i t was n e c e s s a r y t o m e a s u r e t h e s e p r o p e r t i e s a t 4 K f o r s e v e r a l d e v i c e s i n o r d e r t o be a b l e t o s e l e c t " a g o o d one. The o r i g i n a l i n p u t FET TIXM301 #502 p r o v i d e d r e l i a b l e o p e r a t i o n f o r n e a r l y t e n y e a r s . A l l o f t h e m e a s u r e m e n t s i n t h e 4 H e c r y o s t a t and some p r e l i m i n a r y m e s u r e m e n t s i n t h e 3 H e c r y o s t a t were made w i t h i t . E v e n t u a l l y , t h e bond b e t w e e n a w i r e and t h e s i l i c o n c h i p i n s i d e t h e t r a n s i s t o r c a n b r o k e d u r i n g c o o l i n g t o 77 K. A p p a r e n t l y , t h e t h e r m a l c o n t r a c t i o n o f t h e l a c q u e r c o v e r i n g t h e c h i p p u l l e d o f f t h e w i r e . The FET was r e p l a c e d s e v e r a l t i m e s b u t t h e r e p l a c e m e n t s a l l f a i l e d i n a s i m i l a r manner a f t e r one o r two c o o l downs. One o f t h e s e , TIXM301 #13, was r e p a i r e d by r e m o v i n g t h e l a c q u e r w i t h a c e t o n e and b o n d i n g t h e w i r e w i t h s i l v e r p a i n t . I t h a s s u r v i v e d more t h a n a d o z e n t h e r m a l c y c l e s and s p e c t r a o b t a i n e d w i t h i t a r e a s g o o d a s t h o s e f o r TIXM301 #502. The p r o p e r t i e s o f t h e FET's t h a t we m e a s u r e d w e r e t h e t r a n s c o n d u c t a n c e g m w h i c h d e t e r m i n e s t h e g a i n , t h e n o i s e r e s i s t a n c e R . and t h e g a t e - s o u r c e l e a k a g e c u r r e n t I n GS The c u r r e n t n o i s e i„ i s t h e s h o t n o i s e p r o d u c e d by I n GS The n o i s e p r o p e r t i e s o f TIXM301 #13 a r e shown i n T a b l e X I . A n o t h e r FET w i t h R n ( 4 . 2 K) = 89Q and | l | < l n A g a v e d i s t i n c t l y n o i s i e r s p e c t r a . A f t e r a m p l i f i c a t i o n by t h e t u n e d , l i q u i d h e l i u m c o o l e d a m p l i f i e r , t h e a m p l i t u d e m o d u l a t e d RF s i g n a l w e n t t h r o u g h a l o w n o i s e , w i d e b a n d a m p l i f i e r A ? a t room t e m p e r a t u r e . The RF c a r r i e r was t h e n r e m o v e d by a d i o d e p eak d e t e c t o r . The r e s u l t i n g 280 Hz s i g n a l was a m p l i f i e d by a P r i n c e t o n A p p l i e d R e s e a r c h (PAR) m o d e l 185 s i n g l e e n d e d a m p l i f i e r w i t h g r o u n d i s o l a t i o n and a PAR 114 s i g n a l c o n d i t i o n i n g a m p l i f i e r . The s i g n a l t h e n went t o a n I t h a c o D y n a t r a c 391A l o c k - i n a m p l i f i e r and f i n a l l y t o a N i c o l e t 1170 s i g n a l a v e r a g e r w i t h a m o d e l 172/4B i n p u t m o d u l e . The o u t p u t o f t h e PAR 114 a m p l i f i e r was m o n i t o r e d on a T e k t r o n i x 5440 o s c i l l o s c o p e w i t h a 5A22N d i f f e r e n t i a l i n p u t a m p l i f i e r . T h i s was i m p o r t a n t f o r c h e c k i n g f o r c o h e r e n t i n t e r f e r e n c e and i n t e r m i t t e n t m i c r o p h o n i c n o i s e . Power f o r a l l o f t h e n o n - c o m m e r i c a l a m p l i f i e r s was p r o v i d e d by H e w l e t t - P a c k a r d (HP) 6215A power s u p p l i e s f i l t e r e d by N i C d s o l i d g e l b a t t e r i e s . The b a t t e r i e s a l o n e w e re n o t s u i t a b l e b e c a u s e t h e i r v o l t a g e d r o p p e d a s t h e b a t t e r i e s d i s c h a r g e d , c a u s i n g t h e a m p l i f i e r g a i n t o d r i f t . The 60 Hz r i p p l e on t h e power s u p p l i e s was 200 yV p e a k t o p e a k ; i t was r e d u c e d t o 40 yV when f i l t e r e d by t h e b a t t e r i e s . W i t h f i l t e r i n g , t h e r e was no n o t i c e a b l e c h a n g e i n t h e n o i s e o b s e r v e d on t h e o s c i l l o s c o p e when e a c h power s u p p l y was s w i t c h e d on and o f f . S i g n a l l i n e s had t o be r o u t e d c a r e f u l l y t o p r e v e n t m a g n e t i c c o u p l i n g t o s o u r c e s o f 60 Hz i n t e r f e r e n c e . Two p a i r s o f c o i l s clamped t o t h e i n s i d e o f t h e vacuum c a n were u s e d t o p r o d u c e t h e m o d u l a t i o n f i e l d s p a r a l l e l and p e r p e n d i c u l a r t o t h e G r a f o i l . The r e c t a n g u l a r p a r a l l e l c o i l s ( f o r 6=90°) were e a c h 180 t u r n s o f 32 AWG c o p p e r w i r e b o n d e d w i t h GE 7031 v a r n i s h . E a c h c o i l had mean d i m e n s i o n s o f 158 22 mm x 25 mm; t h e y w e re 3.0 mm .square i n c r o s s - s e c t i o n . T h e i r 20 mm s e p a r a t i o n was 30% g r e a t e r t h a n t h e v a l u e t h a t m i n i m i z e s 2 2 d H/dz h a l f w a y b e t w e e n t h e c o i l s , w h ere t h e z a x i s i s t h e a x i s o f t h e c o i l s . An e f f o r t was made t o w i n d t h e c o i l s v e r y e v e n l y and t o a l i g n them a c c u r a t e l y . I f t h e m o d u l a t i o n f i e l d was n o t p a r a l l e l t o H^D, t h e r e w o u l d be an o s c i l l a t i n g t o r q u e on t h e c o i l s . The p e r p e n d i c u l a r c o i l s had t o f i t b e t w e e n t h e p a r a l l e l c o i l s s o t h e f o r m e r were c o n s i d e r a b l y l e s s homogeneous. E a c h was made f r o m 189 t u r n s o f 32 AWG w i r e . The mean c o i l d i m e n s i o n s were 24 mm x 12 mm and t h e y w e re 2.5 mm w i d e by 4.1 mm t h i c k . The mean s e p a r a t i o n was 24 mm. The m o d u l a t i o n f i e l d o f .17 mT peak t o p e a k , u s e d i n b o t h c a s e s , was l a r g e e n o u g h t o b r o a d e n some o f t h e s p e c t r a s l i g h t l y ( s e e S e c t i o n 8.5) b u t s m a l l e r f i e l d s w o u l d h a v e d e c r e a s e d t h e s i g n a l a m p l i t u d e . The s t a t i c m a g n e t i c f i e l d was p r o v i d e d by a M a g n i o n e l e c t r o m a g n e t w i t h a 6.4 cm gap and 23 cm p o l e c a p s . I t was c o n t r o l l e d by an FFC-4 F i e l d R e g u l a t o r w h i c h u s e d a Rawson L u s h 920 MCM r o t a t i n g c o i l p r o b e . The p e a k t o p e a k s e p a r a t i o n o f t h e d e r i v a t i v e o f t h e room t e m p e r a t u r e D 20 a b s o r p t i o n s i g n a l was 2 x l 0 ~ 5 T: t h e l i n e w i d t h was e n t i r e l y due t o t h e m a g n e t i c f i e l d i n h o m o g e n e i t y . The sweep f i e l d H s was p r o v i d e d by a p a i r o f d i s c s h a p e d c o i l s m o u n t e d o n t h e p o l e f a c e s o f t h e magnet. They w e r e d r i v e n by an I n t e r s t a t e E l e c t r o n i c s C o r p . m o d e l F54A f u n c t i o n g e n e r a t o r a n d a K e p c o m o d e l 36-5M b i p o l a r o p e r a t i o n a l a m p l i f i e r , w h i c h was i n t h e c u r r e n t s e n s e mode. The f u n c t i o n g e n e r a t o r a l s o t r i g g e r e d t h e s i g n a l a v e r a g e r . The 159 r o t a t i n g c o i l p r o b e was p o s i t i o n e d s u c h t h a t t h e sweep f i e l d was p r i m a r i l y p e r p e n d i c u l a r t o t h e c o i l a x i s s o a s t o m i n i m i z e c a n c e l l a t i o n o f t h e sweep f i e l d by t h e m a g n e t ; t h e p r o b e d e t e c t e d a b o u t 1% o f t h e sweep f i e l d . 7.3 C o h e r e n t I n t e r f e r e n c e I n t h e r e m a i n i n g s e c t i o n s o f t h i s c h a p t e r , v a r i o u s p r o c e d u r e s n e c e s s a r y f o r o b t a i n i n g g o o d s p e c t r a w i l l be d i s c u s s e d . The f i r s t t o p i c i s t h e c h o i c e o f t h e m o d u l a t i o n f r e q u e n c y and t h e n a t u r e o f t h e c o h e r e n t n o i s e a s s o c i a t e d w i t h i t . N e x t , t h e d e p e n d e n c e o f t h e s i g n a l a m p l i t u d e on t h e t e m p e r a t u r e and o n t h e a m p l i t u d e o f t h e RF e x c i t a t i o n w i l l be c o n s i d e r e d . F i n a l l y , t h e b a c k g r o u n d s i g n a l t h a t had t o be s u b t r a c t e d f r o m t h e H 2 s p e c t r a w i l l be m e n t i o n e d . The m o d u l a t i o n f r e q u e n c y f m had t o be c h o s e n j u d i c i o u s l y i n o r d e r t o a v o i d c o m p r o m i s i n g t h e S:N r a t i o . The b e s t f r e q u e n c y was f o u n d t o be 280 Hz. I d e a l l y , f m s h o u l d be a b o v e t h e t h r e s h o l d v a l u e w here e x c e s s l o w f r e q u e n c y n o i s e becomes s i g n i f i c a n t . The n o i s e i n a 1 Hz b a n d w i d t h was m e a s u r e d a s a f u n c t i o n o f f m a t H Q=1.3 T. The n o i s e was f o u n d t o d e c r e a s e a s f m i n c r e a s e d f r o m 100 Hz t o 700 H z . 4 A t 280 Hz i t was 40% h i g h e r t h a n a t 700 Hz. An u p p e r l i m i t t o f m i s s e t by t h e c o n d i t i o n t h a t i t be s m a l l c o m p a r e d t o t h e l i n e w i d t h o r t h e s p e c t r u m w i l l be d i s t o r t e d . O v e r t h e t e m p e r a t u r e r a n g e c o v e r e d i n t h e s e e x p e r i m e n t s , t h i s l i m i t was a few kHz f o r b o t h H 0 and Dn. 1 6 0 N e a r t h e h a r m o n i c s o f 6 0 Hz, t h e r e was s t r o n g c o h e r e n t i n t e r f e r e n c e s o t h e d i f f e r e n c e b e t w e e n f m and t h e s e h a r m o n i c s h a d t o be s e v e r a l t i m e s t h e l o c k - i n d e t e c t o r b a n d w i d t h , w h i c h was a t m o s t 3 Hz. A l s o , t h e m e c h a n i c a l r e s o n a n c e s o f t h e c r y o s t a t h a d t o be a v o i d e d . T h e s e c o u l d be r e a d i l y o b s e r v e d by s w e e p i n g t h e f r e q u e n c y o f a l a r g e s i n u s o i d a l c u r r e n t r u n n i n g t h r o u g h t h e sweep c o i l s . I n t h e r a n g e o f 2 0 0 - 7 0 0 Hz, t h e r e were o n l y f i v e r e g i o n s r a n g i n g f r o m a b o u t 5 - 2 0 Hz i n w i d t h where c r y o s t a t v i b r a t i o n s w e r e n o t e x c i t e d . A t f i r s t , a m o d u l a t i o n f r e q u e n c y o f 5 7 0 Hz was t r i e d b u t a t H Q = 1 . 3 T c o h e r e n t o s c i l l a t i o n s i n t h e b a s e l i n e s o f t h e d e r i v a t i v e s p e c t r a were o b s e r v e d . The o s c i l l a t i o n s had a p e r i o d o f 1 . 1 mT and a t 1 . 7 K t h e i r a m p l i t u d e was a b o u t one t h i r d t h a t o f t h e p - D 2 « T h i s made t h e i r s t u d y q u i t e d i f f i c u l t b e c a u s e t h e p-D 2 s i g n a l i t s e l f c o u l d o n l y be d i s c e r n e d a f t e r s e v e r a l m i n u t e s o f s i g n a l a v e r a g i n g . The a m p l i t u d e o f t h e o s c i l l a t i o n s was p r o p o r t i o n a l t o t h e m o d u l a t i o n a m p l i t u d e and i n d e p e n d e n t o f t h e sweep r a t e ; t h e a m p l i t u d e d e c r e a s e d a s H Q d e c r e a s e d c a u s i n g t h e o s c i l l a t i o n s t o become u n o b s e r v a b l e a t . 6 T. T h e s e r e s u l t s s u g g e s t e d t h a t t h e o s c i l l a t i o n s m i g h t h a v e a r i s e n f r o m t h e m a g n e t o r e s i s t a n c e o f some m a t e r i a l n e a r t h e NMR c o i l . The de H a a s - v a n A l p h e n e f f e c t was s u s p e c t e d a t f i r s t b u t t h i s seemed u n l i k e l y b e c a u s e i t i s n o r m a l l y o n l y s e e n i n v e r y p u r e m e t a l s . De H a a s - v a n A l p h e n o s c i l l a t i o n s s h o u l d h a v e a p e r i o d t h a t i s p r o p o r t i o n a l t o H 2 b u t t h e o s c i l l a t i o n s t h a t we o b s e r v e d had a b o u t t h e same p e r i o d a t . 8 5 T and 1 . 3 T. F o r t u n a t e l y , i t was 161 n o t n e c e s s a r y t o e s t a b l i s h t h e s o u r c e o f t h e o s c i l l a t i o n s : a t f m = 3 8 5 Hz t h e y w e re q u i t e weak and a t 280 H z , o u r u l t i m a t e c h o i c e f o r f m , t h e y h ad c o n v e n i e n t l y d i s a p p e a r e d . A t t h i s j u n c t u r e , a n o t h e r f o r m o f i n t e r f e r e n c e was r e v e a l e d . The m o d u l a t i o n p i c k - u p had a r e l a t i v e l y l a r g e c o n s t a n t component v Q p l u s a s m a l l c o m p onent Av whose s i g n d e p e n d e d o n t h e d i r e c t i o n o f t h e sweep. F o r t h e t r i a n g u l a r sweep f i e l d H_ shown i n F i g . 3 4 ( a ) , t h e m o d u l a t i o n p i c k - u p i s shown i n F i g . 3 4 ( b ) . I n t h e f o l l o w i n g d i s c u s s i o n , v Q w i l l be i g n o r e d b e c a u s e i t c o u l d e a s i l y be removed by an o f f s e t on t h e l o c k - i n a m p l i f i e r . The sweep d e p e n d e n t component Av was i n d e p e n d e n t o f t h e m o d u l a t i o n a m p l i t u d e A H , t h e sweep a m p l i t u d e A H c , and t h e m a g n i t u d e o f t h e sweep r a t e I H I e x c e p t when t h e s e p a r a m e t e r s were v e r y c l o s e t o z e r o . I f any o f t h e p a r a -m e t e r s w ere e q u a l t o z e r o , t h e n Av was z e r o . The p i c k - u p was o b s e r v e d a t .2 T and 1.3 T b u t was l a r g e r a t t h e h i g h e r f i e l d . The r i s e t i m e o f t h e p i c k - u p was i n d e p e n d e n t o f A H g b u t i n c r e a s e d i n v e r s e l y w i t h |Hg| . S t u d y o f t h e p i c k - u p was q u i t e t e d i o u s b e c a u s e i t s a m p l i t u d e was somewhat l e s s t h a n t h a t o f t h e p - D 2 s i g n a l and b e c a u s e i t f l u c t u a t e d . I t s o u r c e was n o t d e t e r m i n e d b u t i t s b e h a v i o u r s u g g e s t s t h e f o l l o w i n g s c e n a r i o . The t o r q u e s c r e a t e d by t h e i n t e r a c t i o n o f H Q w i t h eddy c u r r e n t s i n d u c e d by H„ c a u s e s some p a r t o f t h e c r y o s t a t t o move b e t w e e n two p o s i t i o n s , x-^  a n d x 2 > The f o r c e r e q u i r e d t o move i t b e t w e e n t h e s e p o s i t i o n s i s v e r y s m a l l b u t a v e r y F i g . 34 (a) T r i a n g u l a r sweep f i e l d . (b) M o d u l a t i o n p i c k - u p p r o d u c e d by t h e t r i a n g u l a r sweep f i e l d . I n r e a l i t y V Q > > A V . ( c ) The sweep f i e l d a c t u a l l y u s e d . The s i g n a l a v e r a g e r s t a r t s c o l l e c t i n g d a t a a t t , s t o p s a t .85 t ^ and r e s t a r t s a t t Q + t ^ . (d) The m o d u l a t i o n p i c k - u p p r o d u c e d by t h e sweep o f ( c ) . The c o n s t a n t component h a s been s u b t r a c t e d . i—1 CTl K) 163 l a r g e f o r c e i s n e e d e d t o move i t o u t s i d e t h a t r a n g e . P r o v i d e d v i b r a t i o n . One c o u l d i m a g i n e t h a t t h e a m p l i t u d e o f t h e v i b r a t i o n w o u l d be d i f f e r e n t a t x-, and x~. A l s o , i t m i g h t p r o v i d e d t h a t JHm| e x c e e d s some v e r y s m a l l v a l u e . W h i c h p a r t o f t h e c r y o s t a t m i g h t be v i b r a t i n g i n t h i s manner i s an o p e n q u e s t i o n : i t i s e v e n p o s s i b l e t h a t i t i s t h e c r y s t a l l i t e s i n t h e G r a f o i l . The p i c k - u p was c o n s i d e r a b l y l a r g e r when H Q ( a n d -V — Hm w e r e p e r p e n d i c u l a r t o t h e G r a f o i l (g=0°) t h a n when t h e y were p a r a l l e l (3=90°). We w e re u n a b l e t o e l i m i n a t e t h e p i c k - u p . I t s f l u c t u a t i o n a l s o p r e c l u d e d o b s e r v i n g i t o f f r e s o n a n c e and s u b s e q u e n t l y s u b t r a c t i n g i t f r o m t h e s p e c t r a . The p i c k - u p c o u l d be a v o i d e d t o a l a r g e e x t e n t by t r i g g e r i n g t h e s i g n a l a v e r a g e r a f t e r t h e p i c k - u p had become r e l a t i v e l y c o n s t a n t [ s e e F i g . 3 4 ( c ) and ( d ) ] . A t t h e end o f e a c h s w eep, H s c o u l d n o t be r e t u r n e d t o i t s i n i t i a l v a l u e t o o q u i c k l y o r t h e r e w o u l d be e x c e s s i v e e d d y c u r r e n t h e a t i n g o f t h e J H e p o t . C o n s e q u e n t l y an 8 5 % / 1 5 % ramp was u s e d a s shown i n F i g . 3 4 ( c ) . D a t a was c o l l e c t e d f o r 40% o f t h e t i m e f o r D 0 and 60% o f t h e t i m e f o r H 0 . be r e s t r i c t e d i n s u c h a way t h a t i t d o e s n o t d e p e n d on H ml 7 . 4 S i g n a l I n t e n s i t y F o r h y d r o g e n m o l e c u l e s i n s t a t e J , t h e n o r m a l i z e d a b s o r p t i o n i n t e g r a l S j was d e f i n e d i n ( V - 9 ) . I t s h o u l d be p r o p o r t i o n a l t o 1/T and t o t h e c o n c e n t r a t i o n o f t h e n u c l e a r s p i n s b e i n g o b s e r v e d b u t i n d e p e n d e n t o f f a c t o r s s u c h a s t h e s i z e o f t h e s a m p l e , t h e g a i n o f t h e m e a s u r i n g s y s t e m , and p r o p e r t i e s o f t h e t u n e d c i r c u i t o t h e r t h a n i t s f r e q u e n c y . P r o v i d e d t h a t t h e RF l e v e l i s l o w e n o u g h t o a v o i d RF h e a t i n g and s a t u r a t i o n , one c a n t a k e t h e s a m p l e t e m p e r a t u r e T t o be e q u a l t o t h e JHe p o t t e m p e r a t u r e and S j t o be i n d e p e n d e n t o f t h e RF l e v e l . I t i s i m p o r t a n t t o a v o i d s a t u r a t i o n b e c a u s e i t may c a u s e d i s t o r t i o n t h a t d e p e n d s on m e a s u r i n g c o n d i t i o n s s u c h a s t h e sweep r a t e . On t h e o t h e r h a n d , t h e RF l e v e l s h o u l d n o t be t o o l o w b e c a u s e , i n t h e a b s e n c e o f s a t u r a t i o n and h e a t i n g , t h e ( u n n o r m a l i z e d ) s i g n a l i s p r o p o r t i o n a l t o t h e RF l e v e l . The s i m p l e s t way t o c h e c k f o r RF h e a t i n g was t o o b s e r v e t h e J=0 s i g n a l f r o m a s a m p l e o f p u r e o-D 2 a s a f u n c t i o n o f t e m p e r a t u r e and RF a m p l i t u d e . n~D2 g a v e a s i n g l e , n a r r o w peak w i t h a s h a p e t h a t was i n d e p e n d e n t o f t e m p e r a t u r e f o r o u r e n t i r e r a n g e . A l t h o u g h t h e a m p l i t u d e o f t h e o-D 2 a b s o r p t i o n s i g n a l i s l e s s t h a n t h a t o f o-H 2 a t 4K, i t i s g r e a t e r a t .3K b e c a u s e t h e l a t t e r b r o a d e n s c o n s i d e r a b l y . The n a r r o w e r o-D 2 s i g n a l a l s o a l l o w e d a s h o r t e r t i m e f o r e a c h s c a n . The p e a k t o p e a k h e i g h t s 0 ^ o f t h e d e r i v a t i v e s i g n a l h a s b e e n p l o t t e d i n F i g . 35. A t h i g h t e m p e r a t u r e s and l o w RF l e v e l s , t h e C u r i e Law i s o b e y e d b u t a s T^ d e c r e a s e s b e l o w I K d e v i a t i o n s o c c u r , p a r t i c u l a r l y a t t h e h i g h e r RF l e v e l s . No d i s t o r t i o n o f t h e s i g n a l was s e e n s o t h e d e v i a t i o n s c o u l d be due t o e i t h e r RF 165 F i g . 35 Peak t o peak h e i g h t o f t h e d e r i v a t i v e o f t h e a b s o r p t i o n s ^ v e r s u s I/T3 f o r o-D 2 sample G14 i n c e l l B. The RF i n p u t a t t e n u a t i o n A was 10 dB ( c r o s s e s ) , 13 dB ( s q u a r e s ) , 19 dB ( d i a m o n d s ) , and 28 dB ( c i r c l e s ) . The s t r a i g h t l i n e i s t h e b e s t f i t t o t h e h i g h t e m p e r a t u r e and low RF l e v e l d a t a . The J = l c o n c e n t r a t i o n was 0 . 0 0 + , u 2 -.00 166 Fig. 36 Graph of s,/c versus 1/T3 for o-H2 sample G19 in c e l l B. B=90°, A=13 dB, and .78<c<!.94. The straight l i n e i s a least squares f i t to the Curie Law for the data with T3>.4 K. 167 h e a t i n g o r s a t u r a t i o n . H e a t i n g i s e x p e c t e d t o i n c r e a s e i n s e v e r i t y a s t h e t e m p e r a t u r e d r o p s b e c a u s e b o t h t h e s p e c i f i c h e a t and t h e r m a l c o n d u c t i v i t y d r o p . The a b s o r p t i o n i n t e g r a l o f o-H 2 n o r m a l i z e d t o c = l h a s b e e n p l o t t e d a g a i n s t T-j""* i n F i g . 36. The RF l e v e l was t h e same a s t h a t u s e d i n t h e m e a s u r e m e n t s d i s c u s s e d i n C h a p t e r s V I I I a n d I X , i . e . , RF a t t e n u a t i o n A=13 dB. A t h i g h e r l e v e l s , e x t r a n o i s e , a p p a r e n t l y f r o m t h e o s c i l l a t o r , became e v i d e n t . The C u r i e Law was o b e y e d down t o .4 K, much l o w e r t h a n f o r o-D 2- But i n p u r e o - H 2 t h e c o n v e r s i o n h e a t i n g i s t e n t i m e s t h e RF h e a t i n g a t A=13dB. T h e r e f o r e , t h e d e v i a t i o n o f t h e o-D 2 d a t a f r o m t h e C u r i e Law s h o u l d be a t t r i b u t e d t o s a t u r a t i o n . F o r o - H 2 , t h e d e v i a t i o n f r o m t h e C u r i e Law a p p e a r s t o be much more s u d d e n t h a n one w o u l d e x p e c t f o r h e a t i n g , a l t h o u g h t h e d a t a i s r a t h e r l i m i t e d . The t e m p e r a t u r e r i s e s h o u l d be i n v e r s e l y p r o p o r t i o n a l t o < t f o r t h e G r a f o i l , w h i c h g o e s a s T , a t l e a s t b e t w e e n 1.5 K and 20 K (Hegde e t a l , 1 9 7 3 ) . The s i g n a l may be s a t u r a t i n g a t l o w t e m p e r a t u r e s b u t i t i s a l s o p o s s i b l e t h a t t h e r e a r e weak w i n g s on t h e l o w t e m p e r a t u r e s p e c t r a t h a t w e re n o t i n t e g r a t e d b e c a u s e t h e y were b u r i e d i n t h e n o i s e . A l e a s t s q u a r e s f i t t o t h e C u r i e Law u s i n g t h e d a t a a t T 3>.4 K g i v e s S j / c = 3.52/T 3 ( V I I - 1 1 ) The a b s o l u t e m a g n i t u d e o f t h e s l o p e h a s no s i g n i f i c a n c e b e c a u s e s-^ c o n t a i n s an a r b i t r a r y n o r m a l i z a t i o n c o n s t a n t . However, t h i s n o r m a l i z a t i o n c o n s t a n t i s t h e same f o r a l l o f t h e 168 a b s o r p t i o n i n t e g r a l s d i s c u s s e d i n t h i s t h e s i s s o t h e r e s u l t s f o r d i f f e r e n t t y p e s o f s p i n s c a n be c o m p a r e d . A p l o t o f s 0 / ( l - c ) v e r s u s T 3 - 1 f o r t h e s m a l l J=0 c o m p o n e n t o f two D 2 s a m p l e s i s g i v e n i n F i g . 37. Sample G7 i n c e l l A was a b s o r b e d a t 23-27 K and t h e J = l p e a k s w e re n o t r e s o l v e d b e l o w 1.4 K f i m p l y i n g a n o n - u n i f o r m f i l m . S ample G18 i n c e l l B was a d s o r b e d a t 40 K and i s b e l i e v e d t o h a v e b e e n u n i f o r m : t h e J = l p e a k s w e r e c l e a r l y r e s o l v e d down t o .85 K. D a t a f r o m t h e two s a m p l e s a g r e e w e l l and t h e C u r i e Law i s o b e y e d o v e r t h e e n t i r e t e m p e r a t u r e r a n g e . T h i s i s r a t h e r s u r p r i s i n g c o n s i d e r i n g t h a t t h e J=0 component o f p u r e o-D 2 showed e v i d e n c e o f s a t u r a t i o n b e l o w 1 K. The 1=2 n u c l e i a r e a p p a r e n t l y r e l a x e d more e f f e c t i v e l y by 1=1 n e i g h b o u r s t h a n by 1=2 n e i g h b o u r s . H o w e v e r , t h i s c o n c l u s i o n s h o u l d n o t be o v e r e m p h a s i z e d b e c a u s e t h e s c a t t e r i n t h e p-D 2 d a t a i s f a i r l y l a r g e , p a r t i c u l a r l y f o r G18 a t l o w t e m p e r a t u r e . A l e a s t s q u a r e s f i t t o t h e C u r i e Law y i e l d s s Q / ( l - c ) = 1.71/T 3 ( V I I - 1 2 ) F i g . 38 i s a g r a p h o f s-^/c v e r s u s T 3 _ 1 f o r t h e same two s a m p l e s o f D 2. A b o v e .9 K, b o t h f o l l o w t h e C u r i e Law a l t h o u g h t h e G7 d a t a i s f a i r l y l i m i t e d i n t h a t r e g i o n . A l e a s t s q u a r e s f i t t o t h e h i g h t e m p e r a t u r e d a t a g i v e s S j / c = . 7 0 5 / T 3 ( V I I - 1 3 ) I n a c c o r d a n c e w i t h ( V - 8 ) , one e x p e c t s t h e r a t i o o f t h e s l o p e s g i v e n i n ( V I I - 1 2 ) a nd ( V I I - 1 3 ) t o be e q u a l t o ( 5 / 6 ) ( 6 / 2 ) = 5 / 2 . The f a c t o r 5/6 i s t h e f r a c t i o n o f J=0 m o l e c u l e s t h a t h a v e 1=2 169 7 _. I (K") F i g . 37 G r a p h o f s Q / ( l - c ) v e r s u s I/T3 f o r t h e J=0 c o m p o n e n t o f D 2. 3=90°. The d a t a i s f r o m D 2 s a m p l e G18 i n c e l l B (.77<c£.98) w i t h A=13 dB ( c i r c l e s ) and A=19 dB ( d i a m o n d ) , a n d f r o m D 2 s a m p l e G7 i n c e l l A (.94<c£. 97) w i t h A=14 dB ( c r o s s e s ) . The l i n e i s a l e a s t s q u a r e s f i t t o t h e C u r i e Law. 170 Fig. 38 Graph of s-^/c versus 1/T3 for the J=l component of T>2' 3 = 90°. The data i s from D 2 sample G18 i n c e l l B (.77j<c<.98) with A=13 dB ( c i r c l e s ) and A=19 dB (diamond), and from D 2 sample G7 in c e l l A (.94£c<.97) with A=14 dB (crosses). The l i n e i s a least squares f i t to the Curie Law, using only the data with To>.9 K . 171 and 6/2 i s t h e r a t i o o f 1(1+1) f o r 1=2 and 1=1. The m e a s u r e d s l o p e s a g r e e w e l l w i t h t h i s r e s u l t s i n c e ( 2 / 5 ) ( J = 0 s l o p e / J = l s l o p e ) = .97 ( V I I - 1 4 ) (V-8) a l s o i m p l i e s t h a t t h e r a t i o o f t h e s l o p e s o f s-^/c v e r s u s f o r o - H 2 and p - D 2 s h o u l d be e q u a l t o t h e r a t i o o f Y ^ H Q = Y ^ O ; U O w a s t * i e s a m e ^ n e a c n c a s e . One f i n d s [ y ( D 2 ) / Y ( H 2 ) 1 ( H 2 s l o p e / D 2 s l o p e ) = .77 ( V I I - 1 5 ) w h i c h i s s i g n i f i c a n t l y l e s s t h a n t h e e x p e c t e d v a l u e o f one. The r e a s o n i s n o t c l e a r . B e l o w .9 K, s ^ / c i s q u i t e d i f f e r e n t f o r G18 and f o r G7. Sample G18 a p p e a r s t o c o n t i n u e t o o b e y t h e C u r i e Law b u t t h e s c a t t e r i n t h e d a t a i s v e r y l a r g e b e c a u s e t h e m o d u l a t i o n p i c k - u p made i t v e r y d i f f i c u l t t o d e t e r m i n e t h e b a s e l i n e . I n f a c t , b e t w e e n .5 K and .7 K i t was i m p o s s i b l e s o s-^  c o u l d n o t be d e t e r m i n e d f o r t h e s p e c t r a i n t h a t r a n g e . F o r G7 t h e t a s k was somewhat e a s i e r ; t h e s u r f a c e a r e a o f t h e G r a f o i l i n c e l l A was t w i c e a s g r e a t a s t h a t i n c e l l B s o t h e r e l a t i v e e f f e c t o f t h e c o h e r e n t i n t e r f e r e n c e was much l e s s . E v en s o , d a t a n e a r .6 K s h o u l d be t r e a t e d c a u t i o u s l y . The d i f f e r e n c e i n t h e l o w t e m p e r a t u r e b e h a v i o u r o f G7 a n d G18 c a n o n l y be a t t r i b u t e d t o t h e n a t u r e o f t h e two s a m p l e s . I t i s known t h a t G7 was n o t u n i f o r m s o p e r h a p s some m o l e c u l e s e x p e r i e n c e d d i f f e r e n t c r y s t a l f i e l d s f r o m m o s t o f t h e o t h e r s and p r o d u c e d a b r o a d a n d weak co m p o n e n t o f t h e a b s o r t i o n s p e c t r u m t h a t was n o t i n c l u d e d i n t h e i n t e g r a l g i v i n g s^. We f o u n d , u s i n g t h e 4 H e c r y o s t a t , t h a t when t h e c o v e r a g e was i n c r e a s e d a b o v e p =1 (/3 x /5 172 c o v e r a g e ) , t h e NMR s p e c t r u m a t 4 K r a p i d l y b r o a d e n e d and d i s a p p e a r e d . The a d s o r b a t e went o u t o f r e g i s t r y w i t h t h e s u b s t r a t e s o t h a t t h e c o n t r i b u t i o n t o t h e c r y s t a l f i e l d a r i s i n g f r o m i n t e r a c t i o n s w i t h t h e s u r f a c e b e g a n t o v a r y f r o m one m o l e c u l e t o t h e n e x t . When t h e J = l c o n c e n t r a t i o n o f G7 was c a l c u l a t e d f r o m S - ^ / S Q / t h i s h y p o t h e t i c a l m i s s i n g c omponent was i n c l u d e d by a s s u m i n g t h a t s-^  o b e y e d t h e C u r i e Law a l l t h e way down t o .3 K. The c o r r e c t i o n t o s ^ / c was i n s i g n i f i c a n t b e c a u s e c ~ l f o r a l l o f t h e G7 d a t a . However, f o r s n / ( l - c ) t h e c o r r e c t i o n was l a r g e . The f a c t t h a t t h e c o r r e c t e d v a l u e s o f S Q / ( 1 - C ) f i t t h e C u r i e Law q u i t e w e l l s u p p o r t s t h e h y p o t h e s i s o f t h e m i s s i n g c o m p onent. 7 . 5 -^ H B a c k g r o u n d The "*"H b a c k g r o u n d s i g n a l c o n t a i n e d t h r e e c o m p o n e n t s . The p e a k t o p e a k w i d t h s o f t h e a b s o r p t i o n d e r i v a t i v e s o f t h e two n a r r o w c o m p o n e n t s were 1.3 mT and a b o u t .1 mT. The b r o a d c o m p o n e n t , w h i c h e x t e n d e d b e y o n d o u r maximum 0-H2 sweep w i d t h o f 12 mT, was v e r y weak and c o u l d o n l y be d i s c e r n e d n e a r .3 K. When o b s e r v i n g t h e o - H 2 s p e c t r a , we u s e d a peak t o p e a k m a g n e t i c f i e l d m o d u l a t i o n o f .17 mT s o t h e n a r r o w e s t c o m p onent o f t h e "''H b a c k g r o u n d s i g n a l was n o t r e s o l v e d . I t s w i d t h was a b o u t t h e same a s t h a t o f t h e "'"H s i g n a l p r e v i o u s l y s e e n by us i n P y r e x . The f a c t t h a t i t i s s o n a r r o w s u g g e s t s t h a t i t i s due t o i s o l a t e d i m p u r i t i e s i n some m a t e r i a l . The medium w i d t h 173 c o m p o n ent was s i m i l a r t o t h e s i g n a l t h a t we had p r e v i o u s l y s e e n i n T e f l o n s o i t p r o b a b l y a r o s e f r o m t h e T e f l o n c o i l h o l d e r and T e f l o n f i l m i n s i d e t h e G r a f o i l c e l l . F i g . 39 shows t h e 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 e c o m b i n e d a b s o r p t i o n i n t e g r a l s o f t h e two n a r r o w c o m p o n e n t s . F o r e a c h RF l e v e l , s v e r s u s T 3~* i s l i n e a r b u t t h e y - i n t e r c e p t i s p o s i t i v e . T h i s c o u l d o c c u r i f p a r t o f t h e s i g n a l were f r o m s p i n s a t T=T^ and p a r t w e re f r o m s p i n s a t some f i x e d t e m p e r -a t u r e . The l a t t e r m i g h t , f o r e x a m p l e , be i n t h e e n a m e l on t h e m o d u l a t i o n c o i l l e a d s , w h i c h w e r e n o t c o m p l e t e l y c o v e r e d by a l u m i n u m f o i l . The s t r a i g h t l i n e i s a l e a s t s q u a r e s f i t t o t h e 13 dB d a t a . I t g i v e s . s = 2.27/T3 + .56 ( V I I - 1 6 ) A t l o w t e m p e r a t u r e s , s i n c r e a s e s a s t h e RF l e v e l d e c r e a s e s , a s one w o u l d e x p e c t i f t h e r e were s a t u r a t i o n . E x a m i n a t i o n o f t h e a b s o r p t i o n s i g n a l shown i n F i g . 40 p r o v i d e s f u r t h e r e v i d e n c e o f s a t u r a t i o n . The r e s p o n s e o f t h e s p i n s t h a t f i r s t come i n t o r e s o n a n c e , i . e , a t t h e l e f t s i d e o f t h e s c a n , i s s t r o n g e r t h a t t h a t f r o m t h o s e t h a t come i n t o r e s o n a n c e l a t e r . A t l o w e r RF l e v e l s t h i s d i s t o r t i o n d i d n o t o c c u r . T h i s e f f e c t c a n be e x p l a i n e d by t h e f o l l o w i n g a r g u m e n t . As t h e f i e l d sweeps up f r o m i t s i n i t i a l v a l u e , t h e f i r s t s p i n s t o come i n t o r e s o n a n c e a r e s l i g h t l y d e p o l a r i z e d . They i n t e r a c t w i t h t h e o t h e r s p i n s , d e p o l a r i z i n g them s l i g h t l y . I f t h e s e s p i n s come i n t o r e s o n a n c e i n a t i m e s h o r t e r t h a n t h e l o n g i t u d i n a l r e l a x a t i o n t i m e T-, , t h e y w i l l p r o d u c e a w e a k e r s i g n a l . T h i s 174 F i g . 39 G r a p h o f s v e r s u s 1/TS f o r t h e -"-H b a c k g r o u n d s i g n a l . The s o l i d s q u a r e s a r e f o r g=0° and a l l o f t h e o t h e r s y m b o l s a r e f o r g = 9 0 ° . The RF a t t e n u a t i o n A was 13 dB ( s q u a r e s ) , 19 dB ( d i a m o n d s ) , and 25 dB ( c i r c l e s ) . The s t r a i g h t l i n e i s a l e a s t s q u a r e s f i t t o t h e 13 dB d a t a . 175 , Z mT F i g . 40 H b a c k g r o u n d s i g n a l , showing d i s t o r t i o n due t o s a t u r a t i o n . The b r o a d component e x t e n d s beyond t h e w i d t h o f t h e s c a n . The .1 mT component o f t h e s i g n a l c a n n o t be s e e n b e c a u s e i t i s o v e r m o d u l a t e d . The t o t a l m e a s u r i n g t i m e t m was 9.1 h, A=13 dB, T 3=.314 K, and g = 9 0 ° . F i g . 41 (a) The e f f e c t i v e g a i n G o f t h e NMR s y s t e m when s w e e p i n g f r o m t o H f and s a t u r a t i n g t h e s p i n s . (b) The u n s a t u r a t e d a b s o r p t i o n s i g n a l A ( H ) . 176 e f f e c t c a n be m o d e l l e d by s a y i n g t h a t t h e e f f e c t i v e g a i n G d e c r e a s e s a s t h e sweep p r o c e e d s a s shown i n F i g . 4 1 ( a ) . I n t h e l i m i t o f s m a l l m o d u l a t i o n , we o b s e r v e a s i g n a l G ( H ) A ' ( H ) where A'(H) i s t h e d e r i v a t i v e o f t h e u n s a t u r a t e d a b s o r p t i o n s i g n a l A ( H ) . I f and a r e r e s p e c t i v e l y w e l l b e l o w and w e l l a b o v e t h e r e s o n a n c e r e g i o n , t h e n A ( ) = A ( ) = 0 . I n t e g r a t i n g by p a r t s , we f i n d t h a t t h e i n t e g r a l o f t h e p r o d u c t G ( H ) A ' ( H ) i s g r e a t e r t h a n z e r o : / f G ( H ) A ' (H)dH=A(H-)G(IL ) - A ( H . ) G ( H . ) - / „ f G» (H) A(H) d H ( V I I " 1 7 a ) T T r t i i H . i =-/ H fG' (H)A(H) dH 1 ( V I I - 1 7 b ) H i A ( H ) i s a p o s i t i v e , e v e n f u n c t i o n o f H-Hg and G'(H) i s a n e g a t i v e f u n c t i o n s o t h e r i g h t hand s i d e o f ( V I I - 1 7 b ) i s p o s i -t i v e , p r o d u c i n g t h e o f f s e t s e e n a t t h e r i g h t s i d e o f F i g . 40. The a b s o r p t i o n i n t e g r a l s o f t h e * H b a c k g r o u n d and p u r e o - H 2 were q u i t e s i m i l a r i n s i z e s o i t was i m p o r t a n t t h a t t h e b a c k g r o u n d be s u b t r a c t e d a c c u r a t e l y . I n p a r t i c u l a r , i f t h e b a c k g r o u n d s i g n a l was n o t c e n t e r e d c o r r e c t l y a s p u r i o u s , n a r r o w c e n t r a l c o m p o nent w o u l d be g e n e r a t e d . C o n s e q u e n t l y , b a c k g r o u n d s p e c t r a were o b t a i n e d a t l e a s t e v e r y .2 K f o r t e m p e r a t u r e s b e l o w 1 K. Above 1.5 K, * H b a c k g r o u n d s p e c t r a were m e a s u r e d a t t h e same t e m p e r a t u r e s a s t h e o-H 2 s p e c t r a . The b a c k g r o u n d s p e c t r a a n d o - H 2 s p e c t r a were r e c o r d e d u n d e r i d e n t i c a l c o n d i t i o n s . Some t y p i c a l d e r i v a t i v e s p e c t r a w i t h and w i t h o u t t h e b a c k g r o u n d a r e shown i n F i g . 42. 1 7 7 Fig. 42 Derivatives of the absorption signal with [(a), (c) , and (e)] and without [(b), (d), and (f)] the *H background s i g n a l . A=13 dB and J3=0°. The horizontal bars are 2 mT. The t o t a l measuring time i s tj_. (a) and (b) : .318 K, ^=1.2 h, c=.95. (c) and (d): .664 K, tm=2.4 h, c=.91. (e) and ( f ) : .860 K, t =1.2 h, c=.89. 178 N o t e s t o C h a p t e r V I I 1. E m e r s o n a n d C u m i n g I n c . , C a n t o n , M a s s . , U . S . A . 2 . M a n u f a c t u r e d b y T e x a s I n s t r u m e n t s b u t n o l o n g e r a v a i l a b l e . 3 . T h i s d e l i c a t e o p e r a t i o n w a s p e r f o r m e d b y J . F . C a r o l a n . 4. T h i s s t a t e m e n t d i s r e g a r d s v a r i o u s p e a k s i n t h e n o i s e a t c e r t a i n v a l u e s o f f m , w h i c h a r e d i s c u s s e d b e l o w . 1 3 5 . We a l s o t r i e d u s i n g t h e C r e s o n a n c e o f G r a f o i l b u t t h a t s i g n a l s a t u r a t e d a t a l l R F a m p l i t u d e s t h a t w e r e l a r g e e n o u g h t o g i v e a u s e f u l s i g n a l . 179 CHAPTER V I I I NMR SPECTRA OF o - H 2 8.1 E v i d e n c e f o r O r i e n t a t i o n a l O r d e r i n g Some NMR a b s o r p t i o n s p e c t r a o b t a i n e d f r o m a h i g h c o n c e n -t r a t i o n s a m p l e o f o - H 2 (G20) w i t h 3=0° and p=.85 a r e shown i n F i g . 43 and 44. The "'"H b a c k g r o u n d h a s b e e n s u b t r a c t e d . A t h i g h t e m p e r a t u r e s , t h e r e i s a d o u b l e t w i t h a s p l i t t i n g Av t h a t i n c r e a s e s s l o w l y a s T d e c r e a s e s . T h i s i s what one w o u l d e x p e c t f o r t h e p a r a r o t a t i o n a l s t a t e , w h i c h i s c h a r a c t e r i z e d by a s i n g l e o r d e r p a r a m e t e r o. A f e a t u r e t h a t i s n o t e x p e c t e d i s t h a t b e l o w 1.2 K t h e c e n t r e o f t h e s p e c t r u m b e g i n s t o f i l l i n . N e a r .6 K, t h e s p l i t t i n g i n c r e a s e s v e r y r a p i d l y and a c e n t r a l p e a k e m e r g e s . B e l o w .6 K, t h e c e n t r a l p e a k d i s a p p e a r s , t h e s p l i t t i n g s l o w l y a p p r o a c h e s a maximum, and two s m a l l , o u t e r p e a k s a p p e a r . A t l o w t e m p e r a t u r e , t h e s p e c t r a a r e s t r o n g f u n c t i o n s o f t h e c o n c e n -t r a t i o n ; t h e r e a r e many s i m i l a r i t i e s i n t h e c h a n g e s i n t h e s p e c t r a c a u s e d by l o w e r i n g c o r r a i s i n g T. Due t o t h e c o n c e n t r a t i o n d e p e n d e n c e , t h e r e was i n s u f f i c i e n t t i m e t o o b t a i n g o o d S:N r a t i o s . I n any e v e n t , e v e n w i t h e x t e n s i v e s i g n a l a v e r a g i n g , t h e b a s e l i n e s o f t h e s p e c t r a w o u l d n o t h a v e b e e n f l a t b e c a u s e o f f l u c t u a t i n g m o d u l a t i o n p i c k - u p . The s h a p e o f t h e b a s e l i n e s t h a t i t p r o d u c e d c h a n g e d w i t h t e m p e r a t u r e . The s p e c t r a shown i n F i g . 45 and 46 were o b t a i n e d f r o m a n o t h e r h i g h c o n c e n t r a t i o n s a m p l e o f o-H 2 (G19) b u t t h i s t i m e w i t h 3=90° and p=.85. The b e h a v i o u r o f t h e s e s p e c t r a i s F i g . 43 A b s o r p t i o n s p e c t r a o f o-H 2 s a m p l e G20 w i t h g =0°. A=13 dB. p =.85. S p e c t r u m (a) (b) ( c ) (d) (e) ( f ) T ( K ) .318 .400 .508 .563 .582 .664 c .95 .94 .94 .93 .92 .91 ^ ( h ) 1.2 1.2 1.2 1.2 2.4 2.4 The v e r t i c a l s c a l e i s t h e same f o r a l l o f t h e s p e c t r a . The a m p l i t u d e s o f t h e s i g n a l s h a v e b e e n m u l t i p l i e d by T s o t h a t i f t h e C u r i e Law were o b e y e d a nd i f c were c o n s t a n t , t h e a r e a u n d e r e a c h o f t h e s p e c t r a w o u l d be e x p e c t e d t o be t h e same. O .5 (v-vn)/36 F i g . 44 A b s o r p t i o n s p e c t r a o f o-H 2 s a m p l e G20 w i t h 3 = 0 ° . A=13 db. D = . 8 5 . S p e c t r u m (a) T ( K ) t m < h > .664 .91 2.4 (b) .752 .90 1.2 ( c ) .860 .89 1.2 (d) 1.19 .88 1.2 (e) 1.76 .87 .90 ( f ) 4.24 .85 1.2 The v e r t i c a l s c a l e i s t w i c e a s g r e a t a s i n F i g . 43 and t h e h o r i z o n t a l s c a l e i s h a l f a s g r e a t . The s p e c t r a h a v e b e e n n o r m a l i z e d i n t h e same manner a s t h o s e i n F i g . 4 3 . N o t e t h a t ( a ) i s t h e same a s F i g . 4 3 ( f ) . 182 F i g . 45 A b s o r p t i o n s p e c t r a o f o-H 2 s a m p l e G19 w i t h 3=90°. A=13 dB. p=.85. S p e c t r u m ( a ) (b) (c)- (d) (e) ( f ) T ( K ) .309 .450 .519 .577 .620 .693 c .94 .92 .91 .90 .87 .86 t m ( h ) 1.2 2.1 2.4 3.7 2.4 3.9 The s p e c t r a h a v e b e e n n o r m a l i z e d i n t h e same manner a s t h o s e i n F i g . 4 3 . The v e r t i c a l s c a l e i s t h e same f o r a l l o f t h e s p e c t r a h e r e . 183 F i g . 46 A b s o r p t i o n s p e c t r a f o r o-H 2 s a m p l e G19 w i t h B=90° A=13 dB. p = . 8 5 . S p e c t r u m ( a ) T( K ) .693 c .86 t m ( h ) 3.9 The s p e c t r a h a v e b e e n n o r m a l i z e d i n t h e same manner a s t h o s e i n F i g . 4 3 . The v e r t i c a l s c a l e i s t w i c e a s g r e a t a s i n F i g . 45 and t h e h o r i z o n t a l s c a l e i s h a l f a s g r e a t . N o t e t h a t ( a ) i s t h e same a s F i g . 4 5 ( f ) . (b) ( c ) (d) (e) .863 1.25 1.98 4.22 .84 .82 .80 .79 .4 2.4 2.8 2.4 184 si m i l a r to that of the 3=0° spectra but the features are not as c l e a r l y resolved. As (IV-14) and (IV-15) show, t h i s i s because i n the case of 3=90° i t i s necessary to average over the azimut-hal angle a of the graphite c r y s t a l axes but for 3=0° i t i s not. In Fig. 47, the temperature dependences of the main peaks of the absorption spectra of G19 and G20 have been plotted. Note that in the disordered state Av/3d= | J | for G19 but Av/3d = 2 ]p\ for G20 because 3=90° and 0° respectively. For G20, the peaks at T - 1=1.51 and 1.72 K - 1 were not resolved but thei r positions could be estimated. The same applies to T _ 1=1.73 K _ 1 for G19. However, for the G19 spectra in the ranges 1.16 K~1<T~1<1.61 K - 1 and T - 1<.331 K-"*" , the peak positions could not even be estimated. The peak s p l i t t i n g s have been corrected for the e f f e c t of broadening which was assumed to be Gaussian. 1 i f two delta function peaks, which have a s p l i t t i n g Av , are subject to Gaussian broadening with a standard deviation c r G , then the broadened peaks w i l l be pulled together. Their s p l i t t i n g Av ' i s given by x = ( A v ' / 2 a G ) 2 coth x ( V l l l - l ) 2 where x = AvAv'/4a_, The broadening was determined from the width of the outer edges of the main peaks: these edges are not affected by the orien-t a t i o n a l d i s t r i b u t i o n of the graphite c r y s t a l s . For G20, the correction was i n s i g n i f i c a n t unless T - 1<.31 K~*. For G19, the correction was i n s i g n i f i c a n t for T - 1>1.7 K - 1 but was about a factor of two for .5 K - 1 < T-1<1.0 i f 1 . For 185 . 8 -• • • . 4 - o CO < •2-O o°o « " ° • a 0 i 1 r 2 3 T " ' ( K " ' ) Fig. 47 S p l i t t i n g of the absorption peaks Av/3d versus 1/T for three o-H2 samples: G20 with .83<.cj<.95 ( c i r c l e s ) , G21 with .21<c_<.26 (diamonds), and G19 with .78<c£.94 (squares). Samples G20 and G21 had B=0° whereas G19 had 6=90°. 186 t h e l a t t e r p o i n t s t h e c o r r e c t i o n i s s o l a r g e t h a t i t c a n n o t be c o n s i d e r e d t o be v e r y r e l i a b l e s i n c e t h e b r o a d e n i n g i s n o t e x a c t l y G a u s s i a n and t h e u n b r o a d e n e d s p e c t r a a r e n o t d e l t a f u n c t i o n l i n e s . F o r b o t h s a m p l e s m e a s u r e m e n t s were i n i t i a t e d a t .3 K and T was i n c r e a s e d s u c c e s s i v e l y f o r s u b s e q u e n t m e a s u r e m e n t s . The t e m p e r a t u r e was n o t i n c r e a s e d m o n o t o n i c a l l y b e c a u s e i t was 3 n e c e s s a r y t o warm t h e s a m p l e t o 4 K when t h e He was r e -c o n d e n s e d . F o r G20, t h i s f i r s t o c c u r r e d a f t e r t h e m e a s u r e m e n t a t .5 K and f o r G19, d u r i n g t h e m e a s u r e m e n t a t .58 K and a g a i n a f t e r t h e m e a s u r e m e n t a t .69 K. The G19 s p e c t r a a t .58 K o b t a i n e d b e f o r e and a f t e r w a r m i n g t o 4 K a p p e a r e d t o be i d e n t i c a l s o t h e y were a d d e d t o g e t h e r . The s p e c t r a a r e v e r y s e n s i t i v e t o t e m p e r a t u r e i n t h i s r e g i o n s o i f t h e r e w e re any h y s t e r e s i s i t must h a v e b e e n v e r y s m a l l . We have i n t e r p r e t e d t h e r a p i d , f a c t o r o f t h r e e i n c r e a s e i n Av n e a r .6 K a s a t r a n s i t i o n t o a n o r i e n t a t i o n a l l y o r d e r e d s t a t e . The two d a t a p o i n t s o f G20 b r a c k e t i n g t h e s h a r p r i s e h a v e c=.91 and c=.92. N o t e t h a t t h e t r a n s i t i o n t e m p e r a t u r e T c w i l l d r o p as c d e c r e a s e s , s o t h a t o n F i g . 47 t h e t r a n s i t i o n w i l l a p p e a r t o be a l i t t l e s h a r p e r , t h a n i t a c t u a l l y i s . A t .3 K, Av a p p e a r s t o be c l o s e t o i t s maximum v a l u e s o t h e s p e c t r u m w o u l d be u n l i k e l y t o c h a n g e much i f T were l o w e r e d any f u r t h e r . C o n s e q u e n t l y , we h a v e f i t t h e .3 K s p e c t r a t o t h e T=0 s p e c t r a e x p e c t e d f o r t h e v a r i o u s o r i e n t a t i o n a l l y o r d e r e d s t a t e s p r e d i c t e d by MFT. T h i s i s d i s c u s s e d i n t h e n e x t s e c t i o n . The r e s u l t s show 187 c o n c l u s i v e l y t h a t t h e p i n w h e e l p h a s e i s t h e o n l y one o f t h e s e s t a t e s t h a t i s c o n s i s t e n t w i t h t h e .3 K s p e c t r a . No a t t e m p t h a s b e e n made t o a n a l y z e t h e s h a p e s o f t h e s p e c t r a i n t e r m e d i a t e b e t w e e n t h e h i g h t e m p e r a t u r e d o u b l e t and t h e .3 K s p e c t r u m . A p a r t i c u l a r l y i n t e r e s t i n g f e a t u r e o f t h e s e s p e c t r a i s t h e e m e r g e n c e o f t h e c e n t r a l p e a k and i t s s u b s e q u e n t d i s a p p e a r a n c e a s T d e c r e a s e s . S p e c t r a i n t h e i n t e r m e d i a t e t e m p e r a t u r e r e g i m e e x h i b i t many o f t h e c h a r a c t e r i s t i c s s e e n i n t h e NMR s p e c t r a o f b u l k o - H 2 f o r ,2<c<.55 and .2 K<T<.7 K ( E s t e v e e t a l , 1 9 8 2 ; Washburn e t a l , 1 9 8 2 ) . B u l k o - H 2 i s known n o t t o ha v e l o n g r a n g e o r i e n t a t i o n a l o r d e r a t s u c h l o w c o n c e n t r a t i o n s b u t , a s m e n t i o n e d i n S e c t i o n 1.3, i t h a s b e e n p r o p o s e d t h a t i t u n d e r g o e s a t r a n s i t i o n t o a q u a d r u p o l a r g l a s s ( S u l l i v a n , 1 9 7 6 ) . R e g a r d l e s s o f w h e t h e r t h e sy m m e t r y o f t h e l o w t e m p e r a t u r e s t a t e i s a c t u a l l y a n y d i f f e r e n t f r o m t h a t o f t h e p a r a r o t a t i o n a l p h a s e , i t i s w e l l e s t a b l i s h e d t h a t t h e NMR l i n e s h a p e s c a n be e x p l a i n e d by c h a n g e s i n t h e p r o b a b i l i t y d i s t r i b u t i o n P( [r| ) o f t h e m a g n i t u d e o f t h e o r d e r p a r a m e t e r a . Some b u l k o-H 2 s p e c t r a and t h e c o r r e s p o n d i n g P( fer| ) 's d e d u c e d f r o m them a r e d e p i c t e d i n F i g . 48. N o t e t h a t i n t h e i n t e r m e d i a t e t e m p e r a t u r e r a n g e , where P( |cr| ) i s v e r y b r o a d , a c e n t r a l c o m p o n e n t i s o b s e r v e d i n t h e s p e c t r a . E v e n a t t h e l o w e s t t e m p e r a t u r e s , t h e c e n t r e o f t h e s p e c t r u m i s n e a r l y f i l l e d i n b e c a u s e o f t h e l o n g t a i l o f P( (cr| ) . Some s u c c e s s i n p r e d i c t i n g t h e t e m p e r a t u r e d e p e n d e n c e o f P( joj ) h a s b e e n a c h i e v e d by u s i n g a mean f i e l d m o d e l i n w h i c h 1 8 8 U (kHz) W Fig. 48 Temperature dependence of (a) the NMR absorption spectrum (only half i s shown) and (b) the order parameter d i s t r i b u t i o n P( ]p\ ) of bulk o-H2 in the disordered state for c=.55 (Washburn et a l , 1982). At each temperature, the experimental spectrum i s v i r t u a l l y i d e n t i c a l to the synthetic spectrum derived from P( |o| ). 189 m o l e c u l e s a r e assumed t o i n t e r a c t w i t h a random d i s t r i b u t i o n o f a x i a l l y s y m m e t r i c q u a d r u p o l a r f i e l d s ( E s t e v e e t a l , 1 9 8 2 ) . T h e s e f i e l d s w o u l d d e p e n d on t h e l o c a l d i s t r i b u t i o n o f J = l m o l e c u l e s . The e n e r g y l e v e l s o f a J = l m o l e c u l e w o u l d be s p l i t i n t o an mj=0 l e v e l and a p a i r o f d e g e n e r a t e mj=+l l e v e l s a s i n t h e c a s e o f a n a d s o r b e d J = l m o l e c u l e w h i c h e x p e r i e n c e s a n a x i a l l y s y m m e t r i c c r y s t a l f i e l d . The m a g n i t u d e o f t h e e n e r g y g a p w o u l d v a r y r a n d o m l y f r o m s i t e t o s i t e . The peak o f P( |a| ) becomes n a r r o w a t l o w t e m p e r a t u r e s b e c a u s e t h e e n e r g y g a p i s l a r g e c o m p a r e d t o kT; v i r t u a l l y a l l o f t h e m o l e c u l e s a r e i n t h e g r o u n d s t a t e r e g a r d l e s s o f t h e w i d t h o f t h e g a p . I n l i g h t o f t h e l o w c o n c e n t r a t i o n , b u l k 0 -H2 m e a s u r e m e n t s , i t a p p e a r s t h a t o u r i n t e r m e d i a t e t e m p e r a t u r e , h i g h c o n c e n t r a t i o n a d s o r b e d o-H 2 s p e c t r a m i g h t w e l l be e x p l a i n e d i n t e r m s o f a d i s t r i b u t i o n o f t h e o r d e r p a r a m e t e r s . F o r 6=0°, p r o b a b l y o n l y a w o u l d be n e e d e d b u t f o r g=90°, i t m i g h t a l s o be n e c e s s a r y t o i n c l u d e r\. I t w o u l d be n e c e s s a r y t o i m p r o v e t h e S:N r a t i o s o f t h e s p e c t r a i n o r d e r t o be a b l e t o p u r s u e t h i s a p p r o a c h . T h e r e a r e some o t h e r a l t e r n a t i v e s f o r e x p l a i n i n g t h e c e n t r a l p e a k s o f t h e a d s o r b e d o - H 2 s p e c t r a b u t t h e y a r e l e s s a p p e a l i n g . S u c h a p e a k m i g h t a r i s e i f some o f t h e m o l e c u l e s were u n d e r g o i n g m o t i o n t h a t was f a s t c o m p a r e d t o t h e i n t r a -m o l e c u l a r d i p o l a r i n t e r a c t i o n . An e x a m p l e w o u l d be m o l e c u l a r r e o r i e n t a t i o n s u c h a s t h a t o c c u r r i n g a t h i g h t e m p e r a t u r e s . C r i t i c a l f l u c t u a t i o n s m i g h t p r o d u c e s u c h m o t i o n b u t t h e y a r e 190 n o r m a l l y s e e n o n l y i n v e r y p u r e s y s t e m s . A n o t h e r e x a m p l e w o u l d be t h e 2D t r a n s l a t i o n a l m o t i o n o f a d s o r b e d m o l e c u l e s i n a f l u i d p h a s e . I t w o u l d o n l y be f u l l y e f f e c t i v e f o r c r y s t a l l i t e s f o r ->• w h i c h t h e p o l a r a n g l e g o f H Q i n t h e c r y s t a l r e f e r e n c e f r a m e was e q u a l t o t h e s o - c a l l e d m a g i c a n g l e o f 55° ( A l e k s a n d r o v , 1 9 6 6 ) . One c a n s e e t h i s by e x a m i n i n g t h e e q u a t i o n f o r t h e NMR f r e q u e n c i e s ( I V - l O b ) . The s e c o n d and t h i r d t e r m s o f t h e r i g h t h and s i d e w i l l a l w a y s be z e r o b e c a u s e t h e m o t i o n makes i t n e c e s s a r y t o a v e r a g e o v e r a l l a z i m u t h a l a n g l e s 6. The f i r s t t e r m 2 w i l l o n l y be z e r o i f 3 c o s £-1=0, i . e . , £=55°. F o r c r y s t a l -l i t e s a t o t h e r a n g l e s , t h e r e d u c t i o n i n t h e s p l i t t i n g w o u l d be l e s s . T h i s h y p o t h e s i s i s c o n s i d e r e d f u r t h e r i n S e c t i o n 8 . 3 . I n t h e 2 - o u t p h a s e , i t i s p o s s i b l e t o g e t a s p e c t r u m c o n s i s t i n g o f a s i n g l e p eak w i t h o u t r e q u i r i n g a n y m o t i o n . F o r V c / r i n t h e r a n g e 3-4, ( 1 1 1 - 2 0 ) shows t h a t |j| <.03. W i t h s u c h a s m a l l v a l u e o f ja| , i f 3 = 0 ° , t h e s p l i t t i n g o f t h e a b s o r p t i o n p e a k s w o u l d be l e s s t h a n t h e b r o a d e n i n g s o o n l y one p e a k w o u l d be r e s o l v e d . T h e s e v a l u e s o f V c / r a r e q u i t e c l o s e t o t h e v a l u e s d e d u c e d f r o m t h e h i g h t e m p e r a t u r e NMR d a t a d i s c u s s e d i n S e c t i o n 7.4. T h e r e f o r e , one m i g h t s u p p o s e t h a t t h e c e n t r a l p e a k o b s e r v e d n e a r .6 K a r i s e s f r o m f o r m a t i o n o f d o m a i n s o f t h e 2 - o u t p h a s e i n t h e t r a n s i t i o n r e g i o n . H owever, t h e r e i s a l s o a c e n t r a l p e a k i n t h e 8=90° s p e c t r a . I n t h a t c a s e ( I V - l O b ) shows t h a t f o r a = 0 , one e x p e c t s p e a k s a t v/3d=+n/2. A t T=0, ( I I I - 1 3 c ) g i v e s 11=1-0 = 1 s o t h e r e w o u l d be a p a i r o f p e a k s a t v / 3 d = + l / 2 c o n t r a r y t o t h e e x p e r i m e n t . 191 W i t h o u r f i n a l 0-H2 s a m p l e , G21, we o b t a i n e d some l i m i t e d d a t a c o n c e r n i n g t h e t e m p e r a t u r e d e p e n d e n c e o f T c o n t h e J = l c o n c e n t r a t i o n . I n t h a t e x p e r i m e n t , NMR s p e c t r a w i t h 3=0° a t t h r e e t e m p e r a t u r e s . 3 3 3 , . 5 7 5 , and 1.81 K w e r e m e a s u r e d a s f u n c t i o n s o f c o v e r t h e r a n g e .2<c^.92 w i t h p=.85, a s u s u a l . F i g . 49 i s a g r a p h o f t h e s p l i t t i n g 6 v / 3 d o f t h e p e a k s o f t h e d e r i v a t i v e o f t h e a b s o r p t i o n s i g n a l a g a i n s t c f o r G20 and G21. The d e r i v a t i v e p e a k s r a t h e r t h a n t h e a b s o r p t i o n p e a k s were p l o t t e d b e c a u s e f o r s e v e r a l s p e c t r a i n t h e t r a n s i t i o n r e g i o n , t h e d e r i v a t i v e p e a k s c o u l d be s e e n c l e a r l y b u t t h e a b s o r p t i o n p e a k s were o b s c u r e d by t h e c e n t r a l c omponent o f t h e s i g n a l . F o r G 2 1 , t h e c r i t i c a l c o n c e n t r a t i o n was a r b i t r a r i l y d e f i n e d t o be t h e one a t w h i c h sv/3d was h a l f w a y b e t w e e n i t s maximum v a l u e and t h e c o n s t a n t v a l u e i t t o o k when c was l o w . A t .3 K , 5 v / 3 d a p p e a r s t o be a p p r o a c h i n g a maximum n e a r c=.9. F o r .58 K, 6^/3d i s a p p a r e n t l y somewhat b e l o w i t s maximum e v e n a t o u r h i g h e s t J = l c o n c e n t r a t i o n . N e v e r t h e l e s s , 6 v / 3 d i s a s t r o n g f u n c t i o n o f c s o t h e e r r o r i n d e t e r m i n i n g t h e c r i t i c a l c o n c e n t r a t i o n w o u l d be s m a l l . F o r G19 and G20, T c was d e t e r m i n e d f r o m t h e a b s o r p t i o n p e a k s p l i t t i n g shown i n F i g . 47. F i g . 50 shows T^ a s a f u n c t i o n o f c f o r t h e t h r e e o-H 2 s a m p l e s . 8.2 The O r d e r e d S t a t e The .3 K o - H 2 s p e c t r a o f G19 and G20 h a v e b e e n c o m p a r e d t o t h e T=0 s p e c t r a c a l c u l a t e d f o r t h e v a r i o u s o r d e r e d s t a t e s . The NMR t r a n s i t i o n f r e q u e n c i e s a s a f u n c t i o n o f c r y s t a l l i t e 192 ~0 oo .9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 ~ .1 -CO < .33 K .58 K o o ° o ° o o o o • • n n n • • n Q • „ • 1.61 K c f o o o o o o o o o o o o o --.1 .2 T A i r .6 T .9 1 1.0 F i g . 49 The s p l i t t i n g o f t h e p e a k s o f t h e a b s o r p t i o n d e r i v a t i v e 6 v / 3 d v e r s u s c f o r o - H 2 s a m p l e s G21 ( o p e n s y m b o l s ) a n d G20 ( s o l i d s y m b o l s ) . A l l m e a s u r e m e n t s were w i t h B=0° and A=13 dB. The t e m p e r a t u r e s were .33 K ( d i a m o n d s ) , .58 K ( s q u a r e s ) , a n d 1.81 K ( c i r c l e s ) . .7 .6 H .4^ •3 H . 2 H i .6 .6 T . 9 1.0 Fig. 50 The t r a n s i t i o n temperature T versus c for thr o-H2 samples: G19 (square), G20 ( c i r c l e ) , and G21 (diamonds). 194 o r i e n t a t i o n a r e g i v e n by ( I V - 1 4 ) and ( I V - 1 5 ) f o r (3 = 0° and 90°. The p r o b a b i l i t y o f e a c h f r e q u e n c y p ( v ) was c a l c u l a t e d n u m e r i c a l l y a s d e s c r i b e d i n S e c t i o n 4.2 u s i n g t h e o r i e n t a t i o n a l d i s t r i b u t i o n f u n c t i o n f o r t h e a d s o r b i n g s u r f a c e s P ( ^ ) d f i . I t i s t h e p r o b a b i l i t y t h a t t h e c - a x i s o f a g r a p h i t e c r y s t a l l i e s i n t h e s o l i d a n g l e d f i , w h i c h makes an a n g l e j; w i t h t h e n o r m a l t o t h e G r a f o i l . The f r e q u e n c y p r o b a b i l i t y f u n c t i o n p ( v ) was t h e n c o n v o l u t e d w i t h a G a u s s i a n f u n c t i o n w i t h s t a n d a r d d e v i a t i o n V I n i t i a l l y , we t o o k P ( S ) t o be a P o i s s o n k e r n e l d e f i n e d by T h i s o r i e n t a t i o n a l d i s t r i b u t i o n f u n c t i o n h a s b e e n u s e d t o f i t t h e n e u t r o n s c a t t e r i n g p e a k s o f C H 4 a d s o r b e d o n P a p y e x ( C o u l o m b e t a l , 1 9 8 1 ) . The b e s t f i t was o b t a i n e d u s i n g q=.6. I f C i s l e s s t h a n t h e h a l f w i d t h a t h a l f maximum (HWHM) o f P ( ? ) , t h e n P ( ? ) i s a l m o s t i d e n t i c a l t o a L o r e n t z i a n . H o w e v e r , P(£) h a s a s h o r t e r t a i l a t l a r g e v a l u e s o f C . We t r i e d t o f i t t h e r e s u l t i n g l i n e s h a p e t o o u r .3 K s p e c t r a u s i n g t h e o r d e r p a r a m e t e r s , q, and o G a s a d j u s t a b l e p a r a m e t e r s . F o r g=0°and q=.6, t h e f i t s w e r e v e r y p o o r , p r i m a r i l y b e c a u s e t h e s m a l l o u t e r p e a k s o f t h e e x p e r i m e n t a l s p e c t r a w e r e n o t r e p r o d u c e d i n t h e s y n e t h e t i c s p e c t r a . T h e s e p e a k s come f r o m m o l e c u l e s t h a t a r e s t a n d i n g up on c r y s t a l s t h a t h a v e C=0. However, i n t h e s y n -t h e t i c s p e c t r a , t h e c o n t r i b u t i o n s f r o m t h e s e m o l e c u l e s were s u p p r e s s e d b e c a u s e P(C ) was m u l t i p l i e d by t h e f a c t o r s i n c P U ) = 2 '• ) {(1+q ) - 4 q c o s c ) ( V I I I - 2 ) 195 c o n t a i n e d i n d°.. As a c o n s e q u e n c e , , t h e c o n t r i b u t i o n f r o m t h e t a i l o f P( O was r e l a t i v e l y l a r g e . The t a i l c o u l d be r e d u c e d by i n c r e a s i n g q b u t t h i s a l s o r e d u c e d t h e w i d t h o f t h e p e a k : i t was s t i l l n o t p o s s i b l e t o o b t a i n a good f i t . I n o r d e r t o f i t t h e n e u t r o n s c a t t e r i n g l i n e s h a p e o f N 2 a d s o r b e d o n G r a f o i l , Kjems e t a l (1976) u s e d an o r i e n t a t i o n a l d i s t r i b u t i o n f u n c t i o n w h i c h was t h e sum o f a G a u s s i a n and i s o t r o p i c c o m p o n e n t s P( t) = A Q + A]_ e x p ( - C 2 / 2 6 2 ) ( V I I I - 3 ) T h e i r b e s t f i t s w e r e o b t a i n e d w i t h A Q / A 1 = . 7 8 and 6=12.7°. T h i s f u n c t i o n h a s t h e a d v a n t a g e t h a t t h e w i d t h o f t h e peak a nd t h e s i z e o f t h e t a i l c a n be a d j u s t e d i n d e p e n d e n t l y p r o v i d e d t h a t 6 i s n o t t o o l a r g e . T h e r e i s an i m p o r t a n t d i f f e r e n c e b e t w e e n t h e o r i e n t a t i o n a l d i s t r i b u t i o n f u n c t i o n s d e f i n e d by Coulomb e t a l and Kjems e t a l . Coulomb e t a l u s e d a 2D d i s t r i b u t i o n f u n c t i o n s o i t was m u l t i p l i e d by s i n j ; . Kjems e t a l u s e d a ID d i s t r i b u t i o n f u n c t i o n s o i t was n o t m u l t i p l i e d by s i n s . F o r t h e NMR s p e c t r a a 2D d i s t r i b u t i o n f u n c t i o n i s r e q u i r e d s o s i n ? s h o u l d be i n c l u d e d . We h a v e d e f i n e d P( r.) t o be PU) = f p + { ( l - f p ) / K 6 ) } e x p ( - z ; 2 / 2 6 2 ) ( V I I I - 4 ) and n o r m a l i z e d i t by f^2 P U ) s i n c d c=l/2* ( V I I I - 5 ) The u p p e r l i m i t o f t h e i n t e g r a l i s TT/2 r a t h e r t h a n f b e c a u s e c r y s t a l l i t e s o r i e n t e d a t an a n g l e C a r e i n d i s t i n g u i s h a b l e f r o m o n e s a t x, +TT . The t e r m f i s t h e f r a c t i o n o f a d s o r b i n g 196 s u r f a c e s f o r m i n g t h e i s o t r o p i c powder c o m p o n e n t . I n t h e s e c o n d t e r m , 1 ( 6 ) i s d e f i n e d by 1 ( 6 ) = / ^ / 2 e x p ( - ? 2 / 2 6 2 ) s i n c dz; ( V I I I - 6 a ) = 6 2 ( l - 6 2 / 3 + 6 4 / 1 5 - 6 6 / 1 0 5 ) i f 6<<1 r a d i a n ( V I I I - 6 b ) S y n t h e t i c s p e c t r a f o r t h e p i n w h e e l p h a s e g e n e r a t e d w i t h t h i s f o r m f o r P ( C ) c o m p a r e d q u i t e w e l l w i t h t h e .3 K e x p e r i -m e n t a l s p e c t r a , p a r t i c u l a r l y f o r 3 = 0 ° , a s shown i n F i g . 5 1 ( a ) and ( b ) . As e x p e c t e d on t h e b a s i s o f t h e 2D powder s p e c t r a d i s c u s s e d i n S e c t i o n 4.2, t h e s y n t h e t i c s p e c t r a f o r t h e o t h e r o r d e r e d p h a s e s were q u i t e d i f f e r e n t f r o m t h e e x p e r i m e n t a l s p e c t r a , e s p e c i a l l y i f 8=0°. A l l o f t h e s e s y n t h e t i c s p e c t r a showed o n l y two p e a k s f o r 6=0°. The e x p e r i m e n t a l l i n e s h a p e i n F i g . 5 1 ( a ) was o b t a i n e d by s u b t r a c t i n g a p a r a b o l a f r o m t h e .318 K s p e c t r u m shown i n F i g . 4 3 ( a ) . The b a s e l i n e s o f a l l o f t h e s p e c t r a w i t h 3=0° a t .3 K were f o u n d t o be a p p r o x i m a t e l y p a r a b o l i c b e c a u s e o f m o d u l a t i o n p i c k - u p . The b e s t f i t s f o r b o t h 3=0° and 90° were o b t a i n e d u s i n g f p = . 2 , 6=15°, a G / 3 d = . 0 4 2 , a/a Q =.78 f o r t h e up s p i n s , and a / a 0 = n / n Q = . 8 7 f o r t h e down s p i n s . a Q and n 0 a r e t h e v a l u e s o f a and n f o r T=0 and c = l , w h i c h a r e g i v e n i n T a b l e I V . The f a c t t h a t a / o Q and r i / n 0 a r e b o t h l e s s t h a n one c o u l d be b e c a u s e c < l o r b e c a u s e o f quantum f l u c t u a t i o n s . The f i t s w ere n o t v e r y s e n s i t i v e t o e i t h e r f p o r 5. The a ' s f o r t h e s p i n s l y i n g down and s t a n d i n g up were d e t e r m i n e d by t h e p o s i t i o n s o f t h e i n n e r and o u t e r p e a k s o f t h e 3=0° s p e c t r a , r e s p e c t i v e l y . G i v e n t h e s e v a l u e s , n was d e t e r m i n e d by t h e p o s i t i o n s o f t h e -1.5 -1.0 -.5 O .5 1.0 1.5 -1.5 -1.0 -.5 O .5 1.0 1.5 U - y J / 3 d (\>-il)/3d F i g . 51 C o m p a r i s o n o f t h e .3 K e x p e r i m e n t a l s p e c t r a ( d a s h e d l i n e s ) w i t h t h e s y n t h e t i c s p e c t r a ( s o l i d l i n e s ) c a l c u l a t e d f o r t h e p i n w h e e l [ ( a ) and ( b ) ] and h e r r i n g b o n e [ ( c ) and ( d ) ] p h a s e s . S p e c t r a i n ( a ) and ( c ) a r e f o r F=0° and t h o s e i n (b) a n d (d) a r e f o r B = 9 0 ° . The e x p e r i m e n t a l p a r a m e t e r s a r e (a) T=.318 K, c=.95, A=13 dB, ^ = 1 . 2 h and (b) T=.309 K, c=.94, A=13 d B , t =1.2 h. The s y n t h e t i c p a r a m e t e r s a r e g i v e n i n t h e t e x t . The s p e c t r a have been n o r m a l i z e d t o t h e same a r e a s . 198 o u t e r p e a k s o f t h e 6=90° s p e c t r u m . S i n c e t h e a ' s f o r t h e s p i n s l y i n g down and t h o s e s t a n d i n g up a r e d i f f e r e n t , t h e i n n e r p e a k s f o r 3 = 90° a c t u a l l y c o n s i s t o f two u n r e s o l v e d p e a k s ; t h a t i s why t h e i r o u t e r e d g e s a r e b r o a d e r t h a n f o r ^ = 0 ° . The p o s i t i o n and w i d t h o f t h e i n n e r p e a k s f o r 6=90° g a v e some r e d u n d a n c y i n t h e d e t e r m i n a t i o n o f t h e o r d e r p a r a m e t e r s . I t s h o u l d be p o s s i b l e t o i m p r o v e t h e f i t by m o d i f y i n g t h e G a u s s i a n f u n c t i o n i n ( V I I I - 4 ) . F o r 6 = 0 ° , t h e m a i n p r o b l e m i s t h a t t h e c e n t r a l minimum o f t h e s y n t h e t i c s p e c t r u m i s t o o s h a l l o w and t h e o u t e r m i n i m a a r e t o o d e e p . The m a i n c o n t r i -b u t i o n t o t h e o u t e r m i n i m a i s f r o m t h e s p i n s s t a n d i n g up on c r y s t a l s w i t h £=25°. The m a i n c o n t r i b u t i o n t o t h e c e n t r a l minimum i s f r o m a l l s p i n s w i t h 5=55°. The p e a k s a r e p r o d u c e d by s p i n s on s u r f a c e s w i t h r, = 0°. T h e r e f o r e , i t seems t h a t t h e number o f s u r f a c e s w i t h £=0° s h o u l d be m a i n t a i n e d , n e a r c=25° t h e y s h o u l d be i n c r e a s e d , and n e a r ?=55° t h e y s h o u l d be d e c r e a s e d . 8.3 I s T h e r e A n o t h e r T r a n s i t i o n ? B e s i d e s o b s e r v i n g t h e s p e c t r a o f 0 - H 2 s a m p l e s w i t h c=.9, we a l s o s t u d i e d a s a m p l e (G21) w i t h c=.2 and 1 = 0 ° . A l l o f t h e s e s p e c t r a c o n s i s t e d o f a d o u b l e t . I n F i g . 4 7 , t h e 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 e s p l i t t i n g A v / 3 d o f t h i s l o w c o n c e n t r a t i o n s a m p l e o f 0 - H 2 h a s b e e n p l o t t e d . The s p l i t t i n g s o f t h e h i g h and l o w c o n c e n t r a t i o n s a m p l e s a g r e e w e l l a t h i g h t e m p e r a t u r e s w h e r e t h e e f f e c t s o f t h e EQQ i n t e r a c t i o n a r e a v e r a g e d o u t by 199 r a p i d r o t a t i o n . As T d e c r e a s e s , t h e s p l i t t i n g o f t h e h i g h c o n c e n t r a t i o n s a m p l e i n c r e a s e s more s l o w l y t h a n t h a t o f t h e l o w c o n c e n t r a t i o n s a m p l e : t h e EQQ i n t e r a c t i o n s o p p o s e t h e a l i g n m e n t t h a t t h e c r y s t a l f i e l d a t t e m p t s t o i m p o s e . The m o s t s t r i k i n g f e a t u r e o f t h e l o w c o n c e n t r a t i o n d a t a i s t h e s m a l l d i s c o n t i n u i t y i n A\>/3d w h i c h o c c u r s a t t h e same t e m p e r a t u r e a s t h e o r i e n t a t i o n a l o r d e r i n g t r a n s i t i o n o f t h e h i g h c o n c e n t r a t i o n s a m p l e . A c o n c e n t r a t i o n o f .2 i s f a r t o o l o w f o r t h e s m a l l d i s c o n t i n u i t y t o be a c o o p e r a t i v e e f f e c t . P e r h a p s t h e r e i s a n o t h e r t r a n s i t i o n a t .6 K i n a d d i t i o n t o t h e o r i e n t a t i o n a l o r d e r i n g t r a n s i t i o n . A c a n d i d a t e f o r s u c h a t r a n s i t i o n i s s u g g e s t e d by c o m p a r i n g t h e t r a n s l a t i o n a l p h a s e d i a g r a m s o f a d s o r b e d h y d r o g e n shown i n F i g . 1 and a d s o r b e d 4 H e shown i n F i g . 52 ( E c k e and D a s h , 1 9 8 3 ) . F i g . 52 c o n t a i n s a g r e a t e r v a r i e t y o f p h a s e s b u t t h i s i s p r o b a b l y s i m p l y a c o n s e q u e n c e o f t h e more i n t e n s i v e s t u d y t h a t 4 H e h a s b e e n s u b j e c t e d t o . I n p a r t i c u l a r , t h e n e u t r o n s c a t t e r i n g m e a s u r e m e n t s o f h y d r o g e n w o u l d n o t h a v e had s u f f i c i e n t S:N r a t i o s t o be a b l e t o d i s -t i n g u i s h b e t w e e n a /J x / T s o l i d and i s l a n d s o f /T x /T s o l i d c o e x i s t i n g w i t h a 2D f l u i d . The p e a k i n t e n s i t i e s and d i f f u s e s c a t t e r i n g w o u l d h a v e b e e n s l i g h t l y d i f f e r e n t i n t h e two c a s e s . C o n s e q u e n t l y , i t i s p o s s i b l e t h a t f o r p < l , H2 u n d e r g o e s a t r a n s i t i o n f r o m a ^ x ^ s o l i d w i t h v a c a n c i e s t o a ^3 x ^3 s o l i d p l u s 2D v a p o u r . I f t h a t w e re t h e c a s e , t h e o r i e n t a t i o n a l o r d e r i n g i n t h e p u r e ^3 x ^3 s o l i d m i g h t be s u p p r e s s e d by t h e v a c a n c i e s and o n l y o c c u r when t h e t e m p e r a t u r e was l o w e n o u g h 2 0 0 1.8 1.6 1.4 L 2 1.0 . 8 . 6 -. 4 -. 2 -~i 1 1 1 r monolayer coverage i r incommensurate solid IC solid -+reg - c r - ^-fluid+ IC solid /LJJ : i : :^reg+fluid ^registereb^ fluid regis tere3/ + vapor/ oriented fluid (?) vapor J L 2 3 4 5 6 T (K) 8 F i g . 5 2 P r o p o s e d p h a s e d i a g r a m o f He o n g r a p h i t e ( E c k e and D a s h , 1 9 8 3 ) . The s o l i d l i n e s r e p r e s e n t r e g i o n s w h e r e t h e d a t a a r e a d e q u a t e t o d e t e r m i n e t h e p h a s e b o u n d a r y . The d a s h e d l i h e s a r e c o n j e c t u r e d p h a s e b o u n d a r i e s . The r e g i s t e r e d p h a s e i s t h e v3x y/J" s o l i d . 201 f o r i s l a n d s o f t h e / F x ^3 s o l i d t o f o r m . T h i s h y p o t h e s i s m i g h t e x p l a i n t h e c e n t r a l p e a k i n t h e NMR s p e c t r u m s e e n n e a r .6 K. T h e r e s h o u l d be c o u p l i n g b e t w e e n t h e t r a n s l a t i o n a l and r o t a t i o n a l m o t i o n o f a d s o r b e d h y d r o g e n m o l e -c u l e s b e c a u s e o r i e n t a t i o n s o f t h e m o l e c u l e s t h a t m i n i m i z e t h e i n t e r a c t i o n e n e r g y w i t h t h e s u r f a c e d e p e n d upon p o s i t i o n ( R i e h l and F i s h e r , 1 973; S t e e l e , 1 9 7 7 ) . The c o m b i n e d r o t a t i o n a l and t r a n s l a t i o n a l m o t i o n o f m o l e c u l e s i n t h e 2D v a p o u r p h a s e c o u l d r e s u l t i n m o t i o n a l n a r r o w i n g o f t h e NMR s p e c t r u m r e g a r d l e s s o f c r y s t a l l i t e o r i e n t a t i o n , p r o d u c i n g t h e c e n t r a l p e a k . The peak w o u l d t h e n d i s a p p e a r a t l o w e r t e m p e r a t u r e s a s t h e 2D v a p o u r p r e s s u r e d r o p p e d . T h e r e a r e some p r o b l e m s w i t h t h i s i n t e r p r e -t a t i o n . The c e n t r e o f t h e s p e c t r u m s t a r t s t o f i l l i n a b o v e I K , a much h i g h e r t e m p e r a t u r e t h a n t h e s u p p o s e d t r a n s i t i o n t e m p e r -a t u r e . A l s o , t h e o r i e n t a t i o n a l o r d e r i n g t e m p e r a t u r e i s s t r o n g l y d e p e n d e n t on c o n c e n t r a t i o n . I f o r i e n t a t i o n a l o r d e r i n g were t r i g g e r e d by a t r a n s l a t i o n a l p h a s e t r a n s i t i o n , t h e t r a n s i t i o n t e m p e r a t u r e s h o u l d n o t d e p e n d on c. E x p e r i m e n t s a t o t h e r c o v e r a g e s , p a r t i c u l a r l y ,85<P_<1 s h o u l d be a b l e t o s e t t l e t h e q u e s t i o n o f w h e t h e r t h e r e i s a t r a n s l a t i o n a l p h a s e t r a n s i t i o n . 8.4 The D i s o r d e r e d S t a t e I n t h e p a r a r o t a t i o n a l s t a t e , t h e s p l i t t i n g o f t h e a b s o r p -t i o n d o u b l e t i s p r o p o r t i o n a l t o j?| . I t s t e m p e r a t u r e d e p e n d e n c e a l l o w s t h e d e t e r m i n a t i o n o f V c and r . We h a v e done t h i s by f i t t i n g t h e h i g h t e m p e r a t u r e e x p a n s i o n o f a c a l c u l a t e d by H a r r i s 202 and B e r l i n s k y ( p r i v a t e c o m m u n i c a t i o n , 1983) and g i v e n i n ( 1 1 1 - 4 4 ) t o o u r e x p e r i m e n t a l d a t a . U s i n g v a r i o u s t r i a l v a l u e s o f V c and r i o was c a l c u l a t e d f o r t h e v a l u e s o f T and c u s e d i n t h e e x p e r i m e n t s . F o r G20, w i t h ,83<c<.89, t h e b e s t r e s u l t s w e r e o b t a i n e d w i t h V c=.65 K a n d r = . 1 2 K . F o r G21, w i t h , 2 K c < , 2 6 , we f o u n d V c=.55 K and r =.19 K. The e x p e r i m e n t a l and f i t t e d v a l u e s o f \p\ h a v e b e e n p l o t t e d as f u n c t i o n s o f T-"*" i n F i g . 53. The f i t s w e r e n o t s e n s i t i v e t o 10% c h a n g e s i n V c i f r was r e a d j u s t e d . F o r t h e h i g h c o n c e n t r a t i o n s a m p l e , i t was n e c e s s a r y t o c h a n g e r by 10% and f o r t h e l o w c o n c e n -t r a t i o n s a m p l e , by 25%. The d i f f e r e n c e s b e t w e e n t h e p a r a m e t e r s f o r t h e two s a m p l e s i s p r o b a b l y n o t s i g n i f i c a n t . P o o r c o n v e r -g e n c e o f t h e h i g h t e m p e r a t u r e e x p a n s i o n a t l o w t e m p e r a t u r e s makes i t d i f f i c u l t t o c h o o s e t h e b e s t p a r a m e t e r s . F o r G20, t h e (gr ) 3 t e r m c o n t r i b u t e d a b o u t 10% t o a a t .75 K and . 5 % a t 4.2 K. F o r G21, c o n v e r g e n c e was p o o r e r b e c a u s e a l a r g e r v a l u e o f r was n e e d e d : t h e ( gr ) J t e r m c o n t r i b u t e d more t h a n 10% b e l o w 1.3 K. N o t e t h a t a c c o r d i n g t o t h e mean f i e l d o r i e n t a t i o n a l p h a s e d i a g r a m shown i n F i g . 6, t h e s e r e s u l t s f o r V c a n d r a r e c o n s i s t e n t w i t h t h e o r d e r e d s t a t e b e i n g t h e p i n w h e e l p h a s e . We a l s o t r i e d u s i n g t h e mean f i e l d r e s u l t f o r a , g i v e n i n ( 1 1 - 1 6 ) , t o f i t t h e NMR d a t a . T h i s was q u i t e s u c c e s s f u l a b o v e 1.3 K b u t b r o k e down a t l o w e r t e m p e r a t u r e s . P o s i t i v e and n e g a t i v e v a l u e s o f V c g a v e e q u a l l y g o o d r e s u l t s , i n c o n t r a s t t o t h e h i g h t e m p e r a t u r e e x p a n s i o n . The p a r a m e t e r s o b t a i n e d f r o m l e a s t s q u a r e s f i t s o f t h e mean f i e l d r e s u l t s t o t h e d a t a a b o v e 203 .15 -. 1 0 -.05-0 + 8 X I i I i r O .5 "1 1 1 1 1 1 1 1.0 1.5" T " ' ( K " ) F i g . 53 Ja| v e r s u s 1/T f o r o-H 2 s a m p l e s G20 w i t h . 8 3<c<_.89 ( c i r c l e s ) and G21 w i t h .21<c<.26 ( d i a m o n d s ) . 0 = 0 ° . The r e s u l t s o f t h e h i g h t e m p e r a t u r e e x p a n s i o n o f o f o r V = . 6 5 K and r=.12 K ( x ) and f o r V c=.55 K and r=.19 K (+) a r e a l s o p l o t t e d . S i n c e V c > 0 , a>0. The c o n c e n t r a t i o n v a r i e s f r o m p o i n t t o p o i n t s o a s i n g l e c u r v e c a n n o t be d r a w n t h r o u g h e a c h o f t h e two s e t s o f d a t a . 204 1.3 K a r e i n c l u d e d i n T a b l e X I I . T a b l e X I I R e s u l t s f o r V C and r f o r O-HQ S a m p l e MFT HT e x p a n s i o n HT e x p a n s i o n (no v a c a n c i e s ) ( w i t h v a c a n c i e s , P=.85) V C ( K ) T ( K ) V Q ( K ) r ( K ) V C ( K ) r ( K ) G20 .44 .09 .65 .12 .30 .11 -.44 .05 -.85 .13 G21 .51 .12 .55 .19 .23 .17 -.51 .02 I f a d s o r b e d h y d r o g e n f o r m e d a p u r e /J x V3~ s o l i d p h a s e a b o v e .6 K, t h e n ( 1 1 1 - 4 3 ) w o u l d no l o n g e r be c o r r e c t i n t h e d i s o r d e r e d s t a t e , e x c e p t a t p = l , b e c a u s e i t a s sumes t h a t a l l s i t e s a r e f i l l e d by h y d r o g e n m o l e c u l e s . B e r l i n s k y ( p r i v a t e c o m m u n i c a t i o n , 1983) h a s made a h i g h t e m p e r a t u r e e x p a n s i o n o f a t o s e c o n d o r d e r i n 3 V c and 6T w h i c h t a k e s i n t o a c c o u n t random v a c a n c i e s w i t h c o n c e n t r a t i o n x f o r t h e c a s e p <1. The f i r s t s t e p was t o s e e how t h e v a c a n c i e s a f f e c t t h e a n i s o t r o p i c H a m i l t o n i a n o f a s y s t e m o f h y d r o g e n m o l e c u l e s g i v e n i n ( I I - 4 ) . The H a m i l t o n i a n o f t h e J = l m o l e c u l e s c a n be e x p r e s s e d a s H = 1 + (1/2) . Z .Hji + . z .H ( V I I I - 7 ) l c 1 ' i , D EQQ 1,3 v The f i r s t sum i s o v e r a l l J = l m o l e c u l e s , t h e s e c o n d i s o v e r a l l n e a r e s t n e i g h b o u r J = l p a i r s , and i n t h e t h i r d sum i i s a J = l m o l e c u l e and j i s a v a c a n c y . n , v 1 - 1 i s t h e c h a n g e i n e n e r g y 205 o f m o l e c u l e i when a m o l e c u l e a t s i t e j i s r e m o v e d . U s i n g ( I I - 4 b ) , t h e v a c a n c y e n e r g y i s H v i j = - ( 8 T t / 5 ) m Y 2 m ^ i ) B ( R i j ) Y 2 m * ( ^ i j ) ( V I I I - 8 ) F o r h y d r o g e n i n t h e /3 x /3 p h a s e o n g r a p h i t e , t a k i n g o n l y n e a r -o e s t n e i g h b o u r i n t e r a c t i o n s , R^j=R 0=4.26 A s o B ( R ^ j ) =-.65 K ( v a n K r a n e n d o n k , 1 9 8 3 ) . T a k i n g t h e z a x i s n o r m a l t o t h e s u r f a c e , t h e p o l a r a n g l e o f i s 90° and t h e a z i m u t h a l a n g l e i s t^y W i t h i n t h e J = l m a n i f o l d , t h e Y 2 m ( J ^ ' s c a n be r e p l a c e d by t h e s p h e r i c a l t e n s o r s d e f i n e d by ( I I - 6 ) . The v a c a n c y e n e r g y t h e n becomes H v l j = - f (?fB(a )Cr 2 0 ( J . ) - ( 3 / 2 ) 1 / 2 { T „ 2 ( ? . ) e - i 2 * i J + T,"2 tf±) e i 2 ^ 1 Z ( V I I I - 9 ) T a k i n g t h e h i g h t e m p e r a t u r e e x p a n s i o n g i v e n i n ( H I - 4 2 c ) t o s e c o n d o r d e r , B e r l i n s k y f o u n d t h a t a = B {V c/3 + (6/5) ( l - x ) B ( R o ) > + |32{ V c 2 / 1 8 - (6/25) (1-x) B 2 ( R ) + (24/25) (1-x) 2 B 2 ( R ) - (2/5) (1-x) V B ( R ) - ( 3 / 2 ) c x V r o c o c + 3 c x ( l - x ) r B ( R Q ) - ( 2 5 / 6 ) c x r 2 > ( V I I I - 1 0 ) The c r y s t a l f i e l d V c i n c l u d e s a c o n t r i b u t i o n f r o m t h e i n t e r -a c t i o n s o f a J = l m o l e c u l e w i t h t h e s u r f a c e a s w e l l a s w i t h n e i g h b o u r i n g h y d r o g e n m o l e c u l e s . U s i n g x = l - p , we c o m p a r e d t h i s e x p a n s i o n f o r v a r i o u s v a l u e s o f V c and r t o t h e e x p e r i m e n t a l s p l i t t i n g s . The f i t s w e r e a b o u t a s g o o d a s w i t h t h e p r e v i o u s e x p a n s i o n w h i c h d i d n o t i n c l u d e v a c a n c i e s , e x c e p t t h a t d i v e r g e n c e o f t h e s e r i e s was a more s e r i o u s p r o b l e m . The 6 t e r m was more t h a n 20% o f t h e B t e r m b e l o w 1.8 K f o r c=.9 and b e l o w 1.1 K f o r c=.2. The b e s t v a l u e s o f V c and r a r e g i v e n i n T a b l e X I I . The i n t r o d u c t i o n o f v a c a n c i e s had no s i g n i f i c a n t e f f e c t on r b e c a u s e , l i k e J=0 m o l e c u l e s , v a c a n c i e s do n o t e n g a g e i n t h e EQQ i n t e r a c t i o n . H o w e v e r , t h e y d i d h a v e a l a r g e e f f e c t on V c - W i t h i n t h e r a n g e s w here t h e h i g h t e m p e r a t u r e e x p a n s i o n s f o r a c o n v e r g e , V c and r c o u l d be c h o s e n s o t h a t e i t h e r t h e p u r e S3 x S3 p h a s e o r S3 x S3 i s l a n d s c o e x i s t i n g w i t h a 2D f l u i d a r e c o n s i s t e n t w i t h t h e 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 e s p l i t t i n g o f t h e NMR d o u b l e t . The v a l u e s o f r g i v e n i n T a b l e X I I a r e a l l s u r p r i s i n g l y l o w c o m p a r e d t o t h e n e a r e s t n e i g h b o u r , r i g i d l a t t i c e v a l u e o f .528 K. 4 T h e r e a r e no c a l c u l a t i o n s o f t h e r e n o r m a l i z a t i o n o f r f o r a d s o r b e d h y d r o g e n b u t i t i s h a r d t o i m a g i n e s u c h a l a r g e c h a n g e . 8.5 B r o a d e n i n g I t was m e n t i o n e d i n C h a p t e r I V t h a t t h e d o m i n a n t s o u r c e o f b r o a d e n i n g f o r p u r e 0-H2 i s e x p e c t e d t o be t h e i n t e r m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n , w h i c h i s t e m p e r a t u r e i n d e p e n d e n t . F i g . 54 shows t h a t t h e m e a s u r e d b r o a d e n i n g 6 i s a c o n s i d e r a b l y more i n t e r e s t i n g f u n c t i o n o f t e m p e r a t u r e . We h a v e d e f i n e d <5 t o be t h e w i d t h a t h a l f maximum o f e a c h o f t h e o u t e r e d g e s o f t h e m a i n a b s o r p t i o n p e a k s . I t d o e s n o t i n c l u d e t h e e f f e c t s o f t h e m i s o r i e n t a t i o n o f t h e g r a p h i t e c r y s t a l s . F o r t h e 90% J = l s p e c t r a a t .58 K and .66 K, w here t h e p e a k s c o u l d n o t be r e s o l v e d b e c a u s e o f t h e c e n t r a l c o m p o n e n t , t h e s h o u l d e r s h a v e CO .12 i .10-- O - 0 6 1 CO .06 .04 H .02H o O o o o o O 0 T " (K" 1) Fig. 54 Broadening 6 / 3 d versus 1/T for o-H, samples G20 with ,83<c_<.95 ( c i r c l e s ) and G21 with . 21<c<_.2 6 (diamonds). B= 0°. 208 b e e n t a k e n i n s t e a d o f t h e p e a k s . As t h e 9 0 % J = l s a m p l e was c o o l e d , i t e x h i b i t e d an i n c r e a s e i n <5/3d p r i o r t o t h e t r a n s i t i o n . T h i s was f o l l o w e d by a s h a r p d r o p , a l m o s t down t o t h e h i g h t e m p e r a t u r e l i m i t , a s o r i e n t a t i o n -5 a l o r d e r i n g o c c u r r e d . F o r T>T C» 6/3d o f t h e 20% J = l s a m p l e a l s o i n c r e a s e d s l o w l y a s T d e c r e a s e d . G o i n g t h r o u g h T c , 6/3d r o s e more r a p i d l y and s e t t l e d a t a v a l u e a b o u t t w i c e a s g r e a t as t h a t o f t h e 9 0 % J = l s a m p l e . I t i s p o s s i b l e t h a t t h e r e i s a l s o a pe a k a t T c i n t h e 20% J = l s a m p l e b u t t h e d a t a i s n o t e x t e n s i v e e n o u g h t o d e t e r m i n e t h i s . I f i t i s assumed t h a t t h e i n t e r m o l e c u l a r d i p o l a r b r o a d e n i n g i s G a u s s i a n , t h e n t h e s t a n d a r d d e v i a t i o n w o u l d be e q u a l t o t h e s q u a r e r o o t o f t h e s e c o n d moment M 2^/ 2. The HWHM w o u l d be 1.17 M 2 1 / / 2 / 3 d , w h i c h , a c c o r d i n g t o ( I V - 3 3 ) , i s .030 f o r c=.90 and .015 f o r c=.23. The m a g n e t i c f i e l d m o d u l a t i o n w o u l d p r o d u c e a d d i t i o n a l ( r e c t a n g u l a r ) b r o a d e n i n g w i t h a HWHM e q u a l t o t h e a m p l i t u d e o f t h e m o d u l a t i o n , w h i c h i s AH m/3d = .018. T h e s e r e s u l t s a r e a p p r o x i m a t e l y c o n s i s t e n t w i t h t h e h i g h t e m p e r a t u r e l i m i t s o f t h e m e a s u r e d v a l u e s o f 6/3d f o r b o t h c o n c e n t r a t i o n s . A f a i r l y c o n v i n c i n g e x p l a n a t i o n o f t h e b r o a d e n i n g m e a s u r e -m e n t s i s a s f o l l o w s . C o n s i d e r an o-H 2 s a m p l e f o r w h i c h c i s somewhat l e s s t h a n o n e . A t h i g h t e m p e r a t u r e s , t h e EQQ i n t e r -a c t i o n w o u l d be a v e r a g e d o u t by m o l e c u l a r r e o r i e n t a t i o n a nd t h e e n e r g y g a p A o f t h e n i j s u b - l e v e l s o f t h e J = l l e v e l w o u l d be e q u a l t o t h e c r y s t a l f i e l d . A l l o f t h e m o l e c u l e s w o u l d 209 e x p e r i e n c e t h e same c r y s t a l f i e l d ( i g n o r i n g v a c a n c i e s and i n t e r s t i t i a l s , i f any) s o t h e y w o u l d h a v e t h e same v a l u e o f a. As T d e c r e a s e d t h e EQQ i n t e r a c t i o n s w o u l d b e g i n t o c o m p e t e w i t h t h e c r y s t a l f i e l d . F l u c t u a t i o n s i n t h e m o l e c u l a r f i e l d , a r i s i n g f r o m d i f f e r e n t l o c a l d i s t r i b u t i o n s o f J = l m o l e c u l e s , w o u l d s t a r t t o become i m p o r t a n t . A d i s t r i b u t i o n o f e n e r g y g a p s A w o u l d be p r o d u c e d , r e s u l t i n g i n i n c r e a s i n g b r o a d e n i n g o f t h e NMR s p e c t r u m . A t T c , t h e m o l e c u l a r f i e l d w o u l d o v e r p o w e r t h e c r y s t a l f i e l d and r a p i d l y w i d e n t h e e n e r g y g a p b e t w e e n t h e g r o u n d s t a t e and f i r s t e x c i t e d s t a t e . L o c a l v a r i a t i o n s i n t h e gap w o u l d no l o n g e r be i m p o r t a n t b e c a u s e t h e g a p w o u l d now be much g r e a t e r t h a n t h e t h e r m a l e n e r g y . T h i s w o u l d mean t h a t t h e b r o a d e n i n g w o u l d d r o p r a p i d l y a t T c , p e r h a p s a s l o w as t h e h i g h t e m p e r a t u r e v a l u e . T h i s s c e n a r i o i s c o n s i s t e n t w i t h t h e g r a p h o f S/3d v e r s u s T-''" f o r t h e c=.9 s a m p l e . I t i s a l s o s u p p o r t e d by t h e f a c t t h a t t h e .3 K NMR s p e c t r a c o u l d be f i t by j u s t two v a l u e s o f 0 , one f o r t h e s p i n s s t a n d i n g up and one f o r t h o s e l y i n g down. F i n a l l y , i t p r e d i c t s t h a t a t i n t e r m e d i a t e t e m p e r a t u r e s t h e r e w o u l d be d i s t r i b u t i o n o f o r d e r p a r a m e t e r s . T h i s was a l r e a d y e x p e c t e d f r o m t h e c o m p a r i s o n w i t h t h e l o w c o n c e n t r a t i o n b u l k o - H 2 s p e c t r a . I f t h e s c e n a r i o were c o r r e c t , t h e p e a k i n 6 s h o u l d d i m i n i s h when c becomes v e r y c l o s e t o 1 b e c a u s e t h e r e w o u l d be l i t t l e v a r i a t i o n o f t h e l o c a l m o l e c u l a r f i e l d s . The s i t u a t i o n f o r c=.2 i s l e s s i n t u i t i v e . A t h i g h t e m p e r -a t u r e s , <$ s h o u l d i n c r e a s e a s T d e c r e a s e s and t h e m o l e c u l a r f i e l d b e g i n s t o a s s e r t i t s e l f . N e a r T c, we h a v e f o u n d t h a t a h a s a s m a l l b u t f a i r l y r a p i d d r o p ( s e e F i g . 4 7 ) . I n t h i s r e g i o n , i n h o m o g e n e i t i e s s u c h a s t h e l o c a l d i s t r i b u t i o n o f J = l m o l e c u l e s w o u l d become more i m p o r t a n t s o t h e r e m i g h t be a s m a l l peak i n $. B e l o w T c , a b e g i n s t o f l a t t e n a s T d e c r e a s e s s o <$ s h o u l d n o t be 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 . 211 N o t e s t o C h a p t e r V I I I 1. I n t h e r e m a i n d e r o f t h i s t h e s i s , t h e t e r m b r o a d e n i n g d o e s n o t i n c l u d e t h e d i s t r i b u t i o n o f s p l i t t i n g s a r i s i n g f r o m t h e d i f f e r e n t c r y s t a l l i t e o r i e n t a t i o n s i n t h e G r a f o i l . The d a s h e d s p e c t r u m o f F i g . 13 w i l l be r e f e r r e d t o a s an u n b r o a d e n e d s p e c t r u m . 2. F o r G20, t h e d e r i v a t i v e p eak s p l i t t i n g g a v e v i r t u a l l y t h e same T c. 3. We have u s e d t h e c o n v e n t i o n a l d e f i n i t i o n o f B ( R ^ ^ ) b u t Van K r a n e n d o n k ' s B ( R ^ j ) i s a f a c t o r o f two l a r g e r . 4. The v a l u e s o f r o b t a i n e d f r o m G20 a r e p r o b a b l y more r e l i a b l e t h a n t h o s e o b t a i n e d f r o m G21. The f i t t o t h e G20 d a t a i s c o n s i d e r a b l y more s e n s i t i v e t o r b e c a u s e c i s much h i g h e r . 5. T c r e f e r s t o t h e o r i e n t a t i o n a l o r d e r i n g t r a n s i t i o n t e m p e r a t u r e o f t h e h i g h c o n c e n t r a t i o n s a m p l e . 212 CHAPTER IX NMR SPECTRA OF p-D 0 9.1 T e m p e r a t u r e Dependence o f t h e p-D^ S p e c t r a NMR a b s o r p t i o n s p e c t r a o f P-D2 h a v e b e e n o b s e r v e d by us i n t h e t e m p e r a t u r e r a n g e o f .3 K t o 8 K. Some o f t h e s e , f r o m s a m p l e G18 w i t h p=.85 and .77<c£.95, a r e shown i n F i g . 55. A t h i g h t e m p e r a t u r e s , t h e r e was a d o u b l e t f r o m t h e J = l m o l e c u l e s and a c e n t r a l p e ak f r o m t h o s e J=0 m o l e c u l e s w i t h 1 = 2. As T d e c r e a s e d , t h e s p l i t t i n g i n c r e a s e d b u t b e l o w a b o u t 1.6 K t h e b r o a d e n i n g o f t h e J = l p e a k s a l s o i n c r e a s e d . A t a b o u t .8 K, t h e d o u b l e t c o u l d no l o n g e r be r e s o l v e d , a s was t h e c a s e w i t h t h e h i g h c o n c e n t r a t i o n o-H2' Upon f u r t h e r c o o l i n g , t h e s h a p e s o f t h e s p e c t r a c o n t r a s t e d s h a r p l y w i t h t h o s e o f 0 - H 2 and o f t h e s y n t h e t i c s p e c t r a c a l c u l a t e d f o r t h e o r d e r e d p h a s e s o f MFT: t h e P-D2 s p e c t r a became v e r y b r o a d and weak. A t .6 K, t h e J = l s p e c t r u m had become t r i a n g u l a r . A t .3 K, i t was somewhat s e m i c i r c u l a r , w i t h a maximum w i d t h s l i g h t l y g r e a t e r t h a n Av/3d = 1. The s p e c t r u m c o u l d , i n p r i n c i p l e , h a v e a maximum w i d t h o f Av/3d = 2. The S:N r a t i o was q u i t e p o o r s o t h e e x i s t e n c e o f weak w i n g s e x t e n d i n g b e y o n d Av/3d = 1 i s n o t p r e c l u d e d . H owever, t h e d a t a f r o m s a m p l e G18 shown i n F i g . 38 i m p l i e s t h a t s u c h w i n g s c o u l d n o t c o n t r i b u t e v e r y much t o t h e t o t a l i n t e n s i t y . I n t h e r e g i o n n e a r .6 K, i t was i m p o s s i b l e t o d e t e r m i n e t h e maximum e x t e n t o f e v e n t h e m a i n component o f t h e s p e c t r u m , b e c a u s e t h e b a s e l i n e p r o d u c e d by t h e m o d u l a t i o n (v-K)/3d (v-v0)/sd F i g . 55 A b s o r p t i o n s p e c t r a o f D 2 s a m p l e G18 w i t h B = 9 0 ° . A=13 dB. p=.85. S p e c t r u m (a) (b) ( c ) (d) (e) T( K ) .318 .580 .892 1.61 3.01 c .93 .91 .90 .83 .86 t m ( h ) 8.52 17.1 5.70 11.0 10.3 The s p e c t r a h a v e b e e n n o r m a l i z e d h t h e same manner a s t h o s e i n F i g . 4 3 . The v e r t i c a l s c a l e i s t h e same f o r a l l o f t h e s p e c t r a . 214 p i c k - u p had t h e same t r i a n g u l a r s h a p e a s t h e J = l s p e c t r u m . M o d u l a t i o n p i c k - u p was much more s e v e r e f o r D 2 t h a n f o r H 2 b e c a u s e t h e m a g n e t i c f i e l d was 1.3 T i n s t e a d o f .2 T. U n f o r t u n a t e l y , t h e p i c k - u p was much w o r s e f o r 8=0°. We h a v e a l r e a d y s e e n t h a t t h e g=90° s p e c t r a a r e e x p e c t e d t o be much more d i f f i c u l t t o i n t e r p r e t b e c a u s e t h e y i n v o l v e a v e r a g e s o v e r t h e a z i m u t h a l a n g l e s a o f t h e c r y s t a l a x e s w h e r e a s t h e 8=0° s p e c t r a a r e i n d e p e n d e n t o f a. The S:N r a t i o s d i d n o t w a r r a n t any q u a n t i t a t i v e a n a l y s i s o f t h e l o w t e m p e r a t u r e s p e c t r a . As r e g a r d s t h e q u a l i t a t i v e f e a t u r e s , one c o n j e c t u r e i s t h a t p-D 2 h a s a w i d e r d i s t r i b u t i o n o f o r d e r p a r a m e t e r s t h a n o-H 2 a t l o w t e m p e r a t u r e s . I n s e c t i o n 8.5, t h e i n c r e a s e i n t h e b r o a d e n i n g o f t h e o - H 2 s p e c t r a a s T d e c r e a s e d t o w a r d s T Q was i n t e r p r e t e d a s b e i n g due t o t h e i n c r e a s i n g i m p o r t a n c e o f s p a t i a l v a r i a t i o n s o f t h e m o l e c u l a r f i e l d . A t T c , t h e o r i e n t a t i o n a l o r d e r i n g c a u s e d t h e b r o a d e n i n g t o d r o p r a p i d l y . I n p-D 2 t h e r e i s no e v i d e n c e o f o r i e n t a t i o n a l o r d e r i n g s o one m i g h t s u p p o s e t h a t a s T d e c r e a s e s t h e b r o a d e n i n g o f t h e p-D 2 s p e c t r a w o u l d c o n t i n u e u n a b a t e d . F u r t h e r e v i d e n c e o f t h i s i s p r e s e n t e d i n s e c t i o n 9.3. 9.2 The J = l D o u b l e t As i n t h e c a s e o f o - H 2 , we a t t e m p t e d t o u s e t h e h i g h t e m p e r a t u r e e x p a n s i o n g i v e n by ( 1 1 1 - 4 4 ) t o f i t t h e 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 e s p l i t t i n g Av/3d o f t h e J = l d o u b l e t . T h i s was n o t s u c c e s s f u l b e c a u s e l a r g e r v a l u e s o f r were n e e d e d t h a n i n 215 t h e c a s e o f 0-H2 s o t h e s e r i e s d i v e r g e d e v e n a t h i g h t e m p e r a t u r e s . C o n s e q u e n t l y , we r e s o r t e d t o MFT. F i g . 56 i s a p l o t o f jcrj v e r s u s T - 1 o b t a i n e d f r o m s a m p l e G18. S i n c e B=90°, ( I V - 1 5 ) shows t h a t \a\ =Av/3d. As i n F i g . 4 7 , c was n o t c o n s t a n t s o t h e d a t a c o u l d n o t be f i t t e d t o a s i n g l e c u r v e . F o r l a r g e v a l u e s o f T _ 1 , t h e d o u b l e t was p o o r l y r e s o l v e d s o t h e e r r o r b a r s o f |a| a r e q u i t e l a r g e . A two p a r a m e t e r l e a s t s q u a r e s f i t g a v e V c=2.3 K and r=.50 K o r V c=-2.4 K and r=.44 K; t h e f o r m e r v a l u e s g a v e a s l i g h t l y b e t t e r f i t b u t t h e d i f f e r e n c e was n o t s i g n i f i c a n t . The v a l u e s o f r a r e a l i t t l e s m a l l e r t h a n t h e r i g i d l a t t i c e v a l u e , r Q = . 5 1 0 K. T h i s i s t o be e x p e c t e d b e c a u s e t h e r e i s l i k e l y t o be some r e n o r m a l i z a t i o n due t o z e r o p o i n t m o t i o n . 9.3 B r o a d e n i n g The b r o a d e n i n g 6 o f t h e J = l d o u b l e t was e x t r a c t e d i n t h e same manner a s f o r 0-H2 i n t h o s e c a s e s w here t h e d o u b l e t c o u l d be r e s o l v e d . A t l o w t e m p e r a t u r e s , t h e r e was o n l y one b r o a d J = l p e a k s o a d i f f e r e n t p r o c e d u r e was n e e d e d . F i g . 56 i n d i c a t e s t h a t t h e s p l i t t i n g Av/3d = |CT| may be a p p r o a c h i n g a l i m i t i n g v a l u e o f a b o u t .25 a t l o w t e m p e r a t u r e s . C o n s e q u e n t l y , Av/3d was t a k e n t o be e q u a l t o .25 f o r t h e p u r p o s e o f c a l c u l a t i n g <$ when T~^>_1.2 . A l t e r n a t i v e l y , we m i g h t h a v e e x t r a p o l a t e d t h e mean f i e l d b e h a v i o u r o f Av/3d t o l o w t e m p e r a t u r e s . However, s i n c e a d o u b l e t i s n o t o b s e r v e d t h e r e , MFT i s p r o b a b l y n o t v a l i d . E v e n i f i t w e r e , s m a l l e r r o r s i n V r and T ( w h i c h a r e 216 • 3 H x <H Ox O x o x O X (X —r-.2 .4-- r -. 8 -1.0 I— 1.2 T " ' ( K " ) F i g . 5 6 U v e r s u s 1 / T f o r p - D 2 . B = 9 0 ° . D a t a f r o m s a m p l e G 1 8 w i t h . 7 7 < c £ . 9 5 ( c i r c l e s ) a n d t h e b e s t . f i t u s i n g M F T ( c r o s s e s ) a r e s h o w n . T h e l a t t e r g i v e s V c = 2 . 3 K a n d r = . 5 0 K . A n e q u a l l y g o o d f i t w a s o b t a i n e d f o r V = - 2 . 4 K a n d r = . 4 4 K . d e t e r m i n e d f r o m t h e h i g h t e m p e r a t u r e m e a s u r e m e n t s ) w o u l d p r o d u c e s u b s t a n t i a l e r r o r s i n |a| a s T d e c r e a s e d . F o r t h e l o w t e m p e r a t u r e s p e c t r a c o n s i s t i n g o f a s i n g l e J = l p e a k , 6 was d e f i n e d by 6 / 3 d = ( l / 2 ) [ ( F W H M ) / 3 d - .25] ( I X - 1 ) F i g . 57 shows <S/3d v e r s u s T-"*". A t h i g h t e m p e r a t u r e s , 6 i n c r e a s e s v e r y s l o w l y as T d e c r e a s e s b u t n e a r 1.1 K, t h e i n c r e a s e becomes more r a p i d and c o n t i n u e s a t l e a s t down t o .3 K. T h i s b e h a v i o u r i s c o n s i s t e n t w i t h t h e c o n c l u s i o n s o f s e c t i o n 9.1, a c c o r d i n g t o w h i c h P-D2 d e v e l o p s an i n c r e a s i n g l y w i d e d i s t r i b u t i o n o f o r d e r p a r a m e t e r s a s T d e c r e a s e s . The h i g h t e m p e r a t u r e l i m i t o f 6/3d i s .011. I n s e c t i o n 4.4 i t was shown t h a t f o r D 2 b r o a d e n i n g by t h e g r a p h i t e d i a m a g n e t i s m was e x p e c t e d t o be l a r g e r t h a n i n t e r m o l e c u l a r d i p o l a r b r o a d e n i n g . B a s e d on t h e room t e m p e r a t u r e d i a m a g n e t i c s u s c e p t i b i l i t y o f G r a f o i l ( w h i c h i s e x p e c t e d t o be t e m p e r a t u r e i n d e p e n d e n t ) , 1 /2 M 2 / 3 d was e s t i m a t e d t o be .009. I f t h e b r o a d e n i n g i s assumed t o be G a u s s i a n , f o r s i m p l i c i t y , t h i s c o r r e s p o n d e n d s t o a HWHM/3d o f .011. The go o d a g r e e m e n t i s somewhat f o r t u i t o u s c o n s i d e r i n g t h e a s s u m p t i o n s t h a t h a v e b e e n made. 218 . 3 H .2 H "O CO c o .1 H o O o o o o o°o T " 1 ( K " 1 ) Fig. 57 Broadening 6/3d versus 1/T for p-D2• 3=90°. The data i s from sample G18 in c e l l B with .77<c£.95 ( c i r c l e s ) and G7 i n c e l l A with .95<c<.97 (diamonds). 219 CHAPTER X CONCLUSIONS 10.1 P r e c i s U n q u e s t i o n a b l y , t h e m o s t i m p o r t a n t r e s u l t o f o u r NMR s t u d i e s o f h y d r o g e n a d s o r b e d on g r a p h i t e i s t h a t 9 0 % 0 - H 2 a t a c o v e r a g e o f p=.85 u n d e r g o e s an o r i e n t a t i o n a l o r d e r i n g t r a n s i t i o n a t .6 K i n t o t h e p i n w h e e l p h a s e . T h i s i s one o f t h e l o w t e m p e r a t u r e o r d e r e d s t a t e s p r e d i c t e d by MFT. A l t h o u g h t h e r e i s g o o d r e a s o n t o e x p e c t MFT t o c o r r e c t l y i d e n t i f y p o s s i b l e o r d e r e d s t a t e s , i t was by no means c l e a r t h a t o r d e r i n g w o u l d i n f a c t o c c u r . F l u c t u a t i o n s due t o quantum e f f e c t s o r t h e 2D n a t u r e o f t h e s y s t e m m i g h t w e l l h a v e p r e v e n t e d i t . The n a t u r e o f t h e t r a n s i t i o n i s s t i l l i n q u e s t i o n . The s m a l l d i s c o n t i n u i t y i n t h e 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 e s p l i t t i n g o f t h e a b s o r p t i o n s i g n a l o f 2 0 % o-H 2 i s p a r t i c u l a r l y p u z z l i n g . We h a v e s u g g e s t e d t h a t i t may be due t o a t r a n s i t i o n f r o m a p u r e S3x S3 s o l i d t o a STx. S3 s o l i d c o e x i s t i n g w i t h a 2D f l u i d a t .6 K. E x p e r i m e n t s a t o t h e r c o v e r a g e s , e s p e c i a l l y .85<p<^l, s h o u l d s e t t l e t h i s q u e s t i o n . G i v e n i m p r o v e d S:N r a t i o s f o r t h e o-H 2 s p e c t r a , i t w o u l d be w o r t h w h i l e t o f i t t h e e x p e r i m e n t a l s p e c t r a f o r .3 K<T<1 K t o s y n t h e t i c s p e c t r a g e n e r a t e d u s i n g a' d i s t r i b u t i o n o f o r d e r p a r a m e t e r s . F o r 8=0°, i t i s p r o b a b l y n e c e s s a r y t o u s e o n l y one o r d e r p a r a m e t e r , a. p - D 2 e x h i b i t e d s u r p r i s i n g l y d i f f e r e n t l o w t e m p e r a t u r e b e h a v i o u r f r o m o - H 2 : t h e r e was no e v i d e n c e o f o r i e n t a t i o n a l 2 2 0 o r d e r i n g down t o .3 K. In c o n t r a s t , p u r e , b u l k p-D 2 o r d e r s a t 3.8 K, a h i g h e r t e m p e r a t u r e t h a n t h e 2.8 K o f p u r e b u l k o - H 2 « We do n o t u n d e r s t a n d t h e r e a s o n u n d e r l y i n g t h e d i f f e r e n t b e h a v i o u r o f a d s o r b e d p - D 2 and o - H 2 « NMR s p e c t r a o f p-D 2 w i t h g=0° m i g h t c l a r i f y t h e s i t u a t i o n ; a s we have s e e n f o r o-H 2 , t h e y a r e much e a s i e r t o i n t e r p r e t t h a n t h e 3=90° s p e c t r a . To o b t a i n good s p e c t r a w i t h 3=0°, i t i s e s s e n t i a l t h a t t h e S:N r a t i o s be i m p r o v e d . M o d u l a t i o n p i c k - u p i s t h e most s e r i o u s impediment t o a c c o m p l i s h i n g t h i s t a s k . A s s u m i n g t h a t t h e s p e c t r a c a n be i m p r o v e d , t h e i n c r e a s e i n t h e b r o a d e n i n g a t low t e m p e r a t u r e s makes i t a p p e a r t h a t i t w i l l be n e c e s s a r y t o use a d i s t r i b u t i o n o f o r d e r p a r a m e t e r s t o g e n e r a t e t h e s y n t h e t i c s p e c t r a b e l o w 1 K. 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 e s p l i t t i n g o f t h e NMR d o u b l e t o b s e r v e d i n t h e p a r a r o t a t i o n a l s t a t e g i v e s V c and r . F o r p < l , i f h y d r o g e n forms i s l a n d s o f t h e /3x / T s o l i d c o e x i s t i n g w i t h a low d e n s i t y 2D f l u i d , t h e r e s h o u l d be few v a c a n c i e s i n t h e s o l i d . I n t h a t c a s e , f i t t i n g a h i g h t e m p e r a t u r e e x p a n s i o n o f a t o t h e NMR d a t a g i v e s V c=.65 K a n d r =.12 K f o r 90% o - H 2 « The e x p a n s i o n i s e x a c t i n 3 V C and t h i r d o r d e r i n g r . F o r a p u r e fix s o l i d , t h e r e would be v a c a n c i e s i f p < l . We t h e n f o u n d t h a t , f o r p=.85, V c=.30 K o r -.85 K w i t h r v i r t u a l l y u n changed. F o r 20% o - H 2 , we f o u n d V c=.55 K and r=.19 K w i t h o u t v a c a n c i e s b u t V c=.23 K and r=.17 K w i t h v a c a n c i e s f o r p=.85. The e x p a n s i o n d e v e l o p e d c o n v e r g e n c e p r o b l e m s n e a r 1 K. T h i s may e x p l a i n t h e d i f f e r e n c e i n t h e 221 r e s u l t s f o r t h e two s a m p l e s . F o r 90% p - D 2 i d i v e r g e n c e o f t h e h i g h t e m p e r a t u r e e x p a n s i o n was more s e r i o u s b e c a u s e a l a r g e r v a l u e o f r was r e q u i r e d . C o n s e q u e n t l y , MFT was u s e d t o f i t t h e 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 e s p l i t t i n g . The f i t was e g u a l l y g o o d u s i n g V = 2 . 3 K and r = . 5 0 K o r V = - 2 . 4 K and r = . 4 4 K. The v a l u e s o f r o b t a i n e d f o r p - D 2 a r e a l i t t l e s m a l l e r t h a n t h e n e a r e s t n e i g h b o u r , r i g i d l a t t i c e v a l u e , r Q = . 5 1 0 K: t r a n s l a t i o n a l z e r o p o i n t m o t i o n i s e x p e c t e d t o r e d u c e r s l i g h t l y . I n t h e b u l k s o l i d , t h e v a l u e o f r d e d u c e d f r o m e x p e r i m e n t s i s . 8 7 r Q f o r b o t h H 2 and D 2 b u t i n 2D t h e r e n o r m a l i z a t i o n i s e x p e c t e d t o be g r e a t e r . A l t h o u g h H 2 m i g h t be e x p e c t e d t o h a v e g r e a t e r z e r o p o i n t m o t i o n t h a n D 2 b e c a u s e H 2 i s a l i g h t e r m o l e c u l e , t h i s c o u l d n o t a c c o u n t f o r t h e a n o m a l o u s l y l o w m e a s u r e d v a l u e o f r f o r a d s o r b e d o - H 2 . The r i g i d l a t t i c e v a l u e i s T Q= . 5 2 8 K. We h a v e no e x p l a n a t i o n f o r t h i s d i s c r e p a n c y . As y e t , t h e r e a r e no r e l i a b l e c a l c u l a t i o n s o f t h e c r y s t a l f i e l d s o f H 2 o r D 2 a d s o r b e d on g r a p h i t e b u t e v e n s o i t i s s u r p r i s i n g t h a t t h e m e a s u r e d v a l u e s d i f f e r s o g r e a t l y . T h e r e a r e two c o n t r i b u t i o n s t o t h e c r y s t a l f i e l d o f an a d s o r b e d J = l h y d r o g e n m o l e c u l e . One i s f r o m i n t e r a c t i o n s w i t h n e i g h b o u r i n g h y d r o g e n m o l e c u l e s and t h e o t h e r i s f r o m i n t e r a c t i o n s w i t h t h e s u r f a c e . The h y d r o g e n c o n t r i b u t i o n V C i s g i v e n by t h e f i r s t t e r m o f ( I I - 5 a ) . F o r h y d r o g e n o n a s u r f a c e w i t h i t s LI n o r m a l a l o n g t h e z a x i s , i f we u s e ( I I - 5 b ) V " becomes 222 V C H - (3/5) ] E(R..) ( x . 1 ) Summing o v e r n e a r e s t n e i g h b o u r s and u s i n g B ( R Q ) = -.65 K ( v a n K r a n e n d o n k , 1 9 8 3 ) 1 , we f i n d V C H = -2.3 K. F u r t h e r n e i g h -b o u r s make a s m a l l a d d i t i o n a l c o n t r i b u t i o n t h a t c o u l d e a s i l y be i n c l u d e d . The s u b s t r a t e c o n t r i b u t i o n t o t h e c r y s t a l f i e l d V c ^ i s more d i f f i c u l t t o e s t i m a t e b u t t h e c a l c u l a t i o n w o u l d be t r a c t -a b l e . I t w o u l d i n v o l v e summing t h e p a i r i n t e r a c t i o n s o f a J = l h y d r o g e n m o l e c u l e w i t h t h e C atoms i n t h e s u b s t r a t e . A s s u m i n g t h e c a r b o n atoms t o be s p h e r i c a l l y s y m m e t r i c , t h e p a i r i n t e r a c t i o n V w o u l d h a v e t h e f o r m V ( R , f t ) = U 1 ( R ) + U 2 ( R ) Y 2 0 ( f t ) (X-2) where R i s t h e v e c t o r b e t w e e n t h e h y d r o g e n m o l e c u l e and t h e c a r b o n atom and Q i s t h e o r i e n t a t i o n o f R w i t h r e s p e c t t o t h e h y d r o g e n i n t e r n u c l e a r a x i s . A p r o c e d u r e f o r summing t h e p a i r i n t e r a c t i o n s h a s been g i v e n by S t e e l e ( 1 9 7 3 ) . The sum c a n be r e p r e s e n t e d by a F o u r i e r s e r i e s i n t h e p o s i t i o n v a r i a b l e s o f a h y d r o g e n m o l e c u l e i n a p l a n e p a r a l l e l t o t h e s u r f a c e . O n l y t h e u n i f o r m component i s r e t a i n e d and t h e sum o v e r a l l g r a p h i t e p l a n e s e x c e p t t h e f i r s t i s r e p l a c e d by an i n t e g r a l . U s i n g c S t e e l ' s p r o c e d u r e , an e s t i m a t e o f V c h a s b e e n made p r e v i o u s l y ( K u b i k and H a r d y , 1978) b u t we now f e e l t h a t i t i s n o t v e r y r e a l i s t i c . The a n i s o t r o p i c p o t e n t i a l U 2 ( R ) was t a k e n t o be p r o p o r t i o n a l t o t h e i s o t r o p i c p o t e n t i a l U]_(R), w h i c h was r e p r e s e n t e d by a L e n n a r d - J o n e s p o t e n t i a l . The p o t e n t i a l p a r a -m e t e r s w ere o b t a i n e d by c o m b i n i n g t h o s e o f t h e H 2 ~ H 2 and C-C 223 L e n n a r d - J o n e s p o t e n t i a l s . The w e l l d e p t h o f t h e C-H p o t e n t i a l was s e t e q u a l t o t h e g e o m e t r i c mean o f t h e w e l l d e p t h s and t h e p o s i t i o n o f t h e w e l l was t h e a r i t h m e t i c mean o f t h e p o s i t i o n s . A g e n e r a l a n i s o t r o p i c h y d r o g e n - s u b s t r a t e i n t e r a c t i o n p o t e n t i a l h a s s i n c e b e e n d e r i v e d (Wang e t a l , 1 9 8 0 ) . P a r a m e t e r s f o r t h e s p h e r i c a l l y a v e r a g e d f o r m o f t h i s p o t e n t i a l w e r e o b t a i n e d f r o m H 2 m o l e c u l a r beam s c a t t e r i n g d a t a ( M a t t e r a e t a l , 1 9 8 0 ) . They a r e s i g n i f i c a n t l y d i f f e r e n t f r o m t h e r e s u l t s o b t a i n e d by c o m b i n i n g t h e H 2 ~ H 2 and C-C L e n n a r d - J o n e s p o t e n t i a l s . I t h a s b e e n c l a i m e d t h a t t h e p r i m a r y r e a s o n i s t h a t t h e a s s u m p t i o n o f s p h e r i c a l s y m metry o f t h e C atoms i n g r a p h i t e i s n o t v a l i d s i n c e t h e g r a p h i t e d i e l e c t r i c c o n s t a n t i s a n i s o t r o p i c ( C r o w e l l and B r o w n , 1 9 8 2 ) . The q u a d r u p o l e c o u p l i n g c o n s t a n t , t h e c r y s t a l f i e l d , a n d t h e o r d e r i n g b e h a v i o u r o f H 2 and D 2 a r e a l l s u r p r i s i n g l y d i f f e r e n t ; i t seems u n l i k e l y t h a t t h e d i f f e r e n c e s i n t h e t r a n s -l a t i o n a l z e r o p o i n t m o t i o n a r e s u f f i c i e n t t o a c c o u n t f o r t h i s . I n c o n t r a s t , t h e i r b u l k s o l i d b e h a v i o u r i s v e r y s i m i l a r . Some o f t h e f u r t h e r e x p e r i m e n t a l work s u g g e s t e d a b o v e s h o u l d i l l u m i n a t e t h e p i c t u r e . U s i n g t h e i n t e n s i t i e s o f t h e NMR s i g n a l s , we h a v e f o u n d c o n v e r s i o n t o be a f i r s t o r d e r p r o c e s s f o r H 2 w i t h a r a t e c o n s t a n t o f . 4 0 ( 2 ) % / h i n o u r c e l l B. The D 2 d a t a had much more s c a t t e r and c o v e r e d a s m a l l e r c o n c e n t r a t i o n r a n g e s o i t was d i f f i c u l t t o a s c e r t a i n w h e t h e r t h e p r o c e s s was f i r s t o r s e c o n d o r d e r . H o w e v e r , a s s u m i n g i t t o be f i r s t o r d e r , we o b t a i n e d r a t e c o n s t a n t s o f . 0 6 9 ( 1 5 ) % / h i n c e l l B and . 0 2 6 ( 7 ) % / h i n c e l l A. The c o n v e r s i o n r a t e was much h i g h e r d u r i n g t h e p r o c e s s o f a d s o r p t i o n b u t n e v e r t h e l e s s , we w e re a b l e t o o b t a i n a d s o r b e d s a m p l e s w i t h i n i t i a l J = l c o n c e n t r a t i o n s o f .99 f o r D 2 and .96 f o r H 2 ' 10.2 F u t u r e Work The e x p e r i m e n t s d e s c r i b e d i n t h i s t h e s i s h a v e shown t h a t h y d r o g e n h a s some u n i q u e o r i e n t a t i o n a l p r o p e r t i e s and t h a t t h e i r e l u c i d a t i o n h a s o n l y begun. To p r o c e e d f u r t h e r , u s i n g NMR, t h e m o s t i m p o r t a n t r e q u i r e m e n t i s t o i m p r o v e t h e S:N r a t i o s . P r e s e n t l y , t h e m e a s u r i n g t i m e s a r e t o o l o n g f o r e x t e n s i v e m e a s u r e m e n t s . Two i m p r o v e m e n t s w o u l d be r e l a t i v e l y s t r a i g h t -f o r w a r d . The m o d u l a t i o n c o i l s f o r H Q p a r a l l e l t o t h e G r a f o i l a r e l a r g e r and c o n s i d e r a b l y more homogeneous t h a n t h o s e f o r t i 0 p e r p e n d i c u l a r t o t h e G r a f o i l . F o r t h e l a t t e r , t h e v a r i a t i o n o f H m o v e r t h e s a m p l e c a u s e s s i g n i f i c a n t l o s s o f s i g n a l i n t e n s i t y . S i n c e t h e c l e a r e s t r e s u l t s c a n be o b t a i n e d w i t h H Q p e r p e n d i c u l a r t o t h e G r a f o i l , i t w o u l d be w o r t h w h i l e t o r o t a t e t h e s a m p l e c e l l by 90°. I n c i d e n t a l l y , i f t h i s w e r e d o n e , i t w o u l d be i n t e r e s t i n g t o s e e w h e t h e r t h e m o d u l a t i o n p i c k - u p was s t i l l w o r s t f o r "$ = 0°. I f i t w e r e , i t w o u l d s t r o n g l y s u g g e s t t h a t t h e G r a f o i l was t h e s o u r c e o f t h e p i c k - u p . I f n o t , t h e s o u r c e must be o u t s i d e t h e s a m p l e c e l l . V i r t u a l l y a l l o f t h e m e a s u r e m e n t s d i s c u s s e d i n C h a p t e r s V I I I and I X e m p l o y e d c e l l B. H o w e v e r , t h e G r a f o i l i n c e l l A had t w i c e t h e s u r f a c e a r e a o f t h a t i n c e l l B e v e n t h o u g h t h e two c e l l s h a d t h e same v o l u m e . I n a d d i t i o n , t h e c o n v e r s i o n r a t e f o r P _ D 2 i n c e l l A was f o u n d t o be .4 t i m e s t h a t i n c e l l B. The d i s a d v a n t a g e o f c e l l A i s t h a t i t i s b e l i e v e d t o h a v e g r e a t e r m i s o r i e n t a t i o n o f t h e a d s o r b i n g s u r f a c e s . H owever, i t i s n o t c l e a r how l a r g e t h i s e f f e c t i s b e c a u s e a l l o f t h e h y d r o g e n s a m p l e s i n c e l l A a r e now b e l i e v e d t o h a v e b e e n n o n - u n i f o r m b e c a u s e t h e a d s o r b i n g t e m p e r -a t u r e was t o o l o w . C o n s i d e r i n g t h e p o t e n t i a l g a i n s i n t h e S:N r a t i o a n d t h e c o n v e r s i o n r a t e , i t w o u l d be u s e f u l t o s w i t c h b a c k t o c e l l A and c h e c k t h e NMR s i g n a l p r o d u c e d b y a u n i f o r m s a m p l e . A d d i t i o n a l m e a s u r e s w o u l d p r o b a b l y s t i l l be n e c e s s a r y , a t -7 l e a s t f o r D 2. U s i n g an RF a m p l i t u d e o f 2x10 T, we d i d n o t f i n d any e v i d e n c e o f s a t u r a t i o n o f t h e NMR s i g n a l e x c e p t i n t h e c a s e o f p u r e o-D 2, s o i t s h o u l d be p o s s i b l e t o i n c r e a s e t h e s i g n a l b y i n c r e a s i n g t h e e x c i t a t i o n . We t r i e d t h i s b u t f o u n d t h a t t h e n o i s e i n c r e a s e d f a s t e r t h a n t h e s i g n a l a t h i g h e r RF l e v e l s a n d t h a t t h e f i r s t s t a g e o f t h e l i q u i d h e l i u m c o o l e d a m p l i f i e r b e g a n t o s a t u r a t e c a u s i n g d i s t o r t i o n . The e x t r a n o i s e must h a v e b e e n f r o m t h e o s c i l l a t o r . U s u a l l y i n NMR, t h e p r o b l e m o f a m p l i f i e r d i s t o r t i o n i s s o l v e d b y u s i n g a b r i d g e t o r e d u c e t h e v o l t a g e a c r o s s t h e a m p l i f i e r i n p u t . H o w e v e r , b r i d g e s t e n d t o i n t r o d u c e m i c r o p h o n i c s a n d e x t r a J o h n s o n n o i s e s o t h i s a p p r o a c h may c r e a t e more p r o b l e m s t h a n i t s o l v e s . A b e t t e r s o l u t i o n i s p r o b a b l y t o m o d i f y t h e a m p l i f i e r i t s e l f s o t h a t i t d o e s n o t s a t u r a t e a t s u c h a l o w l e v e l . U n f o r t u n a t e l y , t h e modu-l a t i o n p i c k - u p c a n be e x p e c t e d t o i n c r e a s e i n p r o p o r t i o n t o t h e 226 RF l e v e l s o t h e p i c k - u p w i l l l i m i t t h e p o t e n t i a l i m p r o v e m e n t s . I n many c a s e s i n NMR, t h e S:N r a t i o c a n be r a i s e d by i n c r e a s i n g H Q b e c a u s e t h e S:N r a t i o i s p r o p o r t i o n a l t o H o ^ / / 2 ( A b r a g a m , 1 9 6 1 ) . I n o u r c a s e , we m i g h t g e t some i m p r o v e m e n t f o r H 2 b u t p e r h a p s n o t f o r D 2 , a t l e a s t a t t e m p e r a t u r e s g r e a t e r t h a n a b o u t 1 K. A t h i g h t e m p e r a t u r e s , t h e b r o a d e n i n g o f t h e D 2 s i g n a l a r i s e s p r i m a r i l y f r o m t h e g r a p h i t e d i a m a g n e t i s m s o i n c r e a s i n g H Q w o u l d r e s u l t i n an i n c r e a s e i n t h e b r o a d e n i n g , r e d u c i n g t h e i n c r e a s e i n t h e a m p l i t u d e o f t h e a b s o r p t i o n s i g n a l . A t l o w t e m p e r a t u r e s t h e r e i s a d d i t i o n a l b r o a d e n i n g , p r o b a b l y f r o m t h e o r d e r p a r a m e t e r d i s t r i b u t i o n , s o some e x t r a b r o a d e n i n g f r o m t h e d i a m a g n e t i s m may n o t be s i g n i f i -c a n t . A more s e r i o u s p r o b l e m i s t h e m o d u l a t i o n p i c k - u p . E x p e r i m e n t a l l y , we have f o u n d i t t o be g r e a t e r a t 1.3 T t h a n a t .2 T and e x p e c t t h a t i t i s p r o p o r t i o n a l t o H Q. I f t h i s i s t h e c a s e , n o t much i m p r o v e m e n t i n t h e S:N r a t i o c o u l d be a c h i e v e d by r a i s i n g H Q. P r e s e n t l y , o n l y a b o u t h a l f o f t h e t o t a l m e a s u r i n g t i m e c a n be u t i l i z e d f o r r e c o r d i n g d a t a b e c a u s e i t i s n e c e s s a r y t o w a i t f o r t h e p i c k - u p t o s t a b i l i z e a t t h e c u l m i n a t i o n o f e a c h sweep o f t h e m a g n e t i c f i e l d . E v en s o , t h e b a s e l i n e s o f t h e s p e c t r a were o f t e n q u i t e p o o r . The d e a d t i m e c o u l d p r o b a b l y be r e d u c e d somewhat i f t h e s i g n a l a v e r a g e r c o u l d c o l l e c t d a t a i n b o t h sweep d i r e c t i o n s u s i n g a s y m m e t r i c , t r i a n g u l a r sweep. We h a v e b e e n u n a b l e t o i s o l a t e t h e s o u r c e o f t h e p i c k - u p b u t i t may w e l l come f r o m t h e G r a f o i l i t s e l f . I n t h a t c a s e , i t m i g h t be d e s i r e a b l e 227 t o a b a n d o n m a g n e t i c f i e l d m o d u l a t i o n a l t o g e t h e r . We h a v e c o n s i d e r e d u s i n g f r e q u e n c y m o d u l a t i o n i n s t e a d b u t i t i s l i k e l y t o i n t r o d u c e new p r o b l e m s . N o r m a l l y , t h e f r e q u e n c y i s m o d u l a t e d by v a r y i n g t h e t u n i n g c a p a c i t o r , w h i c h , i n o u r c a s e , i s i n s i d e t h e c r y o s t a t . T h i s w o u l d b e s t be done u s i n g a v a r a c t o r d i o d e s i n c e m e c h a n i c a l t u n i n g w o u l d i n t r o d u c e m i c r o p h o n i c s . The m a i n p r o b l e m w i t h f r e g u e n c y m o d u l a t i o n i s t h a t t h e m e a s u r i n g s y s t e m has a f r e q u e n c y d e p e n d e n t g a i n s o t h e r e w o u l d be a s p u r i o u s b a c k g r o u n d t h a t w o u l d p r o b a b l y h a v e t o be s u b t r a c t e d . The t e c h n i q u e s d e s c r i b e d i n t h i s t h e s i s c o u l d be u s e d t o s t u d y many v a r i a n t s o f t h e h y d r o g e n on g r a p h i t e s y s t e m . I d e a l l y , one w o u l d l i k e t o be a b l e t o v a r y t h e c r y s t a l f i e l d i n o r d e r t o f u r t h e r e x p l o r e t h e o r i e n t a t i o n a l p h a s e d i a g r a m d e p i c t e d i n F i g . 6. To a c h i e v e t h i s e n d , one w o u l d r e q u i r e a s u b s t r a t e o n w h i c h h y d r o g e n f o r m s a r e g i s t e r e d , h e x a g o n a l l a t t i c e . I f i t were n o t r e g i s t e r e d , t h e r e w o u l d be a d i s t r i -b u t i o n o f c r y s t a l f i e l d s r e s u l t i n g i n b r o a d e n i n g . S u c h a s u b s t r a t e may be d i f f i c u l t t o f i n d i n v i e w o f r e q u i r e m e n t s s u c h a s a h i g h s p e c i f i c s u r f a c e a r e a . We h a v e p r e p l a t e d t h e G r a f o i l w i t h K r a t 9 5 % o f t h e ST x Sf c o v e r a g e and t h e n a d s o r b e d H 2 o n t o i t . The NMR s p e c t r a o f H 2 were q u i t e b r o a d s o we s u s p e c t t h a t t h e K r was c o m p r e s s e d o u t o f r e g i s t r y . B r o a d s p e c t r a were a l s o o b s e r v e d f o r H 2 o n b a r e G r a f o i l i f p > l . MgO p owder p r o m i s e s t o be a g o o d s u b s t r a t e ( B i e n f a i t e t a l , 1 9 8 0 ; V i l c h e s e t a l , 1984; J o r d a n e t a l , 1 9 8 3 ) . I t h a s g o o d u n i f o r m i t y and a h i g h s p e c i f i c s u r f a c e a r e a . The c r y s t a l s t r u c t u r e i s c u b i c s o 228 i f t h e r e w e re a r e g i s t e r e d p h a s e , i t w o u l d n o t be /! x /3. MgO a l s o h a s b e t t e r e l e c t r i c a l and m a g n e t i c p r o p e r t i e s t h a n G r a f o i l f o r NMR work. We have o n l y s t u d i e d t h e o r i e n t a t i o n a l p h a s e t r a n s i t i o n o f a d s o r b e d h y d r o g e n . H o w e v e r , NMR may a l s o p r o v e t o be u s e f u l f o r s t u d y i n g t h e t r a n s l a t i o n a l p h a s e s . I n p a r t i c u l a r a s o l i d - f l u i d t r a n s i t i o n s h o u l d be c h a r a c t e r i z e d by n a r r o w i n g o f t h e NMR s p e c t r u m due t o t h e m o t i o n o f ' t h e m o l e c u l e s . 229 Notes to Chapter X 1. Van Kranendonk's B(R) i s a factor of two greater than ours, which i s the conventional one. B I B L I O G R A P H Y 2 3 0 A b r a g a m , A . , T h e P r i n c i p l e s o f N u c l e a r M a g n e t i c R e s o n a n c e ( O x f o r d u n i v e r s i t y P r e s s , L o n d o n , 1 9 6 1 ) . A l d e r m a n , D . W . , R e v . S c i . I n s t r . , 1 9 2 ( 1 9 7 0 ) . A l e k s a n d r o v , I . V . , T h e o r y o f N u c l e a r M a g n e t i c R e s o n a n c e , t r a n s l a t e d b y S c r i p t a T e c h n i c a , t r a n s l a t i o n e d i t o r C P . P o o l e , J r . ( A c a d e m i c P r e s s , N e w Y o r k , 1 9 6 6 ) , P . 1 0 6 - 1 1 3 . B e r l i n s k y , A . J . , p r i v a t e c o m m u n i c a t i o n ( 1 9 8 3 ) . B e r n a t , T . P . , a n d H . D . C o h e n , J . L o w T e m p . P h y s . , J ^ , 5 9 7 ( 1 9 7 4 ) . B i e n f a i t , M . , J . G . D a s h , a n d J . S t o l t e n b e r g , P h y s . R e v . B , 2 1 , 2 7 6 5 ( 1 9 8 0 ) B i r g e n a u , R . J . , P . A . H e i n e y , a n d J . P . P e l z , P h y s i c a B + C , 1 1 0 , 1 7 8 5 ( 1 9 8 2 ) . B r e t z , M . , a n d T . T . C h u n g , J . L o w T e m p . P h y s . , 17_, 4 7 9 ( 1 9 7 4 ) . B r e t z , M . , J . G . D a s h , D . C . H i c k e r n e l l , E . O . M c L e a n , a n d O . E . V i l c h e s , P h y s . R e v . A , 8, 1 5 8 9 ( 1 9 7 3 ) . C a r n e i r o , K . , W . D . E l l e n s o n , L . P a s s e l l , J . P . M c T a g u e , a n d H . T a u b , P h y s . R e v . L e t t . , 31_, 1 6 9 5 ( 1 9 7 6 ) . C h u n g , T . T . , a n d J . G . D a s h , S u r f . S c i . , 66_, 5 5 9 ( 1 9 7 7 ) . C o l l i n s , R . E . , F l o w o f F l u i d s T h r o u g h P o r o u s M a t e r i a l s ( R h e i n h o l d P u b l i s h i n g C o r p . , N e w Y o r k , 1 9 6 1 ) , C h a p t e r 3 . C o n s t a b a r i s , G . , J . R . S a m s , J r . , a n d G . D . H a l s e y , J r . , J . P h y s . C h e m . , (6_5, 3 6 7 ( 1 9 6 1 ) . C o u l o m b , J . P . , M . B i e n f a i t , a n d P . 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N i e l s e n , P h y s . Rev. L e t t . , 31_, 1275 ( 1 9 7 6 ) . S t e e l e , W.A. , S u r f . S c i . , _3JL' 3 1 7 ( 1 9 7 3 ) . S t e e l e , W.A., J . P h y s . ( P a r i s ) , C±, C4-61 ( 1 9 7 7 ) . S t o c k m e y e r , R., H. S t o r t n i k , and R. Wagner, B e r . B u n s e n g e s . P h y s . Chem., 8_2, 1314 ( 1 9 7 8 ) . S u l l i v a n , N.S., J . P h y s . L e t t . ( P a r i s ) , 37.' L-209 ( 1 9 7 6 ) . S u l l i v a n , N.S., and J.M. V a i s s i e r e , P h y s . Rev. L e t t . , _5_1, 658 ( 1 9 8 3 ) . Thorny, A., and X. D u v a l , J . Chim. P h y s . P h y s i o c h i m . B i o l . , 6 6 , 1966 ( 1 9 6 9 ) ; 67_, 286 ( 1 9 7 0 ) ; 67_, 1101 ( 1 9 7 0 ) . v a n D i j k , H., M. D u r i e u x , J.R. C l e m e n t , and J.K. L o g a n , J . R e s . N a t . B u r . S t a n d . , S e c t . A, 64A, 1 ( 1 9 6 0 ) . V a n K r a n e n d o n k , J . , S o l i d H y d r o g e n ( P l e n u m P r e s s , New Y o r k , 1 9 8 3 ) . Van V l e c k , J . H . , The T h e o r y o f E l e c t r i c a n d M a g n e t i c S u s c e p t i b i l i t i e s ( C l a r e n d o n P r e s s , O x f o r d , 1 9 3 2 ) . V i l c h e s , O.E., J.P. Coulomb, and T.S. Sull i v a n , B u l l . Am. Phy Soc. , 268 (1984) . Walker, E.J. , Rev. S c i . Instrum. , 30_, 854 (1959). Warren, W.H., and W.G. Bader, i b i d . 4_0, 180 (1969). Washburn, S., M. Calkins, H. Meyer, and A.B. Harris, J. Low Temp. Phys. , A9_, 101 (1982). Weber, S., and G. Schmidt, Leiden Comm., 246C (1936). Young, D.M., and A.D. Crowell, Physical Adsorption of Gases. (Butterworths, Washington, D.C, 1962). 235 APPENDIX A HIGH TEMPERATURE EXPANSION OF a We w i l l now g i v e t h e c o e f f i c i e n t s f o r t h e h i g h t e m p e r a t u r e e x p a n s i o n o f a o b t a i n e d by H a r r i s and B e r l i n s k y ( 1 9 8 3 ) . The e x p a n s i o n t o t h i r d o r d e r i n (£T ) i s a=A+B(er) +c ( s r ) 2 + D(BT)3 w h e r e t h e c o e f f i c i e n t s a r e e x a c t i n V=8V(. (11 1 - 4 4 ) I f c i s t h e J = l c o n c e n t r a t i o n , t h e n t h e c o e f f i c i e n t s a r e A B C [1-e V ] / [ l + 2 e " V ] - ( 8 1 / 2 ) c e " V [ l - e " V ] / [ l + 2 e " V ] 3 9c [ - 1 7 e 5 V + e 4 V 1 6 V 2 ( e V + 2 ) 5 x ( - 3 8 0 V 2 c - 1 4 0 V 2 + 6 8 V - 1 0 2 ) + e 3 V ( 3 9 0 0 V 2 c - 6 V 2 + 3 0 6 V - 1 8 7 ) + e 2 V ( - 4 6 8 0 V 2 c - 1 2 6 0 V 2 + 4 0 8 V - 3 4 ) + e V ( 1 1 6 0 V 2 c - 7 0 0 V 2 + 1 3 6 V + 2 0 4 ) + 1 3 6 ] D = 9c 5 1 2 V 3 ( e V + 2 ) 7 [ e 7 V ( 4 89 6 0 V c 2 + 1 6 3 2 0 V c ( A - l ) (A-2) +1632V-9672C+1792) + e 6 V ( - 5 3 3 3 7 6 V 3 c 2 - 8 8 8 V 3 c + 8 6 4 V 3 - 4 8 9 6 0 0 V 2 c 2 - 1 3 7 3 5 2 V 2 c -11424V 2+195840Vc 2+124980Vc+78 40V-67704c +17920) + e 5 V ( - 3 6 9 2 6 3 2 V 3 c 2 - 1 1 2 7 6 8 8 V 3 c - 5 5 3 2 8 V 3 - 1 3 7 0 8 8 0 V 2 c 2 - 5 3 2 8 0 0 V 2 c - 5 4 1 7 6 V 2 +336456Vc-432V-145080c+69888) + e 4 V (A-3) x (269 2 4 0 8 V 3 c 2 + 5 4 1 4 4 3 V 3 c - 2 8 7 6 8 V 3 + 6 8 5 4 4 0 V 2 c 2 236 -385296Vzc-l08640V -685440Vc^+317lOOVc -72576V-9672C+125440) + e 3 V(-18491424V 3c 2 -4521774V3c-56064V3+3133440V2c2+368544V2c -177408V2-244800VC2-104664VC-200816V+309504c +71680) + e2V(6604864V3c2+3027732V3c +3808 64V3-391680V2c2+36 288V2c-225664V2 +881280Vc2-350064Vc-282528V+232128c-86016) + e V x (13420160V3c2+3658 488V3c+409024V3 -15667 20V2c2-354 4 32V2c-109 568V2 +195840Vc2-228768Vc-234048V-l54752C-143360) - (391680Vc2+111360Vc+89984V+154752c+57344) ] (A-4) 237 APPENDIX B CONSTRUCTION OF A QUARTZ-COPPER SEAL The precision bore quartz tube was 13.05 mm OD, .58 mm thick, and 47.2 mm long. It was cleaned using the following process: 1. Boiled for 15 min in a detergent solution.* 2. Rinsed several times in hot water. 3. Etched in 10% HF for 4 min. 4. Rinsed several times in water. 5. Boiled in a solution of 25 ml of KOH p e l l e t s in 100 ml of water. 6. Rinsed several times in hot water. 7. Rinsed in el e c t r o n i c grade methanol. 8. Boiled in methanol for 30 min. 9. Rinsed in methanol. Immediately aft e r cleaning, the tube was placed in an o evaporator and coated with about 500 A of Cr followed by o 1100 A of Cu. Pre-coating with Cr was necessary because Cu does not adhere well to quartz. Attempts to solder d i r e c t l y to the f i l m were unsuccessful because the indium a l l o y solders used appeared to a l l o y with the copper and l i f t e d o f f the f i l m . Consequently the Cu f i l m thickness was increased to .02 mm by el e c t r o p l a t i n g at 50 mA/cm i n a .74 molar solution of CUSO4. The tube was then soldered to a 13.67 mm OD, .25 mm 2 thic k , 11.1 mm long copper tube using Indalloy #5 solder 238 (.375 Sn/.375Pb/.25ln) and Kester 115 3 pure rosin flux. The soldering operation was performed using an iron in a stream of helium gas heated to 180 C. It was very important to use a mild flux in order to prevent damage to the f i l m . The solder had to be non-superconducting in f i e l d s >.15 T and i t was desireable that i t s melting point be greater than 150 C so that the sample c e l l could be baked out when i t was mounted in the cryostat. 239 Notes to Appendix B 1. The water used throughout the cleaning procedure was d i s t i l l e d and demineralized. 2. Indium Corporation of America, Utica, N.Y., U.S.A. 3. Kester Solder, Chicago, 111., U.S.A. 

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