Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Magnetic resonance studies of atomic hydrogen confined by solid molecular hydrogen between 6.4 and 8.2… Steel, Stephen Chris 1985

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1985_A6_7 S74.pdf [ 4.95MB ]
Metadata
JSON: 831-1.0096204.json
JSON-LD: 831-1.0096204-ld.json
RDF/XML (Pretty): 831-1.0096204-rdf.xml
RDF/JSON: 831-1.0096204-rdf.json
Turtle: 831-1.0096204-turtle.txt
N-Triples: 831-1.0096204-rdf-ntriples.txt
Original Record: 831-1.0096204-source.json
Full Text
831-1.0096204-fulltext.txt
Citation
831-1.0096204.ris

Full Text

MAGNETIC RESONANCE STUDIES OF ATOMIC HYDROGEN CONFINED SOLID MOLECULAR HYDROGEN BETWEEN 6.4 AND 8.2 K by STEPHEN CHRIS STEEL^ • - — B . S c , U n i v e r s i t y of Toronto, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June 1985 © Stephen C h r i s S t e e l , 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the The U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: June 1985 A b s t r a c t Magnetic h y p e r f i n e resonance i n zero magnetic f i e l d has been used to study atomic hydrogen gas c o n f i n e d by w a l l s coated with s o l i d molecular hydrogen at temperatures between 6.4 and 8.2 K. The temperature range of the experiments was l i m i t e d by the nature of the atom source used to produce atoms. Measurements of the frequency s h i f t at low atom d e n s i t i e s ( n H 8 x lO" 4" 1 6 n r 3 ) have y i e l d e d a b i n d i n g energy f o r H on H 2 of 34.04 + 0.26 K, and a s u r f a c e frequency s h i f t of -1.16 ± 0.05 MHz. These r e s u l t s are i n e x c e l l e n t agreement with those obtained by Crampton et al between 3.2 and 4.5 K. The p r e s s u r e s h i f t due to the vapour pressure of the s o l i d H 2 was found to be -1.78 ± 0 . 0 1 x 10" 2 < I Hz m3. The low atom d e n s i t y t r a n s v e r s e r e l a x a t i o n r a t e measurements are d i f f i c u l t to i n t e r p r e t . There seems to be a r e l a x a t i o n mechanism on the H 2 s u r f a c e that g i v e s a c o n t r i b u t i o n beyond that a s s o c i a t e d with the d i s p e r s i o n of frequency s h i f t s an atoms see from normal a d s o r p t i o n . The data does show that the s t i c k i n g c o e f f i c e n t of H on H 2 i s g r e a t e r than 0.04. Measurements at higher atom d e n s i t i e s ( n H -> 9 x 10+ 1 8 n r 3 ) gave v a l u e s f o r the s u r f a c e recombination c r o s s l e n g t h which i n c r e a s e d from 0.5 A at 6.4 K to 1.1 A at 8.2 K. The bulk s p i n exchange c r o s s s e c t i o n was found to agree q u i t e w e l l with the - c a l c u l a t i o n s of B e r l i n s k y and S h i z g a l . i i T a b l e of C o n t e n t s L i s t o f T a b l e s v L i s t of F i g u r e s v i Acknowledgements v i i I . I n t r o d u c t i o n 1 I I . M a g n e t i c H y p e r f i n e R e s o n a n c e 4 A. H y p e r f i n e S t r u c t u r e o f t h e G r o u n d S t a t e 4 B. M a g n e t i c R e sonance 8 C. D e n s i t y M a t r i x R e l a x a t i o n 13 D. A b s o l u t e S i g n a l Power 15 1. C a v i t y C o u p l i n g 17 2. The F i l l i n g F a c t o r 21 I I I . P r o p e r t i e s of A t o m i c H y d r o g e n 25 A. S u r f a c e A d s o r p t i o n 25 1. A d s o r p t i o n on t h e H 2 S u r f a c e 25 2. E f f e c t on t h e M a g n e t i c R e sonance S i g n a l 26 3. The S t i c k i n g C o e f f i c e n t 31 B. E f f e c t of Hydrogen V a p o u r 33 C. R e c o m b i n a t i o n 34 1. B u l k R e c o m b i n a t i o n 35 2. S u r f a c e R e c o m b i n a t i o n 37 D. S p i n Exchange 39 IV. Thermometry 44 A. I n t r o d u c t i o n 44 1. V a p o u r P r e s s u r e Thermometry 47 2. M a g n e t i c Thermometry 51 3. R e s i s t a n c e Thermometry 54 B. C a l i b r a t i o n A p p a r a t u s 56 i i i 1 . The C r y o s t a t 56 2. The Vapour P r e s s u r e Thermometer 58 3. The M a g n e t i c Thermometer 59 4. The R e s i s t a n c e Thermometers 65 C. C a l i b r a t i o n R e s u l t s 66 1. M a g n e t i c Thermometer C a l i b r a t i o n ............68 2. R e s i s t a n c e Thermometer C a l i b r a t i o n s 69 V. M a g n e t i c Resonance E x p e r i m e n t s 74 A. The A p p a r a t u s 74 1 . The C r y o s t a t 74 2. The S p e c t r o m e t e r 83 B. Low Atom D e n s i t y R e s u l t s 86 1 . P r o c e d u r e 87 2. F r e q u e n c y S h i f t R e s u l t s 91 3. The T r a n s v e r s e R e l a x a t i o n R a t e 94 C. H i g h D e n s i t y R e s u l t s 101 1. R e c o m b i n a t i o n Measurements 101 2. S p i n Exchange Measurements 109 V I . Summary 117 BIBLIOGRAPHY 121 i v L i s t o f T a b l e s I . H y p e r f i n e C o n s t a n t s 5 I I . H y p e r f i n e S t a t e s a n d E n e r g i e s 8 I I I . E P T - 7 6 R e f e r e n c e P o i n t s 4 8 v L i s t of Figures 1. Hyperfine Energy Levels 7 2. Equivalent Resonant C i r c u i t 16 3. Bulk Spin Exchange Cross Section 40 4. Surface Spin Exchange Cross Length 42 5. Calib r a t i o n Apparatus 57 6. Gas Handling System 60 7. Mutual Inductance Bridge 62 8. Magnetic Thermometer Calibr a t i o n 70 9. GRT Calibration 71 10. CRT Calibration 72 11. Magnetic Resonance Apparatus 76 12. R.F. Discharge C i r c u i t 79 13. 1420 MHz Spectrometer 84 14. A Typical FID 89 15. The Frequency Shift vs. Temperature 92 16. The Transverse Relaxation Rate 95 17. Residual Ti 1 102 18. Inverse Density vs. Time 105 19. E f f e c t i v e Recombination Rate 106 20. Surface Recombination Cross Length 108 21. Relaxation Rates vs. Density 112 22. Bulk Spin Exchange Cross Section 113 v i Acknowleqements I would l i k e t o thank my s u p e r v i s o r , W.N. Hardy, f o r h i s s u g g e s t i o n ' o f t h i s p r o j e c t and h i s s u p p o r t . H i s s u p e r v i s i o n and p a r t i c i p a t i o n were i m p o r t a n t f a c t o r s i n i t s s u c e s s . S. Crampton c o n t r i b u t e d many u s e f u l i d e a s i n d i s c u s s i o n s d u r i n g h i s v i s i t t o U.B.C.. I am i n d e b t e d t o G.A. G a i t of the Dominion Radio A s t r o p h y s i c a l O b s e r v a t o r y , P e n t i c t o n , f o r h i s h e l p w i t h the f r e q u e n c y s t a n d a r d c a l i b r a t i o n s . I g r a t e f u l l y acknowledge the s u p p o r t of the U n i v e r s i t y of B r i t i s h Columbia i n the form of a Graduate F e l l o w s h i p . F i n a l l y , many thanks t o the o t h e r members of the l a b f o r a l l the h e l p f u l h i n t s and d i s c u s s i o n s they p r o v i d e d d u r i n g the c o u r s e of t h i s work, and t o my w i f e , J a n e t , f o r her p a t i e n t encouragement. v i i I . INTRODUCTION T h e r e a r e a number of r e a s o n s f o r t h e i n t e r e s t shown i n s t u d y i n g a t o m i c h y d r o g e n gas a t low t e m p e r a t u r e s . In 1959 Hecht p o i n t e d out t h a t a t o m i c h y d r o g e n would be an i n t e r e s t i n g quantum Bose gas [HECH-59]. I f t h e e l e c t r o n s p i n s were p o l a r i s e d by a l a r g e m a g n e t i c f i e l d , t h e r e c o m b i n a t i o n of a t o m i c H i n t o H 2 would be s u p p r e s s e d , and i t w ould be p o s s i b l e t o a c h i e v e h i g h d e n s i t i e s . B e c a u s e of i t s l a r g e z e r o p o i n t m o t i o n , a t o m i c h y d r o g e n i s t h o u g h t t o r e m a i n a gas down t o T = 0 K. A c o n s i d e r a b l e amount of r e s e a r c h has been done on t h e p o s s i b i l i t y of a c h i e v i n g B o s e - E i n s t e i n c o n d e n s a t i o n of s p i n p o l a r i s e d a t o m i c h y d r o g e n . T h i s would p r o v i d e a c h a n c e t o s t u d y t h e s u p e r f l u i d t r a n s i t i o n i n a weakly i n t e r a c t i n g Bose s y s t e m . S i l v e r a has p u b l i s h e d a r e v i e w o f s p i n p o l a r i s e d h y d r o g e n r e s e a r c h up t o 1982 [ S I L V - 8 2 ] . A s e c o n d m o t i v a t i o n f o r s t u d y i n g a t o m i c h y d r o g e n gas a t low t e m p e r a t u r e s i s t h e p o s s i b i l i t y of making a c r y o g e n i c maser. Such m asers may p r o v e v e r y u s e f u l as t i m e and f r e q u e n c y s t a n d a r d s . S p i n exchange c o l l i s i o n s between t h e H atoms i n h y d r o g e n m a s ers l i m i t t h e i r a c u r a c y as a f r e q u e n c y s t a n d a r d . At low t e m p e r a t u r e s , s p i n exchange d e c r e a s e s d r a m a t i c a l l y , due t o t h e r e d u c e d t h e r m a l v e l o c i t y , and b e c a u s e t h e s p i n e x c h a n g e c r o s s s e c t i o n d e c r e a s e s a t low t e m p e r a t u r e s . In o r d e r t o use a low t e m p e r a t u r e maser as a f r e q u e n c y s t a n d a r d , t h e e f f e c t s o f t h e W9lls u s e d t o c o n t a i n t h e H atoms, on t h e h y p e r f i n e r e s o n a n c e , must be u n d e r s t o o d . 1 2 W a l l s c o a t e d w i t h s o l i d s such as H 2 or Ne have been c o n s i d e r e d [CRAM-81, CRAM-84], as w e l l as l i q u i d h e l i u m c o a t e d w a l l s [HARD-82a]. Astronomy a l s o p r o v i d e s some m o t i v a t i o n f o r s t u d y i n g atomic hydrogen gas a t low t e m p e r a t u r e s . I n t e r s t e l l a r a t o m i c hydrogen e x i s t s o n l y at v e r y low d e n s i t i e s , and i n t e r a c t i o n s between the atoms o c c u r i n g when they a r e adsorbed on the s u r f a c e of d u s t g r a i n s p l a y an i m p o r t a n t r o l e [HOLL-70]. i t i s n e c e s s a r y t o u n d e r s t a n d the a d s o r p t i o n p r o c e s s , as w e l l as s p i n exchange and r e c o m b i n a t i o n i n t o H 2 t h a t occur on t h e s u r f a c e of the dust g r a i n t o e x p l a i n the dynamics of i n t e r s t e l l a r atomic hydrogen. Much work i n t h i s l a b has f o c u s e d on u s i n g magnetic h y p e r f i n e resonance i n z e r o magnetic f i e l d t o study the i n t e r a c t i o n s of atomic hydrogen w i t h v a r i o u s s u r f a c e s used t o c o n f i n e i t . Most of the p r e v i o u s work had been concerned w i t h atoms c o n f i n e d by l i q u i d h e l i u m c o a t e d w a l l s [HARD-82b, JOCH-82, MORR-83a], Some p r e l i m i n a r y work had been done on atomic H c o n f i n e d by s o l i d H 2 w a l l s , but i t was not r e p o r t e d . Crampton et al have a l s o used z e r o f i e l d magnetic resonance t o study H c o n f i n e d by H 2 c o a t e d w a l l s between 3.2 K t o 4 . 6 K [CRAM-82], T h e i r a p p a r a t u s used an open c e l l w i t h a 77 K H atom s o u r c e . The open geometry made i t d i f f i c u l t f o r them t o measure the r e c o m b i n a t i o n r a t e of t h e atomic H. 3 The i n t e n t of the present work was l a r g e l y t o e x p l o r e the c a p a b i l i t i e s of the c l o s e d c e l l technique f o r studying atomic H c o n f i n e d by H 2 coated w a l l s . In p a r t i c u l a r , i t was expected that one c o u l d extend the work of Crampton et al to higher temperatures where, f o r example, the s u r f a c e might be in a b e t t e r s t a t e of anneal and where the e f f e c t s of the H 2 vapour pressure on the h y p e r f i n e resonance c o u l d be seen. In a d d i t i o n , the use of a c l o s e d geometry would make measurements of the recombination r a t e p o s s i b l e . I I . MAGNETIC HYPERFINE RESONANCE M a g n e t i c h y p e r f i n e r e s o n a n c e i s a v e r y v e r s a t i l e p r o b e f o r s t u d y i n g a t o m i c h y d r o g e n a t low t e m p e r a t u r e s . The s i g n a l s t r e n g t h p r o v i d e s a measure o f t h e atom d e n s i t y , w h i l e t h e f r e q u e n c y s h i f t and l i n e w i d t h c an p r o v i d e u s e f u l i n f o r m a t i o n a b o u t atom-atom and a t o m - s u r f a c e i n t e r a c t i o n s . A. HYPERFINE STRUCTURE OF THE GROUND STATE A t t h e t e m p e r a t u r e s o f i n t e r e s t i n t h i s work, t h e o n l y d e g r e e s o f f r e e d o m a v a i l i b l e t o t h e h y d r o g e n atom ( o t h e r t h a n t r a n s l a t i o n a l m o t i o n ) a r e t h o s e of t h e s p i n s . The atom i s a l w a y s i n t h e 1s o r b i t a l s t a t e , and t h e s p i n H a m i l t o n i a n i s H = h B 0 . ( 7 e S - 7 I) + hw 0(S'I) [11- 1 ] where B 0 i s t h e s t a t i c m a g n e t i c f i e l d , 7 e and 7p a r e t h e e l e c t r o n and p r o t o n g y r o m a g n e t i c r a t i o s r e s p e c t i v e l y , S and I a r e t h e e l e c t r o n and p r o t o n s p i n a n g u l a r momentum o p e r a t o r s ( i n u n i t s o f h ) , and ho>0 i s t h e h y p e r f i n e c o u p l i n g c o n s t a n t . T a b l e I g i v e s t h e a c c e p t e d v a l u e s of e a c h of t h e s e c o n s t a n t s . B o t h S and I a r e o f t h e form a/2, where a i s a P a u l i o p e r a t o r f o r s p i n 1/2 p a r t i c l e s . I f we t a k e t h e z a x i s t o be 4 5 Table I. Hyper f i n e Constants Constant • Ref. : u0/2ir = 1 4 2 0 4 0 5 7 5 1 . 7 7 3 ± o . 0 0 1 Hz [PETI- • 8 0 ] u0/2ir = 1 4 2 0 4 0 5 7 5 1 . 7 6 8 ± 0 . 0 0 2 Hz [HELL- 7 0 ] ?e 1 . 7 6 0 8 4 2 ± 0 . 0 0 0 0 2 6 x 1 0 1 1 s' 1 T" 1 [AIPH- 7 2 ] 7 P = 2 . 6 7 5 197 ± 0 . 0 0 0 0 3 9 x 1 0 8 s' 1 T" 1 [AIPH- 7 2 ] - along the d i r e c t i o n of the s t a t i c magnetic f i e l d , H becomes H = h B 0 ( T e S z - 7 p I z ) . + hw 0(S-I) [ I I - 2 ] The e i g e n s t a t e s of S 2 and l 2 (j++>, | +->, ...) pro v i d e a convenient b a s i s set f o r t h i s problem. Two of these b a s i s s t a t e s (|++> and |—>) are a l r e a d y e i g e n s t a t e s of H. In order to determine the remaining two e i g e n s t a t e s , i t i s u s e f u l to look at the s o l u t i o n s i n the l i m i t of l a r g e and small B 0. In the l i m i t of l a r g e B 0 the f i r s t term of H dominates, and a l l of the b a s i s s t a t e s are e i g e n s t a t e s of H, each with a d i f f e r e n t energy. If B 0 i s zero, the t o t a l angular momentum J must commute with the Hamiltonian, s i n c e the system i s r o t a t i o n a l l y i n v a r i a n t . Since the o r b i t a l angular momentum L i s zero, the e i g e n s t a t e s of H must be e i g e n s t a t e s of the t o t a l s p i n angular momentum F 2 and one of i t s components F 6 ( F = S + I ) . D e n o t i n g t h e s e by |a> ... |d> we have: |a> = |0, 0> = {|-+> - |+->}//2 |b> = | 1 ,-l> = |—> |c> = |1, 0> = {|-+> + |+->}/V2 |d> = |1, 1> = |++> [ I I - 3 ] For an a r b i t r a r y s t a t i c f i e l d B 0, |a> and |c> must be of the form |a> = cos 6 |-+> - s i n 8 |+-> |c> = s i n 8 |-+> + cos 8 I+-> [ I I - 4 ] s i n c e a l l the e i g e n s t a t e of H must be o r t h o n o r m a l . 8 i s de t e r m i n e d by s o l v i n g e i t h e r of the e i g e n v a l u e e q u a t i o n s tf|a> = E a|a> H\c> = E c|c> [ I I - 5 ] g i v i n g the r e s u l t t a n 28 = w 0 / B o ( 7 e + 7 p ) [ I I - 6 ] The e i g e n s t a t e s and t h e i r e n e r g i e s a r e summarised i n t a b l e I I , and the energy l e v e l s are shown as a f u n c t i o n of the s t a t i c f i e l d B 0 i n f i g . 1. 7 Magnetic Field BD ( mT) F i g . 1. H y p e r f i n e Energy L e v e l s The energy l e v e l s of the e i g e n s t a t e s i n u n i t s of frequency, as a f u n c t i o n of the f i e l d B 0. 8 Table I I . Hyperfine States and Energies Eigenstates: Energies: |a> = cost? |-+>-sin0 | +-> E a= -h[wg+ (7 e +7p) 2 B g ] l / 2 / 2 - hcj 0/2 |b> = |-> Efa= ha>0/4 " h ( 7 e -7 p)B 0/2 |c> = sine|-+>+cos0|+-> E c= h[cog + ( 7 e + 7 p ) 2 B g ] l / 2 / 2 - hcj 0/2 |d> = |++> E d= hcj 0 / 4 + h ( 7 e -7p)B 0/2 tan 26 = w 0/B 0(7 e+ 7 p) -B. MAGNETIC RESONANCE It i s possible to probe the density ma t r i x describing atomic hydrogen in the ground state by using resonantly induced tr a n s i t i o n s between the hyperfine states. For small values of B 0, less than a few gauss, the angular frequencies corresponding to the various t r a n s i t i o n s are: _ v7 e-7 p)B 0/2 + ( 7 e + 7 p ) 2 B o / 2 c j 0 "o + ( 7 e - 7 p ) B 0 / 2 [II-7] Because the |a> to |c> t r a n s i t i o n frequency i s quadratic in B 0, while the others are linear in B 0, at the f i e l d used in these experiments (B 0 40 mG) the degeneracy between the three tr a n s i t i o n s i s removed enough to allow an unhindered study of the |a> to |c> t r a n s i t i o n , without i t being perturbed much i t s e l f by the f i e l d . w a b " "ac = "ad = 9 S i n c e o n l y t r a n s i t i o n s of t h e t y p e |a> t o |c> a r e of i n t e r e s t , i t w i l l be u s e f u l t o l o o k a t t h e t i m e e v o l u t i o n of t h e d e n s i t y m a t r i x r e s t r i c t e d t o t h e s u b s p a c e spanned by t h e s e two s t a t e s , when t h e r e i s some a p p l i e d ( u s u a l l y t i m e d e p e n d e n t ) f i e l d B_ a l o n g t h e z a x i s . In o r d e r t o do t h i s i t i s u s e f u l t o t a k e a d v a n t a g e of a f o r m a l e q u i v a l e n c e between t h e a c t u a l H a m i l t o n i a n and t h a t of a f i c t i t i o u s s p i n 1/2 s y s t e m [ABRA-61]. The |a>, |c> r e p r e s e n t a t i o n i s t a k e n t o c o r r e s p o n d t o t h e s'z r e p r e s e n t a t i o n o f t h e f i c t i t i o u s s y s t e m . The H a m i l t o n i a n i s t h e n H = - h ( 7 e + 7 p ) B 2 s ; - h c j 0 s ; - 'hu 0/4 [ I I - 8 ] or e q u i v a l e n t l y H = ~hy' (B' - s ' ) - ha)0/4 where 7' = ( 7 e + 7 p ) B' = B x z B'z = 0,0/7' [ I I - 9 ] w h i c h i s t h e H a m i l t o n i a n of a f i c t i t i o u s s p i n 1/2 p a r t i c l e w i t h a g y r o m a g n e t i c r a t i o 7' i n a f i e l d B'. The t i m e e v o l u t i o n o f t h e d e n s i t y m a t r i x i s d e t e r m i n e d by t h e e q u a t i o n / h p = [H, p] [11-10] 10 T h e d e n s i t y m a t r i x c a n b e e x p r e s s e d i n t e r m s o f t h e e x p e c t a t i o n v a l u e o f t h e f i c t i t i o u s v e c t o r m ' : p = m ' . s ' + p . r / 2 [11-11] w h e r e p a c i s t h e p r o b a b i l i t y o f f i n d i n g a n a t o m i n e i t h e r t h e |a> o r |c> s t a t e s . S u b s t i t u t i n g t h i s i n t o [ 1 1 - 1 0 ] g i v e s T h i s i s t h e c l a s s i c a l e q u a t i o n o f m o t i o n o f a m a g n e t i c m o m e n t w i t h a g y r o m a g n e t i c r a t i o 7' i n a f i e l d B ' . T h e d y n a m i c s o f t h i s s y s t e m h a v e b e e n t h o r o u g h l y w o r k e d o u t i n t h e NMR l i t e r a t u r e , a n d m o s t o f w h i c h c a n b e d i r e c t l y a p p l i e d t o t h e h y d r o g e n h y p e r f i n e r e s o n a n c e . T h e r e i s a r e a l m a g n e t i s a t i o n a l o n g t h e z a x i s i s g i v e n w h e r e n H i s t h e c o n c e n t r a t i o n o f h y d r o g e n a t o m s . E q u a t i o n [ 1 1 - 1 2 ] c a n b e u s e d t o d e t e r m i n e t h e m o t i o n o f m ' , a n d h e n c e p . I n t h e a b s e n c e o f a n y a p p l i e d f i e l d , B ' o n l y h a s t h e c o m p o n e n t a l o n g z d u e t o t h e h y p e r f i n e i n t e r a c t i o n , a n d m ' j u s t p r e c e s s e s a b o u t t h e z a x i s w i t h a n a n g u l a r f r e q u e n c y OJ0. T h i s w i l l b e s e e n a s a n o s c i l l a t i n g m a g n e t i s a t i o n a l o n g t h e z a x i s p r o p o r t i o n a l t o t h e m ' = 7' (nT x B ' ) [ 1 1 - 1 2 ] b y M z " n H 7 ' h m z / 2 [ 1 1 - 1 3 ] 11 concentration of atoms and the magnitude of the transverse component of m ' . A small s t a t i c f i e l d along the z axis 1 gauss) doesn't change t h i s except that C J 0 i s replaced by u>ac, given in equation [II-7]. In thermal equilibrium, the density matrix w i l l be p = e'H^T / Tr{e-"/ k T} [11-14] where the trace must be taken over a l l four of the hyperfine states. Expressing t h i s in terms of m ' , and using an approximation v a l i d for kT >> ha>0, we have m ' •» hw0/4kT z [11-15] Thus, in thermal equilibrium, no o s c i l l a t i n g magnetisation w i l l be observed as the transverse components of m ' are both zero. Now consider the eff e c t of an o s c i l l a t i n g magnetic f i e l d along the z axis B = 2B 1cos(cjt). This corresponds to an o s c i l l a t i n g transverse f i e l d in the f i c t i t i o u s system, and can cause t r a n s i t i o n s . Of p a r t i c u l a r interest are the eff e c t s of what are c a l l e d 7r/2 and n pulses. If u = uac' an<^ the o s c i l l a t i n g f i e l d i s applied for a time t , / 2 = » / 2 ( 7 e + 7 p ) B , [11-16] Then, i f m ' only has a component along z to start with, 12 a f t e r the 7r/2 pulse m ' ( t ) = - m'(O)sin(a; t) x + ml(0)cos(u a„t) y [11-17] z oc z ac If m ' had i t s thermal e q u i l i b r i u m value immediately before the TT/2 p u l s e , then an o s c i l l a t i n g m a g n e t i s a t i o n along the z a x i s w i l l be observed M 2 ( t ) = M 0 s i n ( c o a c t ) where M 0 = n Hh 27'oj 0/8kT [ 11 - 1 8 ] Thus the magnetisation immediately a f t e r a TT/2 p u l s e i s a measure of the c o n c e n t r a t i o n of atoms, p r o v i d e d they were in thermal e q u i l i b r i u m before the p u l s e . If the o s c i l l a t i n g f i e l d i s a p p l i e d f o r twice as long, i t i s c a l l e d a n p u l s e . T h i s pulse i n v e r t s the p o p u l a t i o n d i f f e r e n c e between the |a> and |c> s t a t e s . The ir p u l s e can be used to p e r t u r b p so that i t s r e l a x a t i o n back to thermal e q u i l i b r i u m can be s t u d i e d . Thus magnetic resonance of the |a> to |c> t r a n s i t i o n i s very s i m i l a r to magnetic resonance of a s p i n 1/2 p a r t i c l e , except that the r . f . f i e l d s are a p p l i e d and d e t e c t e d along the a x i s of s p i n q u a n t i s a t i o n r a t h e r than t r a n s v e r s e to i t . T h i s i s sometimes r e f e r r e d to as a l o n g i t u d i n a l t r a n s i t i o n . 1 3 C. DENSITY MATRIX RELAXATION The Hamiltonian d e s c r i b i n g the atoms must i n c l u d e terms due to i n t e r a c t i o n s between the sp i n degrees of freedom and the other degrees of freedom of an atom, and i n t e r a c t i o n s between atoms. By inducing t r a n s i t i o n s between the h y p e r f i n e s t a t e s these i n t e r a c t i o n s allow the spin degrees of freedom to exchange energy with the other degrees of freedom and so come i n t o thermal e q u i l i b r i u m . I f t h i s were not the case, magnetic resonance would not be p o s s i b l e , as the i n i t i a l thermal p o p u l a t i o n d i f f e r e n c e i s necessary to induce the o s c i l l a t i n g m a g n e t i s a t i o n . A f u l l d e s c r i p t i o n of these i n t e r a c t i o n s would prove a formidable task, so a phenomenalogical approach introduced by F e l i x Bloch i s u s u a l l y used [ABRA-61]. The e f f e c t s of the i n t e r a c t i o n s are rep l a c e d by two r e l a x a t i o n times, and T 2, best understood i n terms of the f i c t i t i o u s magnetic moment m': m'x = m' x/T 2 « n ' y = n . ' y / T 2 [ 1 1 - 1 9 ] and m'2 = (m' z - <m'z>T)/T [ 1 1 - 2 0 ] where <m'2>T i s the thermal e q u i l i b r i u m value of m'z. In gener a l T, # T 2 , and the mechanisms r e s p o n s i b l e f o r each may be q u i t e d i f f e r e n t . R e l a x a t i o n of the t r a n s v e r s e components 1 4 of m' d o e s n ' t change t h e e n e r g y o f t h e s p i n s y s t e m , i t o n l y i n v o l v e s a l o s s of c o h e r e n c e w i t h i n t h e sam p l e . R e l a x a t i o n of t h e l o n g i t u d i n a l component of m' does change t h e e n e r g y of t h e s p i n s y s t e m , so a f l o w of e n e r g y between t h e s p i n s y s t e m a n d o t h e r d e g r e e s of f r e e d o m i s n e c e s s a r y . N o r m a l l y T, > T 2 , s i n c e most p r o c e s s e s c a p a b l e of c o n t r i b u t i n g t o T, 1 a l s o c o n t r i b u t e t o T i 1 but n o t vice versa. As a r e s u l t o f t h e s e r e l a x a t i o n s , t h e m a g n e t i s a t i o n i m m e d i a t e l y a f t e r a TT/2 p u l s e w i l l be M z ( t ) = M 0 e " t / / T 2 s i n ( o ) a c t ) [11-21 ] where M 0 i s d e f i n e d i n e q u a t i o n [ 1 1 — 1 8 ] , so T 2 c a n be d e t e r m i n e d d i r e c t l y from t h e m a g n e t i c r e s o n a n c e s i g n a l . D e t e r m i n i n g T, i s a l i t t l e more d i f f i c u l t . The s i m p l e s t method r e q u i r e s f i r s t t h a t T 2 be made much s h o r t e r t h a n T, ( w i t h o u t a f f e c t i n g T , ) . T h i s i s u s u a l l y done by a p p l y i n g a m a g n e t i c f i e l d g r a d i e n t a c r o s s t h e sample. T h i s c a u s e s a d i s t r i b u t i o n i n t h e r a t e a t w h i c h m' p r e c e s s e s t h r o u g h o u t t h e s a m p le ( s e e e q u a t i o n [ I I - 7 ] ) , so t h e a v e r a g e v a l u e o f i t s t r a n s v e r s e components q u i c k l y d e c a y s t o z e r o . T, can t h e n be d e t e r m i n e d from t h e s i g n a l a f t e r a s e q u e n c e of two •n/2 p u l s e s . A s s u m i n g t h e s y s t e m i s i n i t i a l l y i n t h e r m a l e q u i l i b r i u m , t h e a m p l i t u d e o f t h e m a g n e t i s a t i o n i m m e d i a t e l y 15 a f t e r t h e s e c o n d ir/2 p u l s e w i l l be M 2 ( T ) = M 0 ( 1 - e ~ T / / T [11-22] where t h e T i s t h e t i m e between t h e t h e two rr/2 p u l s e s . An a l t e r n a t i v e method i n v o l v e s u s i n g a IT p u l s e t o i n v e r t m ' from i t s e q u i l i b r i u m v a l u e , t h e n m e a s u r i n g i t s r e c o v e r y w i t h a 7r/2 p u l s e . A g a i n a s s u m i n g t h e s y s t e m i s i n i t i a l l y i n t h e r m a l e q u i l i b r i u m , t h e a m p l i t u d e o f t h e m a g n e t i s a t i o n i m m e d i a t e l y a f t e r t h e TT/2 p u l s e w i l l be where r i s t h e t i m e between t h e TT and -n/2 p u l s e s . The s p e c i f i c p r o c e s s e s c o n t r i b u t i n g t o T^ 1 and T j 1 i n t h e c a s e o f h y d r o g e n h y p e r f i n e r e s o n a n c e a r e d i s c u s s e d i n c h a p t e r I I I . D. ABSOLUTE SIGNAL POWER In o r d e r t o measure t h e d e n s i t y o f atoms i n t h e c e l l u s i n g m a g n e t i c r e s o n a n c e , we must be a b l e t o r e l a t e t h e m a g n e t i s a t i o n o f t h e atoms a f t e r a m icrowave p u l s e t o t h e power r a d i a t e d out of t h e r e s o n a t o r . M z ( T ) = M 0 ( 1 - 2 e " r / T l ) [11-23] coupling i resonator F i g . 2. E q u i v a l e n t Resonant C i r c u i t T h i s c i r c u i t models the a c t u a l resonant c i r c u i t to some degree of accuracy, and i s u s e f u l f o r d e r i v i n g some of i t s p r o p e r t i e s . 1 7 1. CAVITY COUPLING The f i r s t step i n r e l a t i n g the magnetisation of the atoms to the s i g n a l power determining how the f i e l d s i n the resonator couple to the t r a n s m i s s i o n l i n e . In order to do t h i s , i t i s u s e f u l to c o n s i d e r the e q u i v a l e n t c i r c u i t shown in 2. The resonator i s modeled as an LCR c i r c u i t , coupled by a transformer of t u r n s r a t i o n to a t r a n s m i s s i o n l i n e of c h a r a c t e r i s t i c impedance Z 0. The impedance seen t e r m i n a t i n g the t r a n s m i s s i o n l i n e i s Z L = n 2[R + /(wL - 1/uC)] [11-24] so the v o l t a g e r e f l e c t i o n c o e f f i c e n t w i l l be $ = ( Z L - Z 0) / ( Z L + Z 0) [11-25] Now u>c = 1//LC i s the resonant frequency of the reso n a t o r , Q 0 = o) cL/R i s the q u a l i t y f a c t o r of the resonator alone, and Q c = n 2 c J c L / Z 0 i s the q u a l i t y f a c t o r of the c i r c u i t i f the resonator were l o s s l e s s and none of the power coupled out of the c a v i t y were r e f l e c t e d back. In terms of these . q u a n t i t i e s , the power r e f l e c t i o n c o e f f i c e n t r = $$ i s r = (QS 1 " Q c 1 ) 2 + (OJ/CJC - uc/u)2 [11-26] (Q6 1 + Q ' 1 ) 2 + (u/w c - cj c/ca) 2 T h i s e x p r e s s i o n f o r the power r e f l e c t i o n c o e f f i c e n t i s v a l i d 18 f o r any resonant system i n which the power i s coupled i n t o and out of the system the same way. The f u l l width at h a l f maximum of the resonance i s then A" = " C ( Q 5 1 + Q c 1 ) i f Aco « ^ c [11-27] however, Aw i s u s u a l l y used to d e f i n e Q L = u>c/bu>, the loaded Q of the resonance, which i s what i s u s u a l l y measured. Thus Q L = (Q6 1 + Q c 1 ) " 1 [11-28] C r i t i c a l c o u p l i n g occours when the power r e f l e c t i o n c o e f f i c e n t r vanishes f o r u = o) c. Examaining equation [11-26] g i v e s the c o n d i t i o n f o r c r i t i c a l c o u p l i n g : Q c = Qo [11-29] Thus at c r i t i c a l c o u p l i n g , h a l f of the power l o s t i s d i s s a p a t e d i n the resonator, while the other h a l f i s coupled out i n t o the t r a n s m i s s i o n l i n e . S u b s t i t u t i n g t h i s i n t o equation [11-28] g i v e s the loaded Q at c r i t i c a l c o u p l i n g Q L = Q 0/2 = Q c/2 [11-30] In the magnetic resonance experiments, the atoms i n the c e l l w i l l produce an o s c i l l a t i n g m agnetisation along the a x i s of the resonator M ( r , t ) s i n ( u t ) . The e x t r a time 19 dependance i s i n c l u d e d to account f o r other e f f e c t s such as s p i n d i f f u s i o n . If the resonator i s tuned so that w = u>c, the response f i e l d w i l l be ir/2 r a d i a n s out of phase with the o s c i l l l a t i n g magnetisation and given by B ( r , t ) c o s ( w t ) . The instantaneous power e x t r a c t e d from the atoms i s then P = / _ B ( r , t ) c o s ( c j t ) • 3 M ( r , t ) s i n ( u t ) d 3 r [ 1 1 - 3 1 ] 5 "oT where the i n t e g r a l i s over the volume of the sample. Ignoring the c o n t r i b u t i o n s to the d e r i v a t i v e from the n o n - s i n u s o i d a l time dependence of M, and averaging over one c y c l e g i v e s P = CJ/2 ; s B(r , t ) - M ( r , t ) d 3 r [ 1 1 - 3 2 ] The power e x t r a c t e d from the atoms must be equal to the power l o s s from the resonator ( f o r time s c a l e s longer than the response time of the r e s o n a t o r ) , which can be given i n terms of the loaded Q and the t o t a l s t o r e d energy P = « [ / B 2 ( r ) d 3 r ] / 2 M 0 Q L [ 1 1 - 3 3 ] where the i n t e g r a l i s over a l l space. Equating these two r e s u l t s g i v e s ; s B ( r , t ) - M ( r , t ) d 3 r = [/ B 2 ( r ) d 3 r ] / M 0 Q L [ 1 1 - 3 4 ] 20 I f t h e s y s t e m i s c r i t i c a l l y c o u p l e d , t h e n h a l f of t h e power i s a v a i l a b l e as a s i g n a l , t h e o t h e r h a l f b e i n g d i s s a p a t e d i n t h e r e s o n a t o r . S u b s t i t u t i n g t h e above r e s u l t i n t o e q u a t i o n [ 1 1 - 3 2 ] g i v e s t h e a v a i l a b l e s i g n a l power as P S = O>MOQL Us B ( r , t ) - M ( r , t ) d 3 r ] 2 [ 1 1 - 3 5 ] 4/ B 2 ( r ) d 3 r The two volume i n t e g r a l s , w h i c h depend on g e o m e t r i c f a c t o r s , a r e u s u a l l y u s e d t o d e f i n e a f i l l i n g f a c t o r V = [/ s B ( r , t ) - M ( r , t ) d 3 r ] 2 [ 1 1 - 3 6 ] V_M 0 2 J B 2 ( r ) d 3 r where M 0 i s t h e mean v a l u e of t h e magn i t u d e o f t h e m a g n e t i s a t i o n , g i v e n by e q u a t i o n [ 1 1 - 1 8 ] i n t h e c a s e o f a 7 r/2 p u l s e , and V g i s t h e sample volume. I n t e r m s o f rj t h e s i g n a l power i s P C = uMc.Q rMo 2V c 7j / 4 [ 1 1 - 3 7 ] so t h e v a l u e of n a p p r o p r i a t e t o t h e e x p e r i m e n t a l s i t u a t i o n must be d e t e r m i n e d b e f o r e t h e a c t u a l c o n c e n t r a t i o n o f atoms c a n be c a l c u l a t e d f r o m t h e s i g n a l s t r e n g t h . 21 2. THE FILLING FACTOR The p h y s i c a l s i g n i f i c a n c e of -n as d e f i n e d by [11-36] can be understood by examining a simple case. I f the resona t o r i s a long s o l e n o i d , B i s homogeneous and p a r a l l e l to the a x i s . The magnetisation induced by a it/2 pulse w i l l be p a r a l l e l to B and have a magnitude M 0. In t h i s simple case we have T? = V s/V c [11-38] the r a t i o between the sample volume to the volume of the s o l e n o i d . In any r e a l resonator the B w i l l be inhomogeneous both in magnitude and d i r e c t i o n . If the magnitude i s inhomogeneous, then the angle by which the f i c t i t o u s magnetic moment i s r o t a t e d w i l l vary throughout the sample. For a tt/2 p u l s e , t h i s i s not n e c e s s a r i l y a s i g n i f i c a n t e f f e c t : the magnitude of the induced magnetisation i s at a maximum with r e s p e c t to the f i e l d , so the f i r s t d e r i v a t i v e v a n i s h e s . Since e s s e n t i a l l y a l l the measurements were made usi n g ir/2 p u l s e s , and the f i e l d should be resonably homogeneous over the sample volume, M 0 can be taken to have the value expected f o r a p e r f e c t n/2 p u l s e . In order to determine the e f f e c t s of inhomogeneities i n the d i r e c t i o n of B, i t i s necessary to c o n s i d e r the n o n - s i n u s o i d a l p a r t of the time dependences of the two f i e l d s . I f the time s c a l e of the measurement i s short enough 22 that atoms don't have time to d i f f u s e s i g n i f i c a n t l y , then M(r,t) w i l l be p a r a l l e l t o B ( r , t ) throughout the sample, so M ( r , t ) - B ( r , t ) » M 0 B ( r ) . [11-39] S u b s t i t u t i n g t h i s i n t o equation [11-36] g i v e s V' = [ J s B(r) d 3 r ] 2 [11-40] V s / B 2 ( r ) d 3 r I f , on the other hand, the time s c a l e of the measurement i s long enough f o r the atoms to d i f f u s e f r e e l y , then M(r,t) should be r e p l a c e d by i t s average value througout the sample volume <M> = J s M(r) d 3 r / V g [11-41] If the sample has r o t a t i o n a l symmetry about the z a x i s , then the t r a n s v e r s e components of <M> w i l l v a n i s h . Under these c o n d i t i o n s the f i l l i n g f a c t o r i s then rj" = <MZ>2 [ J s B z ( r ) d 3 r ] 2 [11-42] V_M 0 2 S B 2 ( r ) d 3 r Since n" < T J' the s i g n a l s t r e n g t h i s reduced by the inhomegeneities i f there i s s u f f i c e n t time f o r the spi n s to d i f f u s e . 2 3 The f i e l d B ( r ) induced in the c a v i t y by the o s c i l l a t i n g m agnetisation of the atoms has the same s p a t i a l dependence as the f i e l d of the r . f . pulse used to induce the magnetisation, s i n c e i t i s governed by the resonator mode. Thi s p r o p o r t i o n a l i t y can be used to make a measurement of 17' based on the l e n g t h of the TT/2 p u l s e . Since the n/2 pulse i s very s h o r t , the atoms w i l l not have time to d i f f u s e s i g n i f i c a n t l y , and the magnitude of the m a g n i t i s a t i o n w i l l be determined by the average magnitude of the r . f . f i e l d over the sample volume: If the p u l s e l e n g t h i s v a r i e d to produce a maximum power s i g n a l , then equation [11—16] should be a good estimate of <2B,> Since the r . f . pulse i s a p p l i e d at the c a v i t y resonance, and the c a v i t y i s c r i t i c a l l y coupled, there w i l l be no r e f l e c t e d power. T h i s means that the i n c i d e n t power must be d i s s i p a t e d by the i n t r i n s i c l o s s e s of the c a v i t y , g i v i n g the r e s u l t <2B,> = ; s 2B, ( r ) d 3 r / V g [ 1 1 - 4 3 ] <2B,> - * / ( 7 e + 7 p ) t w / 2 [ 1 1 - 4 4 ] / ( 2 B , ) 2 d 3 r = 2MoQoPiA> [ 1 1 - 4 5 ] 24 If the above r e s u l t s are s u s t i t u t e d i n t o equation [11-40], we get an e x p r e s s i o n f o r r?' i n terms of the ti/2 pulse l e n g t h and the a p p l i e d r . f . power: T ? ' = * 2 « V S [11-46] 2 M o Q o P i ( 7 e + 7 p ) 2 t 2 7 r / 2 Morrow has done more e x t e n s i v e c a l c u l a t i o n s and measurements r e l a t e d to the f i l l i n g f a c t o r f o r the type of resonator used i n these experiments [MORR-83a]. His r e s u l t s suggest that the e r r o r i n t r o d u c e d by using the value of T? from equation [11-46] on the c a l c u l a t e d atom d e n s i t i e s would only be about 4 % even i f B(r) i s 30 % l a r g e r at the c e n t e r of the resonator than at the ends. I l l . PROPERTIES OF ATOMIC HYDROGEN Th i s chapter presents a review of the p r o p e r t i e s of atomic hydrogen r e l e v e n t to the experiments: a d s o r p t i o n on the the s o l i d hydrogen s u r f a c e , the i n t e r a c t i o n with the molecular hydrogen vapour, the recombination of atomic hydrogen, and s p i n exchange c o l l i s i o n s . A. SURFACE ADSORPTION In a l l s t u d i e s of bulk atomic hydrogen, a d s o r p t i o n of the atomic hydrogen on the surface of the c e l l c o n t a i n i n g the atoms p l a y s a s i g n i f i c a n t r o l e . A d s o r p t i o n occurs because van der Waals f o r c e s provide an a t t r a c t i v e p o t e n t i a l between the atoms and the s u r f a c e . These e f f e c t s may be minimized by choosing s u r f a c e s with weak i n t e r a c t i o n s , but no s u r f a c e i s known which doesn't support at l e a s t one bound s t a t e of atomic hydrogen. L i q u i d helium has the lowest b i n d i n g energy of any s u r f a c e s t u d i e d so f a r : Eg = 1.15 ± 0.05 K f o r H on "He, and E f i = 0.41 ± 0.02 K f o r H on 3He [MORR-83a]. 1. ADSORPTION ON THE H 2 SURFACE A T h e o r e c t i c a l c a l c u l a t i o n of the i n t e r a c t i o n of atomic hydrogen with s o l i d hydrogen has been made by Weinrib, which y i e l d e d a b i n d i n g energy of 35. ± 10. K [WEIN-79]. More recent c a l c u l a t i o n s by P i e r r e et a l , t a k i n g i n t o account the extreme quantum nature of the s u r f a c e , give b i n d i n g e n e r g i e s ranging from 25.5 K to 32.7 K. They p r e d i c t a second bound s t a t e on s o l i d hydrogen with a b i n d i n g energy ranging 25 26 between 1.2 K and 2.1 K [PIER-85]. Most of the v a r i a t i o n in t h e i r r e s u l t s i s due to d i f f e r e n c e s i n the H-He p o t e n t i a l s used. They a l s o c a l c u l a t e d the e f f e c t i v e mass of the H atoms in the s t r o n g l y bound s t a t e f o r motion along the s u r f a c e , it and found that m /m = 1.030 ± 0.015. If t h i s r e s u l t i s v a l i d , i t i m p l i e s that the atoms adsorbed on the s u r f a c e are almost as mobile i n the plane of the s u r f a c e as atoms i n a p e r f e c t two dimensional gas. Crampton et al measured the b i n d i n g energy of atomic H on H 2, and obtained the r e s u l t E f i = 35.75 ± 0.31 K, assuming the atoms on the s u r f a c e form a two dimensional gas [CRAM-82]. If the su r f a c e atoms indeed form a two dimensional gas, then the r e l a t i o n s h i p between the bulk and s u r f a c e d e n s i t i e s i n thermal e q u i l i b r i u m can be determined from thermodynamics by equating the chemical p o t e n t i a l s i n the gas and adsorbed phases [DASH-75]. For low s u r f a c e coverages one gets o H = n H A e E B / k T [ I I I - 1 ] where n H i s the bulk H atom d e n s i t y , o H i s the s u r f a c e H atom d e n s i t y , A = h//2~mkT i s the thermal de B r o g l i e wave le n g t h , and E f i i s the b i n d i n g energy of the s u r f a c e s t a t e . 2. EFFECT ON THE MAGNETIC RESONANCE SIGNAL The ground s t a t e of the atoms adsorbed on the s u r f a c e w i l l be pe r t u r b e d by the s u r f a c e , s h i f t i n g the h y p e r f i n e frequency by an amount . T h i s w i l l a l t e r the magnetic 27 resonance s i g n a l from that due to the gas phase atoms alone . In order to c a l c u l a t e the e f f e c t of t h i s on the magnetic resonance s i g n a l , we must look at the a d s o r p t i o n process more c l o s e l y . Each time an atom c o l l i d e s with the s u r f a c e there i s a c e r t a i n p r o b a b i l i t y that i t w i l l be adsorbed. The mean va l u e of t h i s p r o b a b i l i t y i s known as the s t i c k i n g c o e f f i c e n t s. There are three times which are important i n c h a r a c t e r i s i n g the i n t e r a c t i o n of the gas and adsorbed phases: the mean time between c o l l i s i o n s with the s u r f a c e <^c>i the mean time between s t i c k i n g events <7-g> = < T C > / S , and the mean d u r a t i o n of s t i c k i n g events <T S>. The mean time between c o l l i s i o n s with the s u r f a c e can be c a l c u l a t e d , as i t depends only on the v e l o c i t y d i s t r i b u t i o n of the atoms, given by t h e i r temperature, and the geometry of the c e l l . Morrow [MORR-83a] performed a Monte C a r l o c a l c u l a t i o n of <r c> f o r the c e l l geometry used i n these experiments and found <r c> = (7.4 x 10- 5 s K 1 / 2 ) / V T [ I I I - 2 ] The mean f r a c t i o n of time an atom spends on the s u r f a c e (or e q u i v a l e n t l y , the f r a c t i o n of atoms on the surface) x i s determined by the r a t i o between the s u r f a c e and bulk d e n s i t i e s given i n equation [ I I I — 1 ], and the s u r f a c e area to 28 volume r a t i o of the c e l l (A/V) x = ( A / V ) A e E B / k T [ I I I - 3 ] Since i n a l l of the experiments x < 10"", i t can be given i n terms of the v a r i o u s times above as In order to determine the e f f e c t of s u r f a c e a d s o r p t i o n on the magnetic resonance s i g n a l , i t i s necessary to make some reasonable assumptions about the s t a t i s t i c s of a d s o r p t i o n and d e s o r p t i o n . The sim p l e s t assumption i s that the p r o b a b i l i t i e s f o r an atom being adsorbed from the gas, or desorbed from the s u r f a c e , are independent of time. In t h i s case, the times between s t i c k i n g events, and the times of a d s o r p t i o n , w i l l obey Poisson s t a t i s t i c s . T h i s should be a good assumption i n the case of a d s o r p t i o n i f s i s f a i r l y s m a l l : the atoms w i l l u s u a l l y make a f a i r number of c o l l i s i o n s between s t i c k i n g events, and any c o r r e l a t i o n s with p r e v i o u s s t i c k i n g events should be l o s t . The spread i n rQ values should a l s o h e l p smear out any time c o r r e l a t i o n s i n the p r o b a b i l i t y of a b s o r b t i o n . In the case of d e s o r p t i o n , t h i s assumption i s easy to j u s t i f y . S ince the bi n d i n g energy E f i i s much l a r g e r than kT, an atom adsorbed on the s u r f a c e w i l l u s u a l l y experience many c o l l i s i o n s before e x p e r i e n c i n g one s u f f i c e n t l y e n e r g e t i c to x = <r s>/(<r B>+<T S>) * <T S>/<T B> [ I I I - 4 ] 29 desorb i t . T h i s means the s u r f a c e atoms are l i k e l y to the r m a l i s e i n a time short compared to <'"s>» Once the surf a c e atom has t h e r m a l i s e d , the p r o b a b i l i t y of d e s o r p t i o n can depend only on the s u r f a c e temperature and d e n s i t y . Given the above assumptions, the e f f e c t of the s u r f a c e frequency s h i f t on the magnetic resonance s i g n a l can be c a l c u l a t e d . T h i s type of model was f i r s t d e a l t with by Anderson and Weiss [ANDR-54] and i s given a f u l l treatment by Abragam [ABRA-61]. In the case where x i s sma l l , the magnetic resonance s i g n a l c o n s i s t s of a l i n e at a frequency s h i f t e d from the unperturbed gas phase resonance by an amount Aw = <*O/<Tb>( ) [ 111-5 ] with a t r a n s v e r s e decay r a t e T i 1 = ^>§/<Tb> (1 +0§ ) [ 111-6 ] where <£0 = Ao>s<7s> i s the mean phase s h i f t per a d s o r p t i o n . There i s a l s o a broad weak l i n e s h i f t e d by Ao>s, which would be o u t s i d e of the bandwidth of the spectrometer used i n the experiments, and so wouldn't be seen. If the average phase s h i f t per c o l l i s i o n i s small (<p0 « 1), the frequency s h i f t i s l i n e a r i n <j>0, and i s given 30 by Aw ^ A C J S X [ I I I - 7 ] so the s h i f t i s p r o p o r t i o n a l to the f r a c t i o n of atoms on the s u r f a c e . The t r a n s v e r s e r e l a x a t i o n i s then q u a d r a t i c i n <f>0, and i s given by T i 1 <* A C J S 2 X 2 < T B > - [ 111-8 ] I f the frequency s h i f t due to the s u r f a c e atoms can be separated from other frequency s h i f t s , then the s u r f a c e frequency s h i f t A C J s and the b i n d i n g energy E g can be determined from i t s temperature dependence. Once these are known, the s t i c k i n g c o e f f i c e n t s can be determined from the temperature dependence of T 2 . In the case where the mean phase s h i f t per a d s o r p t i o n i s l a r g e (<p0 » 1), then the s i g n a l i s dominated by atoms which have not being dephased by a c o l l i s i o n , so the frequency s h i f t approaches z e r o , while the t r a n s v e r s e r e l a x a t i o n time approaches < T B > . Both Morrow [MORR-83a] and Crampton [CRAM-82] have worked out more complete models f o r the magnetic resonance s i g n a l , d i s c a r d i n g the assumption that the times between s t i c k i n g events obey Poisson s t a t i s t i c s . The r e s u l t s of t h e i r models are very s i m i l a r t o the simple model given above, e s p e c i a l l y i n the case where <p0 « 1. 31 3. THE STICKING COEFFICENT In order for an atom impinging on the s u r f a c e to s t i c k to the s u r f a c e , i t must somehow be abl e t o get r i d of i t s excess energy. T h i s i s u s u a l l y done by c r e a t i n g phonons i n the s u r f a c e . In order to c a l c u l a t e the s t i c k i n g c o e f f i c e n t , one must know the s u r f a c e p o t e n t i a l the atom sees, the phonon spectrum of the s u b s t r a t e , and the c o u p l i n g between atoms c o l l i d i n g with the surface and the phonons. The problem of c a l c u l a t i n g the s t i c k i n g c o e f f i c e n t of a l i g h t atom impinging on a sur f a c e with an a t t r a c t i v e van der Waals i n t e r a c t i o n was t a c k l e d by Lennard-Jones and Devonshire [LENN-36]. They modeled the van der Waals i n t e r a c t i o n as a Morse p o t e n t i a l V(z) = _ D [ e - 2 e ( z - z 0 ) 2 e - e ( z - z 0 ) ] [ U I - 9 ] where D i s the p o t e n t i a l w e l l depth, z 0 i s the p o s i t i o n of the p o t e n t i a l minimum (the su r f a c e i s z = 0), and e determines how steep the p o t e n t i a l w e l l i s . They assumed that the phonon spectrum was d e s c r i b e d by the Debye model. For the c o u p l i n g between the H atoms impinging on the s u r f a c e and the phonons, they assumed i t was j u s t due to the i n t e r a c t i o n energy between the atoms and the s u r f a c e v a r y i n g as the su r f a c e v i b r a t e d beneath them, and would be given by the product of the s u r f a c e displacement (due to phonons) and the d e r i v a t i v e of the morse p o t e n t i a l . T h i s probably o v e r e s t i m a t e s the s t r e n g t h of 32 the c o u p l i n g . Using the above assumptions, they were able to e x p l i c i t l y c a l c u l a t e the s t i c k i n g c o e f f i c e n t as S - 3e*D*h" z Mn I ( n ) [111-10] 32^ 2Mmd«k' ,t9 D 3T n!T(2d-n) where the sum i s over n, the quantum number of the adsorbed s t a t e s of the Morse p o t e n t i a l , d = v/2mD/eh, 8^ i s the Debye temperature of the s u b s t r a t e , m i s the mass of an atom, M i s the mass of s u r f a c e molecule, Mn = d - n -1/2, and I(n) i s the i n t e g r a l K n ) = e~ a' z 2(M 2 + M 2 ) 3 |r(d+l/2 + / M) | 22 Msinh(2Mir) dy [1 - exp(-aM 2-aM 2 )) ] [cosh(2M?r) - cos ( 2u0 it) ] [ 111 - 1 1 ] where a = e 2h 2/2mkT, and (3 = /2mkt9D/eh. T h i s r e s u l t can be used to c a l c u l a t e the s t i c k i n g c o e f f i c e n t of H on H 2, given s u i t a b l e values f o r the parameters. Roberts and Daunt have measured 8^ = 122 K [ROBT-70]. The Morse p o t e n t i a l parameters D and e can be chosen so that they give the same values f o r the H atom bound s t a t e e n e r g i e s as c a l c u l a t e d by P i e r r e et a l , such as those they o b t a i n e d using the p o t e n t i a l of Varandas and Tennyson [VARA-81] ( E 0 = -32.7 K, and E, = -2.1 K) [PIER-85]. T h i s g i v e s D = 61.7 K and e = 0.87 A" 1. S u b s t i t u t i n g these values i n t o equation [111-10 ] gi v e s 33 s =0.38 f o r the s t i c k i n g c o e f f i c e n t at 7.5 K. Th i s i n c l u d e s a c o n t r i b u t i o n of 0.0098 due to atoms s t i c k i n g i n t o the l i g h t l y bound s t a t e . B. EFFECT OF HYDROGEN VAPOUR Because of the f i n i t e vapour pressure of the s o l i d molecular hydrogen s u r f a c e at the temperatures of the experiments, there w i l l be molecular hydrogen present i n the gas phase that has to be taken i n t o account. The low temperature l i m i t of the vapour pressure of a s o l i d can be c a l c u l a t e d from thermodynamics as [DASH-75] where m i s the mass of the molecule, and L 0 i s the l a t e n t heat of f u s i o n at T = 0. I f t h i s e x p r e s s i o n i s used to f i t the low temperature measurements of the vapour pressure of H 2 made by Benvenuti and Calder [BENV-76, BENV-71] L 0 i s found t o have the value 90.62 K. Since the hydrogen vapour i s a low d e n s i t y gas, the i d e a l gas law can be used to r e l a t e t o pressure to the d e n s i t y of molecules i n the gas phase g i v i n g P = (27rm/hM 3 / 2 ( k T ) 5 / 2 e . - L ° / k T [ 111 - 1 2 3 n = (27rmkT/h 2) 3 / 2 e " L ° / k T [ 111 -1 3 ] C o l l i s i o n s between hydrogen atoms and hydrogen molecules i n the gas w i l l p e r t u r b the the h y p e r f i n e 34 Hamiltonian of the atomic hydrogen. T h i s leads to a d e n s i t y dependent s h i f t in the frequency of the h y p e r f i n e resonance Acjfi = 2HKT\^ [ 111 - 1 4 ] The pressure s h i f t doesn't c o n t r i b u t e s i g n i f i c a n t l y to T^ 1 s i n c e i t i s the r e s u l t of a l a r g e number of "weak" c o l l i s i o n s , so the v a r i a n c e i n the d i f f e r e n t phase s h i f t s of the i n d i v i d u a l atoms w i l l be s m a l l . For a b u f f e r gas of 'He, Morrow found K = -11.8 x 10- 2" Hz m3 at 1 K, which i s 16 times s m a l l e r and of opposite sig n to the room temperature value [MORR-83a]. If the pressure s h i f t due to H 2 i s l a r g e enough, i t should be p o s s i b l e to determine K from the temperature dependence of the frequency s h i f t . C. RECOMBINATION Atomic hydrogen i s unstable, and w i l l normally recombine i n t o molecular hydrogen at the f i r s t o p p ortunity by the f o l l o w i n g r e a c t i o n H + H + X - > H 2 + X + 4 . 5 e V [III-15] where X i s some t h i r d body necessary to conserve energy and momentum. The r e a c t i o n i s q u i t e exothermic: the energy r e l e a s e d per recombination corresponds to a temperature of about 52 000 K. 35 In the experiments in t h i s work, recombination occurs i n c o l l i s i o n s between atoms i n the gas phase, and in c o l l i s i o n s i n the gas adsorbed on the s u r f a c e of the experimental c e l l . 1. BULK RECOMBINATION The gas phase c o n t a i n s both the atomic hydrogen being s t u d i e d , and hydrogen molecules due to the vapour pressure of the s o l i d hydrogen s u r f a c e . In the experiments, the d e n s i t y of atomic hydrogen n H ranged from 1 x 1 0 1 7 m~3 to 9 x 1 0 1 8 nr 3 . The d e n s i t y of H 2 molecules i n the gas i s given by equation [111-13]. Over the range of temperatures s t u d i e d , n H z ranged from 6 x 1 0 2 1 n r 3 to 2 x 1 0 2 3 i r r 3 Thus i n the gas, most c o l l i s i o n s i n v o l v i n g two hydrogen atoms w i l l i n v o l v e a hydrogen molecule as the t h i r d body, so the recombination r a t e of atoms i n the gas w i l l be given by nH = " k n H 2 n 2 H [II I - 1 6 ] where k i s the bulk recombination r a t e c o n s t a n t . Greben et al have c a l c u l a t e d the bulk recombination r a t e constant when the t h i r d body i s a helium molecule at T = 1.0 K, 0.1 K, and 0.01 K [GREB-81]. The r a t e constant i s given by k = (2it)7 Z P i | T i > f | 2 8 ( P i - p f ) 6 ( E r E f ) [ 111-17 ] 36 where the sum i s over a l l i n i t i a l s t a t e s i and f i n a l s t a t e s f where the two H atoms are bound, i s the p r o b a b i l i t y of an i n i t i a l s t a t e i , f i s the amplitude f o r s c a t t e r i n g from the i n i t i a l s t a t e i i n t o the f i n a l s t a t e f, and the p's and E's are the t o t a l momenta and energies of the i n i t i a l and f i n a l s t a t e s . The f i n a l s t a t e of the two H atoms can be l a b e l e d by i t s t o t a l e l e c t r o n i c and nuclear s p i n s S and I r e s p e c t i v e l y , and by a r o t a t i o n a l quantum number L. The H-H i n t e r a t o m i c p o t e n t i a l depends on the value of S, but not that of I. I f the S = 0, the two atoms i n t e r a c t by the s i n g l e t p o t e n t i a l , and bound s t a t e s are p o s s i b l e , but i f S = 1, then the atoms i n t e r a c t by the t r i p l e t p o t e n t i a l which i s only weakly a t t r a c t i v e , and doesn't support a bound s t a t e , so only f i n a l s t a t e s with S = 0 l e a d to recombination. Because of the P a u l i p r i n c i p l e f o r protons, L + I must be even. The allowed s t a t e s are r e f e r r e d to as ortho (I = 1, L odd) and para (I = 0, L even) hydrogen. They d i v i d e the recombination rate k i n t o c o n t r i b u t i o n s from recombination i n t o the ortho and para s t a t e s of H 2 k " F ° K p a r a + F i K o r t h o [111-18] where F 0 and F, c o n t a i n the dependence on the v a r i o u s h y p e r f i n e s t a t e s , and can be c a l c u l a t e d in terms of the Clebsch-Gordon c o e f f i c e n t s . In ze r o magnetic f i e l d , with a l l 37 four of the h y p e r f i n e s t a t e s approximately e q u a l l y populated k - d / l 6 ) K p a r a + ( 3 / l 6 ) K o r t h o [ 1 1 1 - 1 9 ] While i n a l a r g e magnetic f i e l d , such that cos 6 =• 1 (where 6 i s given by equation [ I I - 6 ] ) , and only the |a> and |b> s t a t e s are occupied k " < Kpara + K o r t h o ) c o s 2 0 / 4 [111-20] So one must be c a r e f u l comparing recombination r e s u l t s taken at d i f f e r e n t f i e l d s . 2. SURFACE RECOMBINATION The atoms adsorbed on the s u r f a c e of the c e l l w i l l a l s o c o n t r i b u t e to the t o t a l recombination. Since the s u r f a c e can act as the t h i r d body necessary to conserve energy and momentum, recombination on the s u r f a c e i s a second order process 6 H = ~ k s o 2 H [111-21] where k g i s the s u r f a c e recombinatin r a t e . I f the change i n the s u r f a c e d e n s i t y due to recombination i s slow enough, the r a t i o of the s u r f a c e and bulk d e n s i t i e s w i l l stay at the thermal e q u i l l i b r i u m v alue given by equation [111-1 ]. If t h i s i s the case, the s u r f a c e recombination can be d e s c r i b e d 38 by an e f f e c t i v e bulk recombination constant given by k e f f 88 k s ( A / V ) A 2 e 2 E B k T [111-22] In analogy with r e a c t i o n c r o s s s e c t i o n s i n three dimensions, i t i s u s e f u l to d e f i n e a c r o s s l e n g t h for recombination on the su r f a c e X = k s / v s [111-23] where v g = /TrkT/m i s the average r e l a t i v e speed of atoms adsorbed on the s u r f a c e . Any temperature dependence of X i s due to the thermal average of the i n t r i n s i c energy dependence of the recombination r e a c t i o n , and not to f a c t o r s such as s u r f a c e d e n s i t y or c o l l i s i o n r a t e . At s u f f i c e n t l y low temperatures, X should be independent of temperature, and then the the b i n d i n g energy Eg and X can be i n f e r r e d from the temperature dependence of the e f f e c t i v e s u r f a c e recombination r a t e k e f f . In s t u d i e s of H adsorbed on l i q u i d helium below 1 K Morrow [MORR-83a] found X t o be 0.14 ± 0.02 A on "He, and 0.13 A on 3He. Combining both s u r f a c e and bulk recombination e f f e c t s , the d e n s i t y of atoms i n the c e l l should decay as AH " " [ k n H 2 + k e f f ] n 2 H [111-24] The s u r f a c e and bulk c o n t r i b u t i o n s can be e x t r a c t e d by t h e i r 39 d i f f e r e n t temperature dependencies. D. SPIN EXCHANGE Spin exchange c o l l i s i o n s between hydrogen atoms are thought to be the predominant r e l a x a t i o n mechanism for the h y p e r f i n e s t a t e s . Spin exchange c o l l i s i o n s can occour both between atoms i n the gas phase, and atoms adsorbed to the H 2 s u r f a c e . B e r l i n s k y and S h i z g a l [BERL-80] extended the work of B a l l i n g et al [BALL-64] on s p i n exhange, and conclude that i t s e f f e c t on the magnetic resonance s i g n a l can be d e s c r i b e d in terms of spin exchange c r o s s s e c t i o n s . I V = n H<v>c 3 D T i 1 = n H<v>a 3 D / 2 Aa> = n H<v>X, n(p a a - p„„) / 4 [111-25] where <v> = /16kT / 7 r m i s the mean r e l a t i v e v e l o c i t y of two atoms i n the gas phase, and X3JJ are c r o s s s e c t i o n s , and P a a and p c c are d i a g o n a l elements of the d e n s i t y matrix. T h e i r c a l c u l a t e d v a l u e s f o r a 3 D are shown in f i g . 3 as a f u n c t i o n of temperature. Morrow and B e r l i n s k y [MORR-83b] extended these c a l c u l a t i o n s to s p i n exchange o c c u r i n g i n atoms adsorbed i n 40 T (K) F i g . 3. Bulk Spin Exchange Cross S e c t i o n The c a l c u l a t e d values of B e r l i n s k y and S h i z g a l [BERL-80] f o r the bulk spi n exchange c r o s s s e c t i o n as a f u n c t i o n of temperature. 4 1 a two dimensional gas phase. They f i n d T, 1 = o H < v s > a 2 D T i 1 = a H < v s > a 2 D / 2 = a H < v s > X 2 D { ^ a a " ^ c c ) / 4 [ I H - 2 6 ] where <v g> = »/7rkT/m i s the mean r e l a t i v e v e l o c i t y of two atoms on the s u r f a c e , and o 2 D and X 2 r j are r e a c t i o n c r o s s l e n g t h s . T h e i r c a l c u l a t e d values f o r a 2 D a r e shown in f i g . 4 as a f u n c t i o n of temperature. An estimate can be made of the maximum s i z e of the frequency s h i f t due to sp i n exchange by using the l a r g e s t of the c a l c u l a t e d v a l u e s f o r X 2 D and Xg D over the temperature range of i n t e r e s t (Xgp = ~ 1 ^  ^ 2 ' a n ^ X2D = Using the l a r g e s t d e n s i t y of atomic H used i n the experiments (9. x 1 0 1 8 i r r 3 ) , and a s u r f a c e d e n s i t y corresponding to the lowest temperatures used, the frequency s h i f t due to s p i n exchange expected i s only 0.02 Hz, which i s about the p r e c i s i o n of the frequency measurements. Because the e f f e c t i s so s m a l l , i t i s u n l i k e l y i t w i l l be observed. The e f f e c t of s p i n exchange on the r e l a x a t i o n of the d e n s i t y matrix should be l a r g e enough to see. The net e f f e c t of bulk and s u r f a c e s p i n exchange should be given by T, 1 = [ 0 - x ) v o 3 D + ( V / A ) x 2 v s 0 2 D ] n H [111-27] where x i s the f r a c t i o n of atoms on the s u r f a c e given by 42 0.18 0.16 -0.14 0.12 0.10 -r2D 0.08" ( X ) 0.06-0.04" 0.02-0.00 T (K) F i g . 4. S u r f a c e Spin Exchange Cross Length The c a l c u l a t e d v a l u e s of Morrow and B e r l i n s k y [MORR-83b] f o r the s u r f a c e s p i n exchange c r o s s l e n g t h as a f u n c t i o n of temperature. equation [ i l l IV. THERMOMETRY Temperature measurement, or thermometry, i s an important aspect of most c r y o g e n i c experiments, so i t w i l l be worthwhile to d i s c u s s some gen e r a l f a c t s and d e f i n i t i o n s b efore l o o k i n g at the thermometry i n t h i s work i n d e t a i l . A. INTRODUCTION A thermometer i s j u s t a p h y s i c a l system with some property which v a r i e s as a f u n c t i o n of thermodynamic temperature. The r e l a t i o n s h i p between t h i s p r o p e r t y and temperature i s the thermometric f u n c t i o n f o r the thermometer. When t h i s f u n c t i o n i s known, the thermometer i s s a i d to be c a l i b r a t e d . Thermometers can be c l a s s i f i e d by how the thermometric f u n c t i o n i s known. Primary thermometers are those f o r which the thermometric f u n c t i o n can be d e r i v e d from some thermodynamic or s t a t i s t i c a l - m e c h a n i c a l t h e o r y . A l l temperature measurements must u l t i m a t e l y be based on a primary thermometer, e i t h e r d i r e c t l y , or i n d i r e c t l y by using some means of c a l i b r a t i o n . Some common primary thermometers a r e : the gas thermometer (using an equation of s t a t e , u s u a l l y i n the low pressure l i m i t ) , the Johnson-noise thermometer, and the n u c l e a r o r i e n t a t i o n thermometer. Most primary thermometers are d i f f i c u l t to b u i l d and use, so they are mainly found i n n a t i o n a l standards l a b o r a t o r i e s . Secondary standard thermometers are those f o r which the thermometric f u n c t i o n depends only on some w e l l d e f i n e d and 44 45 r e p r o d u c i b l e aspect of the thermometer, u s u a l l y some property of a pure substance. The thermometric f u n c t i o n need only be determined once f o r a p a r t i c u l a r system, by comparison with another thermometer. A l l thermometers based on the same p h y s i c a l system are then c a l i b r a t e d . The most common type of secondary standard thermometer i s the pressure-temperature r e l a t i o n of a phase c o e x i s t a n c e l i n e such as the s a t u r a t e d vapour p r e s s u r e of H 2, "He, or 3He, or the m e l t i n g curve of 3He. Another common type i s the thermal emf of v a r i o u s thermocouple j u n c t i o n s . Most secondary standards are f a i r l y easy to b u i l d and use, but they o f t e n s u f f e r from a f a i r l y l i m i t e d u s e f u l range. Secondary thermometers are those which must be i n d i v i d u a l l y c a l i b r a t e d . They are u s u a l l y simple to use, once they have been c a l i b r a t e d , s i n c e the thermometric pr o p e r t y i s u s u a l l y one that i s easy to measure, such as e l e c t r i c a l r e s i s t a n c e . Some common examples of secondary thermometers a r e : carbon r e s i s t o r s , germanium r e s i s t o r s , platinum r e s i s t o r s , the forward v o l t a g e drop of a s i l i c o n diode, and the magnetic s u s c e p t i b i l i t y of a paramagnetic s a l t . In many cases the thermometric f u n c t i o n i s approximated by some simple power s e r i e s with a l a r g e number of a d j u s t a b l e parameters. However, there are some systems where the mathematical form of t h i s r e l a t i o n can be determined from theory, l e a v i n g j u s t a few parameters to be determined by c a l i b r a t i n g the thermometer. These types of thermometers are very u s e f u l f o r i n t e r p o l a t i n g between 46 widely spaced c a l i b r a t i o n p o i n t s . The best example of t h i s i s the magnetic s u s c e p t i b i l i t y of a paramagnetic s a l t . Because primary thermometers are r a r e and d i f f c u l t to use, few thermometers are c a l i b r a t e d by d i r e c t comparison with a primary thermometer. Instead, a temperature s c a l e i s used. A temperature s c a l e u s u a l l y c o n s i s t s of a s e r i e s of f i x e d p o i n t s , and some type of secondary thermometer f o r i n t e r p o l a t i n g between them. The f i x e d p o i n t s are u s u a l l y phase t r a n s i s t i o n s of pure substances, which occur at r e p r o d u c i b l e temperatures. Temperature s c a l e s p r o v i d e a s o r t of t r a n s f e r standard between the primary thermometers used to e s t a b l i s h them and other thermometers c a l i b r a t e d a c c o r d i n g t o the s c a l e . The two most common temperature s c a l e s i n use today are the I n t e r n a t i o n a l P r a c t i c a l Temperature Scale of 1968 (IPTS-68) [BIPM-76], and the 1976 P r o v i s i o n a l 0.5 K to 30 K Temperature S c a l e (EPT-76) [BIPM-79]. The IPTS-68 s c a l e i s d e f i n e d f o r temperatures above 13.81 K. I t i s d i v i d e d i n t o four ranges, each with a d i f f e r e n t method of i n t e r p o l a t i n g . The lowest two ranges use the platinum r e s i s t a n c e thermometer as an i n t e r p o l a t i n g d e v i c e , up t o 903.89 K. The platinum 10% rhodiuim/platinum thermocouple i s used f o r the range 903.89 K to 1337.58 K. Above 1337.58 K the Planck r a d i a t i o n law f o r the thermal spectrum of a black body i s used. Although the IPTS-68 s c a l e i s widely used, i t i s not very u s e f u l a t cryogenic temperatures f o r s e v e r a l reasons: i t onl y extends down to 47 13.81 K (the t r i p l e p o i n t of hydrogen); i t d e v i a t e s s i g n i f i c a n t l y from true thermodynamic temperature below 27 K [DURX-79]; the platinum r e s i s t a n c e thermometer has a low s e n s i t i v i t y (dR/RdT) at cry o g e n i c temperaures; and the c a l i b r a t i o n procedure f o r the cryogenic range i s q u i t e c o m p l i c a t e d . The EPT-76 s c a l e was developed to s i m p l i f y cryogenic thermometry. The main concern was the d e v i a t i o n s from thermodynamic temperature of the v a r i o u s s c a l e s i n use then. There are s e v e r a l accepted ways of r e a l i s i n g EPT-76: one can use a thermodynamic i n t e r p o l a t i n g instrument, such as a gas thermometer or a magnetic thermometer, c a l i b r a t e d at one or more of the re f e r e n c e p o i n t s ( t a b l e I I I ) ; one can use e a r l i e r temperature s c a l e s , such as the 1958 "He s c a l e , the 1962 3He s c a l e , IPTS-68, or v a r i o u s n a t i o n a l standards l a b o r i t o r y s c a l e s , together with a t a b l e of d e v i a t i o n s [BIPM-79]; and one can use a n a l y t i c a l e xpressions f o r the vapour p r e s s u r e s of 3He and *He which are now part of EPT-76[DURX-83, RSBY-85]. 1. VAPOUR PRESSURE THERMOMETRY Because the s a t u r a t e d vapour pressure of a f l u i d v a r i e s r a p i d l y with temperature (roughly f o l l o w i n g an Arrhenius law), i t i s a f a i r l y s e n s i t i v e thermometer, which doesn't r e q u i r e complicated or expensive apparatus. Since i t i s a secondary standard thermometer, no thermometer c a l i b r a t i o n i s r e q u i r e d . I t s only r e a l disadvantadge i s i t s rather 48 Table I I I . EPT-76 Reference P o i n t s T 7 6 (K) : Reference P o i n t : 27. 1 02 Normal B o i l i n g P oint of Ne 24. 5591 T r i p l e Point of Ne 20. 2734 Normal B o i l i n g Point of e~H 2 17. 0373 B o i l i n g Point of e-H 2 (250 t o r r ) 13. 8044 T r i p l e Point of e-H 2 7. 1999 Superconducting T r a n s i t i o n of Pb 4. 2221 Normal B o i l i n g Point of "He 3. 41 45 Superconducting T r a n s i t i o n of In 1 . 1 796 Superconducting T r a n s i t i o n of A l 0. 851 Superconducting T r a n s i t i o n of Zn 0. 519 Superconducting T r a n s i t i o n of Cd l i m i t e d temperature range (about 1 K to 5 K f o r "He and 13.81 K to 22 K f o r H 2 ) . There are a few problems to be wary of however. A very common p r a c t i c e i s to immerse an apparatus i n l i q u i d helium and.measure the vapour pressure over t h i s bath to determine the temperature of the apparatus. T h i s method i s not r e l i a b l e , as the bath i s r a r e l y i n thermal e q u i l i b r i u m , i n s t e a d having s i z a b l e thermal g r a d i e n t s w i t h i n i t . Even i f the bath were i n e q u i l i b r i u m , there w i l l be a thermal g r a d i e n t due to the h y d r o s t a t i c p r essure of the l i q u i d above the appparatus. To a v o i d t h i s problem a small c e l l , c o n t a i n i n g the a p p r o p r i a t e l i q u i d , can be a t t a c h e d to the apparatus and used f o r vapour pressure measurements. If 49 the volume of t h i s c e l l i s small (a few cm 3), and made of a m a t e r i a l with a high thermal c o n d u c t i v i t y , thermal g r a d i e n t e f f e c t s w i l l be n e g l i g i b l e . The other major problem i s to ensure that the pressure being measured r e a l l y i s the same as the pressure of the l i q u i d - v a p o u r i n t e r f a c e . Pressure g r a d i e n t s along the measuring tube, due to the thermo-molecular e f f e c t , can be e l i m i n a t e d by using a tube with a diameter l a r g e compared with the mean f r e e path of the gas molecules [WHTE-79]. T h i s i s not a problem, as there are no dead volume e f f e c t s i n vapour p r e s s u r e thermometry to make very small diameter tubes d e s i r a b l e . I t i s very important to ensure that the pressure sensing tube i s never c o l d e r than the vapour pressure c e l l , as the measured pressure w i l l correspond to the c o l d e s t p o i n t i n the system. T h i s i s g e n e r a l l y achieved by i n s u l a t i n g the pressure sensing tube with a vacuum j a c k e t u n t i l at a temperature w e l l above that of the c e l l , and p o s s i b l y u s i n g a heater on the tube. Any l a r g e i n f l u x e s of heat down the pressure sensing tube must be avoided, otherwise there may be thermal g r a d i e n t s between the l i q u i d and the apparatus. The heat leak due to d i r e c t thermal conduction can be minimized by using a t h i n w a l l e d tube made from s t a i n l e s s s t e e l or c u p r o - n i c k e l , and heat s i n k i n g i t to some temperature w e l l below room temperature, but above that of the vapour pressure c e l l . The heat leak due to blackbody r a d i a t i o n down the tube can be avoided by p l a c i n g b a f f l e s at one or more 50 i n t e r m e d i a t e p o i n t s i n the p r e s s u r e sensing tube. Since the r a d i a t e d power goes as T \ a b a f f l e at 77 K reduces t h i s heat leak by a f a c t o r of about 200. When us i n g "He below the lambda t r a n s i t i o n there can be s e r i o u s problems with s u p e r f l u i d f i l m flow. Vapour pressure thermometry at these temperatures i s best done with 3He, which avoids t h i s problem, and i s more s e n s i t i v e . Hydrogen p r e s e n t s an a d d i t i o n a l c o m p l i c a t i o n due to the e x i s t a n c e of the ortho and para s t a t e s . At room temperature, H 2 c o n s i s t s of 75% ortho and 25% para hydrogen, while at 20 K i t i s n e a r l y pure para hydrogen. Because the ortho-para c o n v e r s i o n r a t e i s very slow, the ortho-para r a t i o i s o f t e n not at i t s t r u e thermal e q u i l i b r i u m value. Since the vapour pressure curve of hydrogen depends on the ortho-para composition, i t i s important to know the composition of the sample. There are two common approaches to t h i s problem. In the f i r s t , room temperature composition or normal hydrogen (n-H 2) i s used, and the apparatus i s kept f r e e of paramagnetic i m p u r i t i e s , so the ortho-para c o n v e r s i o n r a t e i s slow. If the measurements are done f a i r l y q u i c k l y , i t i s p o s s i b l e to f i n i s h them without the ortho-para r a t i o changing a p p r e c i a b l y . In the second approach, hydrogen with i t s thermal e q u i l i b r i u m ortho-para r a t i o (e-H 2) i s used. A paramagnetic c a t y l i s t i s d e l i b e r a t e l y i n t r o d u c e d to promote r a p i d ortho-para c o n v e r s i o n . The second method i s considered more r e l i a b l e , as i t i s much simpl e r to i n s u r e the presence of a paramagnetic impurity, than to insure i t s absence. 51 2. MAGNETIC THERMOMETRY The magnetic s u s c e p t i b i l i t y of a paramagnetic s a l t i s given by: [CTAS-76] X = Nuc-a + N M o g 2 M B 2 J ( J+1 ) [IV-1] 3k(T + A + 6/T + . . . ) where N i s the number of paramagnetic i o n s , a i s the temperature independent s u s c e p t i b i l i t y , g i s the Lande s p l i t t i n g factor," u% i s the Bohr magneton and, J i s the t o t a l angular momentum quantum number. The parameters A and 6 are due to v a r i o u s higher order i n t e r a c t i o n s , and' are u s u a l l y s m a l l . T h i s form f o r the s u s c e p t i b i l i t y i s v a l i d p rovided that k gT i s l a r g e compared t o the s p l i t t i n g of the angular momentum s u b s t a t e s , but smal l enough that no other s t a t e s are s i g n i f i c a n t l y populated. There are three common methods used t o measure the s u s c e p t i b i l i t y of the s a l t . The most common i s to use an audio frequency (aprox. 20 - 200 Hz) A.C. inductance b r i d g e . The s a l t sample i s p l a c e d i n one of a p a i r of a s t a t i c a l l y wound pickup c o i l s , and the brid g e i s used to measure the mutual inductance between these and a d r i v e c o i l surrounding them [DURX-62, CTAS-72, CTAS-76]. The s a l t may be pl a c e d i n s i d e a c o i l which i s p a r t of an LC resonant c i r c u i t ( f 0 - 1 MHz). The f r a c t i o n a l s h i f t i n the resonant frequency i s then p r o p o r t i o n a l t o the s u s c e p t i b i l t y [BTTS-64, HRLY-70]. An i n c r e a s i n g l y popular method i s to use a squid 52 magnetometer to measure the magnetisation of the s a l t i n a s t a t i c magnetic f i e l d ( o f t e n provided by t r a p p i n g f l u x i n a superconducting tube) [LOUN-74]. Although i n p r i n c i p l e the s u s c e p t i b i l i t y temperature r e l a t i o n c o u l d be used as a secondary standard thermometer, in p r a c t i c e i t i s only used as a secondary thermometer. This i s because determining the absolute s u s c e p t i b i l i t y r e q u i r e s knowledge of f i l l i n g f a c t o r s , and demagnetising f a c t o r s that are d i f f i c u l t to determine with the necessary p r e c i s i o n . A d d i t i o n a l l y , most of the s a l t s used have l a r g e amounts of water of h y d r a t i o n , and are not very r e p r o d u c i b l e as they gain or l o s e water depending on how they are s t o r e d . Because of t h i s , the thermometric r e l a t i o n i s determined i m p l i c i t l y by c a l i b r a t i n g the magnetic thermometer a g a i n s t some other thermometer, or against the f i x e d p o i n t s of some temperature s c a l e . The data i s u s u a l l y f i t to the equation X = A + B/(T + A +• 6/T) [IV-2] where X i s the bridge r e a d i n g i f using a mutual inductance b r i d g e , the f r a c t i o n a l frequency s h i f t i f using the LC resonance method, or the magnetisation i f using a squid magnetometer. The parameters A, B, A, and 6 are determined by the c a l i b r a t i o n . At higher temperatures, i t may be p o s s i b l e to ignore 5 and even A, depending on the s a l t and accuracy d e s i r e d . Because the number of parameters i n the f i t i s small (as few as two), the magnetic thermometer i s a 53 u s e f u l device f o r i n t e r p o l a t i n g between c a l i b r a t i o n p o i n t s . The choice of s a l t depends on the temperature range of i n t e r e s t . The important parameter i s the c o n c e n t r a t i o n of paramagnetic i o n s . A higher c o n c e n t r a t i o n g i v e s a l a r g e r s u s c e p t i b i i t y , which i s e a s i e r t o measure, but the parameters A and 6 w i l l a l s o be l a r g e r , so more c a l i b r a t i o n p o i n t s w i l l be needed. For temperatures down to a few mK, cerium magnesium n i t r a t e , or CMN (2Ce(N0 3) 3•3Mg(N0 3) 2•24H 20) i s the p r e f e r r e d c h o i c e . Above 1 K, the e f f e c t s of A and 6 are completely n e g l i g i b l e . For temperatures above 5 K, manganous ammonium sulphate, or MAS (Mn(NH f t) 2(S0„) 2•6H 20), i s a b e t t e r c h o i c e , s i n c e the s u s c e p t i b i l i t y of CMN i s very small at t h i s temperature. Gadolinium sulphate, or GDS (Gd 2 (S0„) 3 'BH20), because of i t s l a r g e s u s c e p t i b i l i t y has been used up to 83 K [CTAS-76]. Another p o s s i b i l i t y i s s y n t h e t i c ruby, which may be manufactured doped with v a r i o u s c o n c e n t r a t i o n s of chromiuim (although c r y s t a l s with more than 1% chromium tend to crack with thermal c y c l i n g ) [DAUN-58, DAUN-61]. S y n t h e t i c ruby i s c h e m i c a l l y s t a b l e , so u n l i k e paramagnetic s a l t s , i t i s not necessary to worry about changes i n i t s composition with time. And u n l i k e most of the s a l t s , i t i s easy to p r o v i d e a good thermal l i n k between the ruby and the apparatus (Daunt r e p o r t s that the ruby c r y s t a l s can be s u c e s s f u l l y s o l d e r e d to the apparatus [DAUN-58]). 54 3. RESISTANCE THERMOMETRY Germanium and carbon r e s i s t a n c e thermometers are by f a r the most common types of thermometer found i n cr y o g e n i c experiments. T h i s i s because they are simple to use, compact, and f a i r l y cheap. Germanium r e s i s t a n c e thermometers (GRT's) c o n s i s t of a small bar or "bridge" of a r s e n i c or phosphorus doped germanium. Ohmic c o n t a c t s are made to allow a four wire measurement of r e s i s t a n c e ( t h i s prevents lead r e s i s t a n c e s from i n t e r f e r i n g with the measurement). T h i s assembly i s then mounted i n a s t r a i n f r e e holder and encapsulated. The case i s u s u a l l y b a c k f i l l e d with helium to pro v i d e good thermal c o n t a c t with the germanium, although the leads are the major thermal c o n t a c t below a few degrees k e l v i n . The doping l e v e l can be adjus t e d to allow f o r maximum s e n s i t i v i t y (dR/RdT) at d i f f e r e n t temperatures. The d e t a i l s of the R-T r e l a t i o n are not w e l l understood however, so only e m p i r i c a l r e l a t i o n s are a v a i l i b l e to f i t i t . Most of these are chosen p u r e l y f o r ease of computation. The most popular i s to f i t l n ( T ) or 1/T as a polynomial in l n ( R ) , with as many as fourteen terms. The c h i e f advantage of GRT's i s the s t a b i l i t y of t h e i r R-T r e l a t i o n : they have been found to hold t h e i r c a l i b r a t i o n s to w i t h i n l e s s than 1 mK a f t e r repeated c y c l i n g between room temperature and 4K. There are some r e p o r t s of GRT's making i n f r e q u e n t l a r g e jumps i n t h e i r R-T r e a l t i o n , but no evidence of s i g n i f i c a n t d r i f t or aging e f f e c t s . T h i s 55 makes GRT's a very u s e f u l secondary thermometer as they don't need frequent r e c a l i b r a t i o n . Groups of GRT's are now used by most n a t i o n a l standards la b s to maintain t h e i r c r y o g e n i c temperature s c a l e s . Any of the GRT's s u f f e r i n g a c a l i b r a t i o n jump are then r e c a l i b r a t e d a g a i n s t the others [RIJN-72]. The most common form of the carbon r e s i s i t a n c e thermometer (CRT) i s j u s t a carbon composition r e s i s t o r . Besides being cheap, they have the advantage of a very smooth R-T r e l a t i o n , so they don't r e q u i r e an i n o r d i n a n t number of c a l i b r a t i o n p o i n t s . T h e i r c h i e f disadvantage i s t h a t t h e i r c a l i b r a t i o n s h i f t s by s e v e r a l percent on thermal c y c l i n g between room temperature and 4 K. If CRT's are used f o r a c c urate thermometry, they must be r e c a l i b r a t e d each time they are c y c l e d from room temperature. They are u s e f u l when t h e i r f a s t response i s needed, and the standard thermometer used to c a l i b r a t e them i s slow. They can a l s o be u s e f u l without r e c a l i b r a t i n g them in situ each time, p r o v i d e d the temperature need only be known to w i t h i n a few percent. For use above 1 K, r e s i s t o r s manufactured by A l l e n - B r a d l e y or Ohmite are b e s t . Below 1 K t h e i r r e s i s t a n c e becomes i n c o n v e n i e n t l y l a r g e , and those manufactured by Speer or M a t s u s h i t a are p r e f e r r e d [LOUN-74, SAIT-75]. 56 B. CALIBRATION APPARATUS The only type of thermometers s u i t a b l e f o r the magnetic resonance apparatus were e i t h e r GRTs or CRTs, due to l i m i t e d space. Because these thermometers would need c a l i b r a t i n g , i t was decided that some so r t of apparatus should be b u i l t to do t h i s as i t should a l s o prove u s e f u l f o r other experiments. The apparatus was designed so that i t would be p o s s i b l e to r e a l i s e the EPT-76 s c a l e . A magnetic thermometer was chosen as the i n t e r p o l a t i n g d e v i c e , s i n c e gas thermometers are quite.complex at these low temperatures. The r e f e r e n c e temperatures used to c a l i b r a t e the magnetic thermometer were p r o v i d e d by a vapour p r e s s u r e c e l l , which c o u l d be used with e i t h e r hydrogen or helium. 1. THE CRYOSTAT The c a l i b r a t i o n apparatus i s shown i n f i g . 5. The v a r i o u s thermometers are mounted on a copper block, which a l s o c o n t a i n s the vapour pressure c e l l . The copper block i s surrounded by a heat s h i e l d , a l s o made of copper. The block i s suspended from a f l a n g e , kept at 4 K by the helium bath, by three legs which are p a r t of a t h i n w a l l s t a i n l e s s s t e e l tube. The e n t i r e assembly i s then surrounded by a vacuum can, and immersed i n l i q u i d helium. A l l the leads coming i n t o the apparatus are #32 gauge brass wire and are heat sunk to the helium bath, and then to the copper block, to prevent heat l e a k s . The leads are a l l t w i s t e d p a i r s to reduce n o i s e pickup and c r o s s t a l k . The copper block has a 57 THIN W A L L STAINLESS S T E E L TUBING C A P I L L A R Y H E A T E R INDIUM "0" RING 4 K H E A T SINK FOR L E A D S VACUUM C A N WEAK T H E R M A L LINK HEAT SINK FOR L E A D S C R T OR GRT B L O C K HEATER R E G U L A T I N G C R T T H E R M O M E T E R B L O C K VAPOUR P R E S S U R E C E L L HEAT SINK FOR L E A D S Fe(0H) 3 CATALYST INDIUM "O" RING C O P P E R WIRES THERMAL SHIELD SALT S A M P L E PRIMARY C O I L S E C O N D A R Y C O I L S C O P P E R ^ B R A S S PHENOLIC "T 1 1 -2 3 4 cm 1 5 F i g . 5. C a l i b r a t i o n Apparatus The p a r t s of the apparatus i n the l i q u i d helium bath are shown i n c r o s s s e c t i o n . 58 narrow neck and a l l heat leaks i n t o and out of the block come above or i n t o t h i s neck, so there should be no s i g n i f i c a n t thermal g r a d i e n t s below i t . The copper block has s e v e r a l threaded holes provided f o r a t t a c h i n g the v a r i o u s thermometers, to provide good thermal c o n t a c t . The temperature of the block i s c o n t r o l e d by a b i f i l a r wound heater made of manganin wire, wrapped around the neck of the bloc k . A CRT attached adjacent to t h i s heater i s used as the sensor f o r the temperature r e g u l a t o r . Keeping the heater c l o s e to the r e g u l a t i n g thermometer keeps the r e g u l a t i o n time constant s h o r t . A f t e r the f i r s t runs i t was found necessary to add a length of copper wire (5 cm of #16 gauge) to i n c r e a s e the heak leak out of the block at a l l temperatures of i n t e r e s t , so that the temperature c o u l d be r e g u l a t e d by h e a t i n g . 2. THE VAPOUR PRESSURE THERMOMETER The vapour pressure c e l l has a volume of 4.9 cm 3. The bottom of the c e l l i s removable, and i s sea l e d with an indium "0" r i n g . T h i s makes i t p o s s i b l e to add about 1 g of f e r r i c hydroxide (Fe(0H) 3) to c a t a l y s e ortho-para conversion when using H 2 i n the vapour p r e s s u r e c e l l . The pressure sensing tube i s 0.125" diameter t h i n w a l l s t a i n l e s s t u b i n g . The pressure sensing tube i s heat sunk to a p o i n t i n the c r y o s t a t kept at roughly 77 K by the outer l i q u i d n i t r o g e n dewar. A b a f f l e i s a l s o p r o v i d e d here to block any room temperature black body thermal r a d i a t i o n from propagating 59 down the tube. Below t h i s p o i n t , the tube i s t h e r m a l l y i s o l a t e d , as i t i s i n the vacuum space. A b i f i l a r wound heater i s attached to the c a p i l l a r y near the p o i n t where i t enters the vacuum can. T h i s i n s u r e s that the c a p i l l a r y has no spots c o l d e r than the vapour p r e s s u r e c e l l . A s m a l l brass c o u p l i n g (the lower j o i n t hard s o l d e r e d and the upper s o f t soldered) allows removal of the copper block without having to remove the e n t i r e sense tube. T h i s c o u p l i n g a l s o p r o v i d e s an a d d i t i o n a l b a f f l e a g a i n s t black body r a d i a t i o n . The gas handling system used with the vapour pressure thermometer i s shown i n f i g u r e 6. The 2 / r e f e r e n c e volume i s used to allow a known amount of gas to condense i n t o the c e l l . The pressure i n the c e l l was measured with a c a p a c i t a n c e manometer 1. T h i s gauge allowed p r e s s u r e readings to a p r e c i s i o n of about 0.5 t o r r . The pressure sense l i n e had a pressure r e l i e f v a l v e to prevent e x c e s s i v e p r e s s u r e i n case the c e l l unexpectedly warmed up. 3. THE MAGNETIC THERMOMETER A magnetic thermometer using a mutual inductance bridge was designed. The c o n s t r u c t i o n of the c o i l s and sample holder are shown i n f i g . 5. The i n n e r , secondary, c o i l i s a p a i r of a s t a t i c a l l y wound c o i l s , each c o n s i s t i n g of 4500 turns of #38 gauge copper wire, vacuum p o t t e d i n epoxy 2. The primary c o i l c o n s i s t s of 922 turns of #28 gauge copper wire, 1 MKS Baratron model 310BHS1000 0-1000 t o r r sensor, and model 170M-27B readout 2 Emerson and Cummings S t y c a s t LN78058 60 VAPOUR PRESS. CELL VACUUM SPACE F i 9 » 6. Gas H a n d l i n g System T h i s system was used t o i n t r o d u c e hydrogen or h e l i u m i n t o the vapour p r e s s u r e c e l l . 61 a l s o vacuum p o t t e d i n epoxy. The exact number of t u r n s , and p r e c i s e spacing of the f i n a l t u r n s , was a d j u s t e d to make the mutual inductance as c l o s e to zero as p o s s i b l e before p o t t i n g . The c o i l former i s made out of p h e n o l i c , a epoxy and paper laminate, with a small c o e f f i c e n t of thermal c o n t r a c t i o n (a = 1.5 x 10" 5 K " 1 ) . The sample holder c o n s i s t e d of a p h e n o l i c tube, that j u s t f i t s i n s i d e the c o i l s . T h i s tube was d i v i d e d i n t o two s e c t i o n s , corresponding to the two secondary c o i l s , by p h e n o l i c s p a c e r s . One of these s e c t i o n s h e l d the paramagnetic s a l t sample, while the other served as a r e f e r e n c e . The s a l t sample was made i n t o a s l u r r y with s i l i c o n e g r e a s e 3 , and an equal amount of grease was added to the r e f e r e n c e sample space. To provide thermal contact between the s a l t s l u r r y and the copper block , 40 #32 gauge copper wires pass through the sample holder i n a r e g u l a r l y spaced a r r a y , and are connected at one end to the copper b l o c k . Thin wires must be used to a v o i d e x c e s s i v e eddy c u r r e n t h e a t i n g i n the A.C. f i e l d of the primary c o i l . Because the s t r u c t u r e i s symmetric, any magnetic e f f e c t s of the assembly should c a n c e l out between the two s i d e s , except f o r those of the s a l t sample, which i s present only on one s i d e . The block diagram of the mutual inductance b r i d g e " used to measure the s u s c e p t i b i l i t y of the s a l t i s shown i n f i g . 7. The three r a t i o t r a n s f o r m e r s are a d j u s t e d to n u l l the net 3 Dow Corning High Vacuum Grease 4 S.H.E. model BPD A.C. impedance bridge RANGE IN-PHASE QUADRATURE M # paramagnetic salt ^ a reference * inductor r«f. In two-phase detector in-phase quad. S.H.E. mod. BPD F i g . 7. Mutual Inductance Bridge The A.C. b r i d g e used with the magnetic thermometer. The HR-8 l o c k - i n i s only used as a tuned a m p l i f i e r . 6 3 induced v o l t a g e a c r o s s the secondaries of the two mutual inductances. The r a t i o s of the in-phase, quadrature, and range transformers are a, B, and y r e s p e c t i v e l y . I f the o s c i l l a t o r generates an rms s i g n a l of V 0 , then i f R >> the primary r e s i s t a n c e of the two i n d u c t o r s , the c u r r e n t i s V 0 / R and the rms v o l t a g e a c r o s s the secondary of M i s V x = C J M X V 0 / R [ I V - 3 ] S i m i l a r l y , the rms v o l t a g e across the r e f e r e n c e inductor i s V r e f = cjM0a7Vo/R [IV-4] The v o l t a g e w i l l be n u l l e d when M x = ayM0 [IV-5] s i n c e the two secondaries are connected out of phase. The re f e r e n c e i nductor M 0 c o n s i s t s of two c o a x i a l s o l e n o i d s , wound on a p h e n o l i c former, and has an inductance of 2 3 3 jzH. The t h i r d r a t i o transformer was used to n u l l the quadrature p a r t of the s i g n a l . A phase s e n s i t i v e n u l l d e t e c t o r was used. The s i g n a l was f i r s t a m p l i f i e d 5 and then fed i n t o a two phase l o c k - i n d e t e c t o r 6 . T h i s made i t p o s s i b l e to balance the in-phase and 5 using a P.A.R. type B p r e a m p l i f i e r , and the tuned a m p l i f i e r s e c t i o n of a P.A.R. HR-8 l o c k - i n a m p l i f i e r 6 S.H.E. model BPD two phase l o c k - i n 64 quradrature components s e p e r a t e l y , and provided very good s e n s i t i v i t y . The paramagnetic s a l t o r i g i n a l l y chosen f o r the thermometer was MAS. A commercial source of MAS couldn't be found, so the procedure given i n [CTAS-76] was used to produce some. Manganese sulphate (MnSO„-H 20) and ammonium sulphate ((NH«) 2S0 f l) were mixed in a s o l u t i o n i n the c o r r e c t s t o i c h i o m e t r i c r a t i o , and t h i s was allowed to c r y s t a l l i s e . The r e s u l t i n g c r y s t a l s were then r e d i s s o l v e d i n d i s t i l l e d water and r e c r y s t a l l i s e d . T h i s r e c r y s t a l l i s a t i o n procedure was repeated a t o t a l of three times. In theory, any i m p u r i t i e s or s l i g h t excesses of one of the s a l t s should been l e f t i n the supernatant. P r e l i m i n a r y runs however, suggested that the s a l t sample had some s o r t of i m p u r i t y . A p o s s i b l e cause of t h i s i s that manganese sulphate forms c r y s t a l s with v a r i o u s numbers of waters of h y d r a t i o n . I f the h y d r a t i o n s t a t e of the manganese sulphate was s i g n i f i c a n t l y d i f f e r e n t from what was assumed, there would be a l a r g e excess of e i t h e r manganese or ammonium i n the o r i g i n a l s o l u t i o n , and good c r y s t a l s of MAS would be very hard to form. There may have been some other type of i m p u r i t y a l s o . Instead of t r y i n g to f i g u r e out what the impurity problem was, a switch was made to u s i n g gadolinium sulphate as the s a l t . GDS i s l e s s s u s c e p t i b l e to impurity problems as i t i s a simple s a l t ( r a t h e r than a double s a l t ) , and only forms i n one h y d r a t i o n s t a t e . I t was p o s s i b l e to purchase 65 99.999 % pure GDS c o m m e r c i a l l y 7 . The sample that was used c o n s i s t e d of 1.4 g (1.9 mmol) of GDS i n a s l u r r y with s i l i c o n e grease. 4. THE RESISTANCE THERMOMETERS There were three r e s i s t a n c e thermometers used i n t h i s experiment. The f i r s t was the GRT 8 intended as the c e l l thermometer f o r the magnetic resonance experiments. I t was e f f e c t i v e l y u n c a l i b r a t e d : i n another experiment i t had been i n a d v e r t a n t l y heated, changing the c a l i b r a t i o n from that s u p p l i e d with the d e v i c e . The GRT was mounted i n a copper s l e e v e , with s i l i c o n grease to improve thermal c o n t a c t . The leads were thermally grounded to the ou t s i d e of the copper s l e e v e . The remaining thermometers were CRT's 9 encapsulated i n a copper sleeve with epoxy, and leads of #32 brass wire t h e r m a l l y grounded to the o u t s i d e of the s l e e v e . One of the CRT's was used as the sensor f o r temperature c o n t r o l l i n g the thermometer block i n the c a l i b r a t i o n apparatus, while the second was c a l i b r a t e d f o r use i n the magnetic resonance apparatus. The r e s i s t a n c e s of the thermometers was measured with an automatic r e s i s t a n c e b r i d g e 1 0 , which uses A.C. d e t e c t i o n . The e x c i t a t i o n v o l t a g e a c r o s s the thermometers was 1 mV, g i v i n g a power d i s s i p a t i o n i n the thermometers of l e s s than 7 A l d r i c h Chemical Co. 8 S o l i t r o n Devices Inc. model SP1403 9 A l l e n - B r a d l e y 1/4 W 150 R nominal r e s i s t o r s 1 0 RV E l e k t r o n i i k k a Oy. model AVS-45 66 7 nW. No s e l f h e a t i n g of the thermometers was observed at t h i s power l e v e l . The r e g u l a t i n g CRT was measured with a b r i d g e which i s part of a homemade temperature c o n t r o l l e r . T h i s c o n t r o l l e r p r o v i d e s heater power depending on the b r i d g e e r r o r s i g n a l , with terms p r o p o r t i o n a l to the e r r o r s i g n a l , and a l s o i t s time i n t e g r a l and d e r i v a t i v e . C. CALIBRATION RESULTS The procedure f o r a c a l i b r a t i o n run was as f o l l o w s . P r i o r to c o o l i n g down, the vapour pressure c e l l and c a p a c i t a n c e manometer were evacuated and f l u s h e d s e v e r a l times with hydrogen. A f t e r the l a s t e v acuation, the zero of the c a p a c i t a n c e manometer was s e t . The vacuum space was then evacuated and b a c k f i l l e d with about 1 t o r r of helium, to p r o v i d e good thermal contact between the thermometer block and the l i q u i d helium during the cooldown. L i q u i d n i t r o g e n was then t r a n s f e r r e d i n t o the outer dewar, and the apparatus allowed to c o o l to about 77 K. The r e f e r e n c e volume of the gas h a n d l i n g system was then f i l l e d with an amount of hydrogen s u f f i c e n t to h a l f f i l l the vapour p r e s s u r e c e l l when condensed, and l e f t connected to the vapour pressure c e l l . L i q u i d helium was then t r a n s f e r r e d i n t o the inner dewar. During the t r a n s f e r , the hydrogen condensed i n t o the vapour pressure c e l l and then f r o z e t h e r e . A f t e r the helium t r a n s f e r was completed, the vapour p r e s s u r e c e l l was pumped out with a d i f f u s i o n pump to remove 67 any helium that might be pres e n t . The vapour pressure c e l l was then sealed o f f from the gas handling system. The d i f f u s i o n pump was used to evacuate the vacuum space f o r about 4 hours, with the block kept at about 20 K f o r the l a s t h a l f of t h i s p e r i o d . T h i s l e f t the block t h e r m a l l y i s o l a t e d , except f o r the weak thermal l i n k to the bath, and the pressure sensing c a p i l l a r y . C a l i b r a t i o n data was taken by r e g u l a t i n g the thermometer block at some temperature f o r about 10 minutes, a l l o w i n g a l l the thermometers to come i n t o e q u i l i b r i u m . Because the magnetic thermometer was the slowest to come i n t o e q u i l i b r i u m , the inductance bridge s i g n a l was monitored to determine when i t was no longer changing. Readings were taken f o r each thermometer, and a new temperature chosen. A d i f f e r e n t procedure was used to get the thermometer readings at the hydrogen t r i p l e p o i n t . The thermometer block was r e g u l a t e d at a temperature j u s t above the t r i p l e p o i n t . Once the e q u i l i b r i u m heater power had been determined, i t was hel d constant at a s l i g h t l y reduced v a l u e . T h i s caused the thermometer block to slowly c o o l ( t a k i n g about 5 minutes to go through the t r i p l e p o i n t ) . The t r i p l e p o i n t c o u l d be seen by monitoring the block temperature as a f u n c t i o n of time. As the hydrogen s t a r t e d to s o l i d i f y , the l a t e n t heat of f u s i o n p r o v i d e d enough h e a t i n g to keep the temperature of the thermometer block c o n s t a n t . While the thermometer block temperature was const a n t , readings of a l l the thermometers were taken. T h i s was repeated s e v e r a l times. T r i p l e p o i n t 68 data was a l s o taken, using an analogous procedure, slowly warming through the t r i p l e p o i n t . The two s e t s of data agreed w i t h i n the p r e c i s i o n of the measurements. Once a l l the measurements using the hydrogen vapour p r e s s u r e curve had been made, the thermometer block was heated to about 23 K, and the hydrogen pumped out. A s e r i e s of measurements were then made without using the vapour p r e s s u r e theremometer, below the hydrogen t r i p l e p o i n t . In order to get the c e l l to c o o l to 4.2 K, i t was necessary to pump on the helium bath; the weak thermal l i n k to between the thermometer block and the helium bath d i d not prov i d e enough c o o l i n g to r e g u l a t e the temperature r e l i a b l y . I t was not f e a s i b l e t o i n c r e a s e the s i z e of t h i s l i n k , as that would introduce very l a r g e thermal g r a d i e n t s at the higher temperatures. Because of t h i s problem, o n l y a few p o i n t s were taken at temperatures below 4.2 K. 1. MAGNETIC THERMOMETER CALIBRATION There was not enough low temperature data to f i t the A and 6 terms, consequently they were taken to be z e r o . Reported values f o r A vary i n s i g n , but those f o r powdered samples seem to be f a i r l y s m a l l . Reported v a l u e s f o r 6 i n d i c a t e that i t i s about 0.050 K 2 [CTAS-76]. Using worst case v a l u e s from the l i t e r a t u r e , the e r r o r i n t r o d u c e d by i g n o r i n g A and 6 would be about 15 mK. More i m p o r t a n t l y , t h i s e r r o r i s a smooth, slowly v a r y i n g f u n c t i o n of temperature. Such an e r r o r w i l l not have much e f f e c t i n 69 determining temperature dependencies, and i s t y p i c a l l y only about 0.2 % of the temperature. The magnetic thermometer data was f i t to The data used to c a l i b r a t e the magnetic thermometer were: the three EPT-76 r e f e r e n c e p o i n t s using hydrogen (the normal and 250 t o r r b o i l i n g p o i n t s , and the t r i p l e p o i n t ) , and the best low temperature p o i n t (3.560 K). The r e s u l t of t h i s f i t i s shown i n f i g . 8 2. RESISTANCE THERMOMETER .CALIBRATIONS Once the magnetic thermometer had been c a l i b r a t e d , i t was used to provide c a l i b r a t i o n data f o r the r e s i s t a n c e thermometers. The r e s u l t i n g c a l i b r a t i o n f o r the GRT i s shown in f i g . 9. The s o l i d l i n e i s a f i t to the c a l i b r a t i o n data of the form: Nine terms i n the s e r i e s were found s u f f i c e n t to d e s c r i b e the data a c c u r a t e l y . T h i s f i t was then used f o r determining the temperature of the GRT i n the magnetic resonance experiments. The c a l i b r a t i o n data f o r the CRT i s shown i n f i g . 10. The s o l i d l i n e i s j u s t a smooth curve through the da t a . X = A + B/T [IV-6] 1/T = I n a n ( l n R) n [IV-7] F i g . 8. Magnetic Thermometer C a l i b r a t i o n The c i r c l e s are the data p o i n t s which were f i t to p r o v i d e the c a l i b r a t i o n . F i g . 9. GRT C a l i b r a t i o n The s o l i d l i n e i s a f i t to the data g i v i n g 1/T as a polynomial i n ln(R) u s i n g nine terms. R C R T v s . 1 / T 5 0 0 0 — r 2 0 0 0 — RcRT 1000 — (a) : 500 — 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 1 / T ( U T 1 ) F i g . 10. CRT C a l i b r a t i o n The s o l i d l i n e i s a smooth curve through the data. Since the CRT was not used to make accurate temperature measurements, but only f o r temperature r e g u l a t i o n , i t was not necessary to have an a c c u r a t e f i t to i t s c a l i b r a t i o n . V. MAGNETIC RESONANCE EXPERIMENTS T h i s chapter d i s c u s s e s the magnetic resonance experiments. F i r s t the apparatus and procedures used to do the experiments are d e s c r i b e d , and then the r e s u l t s of the measurements. A. THE APPARATUS The d e s c r i p t i o n of the apparatus i s d i v i d e d i n t o two p a r t s : the c r y o s t a t used to maintain the experimental c e l l at the a p p r o p r i a t e temperature, and the microwave spectrometer used to make the magnetic resonance measurements. 1. THE CRYOSTAT The c r y o s t a t used f o r t h i s experiment was o r i g i n a l l y b u i l t by W.N. Hardy and A. Landesman f o r magnetic resonance s t u d i e s of atomic hydrogen [HARD-80], and was a l s o used by M.R. Morrow f o r studying atomic H c o n f i n e d by l i q u i d helium coated w a l l s between 1.0 K and 1.3 K [MORR-83a]. In those experiments, the temperature of the apparatus was c o n t r o l l e d by immersing i t i n the l i q u i d helium bath. For the experiments d e s c r i b e d i n t h i s work i t was m o d i f i e d to i n c l u d e a helium vapour c o o l i n g j a c k e t , so that i t s temperature c o u l d be c o n t r o l l e d above the b o i l i n g p o i n t of l i q u i d helium. Some minor improvements were a l s o made i n the mechanism used to tune the res o n a t o r . The c r y o s t a t f i t s i n t o a 10 cm i n s i d e diameter g l a s s dewar v e s s e l , which holds l i q u i d helium, surrounded by a 74 75 second dewar f i l l e d with l i q u i d n i t r o g e n . I t i s suspended from a f l a n g e at room temperature by the l a r g e c e n t r a l t h i n w a l l e d s t a i n l e s s s t e e l tube, and s i t s j u s t above the l i q u i d helium bath. The support tube has s e v e r a l b a f f l e s to reduce the heat leak from room temperature. The main s e c t i o n of the c r y o s t a t , which i s suspended j u s t above the l i q u i d helium bath, i s shown in f i g . 11. It c o n t a i n s the atom c e l l , the microwave resonator, and the r . f . d i s c h a r g e c o i l used to produce the atoms. At the c e n t r e of the main s e c t i o n of the c r y o s t a t i s the atom c e l l : a pyrex c y l i n d e r roughly 1.5 cm i n diameter and 3.0 cm long, with a t a i l 0.5 cm i n diameter and 2.5 cm long extending i n t o the r . f . d i s c h a r g e c o i l . Using these dimensions and a d i r e c t measurement of the c e l l volume, the area to volume r a t i o i s found to be 3.8 ± 0.1 cm" 1. The c e l l i s f i l l e d w ith about 600 t o r r of hydrogen gas at room temperature, and permanently s e a l e d . The t a i l of the c e l l c o n t a i n s a sm a l l copper f o i l with a 6 0 C o source 10 nO to provide f r e e charges to i n i t i a t e the r . f . d i s c h a r g e used to produce atomic hydrogen. The c e l l j u s t f i t s i n s i d e a 1420 MHz s p l i t r i n g r e s o n a t o r : a tube 2.0 cm i n diameter and 3.2 cm l o n g , with a gap running down the s i d e . The s p l i t r i n g resonator can be thought of as an LC c i r c u i t , with the s p l i t r i n g forming a s i n g l e t u r n c o i l , and the gap forming a c a p a c i t o r [HARD-81]. When c r i t i c a l l y coupled, the loaded Q of the resonator i s about 600 at room temperature, and 2850 at below 10 K. The I 76 R.F DISCHARGE C O A X THIN W A L L S T A I N L E S S S T E E L T U B I N G E L E C T R I C A L L E A D S He VAPOUR PUMPING LINE MICROWAVE C O A X N Y L O N R O D S T E F L O N S U P P O R T C O U P L I N G C A P A C I T O R COUPLING L O O P T E F L O N T R A C K TUNING SLIDE S P L I T RING R E S O N A T O R A T O M C E L L COOLING J A C K E T BIAS F I E L D C O I L GRADIENT C O I L S D I S C H A R G E E N C L O S U R E A T O M C E L L T A I L R.F DISCHARGE C O I L R.F D I S C H A R G E CIRCUIT GERMANIUM T H E R M O M E T E R REGULATING C R T He INTAKE PIPE C O P P E R i r cm 1 3 F i g . 11. Magnetic Resonance Apparatus A c r o s s s e c t i o n view of the resonator housing and c o o l i n g jacket assembly. 77 resonator i s supported by two t e f l o n brackets attached to the top of the resona t o r housing. The resonator can be tuned by means of a copper tuning s l i d e running along a t e f l o n t r a c k mounted i n the gap i n the s p l i t r i n g . Moving the tuning s l i d e v a r i e s the ca p a c i t a n c e a c r o s s the gap, and hence the resonant frequency. The s l i d e i s p o s i t i o n e d by means of a rod, c o n s i s t i n g of a t h i n w a l l s t a i n l e s s s t e e l tube u n t i l j u s t above the resonator housing, then a nylon s e c t i o n which passes through the tuning s l i d e . The upper end of the rod i s attached to a micrometer head at room temperature, which allows f i n e p o s i t i o n i n g . The l i n k a g e has some backlash i n i t which makes i t p o s s i b l e to decouple the tuning s l i d e from the p o s i t i o n i n g rod once the resonator i s tuned, s i n c e the t u n i n g s l i d e i s he l d i n p o s i t i o n by f r i c t i o n . T h i s reduces the e f f e c t of v i b r a t i o n s on the t u n i n g . Two t e f l o n c y l i n d e r s are e c c e n t r i c a l l y mounted on t h i s rod to p r o v i d e very f i n e tuning when the rod i s r o t a t e d r a t h e r than moved v e r t i c a l l y . Microwaves are coupled to the resonator from the s i g n a l coax v i a a s i n g l e t u r n c o u p l i n g loop, and a c a p a c i t o r . The s t r e n g t h of the c o u p l i n g can be v a r i e d by changing the value of t h i s c a p a c i t a n c e . The lower c a p a c i t o r p l a t e i s be moved using an arrangement s i m i l a r to that used with the tuning s l i d e . The resonator housing i s a copper c y l i n d e r 7.6 cm i n diameter which p r o v i d e s a w e l l d e f i n e d boundary c o n d i t i o n f o r the microwaves. Wound around t h i s i s an end c o r r e c t e d 78 s o l e n o i d , used to apply a uniform v e r t i c a l b i a s f i e l d , c o n s i s t i n g of 151.5 turns of copper wire over a l e n g t h of 10.7 cm and two 12 turn end c o r r e c t i o n c o i l s . Two a d d i t i o n a l c o i l s are a v a i l a b l e f o r p r o v i d i n g a v e r t i c a l f i e l d g r a d i e n t . The temperature of the atom c e l l i s measured with a germanium thermometer (GRT) a t t a c h e d to the o u t s i d e of the resonator housing at the bottom. The resonator housing i s f i l l e d with helium gas which serves to thermally connect the atom c e l l and resonator to the resonator housing. I t was not p o s s i b l e to mount the thermometer any c l o s e r to the c e l l , as the e l e c t r i c a l leads would i n t e r a c t with the microwaves i n s i d e the resonator housing. Since the power d i s s i p a t e d i n the c e l l i s f a i r l y s m a l l , the thermal g r a d i e n t s between the c e l l and the resonator housing should be c o r r e s p o n d i n g l y s m a l l . Atoms are produced i n the c e l l using a pulsed r . f . d i s c h a r g e at about 50 MHz i n a c o i l surrounding the t a i l of the c e l l . The c i r c u i t of the d i s c h a r g e system i s shown in f i g . 12. The discharge c o i l i s surrounded by a copper s h i e l d , and the opening f o r the c e l l t a i l i s intended to act as a waveguide beyond c u t o f f , so that the discharge should not i n t e r f e r e with the magnetic resonance s i g n a l s . The r . f . p u l s e s are produced by g a t i n g the s i g n a l from a s i g n a l g e n e r a t o r 1 and then a m p l i f y i n g i t . A d i r e c t i o n a l c o u p ler and c r y s t a l d e t e c t o r are used to observe the r e f l e c t e d s i g n a l in order to tune the r . f . source to match the resonant 1 Hewlett-Packard 8640A 0.5 - 500 MHz S i g n a l Generator 79 10 W R.F AMR - 2 0 dB CRYSTAL DETECTOR! SIGNAL GENERATOR HP 8640A 2 W R.F AMP SCOPE R. F GATE 50 MHz PULSE GENERATOR TEKTRONIX 160 Q + 1420 MHz X/4 TRAP R. F DISCHARGE COIL _ y r m _ 0.01 /zH 0.5 /iH 20 pF F i g . 12. R.F. Discharge C i r c u i t The c i r c u i t used to produce the 50 MHz d i s c h a r g e which produces the atoms. 80 frequency of the d i s c h a r g e c i r c u i t . The resonator housing was surrounded by a copper c o o l i n g j a c k e t used to c o n t r o l the temperature of the apparatus. T h i s ja c k e t has a s e r i e s of channels 1.6 mm deep and 4.8 mm wide, with helium vapour f l o w i n g through them to provide the c o o l i n g . L i q u i d helium i s drawn up from the bath i n t o a flow d i v i d e r where i t b o i l s . The helium vapour produced i s d i v i d e d i n t o four streams, i n a brass flow d i v i d e r , each passing up and down the c o o l i n g j a c k e t nine times. T h i s prevents any s i g n i f i c a n t v e r t i c a l temperature g r a d i e n t s from developing i n the c o o l i n g j a c k e t . The helium vapour flow through the c o o l i n g j a c k e t i s maintained by pumping on the exhaust s i d e of the c o o l i n g j a c k e t . A micrometer needle v a l v e on t h i s l i n e a llows adjustment of the helium flow rate to provide c o n t r o l over the amount of c o o l i n g . P r e c i s e c o n t r o l of the temperature of the apparatus was obtained by r e g u l a t i n g the temperature of a carbon r e s i s t o r thermometer (CRT), mounted on the bottom of the c o o l i n g j a c k e t , using an e l e c t r i c a l h e ater. The heater was a b i f i l a r wound l e n g t h of manganin wire, with a r e s i s t a n c e of about 200 fl. The CRT was mounted i n a copper post, with the leads heat sunk to the post, and was i n s u l a t e d from the helium gas above the l i q u i d bath by a cover made of c l o s e d c e l l foam. The c u r r e n t to the heater was c o n t r o l l e d by a homemade temperature c o n t r o l l e r p r o v i d i n g p r o p o r t i o n a l , i n t e g r a l and d i f f e r e n t i a l c o n t r o l (the same u n i t was used i n thermometer 81 c a l i b r a t i o n a p p a r a t u s ) . To s t a r t a run, the apparatus was f i r s t p r e c o o l e d to 77 K using l i q u i d n i t r o g e n , using the f o l l o w i n g procedure. The helium bath space was f i r s t f l u s h e d and l e f t with about one atmosphere of n i t r o g e n to provide exchange gas (helium c o u l d not be used u n t i l the w a l l s of the inner g l a s s dewar had been c o o l e d w e l l below room temperature, as helium d i f f u s e s through pyrex g l a s s at a s i g n i f i c a n t r a t e at room temperature). The vacuum space of the helium dewar was f l u s h e d s e v e r a l times and l e f t with about 1 t o r r of a i r i n i t , again to p r o v i d e thermal c o n t a c t d u r i n g the p r e c o o l . The outer dewar was then f i l l e d with l i q u i d n i t r o g e n , and the appparatus l e f t f o r s e v e r a l hours t o c o o l to 77 K. Once the apparatus had reached 77 K, the helium t r a n s f e r c o u l d begin. The n i t r o g e n gas was then pumped out of the helium bath space and r e p l a c e d with helium, and the apparatus l e f t t o e q u i l l i b r a t e f o r a short w h i l e . L i q u i d helium was t r a n s f e r r e d i n t o the dewar through the c o o l i n g j a c k e t , so the c o o l i n g power of the helium vapour was used e f f i c e n t l y . The sma l l amount of a i r i n the vacuum space of the inner dewar would fr e e z e out on the w a l l , once the inner w a l l of the helium dewar had been c o o l e d below 77 K, p r o v i d i n g a good vacuum. L i q u i d helium was allowed to c o l l e c t to w i t h i n 7 cm of the bottom of the c o o l i n g j a c k e t . A t y p i c a l t r a n s f e r used 3.5 / of l i q u i d helium to c o o l from 77 K, and p r o v i d e d enough l i q u i d to run f o r about 8 hours. 82 In order to reduce ambient magnetic f i e l d s , the outer n i t r o g e n dewar was wrapped with a sheet of magnetic s h i e l d i n g 2 . The s h i e l d passed through a c o i l , c o n s i s t i n g of ten t u r n s of copper wire, so that the s h i e l d and the c o i l formed two i n t e r l o c k i n g r i n g s . Any c u r r e n t i n the c o i l would induce a magnetic f i e l d c o n f i n e d mainly w i t h i n the s h i e l d because of i t s high magnetic p e r m e a b i l i t y . The s h i e l d was demagnetised by passing a 10 A 60 Hz c u r r e n t through the c o i l , and then slowly reducing the c u r r e n t to zero. T h i s reduced the ambient l o n g i t u d i n a l ( v e r t i c a l ) f i e l d to about 60 mG and the t r a n s v e r s e f i e l d to l e s s than 1 mG. I t was not d e s i r a b l e to remove the magnetic s h i e l d i n g w i t h i n a giv e n run, as the r e s i d u a l f i e l d that r e s u l t e d was always s l i g h t l y d i f f e r e n t . The s h i e l d blocked the u n s i l v e r e d s t r i p down the s i d e of the g l a s s dewar, and made i t necesary to use a helium l e v e l d e t e c t o r to monitor the amount of helium i n the bath. A l e v e l d e t e c t o r was c o n s t r u c t e d using a le n g t h of 50 urn diameter Nb-Ti a l l o y superconducting wire. When a c u r r e n t of 70 mA was passed through the wire, ohmic hea t i n g would keep the that p a r t of the wire i n the helium vapour i n the normal s t a t e , while that immersed i n the l i q u i d s t a y e d superconducting. The vo l t a g e a c r o s s the wire was then p r o p o r t i o n a l to the l e n g t h of wire above the l i q u i d s u r f a c e . 2 Co-Netic AA made by P e r f e c t i o n Mica Co. 83 2. THE SPECTROMETER A block diagram of the spectrometer used to measure the h y p e r f i n e resonance i s shown i n f i g . 13. I t was c o n s t r u c t e d by W.N. Hardy fo r use in e a r l i e r magnetic resonance experiments done in t h i s l a b . In order to provide accurate frequency measurements, a 10 Mhz rubidium frequency s t a n d a r d 3 was used as the time base f o r - t h e spectrometer. T h i s source has a s t a b i l i t y of 3 x 10" 1 1 month" 1, and an absolute accuracy of b e t t e r than 1 x 10" 9. In order to o b t a i n higher accuracy, the rubidium standard was c a l i b r a t e d u s ing s i g n a l s from the Loran-C r a d i o n a v i g a t i o n network". Since the Loran-C s i g n a l s are s l a v e d to cesium atomic c l o c k s , i t was p o s s i b l e to c a l i b r a t e the rubidium standard to b e t t e r than ± 5 x 10" 1 3 . T h i s i s e q u i v a l e n t to a 7 x 10"" Hz change in the h y p e r f i n e frequency, an amount 100 times l e s s than the u n c e r t a i n t y of the frequency measurements. The spectrometer performs two f u n c t i o n s : to produce the 7r/2 and 7r microwave p u l s e s at the h y p e r f i n e frequency used to induce t r a n s i t i o n s ; and to d e t e c t the microwaves r a d i a t e d by the atoms in the c a v i t y . In order to produce a s i g n a l at 1420 MHz, the 10 MHz s i g n a l was doubled, and then fed i n t o a step recovery diode, producing many harmonics. The harmonic at 1420 MHz was f i l t e r e d u s i n g a c o a x i a l resonator and then a m p l i f i e d . 3 Efratom model FRK-L, on loan from the U.S. N a t i o n a l Bureau of Standards * Using an Internav T-201 Loran-C Timing Receiver on loan from the Dominion Radio A s t r o p h y s i c a l O b s e r v i t o r y 84 x 142 MULTIPLIER 10 MHz Rb FREQUENCY STANDARD 10 MHz 1420 MHz .AMR PWR. DIV. 10 dB •»—vNWV-1420 MHz L. 0. IMAGE REJECT MIXER 1420.405 MHz < signal 405 kHz signal ,,'405 kHz reference FINAL MIXER audio - signal •ig. 0SCILLISC0PE SIGNAL AVERAGER FREQUENCY SYNTHESIZER PULSE GENERATOR 405 kHz PHASE SHIFTER gate 90° S. S.B. GENERATOR pulse GATE FREQUENCY SWEEP GENERATOR 1420.405 MHz sweep r 1. F. PHASE Xtal. AMP. SHIFTER det. -20 -20 -TdB cavity cryostat F i g . 13. 1420 MHz Spectrometer A block diagram of the spectrometer used to produce the microwave p u l s e s and observe the h y p e r f i n e resonance. 85 A frequency s y n t h e s i z e r 5 was used to produce a s i g n a l at about 405 kHz, using the 10 MHz from the rubidium standard as a time base. T h i s s i g n a l was then fed i n t o a gated phase s h i f t e r to produce two outputs 90° out of phase. These are mixed with the 1420 MHz s i g n a l to produce a sideband at 1420.405 MHz, which i s then gated, and coupled to the c a v i t y v i a a d i r e c t i o n a l c o u p ler i n the c r y o s t a t mounted j u s t above the s e c t i o n shown in f i g . 11. Because the 90° phase s h i f t e r i s gated, there i s no s i g n a l at the h y p e r f i n e frequency except when a pulse i s being generated. T h i s prevents any leakage i n t e r f e r i n g with the s i g n a l between p u l s e s , when i t i s being recorded. The s i g n a l from the c a v i t y i s a m p l i f i e d by a c r y o g enic low n o i s e a m p l i f i e r c o n s t r u c t e d by W.N. Hardy. The low n o i s e a m p l i f i e r i s a two stage GaAs FET design developed at the U n i t e d S t a t e s N a t i o n a l Radio Astronomy Laboratory, which uses source inductance feedback to achieve good matching, and i s d e s c r i b e d by W i l l i a m s et al [WILL-80]. I t has a gain of about 20 dB, and a n o i s e temperature of 31 ± 7 R under the c o n d i t i o n s i n t h i s experiment (at 4.2 K i t has a n o i s e temperature of 23 K). The low noise a m p l i f i e r i s mounted i n the c r y o s t a t j u s t above the s e c t i o n shown i n f i g . 11. The output of the low noise a m p l i f i e r i s brought out of the c r y o s t a t , a m p l i f i e d again, and mixed with the 1420 MHz s i g n a l to produce an I.F. s i g n a l at 405 kHz, which i s a m p l i f i e d . The I.F. s i g n a l was then mixed with the 405 kHz 5 Hewlett-Packard 3330A 0 - 1 3 MHz Frequency S y n t h e s i z e r 86 output of the s y n t h e s i z e r to produce an audio s i g n a l . The audio s i g n a l was d i s p l a y e d on an o s c i l l o s c o p e 6 . The input a m p l i f i e r of the o s c i l l o s c o p e was used to provide stepwise v a r i a b l e g a i n , and p r o v i d e d a monitor output. T h i s output was fed to a s i g n a l a v e r a g e r 7 , to re c o r d the s i g n a l and allow n o i s e r e d u c t i o n . The d i g i t i s e d and averaged s i g n a l s were then t r a n s f e r e d to a microcomputer 8 and s t o r e d on f l o p p y d i s k s for l a t e r a n a l y s i s . The audio s i g n a l i s at a frequency equal to the d i f f e r e n c e between the frequency determined by the s y n t h e s i z e r and the frequency r a d i a t e d by the atoms. For most of the measurements t h i s was set to be about 300 Hz. In order to a d j u s t the t u n i n g and c o u p l i n g of the reson a t o r , i t was necessary to be able to measure the c a v i t y r e f l e c t i o n c o e f f i c e n t as a f u n c t i o n of frequency. The output of a sweep g e n e r a t o r 9 was coupled i n t o the c a v i t y i n s t e a d of the microwave p u l s e s , and the r e f l e c t e d power was observed as a f u n c t i o n of frequency u s i n g a c r y s t a l d e t e c t o r and an o s c i l l o s c o p e . B. LOW ATOM DENSITY RESULTS T h i s s e c t i o n d e s c r i b e s the data obtained at low d e n s i t i e s ( n H < 8 x 1 0 " 1 6 m" 3), where recombination i s slow and the e f f e c t of s p i n exchange on the s i g n a l i s very s m a l l . T h i s data shows the e f f e c t of the H 2 pressure s h i f t , and the 6 T e k t r o n i x 5440 with a 5A22N d i f f e r e n t i a l a m p l i f i e r p l u g - i n 7 N i c o l e t 1170 with a model 172/4B input p l u g - i n 8 DEC Rainbow-100 9 Hewlett-Packard 8663A 0.1 - 2.5 GHz Frequency S y n t h e s i z e r 87 e f f e c t s of a d s o r p t i o n on the H 2 s u r f a c e . 1. PROCEDURE The procedure used to take the data was as f o l l o w s . The apparatus was f i r s t allowed to come i n t o thermal e q u i l i b r i u m at the d e s i r e d temperature. The sweep generator was used to look at the r e s o n a t o r ' s power r e f l e c t i o n c o e f f i c e n t as a f u n c t i o n of frequency. The tuning and c o u p l i n g were a d j u s t e d so that the resonator was c r i t i c a l l y coupled and tuned to the h y p e r f i n e resonant frequency (1 420.405 752 MHz). Atoms were then c r e a t e d by p u l s i n g the r . f . d i s c h a r g e b r i e f l y (a few p u l s e s , each about 10 us l o n g ) , and the d e n s i t y was allowed to decay due to recombination. A f t e r about a minute, the d e n s i t y was low enough ( n H < 8 x 10" 1 6 n r 3 ) that the d e n s i t y d i d not change a p p r e c i a b l y while the data was being taken and the r e l a x a t i o n processes that i n c r e a s e with the d e n s i t y have become n e g l i g i b l e . Free i n d u c t i o n decays (FIDs) were induced by a p p l y i n g ir/2 microwave p u l s e s . The l e n g t h of the 7r/2 p u l s e s was determined by a d j u s t i n g the length of the p u l s e u n t i l the s i g n a l was a maximum. The time between n/2 was u s u a l l y set to about 250 ms, which i s much longer than T, or T 2, so that the d e n s i t y matrix of the atoms was always i n i t s thermal e q u i l i b r i u m s t a t e . The s i g n a l averager was used to r e c o r d the s i g n a l , averaging about 512 FIDs to improve the s i g n a l to noise r a t i o . I t d i g i t i s e d 1024 p o i n t s at a sampling rate of 20 kHz, so that the f i r s t 50 ms of the FID was recorded. 88 The FIDs were then f i t to a damped c o s i n e wave S(t ) = B + A e " t / / T 2 c o s ( w t + «>) [V-1] where S(t) i s the observed s i g n a l as a f u n c t i o n of time. A t y p i c a l FID and the f i t t e d s i g n a l are shown i n f i g . 14 I n i t i a l guesses f o r the parameters were determined by l o o k i n g at the p o s i t i o n s of the extrema and b a s e l i n e c r o s s i n g s i n the s i g n a l . T h i s was done by the data a q u i s i t i o n program running on the microcomputer, and provided quick approximate r e s u l t s d u r i n g a run. These i n i t i a l v alues were used to o b t a i n b e t t e r v a l u e s f o r the parameters by doing a l e a s t squares f i t of equation [V-1] to the FIDs using a . r o u t i n e c a l l e d NL2S0L from the U.B.C. computer c e n t r e . NL2S0L i s a s o p h i s t i c a t e d , robust n o n - l i n e a r l e a s t squares f i t t i n g r o u t i n e developed at the C o r n e l l U n i v e r s i t y Dept. of Computer Science [DENN-79]. T h i s procedure was then repeated at v a r i o u s temperatures. The range of temperatures was l i m i t e d by the a b i l i t y of the r . f . d i s c h a r g e to produce atoms, without using e x c e s s i v e power. At temperatures above 8.2 K, the d i s c h a r g e would a b r u b t l y cease producing atoms. T h i s i s probably because the vapour p r e s s u r e of H 2 becomes too h i g h to allow the d i s c h a r g e . At temperatures below 6.4 K, the d i s c h a r g e d i d n ' t c r e a t e any atoms u n t i l a l a r g e amount of power had been a p p l i e d and the c e l l heated above 6.4 K. 89 TIME t (ms) F i g . 14. A T y p i c a l FID One of the f r e e i n d u c t i o n decays from the low d e n s i t y data, along with the l e a s t squares f i t to a damped c o s i n e wave. 90 During a run i n which data was being taken, two other s e t s of measurements were a l s o r e q u i r e d : the frequency s h i f t due to the ambient magnetic f i e l d , and the gain of the spectrometer. For small magnetic f i e l d s , the h y p e r f i n e frequency depends q u a d r a t i c a l l y on the magnitude of the f i e l d (see equation [ I I - 7 ] ) . T h i s frequency s h i f t must be accounted f o r i f the s h i f t s due to other e f f e c t s are to be e x t r a c t e d from the data. The amplitude of the magnetic resonance s i g n a l depends on the angle between the magnetic f i e l d and the a x i s of the resonator, g i v i n g maximum s i g n a l amplutude when the two are p a r a l l e l , and v a n i s h i n g when they are p e r p e n d i c u l a r (the amplitude i s p r o p o r t i o n a l to cos0sin[ (7r'/2 )cos0 ] ). The b i a s f i e l d s o l e n o i d was used to n u l l the l o n g i t u d i n a l component of the f i e l d , where the s i g n a l vanished. The width of t h i s n u l l , i n terms of the a p p l i e d l o n g i t u d i n a l f i e l d , was used to determine the magnitude of the t r a n s v e r s e component of the f i e l d . In the experiments the ambient f i e l d was always w i t h i n 1° of the resonator a x i s , and the e f f e c t s of i t s t r a n s v e r s e component were n e g l i g i b l e . Once the value of the s o l e n o i d c u r r e n t needed to n u l l the ambient f i e l d i s known, the frequency s h i f t due to the ambient f i e l d can be determined by p l o t t i n g the the observed frequency as a f u n c t i o n of the square of the d i f f e r e n c e between the s o l e n o i d c u r r e n t and the s o l e n o i d c u r r e n t which n u l l s the f i e l d . T h i s y i e l d s a l i n e a r r e l a t i o n s h i p a l l o w i n g one to determine the d i f f e r e n c e between the fr e q u e n c i e s when 91 the f i e l d i s zero and when the f i e l d has the value used for the r e s t of the measurements. The second measurement r e q u i r e d was the o v e r a l l gain of the spectrometer, so that the a b s o l u t e power of the s i g n a l c o u l d be determined. I f the s i g n a l power i s known, then i t can be used to determine the d e n s i t y of H atoms using equation [11-37]. The microwave sweeper was used to produce a s i g n a l at the h y p e r f i n e frequency. The power of t h i s s i g n a l was measured with a power m e t e r 1 0 . The s i g n a l was then attenuated by 70 dB u s i n g two p r e c i s i o n a t t e n u a t o r s 1 1 , and coupled i n t o the input of the low noise p r e a m p l i f i e r by a 20 dB d i r e c t i o n a l c o u p l e r 1 2 . The a t t e n u a t i o n i n the a t t e n u a t o r s , the coax l i n e s , and the d i r e c t i o n a l c o u p ler were a l l measured to w i t h i n 0.1 dB using the sweeper and the power meter. The r e s u l t i n g s i g n a l was then recorded i n the same manner as the FIDs were, and f i t t e d i n the same manner except that the e x p o n e n t i a l decay f a c t o r was l e f t out. T h i s was done f o r a l l of the g a i n s e t t i n g s on the o s c i l l i s c o p e a m p l i f i e r that were used d u r i n g a run. 2. FREQUENCY SHIFT RESULTS The frequency s h i f t of the low d e n s i t y data i s shown i n f i g . 15 as a f u n c t i o n of i n v e r s e temperature. The frequency s h i f t due to the ambient magnetic f i e l d has been removed 1 0 Hewlett-Packard model 435B power meter and model 8481A power sensor 1 1 Midwest Microwave model 263 30 dB and 40 dB a t t e n u a t o r s 1 2 Midwest Microwave model 5010-20 R 1-2 GHz 20 dB d i r e c t i o n a l c o u p ler 92 Ai/VT V S . I/T 2 0 0 — , 0,12 0.13 0.14 0J5 1/T (K ) F i g . 15. The Frequency S h i f t vs. Temperature The data f o r the frequency s h i f t p l o t t e d as a f u n c t i o n of i n v e r s e temperature. The s o l i d l i n e i s a f i t to the data e x p l a i n e d i n the t e x t . 93 from the data. I f the frequency s h i f t i s due to the combined e f f e c t s of the H 2 pressure s h i f t and the e f f e c t s of the H atoms adsorbed on the H 2 s u r f a c e , and p r o v i d e d the mean phase s h i f t per c o l l i s i o n i s s m a l l , then the observed frequency s h i f t should be given by Aw = 2tiv = (1 - x)2itKT\u + XAOJ [V-2] ti 2 s where K i s the pressure s h i f t c o e f f i c e n t , n H z i s the d e n s i t y of H 2, Awg i s the sur f a c e frequency s h i f t , and x i s the f r a c t i o n of atoms adsorbed on the su r f a c e g i v e n by x = ( A / V ) A e E B / k T [V-3] The s o l i d l i n e i n f i g . 15 i s a f i t to the data, assuming that the frequency s h i f t i s d e s c r i b e d by equation [V-2], The va l u e s of the f i t t e d parameters a r e : the pressure s h i f t c o e f f i c e n t K = -1.78 ± 0.01 x 10 ~ 2 \ Hz m3, the s u r f a c e frequency s h i f t Aa> s/27r = -1. 1 6 + 0.05 x 10 6 Hz, and b i n d i n g energy E f i/k = 34.04 ± 0.26 K. These r e s u l t s f o r the frquency s h i f t data are i n f a i r agreement with the expermental r e s u l t s of Crampton et al f o r atomic H c o n f i n e d by H 2 w a l l s at temperatures below 4.5 K. They measured a su r f a c e frequency s h i f t Aa>s/2ir = -1.12 ± 0.08 x 10 6 Hz, and b i n d i n g energy E f i/k = 35.75 ± 0.31 K [CRAM-82]. The H 2 p r e s s u r e s h i f t i s completely n e g l i g i b l e i n t h e i r experiments, so they couldn't 94 measure K. Comparing the r e s u l t for the b i n d i n g energy with the c a l c u l a t i o n s of P i e r r e et al, the measured value agrees reasonably w e l l t h e i r r e s u l t f o r the b i n d i n g energy on the (1,1,1) s u r f a c e of H 2 using the H-H2 p o t e n t i a l of Varandas and Tennyson [VARA-81] E f i/k = 32.7 K [PIER-85], The value obtained f o r the s u r f a c e frequency s h i f t i s q u i t e low compared to Weinrib's estimate of Ao)s/2rr = -1.7 MHz [WEIN-79], T h i s d i s c r e p r e n c y i s not cause f o r concern. Weinrib obtained h i s estimate by c a l c u l a t i n g the mean d i s t a n c e between the adsorbed atom and the atoms in the H 2 s u r f a c e , and then c a l c u l a t i n g the frequency s h i f t c o r responding to t h i s d i s t a n c e , so h i s r e s u l t i s the frequency s h i f t at the average d i s t a n c e , not the average frequency s h i f t . 3. THE TRANSVERSE RELAXATION RATE The t r a n s v e r s e r e l a x a t i o n r a t e T 2 1 i s shown i n f i g . 16 as a f u n c t i o n of i n v e r s e temperature. As e x p l a i n e d i n chapter I I I , there should be a c o n t r i b u t i o n to T i 1 due to the e f f e c t s of atoms adsorbed on the s u r f a c e . I f the average phase s h i f t per c o l l i s i o n i s s m a l l , i t i s given by T i 1 = A U S 2 X 2 < T B > [V-4] where Ao>s i s the s u r f a c e frequency s h i f t , x i s the f r a c t i o n of atoms given by equation [V-3], and <T r> i s the mean time 95 F i g . 16. The Transverse R e l a x a t i o n Rate The t r a n s v e r s e r e l a x a t i o n r a t e T i 1 p l o t t e d as a f u n c t i o n of i n v e r s e temperature. The s o l i d l i n e i s a t h e o r e t i c a l f i t to the data d e s c r i b e d i n the t e x t . 96 between a d s o r p t i o n s . The mean time between a d s o r p t i o n s can be given in terms of the mean time between c o l l i s i o n s <TC>, which i s known (see equation [ I I I - 2 ] ) , and the s t i c k i n g c o e f f i c e n t s. An examination of f i g . 16 suggests that T^ 1 seems to have a second c o n t r i b u t i o n t h a t i s independent of temperature, or only weakly temperature dependent. A c c o r d i n g l y the data was f i t equation [V-4], with the a d d i t i o n of a temperature independent r e l a x a t i o n (Ti 1) 0» and using the value of ACJ s from the frequency s h i f t measurements. T h i s gave a temperature independent r e l a x a t i o n r a t e ( T i 1 ) 0 = 50.3 ± 1.0 s " 1 , a s t i c k i n g c o e f f i c e n t s = 0.06 ± 0.05, and a b i n d i n g energy Eg/k = 32.2 ± 5.5 K. The l a r g e e r r o r i n the s t i c k i n g c o e f f i c e n t i s because i t has a l a r g e c o v a r i a n c e with the b i n d i n g energy, so i t i s d i f f i c u l t to e x t r a c t e i t h e r one independently. The estimate of the bi n d i n g energy obtained t h i s way i s not p a r t i c u l a r l y a c c u r a t e e i t h e r , but i t i s completely c o n s i s t e n t with the value obtained from the frequency s h i f t data. In order to get a b e t t e r estimate of the s t i c k i n g c o e f f i c e n t , a second f i t was made, with the b i n d i n g energy h e l d f i x e d at the value found by f i t t i n g the frequency s h i f t data (Eg/k = 34.04 K). T h i s o n l y i n c r e a s e d the sum of squares e r r o r f o r the f i t by 1.2 %, and gave a temperature independent r e l a x a t i o n r a t e ( T i 1 ) 0 = 50.94 ± 0.32 s " 1 , and a s t i c k i n g c o e f f i c e n t s = 0.107 ± 0.007. T h i s f i t i s shown as the s o l i d l i n e i n f i g . 16. 97 T h i s value f o r the s t i c k i n g c o e f f i c e n t i s q u i t e a b i t lower than that which was c a l c u l a t e d using the model of Lennard-Jones and Devonshire i n chapter I I I , namely s = 0 . 3 8 at 7.5 K. T h i s i s not too s u r p r i s i n g , as t h i s model probably overestimates the stength of the c o u p l i n g between the atoms impinging on the s u r f a c e and the phonons of the s u b s t r a t e . It should be made c l e a r that the temperature independent r e l a x a t i o n ( T i 1 ) 0 cannot be the r e s u l t of a spread i n resonant f r e q u e n c i e s from inhomogeneities i n the magnetic f i e l d . Any c o n t r i b u t i o n to T~21 due to f i e l d inhomogeneities would in c r e a s e l i n e a r l y with the s t a t i c f i e l d , s i n c e the frequency of the |a> to |c> t r a n s i t i o n i s q u a d r a t i c i n the f i e l d . However, T 2 was found to be e s s e n t i a l l y independent of the s t a t i c f i e l d used f o r the experiments (except when the f i e l d became too small to remove the degeneracy between the three h y p e r f i n e t r a n s i t i o n f r e q u e n c i e s ) . Magnetic i m p u r i t i e s i n the g l a s s c e l l w a l l or the 6 0 C o source are a l s o r u l e d out as sources f o r the e x t r a r e l a x a t i o n . In some experiments done with S.B. Crampton, where neon coated w a l l s were s t u d i e d i n the same apparatus, T i 1 was found to be as low as 6.2 s " 1 . The c e l l used f o r these experiments was made at the same time as the c e l l s used f o r the H-H2 experiments, and the same 6 0 C o source was used. T h i s s t r o n g l y suggests that ( T i 1 ) 0 i s due to the H 2 s u r f a c e . 98 A p o s s i b l e e x p l a n a t i o n f o r the temperature independent r e l a x a t i o n ( T i 1 ) 0 i s that i t i s due to a l i g h t l y bound s t a t e of atomic H on the H 2 s u r f a c e . P i e r r e et al p r e d i c t that such a s t a t e probably e x i s t s , and they c a l c u l a t e b i n d i n g e n e r g i e s ranging between 1.2 K and 2.1 K f o r t h i s s t a t e . If the mean phase s h i f t per a d s o r p t i o n i n t o t h i s s t a t e was l a r g e , then there would be a c o n t r i b u t i o n to the t r a n s v e r s e r e l a x a t i o n r a t e given by T i 1 = l / < r B > , = S 1 / < T c > [v-5] where <r B>, i s the mean time between a d s o r p t i o n s i n the l i g h t l y bound s t a t e , and s, i s the s t i c k i n g c o e f f i c e n t f o r a d s o r p t i o n i n t o the l i g h t l y bound s t a t e . Assuming s, would be a slowly v a r y i n g f u n c t i o n of temperature, then t h i s might account f o r the e x t r a r e l a x a t i o n (Ti 1) 0« If we assume t h i s i s the case, then the t r a n s v e r s e r e l a x a t i o n data g i v e s a value f o r the s t i c k i n g c o e f f i c e n t i n t o the l i g h t l y bound s t a t e of s, = 0.0014 at T = 7.5 K. For comparison, the Lennard-Jones and Davidson model p r e d i c t s s, = 0.0098 at T = K, given the bound s t a t e e n e r g i e s c a l c u l a t e d by P i e r r e et al [PIER-85] using the H-H2 p o t e n t i a l of Varandas and Tennyson [VARA-81]. T h i s i s much l a r g e r than the experimental v a l u e . However, the r a t i o between the two experimental s t i c k i n g c o e f f i c e n t s , and that between the two t h e o r e t i c a l s t i c k i n g c o e f f i c e n t s are i n more reasonable agreement: the experiments give s/s, = 78, while 99 the theory g i v e s s/s, = 38. There i s , u n f o r t u n a t e l y , a problem with t h i s e x p l a n a t i o n . If we know the s t i c k i n g c o e f f i c e n t f o r the l i g h t l y bound s t a t e , and we have an estimate of i t s b i n d i n g energy (from the c a l c u l a t i o n s of P i e r r e et al f o r example), then we can c a l c u l a t e how l a r g e the frequency s h i f t must be to ensure c60 > 1 . The r a t i o between <T"s>, and < r B > i i s given by x , , the f r a c t i o n of atoms adsorbed i n the l i g h t l y bound s t a t e (given by equation [111 —4 ] with the b i n d i n g energy taken to be that i n the l i g h t l y bound s t a t e ) . Taking T = 7.5 K, and the b i n d i n g energy of the the l i g h t l y bound s t a t e to be 5 K, one o b t a i n s the r e s u l t that the s u r f a c e frequency s h i f t i n the l i g h t l y bound s t a t e must be g r e a t e r than 17 MHz f o r <f>0 to be l a r g e r than 1. I f the b i n d i n g energy of the l i g h t l y bound s t a t e i s as small as P i e r r e et al p r e d i c t (< 2.1 K), then the s u r f a c e frequency s h i f t i n the l i g h t l y bound s t a t e would have to be even higher (> 25 MHz). Ad r i a n [ADRN-60] has c a l c u l a t e d the h y p e r f i n e frequency s h i f t f o r a H atom surrounded by s i x H 2 molecules, as a f u n c t i o n of the d i s t a n c e between them. The frequency s h i f t i s p r o p o r t i o n a l to the number of molecules surrounding the atom, so atoms on a s u r f a c e should experience roughly h a l f the s h i f t of those surrounded by H 2. H i s r e s u l t s suggest that the l a r g e s t p o s s i b l e frequency s h i f t f o r an atom adsorbed on an H 2 s u r f a c e would be l e s s than 2 Mhz, so i t seems u n l i k e l y t h at a l i g h t l y bound s t a t e of the H-H2 1 00 p o t e n t i a l can be the source of ( T i 1 ) 0 . Another p o s s i b l e e x p l a n a t i o n f o r the e x t r a t r a n s v e r s e r e l a x a t i o n would be t i g h t b i n d i n g s i t e s on the molecular hydrogen s u r f a c e . Using e l e c t r o n s p i n resonance (ESR) to study H atoms co n f i n e d by H 2 at 4.2 K, Mayer et al found an ESR l i n e that they a t t r i b u t e d to l o c a l i s e d s u r f a c e s t a t e s [MAYR-81]. They c a l c u l a t e d the s u r f a c e d e n s i t y of l o c a l i s e d s i t e s to be between 10 1 * and 1 0 1 5 m~2, which suggests that the s i t e s are probably due to d e f e c t s or g r a i n boundries on the H 2 s u r f a c e . Crampton and co-workers have c o n s i d e r e d l o c a l i s e d t i g h t b i n d i n g s i t e s as a p o s s i b l e e x p l a n a t i o n of non-exponential T, processes [NUNE-83, CRAM-85]. If such s i t e s do e x i s t at d e f e c t s or g r a i n boundries, then they probably have a b i n d i n g energy 2-3 times t h a t of atoms adsorbed on the surface i n the 2D gas phase [CRAM-85]. It would be d i f f i c u l t to determine much about the t i g h t b i n d i n g s i t e s (TBS's) from the T 2 data i n f i g . 16, as some of the temperature dependence of T £ 1 may be due to atoms bound i n TBS's. T h i s l e a v e s a l a r g e number of parameters which have to be understood to e x p l a i n the T 2 data: the exchange r a t e s between the three phases (2D, 3D gas and TBS's), the b i n d i n g energy, frequency s h i f t and s u r f a c e d e n s i t y of the TBS's. The problem would be s i m p l i f i e d i f the s t i c k i n g c o e f f i c e n t s of H on H 2 c o u l d be determined by some other means. Given a value f o r s, and using the b i n d i n g energy and s u r f a c e frequency s h i f t determined by the frequency s h i f t 101 measurements, the c o n t r i b u t i o n to T i 1 due to the atoms adsorbed i n the 2D s u r f a c e gas c o u l d be c a l c u l a t e d . One c o u l d then attempt to e x p l a i n the remaining c o n t r i b u t i o n to T i 1 as being due to atoms i n TBS's. T h i s r e s i d u a l T i 1 r a t e i s shown i n f i g . 17 f o r s e v e r a l p o s s i b l e values of the s t i c k i n g c o e f f i c e n t s. C l e a r l y , s must be g r e a t e r than 0.04, otherwise the atoms adsorbed i n the 2D gas would cause a r e l a x a t i o n r a t e l a r g e r than what was measured. C. H I G H DENSITY RESULTS T h i s s e c t i o n d e s c r i b e s the r e s u l t s of the measurements made with high atom d e n s i t i e s ( n H < 8.5 x 1 0 1 8 m~ 3). 1. RECOMBINATION MEASUREMENTS Because these experiments were done i n a c l o s e d geometry c e l l , i t i s p o s s i b l e to measure the change i n the atom d e n s i t y as a f u n c t i o n of time due to recombination. For these measurements i t i s only the amplitude of the magnetic resonance s i g n a l which i s of i n t e r e s t . Before any of the recombination data was taken, the gai n of the microwave spectrometer was c a l i b r a t e d as d e s c r i b e d i n the p r e v i o u s s e c t i o n . The procedure used to take the data was s t r a i g h t f o r w a r d . The l e n g t h of the 7r/2 p u l s e s was set by a d j u s t i n g the p u l s e l e n g t h u n t i l the amplitude of the FID was a maximum. Then the apparatus was brought to the d e s i r e d temperature and allowed to come i n t o thermal e q u i l i b r i u m . The resonator was checked to make sure 1 02 IOO-90- s = 1.0 80-70H 60-RESIDUAL l/T 2 (s- 1) 50-40-30-20-O s = 0.2 O s = 0.1 o 0 o o 0 o o o °Oooo o • l c P o d • D A 10-—i 1 1 1 1 • r~ 0.12 0.13 0.14 QI5 1/T (K-') 0.16 F i g . 17. R e s i d u a l T j 1 A p l o t of the c o n t r i b u t i o n to the t r a n s v e r s e r e l a x a t i o n r a t e not due to the 2D gas atoms, f o r v a r i o u s v a l u e s of the s t i c k i n g c o e f f i c e n t s. 1 03 that i t was tuned to the h y p e r f i n e frequency, and that i t was c r i t i c a l l y coupled, n/2 p u l s e s were used to induce FIDs every 100 ms. Since the i n t e r v a l between the it/2 p u l s e s was much longer than e i t h e r T, or T 2, the h y p e r f i n e d e n s i t y matrix would always have i t s thermal e q u i l i b r i u m value immediately before a p u l s e . Atoms were c r e a t e d by b r i e f l y p u l s i n g the r . f . d i s c h a r g e . A f t e r a few seconds, the s i g n a l averager recorded and averaged 8 FIDs and s t o r e d them i n memeory. Over a p e r i o d of a minute, four more averaged FIDs were recorded, the time i n t e r v a l between them determined with a stopwatch. As the d e n s i t y , and hence s i g n a l s t r e n g t h , d e c l i n e d with time, the gain of the spectrometer was i n c r e a s e d . Because of t h e i r decreased s i g n a l to n o i s e r a t i o , the number of s i g n a l s averaged was increased to 32, f o r the l a s t few FIDs. T h i s procedure was repeated at r e g u l a r i n t e r v a l s over the temperature range i n which atoms c o u l d be e a s i l y c r e a t e d . The d e n s i t y of atoms c o u l d be determined from the amplitude of the FIDs and the spectrometer gain c a l i b r a t i o n using equation [11-37]. In chapter I I I , i t was shown that recombination should be d e s c r i b e d by a second order r a t e equation AH = - k t O t n 2 H f V " 6 ] where k^ot * s t * i e n e t e ^ f e e t i v e recombination rate (see equation [111-24]). In t h i s case, the time dependence of the 1 04 H atom d e n s i t y should be given by 1 / n H ( t ) = l / n H ( 0 ) + k [V-7] T h i s r e l a t i o n was used to determine a value of k t o t at each temperature by doing a l e a s t squares f i t of l/r> H as a f u n c t i o n of time, with the data p o i n t s weighted by the squared i n v e r s e of t h e i r e r r o r s . A t y p i c a l p l o t of l / n H vs. t i s shown i n f i g . 18 The r e s u l t s of these recombination measurements are shown as a f u n c t i o n of i n v e r s e temperature i n f i g . 19. The data c l e a r l y shows that the t o t a l e f f e c t i v e recombination r a t e i s completely dominated by s u r f a c e recombination over the range of temperatures s t u d i e d , except p o s s i b l y f o r the data taken at the highest two temperature p o i n t s . I f bulk recombination was o c c u r i n g , i t would be at a r a t e p r o p o r t i o n a l to the d e n s i t y of H 2 (see s e c t i o n I I I . C . 1 ) . T h i s would cause a r a p i d i n c r e a s e i n the recombination r a t e at high temperatures because the H 2 vapour p r e s s u r e i s such a r a p i d l y i n c r e a s i n g f u n c t i o n of temperature, and would be e a s i l y seen i n the data ( c f . frequency s h i f t v s . 1/T). Surface recombination can be d e s c r i b e d i n terms of the s u r f a c e recombination c r o s s l e n g t h X d e f i n e d i n equation [111-23]. The s o l i d l i n e i n f i g . 19 i s a f i t made to the recombination data assuming that X i s independent of temperature. T h i s f i t g i v e s X = 10.2 A, and a b i n d i n g energy f o r H on H 2 E R/k = 24.6 K. T h i s estimate of the b i n d i n g 105 l /DENSITY vs. TIME 0 20 40 60 TIME t (s) F i g . 18. Inverse D e n s i t y vs. Time Th i s p l o t shows the decay of the atom d e n s i t y due to recombination. The s o l i d l i n e i s a l e a s t squares f i t to a second order r a t e law. The data was taken at T = 7.225 K. 1 06 k^T v s . 1/T 1 0.12 0.13 0.14 0.15 0.16 1 / T ( K - ' ) F i g . 19. E f f e c t i v e Recombination Rate T h i s p l o t shows the e f f e c t i v e recombination r a t e c o n s t a n t as a f u n c t i o n of i n v e r s e temperature. The s o l i d l i n e i s e x p l a i n e d i n the t e x t . 1 0 7 energy agrees very p o o r l y with that o b tained from the frequency s h i f t data and probably shouldn't be taken very s e r i o u s l y . The assumption that X i s independent of temperature i s rather suspect at these temperatures, although i t should be true at lower temperatures. In experiments with H adsorbed on l i q u i d helium below 1 K, Morrow found that the s u r f a c e recombination c r o s s l e n g t h was X = 0.14 ± 0.02 A f o r H on "He, and X = 0.13 A f o r H on 3He [MORR-83a]. There seems l i t t l e reason to expect X to depend s t r o n g l y on the nature of the s u r f a c e , so the d i s c r e p r e n c y between Morrow's r e s u l t s f o r H adsorbed on l i q u i d helium and the above r e s u l t s suggest that X cannot be taken to be independent of temperature. Another approach to i n t e r p r e t i n g t h i s data i s to assume that the b i n d i n g energy obtained from the frequency s h i f t data i s a c c u r a t e . T h i s makes i t p o s s i b l e to c a l c u l a t e X as a f u n c t i o n of temperature from the data, and the r e s u l t s of doing so are shown i n f i g . 20. The values t h i s g i v e s f o r X seem much more p l a u s i b l e than the pr e v i o u s estimate, and the b i n d i n g energy obtained from the frequency s h i f t data should be q u i t e r e l i a b l e . The i n c r e a s e i n X with temperature i s probably due to an energy resonance i n the intermediate s t a t e averaged over the thermal d i s t r i b u t i o n of c o l l i s i o n e n e r g i e s . 1 0 8 X vs. T 0 . 5 -6.4 6.6 6.8 7.0 7.2 7.4 7.6 7 8 8£> 8.2 T (K) F i g . 20. Surface Recombination Cross Length The s u r f a c e recombination c r o s s l e n g t h X as a f u n c t i o n of temperature, assuming the b i n d i n g energy i s 34.04 K. 109 2. SPIN EXCHANGE MEASUREMENTS Spin exchange c o l l i s i o n s between H atoms are the predominant r e l a x a t i o n mechanism at high d e n s i t i e s . As d i s c u s s e d p r e v i o u s l y (chapter I I I , s e c t i o n D), there are c o n t r i b u t i o n s to the l o n g i t u d i n a l (T,) and t r a n s v e r s e ( T 2 ) r e l a x a t i o n r a t e s due to spin exchange c o l l i s i o n s , between H atoms i n the bulk and those adsorbed on the s u r f a c e , given by T, 1 = [ ( l - x ) v a 3 D + ( V / A ) x 2 v s 0 2 D ] n H T i 1 = T l 1 / 2 [V-8] where x i s the f r a c t i o n of atoms on the s u r f a c e (given by equation [ I I I - 3 ] ) , v and v g are the mean r e l a t i v e v e l o c i t i e s of atoms in the bulk and on the su r f a c e , agjj and a 2 D are the bulk c r o s s s e c t i o n and su r f a c e c r o s s l e n g t h f o r spin exchange r e s p e c t i v e l y , and n H i s the bulk d e n s i t y of H atoms. I f the b i n d i n g energy obtained from the frequency s h i f t measurements (Eg/k = 34.04 K) i s used along with the t h e o r e t i c a l c a l c u l a t i o n s of O^-Q by B e r l i n s k y and S h i z g a l [BERL-80] and those of o 2 D by Morrow and B e r l i n s k y [MORR-83b], the bulk c o n t r i b u t i o n to spin exchange i s found to completely predominate f o r temperatures above 5 K. Assuming t h i s , i t i s p o s s i b l e to determine the bulk s p i n exchange c r o s s s e c t i o n by measuring the two r e l a x a t i o n r a t e s as a f u n c t i o n of H atom d e n s i t y . 1 10 The t r a n s v e r s e r e l a x a t i o n r a t e c o u l d be obtained d i r e c t l y from the FID f o l l o w i n g a it/2 p u l s e . In order to determine the l o n g i t u d i n a l r e l a x a t i o n r a t e , a 7 r - 7 r/2 pulse squence was used (see chapter I I , s e c t i o n C). The procedure used to measure the two r e l a x a t i o n r a t e s as a f u n c t i o n of d e n s i t y was as f o l l o w s . Atoms were c r e a t e d by p u l s i n g the r . f . d i s c h a r g e at the same r a t e as the microwave p u l s e s of the spectrometer, with the d i s c h a r g e f i r i n g about 100 ms bef o r e the microwave p u l s e s and the whole sequence r e p e a t i n g every 250 ms. The d e n s i t y of H atoms immediately a f t e r the microwave p u l s e s remained q u i t e s t a b l e , and c o u l d be v a r i e d by changing the pu l s e l e n g t h of the r . f . d i s c h a r g e . The magnetic f i e l d g r a d i e n t was then turned on, and a rr-7r/2 p u l s e sequence was used to measure T,. The response of the atoms was viewed on the o s c i l l i s c o p e , and the delay between the two p u l s e s a d j u s t e d u n t i l the FID a f t e r the ir/2 pulse changed s i g n . A c c o r d i n g to equation [11-23], t h i s occurs when the delay between the p u l s e s , read o f f the o s c i l l o s c o p e s c r e e n , i s T 1 l n ( 2 ) . Once the measurement of T, was f i n i s h e d , the magnetic f i e l d g r a d i e n t and -v pulses were then turned o f f , and the FID induced by the ir/2 pulse was recorded on the s i g n a l averager. T h i s p r o v i d e d a measurement of T 2 and the H atom d e n s i t y . The l e n g t h of the r . f . d i s c h a r g e p u l s e s was then a d j u s t e d to produce a d i f f e r e n t H atom d e n s i t y , and the process repeated. T h i s procedure y i e l d e d measurements of the two r e l a x a t i o n r a t e s T i 1 and T £ 1 as a f u n c t i o n of H atom d e n s i t y 111 at each of the temperatures s t u d i e d . The r e s u l t s f o r T = 7.067 K are shown i n f i g . 21. A l e a s t squares f i t was made to the data, t a k i n g the r e l a x a t i o n r a t e s to be l i n e a r f u n c t i o n s of the atom d e n s i t y . Assuming that s p i n exchange c o l l i s i o n s i n the bulk were predominating, then the s p i n exhange c r o s s s e c t i o n o^-p c o u l d then be e x t r a c t e d from the data using equation [ V - 8 ] . These experimental r e s u l t s f o r the bulk s p i n exchange c r o s s s e c t i o n s determined by the r e l a x a t i o n r a t e s are shown in f i g . 22 along with the c a l c u l a t e d r e s u l t s of B e r l i n s k y and S h i z g a l [BERL-80] (the T, measurements are denoted by s and the T 2 measurements by O ' s ) . I t i s not s u r p r i s i n g that the r e s u l t s from the T, measurements are somewhat h i g h compared to the c a l c u l a t e d c r o s s s e c t i o n s . If the ir microwave pu l s e was not a d j u s t e d c o r r e c t l y , i t would not completely i n v e r t the l e v e l p o p u l a t i o n s , so the measured r e l a x a t i o n r a t e would be higher than i t a c t u a l l y i s . The f a c t that the T 2 measurements give a l a r g e r value f o r than the T, measurements suggests that there i s another p r o c e s s that c o n t r i b u t e s to T i 1 that a l s o i n c r e a s e s l i n e a r l y w i t h the H atom d e n s i t y , probably r a d i a t i o n damping. R a d i a t i o n damping a r i s e s because the s p i n system i n t e r a c t s with the microwave f i e l d induced i n the resonator d u r i n g the f r e e i n d u c t i o n decay. There are two e f f e c t s that a r i s e because of t h i s . F i r s t , the FID i s damped by an amount p r o p o r t i o n a l to the s t r e n g t h of the s i g n a l (and hence the 180 -r n H ( x 10 m j F i g . . 2 1 . R e l a x a t i o n Rates vs. D e n s i t y 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 r a t e s r a t e s as a f u n c t i o n of d e n s i t y f o r T = 7.067 K. The s o l i d l i n e s are the l e a s t squares f i t s . 1 13 F i g . 22. Bulk Spin Exchange Cross S e c t i o n Experimental values f o r the bulk s p i n exchange c r o s s s e c t i o n . The s o l i d l i n e i s the t h e o r e t i c a l value of B e r l i n s k y and S h i z g a l . 1 14 d e n s i t y ) . Second, i t tends to s h i f t the frequency of the s i g n a l towards the resonant frequency of the c a v i t y i f the c a v i t y i s mistuned. Bloemberger and Pound [BLOE-54] attempted to d e s c r i b e the e f f e c t s of r a d i a t i o n damping by adding an a d d i t i o n a l equation, d e s c i b i n g the c o u p l i n g between the s p i n s and the microwave f i e l d , to the usual Bloch equations used to d e s c r i b e the dynamics of the spin system. Bloom [BLOO-57] obtained s o l u t i o n s to these equations i n c l o s e d form f o r the case when the resonator tuning i s c o r r e c t and T, i s i n f i n i t e . Morrow [MORR-83a] g e n e r a l i s e d Bloom's r e s u l t s f o r the case when the c a v i t y i s mistuned. The e f f e c t s of r a d i a t i o n damping are best d e s c r i b e d i n terms of the instantaneous dampng r a t e where M(t) i s the magnitude of the o s c i l l a t i n g m a g n e t i s a t i o n . If X(t) i s independent of time, then i t d e s c r i b e s an e x p o n e n t i a l decay of M ( t ) . The frequency p u l l i n g caused by the c a v i t y i s where u i s the observed frequency, CJ 0 i s the frequency in the absence of c a v i t y p u l l i n g , uQ i s the resonant frequency of the c a v i t y , and ACJ i s the f u l l width of the c a v i t y X(t) = - dM(t)/M(t)dt [V-9] CJ - CJ 0 = 2 X ( t ) ( c j 0 - CJ )/ACJ [V-10] 115 resonance. Frequency p u l l i n g was not an important e f f e c t in the experiments, s i n c e frequency measurements were only made at low d e n s i t i e s , and the c a v i t y was always tuned much c l o s e r than the width of the c a v i t y resonance. If we take the c a v i t y mistuning to be n e g l i g i b l e , then Morrow's r e s u l t s g i v e X(t) = MOT?" ( 7 e + 7 p ) Q L M ' z ( t ) / 2 + T-21 [V-11] where M' 2 i s the z component of the f i c t i t o u s m a gnetisation, and T i 1 i s the t r a n s v e r s e r e l a x a t i o n r a t e due to other p r o c e s s e s . T h i s means there i s an asymptotic c o n t r i b u t i o n to T i 1 given approximately by ( T i - 1 ) R D = * i 0 T ? " ( 7 e + 7 p)Q LMo/2 [V-12] where M 0 i s the e q u i l i b r i u m value of the z component of the f i c t i t o u s m a g n e t i s a t i o n , given by equation [11-18], In terms of the d e n s i t y of H atoms, t h i s i s ( T 2 1 ) R D = M 0 T ? " Q L h 2 ( 7 e + 7 p > 2 w 0 n H / l 6 k T [V-13] The c o n t r i b u t i o n to the t r a n s v e r s e r e l a x a t i o n given by equation [V-13] was s u b t r a c t e d from the T 2 data, g i v i n g the r e s u l t s f o r the s p i n exchange c r o s s s e c t i o n shown by the £'s i n f i g . 22. These val u e s are i n e x c e l l e n t agreement with the c a l c u l a t i o n s of B e r l i n s k y and S h i z g a l [BERL-80] and are 1 16 lower than those obtained from the T, measurements, as one would expect. VI. SUMMARY The magnetic resonance measurements d e s c r i b e d i n t h i s work have y i e l d e d s e v e r a l u s e f u l r e s u l t s . U n f o r t u n a t e l y , measurements co u l d only be made between 6.4 K and 8.2 K, si n c e the r . f . discharge would not c r e a t e atoms o u t s i d e t h i s range. The frequency s h i f t measurements made at low atom d e n s i t i e s ( n H < 8 x 10" 1 6 n r 3 ) made.it p o s s i b l e to determine s e v e r a l important parameters. The H 2 pressure s h i f t c o e f f i c e n t was found to be K = -1.78 ± 0.01 x IO" 2" Hz m3, about one t e n t h that due to "He at 1 K [MORR-83a]. Assuming the adsorbed atomic H to be a 2D gas gave the s u r f a c e frequency s h i f t as &us/2ir = -1.16 ± 0.05 MHz, and the b i n d i n g energy Eg/k = 34.04 ± 0.26 K. The s u r f a c e frequency s h i f t and b i n d i n g energy agree q u i t e w e l l with the r e s u l t s of Crampton et al obtained from s i m i l a r experiments at temperatures below 4.5 K: Aa>s/27r = -1.12 ± 0.08 MHz, and Eg/k = 35.75 ± 0.31 K. Taken together, these r e s u l t s span the temperature range from 3.2 K to 8.3 K. The good agreement, over t h i s l a r g e temeprature range, i m p l i e s that the model used f o r the adsorbed H atoms (a 2D gas) i s a good d e s c r i p t i o n of the a c t u a l adsorbed s t a t e . The c a l c u l a t i o n s of P i e r r e et al suggest that the b i n d i n g energy should be between 25.5 K and 32.7 K, depending on which of s e v e r a l e q u a l l y l i k e l y H-H2 p o t e n t i a l s they use [PIER-85]. The d i f f e r e n c e between these r e s u l t s and the experimental r e s u l t s i s about the same as the v a r i a t i o n 117 1 18 due to the use of the d i f f e r e n t p o t e n t i a l s , so i t i s not cause f o r concern. The experimental r e s u l t s a l s o c o n firm t h e i r p r e d i c t i o n that the adsorbed atoms would be almost as mobile as a p e r f e c t 2D gas. I n t e r p r e t i n g the low d e n s i t y measurements of the t r a n s v e r s e r e l a x a t i o n r a t e T i 1 , i s rather d i f f i c u l t . In a d d i t i o n to the r e l a x a t i o n due to the frequency s h i f t atoms s u f f e r while adsorbed i n a 2D gas on the s u r f a c e , there appears to be some other process a s s o c i a t e d with the H 2 s u r f a c e which c o n t r i b u t e s to T i 1 . Without understanding the nature of t h i s a d d i t i o n a l process, i t i s d i f f i c u l t to e x t r a c t a good value f o r the s t i c k i n g c o e f f i c e n t s from the temperature dependence of the r e l a x a t i o n r a t e T i 1.. N e v e r t h e l e s s , i t can be s a i d that s must be l a r g e r than 0.04, otherwise the c o n t r i b u t i o n t o T i 1 from the 2D gas w a l l s h i f t would be l a r g e r than what was observed. Assuming the e x t r a c o n t r i b u t i o n to T i 1 i s independent of temperature g i v e s s = 0.107 ± 0.07. Atoms adsorbed i n a l i g h t l y bound s t a t e , as p r e d i c t e d by P i e r r e et al [PIER-85], on the H 2 s u r f a c e cannot account fo r the r e l a x a t i o n r a t e needed to e x p l a i n the data. However, atoms t i g h t l y bound i n l o c a l i s e d s i t e s such as s u r f a c e d e f e c t s or g r a i n boundries might be able t o account f o r i t . Such t i g h t l y bound atoms have been p o s t u l a t e d to account f o r an e x t r a ESR s i g n a l i n experiments by Mayer et al studying H confined.by H 2 at 4.2 K [MAYR-81], and a l s o to e x p l a i n non-exponential l o n g i t u d i n a l r e l a x a t i o n seen by Crampton and 119 co-workers i n magnetic resonance s t u d i e s of H c o n f i n e d by H 2 below 4.5 K [NUNE-83]. U n f o r t u n a t e l y , i t i s not r e a l l y p o s s i b l e to e x t r a c t any parameters from the T 2 data using t h i s model, as there are too many p o o r l y known parameters pre s e n t . In our experiments, the recombination of H atoms was completely dominated by recombination o c c u r i n g among atoms adsorbed on the H 2 s u r f a c e . The s u r f a c e recombination c r o s s l e n g t h X was measured, and found to be a s t r o n g f u n c t i o n of temperature, r i s i n g from X = 0.5 A at 6.4 K to X = 1.1 A at 8.2 K. T h i s i s much l a r g e r than the s u r f a c e recombination c r o s s l e n g t h measured by Morrow f o r H atoms adsorbed on l i q u i d helium below 1 K (X = 0.14 ± 0.02 A f o r H on "He, and X = 0.13 A f o r H on 3He) [MORR-83a]. The sharp i n c r e a s e i n X with temperature i s probably due to an energy resonance i n the i n t e r m e d i a t e s t a t e averaged over the thermal d i s t r i b u t i o n of c o l l i s i o n e n e r g i e s . I t was a l s o p o s s i b l e to measure the e f f e c t s of s p i n exchange c o l l i s i o n s on 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 r a t e s . Values f o r the bulk spi n exchange c r o s s s e c t i o n were determined from the measured r e l a x a t i o n r a t e s by assuming that s p i n exchange o c c u r i n g i n the bulk gas predominated over that o c c u r i n g i n atoms adsorbed on the s u r f a c e . A c c o r d i n g to t h e o r e t i c a l c a l c u l a t i o n s of spi n exchange, t h i s should be true f o r T > 5 K. Atoms couldn't be s t u d i e d at temperatures where s u r f a c e sp i n exchange p l a y s a s i g n i f i c a n t r o l e because the r . f . d i s c h a r g e wouldn't c r e a t e 120 H atoms t h e r e . In order to determine the s p i n exchange c r o s s s e c t i o n from the T 2 measurements, i t was necessary to c o r r e c t f o r r a d i a t i o n damping, which made a c o n t r i b u t i o n roughly equal to that from s p i n exchange. The v a l u e s f o r the s p i n exchange c r o s s s e c t i o n obtained t h i s way agreed q u i t e w e l l with the c a l c u l a t e d values of B e r l i n s k y and S h i z g a l [BERL-80]. T h i s i m p l i e s that the method they used to c a l c u l a t e O g D takes i n t o account the s i g n i f i c a n t e f f e c t s . T h i s p r o v i d e s i n d i r e c t support f o r the c a l c u l a t i o n s of the s u r f a c e s p i n exchange c r o s s l e n g t h done by Morrow and B e r l i n s k y , s i n c e they used the same method extended to two dimensions. BIBLIOGRAPHY [ABRA-61] A. Abragam, The Principles of Nuclear Magnetism, Oxford U n i v e r s i t y Press, Oxford (1961) [ADRN-60] F . J . Ad r i a n , /. Chem. Phys., 32, 972-981 (1960) [AIPH-72] D.E. Gray, E d i t o r , American I n s t i t u t e of Physics Handbook, McGraw-Hill, New York (1972) [ANCS-77] J . Ancsin, Met r ol ogi ca, 13 79-86 (1977) [ANDR-54] P.W. Anderson, P.R. Weiss, J. Phys. Soc. Japan, 9, 316 (1954) [BALL-64] L.C. B a l l i n g , R.J. Hanson, F.M. P i p k i n , Phys. Rev., 133, A607-A626 (1964) [BENV-71] C. Benvenuti, R.S. Calder, Phys. L e t t . , 35A, 291-292 (1971) [BENV-76] C. Benvenuti, R.S. Calder, G. Passard, J. Vac. Sci . Technol. , 13, 1172 (1976) [BERL-80] A.J. B e r l i n s k y , B. S h i z g a l , Can. J. Phys., 58, 881-885 (1980) [BIPM-76] Bureau I n t e r n a t i o n a l de Poids et Measures, Metrologica, 12, 7-17 (1976) [BIPM-79] Bureau I n t e r n a t i o n a l de Poids et Measures, Metrologica, 15, 65-68 ( 1979) [BLOE-54] N. Bloemberger, R.V. Pound, Phys. Rev., 95, 8 (1954) [BLOO-57] S. Bloom, /. Appl. Phys., 28, 800 (1957) [BTTS-64] D.S. BettS, et al , J. Sci. Inst rum. , 41, 515-516 (1964) [CRAM-81] S.B. Crampton, J . J . Krupczak, S.P. Souza, J. de Physique, 42, C8-181 (1981) [CRAM-82] S.B. Crampton, J . J . Krupczak, S.P. Souza, Phys. Rev. B, 25, 4383-4395 (1982) [CRAM-84] S.B. Crampton, et a l , Hydrogen Maser Oscillation at 10 K, submitted to the proceedings of the P r e c i s e Time and Time I n t e r v a l Planning Meeting, Goddard Space Center, (1984) [CRAM-85] S.B. Crampton, p r i v a t e d i s c u s s i o n 121 122 [CTAS-72] T.C. Cetas, C.A. Swenson, Metrologica, 8, 46-64 (1972) [CTAS-76] [DASH-75] [DAUN-58] [DAUN-61] [DENN-79] [DURX-62] [DURX-79] [GREB-81 ] [HARD-81 ] [HARD-82a] [HARD-82b] [HELL-70] [HOLL-70] [HRLY-7 0] [JOCH-82] [LENN-36] T.C. Cetas, Metrologica, 12, 27-40 (1976) J.G. Dash, Films on Solid Surfaces, Academic Press, New York (1975) J.G. Daunt, K. Brugger, Z. Physik Chem. , 16, 203-212 (1958) J.G. Daunt, et al , Proceedings VII International Conference on Low Temperature Physics, U. of T. Pr e s s , Toronto (1961) J.E. Dennis, D.M. Gay, R.E. Welsh, An Adaptive Nonlinear Least Squares Algorilhim, Dept. of Computer Science, C o r n e l l U n i v e r s i t y , Ithaca N.Y. (1979) M. Durieux, et al, Temperature - Its Measurement and Control in Science and Industry, 3, 383-390 (1962) M. Durieux, et al, Metrologica, 15, 57-63 (1979) J.M. Greben, A.W. Thomas, A.J. B e r l i n s k y , Can. J. Phys., 59, 945-954 (1981) W.N. Hardy, L.A. Whitehead, Rev. S c i . Inst rum. , 52, 213-216 (1981) W.N. Hardy, M. Morrow, /. de Physique, 42, C8-171 (1982) W.N. Hardy, et al, Physica, 109-110B, 1964-1977 (1982) H. H e l l w i g , et a l , IEEE Trans, on I n s t r . and Meas. , IM-19, 200 (1970) D. Hollenbach, E.E. S a l p e t e r , /. Chem. Phys., 53, 79-86 (1970) R.T. H a r l e y , J.C. Gustafson, C.T. Walker, Cryogenics, 10, 510-511 (1970) R. Jochemson, et a l , Physica, 109-110B, 2108-2110 (1982) J.E. Lennard-Jones, A.F. Devonshire, Proc. Roy. Soc. Lond. A, 156, 6-28 (1936) 123 [LOUN-74] O.V. Lounasmaa, Experimental Principles and Methods Below 1 K, Academic Press, London (1974) [MAYR-81] R. Mayer, A. Ridner, G. S e i d e l , Physica, 1 0 8 B , 937-938 (1981) [MORR-83a] M.R. Morrow, Magnetic Resonance on Atomic Hydrogen Confined by Liquid Helium Walls, Ph.D. T h e s i s , U n i v e r s i t y of B.C. (1983) [MORR-83b] M.R. Morrow, A.J. B e r l i n s k y , Can. J. Phys. , 6 1 , 1042-1045 (1983) [NUNE-83] G. Nunes J r . , Development and Evaluation of a Theory for Tightly Bound Hydrogen Atoms, B.Sc. T h e s i s , W i l l i a m s C o l l e g e , Mass. (1983) [PETI-80] P. P e t i t , M. D e s a i n f u s e i e n , C. Audoin, Metrologica, 16 , 7 ( 1980) [PIER-85] L. P i e r r e , H. Guignes, C. L h u i l l e r , J. Chem. Phys. , 8 2 , 496-507 (1985) [RIJN-72] C. van R i j n , et a l , Temperature - Its Measurement and Control in Science and Industry, 4 , 815-826 (1972) [ROBT-70] R.J. Roberts, J.G. Daunt, Phys. L e t t . , 3 3 A , 353-354 (1970) [RSBY-85] R.L. Rusby, /. Low Temp. Phys. , 5 8 , 203-205 (1985) [SAIT-75] S. S a i t o , T. Sato, Rev. Sci. Instrum., 4 6 , 1226-1230 (1975) [SILV-82] I.F. S i l v e r a , Physica, 109-110B, 1499-1522 (1982) [VARA-81] A.J.C. Varandas, J . Tennyson, Chem. Phys. L e t t . , 11, 151 (1981) [WEIN-79] A. Weinrib, Hydrogen Atom - Molecular Hydrogen Surface Interactions, B.Sc. T h e s i s , M.I.T. (1979) [WHTE-79] G.K. White, Experiment al Techniques in Low Temperature Physics, 3ed. , Oxford U n i v e r s i t y Press, Oxford (1979) [WILL-80] D.R. W i l l i a m s , W. Lum, S. Weinreb, Microwave Journal, 2 3 , 10, 73-76 (1980) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0096204/manifest

Comment

Related Items