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Magnetic resonance studies of atomic hydrogen gas at liquid helium temperatures Whitehead, Lorne Arthur 1979

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MAGNETIC RESONANCE STUDIES OF ATOMIC HYDROGEN GAS AT LIQUID HELIUM TEMPERATURES by LORNE ARTHUR WHITEHEAD Sc., U n i v e r s i t y of B r i t i s h Columbia, 1977 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1979 © Lorne A r t h u r Whitehead, 19 79 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f p h y s l c s  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 N o v e m b e r 1 5 , 1 9 7 9 ABSTRACT Pulsed magnetic resonance s t u d i e s are r e p o r t e d f o r a gas 1 3 - 3 of hydrogen atoms at d e n s i t i e s of 3-10 X 10 cm and temper-atures of 4.2-77 K. The gas was produced by d i s s o c i a t i o n i n a room temperature R.F. d i s c h a r g e , and p i p e d through g l a s s t u b i n g i n t o the c r y o g e n i c apparatus f o r study. The magnetic resonance t r a n s i t i o n observed i s between the two lowest hyper-f i n e l e v e l s of the Is atom i n a magnetic f i e l d of 6481 Gauss where t h i s s p l i t t i n g has i t s minimum value of about 765.5 MHZ. At 77 K, spin-exchange broadening of the resonance i s observed. By v a r y i n g the number d e n s i t y of hydrogen atoms, the r a t i o of spin-exchange broadening to atomic hydrogen d e n s i t y i s o b t a i n e d , and from t h i s r a t i o the spin-exchange c r o s s s e c t i o n f o r t h i s t r a n s i t i o n i s . c a l c u l a t e d . The c r o s s s e c t i o n obtained i s 60% of the t h e o r e t i c a l v a l u e . At l i q u i d helium temperatures, the spin-exchange c r o s s s e c t i o n i s shown to be a t l e a s t 15 times s m a l l e r than t h a t a t 4 l i q u i d n i t r o g e n temperatures, as. p r e d i c t e d by theory. Fie and H 2 b u f f e r gases . are used to l i m i t the d i f f u s i o n broadening of the resonance, a l l o w i n g the o b s e r v a t i o n of s m a l l frequency s h i f t s o f the f r e e i n d u c t i o n s i g n a l . A model i s proposed i n which i n t e r a c t i o n s . o f the hydrogen atoms with the flow tube w a l l s cause these s h i f t s . . From the d i f f u s i o n broadening of the 4 resonance, the d i f f u s i o n c r o s s s e c t i o n s f o r H i n He a t 4.2 K and H i n H 2 at 5-9 K are i n f e r r e d to be 500 A 2 and 250 A* r e s p e c t i v e l y . i i i TABLE OF CONTENTS Page A b s t r a c t i i Table of Contents i i i L i s t of F i g u r e s v Acknowlegements v i CHAPTER I - INTRODUCTION 1 CHAPTER I I - THE HYPERFINE LEVELS OF THE IS STATE HYDROGEN ATOM 4 2.1 C a l c u l a t i o n of the Energy Eigenvalues 4 2.2 Magnetic F i e l d Homogeneity Requirements 8 2.3 The Technique of P u l s e d Magnetic Resonance 10 CHAPTER I I I - THE PRODUCTION OF ATOMIC HYDROGEN GAS 19 CHAPTER IV - THE 76 5 MHz PULSE SPECTROMETER 2 5 4.1 Block Diagram of the Spectrometer 25 4.2 C h a r a c t e r i s t i c s of the Spectrometer 33 4.3 Resonator Design 35 4.4 Q u a n t i t a t i v e A n a l y s i s of Graphic Output 45 CHAPTER V - EXPERIMENTAL DESIGN 49 5.1 O v e r a l l arrangement 49 5.2 Gas Handling 51 i v 5.3 The Electromagnet System 5 3 5.4 Cryogenic Techniques 55 CHAPTER VI - EXPERIMENTAL RESULTS 58 6.1 Observations a t 77K 58 6.2 Data Taken a t 4.2K With No B u f f e r Gas 66 6.3 A n a l y s i s o f the D i f f u s i o n Problem 67 6.4 Experimental r e s u l t s with B u f f e r Gas at Low Temperature 77 CHAPTER VII - SUMMARY 87 APPENDIX A: POPULATIONS OF STATES RESULTING FROM REPEATED 1Y/2 PULSES WITH SPIN-EXCHANGE RELAXATION 89 REFERENCES 93 V LIST OF FIGURES F i g u r e Page 2.1 H y p e r f i n e l e v e l s of the Is hydrogen atom as a f u n c t i o n of magnetic f i e l d 6 3.1 Hydrogen R.F. d i s c h a r g e assembly 21 4.1 Block diagram of 765 MHz p u l s e spectrometer 27 4.2 765 MHz r e s o n a t o r , t u n i n g rod, and c o u p l i n g loop 36 4.3 A t y p i c a l f r e e i n d u c t i o n s i g n a l 46 4.4 Determination of T 2 and i n i t i a l amplitude, A ( o ) , f o r a f r e e i n d u c t i o n decay 48 5.1 Diagram of probe i n dewar v e s s e l 50 6.1 1/ T2 v s H a t o m d e n s i t y 59 6.2 Free i n d u c t i o n decays a t 4.2K with v a r i o u s helium b u f f e r gas d e n s i t i e s 68 6.3 Computer s i m u l a t i o n r e s u l t s f o r A, and ft vs t, . f o r v a r i o u s v a l u e s of L \ 0 74 6.4 Computer s i m u l a t i o n r e s u l t s f o r T 2 Aco and d vs Ao 76 6.5 Log-log p l o t of T 2 vs h e 78 6.6 Resonance frequency as a f u n c t i o n of magnetic f i e l d 81 6.7 fr -f' •: ' vs 1/T„ 82 o mm 2 6.8 Comparison of experimental decay shapes with computer s i m u l a t i o n r e s u l t s 84 v i ACKNOWLEDGEMENTS I would l i k e t o acknowledge the support of Dr. A. J . B e r l i n s k y i n the s u p e r v i s i o n of t h i s p r o j e c t . I have a l s o b e n e f i t e d g r e a t l y from the help and advic e o f Dr. W. N. Hardy. F i n a l l y I wish to thank the N a t i o n a l Sciences and E n g i n e e r i n g Research C o u n c i l of Canada f o r a Postgraduate S c h o l a r s h i p . CHAPTER I INTRODUCTION The hydrogen atom, by v i r t u e of i t s s i m p l i c i t y , has long been the s u b j e c t of very p r e c i s e t e s t s of p h y s i c a l t h e o r i e s . By analogy one might wonder whether macroscopic q u a n t i t i e s of atomic hydrogen c o u l d p r o v i d e f o r the theory of condensed matter what the i s o l a t e d atom p r o v i d e s f o r quantum e l e c t r o -dynamics - an i d e a l l y simple experimental m a t e r i a l . U n f o r t -u n a t e l y , t h i s does not seem to be the case, mainly because atomic hydrogen i s c h e m i c a l l y unstable - r e a c t i n g to form H^. Atomic H a t h i g h d e n s i t i e s i s t h e r e f o r e not a simple system, and p r e s e n t s formidable experimental d i f f i c u l t i e s . R e c ently, however, i n t e r e s t has developed i n the p o s s i -b i l i t y of p r e v e n t i n g the recombination o f H i n t o by the a p p l i c a t i o n of a very s t r o n g magnetic f i e l d (of order 100 KG) 1-4 at very low t e m p e r a t u r e s - (of order .1 K) . Under these c o n d i t i o n s the e l e c t r o n s p i n s are n e a r l y 100% p o l a r i z e d , and the f a c t t h a t the t r i p l e t - s t a t e p a i r p o t e n t i a l f o r hydro-gen atoms i s mainly r e p u l s i v e i s expected to prevent recombination. Since the hydrogen atom i s a bound s t a t e of two fermions, i t i s a Bose p a r t i c l e , and i t might t h e r e f o r e be p o s s i b l e to observe B o s e - E i n s t e i n condensation i n s p i n p o l a r i z e d atomic hydrogen. For n o n - i n t e r a c t i n g p a r t i c l e s , the temper-ature f o r B o s e - E i n s t e i n condensation i s (1.1) and m and n are the mass and number d e n s i t y of the p a r t i c l e s . -2-For TB(_, = .1 K, the d e n s i t y r e q u i r e d f o r n o n - i n t e r a c t i n g 19 -3 p a r t i c l e s with the mass of a hydrogen atom i s 1.6 X 10 cm ' 22 -3 which i s much s m a l l e r than the d e n s i t y of 2.3 x 10 cm f o r the s u p e r f l u i d t r a n s i t i o n i n l i q u i d helium. I t i s hoped th a t B o s e - E i n s t e i n condensation i n s p i n - p o l a r i z e d atomic hydrogen might occur at such low d e n s i t i e s , i n which case the e f f e c t s of i n t e r a c t i o n s between the atoms would be much sm a l l e r than i n the case of s u p e r f l u i d helium. The system would thus p r o v i d e a very good t e s t of the theory of Bose-E i n s t e i n condensation. To date very l i t t l e e xperimental work has been done on atomic hydrogen gas at low temperatures, although a t room temperature a l a r g e amount of work has been done on the (atomic) hydrogen maser, and data on H-atom recombination and s p i n -7 exchange broadening of magnetic resonances have been o b t a i n e d down to l i q u i d n i t r o g e n temperature. The experimental work re p o r t e d here has been motivated by the need f o r i n f o r m a t i o n on the p r o p e r t i e s of atomic hydrogen at low temperatures which are r e l e v a n t to the p r o d u c t i o n and study of s p i n -p o l a r i z e d atomic H. Pulsed magnetic resonance was used to d e t e c t hydrogen atoms at l i q u i d helium temperatures and l i q u i d n i t r o g e n temp-g e r a t u r e . (This experiment , as w e l l as t h a t of Crampton e t 9 a l . , i s the f i r s t use of magnetic resonance on atomic hydrogen at l i q u i d helium temperatures). Although there are simpler methods of d e t e c t i n g * atomic hydrogen (such as recombination d e t e c t o r s which c a t a l y z e recombination of H i n t o and measure the r e s u l t a n t heat p r o d u c t i o n ) , magnetic resonance was employed -3-because i t i s extremely s p e c i f i c . The H-atom resonance frequency can be both c a l c u l a t e d and measured t o b e t t e r than 7 1 i n 10 . In a d d i t i o n the magnetic resonance s i g n a l p r o v i d e s i n f o r m a t i o n about the i n t e r a c t i o n s of H-atoms with one another, about t h e i r d i f f u s i o n , and about t h e i r recombination r a t e - p r e c i s e l y the type of i n f o r m a t i o n needed f o r f u t u r e " s p i n - p o l a r i z e d " atomic hydrogen s t u d i e s . In Chapter II of t h i s t h e s i s , the d e r i v a t i o n of the h y p e r f i n e l e v e l s of the Is hydrogen atom i n a magnetic f i e l d i s presented, and the technique of p u l s e d magnetic resonance i s d i s c u s s e d . Chapter I I I d e a l s w i t h the p r a c t i c a l problem of producing hydrogen atoms i n a d i s c h a r g e , and d i s c u s s e s the discharge, apparatus used i n t h i s experiment. Chapter IV co n t a i n s a d e s c r i p t i o n of the p u l s e d magnetic resonance system, and the novel r e s o n a t o r employed. In Chapter V, the o v e r - a l l experimental design i s d i s c u s s e d , with emphasis on the c r y o g e n i c , gas h a n d l i n g , and magnetic f i e l d measurement techniques. F i n a l l y , Chapter VI i s a p r e s e n t a t i o n o f the experimental r e s u l t s and t h e i r a n a l y s i s . CHAPTER II THE HYPERFINE LEVELS OF THE IS STATE HYDROGEN ATOM 2.1 C a l c u l a t i o n of the Energy E i g e n v a l u e s . The Hamiltonian f o r the s p i n degrees of freedom of the Is hydrogen atom i s • O l - S - 2H/<?Iz + 2H/<eS2 (2.1) where H i s the magnitude of the magnetic f i e l d (which i s i n the p o s i t i v e z d i r e c t i o n ) , M.i. i s the proton magnetic moment o -23 (M>. = 1.41 X 10 erg/Gauss), X i s the e l e c t r o n magnetic p e -21 -» -» moment (Me = 9.27 X 10 erg/Gauss), I and S are the proton e and e l e c t r o n s p i n o p e r a t o r s , and a i s the h y p e r f i n e c o u p l i n g constant (a = 9.41 X 1 0 ~ 1 8 e r g ) . I t i s convenient to work i n the b a s i s i n which Ir and z S^ are d i a g o n a l . The b a s i s s t a t e s w i l l be r e p r e s e n t e d as If )t* \ \ where f o r ^ r e f e r to the e l e c t r o n s p i n being up or down, and and k r e f e r s i m i l a r l y t o the proton s p i n . In t h i s b a s i s , the matrix r e p r e s e n t a t i o n f o r the operators appearing i n the Hamiltonian are: J.2. - 2. 0 o o \ ( n ) -I 0 o (U) 0 1 o o o I o o (n) it 4) uf) ut) (2.2) -5-0 o o\(**) 1 o o m t ) o -i o ( it) O O - l / U i ) ( f f ) ( t i ) ( i t ) ( 1 * ) 0 o 1 - S z J i o o o \ ( n ) 0 - 1 2 O ( t i ) O Z -I O 1(4*) O O O I / ( i i ) (tt ) ( T i ) ( W ) U i ) and thus the r e p r e s e n t a t i o n o f the Hamiltonian i s o ° \ (n) o (ti) 0 (2.3) 0 HUst^J-aA a/a O 0 O -Hk^HA/ (ii) in) ( T i ) ( 4 * ) I i i ) I t i s immediately e v i d e n t t h a t the s t a t e s |T$ > and | 4 i > are a l s o e i g e n s t a t e s of t h i s Hamiltonian, with energy H(/fe-^>) + a/4 and -RUfe-Mp) + a/4 r e s p e c t i v e l y . These are s t a t e s 4 and 2 as shown i n F i g . 2.1. A f t e r some s t r a i g h t - f o r w a r d a l g e b r a , i t i s found t h a t the other two 6.481 H(kGauss) Figure 2.1 Hyperfine levels of the Is hydrogen atom as a function of magnetic f i e l d . -7-eigenvalues shown i n F i g . 2.1 are with e i g e n s t a t e s <///= O>S0 l * *> - SifnO | T i > l//3= si*9 llt> 4 cosBln) where -Fan 29 = a / 2 t & - b 4 p ) H At zero f i e l d , three of these s t a t e s , (2,3,4), are degenerate, and are the three J=l e i g e n s t a t e s of the t o t a l s p i n o p e r a t o r J = I + S, w h i l e the other, (1), has t o t a l s p i n 0. The energy d i f f e r e n c e , a, between s t a t e 1 and 2, 3, or 4 can be c a l c u l a t e d f a i r l y a c c u r a t e l y from the e l e c t r o n i c wave f u n c t i o n and quantum ele c t r o d y n a m i c s " ^ , and has been measured 12 to 2 p a r t s i n 10 by means of the hydrogen maser which operates a t the zero f i e l d transition^"'". At very high f i e l d s , the energy e i g e n s t a t e s approximate the I?', e i g e n s t a t e s . In the r e g i o n of moderate f i e l d s , there i s a change-over from t o t a l s p i n e i g e n s t a t e s to these h i g h - f i e l d s t a t e s , and i t i s i n t h i s change-over r e g i o n t h a t we f i n d t h a t the energy d i f f e r e n c e E~-E, has a minimum v a l u e . (2.4) (2.5) (2.6) (2.7) (2.8) -8-The value o f E„-E, i s (2.9) which has a minimum value f o r (2.10) at which p o i n t the value o f the energy d i f f e r e n c e i s y\-Ap/MjL. (2.11) The v a l u e of a/h i s 1 420 405 751.768(2) Hz , and of \ -3 i s 1.519 270 335(15) X 10 f o r an e l e c t r o n and proton i n a 12 hydrogen atom. Together these known values g i v e an energy d i f f e r e n c e i n frequency u n i t s of at E9 = 6481 Gauss. Because of the accuracy to which f0 i s known, the magnetic resonance s i g n a l i s a r a t h e r s e n s i t i v e probe of the i n t e r a c t i o n s o f the atoms with t h e i r e n v i r o n -ment (see Chapter VI, S e c t i o n 4). 2.2 Magnetic F i e l d Homogeneity Requirements. There are two advantages to o b s e r v i n g the resonance at a t u r n i n g p o i n t . F i r s t , a minimum frequency can be determined by measurements made around the minimum, thereby removing the requirement f o r high a b s o l u t e accuracy i n the knowledge of the magnetic f i e l d . Thus while the magnetic f i e l d might be measured with an accuracy of onl y one p a r t i n 10 , the value obtained £ = 7 6 5 - H83 207.7(3) Hz. (2.12) -9-f o r fo would be l i m i t e d i n accuracy o n l y by the accuracy of the frequency measurements a t f i e l d s around H o . Second, and more important, i s the f a c t t h a t a t Ho the d e r i v a t i v e o f the resonance frequency with r e s p e c t to magnetic f i e l d i s zero, and hence to f i r s t o rder inhomogeneities i n the magnetic f i e l d do not broaden the resonance l i n e . The a c t u a l degree of broadening of the t r a n s i t i o n due to inhomogeneity can be estimated from a T a y l o r ' s expansion about H o. For inhomogeneities of the order of 1 Gauss i n the sample r e g i o n , which i s t y p i c a l f o r our magnet, onl y the second term i n the expansion i s a p p r e c i a b l e . I f we l e t AH be the d i f f e r e n c e between the a p p l i e d f i e l d and H o , and 5 H be the magnitude of the inhomogeneity of the f i e l d i n the sample r e g i o n , we o b t a i n a broadening i f equal t o : Thus i n the t y p i c a l measurement range of about 10 Gauss to e i t h e r s i d e of H 0, the maximum broadening w i l l be about 25 Hz, and a t H 0 i t w i l l be 1.3 Hz. T h i s i s v e r y s m a l l r e l a t i v e to the measurement accuracy of the magnetic resonance frequency, which was i 3 0 Hz a t b e s t . (2.13) -10-2.3 The Technique of Pulsed Magnetic Resonance Pulsed magnetic resonance i s widely used, and many good treatments of t h i s technique are a v a i l a b l e f o r the case of a s p i n 1/2 p a r t i c l e . In t h i s chapter, the r e l e v a n t r e s u l t s f o r the s p i n 1/2 system are summarized i n a n o t a t i o n which i s a p p l i c a b l e to any two s t a t e system. I t i s then argued t h a t to a good approximation we can t r e a t the hydrogen atom i n our experiment as a two s t a t e system, where i t i s the two lowest h y p e r f i n e s t a t e s which are of i n t e r e s t . The r e s u l t s f o r a two s t a t e system are then a p p l i e d t o t h i s h y p e r f i n e t r a n s i t i o n . In the two s t a t e system, we l a b e l the s t a t e s J l ^ , and such t h a t «)=. ali> + b|2> C2.14) We d e f i n e the v e c t o r o p e r a t o r ^ , whose matrix r e p r e s e n t -a t i o n i s S - 0/2 , where c7 X ) <5"y , and C"z are the P a u l i s p i n m a t r i c e s . The Hamiltonian can be expressed i n the form +t'S (.2.15) where ho and h may be time dependent. L e t , ? be the thermal average of the e x p e c t a t i o n value of s , f o r a system of i d e n t i c a l , n o n - i n t e r a c t i n g p a r t i c l e s o f the type being d i s c u s s e d . By the d e f i n i t i o n of the d e n s i t y matrix, 0^, we have t h a t J/~T\-/oS (2.16) -11-which i m p l i e s t h a t = 1/2 + 2j-S = f{k+Jz M-iJy L A + U y Jfe-J-z I (2.17) Since **/ = [ (.2.18) equations (2.15, 2.17) gi v e (2.19) In a p u l s e d magnetic resonance experiment, the Hamilton-i a n c o n s i s t s of two p a r t s - a constant term of the form U 0Sz and a time dependent term of the form U|(t)(cos(u;t)5x + S'm(ojt)Sy) . (2.20) U-^t) i s zero f o r t<0, a constant f o r 0<(t<T, and zero f o r t>r. For t<0, the system i s i n thermal e q u i l i b r i u m , and hence a l l observables are constant. From elementary s t a t i s t i c a l theory i s given by (.2.21) The Hamiltonian i s d i a g o n a l i n t h i s r e p r e s e n t a t i o n f o r t<0, and we o b t a i n R 0 O Pz (2.22) -12-where P-^  and P 2, the p o p u l a t i o n s of s t a t e s 1 and 2 are giv e n by e x p ( - / ^ ( E l - E 2 W e x p ( ^ 2 ) ^ K + ?x=l ( 2 < 2 3 ) From equation (2.16), we see t h a t J z . = (r-p^/z Jx=o } t<0 J y - 0 (2.24) The s o l u t i o n to (2.19) f o r 0<t<T i s g e n e r a l l y o b t a i n e d by t r a n s f o r m i n g to a c o o r d i n a t e frame r o t a t i n g w i t h angular frequency CO about the z a x i s . The s o l u t i o n i n the l a b o r a t o r y frame i s ^a^R l^ tk lR^o - ^ t l c + 6 0 . J z ( 0 ) (2.25) where R i s the r o t a t i o n m a t r ix, such t h a t R(r) causes a r o t a t i o n of angle | r | about the a x i s r/\r\ , co0 = U>A, and tU| =UiA. When |wo-wl>^60i > ^ ( i ) never d e v i a t e s f a r from the d i r e c t i o n k-', (the maximum angular d e v i a t i o n being 9max- 2CU|/|(Jo-w|) . Since i n p r a c t i s e U-^ i s g e n e r a l l y much small e r than Uo , Wo , and thus the f r a c t i o n a l d i f f e r e n c e between U> and ctfo must be very s m a l l i n order to cause any s u b s t a n t i a l t r a n s v e r s e component of Jf. From the form of the d e n s i t y matrix (2.17), we see t h a t i t i s a l s o e s s e n t i a l l y unchanged when |O)0-U)|» tt>i . In p r a c t i s e , t o - i s chosen to be -13-very c l o s e to ajo, such t h a t IOJo-cu, 1 <&(A)\ , and (2.25) then becomes J t ( i y j * ( o ) c o s ( o U ) ' Jx(t)=JzMsin(co,i)?in(uJbi) >0<t<T (2.26) Jy = sin M cos (Uoi) J If 'r='TTf/ 2 s where r ^ 2 . cot = rr/z ( 2 > 2 7 ) then a t t = Tir/a. , we have: Jz(^/z)=0 (2.28) Jy (rtr/z) = "J z (o) COS (6JoTTf/^  For t ^Trr/i , ( i . e . w i t h the time dependent p a r t o f the Hamiltonian "turned o f f " ) , the s o l u t i o n to (2.19) i s c l e a r l y J « ( ^ = J z ( o ) S i K ( ^ >-t>Tir/i (2.29) J y ^ -Jl(^ COS (C0o^  Since the d e n s i t y matrix f o r t h i s system i s completely determined b y J , the time dependence of the e x p e c t a t i o n value f o r any ope r a t o r can be determined from the above r e s u l t s . A l l of these r e s u l t s are exact and a p p l i c a b l e to any 2-state system. The system u s u a l l y t r e a t e d i n t e x t books i s an ensemble of n o n - i n t e r a c t i n g spin-1/2 p a r t i c l e s , with magnetic moment^, i n a D.C. magnetic f i e l d Ho i n the d i r e c t i o n k. For 0<t<.Tti7a , a time dependent magnetic f i e l d , Hi (t) , i s a p p l i e d : Hi(t) = Hi (t cos(w^ + j sin(oH^ (2.30) The Hamiltonian f o r t h i s system i s #=-2 /< I-H (2.31) and i t i s e a s i l y v e r i f i e d t h a t i n the above n o t a t i o n , J = Pl/lA Uo=2^rlo ^ l = 2^Hi (2.32) UJe= 2jJL Ho A cu, = iji H* A Before a p p l y i n g the above r e s u l t s to the lowest hydrogen h y p e r f i n e t r a n s i t i o n , we must c o n s i d e r three d i f f e r e n c e s between the above treatment and our experimental c o n d i t i o n s . A) R e l a x a t i o n . The system w i l l r e t u r n to e q u i l i b r i u m e v e n t u a l l y , as a r e s u l t of i n t e r a c t i o n s not contained iny¥. The e f f e c t s of these i n t e r a c t i o n s are o f t e n a c c u r a t e l y d e s c r i b e d by a m o d i f i c a t i o n o f (2.19) : J = J x t - J*t+Jr* - Jz-M(o) j (2.33) \ T a . T i where Azi^ i s given by (2.24). -15-Equations (2.33) are c a l l e d the Bloch equations. The e f f e c t of the a d d i t i o n a l terms i s to make the t r a n s v e r s e magnitude decay e x p o n e n t i a l l y to zero with c h a r a c t e r i s t i c time T 2 , of J and to r e t u r n -Jz. to i t s e q u i l i b r i u m value z^.(o} with c h a r a c t e r -i s t i c time T^. E x p e r i m e n t a l l y , U ^  can be made l a r g e enough to ensure t h a t "FTTIX <K ,T 2, and as a r e s u l t equations (2.26) are s t i l l v a l i d . Equations (2.29) become J , ( t H l - e x P ( - t A ) ) - ^ ( o ) Jx Ci) - exp (- i /Ti) Jz to) Si nffjcfi (.2.341 Jy{i)= - e*?{-iIT^JZto) cos(u*L) The e f f e c t of some types of i n t e r a c t i o n s i s t o cause a frequency s h i f t f o r the p r e c e s s i o n o f ^ j . , as w e l l as a decay; t h i s i s d e s c r i b e d by a l l o w i n g T 2 to have an imaginary component, i . e . AO) in ^ 1 (B) The Counter-Rotating F i e l d . In t h i s , (and most), experiments, i t i s most convenient to apply a time dependent Hamiltonian of the form Oiexp(t)= Uiexp Cos(o>t)s* , or etjujvajen+iy (2.36) UiexpW^UilcosC^Sx+Sintoo^Sy) + Ui (coS(-uH)s*+Si'n(-wt)Sy) (2.37) where Ui = U|exp/2 The f i r s t term i s of the same form as (2.20) while the second term has n e g l i g a b l e e f f e c t , s i n c e i t s frequency d i f f e r s by about 2 C0o from U)e. (As d i s c u s s e d e a r l i e r , a time dependent component t h i s f a r o f f resonance w i l l cause an angular d e v i a -t i o n i n ^  of 8— \ctfj.~u)\ — 2Wo .) Thus the e f f e c t of the time - I n -dependent Hamiltonian i n (2.36) i s the same as t h a t i n (2.20), where [)1 = Ut eyp /2. (C) Other St a t e s In many magnetic resonance experiments, the system i n q u e s t i o n has more than two p o s s i b l e s t a t e s , y e t i t i s a v a l i d approximation to t r e a t i t as though the o n l y s t a t e s are the two which d i f f e r by energy "fi (V0, s i n c e the e f f e c t o f the time dependent p a r t of the Hamiltonian on the elements of the d e n s i t y matrix o u t s i d e the m a n i f o l d of these two s t a t e s i s very s m a l l . T h i s i s a r e s u l t of the . f a c t t h a t t h e : frequency of the time dependent p a r t of the Hamiltonian must be very c l o s e to the resonance frequency i n order to s i g n i f i c a n t l y change the d e n s i t y matrix elements f o r a p a i r of s t a t e s . As long as the energy d i f f e r e n c e s between a l l other p a i r s of s t a t e s are s i g n i f i c a n t l y d i f f e r e n t from "KuJ0, the time dependent Hamilton-i a n w i l l be f a r o f f resonance and hence by analogy w i l l not change the d e n s i t y matrix elements corresponding to these p a i r s of s t a t e s . For the purposes of t h i s magnetic resonance experiment, then, i t i s p e r m i s s i b l e to t r e a t the hydrogen atom as a two s t a t e system with energy e i g e n s t a t e s (1) and (2) shown i n F i g . (2.1). The t r a n s i t i o n between these two s t a t e s i s observed by a p p l y i n g an o s c i l l a t i n g magnetic f i e l d i n the t d i r e c t i o n : HiexpU^ t Hiexp COStVt) (2.38) and o b s e r v i n g the v o l t a g e induced i n a r e s o n a t o r by the -17-o s c i l l a t i n g x component o f the ensemble-magnetization, The time dependent Hamiltonian i s (2.39) whererm , the x component of the magnetic moment operator i s , Wx = - 2/teSx (2.40) The ma t r i x elements of ly^ are e a s i l y e v a l u a t e d : Cos(6)Mp+sin(9Ue. Thus O /I2> 12> (2.41) Irrix = 2 (cos(BUf> + sin(e)Mfi)Sx (2.42) and the time dependent Hamiltonian (2.39) g i v e s a value f o r 0i exp i n (2.36) of U (e*j> = 2 (cos (9)jUP + Sin (6)Me) Hi e x P and t h e r e f o r e Ul = (COS(B)MP + Sin(&)Me% exp T77/2 i s given by (2.27): T i r / Z = i j S ^(cos(e)MP+ slnCfiUk) r l i e * P ) ~ (2.43) (2.44) (2.45) The ensemble average of IT) ' i s <mx>= l(cos($U?+Sin(0]U&)<S*> = 2(cos(0ltf/> + Sin (0)Me)J* (2.46) Thus a f t e r the T f/2 p u l s e , ^nf) x) o s c i l l a t e s with frequency U)0 , and i n i t i a l amplitude A (rVfy) = 2 (cos(eUP 4 sin(e)/ie)Jz (o) (2 .47) -18-where Jz(0)=(?\-li)/2 (2.48) These r e s u l t s are the same as one would o b t a i n f o r s p i n 1/2 p a r t i c l e s w i t h magnetic moment cos(B)jUp +sin(6)Me. (2.49) -22 The value of j/etf xs 3.76 X 10 erg/Gauss or 26.6//j>. -19-CHAPTER I I I THE PRODUCTION OF ATOMIC HYDROGEN GAS In t h i s experiment, atomic hydrogen gas was produced by d i s s o c i a t i n g H 2 molecules i n an R.F. d i s c h a r g e . The important design c h a r a c t e r i s t i c s f o r the dis c h a r g e assembly a r e : 1) R.F. frequency, power, and c o u p l i n g 2) Discharge gas pressure 3) Gas flow r a t e 4) Type of w a l l s u r f a c e These w i l l be d i s c u s s e d s e q u e n t i a l l y . In an R.F. d i s c h a r g e , the e l e c t r o d e s are o u t s i d e the di s c h a r g e tube, and the frequency must t h e r e f o r e be h i g h enough t h a t charge does not b u i l d up on the w a l l s o f the discharge tube d u r i n g the R.F. c y c l e and thus screen out the e l e c t r i c f i e l d . We chose t o operate a t 145 MHz because we had a narrowband commercial power a m p l i f i e r i n t h i s range; t h i s frequency was found to be s u i t a b l e . In order to d i s s o c i a t e 18 —1 the H 2 molecules a t the t y p i c a l r a t e o f 7.5 X 10 sec , the d i s s o c i a t i o n energy f o r H 2 of 4.5 eV leads t o a r e q u i r e d power in p u t of about 2- watts. Of course there are many other forms of power d i s s i p a t i o n i n the d i s c h a r g e , and so i t i s a d v i s a b l e to have an R.F. power i n p u t c o n s i d e r a b l y g r e a t e r than t h i s . By f e e d i n g the 4Watt 145 MHz output o f a homemade o s c i l l a t o r - a m p l i f i e r i n t o the power a m p l i f i e r , we o b t a i n e d a maximum output o f 75 Watts. T h i s R.F. power was coupled t o the dis c h a r g e assembly v i a a matching network c o n s i s t i n g of two v a r i a b l e l e n g t h t u n i n g stubs separated by a 1/4 wave l e n g t h of the t r a n s m i s s i o n l i n e used. A B i r d Model 4 3 d i r e c t i o n a l power - 2 0 -meter was employed i n the t r a n s m i s s i o n l i n e to measure the amount of t r a n s m i t t e d and r e f l e c t e d power. F i g . 3.1 shows the d i s c h a r g e tube and R.F. e l e c t r o d e assembly used i n t h i s experiment. The e l e c t r o d e s , A, together with the 8-turn, 1.5 cm diameter c o i l , B, formed a tank c i r c u i t w i t h resonant frequency 145 MHz to match the R.F. power source. The t r a n s m i s s i o n l i n e coupled to the tank c i r c u i t by means o f two turns of wire, C, i n F i g . 3.1, wound on the 8-turn d i s c h a r g e c o i l . R.F. s h i e l d i n g was accomplished by housing the d i s c h a r g e assembly i n a copper box l a r g e enough to a v o i d detuning the r e s o n a t o r . (About 6" X 6" X 6".) C o o l -in g of the dis c h a r g e tube was accomplished by blowing compress-ed a i r over i t . I t was found t h a t the d i s c h a r g e would operate over a range of pres s u r e s from as low as a few m i l l i T o r r to about one T o r r . Under most c o n d i t i o n s , however, the d i s c h a r g e r e q u i r e d some e f f o r t to s t a r t . One method i n v o l v e d adjustment of the t u n i n g network to opt i m i z e the R.F. c o u p l i n g when the disc h a r g e was i n the " o f f " s t a t e , thus producing very h i g h v o l t a g e s i n the d i s c h a r g e r e s o n a t o r c i r c u i t . A f t e r the d i s - : charge s t a r t e d , the network was re-tuned to opt i m i z e c o u p l i n g under o p e r a t i n g c o n d i t i o n . Another method which worked as w e l l was to s t a r t the d i s c h a r g e with a T e s l a c o i l . The g r e a t e s t numbers of atoms were g e n e r a l l y d e t e c t e d i n the c r y o s t a t when the v i s i b l e l i g h t emitted from the d i s c h a r g e was predominantly t h a t of the H<* emission l i n e . T h i s c o i n c i d e d with maximization of the R.F. power d i s s i p a t e d i n the d i s c h a r g e . Under these c o n d i t i o n s the c o l o r of the d i s c h a r g e was an -21-F i g u r e 3 . 1 H y d r o g e n R .F. d i s c h a r g e a s s e m b l y (3/4 f u l l s c a l e ) -22-extremely b r i g h t , deep red. Thus a good measure of the q u a l i t y of the d i s c h a r g e was the r a d i a n t l i g h t i n t e n s i t y as measured by a cadmium s u l f i d e p h o t o - r e s i s t o r p l a c e d i n the copper box near the d i s c h a r g e tube. (This d e v i c e weights the red p o r t i o n of the v i s i b l e spectrum more h e a v i l y than does the human eye.) T h i s " q u a l i t y " measurement was u s e f u l f o r the purpose of o p t i m i z i n g the R.F. c o u p l i n g , and f i n d i n g the most s u i t a b l e gas p r e s s u r e , which was g e n e r a l l y i n the range of 50-300 m i l l i T o r r of E^. 18 —1 The t y p i c a l H 2 flow r a t e chosen was about 7.5 X 10 sec , 3 as t h i s would d e p o s i t 1 cm of s o l i d hydrogen i n the c r y o s t a t per hour of o p e r a t i o n , which was deemed to be the maximum safe r a t e . The o n l y p r a c t i c a l way to l i m i t the flow under these c o n d i t i o n s was by choosing the c o r r e c t diameter f o r the e x i t h o l e , D i n F i g . 3.1, of the d i s c h a r g e tube. The formula f o r the e x i t r a t e of gas from a hole.whose diameter 13 i s s m a l l e r than the mean f r e e path i n the gas i s where A i s the area of the h o l e , and n i s the gas d e n s i t y . With 18 —1 a flow r a t e of 7.5 X 10 sec and t y p i c a l d e n s i t y of 16 — 3 2 X 10 cm , (3.1) i m p l i e s a diameter f o r the e x i t hole of 1.0 mm. Since the mean f r e e path i s of order 1 mm a t t h i s d e n s i t y , t h i s r e s u l t was expected to be roughly c o r r e c t , as was found to be the case. From simple d i f f u s i o n theory, one can estimate the mean r a t e a t which a g i v e n hydrogen atom i n the d i s c h a r g e w i l l c o n t a c t the tube w a l l . At .3 T o r r , i n a tube of t h i s s i z e , (3.1) -23-3 -1 t h i s r a t e i s of order 5 X 10 sec . At a flow r a t e of 1 8 — 1 3 7.5 X 10 sec , a di s c h a r g e volume of 5 cm , and .3 T o r r p r e s s u r e , the mean r e s i d e n c y time f o r an atom i n the flow tube i s 10 msec, and thus i t i s expected t h a t an atom w i l l c o n t a c t the tube w a l l about 50 times before l e a v i n g the d i s -charge. C l e a r l y , then, i t i s d e s i r a b l e to choose a w a l l m a t e r i a l which does not s t r o n g l y c a t a l y z e the recombination of H i n t o E^. ( I t has long been known t h a t metal e l e c t r o d e s are p a r t i c u l a r l y bad i n t h i s r e s p e c t - t h i s i s the main reason f o r employing an R.F. d i s c h a r g e with e x t e r n a l e l e c t r o d e s ) . We used a pyrex discharge tube, and cleaned the i n n e r s u r f a c e as f o l l o w s : I t was f i r s t washed with soap and water, t r i c l o r o e t h y l e n e , acetone, and then water. The tube was then f i l l e d w i t h chromic a c i d , d r a i n e d , and r i n s e d with d i s t i l l e d water. Although t h i s procedure r e s u l t e d i n good q u a l i t y d i s c h a r g e s , t h i s q u a l i t y would d e t e r i o r a t e with time, presumably because of i m p u r i t i e s i n the gas e n t e r i n g the dis c h a r g e or because of degradation of the g l a s s s u r f a c e by the chemical a c t i o n o f the d i s c h a r g e . T h i s problem was s o l v e d with the well-known technique of c o a t i n g the d i s c h a r g e tube with p o l y p h o s p h o r i c a c i d - a very v i s c o u s l i q u i d which remains as a f i l m on the s u r f a c e o f g l a s s and which has a very low vapour p r e s s u r e ( l e s s than a few m i l l i T o r r a f t e r about a day of pumping.) Although we do not understand why the a c i d improves the d i s c h a r g e , i t works so w e l l t h a t we found i t a d v i s a b l e to use i t . In order t o minimize recombination i n the pyrex flow tube l e a d i n g from the di s c h a r g e to the d e t e c t i o n r e g i o n , the i n t e r i o r -24-of the tube was t e f l o n coated. The tube was f i r s t c l e a n e d i n the same manner as the d i s c h a r g e tube, and then coated with duPont FEP Code 120 t e f l o n d i s p e r s i o n d i l u t e d with an equal q u a n t i t y of d i s t i l l e d water. A f t e r the f i l m was d r i e d with a slow flow of helium, i t was heated i n an oven to 390°C under a flow of oxygen, and then c o o l e d s l o w l y ( l e s s than 1°C per minute) to below 200°C. At t h i s p o i n t the oven was turned o f f and allowed to c o o l to room temperature.''" The connection of the d i s c h a r g e tube to the gas h a n d l i n g system i s d i s c u s s e d i n S e c t i o n 5.2. -25-CHAPTER IV THE 765 MHz PULSE SPECTROMETER 4.1 Block Diagram of the Spectrometer As was d i s c u s s e d i n S e c t i o n 2.3, the e l e c t r o n i c system needs f i r s t to pr o v i d e s h o r t p u l s e s of R.F. power a t the resonance frequency, with the c o r r e c t product o f i n t e n s i t y and time to produce aTf/2 p u l s e . In a d d i t i o n , immediately a f t e r the p u l s e i s sent down the t r a n s m i s s i o n l i n e , the system must be able to r e c e i v e and a m p l i f y the extremely weak f r e e -15 i n d u c t i o n decay, ( t y p i c a l l y 5 X 10 watts) a r r i v i n g along the same t r a n s m i s s i o n l i n e . Next, the system must process the r e c i e v e d s i g n a l i n order t o make i t u s e f u l to the experimenter, by t r a n s l a t i n g i t to a low frequency. F i n a l l y , adequate s i g n a l to n o i s e r a t i o s r e q u i r e averaging of many f r e e i n d u c t i o n decays. Since t y p i c a l f r e e i n d u c t i o n decays we observed l a s t e d a time i n the range of 10-1,000>csec, a reasonable c h o i c e f o r the p u l s e d u r a t i o n was a few^Msec, which r e q u i r e d p u l s e power 13 l e v e l s o f tens o f m i l l i w a t t s , some 10 g r e a t e r than the t y p i -c a l s i g n a l s t r e n g t h . Yet the a m p l i f i e r system which measures t h i s very weak s i g n a l must be able to recover from the s a t u r a -t i o n caused by t h i s huge p u l s e i n a time of a few>usec, and a t the same time have an in p u t n o i s e l e v e l as low as p o s s i b l e . The p u l s e spectrometer system which we employed, and which s a t i s f i e d these requirements admirably, was designed by Walter N. Hardy. In the d e s c r i p t i o n o f the design o f t h i s system which f o l l o w s , the manufacturer and model of each p i e c e of equipment w i l l be gi v e n , and where the p i e c e i s "homemade", r e f e r e n c e w i l l be given to a p u b l i c a t i o n d e s c r i b i n g i t s d e s i g n . -26-A b l o c k diagram of the spectrometer i s gi v e n i n F i g . 4.1. The diagram has been d i v i d e d i n t o f o u r s e c t i o n s . The top most s e c t i o n produces the p u l s e s o f R.F. power a t the resonant frequency, and sends them to the d i p l e x e r (the next s e c t i o n down.) In a d d i t i o n , i t sends a continuous R.F. s i g n a l which i s coherent w i t h the p u l s e s to the si g n a l , p r o c e s s o r (bottom s e c t i o n ) , f o r use as a r e f e r e n c e . The d i p l e x e r a c t s as an e l e c t r o n i c r e l a y . In the "on" s t a t e , i t feeds the R.F. p u l s e from the p u l s e generator to the re s o n a t o r i n the experimental apparatus, and i n the " o f f " s t a t e i t connects the output from the re s o n a t o r ( i . e . the f r e e i n d u c t i o n decay and the noise) to the a m p l i f i e r ( t h i r d s e c t i o n down). The a m p l i f i e r s e c t i o n a m p l i f i e s the s i g n a l to the l e v e l o f m i l l i v o l t s , and feeds i t to the s i g n a l p r o c e s s o r , which mixes i t w i t h the r e f e r e n c e s i g n a l from the R.F. p u l s e generator to transform i t down to a moderately low frequency (rJlOO kHz) , and then s i g n a l averages and pr e s e n t s the output on an x-y r e c o r d e r . The d u r a t i o n and r e p e t i t i o n r a t e o f the R.F. p u l s e i s c o n t r o l l e d by the d u r a t i o n and r e p e t i t i o n r a t e of the D.C. p u l s e from the p u l s e generator. The. p u l s e generator a l s o t r i g g e r s the d i p l e x e r , so t h a t i t i s "on" when the R.F. puls e a r r i v e s , as w e l l as the sweep on a mo n i t o r i n g o s c i l l o -scope and the s i g n a l averager. The d e t a i l s of these s e c t i o n s w i l l now be d i s c u s s e d . The R.F. s i g n a l used to produce the R.F. p u l s e s was generated by an HP 8640A s i g n a l generator, used w i t h an out-put of about 100 mW. The frequency of t h i s generator was d i v i d e d by 10 and phase lo c k e d to the Schomandl STO 100M -27-R. F. Pulse Generator synth-e s i z e r A m p l i f i e r 765 amp MHz. amp S i g n a l Processor Figure 4 .1 765 MHz. pulse spectrometer. -28-frequency s y n t h e s i z e r . The d i v i d e - b y - t e n o p e r a t i o n was achieved by modifying the HP 532 7A frequency counter so t h a t the output of i t s p r e s c a l e r was a v a i l a b l e a t the back of the instrument. The counter sampled a p o r t i o n of the power from the R.F. generator v i a a lOdb d i r e c t i o n a l c o u p l e r and d i s p l a y e d the frequency f o r r e f e r e n c e purposes. In a d d i t i o n , the d i v i d e d -by-ten s i g n a l became the l o c a l o s c i l l a t o r i n p u t of a mixer, so as t o produce a beat frequency w i t h the s i g n a l from the s y n t h e s i z e r . T h i s d i f f e r e n c e frequency output of the mixer was then connected to the F.M. i n p u t of the R.F. generator, thus completing the phase-lock loop. T h i s somewhat e l a b o r a t e arrangement was r e q u i r e d to h o l d the frequency d r i f t o f the R.F. generator t o a l e v e l much l e s s than the r e s o l u t i o n of the l i n e c e n t r e , which f o r good s i g n a l s was as small as 30 Hz. As w i l l be d i s c u s s e d s h o r t l y , t h i s s i g n a l g e n e r a t i o n system was set to produce a frequency of approximately one-half the resonance frequency, and was doubled i n l a t e r stages. In order to o b t a i n a r e f e r e n c e s i g n a l a t the resonance frequency, f o r use i n the s i g n a l p r o c e s s i n g s e c t i o n , a lOdb reduced s i g n a l was taken from a second d i r e c t i o n a l c o u p l e r , and f e d through a doubler t o the s i g n a l p r o c e s s i n g s e c t i o n . I t i s convenient at t h i s p o i n t to d i s c u s s two aspects of the s i g n a l generator frequency, the f i r s t being t h a t we operate a t h a l f the resonant frequency and double a t the f i n a l power stage. I n i t i a l l y t h i s method was chosen because our R.F. generator c o u l d not produce a 765 MHz s i g n a l . However, the method a l s o o f f e r e d a d i s t i n c t advantage - the extremely l a r g e o n - o f f r a t i o f o r the r . f . p u l s e s i s more e a s i l y accomplished when the p u l s e i s produced by dou b l i n g another s i g n a l , s i n c e the d o u b l i n g was performed by a component d r i v e n i n t o non-l i n e a r o p e r a t i o n - a c o n d i t i o n which an be r a p i d l y s t a r t e d and stopped with a D.C. c o n t r o l p u l s e . The second aspect of the R.F. frequency warranting d i s c u s s i o n i s t h a t i t was s e t to produce a doubled s i g n a l o f f resonance by an amount A f , so t h a t the r e f e r e n c e t o the s i g n a l p r o c e s s i n g s e c t i o n would y i e l d a beat frequency & f upon mixing with the f r e e i n d u c t i o n s i g n a l . A f , ( u s u a l l y between 10 and 50 kHz.), was chosen so th a t between 5 and 20 beats would occur i n T 2 , i n order to allo w measurement of the envelope of the decay while minimiz-i n g the no i s e bandwidth. As d i s c u s s e d i n S e c t i o n 2.3, the angular frequency o f f s e t , 2trAf, must be small compared t o Wi = TT/2 (TV/a.)"1 , i n order t o have a p u l s e . T h i s was always the case. The R.F. s i g n a l was atte n u a t e d 2 0db and f e d i n t o a mixer which a c t s as a switch which i s " c l o s e d " by a p u l s e from the pu l s e generator. The purpose o f s t a r t i n g with a l a r g e s i g n a l from the R.F. generator and then a t t e n u a t i n g i t b e f o r e u s i n g i t f o r co u n t i n g , r e f e r e n c e , and p u l s e p r o d u c t i o n i s t o make these o p e r a t i o n s as independent of one another as i s reasonably p o s s i b l e . The s i g n a l l e a v i n g the mixer-switch, ( i n the form of R.F. pu l s e s a t h a l f the resonance fre q u e n c y ) , was f e d i n t o one h a l f of an HP 8447F a m p l i f i e r , through an i s o l a t o r , and i n t o the 15 homemade d o u b l e r - a m p l i f r e r . The i s o l a t o r f u r t h e r decreased the p o s s i b i l i t y of the p u l s e - p r o d u c t i o n system i n t e r a c t i n g with the s i g n a l generator,, and hence a f f e c t i n g the continuous -30-r e f e r e n c e s i g n a l f e d to the s i g n a l p r o c e s s i n g s e c t i o n . The doubler a m p l i f i e r i s a l s o switched i n t o o p e r a t i o n only when the h a l f - f r e q u e n c y R.F. p u l s e reaches i t s i n p u t ; t h i s s w i t c h i n g i s produced by a D.C. p u l s e from the p u l s e generator. The 76 5 MHz output was then attenuated to the c o r r e c t l e v e l f o r a TT/2 p u l s e , and was f e d to the d i p l e x e r . 16 The purpose of the d i p l e x e r , (a homemade u n i t ) , i s to e l e c t r o n i c a l l y connect the output of the R.F. p u l s e g e n e r a t i o n s e c t i o n t o the t r a n s m i s s i o n l i n e l e a d i n g to the r e s o n a t o r i n the experimental apparatus, d u r i n g the R.F. p u l s e o n l y . At the same time, the d i p l e x e r p r o v i d e s high a t t e n u a t i o n between the t r a n s m i s s i o n l i n e and s i g n a l a m p l i f i e r s e c t i o n , so as to minimize the s a t u r a t i o n from the R.F. p u l s e . In the i n t e r v a l between p u l s e s , the d i p l e x e r does the r e v e r s e - the a m p l i f i e r s e c t i o n i s connected d i r e c t l y to the t r a n s m i s s i o n l i n e and i s i s o l a t e d from the p u l s e g e n e r a t i o n s e c t i o n . T h i s a c t i o n i s c o n t r o l l e d by a D.C. p u l s e from the p u l s e generator which changes the impedance of PIN diodes at key p o i n t s i n a c o a x i a l r e s o n a t o r system, as d e s c r i b e d i n r e f e r e n c e (16). A 20db d i r e c t i o n a l c o u p l e r was connected between, the d i p l e x e r and the t r a n s m i s s i o n l i n e l e a d i n g t o the r e s o n a t o r , and each output was passed through a lOdb a t t e n u a t o r and i n t o a diode d e t e c t o r . Thus one d e t e c t o r measured the i n t e n s i t y of the i n c i d e n t R.F. p u l s e s (as a check on the f u n c t i o n i n g of the puls e g e n e r a t i o n system and d i p l e x e r ) , and the other the i n t e n s i t y of the r e f l e c t e d p u l s e l e a v i n g the r e s o n a t o r , (which should be zero when the re s o n a t o r i s c r i t i c a l l y coupled t o the t r a n s m i s s i o n l i n e . ) -31-The a m p l i f i e r c h a i n r e c e i v i n g the s i g n a l from the resonat o r v i a the d i p l e x e r c o n s i s t e d o f a broadband Avantek UTO-1011 a m p l i f i e r , f e e d i n g i n t o a homemade, narrowband 17 (765 MHz) a m p l i f i e r , which passed through a hig h pass f i l t e r t o remove some low frequency t r a n s i e n t s produced by the d i p l e x e r , and f i n a l l y i n t o the other h a l f of the HP 8447F a m p l i f i e r . The output from t h i s a m p l i f i e r was f e d to the mixer i n the s i g n a l p r o c e s s i n g s e c t i o n . In t h i s mixer, the f r e e i n d u c t i o n s i g n a l beat with the s l i g h t l y d i f f e r e n t frequency r e f e r e n c e s i g n a l from the p u l s e g e n e r a t i o n s e c t i o n so as to produce an output s i g n a l of magnitude p r o p o r t i o n a l t o t h a t o f the f r e e i n d u c t i o n s i g n a l and of frequency equal t o the d i f f e r e n c e between the f r e e i n d u c t i o n and r e f e r e n c e f r e q u e n c i e s . T h i s s i g n a l , which was almost always s m a l l e r than the n o i s e , was monitored and a m p l i f i e d by a T e t r o n i x 541A o s c i l l o s c o p e , t r i g g e r e d by the pu l s e generator. While the s i g n a l c o u l d not always be seen on the o s c i l l o s c o p e , the d i s p l a y was u s e f u l as a monitor of the o v e r - a l l system f u n c t i o n i n g (such as the "appearance" of the no i s e , the a m p l i f i e r recovery time a f t e r the p u l s e , e t c . ) . The f r e q u e n c y - t r a n s l a t e d f r e e i n d u c t i o n s i g n a l , a f t e r ampli-f i c a t i o n by the o s c i l l o s c o p e , was f e d t o the N i c o l e t 1174 s i g n a l averager. T h i s d e v ice would s t o r e the v o l t a g e l e v e l o b t a ined by sampling the s i g n a l a t 1,024 p o i n t s i n time a f t e r each p u l s e , with the i n t e r v a l between p o i n t s as smal l as one microsecond. The r e s u l t of t h i s sample would be d i s p l a y e d on a cathode ray tube, and the r e s u l t s o f sampling a f t e r success-i v e p u l s e s would be added t o the p r e v i o u s measurements, so. -32-t h a t the d i s p l a y e d s i g n a l would grow i n p r o p o r t i o n to the number of p u l s e i n t e r v a l s sampled. Since the n o i s e i s random, however, i t s r.m.s. magnitude grows i n p r o p o r t i o n to the square r o o t of the number of measurements. Thus the net s i g n a l to n o i s e r a t i o f o r the s i g n a l averager d i s p l a y would i n c r e a s e i n p r o p o r t i o n to the square r o o t of the number of measurements. Since the p u l s e r e p e t i t i o n r a t e was as high as 1 kHz, t h i s was a very convenient and u s u a l l y e s s e n t i a l method of improving the net s i g n a l q u a l i t y . ( I t was not unusual to s i g n a l average f o r as many as 262,000 pulses" " w h i c h - w o u l d t a k e about, f i v e minutes.) A paper copy of the CRT d i s p l a y was obtained u s i n g an x-y r e c o r d e r . -33-4.2 C h a r a c t e r i s t i c s of the Spectrometer Before using the completed spectrometer f o r making q u a n t i t a t i v e measurements, fo u r f e a t u r e s of the system had to be determined. The f i r s t was the n o i s e temperature, which i s d e f i n e d to be the temperature of a 50-ft r e s i s t o r which when connected to the i n p u t of the system w i l l double the observed n o i s e power. T h i s was measured by connecting an Aerospace NS-LB no i s e generator to the i n p u t and a d j u s t i n g i t to i n c r e a s e the r.m.s. amplitude of the n o i s e output by J2. The n o i s e tempera-t u r e o b tained was 170K. The next f e a t u r e checked was the r e c o v e r y time of the system - t h a t i s the time taken a f t e r the R.F. p u l s e f o r the system to recover f u l l s e n s i t i v i t y . T h i s time was r e a d i l y observable on the o s c i l l o s c o p e as a r e g i o n of w i l d f l u c t u a t i o n s devoid of h i g h frequency n o i s e . The n o i s e took on a completely normal appearance, i n d i c a t i v e of r e c o v e r y , 3-lOyM.sec a f t e r each p u l s e . The t h i r d check i n v o l v e d the frequency accuracy of the frequency s y n t h e s i z e r , R.F. generator, and counter system. T h i s frequency was s t a b l e t o w i t h i n our frequency counter r e s o l u t i o n of 1 Hz. The a b s o l u t e accuracy of the counter was checked by u s i n g i t to measure the frequency of the c o l o r s u b - c a r r i e r p u l s e used i n an American network t e l e v i s i o n broad-c a s t . T h i s s i g n a l i s t i e d to an atomic c l o c k and i s monitored by the U.S. N a t i o n a l Bureau of Standards. T h i s check e s t a b l i s h -ed the counter e r r o r to be l e s s than 10 Hz. -34-The f o u r t h , and by f a r the most d i f f i c u l t check on the system was the c a l i b r a t i o n of i t s s e n s i t i v i t y . T h i s was accomplished as f o l l o w s . A continuous s i g n a l a t the reson-ance frequency was obtained from the -lOdb output of the d i r e c t i o n a l c o u p l e r p r i o r to the mixer. (The d i r e c t i o n a l c o u p l e r was always i n p l a c e , so t h a t o b t a i n i n g the s i g n a l would i n no way a l t e r the c h a r a c t e r i s t i c s of the system.) T h i s s i g n a l was then t r a n s l a t e d to a frequency d i f f e r i n g by 10 kHz, by means of a s i n g l e - s i d e - b a n d generator, which mixed the incoming s i g n a l w i t h a 10 kHz s i g n a l so as to produce only the two sidebands, and then s e l e c t i v e l y canceled one of them. Upon f e e d i n g the sideband s i g n a l to the d i p l e x e r i n p u t , the spectrometer output would be a 10 kH.z> beat, which c o u l d be c o h e r e n t l y added by t r i g g e r i n g the s i g n a l averager w i t h the o r i g i n a l 10 kHz s i g n a l i n the s i n g l e - s i d e - b a n d generator. By a t t e n u a t i n g the s i g n a l a p p l i e d t o the d i p l e x e r t o a known magnitude (comparable to t h a t of the f r e e i n d u c t i o n decay), i t was p o s s i b l e to perform a c a l i b r a t i o n of the system as a whole. The c h i e f u n c e r t a i n t y i n t h i s procedure r e s u l t e d from the u n c e r t a i n t y i n the a t t e n u a t o r s used, which were c a l i b r a t e d w i t h a General Microwave 460B power meter. With t h i s c a l i b r a t i o n , the a b s o l u t e power of the f r e e i n d u c t i o n s i g n a l c o u l d then be i n f e r r e d from the spectrometer output. -35-4.3 Resonator Design At a frequency of 765 MHz, c o n f l i c t i n g demands are p l a c e d on the resonator to be employed i n d e t e c t i n g the atoms. A c i r c u i t composed of o r d i n a r y c o i l s and c a p a c i t o r s i s i n a p p r o p r i a t e because of the h i g h i n d u c t i v e impedance of wires at these f r e q u e n c i e s , which makes i t d i f f i c u l t to have a uniform magnetic f i e l d c o n f i n e d mainly to the sample r e g i o n . Moreover, because of the concentrated s u r f a c e c u r r e n t s i n such a design, the ohmic l o s s e s are q u i t e h i g h and the Q of the c i r c u i t i s t h e r e f o r e q u i t e low. An a l t e r n a t i v e which over-comes these problems at high f r e q u e n c i e s i s the use of a resonant c a v i t y . U n f o r t u n a t e l y , a 765 MHz^  resonant c a v i t y has a minimum s i z e of order 20cm - f a r too l a r g e t o be convenient i n a cr y o g e n i c apparatus. The somewhat novel r e s o n a t o r design we chose, (see F i g . 4.2), overcomes both of these problems. In essence, the r e s o n a t o r c o n s i s t s of a s i n g l e t u r n c o i l i n p a r a l l e l w i t h a p a r a l l e l p l a t e c a p a c i t o r . The re s o n a t o r was machined out of a s i n g l e b l o c k of copper i n order to a v o i d d i f f e r e n t i a l thermal c o n t r a c t i o n . The c a p a c i t o r gap was cut wit h a t h i n saw blade:'-and then reduced to zero t h i c k n e s s by compression o f the copper i n a h y d r a u l i c p r e s s . The gap was then opened out to the r e q u i r e d s i z e f o r a resonance of 765 MHz- by means of a copper t u n i n g screw which was prevented from " s h o r t i n g out" the c a p a c i t o r gap by means of a t h i n p i e c e of mylar. (See F i g . 4.2). With t h i s d e s ign, the Q, (.the r a t i o of the resonant frequency t o the f u l l width a t h a l f maximum), was g r e a t e r Figure 4.2 765 MHz resonator, tuning rod and coupling loop, (scale 1:2) -37-than 1,000, and i t was p o s s i b l e to tune the resonant frequency from 200-900 MHz by a d j u s t i n g the copper screw. F i n e t u n i n g was obtained u s i n g a t u n i n g loop c o n s i s t i n g of a copper r i n g of about the same diameter as the r e s o n a t o r bore. The r i n g was mounted c o a x i a l l y with the r e s o n a t o r and c o u l d be moved from w i t h i n a m i l l i m e t e r of the r e s o n a t o r to about 15 mm away by means of a moveable d i e l e c t r i c support rod, as shown i n F i g . 4.2. In e f f e c t , the loop a c t s to decrease the i n d u c t -ance of the s i n g l e t u r n c o i l by opposing any change i n magnetic f l u x through i t s i n t e r i o r , i n much the same way as the e f f e c t i v e inductance of the primary winding of a transformer can be reduced by s h o r t i n g the secondary. T h i s method allowed remote tuning over a range of about 20 MHz, which was s a t i s f a c t o r y f o r the purpose of overcoming the frequency s h i f t which occurs on c o o l i n g to c r y o g e n i c temperatures. (This s h i f t was always l e s s than 2 MHz.) T h i s t u n i n g method would s l i g h t l y a f f e c t the Q; t h i s change i n Q was always l e s s than 20%. Coupling the r e s o n a t o r to the 50/1 c o a x i a l t r a n s m i s s i o n l i n e l e a d i n g from the 765 MHz spectrometer was accomplished by mounting a s i n g l e t u r n loop as a t e r m i n a t i o n t o the t r a n s m i s s i o n l i n e , so t h a t i t was c o a x i a l w i t h the r e s o n a t o r , and on the o p p o s i t e s i d e from the r i n g , as shown i n F i g . 4.2. By a d j u s t i n g the d i s t a n c e between the loop and the r e s o n a t o r , ( s e v e r a l mm), i t was p o s s i b l e to o b t a i n c r i t i c a l c o u p l i n g with the t r a n s m i s s i o n l i n e . There were thus two remote p o s i t i o n adjustments to be made to tune the r e s o n a t o r to match the 76 5 MHz spectrometer - frequency adjustment and c o u p l i n g adjustment. While these were not 100% independent, i t was -38-n e v e r t h e l e s s very easy to q u i c k l y o b t a i n c r i t i c a l c o u p l i n g . Although an a n a l y t i c s o l u t i o n to the f i e l d s i n the resonat o r has not, and probably cannot be found, a simple approximation to them serves t o demonstrate the b a s i c behavior of the re s o n a t o r , and allows a reasonably accurate t h e o r e t i c a l c a l c u l a t i o n of the Q. The approximation c o n s i s t s o f i g n o r i n g edge e f f e c t s . F i r s t , the re s o n a t o r i s assumed to be i n f i n i t e l y long, such t h a t spreading of the f i e l d s a t both ends i s not cons i d e r e d . Second, the f r i n g i n g f i e l d s o f the c a p a c i t o r gap are ignored, which i s a reasonable assumption because the gap i s very narrow r e l a t i v e to the s i z e of the p l a t e s . Under these c o n d i t i o n s the magnetic f i e l d i n the bore o f the r e s o n a t o r w i l l be uniform and i n the a x i a l d i r e c t i o n as i n a p e r f e c t s o l e n o i d . Moreover , the e l e c t r i c f i e l d i n the c a p a c i t o r gap w i l l be uniform and p e r p e n d i c u l a r to the p l a t e s as i n a p e r f e c t • p a r a l l e l p l a t e c a p a c i t o r , and w i l l o s c i l l a t e 90° out of phase with r e s p e c t to the bore magnetic f i e l d . Since t h i s uniform changing e l e c t r i c f i e l d r e q u i r e s a uniform c u r l i n the magnetic f i e l d i n the same d i r e c t i o n , and s i n c e the magnetic f i e l d o u t s i d e the resona t o r w i l l be c l o s e to zero, the magnetic f i e l d i n the c a p a c i t o r gap must b e . i n the a x i a l d i r e c t i o n , and must decrease l i n e a r l y from the value i t has i n the bore at the innermost p o i n t i n the gap, to zero a t the outermost p o i n t i n the gap. ( A c t u a l l y , when the re s o n a t o r i s of f i n i t e l e n g t h and w i t h i n a metal e n c l o s u r e as i n our apparatus (see F i g . 5.1), the magnetic f i e l d o u t s i d e the, r e s o n a t o r w i l l not be zero, s i n c e the magnetic f i e l d l i n e s must form c l o s e d loops i n s i d e the e n c l o s u r e . However, s i n c e the c r o s s s e c t i o n -a l area o u t s i d e the re s o n a t o r i s a t l e a s t f i v e times t h a t . i n s i d e , the e x t e r n a l f i e l d i s s m a l l e r by t h a t amount, and t h i s approximation i s f a i r l y good.) With these f i e l d s , the resonant frequency and the Q can be determined. L e t Tr0 be the r a d i u s of the resona t o r bore, 0,ro be the width of the c a p a c i t o r p l a t e s , and br0 the t h i c k n e s s of the c a p a c i t o r gap. L e t the magnetic f i e l d i n the bore be H, (i) = Ht cos(wi) (4.1) which i m p l i e s t h a t i n the c a p a c i t o r gap (fXHjgap -jfJKOSCeirf) (.4.2) A l s o , l e t the e l e c t r i c f i e l d i n the c a p a c i t o r gap be E(i}- E 0 Sin(wt) C4.3) In order f o r the t o t a l e l e c t r o m a g n e t i c energy t o remain const a n t , i t must be the case t h a t (4.4). Jdfii & = / o l A H* (c.3.s} which i m p l i e s ( f o r ab«l ) t h a t ^°~[at) M» (.4.5) We can now employ the Maxwell equation V X H - c at (.4.6) f o r the f i e l d s i n the c a p a c i t o r gap, to o b t a i n •fir cos(coi) = f(£j'ZUi cos(wi) C4.7) which i m p l i e s The c a l c u l a t i o n of the Q i s s t r a i g h t - f o r w a r d but a l i t t l e more complex. From the theory of simple r e s o n a t o r s , the Q i s given by: store-d energy mean pouuo- loss (4.9) (In our case the s t o r e d energy and power l o s s w i l l be determined per u n i t l e n g t h of the a r b i t r a r i l y long i d e a l resonator.) The energy per u n i t l e n g t h i s 18 The power l o s s per u n i t l e n g t h i s given by w h e r e * = [ w . (4.11) and <J i s the e l e c t r i c a l c o n d u c t i v i t y of the metal. Assuming a « | , as i s the case, • H i & l M j t 2 * * " * ] ( 4 - 1 2 » Thu'S-e: from (4.9), (4.10), and (4.12) we o b t a i n -41-S u b s t i t u t i n g f o r & , we have (4.14) Con v e r t i n g to MKS u n i t s , we o b t a i n (4.15) and s u b s t i t u t i n g f o r tt) , we have Q = f fef("oCreo-f (M.K.,^ (4.16) Both the p r e d i c t i o n f o r the resonant frequency and the Q were t e s t e d on a l a r g e model r e s o n a t o r of p r e c i s e l y known dimensions, and were found to agree, w e l l with the experimental v a l u e s . In a d d i t i o n the magnetic f i e l d i n the a c t u a l r e s o n a t o r used was probed w i t h a small loop of wire and was indeed found to be reasonably uniform and i n the a x i a l d i r e c t i o n . We are thus very c o n f i d e n t t h a t we understand the behavior of t h i s r e s o n a t o r . In order t o s e t the R.F. pu l s e l e n g t h i t i s necessary t o know the value of produced i n the reson a t o r by a continuous i n p u t power P a t c r i t i c a l c o u p l i n g . T h i s can be c a l c u l a t e d from equations (4.9) and (.4.10), l e t t i n g i be the l e n g t h of the res o n a t o r : IK p (4.18) -42-and thus U _ 2 llPC ^ (4.19) In a d d i t i o n we need an e x p r e s s i o n f o r the power f l o w i n g out of the c r i t i c a l l y coupled r e s o n a t o r when i t i s d r i v e n by an o s c i l l a t i n g magnetic moment nrc(t) = msinluit) i n the a x i a l d i r e c t i o n . T h i s c a l c u l a t i o n w i l l proceed i n three steps. F i r s t , the power in p u t t o the resona t o r from the o s c i l l a t i n g moment, , w i l l be determined as a f u n c t i o n of H^. Second, the power l o s s of the resona t o r due to ohmic l o s s e s , Pft» and due to power f l o w i n g i n t o the t r a n s m i s s i o n l i n e , P , w i l l be determined. F i n a l l y , the f a c t t h a t a t steady s t a t e P^ n = PJX + P g w i l l be employed to o b t a i n P g as a f u n c t i o n of m. The average value of P^ n i s : (4.20) where B = H + 4TTM, or i n our case B = Ht + 47)^ i n the z d i r e c t i o n , where V i s the volume of the gas i n the resonator, Thus Rn = (ftrr.5mC«tt)(H,Cos(«0^+ ^pS ihM) = (4.22) The value o f P can be found from (4.19), s i n c e a t c r i t i c a l s ' c o u p l i n g the power f l o w i n g out of the re s o n a t o r , when i t i s made to o s c i l l a t e by the o s c i l l a t i n g magnetic moment, w i l l be the same as t h a t which must flow down the t r a n s m i s s i o n l i n e t o make i t o s c i l l a t e to the same extent without any o s c i l l a t i n g magnetic moment. Thus -43-2&Q cuA (.4.2 3) and t h e r e f o r e p _ H?U>JLYO S 8Q ( 4 - 2 4 ) P^. can a l s o be o b t a i n e d from (4.19), because energy c o n s e r v a t i o n r e q u i r e s t h a t the power f l o w i n g down the t r a n s m i s s i o n l i n e to s u s t a i n o s c i l l a t i o n s i n the r e s o n a t o r w i l l equal the ohmic l o s s e s a t c r i t i c a l c o u p l i n g . Thus Since P. = P.O. + P„ = 2P , we o b t a i n from (4.22 and 4.24) mHi - H\acui>oZ (4.26) S o l v i n g f o r H 1 and s u b s t i t u t i n g i n t o (4.24), we o b t a i n p _ m*aiQ / \ (4.27) These r e s u l t s can now be combined wi t h those of s e c t i o n 2.3 to determine the requirements f o r a TT/2 p u l s e , and the expected s i g n a l s t r e n g t h from a t y p i c a l number of hydrogen atoms. The parameters f o r our r e s o n a t o r were as f o l l o w s : Q = 1180 (4.2 K, unloaded Q) Yb = 6.35 mm i = 12.7 mm W= 27T X 765 MHz = 4.81 X 10 9 s e c " 1 -44-Equation (4.19) thus becomes Hi = [\.16 XlO" 3 Gxwss (e^/secf'1] P'* (4.28) and with a t y p i c a l o p e r a t i n g R.F. p u l s e power of P=.0 79 Watts, or 7.9 X 10 5 erg/sec, we o b t a i n = 1.74 Gauss. Using (2.45), we o b t a i n Tvr/x ~ 2.5 /(sec. Let us now c a l c u l a t e the expected s i g n a l power, P , with 14 a t y p i c a l number, N = 10 , of atoms i n the r e s o n a t o r , with the s p i n system i n e q u i l i b r i u m a t 4.2 K. Using the above reson a t o r parameters, (4.27) becomes P s = \f. SH X l O i X (erg /sec)(e«j /OaossY*] m x (4.29) Immediately a f t e r the Tf/2 p u l s e , the value of m i s giv e n by (2.47) and (2.48): m = N (cos(e)^ r + sin(G)xie)lb-?*) (4 .33) At 4.2K, the f r a c t i o n a l p o p u l a t i o n s d i c t a t e d by (2.23) are P± = .27717 P 2 = .27475 P 3 = .22487 P 4 = .22320 so t h a t P^ - P 2 = , .00242, and the val u e o f m i n ( 4 . 3 3 V i s m = fa.iO X\0'1^e.ro)/&ojji%) N (4.34) 14 With N = 10 , the value of P obt a i n e d from C4.29) i s s 4.59 X 1 0 - 8 erg/sec, or 4.59 X 1 0 ~ 1 5 Watts. -45-4.4 Q u a n t i t a t i v e A n a l y s i s of Graphic Output As mentioned i n s e c t i o n 4.1, the output of the specto-meter system i s a graph produced by an x-y r e c o r d e r . The t o t a l x - a x i s extent of the graph corresponds to a known time i n t e r v a l s e t by the c r y s t a l c l o c k i n the s i g n a l averager, and the f u l l s c a l e displacement on t h i s c h a r t i s known, v i a the c a l i b r a t i o n procedure, to correspond to a c e r t a i n i n p u t power to the spectrometer. F i g . 4.3 shows a t y p i c a l output curve. For the purposes of t h i s experiment, such curves c o n t a i n three p i e c e s of inform-a t i o n - the i n i t i a l s i g n a l i n t e n s i t y , (that i s the f r a c t i o n of f u l l s c a l e o f the o s c i l l a t i o n amplitude a t the beginning of the t r a c e ) , the frequency of the o s c i l l a t i o n s , and, f o r exponent-i a l l y decaying o s c i l l a t i o n s (as was normally the c a s e ) , the decay time of the o s c i l l a t i o n magnitude. The frequency of the graph o s c i l l a t i o n s was determined by counting the l a r g e s t p o s s i b l e number of zero c r o s s i n g s of the curve, and d i v i d i n g h a l f t h i s number ( i . e . the number of c y c l e s ) by the time i n t e r v a l , which was obtained from the known time base of the graph. The a c t u a l frequency of the f r e e i n d u c t i o n decay c o u l d then be ob t a i n e d by a p p r o p r i a t e a d d i t i o n or sub-t r a c t i o n o f t h i s frequency to t h a t o f the l o c a l o s c i l l a t o r . The l a r g e s t e r r o r r e s u l t e d from n o i s e i n the s i g n a l which lowered the accuracy of the time i n t e r v a l measured on the graph; t h i s e r r o r could be estimated from the s c a t t e r i n repeated a n a l y s i s of the same s i g n a l . The i n i t i a l amplitude and decay time was determined by measuring the amplitude of the o s c i l l a t i o n as a f u n c t i o n of time. 5 msec Figure 4 . 3 A typical free induction decay. - 4 7 -P r o v i d i n g the decay i s reasonably e x p o n e n t i a l , and the decrease i n amplitude per o s c i l l a t i o n i s not too g r e a t , an a c c u r a t e , and simple way of doing t h i s i s t h a t i l l u s t r a t e d i n F i g . 4 . 3 . A l i n e i s drawn between two adjacent maxima of the o s c i l l a t i o n , and the v e r t i c a l d i s t a n c e from the minimum between them to the l i n e i s measured and i s taken to be twice the amplitude a t the time at which the minimum oc c u r r e d . For an approximately e x p o n e n t i a l decay, the f r a c t i o n a l e r r o r i n t r o d u c e d by t h i s procedure i s independent of the amplitude of the o s c i l l a t i o n s , and hence w i l l not a f f e c t the measure of the decay time. More-over, t h i s f r a c t i o n a l e r r o r i n the amplitude obtained i s of 2 order a where a i s the f r a c t i o n a l decrease i n amplitude per o s c i l l a t i o n . The e r r o r thus c o n t r i b u t e d to the i n i t i a l amplitude value obtained was of order 1%. The advantage of t h i s technique i s i t s s i m p l i c i t y , q uickness, and i n s e n s i t i v i t y to slow background d r i f t o f the center p o i n t of the o s c i l l a t i o n . The data o b t a i n e d f o r the amplitude as a f u n c t i o n of time was then p l o t t e d on l o g - l i n paper, as shown i n F i g . 4 . 4 , and the slope and i n t e r c e p t of the r e s u l t i n g s t r a i g h t l i n e p r o v i d e d the decay time and i n i t i a l amplitude of the o s c i l l a t i o n . - 4 8 -Figure 4 . 4 Determination of T2 and i n i t i a l amplitude A ( 0 ) for a free induction decay. -49-CHAPTER V EXPERIMENTAL DESIGN 5.1 O v e r a l l arrangement. F i g . 5.1 shows the arrangement of the H-atom flow tube, r e s o n a t o r , r e s o n a t o r t u n i n g rod and magnetic resonance t r a n s -m i s s i o n l i n e i n s i d e the l i q u i d helium dewar, w i t h i n the p o l e caps of the electromagnet. The mixture of molecular and atomic hydrogen gas, a f t e r l e a v i n g the d i s c h a r g e tube, passes down the 1.0 meter long t e f l o n coated 12mmOD pyrex flow tube to the d e t e c t i o n r e g i o n . T h i s flow tube i s s e a l e d w i t h an O-ring s e a l as i t passes through the f l a n g e i n t o the evacuated r e g i o n o f the system. The flow tube terminates i n s i d e the copper can which l i e s between the 12" pole s of the electomagnet. A 1/2" OD s t a i n l e s s s t e e l tube surrounds the flow tube between the f l a n g e and the can to maintain the vacuum s e a l . The r e s o n a t o r i t s e l f i s supported by a f i b e r g l a s s support p l a t e above the end of the flow tube, with i t s bore c o a x i a l with the tube. At l i q u i d helium temperatures, a l l of the H 2 gas would e v e n t u a l l y f r e e z e on the w a l l s of the copper can, whereas at hig h e r temperatures i t was pumped out of the system w i t h a d i f f u s i o n pump connected t o the copper e n c l o s u r e by means of a pipe which f o r reasons of c l a r i t y i s not shown i n F i g . 5.2. (See s e c t i o n 2 of t h i s c h a p t e r ) . The t u n i n g of the re s o n a t o r was accomplished by h o l d i n g the t u n i n g r i n g i n p l a c e beneath the reson a t o r w i t h a rod of m i c a r t a , which was att a c h e d to a 1/8" OD t h i n w a l l s t a i n l e s s s t e e l tube which passed out of the copper can and through the top f l a n g e so t h a t i t s v e r t i c a l p o s i t i o n c o u l d be a d j u s t e d . The vacuum s e a l of the system was maintained by surrounding t h i s -50-p y r e x f l o w t u b e 1/8 S . S . t u n i n g r o d r e s o n a t o r ± f e m a l e - OSM c o n n e c t o r o - r i n g s e a l s h e l i u m p u m p i n g p o r t SMA. c o a x i a l c a b l e d e w a r v e s s e l ( d o u b l e v a c u u m s p a c e , 7 7 K s h i e l d ) c o u p l i n g l o o p t u n i n g l o o p • c o p p e r v a c u u m c a n F i g u r e 5 . 1 P r o b e i n d e w a r v e s s e l , ( s c a l e 1:3) 12 p o l e c a p s o f e l e c t r o m a g n e t s t a i n l e s s s t e a l tube with a 3/16" t h i n w a l l tube which s e a l e d to the re s o n a t o r e n c l o s u r e can and to the top f l a n g e . it Above the f l a n g e , the 1/8 s.s.tube passed through an O-ring s e a l , which served both as a vacuum s e a l and as a means of h o l d i n g the t u n i n g rod i n a f i x e d p o s i t i o n . The c o u p l i n g loop was p o s i t i o n e d above the r e s o n a t o r (hence around the flow t u b e ) , and was h e l d i n p l a c e by the SMA s e m i - r i g i d 50JT. c o a x i a l t r a n s -m i s s i o n l i n e which i t terminated. T h i s t r a n s m i s s i o n l i n e passed out of the copper can and through the f l a n g e w i t h the same 3/16" tube and O-ring s e a l arrangement as t h a t which vacuum-sealed the 1/8" OD t u n i n g rod. Above the f l a n g e , the t r a n s m i s s i o n l i n e coupled to the 765 MHz p u l s e spectrometer system. Both the t r a n s m i s s i o n l i n e and the t u n i n g rod had square p i e c e s of brass attached to them which moved i n brass s l o t s i n such a way as to prevent r o t a t i o n of the rods, but which allowed v e r t i c a l motion, so t h a t the r o t a t i o n a l alignment of the c o u p l i n g loop and t u n i n g r i n g was maintained at a l l times. In the remainder of t h i s chapter, f u r t h e r d e t a i l s o f de s i g n w i l l be presented i n the f o l l o w i n g three c a t e g o r i e s - gas h a n d l i n g , the electromagnet system, and c r y o g e n i c techniques. 5.2 Gas Handling The molecular hydrogen gas s u p p l i e d to the d i s c h a r g e tube was ob t a i n e d from an u l t r a - h i g h p u r i t y (99.99 9%) h i g h pressure gas c y l i n d e r , by means of a r e g u l a t o r which reduced the pr e s s u r e to about 5 p s i . T h i s gas then passed through a f i n e metering v a l v e , which acted as the flow c o n t r o l f o r the H 2 gas. T h i s v a l v e was a d j u s t e d so as to c r e a t e the c o r r e c t o p e r a t i n g -52-p r e s s u r e s o f about 350 mTorr i n the di s c h a r g e tube. The pres s u r e was measured immediately a f t e r the metering v a l v e by means of a 20mm Hg f u l l s c a l e Wallace & T i e r n a n gauge and a l s o by a 200 mTorr mid-scale thermocouple gauge. (The thermo-couple gauge had to be used i n c o n j u c t i o n with a c a l i b r a t i o n t a b l e f o r use with H 2 gas - the re a d i n g on the gauge i s approximately twice the t r u e p r e s s u r e of H2.) I t was assumed t h a t the pr e s s u r e measured by these gauges was a good measure of the pr e s s u r e i n the di s c h a r g e tube, s i n c e the main r e s t r i c -t i o n to the flow beyond the metering v a l v e was the smal l h o l e at the o u t l e t of the d i s c h a r g e tube. In the p r e s s u r e range of lOOmTorr to s e v e r a l T o r r , the flow r a t e was roughly p r o p o r t i o n a l to p r e s s u r e . The hydrogen gas was pi p e d from the metering v a l v e and pres s u r e measurement assembly to the di s c h a r g e tube through 1 meter of 3/8" OD P o l y - f l o t u b i n g , at which p o i n t the t u b i n g connected, v i a a P o l y - f l o f i t t i n g , to an O-ring s e a l s e a l i n g to the g l a s s d i s c h a r g e tube. The d i s c h a r g e bulb i t s e l f was p o s i t i o n e d about 15 cm above and 15 cm o f f - c e n t e r from the fl a n g e on the dewar, i n the same o r i e n t a t i o n as shown i n F i g . 3.1. T h i s p o s i t i o n i n g was chosen t o f a c i l i t a t e access t o the dewar, and at the same time minimize the path l e n g t h from the d i s c h a r g e to the d e t e c t i o n r e g i o n . The d i s c h a r g e bulb was connected to the flow tube p a s s i n g i n t o the dewar by means of a 12mm OD pyrex tube which coupled to the d i s c h a r g e v i a a s:ilicones:greased s p h e r i c a l , ground g l a s s j o i n t , and connected to the main flow tube by a double O-ring s e a l i n g brass p i e c e . T h i s i n t e r m e d i a t e flow tube was i n t e r n a l l y coated w i t h p o l y --53-phosphoric a c i d . The t o t a l path l e n g t h of t u b i n g l e a d i n g to the d e t e c t i o n r e g i o n was thus 1.3m. The copper can surrounding the re s o n a t o r was pumped by means of a 20 cm long 1/4" OD t h i n w a l l s t a i n l e s s s t e e l tube connecting to an 60 cm long 3/8" OD t h i n w a l l s t a i n l e s s s t e e l tube l e a d i n g to a Leybold connector on the dewar f l a n g e . A 2.5 meter long 1 1/8" OD copper pumping l i n e l e a d from t h i s f l a n g e to a d i f f u s i o n pumping s t a t i o n . When making measurements at l i q u i d helium temperatures, t h i s pumping - l i n e was not needed f o r pumping molecular hydrogen out of the system, because the H 2 would s o l i d i f y a t these temperatures, but i t was n e v e r t h e l e s s u s e f u l f o r the purpose of adding or removing helium b u f f e r gas to the vacuum system. For t h i s purpose a c y l i n d e r o f u l t r a -h i g h p u r i t y helium with a r e g u l a t o r and v a l v e was connected t o the pumping l i n e near the d i f f u s i o n pump. The pre s s u r e of helium gas c o u l d then be monitored with the i o n i z a t i o n p r e s s u r e gauge on the pumping s t a t i o n . 5.3 The Electromagnet System. The electromagnet system used i n t h i s experiment c o n s i s t e d of a V a r i a n V012-3B electromagnet used with 12" po l e caps, and powered with a V a r i a n V2100-B power supply. The magnetic f i e l d of 6,481 Gauss was measured w i t h a home-made NMR magnetometer 3 system. The NMR sample c o n s i s t e d of about .25 cm of a s a t u r a t e d CuSO^ i n water s o l u t i o n , s u b j e c t to a modulation f i e l d o f about 1 Gauss at 17 Hz, and the NMR spectrometer was a home-made Robinson o s c i l l a t o r o p e r a t i n g a t 27.590 MHz. The amplitude modulation was d e t e c t e d w i t h a PAR model HR8 phase s e n s i t i v e - 5 4 -d e t e c t o r u s i n g the magnet modulation s i g n a l as a r e f e r e n c e and o p e r a t i n g with a time constant of 3 seconds. The output of the phase s e n s i t i v e d e t e c t o r , which was p r o p o r t i o n a l to the d e r i v a t i v e of the NMR a b s o r p t i o n w i t h r e s p e c t to f i e l d , was fe d with the a p p r o p r i a t e s i g n to a v a r i c a p diode i n the Robinson o s c i l l a t o r tuned c i r c u i t , such t h a t the o s c i l l a t o r frequency would " l o c k on" to the c e n t r e of the NMR a b s o r p t i o n l i n e . The r e s u l t a n t frequency was measured with an HP model 52 4 5Jt frequency c o u n t e r a n d u s i n g the known'.gyromagnetic . . . r a t i o f o r protons i n water of 42 .576 MHz/kGauss,. the magnetic f i e l d c o u l d be c a l c u l a t e d . Using t h i s f i e l d measurement system, the inhomogeneity of 3 the magnet i n the 1 cm d e t e c t i o n r e g i o n was found to be about 1 Gauss, and i n a d d i t i o n the f i e l d p r o f i l e of the magnet was measured. The magnet power supply was s t a b l e to b e t t e r than 5 1 p a r t i n 10 over p e r i o d s of hours, although on s e v e r a l 3 o c c a s i o n s i t s output s h i f t e d by about 1 p a r t i n 10 , and remained s t a b l e at t h i s new v a l u e . T h i s s h i f t never o c c u r r e d d u r i n g a run, and we concluded t h a t i t had a d i f f e r e n t o r i g i n than d i d the u s u a l very slow d r i f t . For r o u t i n e m o n i t o r i n g of the f i e l d d u r i n g the experiment, the water sample was l o c a t e d 6 cm from the r e s o n a t o r i n the dewar, where the magnetic f i e l d was w i t h i n 3 Gauss of the value i n the r e s o n a t o r . The t o t a l u n c e r t a i n t y i n the f i e l d was of the order of 3 Gauss, which was l e s s than the u n c e r t a i n t y i n our measurement of H e. .• . -55-5.4 Cryogenic Techniques. The s t a i n l e s s s t e e l dewar v e s s e l employed i n t h i s e x p e r i -ment was of b a s i c a l l y c o n v e n t i o n a l d e s i g n . (See Fig.5.1) The l i q u i d n i t r o g e n v e s s e l d i d not completely surround the l i q u i d helium v e s s e l , due to l a c k of space i n the t a i l of the dewar. Instead, the bottom p o r t i o n of the helium v e s s e l i s surrounded by a copper can which i s h e l d a t about 77K by thermal c o n t a c t to the l i q u i d n i t r o g e n above. The vacuum space between the outer j a c k e t and the l i q u i d n i t r o g e n space 3 contained about 100 cm of molecular s e i v e m a t e r i a l to maintain a hard vacuum, while the vacuum space between the l i q u i d n i t r o g e n and l i q u i d helium contained about 1 T o r r of a i r which p r o v i d e d thermal c o n d u c t i v i t y f o r c o o l i n g t o 77K, and which would f r e e z e out upon t r a n s f e r r i n g l i q u i d helium to the i n n e r v e s s e l . The l i q u i d n i t r o g e n would l a s t f o r about 12 hours (the dewar was not s u p e r - i n s u l a t e d ) , and the 6 l i t e r s of l i q u i d helium s t o r e d by the i n n e r v e s s e l would l a s t about the same l e n g t h of time w i t h no a d d i t i o n a l energy i n p u t to the system. The l i q u i d helium l e v e l was d e t e c t e d by m o n i t o r i n g the c u r r e n t flow i n each of three 1 0 0 J l , 1/4 Watt carbon-composition r e s i s t o r s , which were mounted i n s i d e the dewar a t the bottom, middle, and top of the " b e l l y " of the l i q u i d helium v e s s e l . These r e s i s t o r s had a v o l t a g e of about 10 v o l t s a p p l i e d to them, as t h i s was found to be optimal f o r o b s e r v i n g the sudden drop i n c o n d u c t i v i t y which these r e s i s t o r s undergo upon being covered i n l i q u i d helium as opposed to 4.2 K helium vapour. The p r i n c i p l e behind t h i s phenomenon l i e s i n the f a c t t h a t the -56-l i q u i d helium i s s l i g h t l y more e f f i c i e n t a t c o o l i n g the r e s i s t o r s than i s the vapour, and t h a t the carbon composition r e s i s t o r s i n c r e a s e r e s i s t a n c e d r a m a t i c a l l y as t h e i r temperature drops. In our apparatus, the c u r r e n t through the r e s i s t o r s would drop very suddenly by about 30% upon immersion when t r a n s f e r r i n g l i q u i d helium i n t o the v e s s e l , and would r e t u r n to the higher value when the l i q u i d l e v e l dropped beneath them. One disadvantage to t h i s technique was t h a t the power d i s s i p a t i o n (about 200 mwatt) was s u f f i c i e n t l y high t h a t continuous use of t h i s d e t e c t i o n system was i m p r a c t i c a l . While t a k i n g measurements i t was important to know the temperature of the gas i n the d e t e c t i o n r e g i o n , p a r t i c u l a r l y d u r i n g measurements which were performed as the temperature of the system was r i s i n g above l i q u i d helium temperatures. T h i s temperature was determined by m o n i t o r i n g the r e s i s t a n c e of a 100/2 A l l e n - B r a d l e y carbon composition r e s i s t o r which was mounted on the support which h e l d the r e s o n a t o r i n p l a c e . The r e s i s t a n c e of t h i s r e s i s t o r was measured a t room temperature, l i q u i d n i t r o g e n temperature, and l i q u i d helium temperature, and these data were used t o f i n d the constants i n the assumed r e l a t i o n s h i p : T B l n l R l + k / i r » M - A (.5.1) T h i s temperature sensor was found to have another use i n the experiment - when o p e r a t i n g a t l i q u i d helium temperatures with no helium b u f f e r gas, the r e s i s t o r was s u b s t a n t i a l l y heated by the recombination of hydrogen atoms on i t s s u r f a c e . In f a c t , the recombination h e a t i n g of the r e s i s t o r was a f a r more -57-s e n s i t i v e d e t e c t i o n method f o r hydrogen atoms than was the magnetic resonance system. Using t h i s d e t e c t o r , we were ab l e to observe an i n t e r e s t i n g behaviour which o c c u r r e d every time we t r a n s f e r r e d l i q u i d helium i n t o the system - the magnetic resonance s i g n a l , which had been v i s i b l e a t 77 K, would dissappear upon c o o l i n g t o 4.2 K, and the r e s i s t o r - d e t e c t o r would i n d i c a t e no recombination h e a t i n g . A f t e r a p e r i o d of about 5 minutes, however, some atoms would be d e t e c t e d by the r e s i s t o r , and a f t e r a few more minutes the h e a t i n g e f f e c t would reach i t s maximum v a l u e , whereupon the u s u a l s t r e n g t h of magnetic-resonance s i g n a l would be observed. We concluded from t h i s t h a t the c o o l e d flow tube was s t r o n g l y c a t a l y z i n g recombination of the atoms immediately a f t e r t r a n s f e r r i n g , but t h a t some chemical or thermal a c t i o n of the atomic hydrogen gas would g r a d u a l l y change the nature of the tube so as to decrease the c a t a l y t i c e f f e c t . T h i s concludes the d e s c r i p t i o n of the experimental set-up. Some d e t a i l s of the o p e r a t i o n of the system have not been mentioned i n t h i s chapter as they f i t more n a t u r a l l y with a d e s c r i p t i o n of the data which they were used to o b t a i n . These d e t a i l s w i l l appear i n the a p p r o p r i a t e p l a c e s i n the next chapter. - 5 8 -CHAPTER VI EXPERIMENTAL RESULTS. 6.1 Observations a t 77K. By f i l l i n g the outer dewar w i t h l i q u i d n i t r o g e n , and p l a c i n g helium gas i n the i n n e r dewar, the atom d e t e c t i o n system would be maintained at 77K . Under these c o n d i t i o n s the f r e e i n d u c t i o n s i g n a l c o u l d be observed, wi t h a s i g n a l to n o i s e r a t i o of about 10, a f t e r 16,000 p u l s e s were averaged. Measurements were taken w i t h v a r i o u s power l e v e l s i n the R.F. d i s c h a r g e , which r e s u l t e d i n a change i n s i g n a l s t r e n g t h , presumably due to a change i n the number of hydrogen atoms rea c h i n g the d e t e c t i o n r e g i o n . F i g . 6.1 i s a graph of the observed decay r a t e s , ( 1 / T 2 ) , as a f u n c t i o n of the d e n s i t y of hydrogen a t o m s , h H , where t h i s d e n s i t y i s c a l c u l a t e d from the s i g n a l s t r e n g t h i n the manner o u t l i n e d i n s e c t i o n 4.3, assuming t h a t the p o p u l a t i o n s of the s p i n system are i n thermal e q u i l i b r i u m at 77K. The l i n e shown i s a l e a s t squares f i t to the data of the form I/TQ. = of +/S n H w i t h cK =(127 sec) and /3= 1.24 XlO cm 3/sec. T h i s suggests t h a t there i s a t r a n s v e r s e r e l a x a t i o n mechanism independent of lri H, and one which i s p r o p o r t i o n a l t o n^ . The former probably r e s u l t s from i n t e r a c t i o n s between hydrogen atoms and p a r a -magnetic s i t e s on the w a l l s and/or paramagnetic i m p u r i t i e s i n the gas, while the l a t t e r i s the r e s u l t of s p i n exchange between hydrogen atoms. I f one w r i t e s — — o e x V H. H/Z ( 6 > 1 ) a n H -59-- 6 0 -w h e re VH-H i s t h e a v e r a g e r e l a t i v e s p e e d o f two a t o m s , g i v e n and <Sey i s 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 f o r H-H s c a t t e r i n g , t h e n t h e v a l u e f o r o"ex i m p l i e d by t h e d a t a i n F i g . . 6 . 1 . ' i s T h i s v a l u e i s 60% o f t h e a n a l o g o u s v a l u e f o r o"e* f o u n d by 7 de S a i n t f u s c i e n a n d A u d o i n , f o r t h e z e r o f i e l d h y p e r f i n e t r a n -19 s i t i o n a t 77K, w h i c h a g r e e d w i t h t h e c a l c u l a t i o n s o f A l l i s o n A l t h o u g h we a r e o b s e r v i n g a d i f f e r e n t h y p e r f i n e t r a n s i t i o n , 20 t h e v a l u e o f 6*ex s h o u l d be e s s e n t i a l l y t h e same, a n d i t i s t h e r e f o r e d i f f i c u l t t o e x p l a i n t h i s d i s c r e p a n c y . One p o s s i b l e e x p l a n a t i o n i s t h a t t h e s p i n s y s t e m may n o t be i n e q u i l i b r i u m a t 77 K, so t h a t t h e v a l u e s o f r»H f o r t h e d a t a i n F i g . 6.1 w o u l d be i n c o r r e c t . I t i s t h e r e f o r e w o r t h -w h i l e t o c o n s i d e r t h e m e c h a n i s m s by w h i c h t h e s p i n s y s t e m may r e a c h t h e r m a l e q u i l i b r i u m . One s u c h m e c h a n i s m i s t h e s p i n e x c h a n g e i n t e r a c t i o n i t s e l f . 21 T h i s m e c h a n i s m h a s b e e n m o d e l l e d by Brown i n a h i g h t e m p e r a -t u r e a p p r o x i m a t i o n w h i c h i s v a l i d a t 77K. I n t h i s t r e a t m e n t , one e x p r e s s e s t h e p r o d u c t s t a t e o f two atoms a p p r o a c h i n g e a c h o t h e r a s a l i n e a r c o m b i n a t i o n o f s t a t e s i n w h i c h t h e e l e c t r o n s a r e i n t h e s i n g l e t o r t r i p l e t s t a t e ( s i n c e t h e s e a r e t h e e i g e n s t a t e s o f t h e e x c h a n g e i n t e r a c t i o n b e t w e e n t h e two h y d r o g e n a t o m s ) . I n t h e c o u r s e o f t h e c o l l i s i o n , t h e s i n g l e t e l e c t r o n i c s t a t e s u n d e r g o a d i f f e r e n t p h a s e s h i f t t h a n t h e t r i p l e t e l e c t r o n i c s t a t e s , a n d a s a r e s u l t t h e p a i r o f atoms by (6.2) 14 A -61-are i n a l i n e a r combination of product s t a t e s a f t e r the c o l l i s i o n . By assuming t h a t the d i f f e r e n c e i n phase s h i f t s i s s u f f i c i e n t l y l a r g e t h a t i t can be modelled as a random angle, the p r o b a b i l i t y can be c a l c u l a t e d f o r two atoms which were i n h y p e r f i n e s t a t e s fti , (f/j before the i n t e r a c t i o n being i n s t a t e s (J}K> [fa afterward. Thus Brown d e r i v e s a r a t e equation f o r the po p u l a t i o n s of the h y p e r f i n e s t a t e s . I f we l e t C-c be the number d e n s i t y of atoms i n the Ith energy e i g e n s t a t e (where the s t a t e s are numbered i n order of i n c r e a s i n g energy as i n s e c t i o n 2.1), and d e f i n e fii by (.6.3) where Ci.e i s the expected number d e n s i t y a t thermal e q u i l i b r i u m , then t h i s r a t e equation i s ItS] h i - i » \ A 1 -(i+ri 1 -d-x) -1 1 -1 1 n \ 1 - 0 - x ) | - ( n * ) J where S i s a constant depending on the c r o s s s e c t i o n f o r a c o l l i s i o n r e s u l t i n g i n a l a r g e phase s h i f t , g i v e n by S=(5e.x7H.H (6.5) and X i s giv e n by X^8cos a(0)(l-cos 1^)) (6.6) where $ i s d e f i n e d i n equation (2.8) (at 6481 Gauss, X = .012). The general solution to t h i s rate equation i s /A(±V\ / I o i A W . o l - l A W -1 o \ W)/ \ 0 - i -I \ / \ C2exp(-SnMXi/2) / (6.7) where C2=(/030-/°i«>)/2. (6.8) and yO £ o i s the value of ^  at t=0. At t y p i c a l flow rates of 10±O molecules E^ sec , and 16 - 3 t y p i c a l H 2 densities i n the flow tube of 10 cm , the gas flow v e l o c i t y was of order 100 cm/sec, and thus the gas would take about 150 msec to flow from the edge of the electromagnet pole to the resonator. By comparison, the eq u i l i b r a t i o n times (SnHX/2) ^, and (Sty) - 1, at H densities of 14 -3 about 2X10 cm , are about 3 msec, and 20M sec respectively. Thus i t i s expected that by the time the gas reaches the resonator, the spin exchange relaxation of the population levels w i l l have proceeded to i t s f u l l e s t extent. These f i n a l populations w i l l be: fist O C6.9) o -63-Thus the p o p u l a t i o n s of l e v e l s one and three reach t h e i r e q u i l i b r i u m v a l u e s , w h i l e those of l e v e l s two and four do not. I f we assume t h a t the va l u e s of C-^ , C^, C^, and C^, are a t t=0, then the f i n a l p o p u l a t i o n s are (6.10) C,f = Cie. The f i n a l p o p u l a t i o n d i f f e r e n c e between l e v e l s one and two i s thus Cif-C* = (C2e-C^) -V (cH(,-Cze)/2 (6.11) At 77K, the e q u i l i b r i u m v a l u e s f o r the Cj's a r e : C a e = ( l - 5: HO xid*) ri„// C (e = (l+ 6.26 x 16^) K.M (.6.12) and thus ( C 2 f ~ C l f ) / ( C 2 e ~ C l e ^ = 1 3 - ° ' t n a t i s t h e f i n a l p o p u l a t i o n d i f f e r e n c e between l e v e l s one and two i s 13 times g r e a t e r than i t would be a t thermal e q u i l i b r i u m . T h i s enhanced p o p u l a t i o n would cause an enhanced magnetic resonance s i g n a l i n the same p r o p o r t i o n . However, as i s shown i n -64-Appendix A," t h i s enchancement f f l i s a p p e a r s a f t e r r e p e t i -t i v e Tf/2 "pulses, T h i s occurs because the TT/2 p u l s e averages the p o p u l a t i o n s of s t a t e s one and two, and the s p i n exchange r e l a x a t i o n then r e t u r n s the p o p u l a t i o n of l e v e l one t o e q u i l i b r i u m , l e a v i n g the p o p u l a t i o n of l e v e l two wit h about h a l f i t s o r i g i n a l d i f f e r e n c e from e q u i l i b r i u m . The end r e s u l t of t h i s process i s the f o l l o w i n g : That i s , when one averages the above C , • and C 0 v a l u e s with a to i t s f u l l e s t e xtent, the va l u e s of the Ci are back to what they were bef o r e the Tf/2 p u l s e . The d i f f e r e n c e between l e v e l s one and two bef o r e the tf/2 p u l s e under these c o n d i t i o n s i s Thus although there i s c l e a r l y the p o s s i b i l i t y o f a s i g n a l enhancement r e s u l t i n g from s p i n exchange, the degree of t h i s enhancement should be s t r o n g l y dependent on the p u l s e r e p e t i t i o n r a t e . When the r a t e i s slow enough t h a t every atom f l o w i n g down the tube experiences only one Tf/2 p u l s e (which r e q u i r e s a p u l s e i n t e r v a l o f 10 mses), the s i g n a l should be enhanced 13 times, while at r e p e t i t i o n " r a t e s c a using s e v e r a l Tf/2 p u l s e s (6.13) ft/2 p u l s e , and allows s p i n exchange r e l a x a t i o n to proceed 1 6 . 1 4 ) -65-per atom, the enhancement should r a p i d l y disappear. Experimen-t a l l y , no s i g n i f i c a n t v a r i a t i o n i n s i g n a l s t r e n g t h was found i n v a r y i n g the p u l s e i n t e r v a l from 200^csec to 10 msec. Thus i t seems t h a t s p i n exchange i s not the o n l y r e l a x a t i o n mechanism prese n t . The o n l y other mechanisms we have thought of are i n t e r a c t -ions between the hydrogen atoms and paramagnetic s i t e s on the w a l l s of the flow tube. Such i n t e r a c t i o n s w i t h u n l i k e s p i n s would cause complete e q u i l i b r a t i o n of the s p i n system. However, i t seems d o u b t f u l t h a t t h i s process c o u l d s i g n i f i c a n t l y c o n t r i -bute to the observed r e l a x a t i o n r a t e s . The most probable p a r a -magnetic gas i m p u r i t y i s / which has s p i n exchange c r o s s sec-"™ 16 2 21 t i o n 25.1 X 10 cm , and would t h e r e f o r e have to be about as abundant as hydrogen atoms, ( i . e . s e v e r a l molecules per thousand molecules E^) , to s i g n i f i c a n t l y c o n t r i b u t e . Since we have no reason to expect i m p u r i t i e s a t t h i s very h i g h l e v e l , t h i s p o s s i b i l i t y can be r u l e d out. A s i m i l a r problem e x i s t s with paramagnetic w a l l s i t e s . Since the atoms d i f f u s e t o the t e f l o n coated w a l l i n times somewhat g r e a t e r than , the t e f l o n s u r f a c e would have to be almost completely covered w i t h paramag-net s i t e s t o s i g n i f i c a n t l y c o n t r i b u t e , which seems very u n l i k e l y . I t i s p o s s i b l e t h a t such i n t e r a c t i o n s w i t h u n l i k e s p i n s would weakly c o n t r i b u t e to the r e l a x a t i o n , and t h a t i n combina-t i o n w i t h s p i n exchange r e l a x a t i o n t h e r e c o u l d be a net s i g n a l enhancement less.dependents on p u l s e r a t e . Another explan-a t i o n f o r our s m a l l value f o r o*ex i s an e r r o r i n the s p e c t r o -meter c a l i b r a t i o n procedure. At p r e s e n t the problem remains unresolved. -66-6.2 Data Taken at 4.2 K With No B u f f e r Gas. Upon t r a n s f e r r i n g l i q u i d helium i n t o the i n n e r dewar, the magnetic resonance s i g n a l would disappear and then a f t e r a few minutes i t would g r a d u a l l y r e t u r n , as was d e s c r i b e d i n s e c t i o n 5.4. Although the s i g n a l was s t r o n g e r a t l i q u i d helium temperature, the a c t u a l number d e n s i t y c a l c u l a t e d , assuming thermal e q u i l i b r i u m of the s p i n system at 4.2K, 13 -3 had decreased to 3 X 10 cm . T 2 f o r t h i s s i g n a l was l e s s than 50>«.sec. Although t h i s s h o r t T 2 c o u l d a r i s e from s p i n exchange, the necessary c r o s s s e c t i o n would have to be very l a r g e , which would c o n t r a d i c t the t h e o r e t i c a l p r e d i c t i o n 19 of A l l i s o n . Another e x p l a n a t i o n was t h a t the atoms were d i f f u s i n g out of the r e s o n a t o r or a g a i n s t the w a l l of the flow tube, s i n c e a t these temperatures the H 2 gas i s f r o z e n out on the w a l l of the flow tube, and such d i f f u s i o n times would t h e r e f o r e be q u i t e s h o r t . Since the mean f r e e path of the atoms at d e n s i t i e s of 13 -3 3 X 10 cm i s of the order of 30 cm, and thus much longer than the sample r e g i o n , the c h a r a c t e r i s t i c time, t , f o r them to leave the r e s o n a t o r or c o l l i d e with the flow tube w a l l i s g i v e n roughly by (6.15) where i i s a c h a r a c t e r i s t i c l e n g t h of the sample r e g i o n , and v" i s the average speed of the atoms, which i s (.6.16) -67-L e t t i n g ^ = l c m , the value of t o b t a i n e d i s about 50/(sec, which s t r o n g l y suggested t h a t such d i f f u s i o n was the mechansim behind the observed T 2-To t e s t t h i s h y p o t h e s i s , a s m a l l q u a n t i t y of helium gas was added to the sample r e g i o n v i a the pumping l i n e , with the d i f f u s i o n pump c l o s e d o f f . The r e s u l t was a l a r g e i n c r e a s e i n T 2, as would be expected i f d i f f u s i o n was the decay mechanism, s i n c e the mean f r e e path f o r the atoms would decrease upon a d d i t i o n of helium " b u f f e r " gas, thus length e n i n g d i f f u s i o n times. F i g . 6.2 shows a s e r i e s of f r e e i n d u c t i o n decays at v a r i o u s b u f f e r gas d e n s i t i e s . In s e c t i o n 4, d e t a i l e d d i s c u s s i o n of the r e s u l t s o b t a i n e d by v a r y i n g b u f f e r gas d e n s i t y are presented. Before p r e s e n t i n g these, however, i t i s u s e f u l to d e s c r i b e the a n a l y s i s which was performed of the expected r e l a t i o n s h i p between T 2 and b u f f e r gas d e n s i t y . 6.3 A n a l y s i s of the D i f f u s i o n Problem. In the next s e c t i o n , i t w i l l be shown t h a t the decay r a t e of the f r e e i n d u c t i o n s i g n a l , 1/T 2, and a frequency s h i f t of the s i g n a l , Aou, both o c c u r r e d i n d i r e c t p r o p o r t i o n to the d i f f u s i v i t y of the hydrogen atoms i n the b u f f e r gas. In t h i s s e c t i o n a model i s presented to e x p l a i n t h i s behaviour, and a d e s c r i p t i o n i s g i v e n of a computer s i m u l a t i o n of t h a t model. The r e s u l t s of t h i s s i m u l a t i o n w i l l be employed i n the a n a l y s i s i n s e c t i o n 4. In the model, the atoms d i f f u s e through the b u f f e r gas, out of the r e s o n a t o r r e g i o n and/or a g a i n s t the w a l l of the flow F i g u r e 6 . 2 F r e e i n d u c t i o n d e c a y s a t 4 . 2 K w i t h v a r i o u s h e l i u m b u f f e r g a s d e n s i t i e s -69-tube. I t i s p o s t u l a t e d t h a t d u r i n g the time an atom i s i n c o n t a c t w i t h the w a l l of the flow tube (which i s probably phase s h i f t can produce an e f f e c t i v e frequency s h i f t o f the t o t a l f r e e i n d u c t i o n s i g n a l , and w i l l a l s o cause a decay of t h a t s i g n a l s i n c e atoms which have accumulated d i f f e r e n t amounts of phase s h i f t become out of phase with one another. These phase s h i f t s r e s u l t from the change i n the c o n t a c t h y p e r f i n e i n t e r a c t i o n due to c o l l i s i o n s w i t h the H 2 coated w a l l s . (This same e f f e c t leads to a lowering of the h y p e r f i n e frequency f o r H trapped i n s o l i d H,,.) The c o l l i s i o n with the w a l l i s not instantaneous s i n c e the atom can " s t i c k " or stay c l o s e t o the w a l l f o r some time. We model t h i s e f f e c t by saying t h a t the atom has an equal p r o b a b i l i t y of l e a v i n g the s u r f a c e per u n i t time, and [thus t h a t the p r o b a b i l i t y d i s t r i b u - -t i o n f o r the magnitude of the phase s h i f t i s given by In t h i s model, the v a l u e s f o r T 2 and Aware determined as f o l l o w s . L e t $i (t) be the t o t a l accumulated phase s h i f t f o r t h the i atom, and l e t 0.(t) be a f u n c t i o n which i s 0 i f the ' r i atom i s not i n the r e s o n a t o r a t time t , and 1 i f i t i s . Then the r e l a t i v e amplitude of the f r e e i n d u c t i o n s i g n a l , A ( t ) , i s g i v e n by coated with s o l i d H„) , i t undergoes a phase s h i f t - A ' . T h i s (6.17) AM (6.18) t -70-and the net phase s h i f t of the f r e e i n d u c t i o n s i g n a l , ( t ) , i s g i v e n by: 0(-fc)=tan _X 1 C6.19) T 2 i s then taken to be the time at which the amplitude has decayed to 1/e, t h a t i s A ( T x ) = e x P ( - r ) (6.20) Now i t w i l l be seen t h a t i n the computer s i m u l a t i o n it) i n c r e a s e s q u i t e l i n e a r l y w i t h time, (see F i g . 6.3) and i t i s thus reasonable to speak of the e f f e c t i v e frequency s h i f t , Aou, as dt ( 6 - 2 1 ) More p r e c i s e l y , s i n c e we measured frequency by counting the t o t a l phase s h i f t o c c u r r i n g over a time of order T 2 (see s e c t i o n 4.4) , the best d e f i n i t i o n of Auu i s g i v e n by: A w = = i j M (6.22) In s i m u l a t i n g the above model a g r e a t s i m p l i f i c a t i o n can be made by assuming t h a t the mean f r e e path of the hydrogen atoms i s q u i t e a b i t s m a l l e r than the sample r e g i o n , so t h a t the movement of the atoms i s governed by the d i f f u s i o n equation. T h i s assumption i s v a l i d f o r a l l of the measurements presented i n the next s e c t i o n . I f r j i s the number d e n s i t y of hydrogen atoms which r e c e i v e d the Tt/2 p u l s e , the d i f f u s i o n equation i s -71-h* = D V 2 f i * ( 6 . 2 3 ) where D i s the d i f f u s i v i t y of the hydrogen atoms i n the b u f f e r gas, and the boundary c o n d i t i o n s are ( V n * ) - f i - 0 <6-24) where n i s the normal v e c t o r to any impermeable s u r f a c e . The value of D may be expressed i n terms of a d i f f u s i o n c r o s s s e c t i o n , Q 0, as b = ( i ? ) V R / n B Q 0 ( 6 . 2 5 ) where V R i s the average r e l a t i v e speed of hydrogen atoms and b u f f e r gas p a r t i c l e s , and h&is the b u f f e r gas d e n s i t y . Although we do not s o l v e the d i f f u s i o n e quation d i r e c t l y , ( i n s t e a d we model the a c t u a l motion of the atoms), i t i s i n s t r u c t i v e to study the form of t h i s equation. Since the equation i s l i n e a r , i t i s p o s s i b l e to s c a l e a s o l u t i o n f o r one set of i n i t i a l c o n d i t i o n s and d i f f u s i o n c onstant to a s e t of i n i t i a l c o n d i t i o n s with the same geometry, but d i f f e r e n t s i z e , and/or with d i f f e r e n t d i f f u s i o n c onstant. I f we l e t tt^CiTji) be the s o l u t i o n to the problem with i n i t i a l c o n d i t i o n n*0%O) , boundary d e f i n e d by f ^ ( r ) = 0 , and d i f f u s i o n constant D^, and we seek a s o l u t i o n t o nj.*(?\o) , f 2 ( r ) = 0 , D 2, then i f n*lr>o) = ni*(0WSo) and f 2 (r) = f x ( a r ) ; the s o l u t i o n n * C ^ O i s g i v e n by h z * ( R t ) = n , * ( a r , - ^ ^ ( 6 . 2 6 ) In other words, i f one s c a l e s every f e a t u r e of the system by t h e same f a c t o r , a n d / o r c h a n g e s t h e d i f f u s i o n c o n s t a n t , t h e c h a r a c t e r i s t i c t i m e s a n d r a t e s s i m p l y c h a n g e b y an a p p r o p r i a t e f a c t o r , a n d t h e e v o l u t i o n o f h* i s o f t h e same b a s i c f o r m . Thus we c a n e m p l o y a c o m p u t e r s i m u l a t i o n i n a r e g i o n w i t h t h e same g e o m e t r y a s t h a t o f t h e e x p e r i m e n t a l s a m p l e r e g i o n , b u t w i t h a r b i t r a r y s i z e and d i f f u s i o n c o n s t a n t , a n d a p p l y t h e r e s u l t t o t h e e x p e r i m e n t a l r e g i o n . I n p a r t i c u l a r , t h e a b o v e c o n s i d e r a t i o n s i m p l y t h a t t h e d i f f u s i o n c o n t r o l l e d T 2 f o r a g i v e n g e o m e t r y s h o u l d be o f t h e f o r m ( 6 . 2 7 ) where H i s a c h a r a c t e r i s t i c s i z e , ( w h i c h we w i l l t a k e f o r c o n -v e n i e n c e t o be t h e h e i g h t o f t h e r e s o n a t o r ) , a n d of i s a d i m e n s i o n l e s s f a c t o r w h i c h may d e p e n d on t h e v a l u e o f Ao . The v a l u e o f <% c a n t h u s be o b t a i n e d f r o m t h e c o m p u t e r s i m u l a -t i o n , a n d a p p l i e d t o t h e e x p e r i m e n t a l r e s u l t s . I n t h e c o m p u t e r s i m u l a t i o n t e c h n i q u e , e a c h atom i s r e p r e s e n t e d a s a p o i n t w i t h i n t e g e r c o o r d i n a t e s i n t h r e e d i m e n s i o n s , a n d h a s an a d d i t i o n a l r e a l v a r i a b l e t o r e c o r d i t s t o t a l a c c u m u l a t e d p h a s e s h i f t . The c o o r d i n a t e s o f e a c h atom u n d e r g o a random c h a n g e a t e a c h i t e r a t i o n o f t h e p r o g r a m . The c h a n g e c o n s i s t s o f m o v i n g e i t h e r p l u s o r m i n u s one u n i t i n t h e x, y, a n d z d i r e c t i o n s , t h e c h o i c e i n e a c h d i r e c t i o n b e i n g made by a random number g e n e r a t o r . The f l o w t u b e was m o d e l l e d a s a s e m i - i n f i n i t e t u b e e x t e n d i n g i n t h e p o s i t i v e % d i r e c t i o n , w h i c h h a d a c r o s s s e c t i o n o f Nx'.N u n i t s , w h e r e N c o u l d be a n y 3 odd i n t e g e r . N atoms were i n i t i a l l y " placed", one on each of the s i t e s i n the N bottom l a y e r s of the flow tube, so as to model the o r i g i n a l d i s t r i b u t i o n of atoms i n the res o n a t o r which r e c e i v e a TT/2 p u l s e . (The value of N was chosen t o be l a r g e enough t o reduce random n o i s e and e l l i m i n a t e the e f f e c t of d i s c r e t e steps. N=13 was found t o be s u f f i c i e n t . ) Whenever a random step would cause an atom t o pass through the w a l l of the flow tube, the component of the step r e s p o n s i b l e f o r t h i s would not be taken. In a d d i t i o n , the phase of the atom would be changed an amount,-A , where A was randomly chosen a c c o r d i n g t o the formula where RND i s a random number between 0 and 1 .- I t i s e a s i l y shown t h a t the p r o b a b i l i t y d i s t r i b u t i o n f o r t h i s A i s t h a t given i n ( 6 . 1 7 ) . At the end of each i t e r a t i o n , the computer would c a l c u l a t e A and p as d e s c r i b e d i n (.,6"t 18f):~ahda-.(6.19)', and p r i n t them. F i g u r e 6 .3 shows a p l o t of A(t) and 0 (t) f o r v a r i o u s v a l u e s of A 0. The e f f e c t i v e d i f f u s i o n constant corresponding t o the random walk procedure used i n t h i s program i s e a s i l y determined, by c o n s i d e r i n g the mean square d i s t a n c e an atom would t r a v e l a f t e r h random steps w i t h no impermeable boundaries, and comparing t o the r e s u l t p r e d i c t e d from the d i f f u s i o n equation. Since A=-A 0^(RND) (6.28) < l-*>=<X*> + < i f > + <2L*> (6.29) Figure %<o. 3 Computer simulation r e s u l t s f or Anr\ and vs t, for various values of A 0 • -75-and the walk procedure i s symmetrical and independent i n the t h r e e c o o r d i n a t e s , <r * >=3<X 2> (6.30) The value of i s most e a s i l y o b t a i n e d by means of the c e n t r a l l i m i t theorem, which s t a t e s t h a t i f one adds n random v a r i a b l e s , each w i t h v a r i a n c e (7X, then the v a r i a n c e of the sum w i l l be nff 1. In our case, the p r o b a b i l i t y d i s t r i b u t i o n f o r the random v a r i a b l e i s P(Ax)=±*(Ax+l) + ^ ( A x - l ) (6.31) which has v a r i a n c e 6**=1. Thus the v a l u e f o r <Xa> i s r» , and n. Now i n the s o l u t i o n t o the d i f f u s i o n equation w i t h a d e l t a f u n c t i o n d e n s i t y d i s t r i b u t i o n at the o r i g i n at t=0, one f i n d s <r 2>=6Dt ( . 6 . 3 2 ) and s i n c e < v * > = 3 n , and t=n, the e f f e c t i v e d i f f u s i o n constant i n t h i s computer s i m u l a t i o n i s D=l/2. For the purpose of d e t e r m i n i n g c * , we note t h a t the h e i g h t of the r e s o n a t o r , H, i s N u n i t s i n t h i s s i m u l a t i o n , and hence . HP _ J5- (6.33) *" Ha " 2N3" where i s measured i n number of i t e r a t i o n s . As can be seen i n F i g . 6.3 (p (t) i n c r e a s e s q u i t e l i n e a r l y i n time, and so i t i s meaningful to speak of a frequency s h i f t Au^ d e f i n e d as F i g u r e 6.4 C o m p u t e r s i m u l a t i o n v a l u e s f o r T 2 A c o a n d c\ v s Ao. - 7 7 -( 6 . 3 4 ) T L F o r comparison t o experimental data, i t i s u s e f u l t o determine the dimensionless q u a n t i t y T to I t would seem reasonable t h a t t h i s q u a n t i t y would depend only on the nature of the phase s h i f t on the w a l l s , s i n c e the number of such phase s h i f t s i n a time T 2 should be independent of the d i f f u s i o n r a t e and ab s o l u t e s i z e of the system. T h i s f a c t was v e r i f i e d by changing the value of N (hence lengthen-i n g T 2 ) , and f i n d i n g t h a t n e i t h e r 0(^2^ n o r °* w e r e s i g n i f i -c a n t l y changed. Thus our computer s i m u l a t i o n had one v a r i a b l e parameter, Ao* and c o u l d be used t o p r e d i c t the dimensionless q u a n t i t i e s ^(Ta^ and T2/_yi> f o r t h i s geometry. F i g . 6.4 shows a p l o t of o( and T 2Au) vs Ao . These r e s u l t s w i l l be compared t o the e x p e r i -mental data i n the next s e c t i o n . 6.4 Experimental R e s u l t s w i t h B u f f e r Gas at Low Temperatures. The v a l u e s o f T 2 o b t a i n e d from the f r e e i n d u c t i o n decays are shown as a f u n c t i o n of b u f f e r gas d e n s i t y i n the l o g - l o g p l o t i n F i g . 6.-5. The c i r c l e s i n t h i s graph i n d i c a t e data which were taken with helium b u f f e r gas, which was added v i a the pumping l i n e with the d i f f u s i o n pump c l o s e d o f f , and the c r o s s e s i n d i c a t e data taken w i t h H 2 b u f f e r gas, obt a i n e d by a l l o w i n g the s o l i d H 9 coated d e t e c t i o n r e g i o n to warm TiAtu = <p (Tx) (6.35) -78-L_ I I I 1 4 I015 I016 I017 I018 nB(cm"3) .Figure 6.5 L o g - l o g , p l o t o f vs rig. C i r c l e s represent measurements w i t h He b u f f e r gas a t 4.2K, the crosses are f o r H 2 buf fe r gas f o r 5.6<T<8.4K; the s o l i d l i n e represents a l i n e a r dependence on n ^ . -79-slowly between 5 and 9 K Cafter a l l the l i q u i d helium i n the dewar had b o i l e d o f f ) , so as to produce a vapor p r e s s u r e o f H 2- For both the He and H 2 b u f f e r gases, the maximum d e n s i t y which c o u l d be used was determined by the f a c t t h a t the magnetic resonance s i g n a l would disappear above a c e r t a i n d e n s i t y , presumably because the atoms were unable t o d i f f u s e through the b u f f e r gas t o the re s o n a t o r b e f o r e recombining. In the case of He b u f f e r gas, the d e n s i t y , he , was determined by measuring the gas pr e s s u r e i n the pumping l i n e w i t h the i o n gauge on the d i f f u s i o n pump (which had been c a l i b r a t e d a g a i n s t a V a r i a n 843 gauge of 2.2% a c c u r a c y ) , and c a l c u l a t i n g the corresponding d e n s i t y i n the d e t e c t i o n r e g i o n . In performing t h i s c a l c u l a t i o n , one must take i n t o account the 22 thermomolecular e f f e c t , which f o r a pumping l i n e o f our dimension, takes the form where we have assumed t h a t the He behaves l i k e an i d e a l gas. The d e n s i t y of H 2 b u f f e r gas was determined by measuring the temperature of the d e t e c t i o n r e g i o n v i a the c a l i b r a t e d carbon r e s i s t o r thermometer (see s e c t i o n 5.4), and i n f e r r i n g H B 23 from the known vapour pressure of H,,. Acc o r d i n g to the d i f f u s i o n d i s c u s s i o n of the l a s t s e c t i o n , we expect t h a t T„ should be i n v e r s e l y p r o p o r t i o n a l to D, and (6.36) (where T=* and Th are the temperatures i n the two regions) and thus (6.37) -80-hence p r o p o r t i o n a l t o n.B . For both the He and H 2 b u f f e r gas r e s u l t s i n F i g u r e 1, i t can be seen t h a t l i n e s of u n i t slope on the l o g - l o g p l o t f i t the data w e l l over two decades of b u f f e r gas d e n s i t y , i n good agreement wi t h the d i f f u s i o n model. (The apparent d e v i a t i o n from t h i s r e l a t i o n s h i p o f the hi g h e s t d e n s i t y H 2 b u f f e r gas p o i n t s w i l l be d i s c u s s e d below.) Frequency s h i f t data were ob t a i n e d by v a r y i n g the magnetic f i e l d about the va l u e H 0, i n order to determine the experimental minimum frequency, f ; . Some of these r e s u l t s are shown i n min 14 -3 F i g . 6.6, f o r h 6(He)=(2-3)XlO cm ( c i r c l e s ) , and 1 5 - 3 n B(He)=10 cm ( c r o s s e s ) . Both s e t s of data f i t a p a r a b o l a of the p r e d i c t e d c u r v a t u r e , centered about e s s e n t i a l l y the same value of magnetic f i e l d . I t i s c l e a r from t h i s data t h a t the experimental minimum frequency, f g ^ ^ / i s n e g a t i v e l y s h i f t e d from the t h e o r e t i c a l v alue f u , by an amount which i s pre s s u r e dependent. The pr e s s u r e dependence of t h i s s h i f t was found, w i t h i n experimental e r r o r , to be the same as t h a t of the decay r a t e , as i n d i c a t e d i n F i g . 6.3, where the d i f f e r e n c e f -f:... ' ^ ' o mm i s p l o t t e d as a f u n c t i o n of 1/T 2. The data can be f i t with a s t r a i g h t l i n e through the o r i g i n w i t h slope such t h a t 2 Tr (fo-f:.- .. 0 T„=T„Au> =1. 3 dh . 4 . The f a c t t h a t T„A<o i s constant o mm 1 1 A f o r t h i s data, and t h a t T 2 o( n B s t r o n g l y suggests t h a t the d i f f u s i o n model of the pr e v i o u s s e c t i o n i s c o r r e c t . In F i g . 6.4, i t can be seen t h a t T 2 Aco i n the s i m u l a t i o n has a broad maximum at Ao—.2. The experimental value of 1 . 3 ± . 4 would be c o n s i s t e n t w i t h . I < Ao <. 3. F u r t h e r i n f o r m a t i o n on the value of Ao was obt a i n e d by ob s e r v i n g the shape of the decay curve Jin hit) vs t . W i t h A o = . l , the s i m u l a t i o n decay curve was r a t h e r H - H 0 (Gauss) Figure 6.6 Resonance frequency as a func t ion o f magnetic k f i e l d , p l o t t e d as f - f . vs H-Ho where f.=765 483 208 Hz. and H.-6481.9 G. - 8 2 -Figure 6.7 fo-fmir» vs 1/T2 for (dots) and He (squares) buffer gas data. -83-non-exponential, as would be expected f o r a pure d i f f u s i o n problem, while f o r Ao=.3, the decay was e s s e n t i a l l y e x p o n e n t i a l . In the experimental data, i t was found t h a t the decays with He b u f f e r gas d i s p l a y e d the non-exponential appearance of the s i m u l a t i o n s w i t h A o = . l , while those with H 2 b u f f e r gas were q u i t e e x p o n e n t i a l . Examples are shown i n F i g . 6.8. Thus i t appears t h a t the nature of the w a l l of the flow tube was d i f f e r e n t i n the two cases ( p o s s i b l y because of the s l i g h t l y d i f f e r e n t temperatures, l e a d i n g to d i f f e r e n t v a l u e s of Ao. Knowing A 0, can be determined from the data i n F i g . 6.4:, which can be employed wi t h equations (6.25) and (6.2 7) to o b t a i n Q - 3TT V~K_ Ta. (6.38) 32 o^ H* n B where Va , the average r e l a t i v e speed of a hydrogen atom and b u f f e r gas p a r t i c l e , i s given by V * - ^ / ™ , > mH + r«B ( 6 ' 3 9 ) Using A 0 = . l f o r the helium data, which yieldso( =.25, and -18 3 the value T 2/n$ =1. 4X10 cm sec f o r the top l i n e i n F i g . 6.5, we o b t a i n a value f o r tyo f o r H i n He a t 4.2 K of about 500 A. For H 2, l e t t i n g Ao=.3, we found e( =.09, and u s i n g the value -19 3 T 2/n B=2.6X10 cm sec f o r the bottom curve i n F i g . 6.5, we o b t a i n a v a l u e f o r Q 0 f o r H i n H 2 i n the range of 5-9 K of about 250.'A* In view of the u n c e r t a i n t y i n the p r e c i s e nature of the - 8 4 -8 Comparison of experimental decay shapes with computer simulation results. -85-phase s h i f t s at the w a l l s , and s l i g h t d i f f e r e n c e s between the a c t u a l flow tube geometry and t h a t of the s i m u l a t i o n , these r e s u l t s are not expected to be very a c c u r a t e . However i t would seem u n l i k e l y t h a t our c a l c u l a t e d v a l u e s c o u l d be i n c o r r e c t by more than a f a c t o r of two, and i n t h i s l i g h t the very l a r g e v a l u e s of we o b t a i n e d are extremely s u p r i s i n g , and would imply resonances i n the H-He and H-H2 c r o s s s e c t i o n s 4 a t low e n e r g i e s . However, at l e a s t f o r H-He , the e x i s t a n c e of resonances i s i n c o n s i s t e n t with the measurements of Toennies, Welz, and Wolf , who o b t a i n about 40 A f o r the t o t a l c r o s s s e c t i o n at low e n e r g i e s . I t i s very u n l i k e l y t h a t the p r o p e r l y averaged thermal c r o s s s e c t i o n would be more than twice t h i s v a l u e . N e v e r t h e l e s s , we have been unable to f i n d i n c o n s i s t - ; e n c i e s i n our data, or any a l t e r n a t i v e e x p l a n a t i o n f o r the long T 2 1 s which occur i n p r o p o r t i o n to a p p a r e n t l y s m a l l q u a n t i t i e s of b u f f e r gas. The e f f e c t of s p i n exchange, which should produce a decay r a t e independent of b u f f e r gas d e n s i t y , was not apparent f o r most of the low temperature data. In F i g . 6.5, however, i t can be seen t h a t a t the h i g h e s t b u f f e r gas d e n s i t i e s , (with H 2 b u f f e r gas), the data curve away from the l i n e a r r e l a t -i o n s h i p . The dotted curve i s of the form 1 / T x = l / a n B + l / T x e x (6.40) where a i s the value of T 2 / n a f o r the l i n e p a s s i n g through the H 2 b u f f e r gas data, and T 2* =12 msec. The reasonably good f i t suggests t h a t the q u a n t i t y T®* r e p r e s e n t s the T? which would - 8 6 -r e s u l t p u r e l y from s p i n exchange. The value of n H, determined 13 -3 from the s i g n a l s t r e n g t h , was 3X10 cm , and us i n g the same procedure as i n s e c t i o n 6.1 we f i n d the s p i n exchange c r o s s s e c t i o n 6"ex a t the temperature i n v o l v e d (~8K) , to be 0.9 A. Th i s r e s u l t i s c o n s i s t e n t w i t h e x t r a p o l a t i o n s of the r e s u l t s 19 of A l l i s o n , and i s i n good agreement with the c a l c u l a t i o n of 20 B e r l i n s k y . Of course i t i s p o s s i b l e t h a t there i s some other b u f f e r gas d e n s i t y independent decay mechanism and t h a t <5~eiC i s t h e r e f o r e s m a l l e r . In any case, i t i s c l e a r t h a t 6*e<'1 decreases by a t l e a s t a f a c t o r of 15 as the temperature i s lowered from 77 to 8 K. -87-CHAPTER VII SUMMARY Atomic hydrogen gas was produced by d i s s o c i a t i n g H 2 i n a room temperature R.F. d i s c h a r g e , and was p i p e d 1 meter i n t o a c r y o g e n i c system which c o u l d c o o l the gas t o temperatures i n the range of 77-1 K. The atoms were d e t e c t e d by means of p u l s e d magnetic resonance. The t r a n s i t i o n observed was between the two lowest h y p e r f i n e l e v e l s of the Is atom, i n a f i e l d of 6.5 kG, where the resonance frequency has a minimum va l u e of 765 MHz. At 77 K, the e f f e c t s of s p i n exchange r e l a x a t i o n were observed. The s p i n exchange c r o s s s e c t i o n was c a l c u l a t e d from the observed p r o p o r t i o n a l i t y constant between the s i g n a l decay r a t e and the hydrogen atom number d e n s i t y . The v a l u e of 14 A 19 obtained i s 60% of the t h e o r e t i c a l v alue , which has p r e v i o u s l y 7 been e x p e r i m e n t a l l y confirmed/. I t i s p o s s i b l e t h a t t h i s d i s c r e p a n c y r e s u l t e d from the p o p u l a t i o n s of the h y p e r f i n e s t a t e s d i f f e r i n g from t h e i r thermal e q u i l i b r i u m v a l u e s , as i s p o s s i b l e i f spin-exchange between hydrogen atoms i s the dominant r e l a x a t i o n mechanism. However, such an e x p l a n a t i o n would appear to c o n t r a d i c t the observed independence of the s i g n a l s t r e n g t h and p u l s e r e p e t i t i o n r a t e f o r r a t e s l e s s than about 1/2T 2. At present the problem remains unresolved. At l i q u i d helium temperatures the e f f e c t s of spin-exchange were much s m a l l e r , and d i f f u s i o n out of the r e s o n a t o r was the dominant T 2 mechanism. Decay r a t e s were measured as a f u n c t i o n 4 of the d e n s i t y of He b u f f e r gas a t 4.2 K and H 2 b u f f e r gas i n the range of 5-9 K. In a d d i t i o n a s h i f t of the minimum frequency was observed, and was found t o be p r o p o r t i o n a l t o the decay r a t e . These r e s u l t s were e x p l a i n e d by a model i n which the atoms d i f f u s e d through the b u f f e r gas, out of the r e s o n a t o r and/or a g a i n s t the w a l l of the flow tube, i n which case they would undergo a random phase s h i f t . The one f r e e parameter i n t h i s model was the mean magnitude of the phase s h i f t . In a computer s i m u l a t i o n of the model, we o b t a i n e d good f i t s t o the decay shapes of the experimental s i g n a l s , as w e l l as the c o r r e c t r a t i o of frequency s h i f t t o decay r a t e . 4 T h i s a n a l y s i s i m p l i e d very low d i f f u s i v i t i e s f o r H i n He and H 2, and hence very l a r g e d i f f u s i o n c r o s s s e c t i o n s . At 4 92. 4.2 K i n He b u f f e r gas, the c r o s s s e c t i o n obtained was 500 A, 02. while a t 5-9 K i n H 2, i t was 250 A. Although these r e s u l t s are not expected to be extremely a c c u r a t e , i t seems u n l i k e l y t h a t they are o f f by more than a f a c t o r of two, and these l a r g e v a l u e s are t h e r e f o r e q u i t e s u r p r i s i n g , and warrant f u r t h e r study The e f f e c t s of spin-exchange were observed at 8 K w i t h our h i g h e s t d e n s i t i e s of b u f f e r gas. The c r o s s s e c t i o n c a l c u l a t e d was .9 A , which i s c o n s i s t e n t w i t h e x t r a p o l a t i o n s of the r e s u l t 19 of A l l i s o n , and i s m good 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 From a p r a c t i c a l p o i n t of view, i t has been e s t a b l i s h e d t h a t at low d e n s i t i e s atomic hydrogen gas can be t r a n s p o r t e d d i s t a n c e s of order 1 meter, c o o l e d to l i q u i d helium temperatures and s t u d i e d by means of magnetic resonance. I t i s hoped t h i s i n f o r m a t i o n w i l l be u s e f u l f o r f u t u r e work i n the f i e l d of s p i n - p o l a r i z e d hydrogen. -89-APPENDIX A: POPULATIONS OF STATES RESULTING FROM REPEATED ft/2 PULSES WITH SPIN-EXCHANGE RELAXATION. We wish t o c o n s i d e r the r e s u l t of performing the f o l l o w i n g o p e r a t i o n N times: a Tf/2 p u l s e i s a p p l i e d , e q u a l i z i n g the p o p u l a t i o n s of s t a t e s 1 and 2, and the system then e q u i l i b r a t e s t o the f u l l e s t extent p o s s i b l e v i a spin-exchange r e l a x a t i o n . These o p e r a t i o n s can be c o n v e n i e n t l y discussed i n matrix n o t a t i o n . We express the s t a t e p o p u l a t i o n s i n terms of the q u a n t i t i e s pi d e f i n e d i n equation ( 6.3). The o p e r a t i o n of a TT/2 p u l s e on s t a t e s 1 and 2 can be expressed as /°3 n-u /°x m i For convenience, we w i l l d e f i n e the ope r a t o r c a u s i n g t h i s o p e r a t i o n t o be Otr/a , expressed as \ 1 \ o / i 0 0 o) 0 o 1 0 o /°3n -a 0 0 lk h 0 % 'kl (A.l) 0 i o o o -a + 0 0 o 0 Hi > a I 0 0 0 ^ = I t i s understood the ope r a t o r i s to be a p p l i d d as i n A . l . i s e a s i l y v e r i f i e d t h a t (Oxrfa^ - OTT/2 ~ e q u a l i z i n g the n p o p u l a t i o n s of s t a t e s 1 and 2 N times i s e q u i v a l e n t to e q u a l i z i n g them once, as expected.) S i m i l a r l y , the o p e r a t i o n of e q u i l i b r a t i o n v i a s p i n -exchange t o the f u l l e s t p o s s i b l e e x t e n t , as d e s c r i b e d i n (A.2) ( I t - 9 0 -equations ( 6 . 9 ) , can be expressed as / % o -'A o \ oooo 1fe o Vi o \o o o o i (Again i t i s e a s i l y v e r i f i e d t h a t (Osoinx^ — OSJ>I*X expected.) Le t us eva l u a t e e x p l i c i t l y our combined o p e r a t i o n , (A.3 ) as / Hi O -% O \ o o o -% 0 sk \0 o o 0 -% + \ o 1 \ / \ 0 1 / o -a a \ o o o Vz o -ft -ft o o o o % o % in oooo 1 o o o 0 1 0 0 o o \ Hz \ /°3K (A.4) 1 /°3n (A.5) Define M, V as fo l l o w s ; o o o o \o o o o / , Q/2 o (A. 6) I 0 so t h a t (Ospinx 0n/2) =• V + M . The a p p l i c a t i o n of t h i s o p e r a t i o n s e v e r a l times can be e a s i l y expressed i n t h i s n o t a t i o n : -91-(Ospx Oif^)2 = \/+ M(^ + M) « V -f M\? + M 1 (Ojp^CWf = V + M(?+M(tf + M))« +M2V -I M3 (A. 8) e t c . C l e a r l y , (A. 9) T h i s e x p r e s s i o n i s e a s i l y e v a l u a t e d , s i n c e M has the p r o p e r t y t h a t M = h M (A.10) Thus ( < v , o*j"- v + (| i%r) M V + ' M Summing the geometric s e r i e s , we o b t a i n (A.11) T h i s e x p r e s s i o n r a p i d l y converges as N-^ Oo t o / \ N . Liw (OsoShxOn&j » 4 M V + V H->oo ' S u b s t i t u t i n g the e x p r e s s i o n s f o r M, V i n (A.6), we o b t a i n \ (A.12) (A. 13) N-*oo 2a 0 -2a 0 (A. 14) / Thus f (irep. poise) 0 -2a \ © ^ (independent of y 0 o ) (A.15) That i s , r e g a r d l e s s of what the pt v a l u e s were i n i t i a l l y , the r e s u l t of r e p e a t i n g t h i s IT/2 p u l s e - s p i n exchange r e l a x a t i o n o p e r a t i o n many times w i l l be t h a t expressed i n (A.15). I t i s i n t e r e s t i n g t h a t n e i t h e r Tf/2 p u l s i n g nor s p i n exchange r e l a x -a t i o n alone "erase the past" i n t h i s manner. Let us now e v a l u a t e the p o p u l a t i o n d i f f e r e n c e between s t a t e s 1 and 2 corresponding to jo ( r e p p U i s e ) " Employing equation (6.3), we have (A.16) and thus fc2-C^.ri^= f " l / V f ' W r l t f c + teae-Cw) ( A . 17) Employing (A.15) and the d e f i n i t i o n of a i n ( A . l ) , we o b t a i n ( C z - C , ) ^ . r l s e=0 (A. 18) -93-REFERENCES 1. W.C. Stw a l l e y and L. H. Nosanow; Phys. Rev. L e t t . ; 36_, 910 (1976) . 2. W.C. St w a l l e y ; Phys. Rev. L e t t . ; 3J7, 1628 (1976). 3. A. J . B e r l i n s k y , R.D. E t t e r s , V.V. Goldman, and I.F. S i l v e r a ; Phys. Rev. L e t t . : 39_, 356 (1977). 4. A. J . B e r l i n s k y ; Phys. Rev. L e t t . ; 3_9, 359 (1977). 5. D. Kleppner, H.M. Goldenberg, and N.F. Ramsey; Phys. Rev.; 126, 603 (1962) . 6. P.W. T r a i n o r , D.O. Ham, and F. Kaufman; J . Chem. Phys. 58, 4599 (1973) . 7. M. D e s a i n t f u s c i e n and C. Audoin; Phys. Rev.; A 13, 2070, (1976). 8. W. N. Hardy, A. J . B e r l i n s k y , and L.A. Whitehead; Phys. Rev. L e t t . ; 4_2, 1042 (1979). 9. S. Crampton, T. Greytak, D. Kleppner, W. P h i l l i p s , D.A. Smith, and A. Weinrib; Phys. Rev. L e t t . ; 42, 1039 (1979) 10. G.W. S e r i e s ; The Spectrum of Atomic Hydrogen; (Oxford U n i v e r s i t y P r e s s , 1957); Chapter 11. 11. H. H e l l w i g , R. F. C. Vessot, M.S. L e c i n e , P.W. Z i t z e w i t z , D.W. A l l e n , and J.W. Glaze; IEEE Trans. Instrum. Meas.; 1M-9, 200 (1970). 12. P.F. Winkler, D. Kleppner, T. Myint, and F.G. Walther; Phys. Rev.; A 5, 83 (1972). 13. F.W. Sears; An I n t r o d u c t i o n to Thermodynamics, The  K i n e t i c Theory of Gases, and S t a t i s t i c a l Mechanics; (Addison-Wesley Inc., Reading, Mass.); Equations (11-8) and (12-32). The author i s indebted to J.Th.M. Walraven f o r sending unpublished notes t o M.D. P h i l l i p s d e s c r i b i n g t h i s t e f l o n c o a t i n g procedure. Designed by W.N. Hardy, u s i n g RCA 2N5108 R.F. power t r a n s i s t o r , d e s c r i b e d i n RCA s o l i d s t a t e d i v i s i o n f i l e #2 80. D. I. Hoult and R.E. Richards; J . Magnetic Res.; 22, 561 (1976). 14. 15. 16. -94-17. Designed by W.N. Hardy. 18. J.D. Jackson; C l a s s i c a l E l e c t r o d y n a m i c s ; (John Wiley and Sons, Inc. New York, 1975) Equations (8.58) and (8.8) . 19. A.C. A l l i s o n ; Phys. Rev.; A 5, 2695 (1972). 20 A. J . B e r l i n s k y , t o be p u b l i s h e d . 21. R.L. Brown; J . Res. Nat. Bur. Stand.; 76A, 103 (1971). 22. T.R. Robrts and S.G. Sydoriak; Phys. Rev.; 102, 304 (1956). 23. W.H. Wooley, R.B. S c o t t , and F.G. Brickwedde; J . Res. Nat. Bur. Stand.; 4JL, 454 (1948) (Research Paper No. RP1932.) 24. J.P. Toennies, W. Welz, and G. Wolf; Chem Phys. L e t t . ; 44:, 5 (1976) . 

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