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Spin relaxation and recombination in atomic hydrogen gas at temperatures around 1 K Marsolais, Richard 1980

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SPIN RELAXATION AND RECOMBINATION IN ATOMIC HYDROGEN GAS AT TEMPERATURES AROUND 1 K by RICHARD MARSOLAIS B . S c , U n i v e r s i t e de Mont rea l , 1978 A 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 requ i red standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1980 (c) R ichard M a r s o l a i s , 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS Date -PEC. ABSTRACT An experiment f o r s tudy ing a gas of atomic hydrogen a t temperatures between 0.9 K and 1.3 K i s descr ibed.The f i e l d dependence of the recom-b i n a t i o n r a t e of H atoms i n t o fl^ molecu les and the spin-exchange c r o s s -s e c t i o n are inves t igated.Atoms a re produced i n s i t u i n a low temperature d i scharge and are conf ined i n a pyrex c e l l . T h e w a l l s of t h i s c e l l are 4 covered w i t h a f i l m of s u p e r f l u i d He f o r p revent i ng recombinat ion . The d e t e c t i o n and the study of the hydrogen atoms i s done u s ing the technique of pu l sed magnetic resonance between the lowest two h ype r f i n e l e v e l s of the Is hydrogen atoms.The frequency i s about 765.5. MHz and corresponds to a sha l low minimum i n the d i f f e r e n c e between the lowest two h ype r f i n e l e v e l s , w h i c h occurs a t 6481 kGauss. A s imple model i s a l s o de r i ved f o r e x p l a i n i n g the f i e l d dependence 4 of the recombinat ion i n the presence of a r e l a t i v e l y l a r g e He d e n s i t y . In f a c t , d u e to the vapour p re s su re , the he l ium den s i t y w i l l be :-above 18 3 10 atoms/cm : in the ' r ange o f ; temperatures i n which we a re i n t e r e s t e d , 1 1 1 2 3 w h i l e the d e n s i t y of hydrogen atoms w i l l be about 10 -10 atoms/cm . The r e l a x a t i o n mechanism assumed i s spin-exchange.The parameters of the model a re f i t to the e x i s t i n g d a t a . I t i s found i n p a r t i c u l a r that the asymptot ic behav ior of the h ype r f i ne popu la t i on s depends s e n s i t i v e l y on the i n i t i a l popu la t i on s . . i i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i CHAPTER ONE: INTRODUCTION 1 CHAPTER TWO: BASIC THEORY OF THE HYDROGEN ATOM 7 (2-1) The Hami l ton ian 7 (2-2) The Hyper f ine Leve l s 10 (2-3) The Hyper f i ne Leve l s a t High and Low F i e l d s 11 CHAPTER THREE: MAGNETIC RESONANCE OF HYDROGEN 14 (3-1) General Two L e v e l System 14 (3-2) Case of the Hydrogen Atom 19 (3-3) Pu l sed Magnetic Resonance 20 CHAPTER FOUR: SIGNAL THEORY AND LEVEL DYNAMICS 22 (4-1) P r i n c i p l e of the Resonator 22 (4-2) Spin Ensemble and S t rength of ' the S i g na l 24 (4-3) M u l t i l e v e l Dynamics 26 CHAPTER FIVE: BULK RECOMBINATION 35 (5-1) P r e l i m i n a r i e s 35 (5-2) Asymptot ic S ta tes 36 (5-3) The Model 39 (5-4) Spin-Exchange 41 (5-5) Re su l t s 45 CHAPTER SIX: THE 765 MHz EXPERIMENT 54 (6-1) The Two T r a n s i t i o n s 55 (6-2) S e n s i t i v i t y to F i e l d Homogeneity 55 (6-3) E f f e c t i v e Magnetic Moments 57 (6-4) S t rength of the S i gna l 57 (6-5) Expected \ - Pu l se 62 i v TABLE OF CONTENTS (Continued) Page CHAPTER SEVEN: EXPERIMENTAL APPARATUS 63 (7-1) P r i n c i p l e 63 (7-2) The C ryos ta t 64 (7-3) The Resonator 69 (7-4) The Low Temperature Discharge 74 (7-5) Temperature Measurement and C o n t r o l 75 (7-6) Indium Sea l 76 (7-7) Exper imenta l Procedure 76 (7-8) Performance 78 APPENDIX A: SHIM COILS 8 0 REFERENCES 102 LIST OF TABLES Recombination Ma t r i x f o r 1 = 0 Recombination Ma t r i x f o r 1 = 1 Popu l a t i on of the Hyper f i ne Leve l s 1°K and 1.5°K and f o r H = 405 G and H = 6.5 kG. ° o v i LIST OF FIGURES F i gu re Page 2-1 Hyper f i ne l e v e l s of a I s hydrogen atom as a f u n c t i o n of the magnetic f i e l d , 13 5-1 E v o l u t i o n of the popu la t i on of the hype r f i ne l e v e l s s t a r t i n g w i t h a l l l e v e l s equa l l y populated and i n zero f i e l d 49 5-2 E v o l u t i o n of the hydrogen atom den s i t y f o r d i f f e r e n t f i e l d s i n case A 50 5-3 E v o l u t i o n of the hydrogen atom den s i t y f o r d i f f e r e n t f i e l d s i n case B 51 5-4 I l l u s t r a t i o n of the r educ t i on of the recombinat ion r a t e when thermal e q u i l i b r i u m c o n d i t i o n s a re assumed 53 7-1 O v e r a l l v iew of the c r y o s t a t 67 7-2 D e t a i l of the bottom par t of the c r y o s t a t 68 7-3 Current path f o r the even and the odd modes of the resonator 72 7-4 Resonator des ign 73 2 A - l Shim c o i l c o n f i g u r a t i o n : Z , and s i n g l e c o i l 85 A-2 Shim c o i l c o n f i g u r a t i o n : ZX and ZY 88 2 2 A-3 Shim c o i l c o n f i g u r a t i o n : XY and X -Y 90 A-4 Sketch of shim c o i l dev i ce 93 A-5 Superconducting switches 96 A-6 C i r c u i t r y of the shimming dev ice 98 A-7 Sketches of f r e e i n d u c t i o n decays 101 CHAPTER ONE - INTRODUCTION In low temperature phy s i c s , i t i s p o s s i b l e to observe s t r i k i n g phenomena, such as s upe r conduc t i v i t y or s u p e r f l u i d i t y , which g i v e d i r e c t evidence of the quantum nature of the system and which become ev ident on l y when most of the thermal energy has been removed. A good example of such a system i s he l ium, f o r which the re e x i s t two i s o tope s , namely 3He and ^He. The former i s a fermion and, thus, obeys Fe rmi -D i rac s t a t i s t i c s , and the l a t t e r i s a boson and obeys Bo se - E i n s t e i n s t a t i s t i c s . There fo re , the c o l l e c t i v e behavior i s q u i t e d i f f e r e n t i n the two cases . In f a c t , l i q u i d ^He has a s u p e r f l u i d s t a t e , which occurs a t temperatures below 2.17°K, the A p o i n t , whereas 3He remains a normal l i q u i d down to about 2 mK where i t becomes a s u p e r f l u i d due to Cooper p a i r i n g (Legget 1980). S u p e r f l u i d i t y i s the s i gna tu re of Bo se - E i n s t e i n condensat ion, a consequence of Bo s e - E i n s t e i n s t a t i s t i c s . I t occurs when the p a r t i c l e s s t a r t to popu late the lowest energy s t a t e , whence the word condensa-t i o n . In 1959, i t was po inted out by Hecht (Hecht 1959) that he l ium was not unique i n that r e spec t , and that hydrogen (H) and t r i t i u m (T) atoms cou ld a l s o show s u p e r f l u i d p r o p e r t i e s , s i n ce they a re bosons. By c on t r a s t to he l ium, however, a gas of hydrogen atoms i s h i g h l y r e a c t i v e . I t w i l l not on l y r e a c t w i t h the w a l l s of most c o n t a i n e r s , but i t i s a l s o un s tab le aga in s t recombinat ion i n t o ^ mo lecu le s . Hydrogen atoms w i l l b ind to any m a t e r i a l and, i f the b ind ing i s not chemica l , recombinat ion i n t o H w i l l q u i c k l y 2 f o l l o w . This h i gh r e a c t i v i t y has always compl icated the problem of expe r imenta l l y observ ing the thermodynamic p r o p e r t i e s of H atom gas. Only r e c e n t l y has i t been p o s s i b l e to s t a b i l i z e an atomic H gas a t reasonab le d e n s i t i e s , where " s t a b i l i z e d " means that the atom d e n s i t y decays s l ow l y enough that there i s time f o r making experiments (see Nature, Sept. 1979, or Phy s i c s Today, June 1980). B ind ing to the w a l l can be reduced by coa t i n g the inner su r face of the c on t a i n i n g v e s s e l w i t h a m a t e r i a l f o r which the b ind ing energy i s much sma l le r than the average k i n e t i c energy of the atoms. For i n s t ance , t e f l o n i s f r e q u e n t l y used as a c oa t i n g a t h i gh temperature s i n ce i t s b i nd ing energy i s about 255°K ( Z i t z e w i t z 1970). The l e a s t b i nd ing s o l i d s u r f ace i s e i t h e r mo lecu la r hydrogen (Crampton 1980) or p o s s i b l y s o l i d neon. However, the b i nd ing energy of H atoms on s o l i d ^ i s s t i l l h i gh by c ryogen ic standards (^ 35°K). For s tudy ing H atom gas i n a much lower range of temperatures, i t i s necessary to have a f i l m of s u p e r f l u i d 4He cover ing the su r faces (Hardy et a l . 1980, or S i l v e r a and Walraven 1980). The b ind ing energy on ^He has r e c e n t l y been measured to be 0.93°K ± 0.05°K (Morrow et a l . ) , and t h i s r e s u l t i s c on s i s t en t w i t h the c a l c u l a t i o n s of Edwards and Mantz (Edwards and Mantz 1980). Thus, the e f f e c t of the b i nd ing w i l l become important on l y w e l l below 0.9°K, and i t i s now a standard procedure to use s u p e r f l u i d ^He f o r p revent ing b i nd ing to the w a l l s . However, un le s s the temper-a tu re i s w e l l below 1°K, there remains a s u b s t a n t i a l amount of he l ium gas due to the vapor pressure and t h i s w i l l cause bu lk recombinat ion as we s h a l l see l a t e r . Bu lk recombinat ion i s reduced a t low temperatures by the 3 a p p l i c a t i o n of a l a r g e magnetic f i e l d . Th is was evidenced exper imen-t a l l y by the work of the Amsterdam group where f i e l d s up to 70 kGauss a t temperatures around 0.3°K were used (see S i l v e r a and Walraven 1980). Th is approach f o r s t a b i l i z i n g atomic hydrogen i s based on the f a c t t ha t the i n t e r a c t i o n between two hydrogen atoms depends s e n s i t i v e l y on t h e i r t o t a l e l e c t r o n i c s p i n , denoted S. For S=l, the p o t e n t i a l U(R), where R i s the i n t e r n u c l e a r d i s t a n c e , i s b a s i c a l l y r e p u l s i v e (except f o r a sha l low minimum a t R = 4A* due to the d i s p e r -s ion f o r c e s ) , w h i l e f o r S=0 the p o t e n t i a l has a r e p u l s i v e core and a s t rong a t t r a c t i v e t a i l , w i t h a deep minimum (4.7 eV) a t R = 0.74 The s i n g l e t p o t e n t i a l (S=0) supports about 330 bound and q u a s i -bound energy l e v e l s which correspond to d i f f e r e n t v i b r a t i o n a l and r o t a t i o n a l s t a t e s . On the other hand, the 3 E ^ t r i p l e t p o t e n t i a l (S=l) supports no bound s t a t e s . Thus, i f one cou ld p o l a r i z e a l l e l e c t r o n i c sp ins of the atoms, then one would f o r c e a l l the c o l l i d -ing H atoms to i n t e r a c t v i a the t r i p l e t p o t e n t i a l , thereby s t a b i l i z -ing the gas aga in s t recombinat ion . P o l a r i z a t i o n i s achieved e x p e r i -menta l l y by app l y i ng a l a r g e magnetic f i e l d . This technique i s u s e f u l however on l y a t ve ry low temperatures s i nce the Zeeman energy i s very s m a l l . The idea of s p i n - p o l a r i z e d hydrogen (denoted H+) has f a s c i n a t ed t h e o r i s t s f o r a long t ime, and t h i s has r e s u l t e d i n a long s e r i e s of papers on the s ub j e c t . Some of the papers i n v e s t i g a t e the s t a b i l -i t y , o thers are about the thermodynamics of the gas. In t h i s t h e s i s , we are concerned w i t h the former, but l e t us f i r s t go b r i e f l y over the l a t t e r , s i n ce the u l t i m a t e aim of our e f f o r t s i n t h i s area of low temperature phys ic s i s to study the thermodynamic p r o p e r t i e s of 4 a gas of H i atoms. An Important t h e o r e t i c a l p r e d i c t i o n i s that a gas of H+ w i l l remain i n the gaseous phase a t a l l temperatures, f o r pressures l e s s than M.00 bar (see M i l l e r e t a l . 1977). B o s e - E i n s t e i n c on -densat ion i s a l s o p r ed i c t ed f o r H4- (and T+) i n the bu l k . The c o n d i t i o n f o r 3-D Bose condensat ion f o r a weakly i n t e r a c t i n g Bose gas (which a p p l i e s w e l l to H+) i s (see L i f c h i t z et a l . 1980): = 3.31 -n 2 2/3 C mkg °H where n^ i s the den s i t y of H atoms and m i s the mass of one hydrogen atom. For the h i ghes t repor ted d e n s i t i e s , which a re around 1.5 x 1 0 1 6 c m - 3 (Walraven 1980, Kleppner and Greytak 1980), the c r i t i c a l temperature would be T = 1 mK, w h i l e f o r the lowest temperature a t which experiments have been performed, namely T = 0.1 K, the c r i t i c a l d en s i t y would be n = 1.6 x 1 0 1 9 c m - 3 . Thus, the e x p e r i -m e n t a l i s t s a re c l o s e t o , but not yet a t the po i n t where they can observe Bose-condensat ion. The problem i s , however, .-.complicated by the presence of sur faces on which hydrogen b inds,and i t has been suggested tha t su r face s u p e r f l u i d i t y can a l s o take p l ace even be fo re 3D-condensation (see, f o r i n s t ance , Edwards and Mantz 1980). The s t a b i l i t y of s p i n - p o l a r i z e d hydrogen i s s t i l l a h o t l y debated sub jec t and, a l though H-l- i s c l e a r l y much more s t a b l e than unpo l a r i zed H, the extent of the improvement as a f u n c t i o n of the magnetic f i e l d and temperature i s not ye t understood i n d e t a i l . C a l c u l a t i o n s a re d i f f i c u l t because recombinat ion of hydrogen atoms i n t o H ? molecules i s i n t r i n s i c a l l y a three-body process : 5 H + H + X - > H 2 + X , where X = He, H or w a l l . The t h i r d body i s there on l y f o r conserv ing energy and momentum. I n the e a r l y c a l c u -l a t i o n s , the recombinat ion r a t e was p r ed i c t ed to have a f i e l d depen-- P e H 0 / k B T dence R ^ e , where i s the magnetic moment of the e l e c t r o n and H q i s the a p p l i e d s t a t i c magnetic f i e l d (see Jones e t a l . 1958). Th i s imp l i e s a l a r g e decrease o f the recombinat ion r a t e w i t h i n c r e a s -ing H Q/T. For i n s t an ce , t ak i ng H q = 100 kG (a conven ien t l y a c h i e v -ab l e l a b o r a t o r y f i e l d ) and T = 0.1°K, we see t h a t the r a t e i s 10 60 times sma l le r than the r a t e w i t h zero f i e l d . Such a f i e l d dependence i s based on the assumption tha t the sp i n s t a t e s w i l l reach thermal e q u i l i b r i u m e f f i c i e n t l y and r a p i d l y . However, the s t r eng th of the r e l a x a t i o n mechanisms present , such as spin-exchange or d i p o l e - d i p o l e i n t e r a c t i o n s , do not f u l l y j u s t i f y such an assump-t i o n s i n ce recombinat ion i t s e l f d i s t u r b s the sp i n s t a t e s from e q u i l i b r i u m . More impo r t an t l y , the s i t u a t i o n i s compl icated by the f a c t t ha t there i s a coup l i n g between the nuc lea r s p i n and the e l e c t r o n i c s p i n which g i ve s r i s e to a se t o f fou r s p i n s t a t e s , the h ype r f i n e s t a t e s (Chapter Two) and which e f f e c t i v e l y prevents the achievement of complete e l e c t r o n s p i n p o l a r i z a t i o n w i t h a v a i l a b l e l a b o r a t o r y f i e l d s . I t i s now suspected t ha t the recombinat ion r a t e w i l l r a the r a 2 go as R ^ (-z—tt~) , where " a " i s the coup l i ng constant of the hyper-e o f i n e i n t e r a c t i o n i n the hydrogen atom. This means tha t we need to apply a very l a r g e f i e l d i n order to improve s i g n i f i c a n t l y on the recombinat ion r a t e , but that s t a b i l i z a t i o n may s t i l l be f e a s i b l e . However, there s t i l l remains the p o s s i b i l i t y of having R ^ RQ+R^(H), where R i s some constant independent of the f i e l d and where R, (H) o 1 6 goes to zero a t h i gh f i e l d . In such a case, there i s no p o s s i b i l i t y of e l i m i n a t i n g recombinat ion a l though one can reduce the r a t e down to about R q by app l y i ng a st rong magnetic f i e l d . In t h i s t h e s i s , we s h a l l i n v e s t i g a t e the f i e l d dependence of the recombinat ion v i a H + H + He -> + He which i s dominant when a f i l m of s u p e r f l u i d Hel ium i s used to prevent b ind ing to the w a l l s a t temperatures above 0.1°K. We s h a l l r e s t r i c t our i n v e s t i g a t i o n to the range of temperatures from 0.9°K to 1.3°K. The lower l i m i t i s imposed by p r a c t i c a l c on s i de ra t i on s s i nce temperatures below 0.9°K are not a ch i evab le i n our experiment. The upper l i m i t i s due to the f a c t that the ^He vapor pressure inc reases r a p i d l y w i t h temperature, l e a d i n g to a r a p i d l y i n c r ea s i n g recombinat ion r a t e (Chapter F i v e ) . The t he s i s w i l l be organ ized as f o l l o w s . In Chapter Two we s h a l l g i ve a b r i e f rev iew of the bas i c theory of the hydrogen atom, e s p e c i a l l y i t s hype r f i ne l e v e l s . Because magnetic resonance i s used f o r d e t e c t i n g and s tudy ing the gas of hydrogen atoms, we s h a l l d e s c r i b e t h i s techn ique, and how i t can be app l i ed to the hydrogen atoms, i n Chapter Three. In Chapter Four i s l i s t e d a s e r i e s of equat ions and concepts u s e f u l f o r understanding the s i g n a l s . Chapter F i v e dea l s w i t h the d e r i v a t i o n of a s imple model f o r recombinat ion v i a H + H + He YL + He. Chapter S i x conta in s i n f o rmat i on r e l e v a n t to the experiment w h i l e Chapter Seven conta in s a d e s c r i p t i o n of the aparatus . 7 CHAPTER TWO BASIC THEORY OF THE HYDROGEN ATOM The hydrogen atom i s a w e l l known t h e o r e t i c a l e n t i t y . In t h i s chapter , i n f o rmat i on of i n t e r e s t to t h i s work w i l l be g i ven and the n o t a t i o n used throughout the t e x t w i l l be i n t roduced . 2 -1 . The Hami l ton ian The Hami l ton ian of a s i n g l e hydrogen atom c o n s i s t s of a pa r t which depends on the p o s i t i o n of both the proton (R) and the e l e c t r o n ( r ) , and one s p i n dependent p a r t . The former, denoted H i s the dominant one and the l a t t e r , denoted H , i s handled p e r t u r b a t i v e l y . s 2 -1 -1 . The S p a t i a l Hami l ton ian Th i s Hami l ton ian g i ves the a c t u a l wave f u n c t i o n of the hydrogen atom, denoted r . , which obeys the f o l l o w i n g e i g en func t i on equat ion: where r and R denote r e s p e c t i v e l y the p o s i t i o n of the e l e c t r o n and the p ro ton , and i s the energy of s t a t e T^. Th i s wave f u n c t i o n i s de r i ved i n many textbooks on elementary quantum mechanics (see L a l o e et a l . 1977). Under the exper imenta l c o n d i t i o n s which we have, a l l atoms can be cons idered to be i n t h e i r ground s t a t e , namely the I s s t a t e . There fo re , i = I s , and I\ becomes: I H r . ( r J t ) = E. r . ( r O X I X (2-1) r l e-|R - r|/a 0 (2-2) I s e /ira 3 h o where P i s the momentum of the p ro ton , and a Q the Bohr r a d i u s . 2-1-2. The Spin Hami l ton ian In the I s s t a t e of the hydrogen atom, the f i n e s t r u c t u r e term g i ves no c o r r e c t i o n , and on l y the contac t term of the hype r f i ne H a m i l -•+ • -> t on i an remains (as a second order c o r r e c t i o n te rm) . I f M and M are e p r e s p e c t i v e l y the moment operator s of the e l e c t r o n and the p ro ton , then the expres s i on of t h i s Hami l t on i an i s : H, . = - A M *M (2-3) h . f . e p where A i s j u s t a constant to be g i ven below. When a magnetic f i e l d , H, i s p resent , we have to add the term d e s c r i b i n g the coup l i ng to t h i s f i e l d : tf_ = - M «H - M -H (2-4) The moment operator s can be r e w r i t t e n i n terms of the sp i n ope ra to r s : M = -hv S and M = tty I (2-5) e e p p where y £ a n ^ Yp a r e the gyromagnetic r a t i o . In order to work w i t h p o s i t i v e q u a n t i t i e s , we use the convent ion t h a t : •fry = - u . fiy = V a n ( i - At\ 2Y Y = a . (2-6) ' e H e ' ' p K p ' e ' p That way, the t o t a l s p i n Hami l ton ian becomes: H = H. . + HN = y S-H - y I-H + a l - S (2-7) s h . f . C e p Note tha t the operator s S and I a r e de f i ned such tha t S = % a and I = % t (2-8) -y where a and T a re the u sua l P a u l i m a t r i c e s . The a c t u a l va lues of the constants a re : V = - fiye = 1.8570 x 1 0 " 2 0 erg/Gauss y = - -ny = 2.8213 x 1 0 ~ 2 3 erg/Gauss (2-9) P P a = - A f i 2 Y e Y p = 9.4119 x 1 0 " 1 8 erg, The constant " a " and the r a t i o Yp/Y e a r ^ known to a ve ry h igh accuracy: ^ = 1420.405 751 768 (2) MHz (He l lw i g e t a l . 1970) Y n n = R = 1.519 270 335 (15) x 1 0 " 3 (Winkler et a l . 1972) "^ e -hy or — £ • = 1.521 032 181 (15) x 1 0 " 3 (idem) ^B where i s the Bohr magneton. Instead of u s ing energy, one may want to use f r equenc i e s . The cons tant s a re then: y = 2.80247 MHz/Gauss e y p = 4.25771 x 10~ 3 MHz/Gauss (2-10) a = 1420.406 MHz 10 2-2. Hyper f i ne Leve l s Cons ider a hydrogen atom i n a s t a t i c and un i fo rm magnetic f i e l d a long the z - a x i s : H = H Then the Hami l ton ian (2-7) becomes: H = (u S -y I )H + a l - S (2-11) s e z p z o Our s t a r t i n g ba s i s w i l l be the one obta ined by the d i r e c t product of the e i gen s ta te s of S and I : |nu ,m >, where m = ± % and m = ± %. Z Z o J_ S -L. I t i s e a s i l y v e r i f i e d that |++> and [—> a re a l ready e i gens ta te s of . There remains a submatr ix, mix ing the other two s t a t e s , which i s e a s i l y d i a g o n a l i z e d . The fou r e i gens ta te s and t h e i r corresponding energ ies a re g i ven below i n order of i n c r ea s i n g energy: |l> = - s i n 0 |+-> + cos |-+> (2-12a) |2> E I—> b) |3> = cos 6 |+-> + s i n 0 |-+> c) |4> E |++> d) where tan 26 = 7 , a (2-13) (.y +y )H " e K p o E n = - f - % /a2+ (y +y ) 2 H2 (2 - l4a) 1 4 e p o E 2 " f " < * e V ^ b ) E = - f + h /a2+ ( v +y ) 2 H 2 c ) 3 4 *e p o 11 2-3. The Hyper f i ne Leve l s at High and Low F i e l d s 2 -4 -1 . At Low F i e l d When the s t a t i c magnetic f i e l d i s ve ry low, tha t i s when H o ( u e +y p ) /a << 1, we see the th ree upper l e v e l s c o l l a p s i n g i n t o the same l e v e l , w h i l e the bottom one s tays a p a r t . I f we d e f i n e F = S+I, then F and a re good quantum numbers i n the low f i e l d l i m i t . Taking the l i m i t Hq-*0 i n equat ion (2-13), we o b t a i n cos 0 -> 1//2 and s i n 6 l//2~, so that the e i gens ta te s (2-14) become: |l> + -/=• l+~> + ^ l"+> = 1-0,0>F (2-15a) |2> + J—> = |1,-1> F b) |3> + k |+_> + ^ |_+> = ] l 5 o > F c ) 4> |++> = 11,1 F d) where [ > p r e f e r s to the coupled ba s i s I f .M^. Taking the l i m i t where H o ( y e + y p ^ a i s v e r y s ™ 3 1 1 . the express ions f o r E 3 ( H q ) and \(R ) become: (u -Hi ) 2 H 2 h «. - a [ 3 + e j, ] ( 2 _ l 6 a ) (u +V ) 2 H 2 Whi le l e v e l s E 2 and E 4 a re s t i l l g i ven by express ions ( 2 - l 4 b , d ) . 2-4-2. A t High F i e l d In the l i m i t of h i gh f i e l d , g i ven by the c o n d i t i o n HQ(ye+Vp)IA>>%, we o b t a i n the almost decoupled b a s i s : |l> -> - e |+-> + |-+> (2-17a) |3> ->- |+-> + e |-+> b) S ince cos 6 1 and sinO->--r^—7 — r r - H e r o _I » ^  2H (y +y ) v ^ i o ; o e p where e i s c a l l e d the admixture c o e f f i c i e n t . The corresponding energ ies a re : (y +y )H E l * " t " 6 2 P ° ( 2 " 1 9 a ) (y +y )H • Note that s t a t e s |2> and |4> a re independent of f i e l d and t h e i r ener -g i e s a re g i ven by (2-14b,d) f o r a l l f i e l d s . CHAPTER THREE MAGNETIC RESONANCE OF HYDROGEN Magnetic resonance i s a p a r t i c u l a r l y u s e f u l technique f o r s tudy ing H-atoms because the magnetic resonance spectrum of hydrogen i s q u i t e d i s t i n c t i v e . As we saw i n Chapter Two, the hydrogen atom has four h ype r f i n e l e v e l s i n i t s ground s t a t e , and i t would be a f o rm idab le t a sk to so l ve a n a l y t i c a l l y the equat ions g i v i n g the t ime e v o l u t i o n of t h i s system when a t ime dependent magnetic f i e l d i s a p p l i e d . However, the h ype r f i ne l e v e l s a re ve ry sharp, and i f none of the s i x energy d i f f e r e n c e s a re ve ry c l o s e to each o the r , then we can assume that we a re l o o k i n g a t on l y two l e v e l s when the atoms are e x c i t ed w i t h an RF magnetic f i e l d . In other words, the RF f i e l d couples e f f e c t i v e l y on l y two l e v e l s and, as f a r as magnet ic resonance i s concerned, i t i s a t w o - l e v e l system. S ince the motion of a s p i n - % i n a magnetic f i e l d i s desc r ibed i n a l l textbooks on magnetic resonance, we s h a l l take a more genera l approach here . S t a r t i n g w i t h the d e f i n i t i o n of a genera l t w o - l e v e l system ( f o r which a s p i n -% i s j u s t a s p e c i a l c a s e ) , we s h a l l f i n d i t s equat ion of motion and show that i t can be r e w r i t t e n i n a form s i m i l a r to that of a r e a l s p i n - % . Then we w i l l d i s cu s s the technique of pulsed magnet ic resonance. 3-1. Genera l Two-Level System Consider a quantum system possess ing two, or more, energy l e v e l s which are e i gens ta te s of some Hami l t on i an , H. We s h a l l say tha t we have a t w o - l e v e l system i f there e x i s t s a second (time-dependent) Hami l ton i an , h " , which couples on ly one p a i r of l e v e l s t oge the r . We s h a l l assume that these two l e v e l s a re non-degenerate and h" possesses on l y non-d iagonal elements i n the sub-space of these two l e v e l s . I f these two l e v e l s a re denoted a and b, f o r which the energ ies a re E and E, , and i f the system i s i n s t a t e a b tp, then we can d e f i n e the den s i t y ma t r i x as (see Merzbacher 1970): p a a P a b W P a a Pab }  pba p b b / \ P a b 1 _ p a a , where p = <i|i|»<ip|j> i , j = a ,b . Cons ider now a mat r i x operator 0, and the ma t r i x operator S = \ a which was in t roduced i n Chapter Two. Because i t i s a two by two m a t r i x , we can r e w r i t e 0 as: 0 = A + 2l - S (3-2) o where A = k Tr(0) and A = Tr(OS) . (3-3) o For i n s t ance , i f 0 i s the t o t a l Hami l ton ian H = H+H' then: t o t H = h + 2&'t (3-4) t o t o where h = \ Tr(H) =• E and t. •= Tr ( f/ ' S ) . o Suppose now that 0 = p, then a s i m i l a r procedure g i ve s : p = \ + 2<S>-S (3-5) where <S> = Tr(pS) i s now the expec ta t i on of the operator S. I t comes i n f a c t from the d e f i n i t i o n of the d e n s i t y m a t r i x t h a t , f o r any operator 6, the expec ta t i on va l ue i s g i ven by: <0> = Tr(pO) (3-6) I t i s a u s e f u l f a c t that the den s i t y m a t r i x can be r e w r i t t e n i n terms of <S>, a v e c t o r , which w i l l now c h a r a c t e r i z e the system. We • s h a l l " denote S a ,Sg and the components of S i n the f i c t i t i o u s space ; corresponding r e s p e c t i v e l y to , and S z . P h y s i c a l l y , <Sa> and <S_> are a measure of the c o r r e l a t i o n between the s t a t e s , w h i l e <S > 8 y i s a measure of the energy. <ff> i s g i ven by: E +E <H> = a 0 b + <S > (E -E, ) 2 y a b or <AH> = <H> - E = <S >«AE (3-7) Y where A'E = E - E t and E = (E +E,.)/2 . a b a b under c o n s i d e r a t i o n , say N, |a> E Ib> E When-more then one system are a \ E^ . and i f they are weakly i n t e r - ^ a c t i n g , then we can j u s t use E^ the average d e n s i t y m a t r i x which i s de f ined as: ~ 1 N p r: I y p. (3-8) i=l Thus, i n s t ead of (3-5) we have: p = 2 <S>T^ S (3-9) where the s u b s c r i p t T s p e c i f i e s the " t h e r m a l " average, i . e . we use p r a t h e r than p. We can now make the analogy between a s p i n - % and our genera l t w o - l e v e l system. As mentioned i n Chapter Two, the Hami l ton ian of a s p i n - % i n a magnetic f i e l d i s : H = -M«H = -yS«H. We then see that (3-4) has t h i s form i f we put : 2h E - y ' H ' , (3vl0) (see Abragam 1961, p. 36), where the prime s p e c i f i e s t ha t both u and H now r e f e r to a f i c t i t i o u s s p i n f o r wh ich the operator i s S. The analogy can be v i s u a l i z e d more c l e a r l y by s u b s t i t u t i n g equat ion (3-4) i n t o the Schrbdinger equat ion f o r den s i t y m a t r i c e s , namely: We then o b t a i n : d<|> = 2 j x< !> ; = _ ^ x < t > at Tl TI which i s e x a c t l y the equat ion of motion of a s p i n - % i n a magnetic f i e l d (see S l i c h t e r 1978). Th i s y i e l d s a set of th ree coupled equat ions g i v i n g the e v o l u -t i o n of a s p i n l i k e system when no r e l a x a t i o n mechanisms a re present . Note that the term h , the average energy of the two l e v e l s , has no e f f e c t on the e v o l u t i o n of the system, and tha t h may w e l l be t ime dependent. N a t u r a l l y , the system tends toward e q u i l i b r i u m c o n d i t i o n s , due to r e l a x a t i o n mechanisms. In the u sua l treatment of Magnetic Resonance, one in t roduces e m p i r i c a l r e l a x a t i o n terms i n the equat ion of e v o l u t i o n of the sp ins i n such a way that we o b t a i n two r e l a x a t i o n t imes T^ and T . T^ i s the " s p i n - l a t t i c e " r e l a x a t i o n time f o r the component of S a long the s t a t i c f i e l d , H q , u s u a l l y . T^ i s the " s p i n - s p i n " r e l a x a t i o n t ime and belongs to the components of ^ i n the plane normal to H . The r e s u l t i n g equations a re known as the "B loch equat i on s " : d<S > S -<S > -T- = - y ' ( S ' x < S > ) Y .+ • % * (3-13a) d<S > <S > = - y ' (H ' x < ! > ) ( j - 1 - b) d<S > <S > — = - y ' ( H ' x < S » a - -JSL- . e) where S q i s the e q u i l i b r i u m va lue f o r S i n the s t a t i c f i e l d . We can d e r i v e the r e l a x a t i o n terms from the knowledge of the system a t the mic ro scop i c l e v e l . Cons ider the f o l l o w i n g equat ion: dp I . aa aa 1 ab 2 , /o i / \ Using equat ion ( 3 -5 ) , t h i s becomes: (3-15) where S + - ± i S ^ . These two express ions a re merely the r e l a x a -t i o n terms of the Bloc % equat ions . _ iWe then see that the actual calculation of the relaxation times, and i s a microscopic quantum mechanical problem. 3-2. Case of the Hydrogen Atom In our experiment, the exciting f i e l d , H^  coswt, i s applied at right angle to the static magnetic f i e l d , say the x-axis, so that the interaction Hamiltonian, tf',will become (using equations (2-4), (2-5) and (2-6)): h" = (u S -u I )H- coswt = -M H.. cosut (3-16) e x p x 1 x 1 Consider now a pair of hyperfine states (see Chapter Two) of the hydrogen atom: |a> and |b>. In this basis, h" has no diagonal elements for any pair of hyperfine states, and, since <a|M |b>-; = <b|M |a>, the interaction Hamiltonian i s proportional to S , the equivalent of in this f i c t i t i o u s space. From (3-10) and (3-4), we then obtain: - u'H' = 2Tr(n"S) = -2<a|M |b>H. cosut a (3-17) 1 x 1 1 It i s then legitimate to define: u' = 2<a|Mx|b> - u a b (3-18) The hydrogen atom w i l l consequently behave like a spin-% of ab gyromagnetic ratio u excited by a f i e l d H^  coscot a. 20 3-3. Pu l sed Magnetic Resonance Pulsed MR proceeds i n two s teps . One f i r s t i r r a d i a t e s the sample (which s i t s i n a s t a t i c homogeneous magnetic f i e l d ) w i t h a t ime dependent magnetic f i e l d , H^ ( t ) , f o r a per iod of t ime, T , which i s s u f f i c i e n t l y short that the system does not have enough t ime to r e l a x . Second, the e x c i t i n g f i e l d i s turned o f f and the system can evo lve accord ing to i t s own Hami l t on i an . A c a v i t y tuned a t the app rop r i a t e frequency i s used as a probe f o r e x c i t i n g the system dur ing the t ime T , and f o r d e t e c t i n g the r e s u l t a n t s i g n a l which i s due to the f l u c t u a t i n g magnetic moment of the sample. Th i s c a v i t y i s pa r t of our d e t e c t i o n d e v i c e , which a l s o i n c l ude s a t r an sm i s s i on l i n e and a spectrometer. Th is spectrometer i s thoroughly desc r ibed i n Lorne Whitehead ' s t h e s i s (Whitehead 1979). The c a v i t y w i l l be desc r ibed i n Chapter Steven. We are i n t e r e s t e d i n the s o l u t i o n of the B loch equations i n both regimes of pulsed MR. We w i l l on l y quote the r e s u l t s here, s i n ce they a re de r i ved i n most textbooks on magnetic resonance ( S l i c h t e r 1978) . Suppose that we s t a r t from the e q u i l i b r i u m c o n d i t i o n : <S(0)> = ( 0 , 0 , S Q ) , i . e . no c o r r e l a t i o n s and a f u l l y r e l a xed system, then the e v o l u t i o n of the system dur ing the pu l se i s g i ven by the f o l l o w i n g equat ions i f the pu l se i s ve ry sho r t : <S >(t) (3-19a) Y <S>(t) b) <S. >(t) - S sin(oj, t)cos(o) t ) o 1 o c) a 21 y * ^ y 'H where to, = ——— and 'OJ = — - — 1 2fi o Normal ly , we w i l l app ly what are c a l l e d " — - p u l s e s " which b r i ng <S> from the y - ax i s down i n to the :a^ _6 p lane . From equat ions (3-19) , we see that t h i s r e q u i r e s t ha t : u^x = T T / 2 (3-20) Fo l l ow ing such a pu l s e , the e v o l u t i o n of the system i s g i ven by: - t / T <S >(t). = S (1-e ) (3-21a) Y o - t /T <S >(t) = -S e cosw t b) g o o - t /T <S >(t) = • S e sinto t c) o o o Th i s i s the ba s i c scheme i n pulsed MR, but there e x i s t others which have s p e c i a l purposes. " S p i n echo" i s one of them, and one v e r s i o n of i t c o n s i s t s i n the a p p l i c a t i o n of a T T / 2 pu l se , f o l l owed a f t e r a t ime T by a 7r-pulse (=2x . ) . An echo s i g n a l w i l l appear -2T/T 2 a t t = 2T, the ampl i tude of which i s g i ven by S Q e ( S l i c h t e r 1978, p. 42 ) . Th i s i s a u s e f u l scheme f o r measuring T 2 when the magnet has a poor homogeneity which r e s u l t s i n short decay t imes (see Chapter F o u r ) . CHAPTER FOUR SIGNAL THEORY AND LEVEL DYNAMICS The signal to be detected i s expected to be very weak, compelling us to a very careful i d e n t i f i c a t i o n and estimate of a l l the parameters involved. We need to understand the effects of the resonator, which i s one of the key parts of the detection system,and .to know the . expected strength of the signal. 4-1. P r i n c i p l e of the Resonator The resonator which we use i s designed to detect the o s c i l l a t i n g magnetic moment produced by the spin ensemble under study. Since we have a fluctuating magnetic moment, energy can be trans-ferred to the resonator .Let Pg be-^the-power delivered- to the resonator by the spins. We have ohmnie power loss in-the resonator; P.-,and power P^ flowing i n the transmission l i n e to the spectrometer; At steady state, we must have P = P + P . At c r i t i c a l coupling, we S K L have P = P . L is. Now, i n a quite general way, the quality factor, Q, of any resonator i s given by the expression: n = Stored Energy _ t» U (4-1) mean power loss where u> i s the angular frequency, U the stored energy and <PT)> the mean power loss per cycle. These quantities can i n p r i n c i p l e be calculated, but any l e v e l of accuracy requires a careful measurement De f i n i n g H^(0) as the v a l ue of the ampl i tude of the o s c i l l a t i n g magnetic f i e l d a t r 5 0, the " c e n t r e " of the re sonato r , we o b t a i n : U = gH^CO) (4-2) where 3 i s a normal i zed f i e l d energy f a c t o r , which has u n i t s of volume. I t s exact exp re s s i on i s : B J ~ - — (4-3) VR ^ 2 ( 0 ) For a homogeneous f i e l d f i l l i n g the volume V of the resonator i t R reduces t o : 3 = T £ (4-4) On the other hand, we can w r i t e : where a i s the c o n d u c t i v i t y of the m e t a l , 6 the s k i n depth and y some geomet r i ca l f a c t o r having the f o l l o w i n g a n a l y t i c a l expres s ion (see Jackson 1975): 2 c Y = - T j da |nxH,,|2 (4-6) 32-rr2 S R « ->• -> where n i s the normal to the su r f ace , the magnetic f i e l d p a r a l l e l to the su r face and S the t o t a l s u r f a ce . R Thence the expres s i on (4-1) becomes: Q = — Sow (4-7) Y where the s k i n depth, <5, i s g i ven by: 6 = 7= \±) (4-8) Therefore , f o r a g i ven resonator w i t h ohmnic l o s s e s a t the meta l s u r f ace s , Q takes the form:. Q = A(a>a)% (4-9) where A i s a cons tant of the r e sona to r . Our a im, here, i s not on ly to f i n d express ions which g i v e us a good understanding of the re sonato r , but a l s o to r e l a t e the f i e l d i n the resonator to whatever we can measure from the o u t s i d e . Equat ion (4-5) can be r e w r i t t e n as : (4-10) S i nce , a t c r i t i c a l coup l i ng Q = Q , P .= P .and P = P + P , C K L b K • L we can furthermore w r i t e (4-10) as: (4-11) F i n a l l y , u s ing equat ion (4 -7 ) , we ge t : rQ <P >\% (Q < P T A ^ 4-2 . Spin Ensemble and Strength of the S i g n a l As mentioned p r e v i o u s l y , the resonator i s merely a probe f o r magnetic moments which r e s u l t from the t o t a l c o n t r i b u t i o n of a l l " s p i n s " i n the sample. On the other hand, every s p i n f e e l s the l o c a l f i e l d and responds to I t . So, i n g ene r a l , the power i nvo l ved when sp ins evolve i n a magnetic f i e l d i s g i ven by: = I <M«B> d 3 r (4-13) S. V R t where B = H + 4irM, (4-14) and < > t means t ime averag ing . Assume f i r s t that H = cos(wt+<|>p i i s the f i e l d produced by the sample and that i t i s un i form throughout the resonator and the sample, which occupies a f i n i t e volume . I f every moment, y, precesses around the s t a t i c and l o c a l l y homo-geneous f i e l d = H^k, and i f the d e n s i t y of the sp ins i s p^, then: y.u = V sinuJt (4-15) M x = p s y x (4-16) Hence, the power < P g > which we can de tec t w i l l be: <p > = 3 \ <fL_ (p Tisincot) *{;H\ cos (cot + <f> ) + Arrp ysiruot}> d 3 r d t X J. o t V (-p) OJH cos <j> (4-17) V R where N i s the t o t a l number of s p i n s . At resonance, <j)j. goes to ze ro , and, d e f i n i n g Ny = m and V /V = n, the f i l l i n g f a c t o r , • S R equat ion (4-17) becomes: <PS> = r, I c o ^ (4-18) we can now combine t h i s expres s ion w i t h equat ion (4-12) and o b t a i n a r e l a t i o n s h i p between the moment of the sp ins and t h e i r power output : . m2o)Q ^ = ^ - i 6 F ( 4 " 1 9 ) Because we assumed homogeneity, the f a c t o r g i s merely equal to V / 8 T T and we f i n a l l y get : K TTTl2m2U)Q <P > = S. (4-20) L 2V D ^ ' R Obv ious ly , t h i s de sc r i be s an i d e a l resonator and the equat ions have to be gene r a l i z ed to a more r e a l i s t i c case. We s h a l l not go i n t o tha t here, but we s h a l l mention that (4-20) does not change ve ry much, and tha t the c a l c u l a t i o n of n would r e q u i r e an accu ra te knowledge of the magnetic f i e l d (see Bloembergen and Pound 1954). 4 -3 . M u l t i l e v e l Dynamics We use a resonator to probe the s p i n , or moment, ensemble which i s our sample. Each of these sma l l moments i s i n f a c t one hydrogen atom, and each atom has some c a l c u l a b l e e f f e c t i v e magnetic moment which depends on the s t a t i c magnetic f i e l d , H q , and on which set of two l e v e l s we a re l ook i ng a t . The s t r eng th of the s i g n a l , which we d e t e c t , i s c l o s e l y r e l a t e d to the popu l a t i on of these two l e v e l s , and the popu l a t i on d i s t r i -b u t i o n i s i t s e l f governed by the dynamics of these l e v e l s , which, i n t u r n , can be a f f e c t e d by the probing re sona to r . The dynamics i s g i ven by a s e r i e s of mechanisms which can be s p l i t i n t o two broad c l a s s e s : R e l a x a t i o n mechanisms and recombinat ion 27 mechanisms. In the f i r s t c l a s s we put those mechanisms which do not a l t e r the t o t a l popu l a t i on of the atoms d i r e c t l y . These mechanisms a re more or l e s s standard i n magnetic resonance. Because the t o t a l p o p u l a t i o n i s t ime-dependent, we have to cons ider r e l a x a t i o n due to the recombinat ion of atoms i n t o molecu les , which we s h a l l t h e r e -f o r e put i n a separate c l a s s . I t i s worth n o t i c i n g that t h i s l i s t i s not exhaus t i ve , and some mechanisms, such as " p re s su re s h i f t " , w i l l not be d i s cu s sed because they a r e not important f o r the present exper iments. 4 - 3 - 1 . R e l a x a t i o n Mechanisms Fo l l ow ing a i r/2-pulse, the s i g n a l s t a r t s a t some ampl i tude and then decays. We can o b t a i n some i n f o rmat i on on the r e l a x a t i o n mechanisms i nvo l ved by s tudy ing t h i s decay. Never the le s s , some u n d e s i r a b l e mechanisms w i l l modi fy, or quench, those mechanisms i n which we a re i n t e r e s t e d . In the f o l l o w i n g , we s h a l l f i r s t d e s c r i b e the unwanted mechanisms which we t h i n k w i l l be the most impor tant . Spin exchange, which i s the main mechanism under study, w i l l then be d e s c r i b e d . ( i ) Inhomogeneity of the S t a t i c F i e l d When the s t a t i c magnetic f i e l d i s inhomogeneous, the p reces s ing sp ins l o s e t h e i r c o r r e l a t i o n and they get out of phase, r e s u l t i n g i n a superimposed T^ mechanism. The corresponding T^ i s g i ven by (see Abragam 1961, p. 50): (4-21) where the star indicates that i t i s not a real relaxation mechanism. Aw = 2irAf is the spread in frequency due to inhomogeneity as i t i s calculated in Chapter Six. Over the volume of our sample.which i s about 3:! —6 1.5 cm , the magnet used has a homogeneity AH/H 'v 10 at 42 kG (and this is expected to be improved by the new set of shim c o i l s , see Appendix A). As we shall also see in Chapter Six, two fi e l d s , corres-ponding to two transitions, are of interest to us, and scaling down the above inhomogeneity, we obtain from equations (6-4),(6-5) and (4-21) that: T2*(405G) = 1.6 x l O - 4 and T2*(6.5kG) = 3.2 x 103 sec. Although the magnetic moment at 405 G i s about twenty times larger than at. 6.5 kG, the homogeneity of the magnet i s much more important, and may easily make the signal d i f f i c u l t to see. If the atoms did not move in the inhomogeneous f i e l d then the signal might s t i l l be recovered by the spin-echo technique. However, this i s not possible because the atoms in the sample diffuse rapidly. ( i i ) Diffusion The 'sample' consists of a closed tube f i l l e d with a mixture of hydrogen and helium gas. Therefore, atoms w i l l diffuse,especially at 4 very low temperatures (^ 1 K) where the He density, drops dras-t i c a l l y . Following a TT/2-pulse, the movement of the atoms w i l l destroy the correlation among the phases i f the f i e l d i s not homogeneous. Calculations of the effect of diffusion on the shape of the signal are given by Carr and Purcell (Carr and Purcell 1954). Considering a simple gradient in H^ , they found that the induced decay w i l l be of the form: 29 M(t) = K exp { - y 2 ( | ^ ) 2 ^ 1 } (4-22) o 9z 3 where D i s the d i f f u s i o n cons tant . Th i s shape, which i s superimposed on the u sua l exponent i a l decay, resembles a Gauss ian shape. Measurements were done on the d i f f u s i o n constant f o r the d i f f u -s i on of hydrogen atoms i n hel ium gas near 1°K (see Hardy et a l . 1980). Using the r e l a t i o n : where v i s a t he rma l l y averaged r e l a t i v e speed of H-He p a i r and Q i s the the rma l l y averaged d i f f u s i o n c r o s s - s e c t i o n , they found t ha t Q = 20 ± 1 & f o r a temperature range 0.98<T<1.24°K. Th i s i s an important number i n t h i s experiment. ( i i i ) R a d i a t i o n Damping As the r e r a d i a t i n g atoms e x c i t e the re sonato r , the consequences of Jou le heat ing can damp the ve ry weak s i g n a l which i s be ing em i t ted . S u r p r i s i n g l y , a h i gh Q resonator w i l l damp the s i g n a l more than a low Q one, a l though i t o f f e r s l e s s ohmnic l o s s e s . In f a c t r a d i a t i o n damping comes from the i n f l u e n c e of the r a d i a t e d f i e l d on the sample i t s e l f (Bloom 1957) . Th i s was the main decay mechanism i n the zero f i e l d experiment, and the Q of the resonator .had to be reduced f o r c e r t a i n of the experiments (Hardy et a l . 1980). In our experiment, i t i s p o s s i b l e to f a ce s i m i l a r problems, thereby r e q u i r i n g some knowledge of the e f f e c t of r a d i a t i o n damping on the s i g n a l . We s h a l l concent ra te on the major consequence of r a d i a t i o n damping, namely the s h o r t - c i r c u i t i n g of the i n t r i n s i c l^-mechanisms of the sample. I f Q i s the q u a l i t y f a c t o r of the re sonato r , n the f i l l i n g f a c t o r , y the " e f f e c t i v e " magnetic moment and M(0) the i n i t i a l magnet i za t i on of the sample, then the c h a r a c t e r i s t i c t ime f o r r a d i a t i o n damping i s approx imate ly g i ven by: 2nQnyM(0) (4-24) In Chapter s i x , i t w i l l be demonstrated that M(0) - ny(P_^-P ) where n i s the atom d e n s i t y and ^ ^ " P j 1 S t n e popu l a t i on d i f f e r e n c e . Therefore equat ion (4-24) becomes: T ^ Qnny2(P,-P,) ( 4 " 2 5 ) In our experiment, the re are two p o s s i b l e t r a n s i t i o n s f o r which the y ' s a re : y 2 3 = 7.53 x l C T 2 0 erg/Gauss (at 6.5 kG) and y 2 3 = 1.67 x 1 0 ~ 2 2 erg/Gauss (at 405 G ) . I f we assume Q = 1000, n = 0.1, T = 1°K and n = 5 x 1 0 1 2 c m - 3 , then we o b t a i n : T(405G) = 5 . 2 sec T (6.5kG) = 1.8 x 10 3 sec Rad i a t i on damping i s c l e a r l y not important a t 6.5 kG, and, even i f i t i s much sma l l e r a t 405 G, the homogeneity problem w i l l dominate. Note that r a d i a t i o n damping i s much more important f o r the zero f i e l d experiments because y i s much l a r g e r . 31 ( i v ) Sp in ^Exchange I n t r i n s i c a l l y , a gas of hydrogen atoms possesses r e l a x a t i o n mechanisms. The two most important ones are spin-exchange and d i p o l e -d i p o l e i n t e r a c t i o n s . A l though the l a t t e r i s a r a t h e r weak p roces s , i t w i l l dominate when spin-exchange becomes i n e f f e c t i v e ( S t a t t and B e r -l i n s k y ) . T h e o r i t i c a l c a l c u l a t i o n s have been done f o r both processes and we s h a l l on ly quote the main r e s u l t s s i n ce they w i l l be d i scus sed f u r t h e r i n Chapter F i v e i n the context of the recombinat ion problem. We s h a l l s t a r t w i t h sp in-exchange. Accord ing to B e r l i n s k y and S h i z g a l ( B e r l i n s k y and S h i z g a l 1980), a l l spin-exchange t r a n s i t i o n r a t e s can be w r i t t e n i n terms of three q u a n t i t i e s : two temperature-dependent c r o s s - s e c t i o n s , a + and a , each r e f e r r i n g to a sum over even (+) and odd (-) angular momentum s t a t e s , 2 and one f i e l d dependent f a c t o r , X = s i n 29, 6 be ing the angle de f i ned i n Chapter- Two. At low temperatures <a >^ , becomes n e g l i g i b l y sma l l + 62 whereas <a reaches some l i m i t around 1 A . I t i s found t h a t , a t a l l temperatures and f i e l d s , l e v e l 2 and 4 have same r a t e : p 2 = p^ » where p i s the p o p u l a t i o n of the l e v e l . Then, spin-exchange does not r e l a x the d i f f e r e n c e of popu la t i on s of the Zeeman s t a t e s . Moreover, s p i n -exchange cannot cause t r a n s i t i o n s w i t h i n the man i fo ld of s t a t e s 1 and 2. Th i s means tha t no t r a n s i t i o n between l e v e l s 1 and 2 w i l l take p l a ce i f the c o l l i d i n g atoms are i n a combinat ion of s t a t e s 1 and/or 2. Th i s i s a s t rong s e l e c t i o n r u l e which says tha t there w i l l be no e q u i l i b r a t i o n of these two l e v e l s i f the o ther two hype r f i ne s t a t e s (3 and 4) a re not popu la ted . Th i s happens f o r h i gh magnetic f i e l d s and low temperatures. F i n a l l y , we can mention t h a t , i f we d e f i n e the r e l a x a t i o n time T j as the r a t e a t which Pg-p^ r e l a x e s , then we can r e l a t e the T ^ ' s . a t a l l f i e l d s by: T^H) 1^(0) ( 4 2 6 ) I t i s i n t e r e s t i n g to l ook a t some numbers. For a den s i t y n = 5 x 1 0 1 3 c m - 3 and a temperature T = 4.2°K i t was measured tha t n T - 1 = a i ^v = 110-180 s - 1 (Crampton e t a l . 1979), and c a l c u l a t e d tha t T 1 " 1 = 72 s e c - 1 ( Be r l i n s k y et a l . 1980), a l l i n zero f i e l d . E x t r a -p o l a t i n g the c a l c u l a t i o n s of B e r l i n s k y et a l . to T = 1°K and n = H 5 x 1 0 1 2 cm 3 , which i s more i n our range of i n t e r e s t , we ob ta i n 1 - 5.'8 s e c - 1 , or = 0.17 sec . As the magnetic f i e l d i s i n c rea sed , X decreases, thereby i n c r ea s i n g T^. For H = 6.5 kG, w i l l be = 28 sec. Thus, we are working w i t h very long sp i n l a t t i c e r e l a x a t i o n t imes , and t h i s w i l l hamper the usage of s i g n a l averag ing , which i s a u s e f u l t o o l f o r d e t e c t i n g weak s i g n a l s . (v) D i p o l e - D i p o l e I n t e r a c t i o n s Assuming that on ly l e v e l s 1 and 2 were s i g n i f i c a n t l y popu lated, S t a t t and B e r l i n s k y ( S t a t t and B e r l i n s k y ) c a l c u l a t e d T^ due to d i p o l e - d i p o l e i n t e r a c t i o n s i n h igh magnetic f i e l d . They found tha t "Y — i e T^ x = Ujj[1 + e —1 A, where e i s the admixture c o e f f i c i e n t de f i ned i n Chapter Two and A i s a f a c t o r independent of n^ or e. Using t h i s r e l a t i o n , we can e x t r a p o l a t e t h e i r equat ion (10) which was c a l c u -l a t e d f o r H q = 100 kG and n^ = 1 0 1 6 c m " 3 , and we o b t a i n : Y T - 1 - n j l + e — ] ( 6 . 7 9 T % - 0 . 7 4 6 T 3 / V l 0 - 2 0 s e c " 1 (4-27) 1 H Y_ Consider a den s i t y of 5 x 1 0 1 2 cm 3 , a f i e l d H = 6.5 kG and a temperature T = 1°K, then equat ion (4-29) y i e l d s : T ^ - 1 - 2.14 x 10~h s - 1 or ^ - 4670 sec. We can then conclude that d i p o l e - d i p o l e i n t e r a c t i o n s w i l l be important i n our experiment on ly i f the d e n s i t y i nc reases c on s i de r ab l y . 4 -3 -2 . Recomb i na t ion There e x i s t many routes f o r the recombinat ion of hydrogen atoms i n t o a hydrogen mo lecu le . In a l l cases, a t h i r d body i s r equ i red f o r conserv ing energy and momentum. Depending on the exper imenta l c o n d i t i o n s , c e r t a i n schemes w i l l dominate. In t h i s experiment, recombinat ion i n the bu l k by the process H + H + He + He 6 7 w i l l dominate, s i n ce the hel ium den s i t y i s about 10 -10 times l a r g e r than the hydrogen d e n s i t y and s i nce the b i nd ing to the w a l l i s n e g l i g i b l e i n the range of temperatures a ch i evab le w i t h our aparatus . A model f o r recombinat ion w i l l be desc r ibed i n the next chapte r . The most important f e a t u r e of t h i s model i s t ha t recombinat ion depends on the hype r f i ne s t a t e s of the c o l l i d i n g atoms, and, conse-quent l y , the recombinat ion r a t e w i l l depend on the popu l a t i on d i s t r i -bu t i on of the atoms among the hype r f i ne s t a t e s . An extreme case would be a gas of hydrogen atoms a l l i n the h = 2 hype r f i ne s t a t e . In t ha t case, there w i l l be no recombinat ion a t a l l and the gas w i l l be s t a b l e ( i f we neg lec t s p i n - s p i n i n t e r a c t i o n s and w a l l e f f e c t s ) . Hence, r e l a x a t i o n mechanisms and recombinat ion are deeply i n te rconnected and the dynamics of the h ype r f i ne l e v e l s i s the r e s u l t of both of them. For t h i s reason, the model de r i ved i n the next chapter w i l l i n c l ude spin-exchange as the r e l a x a t i o n mechanism. CHAPTER FIVE BULK RECOMBINATION Bulk recombinat ion i n v o l v i n g the process H + H + He W + He w i l l be cons idered i n t h i s chapter and a s imple model w i l l be der ived.) ; 5-1. P r e l i m i n a r i e s The process H + H + He -* H^ + He has been i n v e s t i g a t e d e x p e r i -menta l l y i n zero f i e l d , and f o r temperatures ranging from about 1°K to 1.25°K, (Hardy et a l . 1980). A s imple model was used to e x p l a i n the da ta . When no e x p l i c i t c o n s i d e r a t i o n i s taken of the hype r f i ne s t r u c t u r e of the hydrogen atoms, we can regard the recombinat ion problem as a chemica l process where the r e a c t i n g p a r t i c l e s have some p r o b a b i l i t y of c o l l i d i n g together and r e a c t i n g . This leads to a r a t e equat ion of the type: °H " " k "He ^ ( 5 - 1 } where n^ and n^ e are r e s p e c t i v e l y the hydrogen atom and hel ium atom d e n s i t i e s . The f a c t o r k i s j u s t a measure of the e f f i c i e n c y of the c o l l i s i o n . An exact s o l u t i o n can be found to t h i s equat ion , and i t xs: n„(0) V f c ) = TTTTt  ( 5 " 2 ) C R where we d e f i n e t_, 1 = k n_T n (0) . Note that i n t h i s model, the R Tie H c h a r a c t e r i s t i c t ime, t , depends on the i n i t i a l d e n s i t i e s . R Although elementary, t h i s model succeeded i n e x p l a i n i n g the da ta . Indeed, by p l o t t i n g d ( l / r i ^ )/dt aga in s t n ^ (which i s d i r e c t l y r e l a t e d to the temperature) , we ob ta i n a s t r a i g h t l i n e whose s lope i s k. The average k obta ined i s : k = ( 0 . 2 8 ± 0 . 0 3 ) x l C ~ 3 2 c i n 6 s _ 1 . Th i s s imple model makes no p r e d i c t i o n about the f i e l d dependence of the recombinat ion . In order to f i n d the e f f e c t of a f i e l d , we have to i n c l ude the s p i n degrees of freedom i n the problem s i n ce the i n t e r a c t i o n between hydrogen atoms depends on the hype r f i ne s t a t e of the atoms. Obv ious l y , then, the recombinat ion r a t e w i l l depend on the popu l a t i on d i s t r i b u t i o n among the h ype r f i ne s t a te s and, conse-quent l y , the r e l a x a t i o n mechanisms w i l l p l ay a c r u c i a l r o l e i n the recombinat ion problem. Therefore, any r e a l i s t i c model f o r recombina-t i o n i nc ludes r e l a x a t i o n mechanisms. As mentioned i n Chapter Four, spin-exchange i s the dominant r e l a x a t i o n mechanism i n low f i e l d (6.5 kG i s cons idered sma l l when the temperature i s around 1°K s i n ce y H <kT). We s h a l l t he re f o re e a. take spin-exchange f o r the r e l a x a t i o n of the h ype r f i ne s t a te s i n our model. The s t r a tegy w i l l then be to d e r i v e the model f o r the recombinat ion r a t e and t h e r e a f t e r to i n c l ude the equat ions d e s c r i b i n g spin-exchange ( B e r l i n s k y 1979). F i n a l l y , the parameters w i l l be matched to the e x i s t i n g da ta . 5-2. Asymptot ic S ta tes In a s c a t t e r i n g problem, the u sua l approach i s to l ook a t the asymptot ic wave f unc t i on s of the c o l l i d e r s , f a r from the s c a t t e r i n g s i t e . In that l i m i t , the atoms a re f r e e and the t o t a l wave f u n c t i o n i s g i ven by the d i r e c t product of the r e s p e c t i v e wave f u n c t i o n s , or s t a t e s . In our problem, three atoms come i n (two hydrogen atoms and a he l ium atom) and one molecu le p lus one atom come out (a hydro-gen molecule and the he l ium atom). The he l ium atom i s there on ly f o r conserv ing energy and momentum. ^He behaves l i k e a n e u t r a l body s i nce i t has a t o t a l e l e c t r o n i c sp in equal to zero (S=0) and has zero nuc l ea r sp i n (1=0). Hence, the sp i n pa r t does not enter d i r e c t l y i n t o the s c a t t e r i n g problem, so that the asymptot ic he l ium s t a t e can be denoted s imply by: |He> = |p3> (5-3) where p^ i s the momentum. The hydrogen molecu le r e s u l t i n g from the c o l l i s i o n has, obv i ou s l y , S = M = 0, and can be i n • any of i t s v i b r a t i o n a l ( v ) , r o t a t i o n a l (£) and nuc l ea r sp i n (I,M ) s t a t e s . The asymptot ic s t a t e i s then: |H2> = |v ,£ , I ,M r k~> (5-4) where lc i s the momentum. However, I + I must be even. The case of the hydrogen atoms i s somewhat more compl icated because, f i r s t , the nuc lear and e l e c t r o n i c sp ins a re coupled by the hype r f i ne i n t e r a c t i o n (see Chapter Two) and, second, these two p a r t i c l e s a re i d e n t i c a l , r e q u i r i n g an ant i symmetr ized s t a t e . The i n t e r a c t i o n between two hydrogen atoms depends on t h e i r t o t a l e l e c -t r o n i c s p i n , S (see Chapter One). Thus we have to r e l a t e the hyper -f i n e s t a t e s to the s imply -coup led b a s i s : |SMQ, IM T>. This r e l a t i o n s h i p i s e a s i l y found and i s f o rma l l y w r i t t e n as: | h 1 f h > = I I A ( h . , h ;IM SM )|SM IM > (5-5) 1 2 SM S IM T 1 2 1 S S I where A(h..h ;IM ,SM ) = <SM ,IM |h h > i s a 16 x 16 u n i t a r y m a t r i x . J- Z. J_ O O J- J- £ The an t i s ymmet r i z a t i on i s more i nvo l ved but s t r a i g h t f o r w a r d i f one cons ider s the s imply -coup led ba s i s and goes i n t o the cent re of mass of the two hydrogen atoms frame of r e f e r e n c e . Quoting B e r l i n s k y ( B e r l i n s k y ) , the r e s u l t i n g ant i symmetr ized wave f u n c t i o n i s : M M i qWt - R )/« -iq - (R -R )/* * ( r 2 - R 1 ) } { e 1 1 + ( - l ) i + b e 1 2 } (5-6) M I M s -where <j> , <f> are s p i no r s , ±q a re the momenta of a hydrogen atom i n J. o -> the new frame, and $(r-R) i s a I s hydrogen wave f u n c t i o n (stands f o r (•rra 3 ) ^ e ^ r l ^ a 6 in equat ion ( 2 - 2 ) ) . In a fo rmal d e r i v a t i o n i t i s o convenient to use the p r o j e c t i o n ope ra to r , P which e l im i na t e s the undes i red s t a t e s from a gene ra l non-symmetrized s t a t e , and to sum over a l l the s t a t e s . E x p l i c i t l y , we have: ^ i - V V V = s £ s l q ; s M s , i M l > A A <q ; SM s , IM I |p 1 , p 2 ; h 1 h 2 > (5-7) where IqjSMg.IM^^ stands f o r |.^ > as g i ven by (5 -6 ) , and I p ^ j p ^ h ^ h , ^ i s the non-symmetrized two-hydrogen atoms s t a t e ( h^ ,h 2 are the hyper -f i n e s t a t e s ) . The ma t r i x element i n (5-7) can e a s i l y be eva luated and i t i s approx imate ly g i ven by: A <q ; SM s , IM I |p 1 , p 2 ; h 1 , r i 2 > = \ ACh^h^SMg.IMj)-•[6q,'q + ( - l ) S + I 6 q ' , - q ] (5-8) where q ' = q^ = - q 2 i s the momentum i n the s p e c i a l frame mentioned above. The asymptot ic s t a te s w i l l then be: |i> = P s | p 1 , P 2 ; h 1 , h 2 ; p 3 > (5-9) |f> = | v , £ , I , M I , ^ ; p 3 , > (5-10) 5-3. The Model From s c a t t e r i n g theory, the t o t a l t r a n s i t i o n r a t e per u n i t volume i s (see Goldberger and Watson 1964, p. 185): d R.ad3 k d 3 p ' 3 d ^ d 3 p 2 d 3 p 3 V V V ^ H e ^ T f . | 2 (5-11) where f ' s a re Boltzmann d i s t r i b u t i o n s f o r the momenta of the c o l l i d i n g p a r t i c l e s , and l T f . J 2 1 S the t r a n s i t i o n m a t r i x : ^f± = < f | T | i > -We s h a l l now make the reasonable assumption that the s c a t t e r i n g ope ra to r , T, does not change any of the sp i n quantum numbers- The sp i n pa r t can then be f a c t o r e d out and, s i nce the bound s t a t e f o r H requ i r e s t ha t S = M = 0, equat ion (5-7) and (5-8) y i e l d : |<v ,J l , I ,M 1 ,k ;p 3 |T|q , ,0 ,0 ; I ,M I ;p 3 >| 2 -• A 2 ( h 1 , h 2 ; 0 0 , I , M I ) - r ( v , ^ , k ; q 3 / q , , I , M i ; p 3 ) . A 2 ( h 1 h 2 ; 0 0 , I M I ) (5-12) where r ( v ,l,k;q^/q ' ; I M ^ ; p 3 ) has been in t roduced to s i m p l i f y the equat ions s i nce i t w i l l become a parameter of the model. Note that there i s no s p e c i a l requirement on the nuc lea r sp i n s t a t e f o r recombinat ion to occur , a l though t h e . r o t a t i o n a l quantum number, I, of the r e s u l t a n t molecule i s r e l a t e d to the nuc lea r sp i n s t a t e by: I = \ ( l - ( - l ) ^ ) , where 1 = 1 corresponds to ortho-hydrogen and 1 = 0 to para-hydrogen. The expres s ion i n (5-12) possesses two important f e a t u r e s . F i r s t , a l l the f i e l d dependence i s i n the A 2 term, which i s a m a t r i x . Second, the r a t e f a c t o r , r , w i l l e x p l i c i t l y depend on the f i n a l s t a t e of the mo lecu le . In p a r t i c u l a r , i t depends on I, and B e r l i n s k y , Greben and Thomas, who have done p r e l i m i n a r y c a l c u l a t i o n s on the r a t e f a c t o r , b e l i e v e tha t i t i s much l a r g e r f o r ortho-hydrogen than f o r para-hydrogen. Equat ions (5-11) and (5-12) can now g i ve us the r a t e equat ion which we need. Because we are not i n t e r e s t e d i n any s p e c i f i c momentum, we s h a l l perform the i n t e g r a t i o n over the momenta. We s h a l l assume that | T^_J 2 , depends on ly on I,h^ and ^ ( M j . ^ e t e r _ mined by h 1 and h ). I f n i s the den s i t y of he l ium atoms, n, i s i 2 He ti-the den s i t y of hydrogen atoms i n s t a t e h, then the recombinat ion equat ion i s : dn d T = - n H e I B(I> b R h h ' n h \ ' ( 5 " 1 3 ) where B(I) r ep l ace s the i n t e g r a t i o n over momenta of the Boltzmann d i s t r i b u t i o n , the d e l t a f unc t i on s and the r a t e f a c t o r , and where R* i s A 2 (see Tables I and I I ) . hh A f i r s t order s o l u t i o n can be found to (5-13) and i t i s : t n h ( 0 ) : n. = n u ( 0 ) ( l - f - + . . . ) = ^ T 7 - (5-14) *h " h v " / v ^ x T V t / n 1+ ' where T , 1 = nT, E B(I) E R,1, ,n, . ( 0 ) . Th is e s t a b l i s h e s the approx-h He j h 1 hh ' h ' ^ imat i ve c h a r a c t e r i s t i c t ime of the recombinat ion process . 5-4. S p in-exchang e Rate equat ions f o r s p i n exchange were der i ved i n zero f i e l d by B a l l i n g e t a l . ( B a l l i n g e t a l . 1964) and gene ra l i z ed by B e r l i n s k y to non-zero f i e l d ( Be r l i n s k y 1979; B e r l i n s k y and S h i z g a l 1980). We s h a l l on l y quote these equat ions here: n± = % { - X ( l - X ) Y e ( n 2 e " - n ^ ) - 2 X Y ( n 2 e ' ^ ^ ^ - n 9 n . ) e l I H - X 2 Y ( n 2 e " 2 3 E 1 3 - n 2 ) - 2 X Y ( n . n . e ^ ^ - n n . ) e • 1 3 o 1 <d 15 - g ( E -(.g ) - 2 X Y (n n e ^ - n n )} 1 4 W ' (5-15a) n 2 = % { - X Y e ( r l 2 n 4 - n 1 2 e " E ( E 1 2 + E 1 4 ) ) - 2 [ ( l - X ) Y e + Y o ] ( n 2 n 4 - n 1 n 3 e ~ B ( E 1 2 + E 3 ^ ) ) -XY o(n 0n /e" 6 ( E23- E3<+ ) _2 n e 2 4 n 3 3 ^> (5-15b) n 3 = % { - X ( l - X ) Y e ( n 1 n 3 - n 1 2 e ~ B E 1 3 ) - 2 [ ( l - X ) Y E + Y 0 ] ( n i n 3 e - 8 ( E l 2 + E 3 , ) + X ( l - X ) Y e ( n 3 n 1 e " 3 E 1 3 _ n 2 ) - 2 X Y o ( n 3 n 2 - n i n 2 e - E E 1 3 ) - X 2 Y (n 2_ n 2 e - 2 e E 1 3 e 3 1 ' - 2 X Y e ( n 3 2 - n 2 n 4 e - e ( E 2 3 - E 3 i + ) ) - 2 X Y (n_n -n.n.e ^ E l 3 , ) l o 3 4 1 4 n (5-15c) n 4 = n 2 (5-15d) where E„ = E^ . - E_^  (E^ . i s the energy of hyperfine l e v e l j ) and where X , Y £ and Y q a r e parameters. X = sin 226 was introduced e a r l i e r i n Chapter Four, and y , y are rate constants for even and odd e o angular momentum states. The effect of temperature i s contained i n -BE the factors e introduced by d e t a i l balance considerations. TABLE - I RECOMBINATION MATRIX FOR I = 0 : R° h ' h i 2 3 4 1 s i n 2 9 c o s 2 0 0 ^ ( c o s 2 0 - s i n 2 0 ) 2 0 2 0 0 0 h 3 % ( c o s 2 6 - s i n 2 0 ) 2 0 s i n 2 0 co s 2 6 0 4 0 h 0 0 TABLE - I I 1 RECOMBINATION MATRIX FOR 1 = 1 : 'R, , hh h ' h 1 2 3 4 1 0 s i n 2 6 h h co s 2 8 2 s i n 2 9 0 h c o s 2 e 3 h \ c o s 2 e 0 \ s i n 2 6 4 c o s 2 e h \ s i n 2 0 0 We can determine Y E and y Q by l i n e a r i z i n g equat ions (5-15) w i t h exponen t i a l f a c t o r s exc luded. I f we put n. = (<$ + <5.)n , 1 1 H • * and take the d i f f e r e n c e n^ - n^, then we can o b t a i n an equat ion s i m i l a r to equat ion (2a) of B e r l i n s k y and S ch i z ga l ( Be r l i n s k y and Sh i z g a l 1980). The r a t e constants a r e : n y = o + v n /2 and I L Y = H e H H o a~ v n /2, where v i s the mean v e l o c i t y and a~ the thermal average H of the c ros s s e c t i o n s . Using t h e i r t a b l e I, we can c a l c u l a t e y^ and Y q a t d i f f e r e n t temperatures: T(°K) y (cm 3 /sec) y (cm 3/sec) e o 0.75 8.081 x 1 0 ~ 1 3 1.067 x 1 0 " 1 4 1.00 8.172 x 1 0 - 1 3 1.594 x 10~lh 1.25 8.140 x 1 0 ~ 1 3 2.025 x 10~lh 1.50 8.056 x 1 0 ~ 1 3 2.484 x 1 0 " 1 4 The range of temperatures of i n t e r e s t to us i s from 0.9°K to 1.3°K, so that we can assume that y^ and y^ are e f f e c t i v e l y i n d e p e n -dent of the temperature, and we can put : y = 8.15 x 1 0 " 1 3 cm 3/sec e y = 1.8 x l O - 1 4 cm 3 /sec 5-5. Re su l t s One important conc l u s i on can be drawn from the approx imat ive s o l u t i o n (5-15): the i n i t i a l c o n d i t i o n s p l a y a c r u c i a l r o l e i n the recombinat ion process ,and,un less the r e l a x a t i o n mechanism i s e f f i c i e n t ( i . e . f a s t and n o n - s e l e c t i v e ) , t h e y w i l l determine the e v o l u t i o n of the hydrogen gas. The characteristic times w i l l depend on the experimental conditions, and the starting point i s then to investigate these condi-tions . In our experiment, the atoms w i l l be produced in a low temperature discharge (Chapter Seven) and i t i s then reasonable to assume that the hyperfine states are equally populated. In other words, we assume in f i n i t e temperature. Also, the achievable densities in a discharge 11 3 11 12 -3 are above 10 atoms/cm , let us say between 10 and 10 cm , and the He density w i l l be about 10 -10 cm . The characteristic time for recombination, as given by (5-2), w i l l be: t R = (k n H e n H(0))" 1 = 100-4000 sec. On the other hand, spin-exchange has a characteristic time: tS ~ Y nH^ _ 1 = 1 " 5 0 0 S e c* Thus, spin-exchange tends to be much faster than recombination. Let us f i r s t analyse the zero f i e l d case. We note that n^ = (n^,n2,n3,n^) = (1,1,1,1) i s an eigenvector of recombination (this i s true at a l l f i e l d s ) , but not of spin-exchange. Thus, the recombination process does not perturb the i n i t i a l population distribution, although the total density decreases. However, spin-exchange changes the population distribution very rapidly (see figure 5-1) and, subsequently, recombination proceeds with the levels always at equilibrium. The numerical solution i s easily understood in terms of a thermal e q u i l i -brium situation. Let us c a l l A the case where B(0) ^ 0 with B(l) = 0 in equation (5-13) and B the case where B(l) / 0 with B(0) = 0. Case A corresponds to the production of para-hydrogen only, while case B corresponds to the production of ortho^-hydrogen only. In case A, equat ion (5-13) y i e l d s ; . ,_ 1 2 , 1 2 , 1 V B n H e = 4 n l + 5 n 3 + 2 n 2 n 4 (5-16) I f we assume that thermal e q u i l i b r i u m p r e v a i l s , then (5-16) becomes: 3„ 1„ 1, •VBnHe = 2 pa - i g a e + 3e 2»a -^ fcia + + 1 16 -9 2 2 1 + B a 2 A s i m i l i l a r procedure f o r case B g i ve s : ^ H / B n H e = 16 29 2 2 1 1 + ^ a 2 (5-17) (5-18) We observe, that the e f f e c t of the temperature*;appears on l y to second 2 2 ^ 3 order and i s almost n e g l i g i b l e i n both cases (.3 a - 5 x 10 a t 1 K) . Consequent ly, recombinat ion w i l l proceed as i f the r e l a x a t i o n mecha-nism were i n e f f e c t i v e , and l /h j j w i l l change l i n e a r l y i n t ime (see f i g u r e 5^ -2 and 5-3) . The s lope of the curve i s Bn^. /16 i n case A, He and 3Bnjj e/16 i n case B. The exper imenta l va lue i s matched f o r B(0) = 16k = 4,48 x 1 0 " 3 2 c m 6 s _ 1 or B ( l ) = 16k/3 = 1.49 x 1 0 " 3 2 c m V " 1 , and t h i s i s conf i rmed by the numer ica l s o l u t i o n . Thus , the s t r i k i n g r e -s u l t s are that recombinat ion behaves as i f there were no r e l a x a t i o n mechanism as long as the ( i n i t i a l ) p opu l a t i on d i s t r i b u t i o n does not dev i a t e too much from the i n f i n i t e temperature d i s t r i b u t i o n , and that the h ype r f i ne s t r u c t u r e does not make the s o l u t i o n very d i f f e r e n t from the one g i ven i n s e c t i o n 5-1. I t i s , however, not p o s s i b l e to s o r t out what i s the app rop r i a te c o n t r i b u t i o n of each case (A and B) because b o t h of them produce the same r e s u l t i n zero f i e l d . Thus, i t i s u s e f u l to l ook a t the f i e l d dependence of the recoirb, ination rate for each case (and, eventually, a mixture of the two cases) . We first adjusted the two constants so that the zero field slope corresponded to the experimental value as derived above. Then we looked at the recombination for both cases at H = 6481 Gauss and H = o o 10 kGauss, (see figures 5-2 and 5-3). In both cases, the field sup--im-presses the recombination, and we obtained similar curves when l/n u rl was plotted against time. The curve starts with the zero field slope and subsequently, curves dcwTward, indicating a reduced rate constant. However, the reduction of the recomS ination rate was larger at 6481 G than i t was at lOkG. This unexpected b ehaviour comes from the dynamics of the hyperfine levels, and the selectivity of the relaxation mecha-nism. It is not straightforward to sort out the contribution of each case and we would have to investigate their special features, such as the field at which the slowest reconb Ination rate takes place, in order to make the difference between case A and B, We shall not enter into those considerations. There are a few- important conclusions to draw here. First, although the i n i t i a l conditions are crucial, the dynamics induced by the relaxation mechanism is also important and is intimately related to the recoirb ination problem. Second, i t is difficult to separate the effect of each case (production of ortho-hydrogen or production of para—hydrogen) when the hyperfine levels are equally populated i n i t i a l -ly. Third, the assumed i n i t i a l conditions give rise to simple behaviour for the recombination. In zero field, the recombination rate is weakly affected by the relaxation mechanism, and, as the field is increased, recoirbination tends toward a reduced rate. There is more work to b e done on this model, but we need a better i 1 , H 1 0 0.5 .1.0 1.5 2,0 t (sec) F i g u r e 5-d: E v o l u t i o n of the popu l a t i on P f - t h e h y p e r f i n e l e v e l s s t a r -t i n g w i t h , a l l l e v e l s - e q u a l l y populated and , in zero f i e l d . The b roken l i n e the thermal- e q u i l i b r i u m d i s t r i b u t i o n . i 1 1-o I 1 \ \ •» 0 100 200 300 400 t (sec) Figure 5-2: Evolution of the hydrogen atom density for different fields in case A. 52 understanding - of "the exper imenta l c o n d i t i o n s . We s h a l l c l o s e t h i s chapter w i t h an i l l u s t r a t i o n of the e f f e c t of the i n i t i a l c o n d i t i o n s . Cons ider the case i n wh i ch one s t a r t s w i t h thermal e q u i l i b r i u m popu la t i on s of the h ype r f i ne l e v e l s , and cons ide r two f i e l d s : H o = 405 Gauss and H o = 6.5 k feu s s , w h i l e l e t t i n g B(0) = 0, (Case B ) . The e f f e c t of the a p p l i c a t i o n of a f i e l d i s much l a r g e r (see f i g u r e 5-4). Th i s case corresponds to most p r e d i c t i o n s , s i n ce one u s u a l l y expects that the h ype r f i ne l e v e l s w i l l reach t h e i r lew tempe-r a t u r e popu la t i on s f o r w h i c h recombinat ion i s slew a t h i gh f i e l d . RecomSination i s suppressed Because a h=2 s t a t e atom w i l l not reconb i n e w i t h another h=2 s t a t e atom, and w i l l reconb i ne w i t h a h=l atom on ly because there i s an admixture of up e l e c t r o n i c s p i n i n the h=l s t a t e w h i c h decreases as the magnetic f i e l d i s i n c rea sed . F i gu re 5-4: I l l u s t r a t i o n of the r e d u c t i o n of the recombinat ion r a t e when thermal e q u i l i b r i u m c o n d i t i o n s are assumed. CHAPTER SIX THE 765 MHz EXPERIMENT Th i s experiment was designed f o r the study of the i n f l u e n c e of the magnetic f i e l d on the recombinat ion r a t e s and s p i n exchange, and MR i s used as a probing techn ique (see Chapter Three) . When we l ook a t the h ype r f i ne l e v e l s of the hydrogen atom (see F i g . 2-11, we see a ve ry sha l low minimum i n the sepa ra t i on between the two bottom l e v e l s occur ing a t 6,481 G. The idea i s to use that set of l e v e l s f o r MR s tud ie s s i nce i t i s not ve ry s e n s i t i v e to the homogeneity or the exact va lue of the s t a t i c magnetic f i e l d . Th i s i s an important c o n s i d e r a t i o n because we a re i n f a c t observ ing a s i g n a l which, as we s h a l l see below, i s very weak and because the magnetic f i e l d i s not shimmed when we f i r s t l o o k f o r a s i g n a l . The frequency of t h i s t r a n s i t i o n i s 765 MHz, and r e q u i r e s a t ran sve r se magnetic f i e l d . Even i f 6.5 kG i s not an i d e a l l y s t rong magnetic f i e l d , i t i s s t i l l p o s s i b l e to b r i n g the f i e l d to h igher v a l ue s , a t the expense of l o s i n g t empo ra r i l y the s i g n a l , and then come back to see what happened i n the i n te r ven i ng t ime. We a l s o f i n d that there i s another t r a n s i t i o n a t 765 MHz which occurs between l e v e l s 2 and 3 a t a f i e l d of 405 G. This t r a n s i t i o n possesses an e f f e c t i v e magnetic moment much h igher than f o r the 1-2 t r a n s i t i o n ( y 2 3 = 22.2 y 1 2 ) but i s much more s e n s i t i v e to the exper imenta l c o n d i t i o n s , such as the homogeneity of the s t a t i c f i e l d . I t neve r the le s s p rov ides one ex t ra f i e l d where the atoms can be " s e e n " . We s h a l l now go i n t o the l i s t i n g of a set of data concern ing t h i s experiment, together w i t h some p r e d i c t i o n s . .6-1. The Two T r a n s i t i o n s I t i s e a s i l y de r i ved that the sha l low minimum between l e v e l s 1 and 2 occurs a t : a(y -u ) / i \ H 6 P = - * - (6-1) ° 2/y~y (y +y ) 2^T (l+n)n" z e p e p e where n = y /y (see Chapter Two). p e S u b s t i t u t i n g the numbers i n t h i s expres s i on , we get : H = 6481 G. o S u b s t i t u t i n g equat ion (6-1) i n t o the expres s ion f o r the energy d i f f e r e n c e , we o b t a i n that the corresponding frequency i s : f = fir t 1 + f r ^ <6-2) o -2h l+n = 765 483 207.7 Hz. The low f i e l d corresponding to the other 765 MHz t r a n s i t i o n , which occurs between l e v e l s 2 and 3, i s g i ven by: 7,-H ' = T T 9 - { - ( l - n ) d + ^ - ) + (1 + A (1 + r , A (6-3) o 2y n l+n e = 404.76 G. 6-2. S e n s i t i v i t y to F i e l d Homogeneity Any inhomogeneity of the s t a t i c magnetic f i e l d w i l l r e s u l t i n a spreading of the resonance frequency of the atoms. I t i s t he r e f o r e worthwhi le l o o k i n g a t the s e n s i t i v i t y of the resonance frequency to the f i e l d f o r each t r a n s i t i o n . I t was mentioned above that the 765 MHz t r a n s i t i o n a t 6.5 kG i s r a t he r i n s e n s i t i v e to changes i n the s t a t i c f i e l d because i t corresponds to a sha l low minimum. The broadening, 6f , of the r e son -ance frequency i s e a s i l y obta ined by a T a y l o r ' s s e r i e s expansion about H + AH, where AH i s the d i f f e r e n c e between the a c t u a l f i e l d . o and the resonance H q (see Whitehead 1979), and we keep on ly the lowest order terms: <Sf = 1.31 (HzG~ 2) (2AHSH + (<5H)2) (6-4) where 6H i s the magnitude of the inhomegeneity. I f AH = 0 and i f 6H ^ 1 0 _ 6 H , then 6f - 5.5 x 10~ 5 Hz, a broadening which i s n e g l i g i b l e i n our exper iment. The s i t u a t i o n w i l l be s i g n i f i c a n t l y d i f f e r e n t i n the case of the t r a n s i t i o n a t 405 G, f o r two reasons: the e f f e c t i v e magnetic moment i s l a r g e r , and one i s not a t a t u rn ing p o i n t f o r the f requency. App ly ing the same procedure as before to t h i s t r a n s i t i o n we o b t a i n : 6f = [ 1 . 32x l 0~ 3 ( - ^ f )AH + 2. 2 7 5 7 ( ^ - ) ]6H (6-5) G G I f AH = 0 and 6H = 10~ 6 H, then Sf - 1 kHz. Th i s i s more than n o t i c e a b l e , and great care w i l l have to be taken i n des i gn ing and b u i l d i n g the experiment i n o rder to min imize the e f f e c t of t h i s l i m i t a t i o n . 57 6-3. E f f e c t i v e Magnetic Moments In Chapter Three, we de r i ved an expres s ion f o r the e f f e c t i v e magnetic moment f o r the va r i ou s p a i r s of h ype r f i ne l e v e l s between which t r a n s i t i o n s can be induced by a t ransver se o s c i l l a t i n g magnetic f i e l d , (equat ion (3 -19 ) ) . The two e f f e c t i v e moments which we need a re : u 1 2 = u s in6 + u cos0 e p = 7.5146 x 1 0 ~ 2 2 erg/G (6-6) = 0.11341 MHz/G ( a t 6.5 kg : 6 = 2.232°) and u 2 3 = -u cos0 + u sin8 e p = -1.6725 x 1 0 " 2 0 erg/G (6-7) = -2.5240 MHz/G ( a t 405 G : 6 = 25.67°) The r a t i o between these two moments i s y 2 3 / ] i 1 2 = -22.2 , which exp la i n s why the t r a n s i t i o n a t 405 G i s s t i l l appea l ing d e s p i t e a l l the o ther d i f f i c u l t i e s . 6-4. S t rength of the S i g na l I t i s very important to es t imate the s t reng th of the s i g n a l to be detected by the spectrometer, because the s e n s i t i v i t y of the spectrometer i s f i n i t e and because s i g n a l averag ing may be d i f f i c u l t f o r these measurements s i n ce T 1 may be very l ong . In Chapter Four we de r i ved an expres s ion (equat ion (4-20)) which r e l a t e d the power t r an sm i t ted i n the l i n e by the atoms to t h e i r o s c i l l a t i n g magne t i za t i on : ,n2m2u)Q < pL> = -nr- (6-8) As we know, the t o t a l magnet i za t ion decays and so does the power em i t t ed . We s ha l l , t he re f o re r e s t r i c t our i n v e s t i g a t i o n to the i n i t i a l s i g n a l . The i n i t i a l magnet i za t ion a f t e r a n/2-pulse i s g i ven by equat ion (3-14): m(0) = N R u e f f Tr[p(0.)S x] (6-9) where N i s the t o t a l number of atoms. The quest ion i s now: what H i s the i n i t i a l average den s i t y matr i x ? In t h i s experiment the atoms a re produced by a low temperature d i scharge which e s s e n t i a l l y l eaves them w i t h an i n f i n i t e temperature. This leads to no s i g n a l a t a l l because we need a popu l a t i on d i f f e r e n c e i n order to have a non-zero v a l u e of <S > a f t e r a T r / 2 - p u l s e . We t he re f o re have to wa i t x c u n t i l the gas has coo led down, c r e a t i n g the popu l a t i on d i f f e r e n c e which i s necessary f o r any s i g n a l . The mechanisms which perform the c o o l i n g are c a l l e d the s p i n l a t t i c e r e l a x a t i o n mechanisms or s imply T^-mechanisms. We saw i n Chapter Four that thermal e q u i l i b r i u m i s not e a s i l y a t t a i n e d i n a gas of hydrogen atoms o r , i n o ther words, T^ i s v e r y l o n g . Never the le s s , suppose that thermal e q u i l i b r i u m i s f u l l y a t t a i n e d , and tha t a l l atoms a re u n c o r r e l a t e d , which i s g e n e r a l l y t r u e . Then: (6-10) where P^ and P^ are the thermal popu la t ions as g i ven by the Boltzmann d i s t r i b u t i o n . Therefore the t o t a l magnet i za t ion i s : y e f f m(0) = N H - ( P a - P b ) (6-11) The Boltzmann d i s t r i b u t i o n i s mere ly: - v v e P i = -E./kT (6"12> E e 1 By making use o f equat ion (6-12) we can make a t a b l e of the popu la t i on d i s t r i b u t i o n among the four h ype r f i n e l e v e l s f o r d i f f e r e n t tempera-tures and magnetic f i e l d s . I n t a b l e I I I i s l i s t e d the popu la t i on d i s t r i b u t i o n a t 1°K and 1.5°K, f o r the two f i e l d s of i n t e r e s t to us . From t h a t , we can e x t r a c t t he va lues of the maximum magnet i za t i on per atom, m/N , which w i l l be used together w i t h (6-11) and (6-8) f o r determin ing the s t reng th of the s i g n a l . This i s l i s t e d i n the f o l l o w i n g : T = 1°K 1.5°K H = 6.5 kG 4.870 2.953 o H = 405 G -76.10 -50.93 o i n 10 2 t f erg/Gauss. I f we assume the f o l l o w i n g q u a n t i t i e s : Q £ = 1000, n = 0.2, a) = 2TT 765 MHz, g = V_/8TT - 0.1 cm 3 and N = 5 x 1 0 1 2 , and s u b s t i t u t e them in to equat ion (6-8) we o b t a i n f o r <P >: T = 1°K 1.5°K H = 6.5 kG 0.0071 0.0026 <P > o I L H = 405 G 1.74 0.779 o i n 10~ L T F wa t t . The d i f f e r e n c e between the two t r a n s i t i o n s a v a i l a b l e i s s t r i k i n g , and i t i s not necessary to emphasize t h i s po i n t anymore. The spectrometer which we s h a l l use (Whitehead 1979) has a mea-sured no i se temperature, T , of about 100°K. ' I n o rder to see the s i g n a l w i thout s i g n a l averag ing , we need a s i g n a l to no i s e r a t i o , S/N, of about 5. We then need to have: (S/N), = 5 (6-13) Suppose that the s i g n a l decays i n about 1 msec, which i s a good es t imate when the f i e l d i s shimmed, then Af ^ 160 Hz and P ^ 1 0 _ I E w a t t . Thus, the spectrometer can de tec t the atoms, but we r e l y on a s u f f i c i e n t shimming of the magnetic f i e l d i n order to see our f i r s t s i g n a l . However, the den s i t y i s a l s o a d e c i s i v e f a c t o r s i nce the power s ca l e s l i k e n 2.. I t i s u n l i k e l y that n^ w i l l exceed l O 1 ^ c m - 3 , but t h i s i s more than enough to make i t e a s i l y ob se r vab le . 61 TABLE - I I I POPULATION OF THE HYPERFINE LEVELS AT 1°K AND 1.5°K AND FOR H = 405 G o AND H = 6 . 5 kG. o P, (H =405G) n o P^CH =6.5kG) h o 1°K 1 2 3 4 0.26543 0.25234 0.24324 0.23899 0.35919 0.34623 0.14960 0.14497 1.5°K 1 2 3 4 0.26023 0.25161 0.24552 0.24265 0.32471 0.31685 0.18110 0.17734 7T The w id th of a — - pu l s e i s g i ven by equat ion (3-20) IT/2 2OJ1 where OJ^ = yH^/2h. We know everyth ing but H^. However, i n Chapter Four, we de r i ved a r e l a t i o n s h i p between H ,<P > and the parameters of the resonator (equat ion ( 4 -12 ) ) . S u b s t i t u t i n g tha t i n t o the express ion f o r T we o b t a i n the u s e f u l fo rmula : TT/ I _ TTh , BO) s \ ,r \/2 - ~ ( Q ^ p - 7 ) ( 6 " 1 4 ) C Lt During a i r/2-pulse, the energy i s s upp l i ed by the spectrometer which d e l i v e r s about 6.3 mW (measured). Assuming t h i s number and the q u a n t i t i e s of s e c t i o n 6-4, we can es t imate the r equ i r ed l e n g t h of the iT / 2 - p u l s e s : T ,„ (405G) = 0.45 ysec IT/ Z T . „ (6.5kG) - 10 y sec . TT/2 CHAPTER SEVEN EXPERIMENTAL APPARATUS 7 -1 . P r i n c i p l e The aim of t h i s experiment i s to measure the f i e l d and temperature dependence of the bu l k recombinat ion r a t e and the spin-exchange c r o s s -s e c t i o n of the hydrogen atom gas. The p r i n c i p l e of the experiment can be summarized as f o l l o w s . Upon the a p p l i c a t i o n of a s t a t i c and homogeneous magnetic f i e l d , H , the hype r f i ne l e v e l s of the hydrogen atom s p l i t (see Chapter Two), thereby changing both the a v a i l a b l e t r a n s i t i o n f r equen -c i e s and the recombinat ion r a t e s . With the a v a i l a b l e 765 MHz s p e c t r o -meter, there a re t r a n s i t i o n s a c c e s s i b l e a t 405 Gauss and 6.5 kGauss s u i t a b l e f o r magnetic resonance s t u d i e s , which p rov ide i n f o rmat i on on r e l a x a t i o n mechanisms, recombinat ion r a te s and abso lu te d e n s i t i e s of atoms. The range of temperatures i n which we a re f i r s t i n t e r e s t e d i s from about 0.9°K to 1.3°K. In t h i s range, the he l ium atom den s i t y i s h i gh enough to make H+H+He-^ H^+He the dominant recombinat ion mechanism wi thout i nduc ing too h igh a r a t e . The w a l l s of the c l o sed c e l l c o n t a i n i n g the sample w i l l be coated w i t h a f i l m of s u p e r f l u i d ^He i n order to prevent w a l l recombinat ion . The b i nd ing energy of the hydrogen atoms on ^He i s 0.9±0.05°K (see Morrow et a l . ) which i s low enough to have a sma l l e f f e c t on t h i s experiment. The hydrogen atoms w i l l be produced i n s i t u by a low temperature RF d i s cha rge which can produce d e n s i t i e s above 1 0 1 2 c m - 3 . The p r o t e c t i v e f i l m i s a f f e c t e d by the d i s cha rge s , but the extent of the damage can be c o n t r o l l e d by an app rop r i a t e cho i ce of the r e p e t i t i o n r a t e and of the power i n the d i s cha rge p u l s e . Moreover, i f the l o c a l r i s e i n temperature i s not too h i gh , the f i l m w i l l q u i c k l y run towards the warm area and e q u i l i b r a t e the temperature. Temperatures below 4.2°K a re obta ined i n t h i s experiment by pumping on a bath of ' l i q u i d ^He i n which are immersed the sample and the re sona to r . The pot , which conta in s the l i q u i d he l ium bath , i s p laced i n s i d e a vacuum can and a ( c o n t r o l l a b l e ) vacuum i s e s t a b l i s h e d . Th is i s o l a t e s the rma l l y the inner pot from the e x t e r na l bath of l i q u i d he l ium which i s used f o r ma in ta i n i ng the superconduct ing magnet i n i t s superconduct ing s t a t e , and which i s a l s o used as a supply f o r the inner pot . Th i s magnet produces the s t a t i c magnetic f i e l d which can be as h i gh as 45 kG. 7-2. C ryos ta t The f o l l o w i n g d e s c r i p t i o n r e f e r s to f i g u r e s 7-1 and 7-2. The c r y o s t a t c o n s i s t s of two pots i n the r eg i on of the magnet, a l i q u i d he l ium v a l v e and a s e r i e s of tubes l e ad i n g to room tempera-t u r e , where connect ions are made to the e l e c t r o n i c s and the pumps. The pots are s imply two c l o sed aluminum c y l i n d e r s . The outer one f i t s n i c e l y i n the bore of the magnet ( i n f a c t , the bore of the shim c o i l dev ice ) and the inner one i s l a r g e enough to c o n t a i n a r a t he r l a r g e r e sona to r . Aluminum has been chosen as a c o n s t r u c t i o n m a t e r i a l because of i t s ve ry low magnetic s u s c e p t i b i l i t y which i s necessary i f we are to ach ieve a very h i gh f i e l d homogeneity. Each pot c o n s i s t s of a f l ange and a removable p a r t . Both f l ange s are r i g i d l y he ld together by most of the t u b i n g . Indium w i r e i s used as gaskets f o r s e a l i n g the removable par t to the f l a n g e . Brass screws a re used f o r making these j o i n t s s i n ce t h e i r thermal c o n t r a c t i o n i s s i m i l a r to aluminum, wh i l e a t the same time they have s t r eng th and a re reasonably non-magnetic. The procedure f o r making a s e a l w i l l be desc r ibed l a t e r . A l l permanent j o i n t s are epoxied w i t h S t yca s t 2850-FT whose thermal c o n t r a c t i o n i s matched to brass but which a l s o works w e l l w i t h aluminum. The sur faces of both the tubing and the aluminum had to be sand-b la s ted p r i o r to epoxying f o r r e l i a b l e bonding. There, are s i x s t a i n l e s s s t e e l tubes going through both f l ange s and an a d d i t i o n a l one going through the outer p o t ' s f l a n ge o n l y . The l a t t e r i s 3/8 i n . diameter tubing f o r making a vacuum between the two pots and terminates w i t h a SV8 v a l v e on the top of the c r y o s t a t . The c e n t r a l tube i s f o r pumping on the l i q u i d he l ium bath i n the inner pot . Th i s tub ing was o r i g i n a l l y V diameter a l l the way up to the top of the c r y o s t a t , but has now been enlarged over most of i t s l e n g t h i n order to i nc rease the pumping speed. I t i s covered i n the bottom w i t h a f l u x l i m i t i n g ho l low cap which serves to l i m i t the f l ow of the s u p e r f l u i d he l ium w h i l e a l l o w i n g f a s t pumping speed. Two of the remaining tubes, which are 3/16" i n d iameter , a re f o r the re sona to r . One conta ins a s t a i n l e s s s t e e l c o a x i a l c ab l e which c a r r i e s the s i g n a l and which can be moved up and down i n o rder to ad ju s t the c o u p l i n g . The other one conta ins a 1/8" diameter s t a i n l e s s s t e e l tube which can a l s o be moved up and down and which i s f o r the tun ing of the re sona to r . On the top f l a n ge , both of them a re he ld by an adjustment dev i ce which can move the c o a x i a l c a b l e , on the inner tube, up and down very smoothly. Another 3/16" diameter tube con ta i n s an o p t i c a l f i b e r which i s used f o r observ ing the d i s cha r ge . I t i s te rminated, a t the top, by a s e n s i t i v e l i g h t de tec to r (a MFPD100 diode) which i s connected to a p r e a m p l i f i e r . One of the tubes i s i n f a c t a c o a x i a l c ab l e which c a r r i e s the RF-pu l ses f o r the low temperature d i s cha rge . I t stops about 1" below the f l ange of the inner can (because the s t a i n l e s s s t e e l c o a x i a l cab le i s magnetic) and a copper c o a x i a l cab le i s connected to i t f o r the remain ing d i s t ance to the d i scharge c i r c u i t . F i n a l l y , there i s a short V ' d iameter tube which i s connected to the l i q u i d he l ium v a l v e and which a l l ows r e f i l l i n g of the inner pot w i t h l i q u i d he l i um. The v a l v e i t s e l f s i t s 3 inches above the outer c a n ' s f l a n g e . Care was taken to shor ten the tubes extending i n s i d e the inner can as much as p o s s i b l e , because of the f i e l d homogeneity problem. Moreover, t h i n w a l l tub ing was used wherever p o s s i b l e , i n order to reduce the heat l oad on the main l i q u i d hel ium bath and e s p e c i a l l y on the inner pot which has a sma l l c a p a c i t y . A s e t of seven w i re s are fed through both f l a n ge s , a l l o w i n g e l e c t r i c a l connect ions to the r e s i s t o r s i n s i d e . These feed throughs a re #18 copper w i r e about 1" l o n g . The v a r n i s h was removed a t each end of the w i r e s , which were then epoxied i n t o ho les through the f l a n g e s . On the upper f l a n g e , these w i re s are connected to a 9 -p in connector (Amphenol 126 s e r i e s Hex . ) , i n order to avo id f requent s o l d e r i n g . The l i q u i d he l ium va l v e i s b a s i c a l l y j u s t a s t a i n l e s s - s t e e l need le and a round ho le w i t h square edges. A s i n t e r element (#7) i s used as a f i l t e r to prevent s o l i d matter from en te r i ng i n t o the v a l v e and damaging i t . A l l the body, i n c l u d i n g the v a l v e sea t , i s made LIGHT DETECTOR OUTER CAN PU1PING LINE TUNING RF-DISCHARGE (INPUT) INDIUM SEAL mQMET Jl-He VALVE CONTROL £-He BATH PU1PIN G LINE SIGNAL AND COUPLING £-He VALVE BOTTOM PART OF THE CRYOSTAT F i gu re : 7 - l : O v e r a l l v i e r of .the c r y o s t a t . 68 [ * — 3" 0 17|" I » Tain pumping l i n e ndium s e a l Outer can Indium s e a l R e s t r i c t i o n F l u x l i m i t i n g s l eeve ' Heater Edge of the magnet -Inner pot L o c a t i o n of the resonator Center of magnet Epoxy j o i n t s F i g u r e 7-2: D e t a i l of the Bottom p a r t of the c r y o s t a t . out of b ras s . I t i s so ldered to the s t a i n l e s s s t e e l tube and i s h e l d r i g i d l y by the c e n t r a l tube i n o rder to prevent exces s i ve torque on the epoxy j o i n t a t the f l a n g e . A l l j o i n t s to the inner pot are vacuum t i g h t . In p a r t i c u l a r , the l i g h t de tec t o r and the a d j u s t i n g dev ices i n c l ude 0 - r i n g s e a l s . We p r e v i ou s l y mentioned one p recaut i on f o r l i m i t i n g the f l u x of s u p e r f l u i d he l i um. The short tube l e a d i n g to the l i q u i d he l ium v a l v e a l s o has a r e s t r i c t i o n i n the bottom f o r t ha t reason. More-over, both of the s l i d i n g p a r t s , which were desc r ibed above, have s leeves around them a t the bottom where they extend out of the tubes 7-3. The Resonator The geometry of the c r y o s t a t does not permit the use of the u sua l c o i l - c a p a c i t o r re sonato r , because the e x c i t i n g f i e l d has to be pe rpend i cu la r to the s t a t i c f i e l d . We t he re f o re decided to use the " S l o t ted -Tube Resonator " or STR (see Schneider e t a l . 1977). I t i s , b a s i c a l l y , j u s t a quar ter -wave length balanced t ransmi s s i on l i n e sho r t c i r c u i t e d a t one end. Th i s t r an sm i s s i on l i n e c o n s i s t s of an ou te r , " f i e l d l i m i t i n g " , c y l i n d r i c a l conductor and an inner s l o t t e d tube (see f i g u r e 7 -4 ) . This c a v i t y has two modes which are almost degenerate. The f i r s t one, which we do not want, i s the " e ven " mode which correspond to the normal mode of a c o a x i a l t r an sm i s s i on l i n e . The second one, which i s the de s i r ed one, i s the "odd." mode which y i e l d s a magnetic f i e l d i n the cent re of the s l o t t e d tube. In the even mode, the cu r r en t f l ows a long both "w ings " or "arms" of the s l o t t e d tube s ymmet r i ca l l y . A cu r r en t s t a r t i n g a t the f r e e end of the 70 s l o t t e d tube f lows down and then creeps a long the s h i e l d (see f i g u r e 7 - 3 - a ) . As we can see, t h i s mode i s e a s i l y e l im i na ted by removing the c y l i n d r i c a l s h i e l d . The odd mode i s very d i f f e r e n t . The c u r r e n t i n each arm f l ows i n o p p o s i t i o n . I f i t goes up i n one wing, then i t goes down i n the o the r . The ex i s tence o f t h i s mode does not depend on the presence of the c y l i n d r i c a l s h i e l d which i s there on l y f o r l i m i t i n g the magnetic f i e l d to a sma l l e r volume and f o r ma i n t a i n i n g a l a r g e Q f a c t o r (see f i g u r e 7 -3 -b ) . S ince the modes a re almost degenerate, i t i s d i f f i c u l t to coup le to one mode w i thout c oup l i n g to the o t h e r . In p r a c t i c e one-observes a m ix tu re of the two modes, i . e . there a re s t i l l two modes, but they are no longer pure. I t i s important to decouple the two modes as much as p o s s i b l e , not so much because the homogeneity of the r e s u l t i n g magnetic f i e l d i s a f f e c t e d , but because the f i l l i n g f a c t o r i s s i g n i f i c a n t l y reduced s i nce the even mode has most of i t s f i e l d between the s h i e l d and the s l o t t e d tube. Accord ing to Schneider e t a l . , the homogeneity of the r f -magnet i c f i e l d i s opt imized when the s l o t i n the inner tube obeys the r e l a t i o n -s h i p : S = 0.77 D, where S i s the s i z e of the s l o t and D the diameter o f the s l o t t e d tube. They a l s o suggest to use a s h i e l d w i t h diameter 2D. Many t r i a l s were made to ach ieve good decoup l ing of the two modes, and a convenient des ign was f i n a l l y reached, which o f f e r s the advan-tage tha t the resonator i s s imp le and compact. The f i r s t t h i n g to n o t i c e i s t ha t the inner tube i s not s l o t t e d a l l the way down. This w i l l have the e f f e c t of s h i f t i n g the frequency of the even mode w i t h r e spec t to the odd mode. Neg lec t i ng end e f f e c t s , we can make a crude es t imate of t h i s s h i f t by c on s i de r i n g the quarter -wave-l e n g t h f o r each mode, and t h i s g i ves a s h i f t of about 140 MHz. The tun ing i s done by moving a t h i c k m e t a l l i c (aluminum) c y l i n d e r between the inner and the outer tube. S i m i l a r l y , the coup l i ng i s done by moving the c o a x i a l c ab l e which c a r r i e s the s i g n a l . No t i c e , a l s o , t ha t the whole resonator i s made out of aluminum and that a minimum number of screws have been used f o r f i x i n g removable p i e ce s . The r f -magnet i c f i e l d i n the s l o t t e d tube v a r i e s as cos 2(2irx/?c) , where x i s the d i s t ance from the shor t i n the bottom of the s l o t t e d tube, and X i s the wavelength. Therefore , the " c e n t r e " of the resonator i s near the s ho r t . For reasons to be exp la ined i n the next s e c t i o n , there i s a ho le i n t h i s s ho r t . This has the e f f e c t of mod i fy ing the magnetic f i e l d i n that r e g i o n , w h i l e reduc ing the Q of the resonator s i n ce the cu r r en t i s compelled to f l o w i n the corners where the s l o t t e d tube meets the s ho r t . I t neve r the le s s should not be such a l i m i t i n g f a c t o r compared to the problems caused by the mix ing of the modes and the achievement of reasonably h i gh atom d e n s i t i e s . I t i s easy to o b t a i n a q u a l i t y f a c t o r of about 500 a t room temperature, and the f i n a l des ign ach ieves a Q above 700. This resonator has been designed and b u i l t c a r e f u l l y f o r maximiz ing i t s Q. For that reason, a l l su r faces were smoothed and p o l i s h e d . The resonator i s he ld by two supports , which a l l o w some a d j u s t -ment, and g reat care was taken i n the p o s i t i o n i n g of the center of the resonator w i t h re spect to the cen t re of the magnet. Furthermore, we est imated to what extent the va r i ou s p ieces of the assembly would c o n t r a c t on c o o l i n g , because t h i s can s i g n i f i c a n t l y change the 72 t 1 ] 1 , I 1 1 1 1 1 1 1 1 1 1 ] '. 1 a* 1 1 • k I 1 11 1 1 • (a) (b) F i gu r e 7-3: a) Current path f o r the even mode, b ) Current path f o r the odd mode. 73 4.8" -x. T 1.8" 1 3.50" 0.80" J ,Coaxial cab le for coupling ID . to the resonator X Capacitor l I i I < I I v. _/ Tuning cylindre Slotted tuSe •Shield Centre of resonator Cell Coil for RF—discharge •Coaxial cable for RF-discharge Figure 7-4: Resonator design, s e t t i n g . The t o t a l c o n t r a c t i o n turns out to be about 5 mm, and the cen t re of the magnet has to be l o c a t e d a t l e a s t t ha t amount above the de s i r ed cent re of the resonator which we chose to be 0 .6 " above the bottom of the s l o t t e d tube. 7-4. Low Temperature Discharge A low temperature d i s charge i s used f o r producing the atoms i n t h i s experiment. The d i scharge re sona to r , which operates a t 70 MHz, i s j u s t a c o i l and a c a p a c i t o r i n p a r a l l e l , and the c e l l f i t s i n the bore of the c o i l . Th i s c e l l conta in s a mixture of he l ium and molecu la r hydrogen. I t i s f i l l e d a t room temperature w i t h about h a l f an atmosphere of he l ium and h a l f an atmosphere of hydrogen gas. I t -was not a t r i v i a l matter to des ign the d i s cha rge resonator because i t has to work i n a r a t h e r i n ten se magnetic f i e l d and because i t has to s t a r t by i t s e l f w i thout e x t e r n a l t r i g g e r i n g source such as a Te l sa c o i l . I t was found i n the zero f i e l d experiment that t r a n s -former coup l i ng to t h i s resonator seemed to work w e l l . Th i s i s e a s i l y r e a l i z e d by wind ing about one t u rn of pr imary c o i l around the resonator c o i l which conta in s about ten tu rn s . We never the le s s have to watch f o r p o s s i b l e a r c i n g which can damage the c a p a c i t o r o r the c o i l . I t was a l s o v e r i f i e d , a t room temperature, t h a t f o r best performance the a x i s of the c o i l should be p a r a l l e l to the a x i s of the magnet, a c h a r a c t e r i s t i c which s u i t s us . The d i s cha rge c i r c u i t i s l o c a t e d beneath the 765 MHz re sonato r , i n a c lo sed m e t a l l i c c o n t a i n e r . We can see on f i g u r e 7-4 tha t a channe l , about 0.9" l o n g , separates the s l o t t e d tube from the d i s cha rge . This channel has two purposes. F i r s t , the c e l l can s l i d e i n , and atoms c reated by the d i scharge w i l l d i f f u s e to the prob ing r e g i o n . Second, i t i s w e l l beyond c u t - o f f f o r the RF of the d i s charge and helps to reduce i t s propagat ion i n t o the r e sona to r . The source of RF i s an Arenberg pulsed RF generator which possesses the u s e f u l f e a t u r e that the frequency can be v a r i e d over a wide range, which a l l ows us to use a f i x e d frequency d i s charge c i r c u i t . I t i s worth mentioning that a l l e l e c t r i c a l so ldered j o i n t s were made out of indium, a l though they were t i n - l e a d p r e v i o u s l y . Th is i s because the c r i t i c a l f i e l d f o r indium i s below 405 Gauss where our low f i e l d , homogeneity s e n s i t i v e , t r a n s i t i o n takes p l a c e . I t i s i n f a c t h i g h l y unde s i r ab l e to have a superconductor c l o s e to the prob ing r eg i on because of i t s diamagnetism. 7-5. Temperature Measurement and Con t r o l In t h i s experiment, c a l i b r a t e d r e s i s t o r s a re used f o r measuring the temperature and f o r sens ing the l i q u i d he l ium. One of them (120 nominal) i s l o c a t e d on one of the rods , which support the magnet, about 2 cm above the top of the magnet. I t serves two purposes. One, i t g i ves the ambient temperature i n the bore of the dewar. Two, i t i s u s e f u l f o r determin ing the l e v e l of the l i q u i d he l ium which should never be too c l o s e to the magnet. Another one (100 Q nominal) i s l o ca ted on the l i q u i d hel ium v a l v e . I t serves the same purposes as the other except that i t i s cm h i ghe r . A t h i r d one (210 Q nominal) i s p r e c i s e l y c a l i b r a t e d i n zero f i e l d and i s l o c a t e d i n s i d e the inner pot , above the r e sona to r . This one g i ves the temperature of the inner pot and a l l ows us to c a l c u l a t e the hel ium gas den s i t y i n the c e l l , n^ , s i n ce n^ i s p r e c i s e l y r e l a t e d to the temperature. Another p r e c i s e l y c a l i b r a t e d r e s i s t o r w i l l be added i n the inner pot f o r comparison purposes i n the next r u n . The temperature i s c o n t r o l l e d i n two ways. F i r s t , the c o o l i n g i s done i n the pot by pumping on l i q u i d he l ium. This should b r i n g i t down to almost 1°K. Second, a heater (90 fi a t 1°K) can be used f o r r a i s i n g the temperature or f o r s tudy ing the behaviour of the i nner pot under f i n i t e heat l o a d . This heater i s .003" brass w i r e f o l d e d , tw i s t ed and then wound around the outer su r face of the inner can. The w i re i s f o l ded i n order to avo id the i nduc t i o n of a magnetic f i e l d when the heater i s on. Armstrong A-12 epoxy; i s used f o r g l u i n g . 7-6. Indium Sea l The indium s e a l s , de sc r i bed above, a re made as f o l l o w s . The su r faces of both the f l ange and the removable can a re c leaned f i r s t w i t h t e t r a c h l o r o e t h y l e n e which removes the grease and pa r t of the o x i de . Then, a sma l l amount of Apiezon-M grease i s a p p l i e d to both s u r f a ce s , e s p e c i a l l y i n the corner where the indium w i re s i t s , to prevent s t i c k i n g to the aluminum. Two indium w i re s a re then wound around the j o i n t , one on top of the o the r , and the ends a re c ro s sed . One can now s l i d e the removable can i n the f l ange and screw i t t i g h t l y . 7-7. Exper imenta l Procedure The dewar, which we use i n t h i s experiment, i s a l a r g e volume s t a i n l e s s s t e e l dewar whose bore i s 9 inches i n diameter and whose depth i s 4 f e e t . I t t he r e f o r e has a l a r g e thermal i n e r t i a and a s p e c i a l procedure has been dev i sed f o r c o o l i n g i t . I t i s supported by a wooden rack which possesses a p u l l e y assembly a l l o w i n g r a i s i n g and lower ing of the dewar. The magnet i s supported by three rods which are screwed on an aluminum f l a n g e , which s i t s on the top of the dewar. F i n a l l y , the c r y o s t a t s l i d e s i n the midd le of that f l ange which has a l a r g e ho le i n i t s c e n t r e . The c r y o s t a t can be taken out independent ly from every th ing e l s e , even when the dewar i s mounted and c o l d . This i s an advantageous f e a t u r e which a l l ows m o d i f i c a t i o n s on the c r y o s t a t w h i l e the c r y o s t a t i s s t i l l c o l d o r w h i l e i t con ta i n s l i q u i d n i t r o g e n . The c o o l i n g procedure goes i n a few s teps . F i r s t , one f l u she s t he bore w i t h gaseous n i t r o g e n a few times and then t r a n s f e r s l i q u i d n i t r o g e n i n the l i q u i d n i t r o g e n j a c k e t . A f t e r s e ve r a l hours, the i n s i d e i s very c o l d and one adds l i q u i d n i t r o g e n to the c e n t r a l r eg i on of the dewar. This f i r s t step i s there on ly to avo id thermal shock and s t r e s s on the c r y o s t a t . One wa i t s aga in f o r about twelve hours and then s t a r t s to t r a n s f e r the l i q u i d n i t r o gen out of the dewar by app l y i ng pressure i n s i d e . Then, one pumps on the remaining l i q u i d n i t r o g e n and obta in s f u r t h e r c o o l i n g of the magnet. F i n a l l y , one can t r a n s f e r l i q u i d he l ium i n t o the dewar. The l i q u i d he l ium va l ve i s now open so tha t c o l d he l ium gas can f l ow i n and coo l the inner pot down. When the l i q u i d he l ium l e v e l i s h i gh enough, i t s t a r t s to f i l l the inner pot i f one a p p l i e s a s l i g h t under pressure i n s i d e . R e s i s t o r s a re used (see s e c t i o n 7-6) f o r d e t e c t i n g the l i q u i d he l ium l e v e l . There i s one on the v a l v e and one i n s i d e , above the re sona to r . The next step i s to pump on the l i q u i d he l ium bath i n the pot f o r c o o l i n g i t down to 1°K. 7-8. Performance Because of t h e i r f r a g i l i t y , i t took s e ve r a l t r i a l s before we succeeded i n a ch i e v i n g permanent epoxy j o i n t s , but the c r y o s t a t can now be c y c l ed w i thout any mechanica l problem. However, we cou ld not lower the temperature below 1.15°K by pumping on the l i q u i d he l ium bath, and there are a few reasons f o r t h i s . We have est imated a maximum heat l oad of 30 mW. This i s f a i r l y s m a l l , but c o n t r i b u t e s to the l i m i t a t i o n of the c o o l i n g . The c reep ing s u p e r f l u i d he l ium i s the source of another heat l e a k , and t h i s i s why we have added p iece s wh ich l i m i t the f l ow of super-f l u i d to the o u t s i d e . We a l s o est imated the pumping speed and we can conclude t h a t , f i r s t , the r e s t r i c t i o n i n the bottom of the pumping l i n e , which p r e v i o u s l y was made out of t e f l o n , added too much impedance on the pumping l i n e , and, second, the l i n e i t s e l f was too s m a l l . A l though the main problem was the r e s t r i c t i o n , we decided to modify both of them. We have not yet t r i e d the c r y o s t a t w i t h these improvements., A l though we cannot be d e f i n i t e a t t h i s p o i n t , i t seemed that the l i q u i d Helium va l v e was behaving normal ly but tha t there was i n t e r m i t t e n t leakage through the s e a l . However, i f there was a l e a k , i t was ve ry sma l l most of the t ime and d i d not pe r tu rb the experiment. The low temperature d i s charge worked w i thout any d i f f i c u l t y , even at low power. We observed a narrow resonance frequency (about 1 MHz) which i n d i c a t e s a h i gh Q. The l i g h t de tec to r was e a s i l y d e t e c t i n g the s i g n a l from the d i scharge even i f the geometry was not opt imized i n t ha t r e spec t . We cou ld observe the e f f e c t of the d i s charge i n t e n s i t y , e i t h e r more power or f a s t e r r e p e t i t i o n r a t e , on the temperature i n the i nne r pot which s t a r t e d to r i s e s l ow l y f o l l o w i n g an i n c rea se i n i n t e n s i t y . As expected, the Q of the 765 MHz resonator i nc reased as the temperature was lowered, and we measured a Q of about 800 a t 4.2°K. We hope that the new resonator w i l l o f f e r a Q above 1000 a t t h i s same temperature. The tun ing and the coup l ing o f f e r e d no d i f f i c u l t y . We d i d not see a s i g n a l , but t h i s was not a complete s u r p r i s e because one r equ i r e s the best exper imenta l c ond i t i on s which we d i d not have. We never the le s s hope that we w i l l be ab l e to de tec t the hydrogen atoms i n the next run, once a l l the necessary m o d i f i -c a t i o n s have been made. APPENDIX A In t h i s experiment, i t i s e s s e n t i a l to ach ieve a s i g n a l to no i s e r a t i o l a r g e r than one f o r a s i n g l e f r e e i nduc t i o n decay, because s i g n a l averag ing may not be p o s s i b l e (Chapter S i x and Fou r ) . Many f a c t o r s c o n t r i b u t e to l i m i t the S/N r a t i o , such as the no i se f i g u r e of the spectrometer, the Q and the f i l l i n g f a c t o r s of the re sonato r , the d e n s i t y of hydrogen atoms and the homogeneity of the magnet. In t h i s appendix, we s h a l l be concerned with,.the magnet homogeneity. The S/N r a t i o i s i n v e r s e l y p r o p o r t i o n a l to the homogeneity of the magnet. Indeed, an inhomogeneous magnetic f i e l d w i l l induce a superimposed T^ mechanism which w i l l a c c e l e r a t e the decay of the s i g n a l (Chapter Four) and t h i s compel l s us to use a wider frequency band f o r observ ing the s i g n a l , thereby reduc ing the s i g n a l to no i s e r a t i o . Furthermore, t h i s f a s t e r decay of the s i g n a l quenches the r e l a x a t i o n mechanisms which we want to study. For these reasons, i t i s a b a s i c requirement to achieve a h i g h l y homo-geneous s t a t i c magnetic f i e l d . The superconduct ing magnet used i n t h i s experiment has a r a ted r e s o l u t i o n of 1 pa r t s i n 1 0 7 f o r a 5 mm sample tube. This quan t i t y has been c a l c u l a t e d u s ing the h a l f w i d t h of the l i ne shape of a magnetic resonance s i g n a l . However, the manufacturer obta ined a s t r ong l y peaked l i ne shape w i t h s u b s t a n t i a l wings and used i t f o r c a l c u l a t i n g the r e s o l u t i o n . Such a l i ne shape i s equ i va l en t to having a h i gh homogeneity i n a sma l l reg ion around the cent re of the magnet and a poor homogeneity o u t s i d e . Our aim i s , however, to ob t a i n h i gh homogeneity over a l a r g e r volume, and t h i s cannot be achieved by a d j u s t i n g the f i r s t order shim c o i l s b u i l t i n t o the magnet. We can improve on the homogeneity i n two ways: e i t h e r by averaging the f i e l d by sp inn ing the sample around the main a x i s , or by shimming the f i e l d u s ing h i ghe r - o rde r c o r r e c t i o n c o i l s appro-p r i a t e l y des igned. The former approach i s not p r a c t i c a l f o r our gaseous samples, and we s h a l l adopt the l a t t e r approach. PRINCIPLE OF THE SHIM COILS , Any f i e l d can be expressed mathemat ica l l y i n terms of a complete set of f u n c t i o n s , each one of which has a s p e c i f i c symmetry. I n i t i a l l y , the f i e l d i n a h igh r e s o l u t i o n magnet i s f a i r l y homogeneous and impe r fec t i on s to i t a re u s u a l l y very s m a l l . In p r a c t i c e , when one expands the observed d e r i v a t i o n i n a s e r i e s , one f i n d s that the importance of a term goes i n v e r s e l y w i t h i t s o r de r . The i dea of shimming the f i e l d r e l i e s on the f a c t t h a t we can e l i m i n a t e the f i r s t few terms by adding f i e l d s of the same symmetry but oppos i te s i g n , l e a v i n g on ly the homogeneous term p lus terms of h igher o rder which should be weaker. I t i s not p r a c t i c a l to add c o r r e c t i n g f i e l d s f o r each s p a t i a l component of the f i e l d . For example, t h i s would r e q u i r e n ine c o i l s j u s t f o r the f i r s t order s e t . What i s u s u a l l y done i s to c o r r e c t f o r the a x i a l or z-component of the f i e l d , i n which we a re i n t e r e s t e d , i f the main i s a long t h a t a x i s . We can j u s t i f y t h i s cho i ce i n two ways. On the one hand, one can argue that c o r r e c t i n g f o r one component s u f f i c e s to b r i n g the other two in the right d i r e c t i o n . On the other hand, one can compare the importance of the r a d i a l components of the imperfections to the a x i a l one, and find that the l a t t e r contributes more to the homogeneity than the former. To see this arguement, l e t us write the actual f i e l d i n the bore of the magnet as B'='B + B', where ->• ~ ->-B q i s the perfectly homogeneous f i e l d along z, and B' i s a small imperfection to i t . Thus, remembering that B' << B Q , |b| = IB + B * l = [ ( B + B') 2 + B' 2 + B ' 2 ] ^ 1 1 1 o 1 o z y x = B + B' + B'2 + B'2 x o ' z 2B o Thus, components B^ and B^ do not contribute s i g n i f i c a n t l y to the f i e l d inhomogeneity compared to B' and i t suffices to shim only z > the a x i a l component. Furthermore, i f one cannot achieve a f i e l d of a pure symmetry, or order, one can at least design a c o i l which contains that desired term as the dominant one, plus higher order terms among which one t r i e s to eliminate the f i r s t few. In other words, the shiming device consists of a properly designed c o i l which possesses the desired "term" plus some high order terms which we do not want, of course, but which cannot be eliminated. ZEROTH, FIRST AND SECOND ORDER COILS The general solution for the potential around the o r i g i n i s : V = • Z m E n rn P m (cos0){A m cos m<|> + B™ s i n m<J>}, n=l m=o n n n where A u l and B1" a re constants and p m i n the n th order a s soc i a ted n n n Legendre po l ynomia l . Th i s g i ves f o r B^: 3V 5 9 m=n n _ i m m + l B = - 4r = ^ £ r [(n-m)cose P (cos6) + s ine P ( co se ) J . z 8 z n=l m=o n n •:[Am cos (md>) + B™ sin(md>)], n n which, i n C a r t e s i a n coo rd i na te s , becomes: B z = A° + 2A°z + 3A^x + 3B^y + | A ° ( 2 z 2 - x 2 - y 2 ) + 12A*zx + 12B*zy + + 1 5 A3 ( x 2 - y 2 ) +30B 2 xy + 4 A ° z [ z 2 - | ( x 2 +y 2 ) ] A j x ( 4 z 2 - x 2 - y 2 ) + X B 4 y [ 4 z 2 - * 2 - y 2 ] + ••• (see Anderson 1961). Th is i s our ba s i c equat i on . We note the important property tha t we have a s e r i e s of independent terms. The f i r s t term i s s imply the homogeneous f i e l d i t s e l f , a cons tant , which we want to keep. The f o l l o w i n g three terms, x , y and z , a re the f i r s t order imper-f e c t i o n terms which can be co r rec ted f o r by the c o i l s bear ing the same name, namely the x, y and z shim c o i l s . These are a l ready a v a i l a b l e i n the magnet, and w i l l not be d i scus sed any f u r t h e r . The second order terms which a re f i v e i n number a r e : 2 z 2 - x 2 - y 2 (des ignated by z 2 be low), x z , y z , xy and x 2 - y 2 . We s h a l l now look a t a l l these f i v e terms and how we c o r r e c t f o r them. Z 2 - SHIM COIL This i s the s imp le s t one and i t i s d i scussed f i r s t because i t e x e m p l i f i e s the method q u i t e w e l l . Together w i t h the z - c o i l , i t i s termed an a x i a l c o r r e c t i o n term because of i t s s p e c i a l symmetry. A z 2 (o r , r a ther , 2 z 2 - x 2 - y 2 ) term appears i n the s e r i e s expan-s i on of the f i e l d c reated by a c i r c u l a r loop whose symmetry a x i s i s a long the z - a x i s . P r a c t i c a l c on s i de ra t i on s d i c t a t e i t s use, because i t i s s imple and c i r c u l a r . In f a c t , i t i s very easy to wind a w i r e around a c y l i n d e r and i t i s mechan ica l l y s t r ong . F u r t h e r -more, we want to take as l i t t l e room as p o s s i b l e f o r the c o i l s , and that they are l y i n g c l o s e to the b o r e ' s su r face i s g reat advantage. The o rde r i n g of loops which a l l ows one to e l i m i n a t e up to the f o u r t h order term, keeping on ly the z 2 dependence, i s shown i n F i g . A - l - b . I t c o n s i s t s of f ou r l o op s , arranged s ymmet r i ca l l y about the x y -p l ane . Equal cu r ren t s a re f l ow ing i n the two outer loops and s i m i l a r l y f o r the two inner l oop s , even though i t i s not the same i n both set s of l oop s . This way, we ga in r e f l e c t i o n symmetry. The z component of a f i e l d due to a c i r c u l a r r i n g , whose p r i n c i p a l a x i s i s a long the z - a x i s and whose cent re i s l o ca ted a t z^, i s g iven by: -u i s i n 2 a m n-1 B ( r ,6 ) = — 2 - T Z (-) P ' ( co sa ) P , (cos6) . ( A - l ) z 2a n=X c n n-1 (Sauzade and Kan 1973, p. 45 ) . In t h i s expres s i on (the equat ion i s i n SI u n i t s ) , i i s the cu r ren t c i r c u l a t i n g around the l oop . The other parameters a re i l l u s t r a t e d 85 a) T a L z=z b) x 0.3 a = 0.482" (0.437" - 0.527") 1.2 a = 1.950" (1.800" - 2.100") F i g u r e A - l : a) D e f i n i t i o n s of the parameters f o r the d e s c r i p t i o n of a f i e l d c reated by a l oop . 2 b) Z -sh im c o i l c o n f i g u r a t i o n . In paratheses are g i ven the a c t u a l span of the w i r i n g , co r re spond ing to the w i d t h . i n F i g . A - l - a . The r e f l e c t i o n symmetry enables us to e l i m i n a t e the terms which change s i gn when z^ goes to ~ Z Q - l n o ther words, on ly the even terms remain, namely terms con ta i n i n g P , V^, Of those, we want to keep P^CcosO), which i s j u s t : k ( 3 c o s 2 0 - l ) = - \ ( 2 z 2 - x 2 - y 2 ) 2 r 2 The term compr i s ing P^ ( c o s 0 ) ( i . e . , n=5) van i shes i d e n t i c a l l y when cosa i s one of the r oo t s of P^ ( x ) . These r oo t s a re : x = ± 0.77, ± 0.29, l ead i ng to a = 4 0 . 1 ° , 73 .4° . Conven ient l y , the re are two se t s of l oops , each t a k i n g one of these ang les . In terms o f the z - p o s i t i o n of the cen t re of the loop these correspond r e s p e c t i v e l y to : 1.19a and 0.30a. The constant term, con ta i n i n g P q (COS0 ) , goes away a f t e r an app rop r i a t e cho i ce of the cu r r en t i n t e n s i t i e s . I f 1^ and I are the cu r r en t f l ow ing i n the i nne r and the outer loop r e s p e c t i v e l y , and s i m i l a r l y f o r t h e i r angles and otQ, then we need to f u l f i l l the f o l l o w i n g c o n d i t i o n : u I . s i n 2 a . u I s i n 2 a O 1 1 ^ O O O , f s P 1 ( c o s a ± ) = r a P 1 ( c o s a o ) o r , a r rang ing the terms: 87 I. 1 s i n2 a P ' ( c o s a ) 6 1 o . s i n a I o - = - 0.30. s i n a. P ' ( co sa . ) i 1 i s i n ' ' a . The r e l a t i v e cu r ren t i s ad jus ted by p i c k i n g the app rop r i a te r a t i o f o r the number of t u r n s . A c t u a l l y we have to run w i re s from one loop to the other i n order to feed them us ing on ly one c i r c u i t . Th i s i s done e a s i l y i f one used the same path f o r cu r ren t s of oppos i te d i r e c t i o n s and equal i n t e n s i t y . The topology a l l ows i t and t h i s i s t r ue a l s o f o r the next f ou r c o i l s . The on ly problem might be to reduce as much as p o s s i b l e the number of such w i r e p a i r s . ZX AND ZY COILS These two c o i l s go together and they are r e l a t e d by a 90° r o t a t i o n around the z - a x i s . Thus i t s u f f i c e s to ana ly se one of them, say xy. Here, we use c i r c u l a r a rc s as dep i c ted i n F i g . A - 2 - a , which a l s o f i t on a c y l i n d r i c a l base. The f i e l d c reated by a c i r c u l a r a r c i s very comp l i ca ted , and i t s form, i n c l u d i n g terms up to the f o u r t h o rde r , i s g iven i n Sauzade and Kan, p. 48-49 (Sauzade and Kan 1973), but there are some mistakes i n the expres s ion g i v e n . S tack ing e i gh t of these a rc s and app l y i ng c u r r e n t s as shown i n F i g . A -2 -b , we ga in symmetries and most of the terms drop out . We s h a l l summarize how i t proceeds. The terms not changing s i gn when Z q -> z^ are e l im i na t ed because of the r e f l e c t i o n ant i - symmetry. Therefore , up to the f o u r t h o rde r , the terms l e f t a r e : z , x z , z 3 , x 2 z , y 2 z , x z 3 , x 3 z 88 z 2 = 3.12 a = 4.867" (4.807" - 4.927") z,= 0.68 a = 1.061" (1.021" - 1.101") F i g u r e A -2 : a) Parameters of the a rc which produces a f i e l d around z^. b) ZY and ZX shim c o i l s c o n f i g u r a t i o n . Here i s g iven the c o n f i g u r a t i o n f o r ZX,but a r o t a t i o n by 90 g i ve s the ZY c o n f i g u r a t i o n . I n paratheses a re g i ven the w id th of the w i r i n g . and x y z z . S ince we have r e f l e c t i o n symmetry i n the zy p lane, on ly those terms remain which do not change s i gn as i + i , and x .-»- - x . Th is reduces the se t t o : x z , x z 3 , x 3 z and x y 2 z . F i n a l l y , t ak i ng 0 = 60° , a l l th ree remaining f o u r t h order terms a re p o r p o r t i o n a l to the po l ynomia l : 4Z 4 - 4 l z 2 a 2 + 18ah, o o which van i shes f o r z = ± 0.68a or ± 3.13a. o The cu r r en t s i n a l l branches are then equa l . The cu r ren t s running v e r t i c a l l y c ance l each other by p a i r s a long the z - a x i s , l e a v i n g on l y a very weak f i e l d a t the o r i g i n . XY AND X 2 - Y 2 SHIM COILS Once aga in , they are r e l a t e d to each other by a r o t a t i o n , the r o t a t i o n angle being 4 5 ° . We s h a l l then ana lyse on l y one of them: 2 2 We a l s o make use of c i r c u l a r a r c s , but the arrangement i s more comp l i ca ted . The s t ack i ng and the d i r e c t i o n s of the cu r ren t s a re shown i n F i g . A -3 . Summarizing, the r e f l e c t i o n symmetry about the xy p lane leaves o n l y : 1, x , x 2 , y 2 , x 3 , x z 2 , x y 2 , , x 4 , y 1*, x 2 z 2 , x 2 y 2 , and y 2 z 2 . S ince we have oppos i te a rcs about the zy p l ane , on ly terms not changing s i gn when x -> - x s tay ; so x , x 3 , x z 2 and x y 2 d i s appea r . App ly ing a s i m i l a r procedure f o r the arcs oppos i te about the z - x p lane (exchanging i f o r - i , x f o r y and y f o r -x) we get as i 9 2 U U U 9 9 9 9 9 9 remaining terms: 1, y , x , z > y » x , y z , y x , x ' z ^ . 90 F i gu re A -3 : XY and X^-Y^ shim c o i l c on f i gu r a t i on .He re i s g i ven the 2 2 c o n f i g u r a t i o n f o r XY,and the X -Y c o n f i g u r a t i o n i s ob-ta i ned by a 90 r o t a t i o n aroud the z - a x i s . I n parentheses are g i ven the a c t u a l w id th of the w i r i n g . 91 We see t h a t , because we changed i f o r - i i n going from the set a long x to the set of a rc s a long y a x i s , a l l common terms w i l l c ance l i d e n t i c a l l y , namely 1, z 4 and y 2 x 2 , l e a v i n g the d i f f e r e n c e between the other ones. The unde s i r ab l e terms, x^-y^ are x 2 z 2 - y 2 z 2 , are both p r o p o r t i o n a l to the po l ynomia l : 1 2 z 2 - 4 6 z 2 a 2 + 40a o o which van i shes f o r z = ± 1.93 a or ± 0.335a. o F i n a l l y , the f a c t o r s i n 28 m u l t i p l y i n g the remaining x 2 - y 2 term i s maximized when 6 = 4 5 ° . This i s the angle chosen he re . The v e r t i c a l w i re s a l s o have very l i t t l e e f f e c t he re . SHIM COILS AS A PRACTICAL DEVICE" I n v e s t i g a t e the Problem In des i gn ing t h i s d e v i c e , we had to con s ide r a set of c o n s t r a i n t s because we want to keep as many as p o s s i b l e of the advantageous f ea tu re s of t h i s magnet. The magnet has, f o r i n s t an ce , a r a t h e r l a r g e bore r ad iu s (3 V ) , and we wanted to ma in ta in t h i s c h a r a c t e r i s t i c . The shim c o i l s de sc r i bed above have a s u i t a b l e geometry to meet t h i s requi rement. Thence we used a p i e ce of aluminum p ipe which t i g h t l y f i t i n the bore and which has an inner diameter of 3 inches . Channels were grooved on the ou t s i de f o r accommodating the w i r e s . The cho ice of the m a t e r i a l i s a l s o important here because of the magnetic i m p u r i t i e s one might f i n d i n i t . From C u r i e ' s Law we know that these i m p u r i t i e s show up very s t r o n g l y a t low tempera-t u re s . I t seems tha t the best cho ices a re e i t h e r aluminum or copper, a l b e i t aluminum i s supposedly b e t t e r . In p r a c t i c e , both of these m a t e r i a l s a re found as a l l o y s and one must l ook a t the compos i t i on . In s p i t e of t h i s , and f o r some unexpla ined reasons, aluminum seems to quench the e f f e c t of the magnetic i m p u r i t i e s i n i t . Moreover, s i n ce the frame of the magnet i s a l s o made of aluminum and s i nce there i s a t i g h t f i t between the l a t t e r and the dev i ce , i t i s p r e f e r a b l e to use the same composite m a t e r i a l f o r a vo id i ng c o n t r a c t i o n problems. F i ve shim c o i l s have to s i t i n very l i t t l e space, s i nce the support ing c y l i n d e r has a w a l l t h i cknes s of on ly 1/8". F u r t h e r -more, the shape of the c o i l s compel ls us to superimpose them one on top of the o the r , over laps being unavo idab le , and t h i s i s not an easy task . F i n a l l y , we can e i t h e r run the c o i l s c on t i nuou s l y , feed ing cu r r en t through them from the ou t s i de a l l the t ime, or use super-conduct ing w i re s and leave them i n t o p e r s i s t e n t mode when the r equ i r ed cu r ren t i s f l ow ing i n . We chose the second s o l u t i o n , expect ing a reduced b o i l o f f of l i q u i d he l i um, a l though t h i s technique i s more comp l i ca ted . For i n s t ance , we had to make use of superconduct ing switches which we s h a l l d e s c r i be below. Design A ske tch of the dev ice i s g iven i n F i g . A-4. I t i s s i t t i n g on top of the magnet which, i n pa s s ing , i s used ups ide down. This way, switches and w i re s are f u r t h e r away from the cen t re of the magnet, reduc ing pe r t u rba t i on s to the f i e l d a t that s i t e . The aluminum tubing covers the whole l e n g t h of the 93 7.25 17.375 k - 3 . 2 5 " * l _ * | 3.00" ( < -Y Epoxy j o i n t s 1.625" T Switch © + C M . ^ Sec t i on c on t a i n i n g c o i l s © P r o t e c t i v e Box 6.0" F i gu re A-4: Shim c o i l d e v i c e . A schematic drawing of the aluminum i n s e r t c on t a i n i n g the shim c o i l s and the superconduct ing sw i t ches . C M . i s the cen t re of the magnet. magnet, ma in ly because some c o i l s a re ve ry extended. This p ipe has been th inned down a l l a long to f i t n i c e l y i n the bore and an e x t r a l a y e r (^0.01") has been taken o f f i n the reg i on where the c o i l s s i t , a l l o w i n g f u r t h e r coverage of the work. The superconduct ing w i re used f o r the c o i l s i s the same as the one i n the magnet. I t thus has the same fea tu re s and, f o r i n s t ance , h i gh enough a c r i t i c a l f i e l d . The d iameter , i n c l u d i n g c o a t i n g , i s 0.010", and, i n p r i n c i p l e , f i v e l a y e r s cou ld f i t one on top of the other wi thout any problem. The w i re s s i t i n an a r ray of grooves which have a s p e c i a l depth f o r each c o i l . The grooves are coated w i t h a double s i de tape to he lp i n the w ind ing . S ince the c o i l s have to ove r l ap we arranged the order to ach ieve bes t mechanica l s t a b i l i t y . A p i e ce of aluminum f o i l was wrapped around the c o i l s , p r o v i d i n g f u r t h e r s t r eng th and p r o t e c t i o n . We should mention here that we have to wind many wi red turns i n o rder to get the app rop r i a te cu r ren t r a t i o and i n t e n s i t i e s , thereby adding w id th to a l l l i n e s of the skim c o i l s . Th i s reduces the accuracy of the c o r r e c t i o n , a l though by a very sma l l amount. The e l e c t r i c a l connect ions and the switches are l o c a t e d i n a sma l l box around one end of the c y l i n d e r . The superconduct ing w i re s are spot-welded together a f t e r removal of the copper l a y e r w i t h n i t r i c a c i d . For convenience, a l l connect ions to w i re s going up to the top of the dewar are done v i a a 9 -p in connector , on the s i de of the box. The c u r r e n t feed ing of the c o i l s i s c o n t r o l l e d by superconduct ing switches which we s h a l l d e s c r i be i n the next s e c t i o n . Most p ieces are made of aluminum, even the screws, and the permanent j o i n t s a re done w i t h epoxy (Armstrong A -12 ) . Switches The use of superconduct ing w i re s prov ides us w i t h the p o s s i -b i l i t y of going i n t o p e r s i s t e n t mode. This procedure r equ i r e s superconduct ing sw i tches . The i r p r i n c i p l e i s s imp le . Above a c r i t i c a l temperature, T , a superconduct ing m a t e r i a l becomes normal, t ha t i s i t becomes a s imple conductor w i t h a f i n i t e c o n d u c t i v i t y . M a t e r i a l s can be found w i t h T^ a few degrees above 4.2°K, thereby r e q u i r i n g a sma l l amount of heat to change t h e i r s t a t e (or mode). As shown i n F i g . A - 5 - a , a sma l l l e ng th of superconduct ing w i r e i s s h o r t -c i r c u i t i n g a c o i l (one of the shim c o i l s ) and a heater i s next to i t . The whole assembly i s immersed i n l i q u i d Hel ium, keeping the sma l l w i r e i n i t s superconduct ing s t a t e . A cu r ren t can f l ow f r e e l y around the c o i l and t h i s sma l l w i r e and t h i s corresponds to the p e r s i s t e n t mode. Converse ly , when a l i t t l e b i t of heat i s r e l e a s e d , f o l l o w i n g the a p p l i c a t i o n of a v o l t a g e , the sma l l w i re becomes normal and the cu r ren t s can r e l a x s i n ce the c o i l i s now i n s e r i e s w i t h r e s i s t i v e elements. I t can thus be adjus ted from the o u t s i d e . Turning the heater o f f j u s t leaves the s e l e c ted cu r r en t i n the c o i l . In p r a c t i c e , . w e have to make sure that the heat ing i s e f f i c i e n t , w i thout b o i l i n g o f f too much he l ium. What we need, i n f a c t , i s to reach a peak temperature above T^ i n a r a the r sma l l r e g i o n . On the other hand, the c o o l i n g must be f a s t enough i n going back to the 96 — Superconducting w i r e — Copper w i r e w 1 Heate 200 C i 1 13 C o i l Switch F i gu re A -5 : a) Schematic drawing of a superconduct ing s w i t c h , b) P h y s i c a l aspect of a s w i t c h . p e r s i s t e n t mode. What we d i d i s to wind a ( tw i s ted ) superconduct ing w i r e , rough ly 20 cm l ong , around a meta l g l a ze r e s i s t o r (about 200 ohms) which keeps a constant r e s i s t i v i t y when the temperature i s lowered. Note that the copper l a y e r on the superconduct ing sw i t ch w i r e was p r e v i o u s l y removed so tha t the normal s t a t e r e s i s t a n c e of the w i r e i s l a r g e . The r e s i s t o r was then enclosed i n a t e f l o n capsu le , and the remaining space was f i l l e d w i t h p a r a f f i n wax (see F i g . 4 - 5 -b ) . The power r equ i r ed to t u rn the sw i t ch on i s on ly 0.3W and the sw i t ch i ng time i s of the order of a second. C i r c u i t r y A l l f i v e c o i l s are wound i n s e r i e s and the cho i ce of the c o i l , which one wants to change the cu r r en t f l ow ing i n , i s done by the sw i t ch se l ec ted to be turned on ( F i g . A-6) . That way, on ly two w i re s a re necessary to c a r r y down the c o i l s ' c u r r e n t s . This i s important s i n ce these w i re s must c a r r y about 1 amp and t he re f o re have to be made of copper. This procedure has the inconvenience tha t one cannot r a p i d l y go from one c o i l to the o the r , or even ad ju s t two c o i l s a t the same t ime . This would have been an advan-tage because second order c o i l s make very f i n e adjustments which are hard to observe. S ince the cu r r en t f o r the heater i n the sw i t ch i s sma l l (40 mA), we used #32 brass w i re i n order to reduce the heat load on the he l ium bath . 98 IN OUT COM. Z ZX ZY XY X -Y F i gu re A-6: C i r c u i t r y of the c o i l s and the sw i t ches . PERFORMANCES Using B^O and probes, designed and made by W.N. Hardy and B.W. S t a t t , we were ab l e to study the i n f l u e n c e of the shim c o i l s on the f i e l d homogeneity. As a l ready mentioned, they have very l i t t l e e f f e c t , and o p t i m i z a t i o n of the f i e l d homogeneity i s r a t he r t r i c k y . This i s a c t u a l l y a good a p p l i c a t i o n of the t r i a l and e r r o r method, and i t takes hours to reach the optimum. In p r i n c i p l e , each c o i l i s independent from the o the r . However, on a p r a c t i c a l b a s i s , they induce cu r ren t s i n each other every time we change the va l ue of one of the c u r r e n t s . The mutual inductance between the second order c o i l s i s n e g l i g i b l e but the f i r s t o rder c o i l s do induce a cu r r en t i n them. Neverthe-l e s s , the procedure i s i t e r a t i v e i f one s t a r t s w i t h the f i r s t order ones, the cu r ren t s induced being ve ry sma l l from the second to the f i r s t o r d e r . Another i n t e r e s t i n g , though not d e s i r a b l e , phenomenon occurred when the cu r r en t was inc reased up to about one ampere i n the second order shimming c i r c u i t . We observed tha t the c i r c u i t r y i n the p r o t e c t i v e box cou ld induce a n o t i c e a b l e f i e l d a t the cen t re of the magnet, which u s u a l l y went back to zero when the cu r r en t was decreased. Sometimes, u n f o r t u n a t e l y , i t d i d not go back to ze ro , c e r t a i n l y because the magnetic f o r ce s had moved the w i re s around. We s h a l l mention tha t these shimming c o i l s a re ve ry s e n s i t i v e to the " c h o i c e " of the cent re of the magnet (o r , e q u i v a l e n t l y , the sample). This i s p a r t i c u l a r l y t rue of the z z - s h i m which of a l l the s h imco i l s has the g rea te s t i n f l u e n c e onto the f i e l d . We cou ld not thoroughly check the e f f e c t of each c o i l on the magnetic f i e l d . However, every c o i l ac t s i t s own way, but on ly z z showed up c l e a r l y the e f f e c t which i t has on the f r e e i n d u c t i o n decay (FID). The z 2 - s h i m g ives bu lk to the f i r s t p a r t of the FID w h i l e dep l e t i n g the end of i t (see F i g . A -7a ) . Th is i s a good f e a t u r e because the observed tendency o f the FID was a f a s t decay i n the f i r s t few m i l l i s e c o n d s f o l l owed by a ve ry long slow decaying t a i l (see F i g . A -7b) . A l though the other shim c o i l s brought s l i g h t improvement i n the FID, t h e i r s p e c i f i c r o l e cou ld not be so r ted out . The best T^ which we obta ined was t y p i c a l l y 80-100 msec, which can be cons idered u n s a t i s f a c t o r y and there seemed to be spur ious T 0 mechanisms which s h o r t - c i r c u i t e d the i n t r i n s i c one. F i g u r e A -7 : a)Envelope of a fjjee i n d u c t i o n decay which shows the e f f e c t of the z shim. b ) I n i t i a l f r e e i n d u c t i o n decay w i thout second order sh iming. REFERENCES Abragam 1961: A. Abragam, The P r i n c i p l e s of Nuclear Magnetism, Oxford U n i v e r s i t y Press (1961). Anderson 1961: W.A. Anderson, Rev. S c i . In s t rum., 241 (1961). B a l l i n g e t a l . 1964: L.C. B a l l i n g , R . J . Hanson and F.M. P i p k i n , Phys. Rev., 133, A607 (1964). B e r l i n s k y 1979: A . J . 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Schneider et al. 1977: H.J. Schneider and P. Dullenkopf, Rev. Sci. Instrum., Vol. 48, #1, (1977). Silvera and Walraven 1980: I.F. Silvera and J.T.M. Walraven, Phys. Rev. Lett. 44_, 164 (1980) . Slichter 1978: CP. Slichter, Principles of Magnetic Resonance, 2nd Ed., Springer-Verlag, N.Y. (1978). Statt and Berlinsky: B.W. Statt and A.J. Berlinsky (submitted). Walraven et al. 1980: J.T.M. Walraven, I.F. Silvera and A.P.M. Matthey, Phys. Rev. Lett. 4_5, 449 (1980). Whitehead 1979: L.A. Whitehead, Magnetic Resonance Studies of Atomic Hydrogen Gas at Liquid Helium Temperatures, M.Sc. Thesis (1979). Winkler et al. 1972: P.F. Winkler, D. Kleppner, T. Myint and F.G. Walther, Phys. Rev. A, 5, 83 (1972). Zitzewitz 1970: P.W. Zitzewitz, Harvard Ph.D. Thesis (1970); also Rev. Scient. Instrum. 41, 81 (1970). 

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