UBC Theses and Dissertations

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UBC Theses and Dissertations

Many-quantum transitions in the conduction electron spin system of lithium metal. Koss, Terry A. 1968

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( i ) . M A N Y - Q U A N T U M T R A N S I T I O N S I N T H E C O N D U C T I O N E L E C T R O N S P I N S Y S T E M O F L I T H I U M M E T A L by T E R R Y K O S S B . S c , U n i v e r s i t y o f W a s h i n g t o n , 1 9 6 6 A T H E S I S . - . S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n t h e d e p a r t m e n t o f P H Y S I C S We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d . T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A J u l y 1 9 6 8 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h its r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f P h y s i c s The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e J u l y 24, 1968 ( i i ) ABSTRACT Resonance a b s o r p t i o n s p e c t r a have been o b s e r v e d which may be i n t e r p r e t e d as many-quantum t r a n s i t i o n s where the a x i s o f q u a n t i z a t i o n i s a l o n g the e f f e c t i v e magnetic f i e l d i n the r o t a t i n g frame. By t h i s d e s c r i p t i o n , r e s o n a n c e s which r e q u i r e d up t o f i v e quanta were o b s e r v e d . The s p i n system used was t h a t o f c o n d u c t i o n e l e c t r o n s o f l i t h i u m m e t a l i n n e u t r o n i r r a d i a t e d l i t h i u m f l u o r i d e c r y s t a l s . An a n a l y s i s o f the e x p e r i m e n t a l r e s u l t s u s i n g a m o d i f i e d B l o c h - 7 e q u a t i o n under the a s s u m p t i o n t h a t r v = 7^7^1.5x10 s e c . and t h a t the s p i n system r e l a x e s toward the i n s t a n t a n e o u s f i e l d i s p r e s e n t e d . A b r i e f o u t l i n e o f the con c e p t o f s p i n t e m p e r a t u r e i s i n c l u d e d . A comparison i s made between some of the p r e d i c t i o n s of the s p i n t e m p e r a t u r e concept and the s i m p l e B l o c h t h e o r y used i n a n a l y z i n g the e x p e r i m e n t a l d a t a . "x ( i i i ) T A B L E OF CONTENTS A B S T R A C T i i T A B L E O F CONTENTS i i i L I S T O F F I G U R E S . . . .iv, ACKNOWLEDGEMENT v i I N T R O D U C T I O N 1 THEORY 8 A P P A R A T U S . 13 E X P E R I M E N T A L PROCEDURE AND R E S U L T S 23 T H E MEASUREMENT OF THE R E L E V A N T M A G N E T I C F I E L D S 4 7 A P P E N D I X 5 2 B I B L I O G R A P P I Y 5 7 (iv) LIST OF FIGURES Figure Page 1. Fields in the Lab. and 2 Rotating Frames ,2. Block Diagram of Experimental Apparatus 17 3. Spectrometer Circuit 18 4. TE104 Mode Cavity 19 5. Klystron Frequency Stabilizer 20 6. Variable Coupling Device 21 7. Block Diagram of Klystron 22 Power Supply 8. Preliminary Derivative Traces at 23 7.25 and 23 MHz 9. Qualitative Effect on Derivative of Increasing H,Through Cavity Coupling 24 10. Qualitative Effect on Derivative Traces of Increasing H ^  25 11. The Effect of Increasing Hvi on the 29. Relative Peak Strength at 2.6 MHz 12. The Effect of Increasing H^on the 30 Relative Peak Strength at 3.0 MHz 13. Plot of vs In of Relative Peak Strength 31 14. Theoretical Derivative Traces 34 at different T'S 15. The Theoretical Effect of Increasing 35, 36 H,While the Other Parameters are held constant 16. The theoretical effect of Increasing 37,38 and Hr$ while the Other Parameters are held constant 17. Comparison of Theory and Experiment 40 for two frequencies (v) LIST OF FIGURES (continued) Figure ' Page 18. Comparison of Theory and Experiment for 41 D i f f e r e n t Values of H^and II, 19. The N=l resonance Line Run for , 43 Four Values of 20. Comparison of Di f f e r e n t Predictions 44 of the Line Width as Measured i n the Laboratory Frame 21. Separation i n the Lab. Frarae of the 46 Two N=l Resonance Lines as a Function of 9 22. C a l i b r a t i o n Data to Find the Magnetic F i e l d Produced by the RF Spectrosieter c o i l 49 ( v i ) ACKNOWLEDGEMENT The e x p e r i m e n t s d e s c r i b e d h e r e i n were m o t i v a t e d by the p r e v i o u s work o f E r i c Enga t o whom I am g r a t e f u l f o r the c o n s t r u c t i o n o f most o f the n e c e s s a r y equipment. V a l u a b l e a s s i s t e n c e was g i v e n by Bob P a r s o n s and P a t R y a l l who wrote and r e f i n e d the computer program used i n the a n a l y s i s . S p e c i a l t h a n k s i s g i v e n t o Dr. S. A l e x a n d e r who gave i n v a l u a b l e a s s i s t e n c e i n the p r e s e n t a t i o n o f the t h e o r y . The p r o j e c t was g u i d e d and s u p p o r t e d by Dzv C h a r l e s S c h w e r d t f e g e r w i t h o u t whose a s s i s t e n c e and p a t i e n c e the ex p e r i m e n t would never have been c o m p l e t e d . The r e s e a r c h was s u p p o r t e d through a N a t i o n a l R e s e a r c h C o u n c i l o p e r a t i n g grant, NRC-A-2226. -1-1. INTRODUCTION The c l a s s i c a l e q u a t i o n d e s c r i b i n g the mot i o n o f a f r e e magnetic d i p o l e w i t h a d i p o l e moment y. i n a s t a t i c magnetic f i e l d H. i s j £ = ^ x ^ ( M ) where y i s d e f i n e d by the r e l a t i o n [i=yJ and J i s the a n g u l a r momentum o f the d i p o l e . y i s c a l l e d the gyroruagnetic r a t i o . T h i s e q u a t i o n i s g e n e r a l l y s o l v e d by n o t i n g t h a t i n a c o o r d i n a t e system r o t a t i n g a t an a n g u l a r f r e q u e n c y u around the a x i s which i s p a r a l l e l t o H 0 t h e e q u a t i o n of motion of the d i p o l e moment becomes The magnetic moment i s thus s t a t i o n a r y i n a frarae r o t a t i n g a t u=-yl\. T h i s s o l u t i o n then i s e a s i l y t r a n s f o r m e d back t o the l a b o r a t o r y frame. L i k e w i s e i t i s easy t o show t h a t i n the quantum t h e o r y the e q u a t i o n f o r the e x p e c t i o n v a l u e o f the magnetic niOEient f o r an i s o l a t e d s p i n i s g i v e n by i < * > = <M.>-x yff0 I /. 3) which i s j u s t the c l a s s i c a l e q u a t i o n . I n a d d i t i o n i f the s p i n s a r e n o n - i n t e r a c t i n g the e x p e c t a t i o n v a l u e of the t o t a l m a g n e t i z a t i o n obeys the same e q u a t i o n . The r o t a t i n g r e f e r e n c e fratae mentioned above assumes s p e c i a l i m p o r t a n c e when a r o t a t i n g magnetic f i e l d o f f r e q u e n c y y i and magnitude H , i s a p p l i e d p e r p e n d i c u l a r t o the s t a t i c f i e l d H e. I n t h i s case t h e r e i s a s t a t i c f i e l d H e f i n the r o t a t i n g frame and an a n a l y s i s s i m i l a r t o t h a t above shows t h a t the m a g n e t i z a t i o n s h o u l d p r e c e s s about viewed i n t h e r o t a t i n g frame. Thus i t i s seen t h a t t h e r o t a t i n g frame i s a u s e f u l d e v i c e f o r v i s u a l i z i n g the resonance p r o c e s s . The a n g u l a r f r e q u e n c y of t h e p r e c e s s i o n o f t h e s p i n s i n the r o t a t i n g frame i s g i v e n A H . C«0 LAB Fmne H* - Hi cos Hi 8 H B = by w e f f = y H e f f . F i g u r e 1 shows the magnetic f i e l d s as viewed i n the l a b frame and the r o t a t i n g frame. It i s o b v i o u s from the above d i s c u s s i o n t h a t when ( H 0 - / )=0 t h e e f f e c t i v e f i e l d i s j u s t H,and as the m a g n e t i z a t i o n p r e c e s s e s about this f i e l d i t s time average i n the z d i r e c t i o n i s z e r o . So f a r i n the d i s c u s s i o n a l l i n t e r a c t i o n s o f the s p i n s w i t h the l a t t i c e o r themselves has been n e g l e c t e d . In r e a l systems t h e r e a r e s p i n - l a t t i c e i n t e r a c t i o n s which cause the s p i n system t o t e n d toward the t h e r m a l e q u i l i b r i u m v a l u e i t v/ould have i n the absence o f a r o t a t i n g f i e l d . There a r e t h u s two competing p r o c e s s e s , the r o t a t i n g f i e l d c a u s i n g the m a g n e t i z a t i o n t o p r o c e s s about H e^ 5 and the s p i n l a t t i c e r e l a x a t i o n p r o c e s s e s c a u s i n g the m a g n e t i z a t i o n t o r e l a x toward H 0. However even i n the case o f v e r y s t r o n g s p i n - l a t t i c e c o u p l i n g c o r r e s p o n d i n g t o c h a r a c t e r i s t i c r e l a x a t i o n t i m e s T s h o r t e r t h a n the s p i n Larmor p r e c e s s i o n p e r i o d t h e r e s h o u l d be, i n the r o t a t i n g frame, a n e t component of m a g n e t i z a t i o n a l o n g H e ?£ g i v e n by the u s u a l e x p r e s s i o n f o r the t h e r m a l e q u i l i b r i u m of the m a g n e t i z a t i o n o f a s p i n system i n a s t a t i c 1 magnetic f i e l d . I f on the o t h e r hand r i s comparable t o o r l a r g e r than the Larmor p r e c e s s i o n p e r i o d the magnitude of the m a g n e t i z a t i o n a l o n g H e j f can be much l a r g e r . T h i s s i t u a t i o n can be d e s c r i b e d by d e f i n i n g a new t e m p e r a t u r e T / / 0 / v such t h a t e q u a t i o n 1.4 remains v a l i d f o r the g i v e n m a g n e t i z a t i o n . In any case i t i s p o s s i b l e t o cause a resonance i n the r o t a t i n g frame by a p p l y i n g a p e r t u r b i n g f i e l d of the c o r r e c t f r e q u e n c y perpen-d i c u l a r t o H Qf.f . 2 R e d f i e l d d e v e l o p e d an e s s e n t i a l l y thermodynamic d e s c r i p t i o n o f r esonance e x p e r i m e n t s i n the r o t a t i n g frame. One of the 3 a s s u m p t i o n s of the t h e o r y as o u t l i n e d i n M i s s F r a n z ' s t h e s i s i s t h a t the s p i n - s p i n i n t e r a c t i o n i s l a r g e compared t o the s p i n l a t t i c e i n t e r a c t i o n . T h i s i m p l i e s t h a t the s p i n system can have a t e m p e r a t u r e d i f f e r e n t from t h a t o f the l a t t i c e . The - 4 -s t a r t i n g p o i n t f o r t h i s t h e o r y i s the a s s u m p t i o n o f a system H a m i l t o n i a n o f the form H=H2 +llss where Hz i s the Zeeman i n t e r a c t i o n energy and Rssis the s p i n - s p i n c o u p l i n g energy. Note t h a t t h i s H a m i l t o n i a n i g n o r e s any s p i n - l a t t i c e i n t e r a c t i o n . The H a m i l t o n i a n i s t r a n s f o r m e d t o the r o t a t i n g frame t h r o u g h two t r a n s f o r m a t i o n o p e r a t o r s . !H*'= e'*G l y H V t > G l * (I**) The f i r s t t a k e s the system i n t o a frame r o t a t i n g a t a n g u l a r f r e q u e n c y u around the z a x i s . The second a l i g n s the a x i s of q u a n t i z a t i o n a l o n g the e f f e c t i v e f i e l d a x i s . The r e s u l t i n g H a m i l t o n i a n i s : *»H*'--v*ffcw:iy *Z $ AiKf,T*T+ 0i*rZ^Xn* *Oinpr^ Jv3 * '4 (3 Cos1©-/) Ej*e =f lA e> c o s e Sj,/ # There a r e two p o i n t s o f immediate i n t e r e s t . The m a g n e t i -z a t i o n o f a s p i n system i n a magnetic f i e l d d e f i n e s a s p i n t e m p e r a t u r e based on C u r i e ' s law. I f a s p i n system i s i n i t i a l l y i n e q u i l i b r i u m with a s t a t i c f i e l d a t some temperature T and i f t he s t a t i c f i e l d can be red u c e d w i t h o u t c h a n g i n g the The c o u p l i n g c o e f f i c i e n t s ^ ^ and tfjtf a r e d e f i n e d by the s p i n - s p i n H a m i l t o n i a n i n the l a b frame. The c o n s t a n t s Aj\i and B J K a r e d e f i n e d i n terms o f the s e by the r e l a t i o n s , /Ci ^ '/a ^ c e s * a ; K - l ) , * - % & ( I W * - * ! * - ! ) F o r a complete d i s c u s s i o n , a r e f e r e n c e i s Abragam pp. 546-548. - 5 -m a g n e t i z a t i o n we say t h a t the temp e r a t u r e i s lo w e r than b e f o r e . I t happens t h a t i f t h e r e i s no s p i n - l a t t i c e r e l a x a t i o n the t o t a l s t a t i c m a g n e t i z a t i o n can be made t o p r e c e s s about the much s m a l l e r i n the r o t a t i n g frame, a p r o c e s s known as a d i a b a t i c f a s t passage. T h i s means t h a t the r a t i o o f the u l a t t i c e t o s p i n t e m p e r a t u r e i s o f the o r d e r of magnitude ~f— . 4 nzis An e x a c t c a l c u l a t i o n shows: where H ^ i s the l o c a l f i e l d i n the r o t a t i n g frame. From C u r i e ' s Law the m a g n e t i z a t i o n i n the r o t a t i n g frame i s fl * H t u j l I 1.7) The second p o i n t i s t h a t the r e s o n a n t l i n e w i d t h i n the 5 r o t a t i n g frame s h o u l d be dependent on the term ( 3 c o s ^ - l ) The dependency i s such t h a t the l i n e w i d t h s h o u l d narrow as ( 3 c o s ^ - l ) goes t o z e r o o r 54.7°. T h i s e f f e c t s h o u l d s e r v e as a t e s t f o r the a p p l i c a b i l i t y of the s p i n temperature t h e o r y i n t h i s form t o i n d i v i d u a l systems. The i n i t i a l e x p e r i m e n t s o f R e d f i e l d d e t e c t e d l a r g e d i s p e r s i o n s i g n a l s a t r f f i e l d s t r e n g t h s h i g h e r than needed t o e n sure the s a t u r a t i o n o f the a b s o r p t i o n s i g n a l . The s p i n 23 6 system was the n u c l e i o f Na i n N a C l . He a l s o o b s e r v e d what he termed " r o t a r y s a t u r a t i o n " by a p p l y i n g an a u d i o f r e q u e n c y f i e l d p e r p e n d i c u l a r t o H c^ f i n " t h e r o t a t i n g frame. When the n u c l e a r s p i n system was a t resonance a d e c r e a s e i n the d i s p e r s i o n d e r i v a t i v e s i g n a l was o b s e r v e d when the a u d i o f r e q u e n c y s a t i s f i e d t he c o n d i t i o n u=y8.iff . I n t h i s - 6 -7 e x p e r i m e n t the H , f i e l d was s t r o n g enough t o ensure s a t u r a t i o n . S i n c e the r e s u l t s o f t h e s e e x p e r i m e n t s c o u l d not be s a t i s -f a c t o r i l y e x p l a i n e d by the u s u a l phenomemological B l o c h e q u a t i o n s R e d f i e l d i n t r o d u c e d the concept o f s p i n t e m p e r a t u r e . S u b s e q u e n t l y R e d f i e l d ' s t h e o r y has been a p p l i e d s u c c e s s f u l l y t o many n u c l e a r resonance e x p e r i m e n t s , e s p e c i a l l y t h o s e i n v o l v i n g h i g h enough H, f i e l d s t o cause s a t u r a t i o n . Enga a p p l i e d t h e s e i d e a s t o e l e c t r o n s p i n r e s o n a n c e . As the energy o f i n t e r a c t i o n o f an e l e c t r o n s p i n w i t h a magnetic f i e l d i s about 1 0 0 0 t i m e s g r e a t e r than the i n t e r a c t i o n of a t y p i c a l n u c l e a r s p i n i n the same f i e l d , the f r e q u e n c y o f the a n a l o g o u s f i e l d s used i n Enga's ex p e r i m e n t were t h r e e o r d e r s of magnitude l a r g e r than those used by R e d f i e l d . T h i s f a c t e n a b l e d Enga t o use a m a r g i n a l o s c i l l a t o r o p e r a t i n g i n the megacycle r e g i o n t o d i r e c t l y m o n i t o r changes of m a g n e t i z a t i o n a l o n g H e ^ $ . The s p i n system used was the f r e e r a d i c a l i n Dpph, an o r g a n i c compound w i t h w e l l known p a r a -magnetic resonance p r o p e r t i e s . Dpph has a u s u a l e . s . r . l i n e w i d t h o f about f o u r gauss. T h e r e f o r e h i s e x p e r i m e n t s were c o n f i n e d t o f i e l d s (He$* ) i n e x c e s s o f t h r e e gauss. T h i s c o r r e s p o n d s t o a r a d i o f r e q u e n c y ( r f ) o f a p p r o x i m a t e l y t e n megacycles. Y/hen he a p p l i e d an r f f i e l d a l o n g H„in a d d i t i o n t o the microwave f i e l d p e r p e n d i c u l a r t o Ho he o b s e r v e d a resonance when|H £^[ T h i s has two s o l u t i o n s a s : 8 'k and t h i s i s s a t i s f i e d f o r , -7-These two l i n e s were obse r v e d w i t h the e x p e c t e d symmetry. The l i n e f o r h i g h e s t H ewas a b s o r p t i v e and t h a t f o r l o w e s t H^was e m i s s i v e . T h i s would c o r r e s p o n d t o the f a c t t h a t the magnet-i z a t i o n s h o u l d be p a r a l l e l and a n t i p a r a l l e l t o the H 0 f i e l d d i r e c t i o n i n the two c a s e s r e s p e c t i v e l y . I n a d d i t i o n t o th e s e two l i n e s an i n t e r e s t i n g c e n t e r l i n e was o b s e r v e d c o r r e s p o n d i n g t o He£$ =H». As t h i s c e n t e r l i n e had no r e a d y i n t e r p r e t a t i o n i n terms o f the t h e o r y proposed i t was d e c i d e d t o i n v e s t i g a t e t h i s r o t a t i n g frame e x p e r i m e n t u s i n g a d i f f e r e n t s p i n system. N e u t r o n i r r a d i a t e d l i t h i u m f l u o r i d e c r y s t a l s seemed a good system because v e r y narrow c o n d u c t i o n e . s . r . 9 l i n e s had been o b s e r v e d and the c r y s t a l s had a h i g h m e l t i n g t e m p e r a t u r e . Dpph c r y s t a l s would melt under the s t r o n g a b s o r p t i o n of power from the k l y s t r o n used i n t h i s e x p e r i m e n t and i t was proposed t h a t the m e l t i n g was i n some way r e s p o n -s i b l e f o r the anomalous c e n t e r l i n e . - 8 -2. THEORY The L i F samples used had a peak to peak derivative e.s.r. linewidth of approximately .35 gauss. It i s well known that the peak to peak derivative width i s related to the half power resonance curve width by ^^ff/i' The c h a r a c t e r i s t i c spin-l a t t i c e relaxation time can be found from the half width of the absorption peak. Using the uncertainty p r i n c i p l e At&T-ii and i n t e r p r e t i n g kT as the c h a r a c t e r i s t i c time for relaxation of a spin to the l a t t i c e gives: where p. i s the magnetic dipole moment of the electron. Thus: T'/a * 'A 2. 7 _ I'll YJ£Z^~ ( ^ * "2-) TiL - I' s'hJj — 1 <* where r i s i n seconds and H i s i n gauss. These expressions -7 give a relaxation time of approximately 1.9x10 seconds for the measured peak to peak width of the sample. This means that the spin l a t t i c e relaxation time i s shorter than the Larmor precession time. This immediately throws serious doubt upon the v a l i d i t y of using the conclusions of the spin temperature theory to describe the r e s u l t s of t h i s experiment. The simplest and most standard approach for most resonance experiments has been to s t a r t with the phenomenalogical equations of Bloch. Bloch's equations are: ~ ( ^ - ^ } 'p.3) where M i s the magnetization, H i s the t o t a l instantaneous -9-7 -1 -1 f i e l d and ^=1.76x10 oe s e c . i s t h e gyroraagnetic r a t i o o f the e l e c t r o n . When one has l a r g e p e r t u r b i n g f i e l d s and the sample i s such t h a t 7~Jl =r a , the B l o c h e q u a t i o n s a r e m o d i f i e d t o the form, J T 7 The microwave f i e l d i s l i n e a r l y o s c i l l a t i n g and i s p e r p e n -d i c u l a r t o H e. The r a d i o f r e q u e n c y f i e l d i s p a r a l l e l t o Ho and i s a l s o l i n e a r l y o s c i l l a t i n g , thus. H= ce* ut + k(lU *KfCo$jn) T h i s form o f the B l o c h e q u a t i o n s has been used b e f o r e i n 10,11 i n t e r p r e t i n g low f i e l d e . s . r . r e s u l t s . The resonance i s d e t e c t e d a l o n g H e j $ i n the r o t a t i n g frame so we must t r a n s f o r m t h i s e q u a t i o n i n t o the r o t a t i n g frame by t a k i n g H 2 a s ( H ^ - ^ - j f r and Hx as c H, . R e l a x a t i o n i s s t i l l assumed towards H ebecause o f the s t r o n g s p i n l a t t i c e c o u p l i n g . The e q u a t i o n t o be s o l v e d i s : ~~- i y(ATKfBfH,-^+*H , 7 ) -^ft-*9l&(HtiH,fCesut+i(ll,t>9sJlty]'^ ^ ^ T h i s e q u a t i o n can not be s o l v e d e x a c t l y and s i n c e the magnet-ic i z a t i o n i s p e r i o d i c i t i s e f f a c a c i o u s t o use a F o u r i e r expansion; S e t t i n g ft sr X S1h e * n J l t (<2.0 v/here o b v i o u s l y /7n - rl-h and s u b s t i t u t i n g 2.6 i n t o 2.5 g i v e s the i t e r i t i v e eq. ( LhJl-A) Mn - 8 (/?*»-! * /? h t,) + Cn =0 (5.?) where the m a t r i c e s A and B are d e f i n e d so t h a t a /Tn r. y /?"„ x (a . ?; and where IL,^ = £ W, */? (w„ --10-To use the s e e q u a t i o n s t h e y must be w r i t t e n out e x p l i c i t l y . I g i v e enough here so t h a t a l l the o t h e r s a r e a p p a r e n t . There a r e c l e a r l y 6n e q u a t i o n s as t h e r e i s a r e a l and i m a g i n a r y component o f the m a g n e t i z a t i o n j e.g., The e q u a t i o n s a r e : 3 . ^ R ^ W ' / i e •ayr% A i. , [ / r-yr^*' ,^»o 10. fl,1*+si7fi,** ^rfon** * t^t/iMF +v rH** {>  0. I I ri,**-jiT Ai,*r +* r •= 18. ^ . r / l / S In t h e above e q u a t i o n s ^ =• (^'''¥'), -11-T h i s s u g g e s t s s o l v i n g the 6n x 6n m a t r i x e q u a t i o n AM=C where A i s t h e m a t r i x d e f i n e d by the c o e f f i c i e n t s o f the m a g n e t i z a t i o n and M i s a 6n d i m e n s i o n a l v e c t o r r e p r e s e n t i n g the components of m a g n e t i z a t i o n , C i s a n o t h e r column v e c t o r r e p r e s e n t i n g the inhomogeneous terms i n the e q u a t i o n s . There a r e o n l y t h r e e non z e r o terms C ,C , and C . C and C a r e 1 5 11 1 11 t h r e e o r d e r s o f magnitude s m a l l e r than C and i t proved t o 5 _make no n o t i c e a b l e d i f f e r e n c e i n the r e s u l t s t o n e g l e c t t h e s e terms and r e t a i n o n l y C . T h i s i s r e a s o n a b l e s i n c e H. ?^ H, °* Hn. 5 The m a t r i x e q u a t i o n was s o l v e d n u m e r i c a l l y on an IBM 7040 computer. T e r m i n a t i n g the s e r i e s a t n=5 and c h o s i n g f i x e d v a l u e s f o r a l l the parameters e x c e p t one which assumed f i v e v a l u e s , one machine r u n took 10 m i n u t e s . T h i s was found t o be s u f f i c i e n t f o r our needs as we needed o n l y t o d e t e r m i n e M vor more p r e -c i s e l y the d e r i v a t i v e o f M * r w i t h r e s p e c t t o H „. I n c l u d i n g the n=6 terms s e r v e d t o change the c a l c u l a t e d d e r i v a t i v e a t 4 n=5 by o n l y one p a r t i n 10, r e g a r d l e s s o f the f a c t t h a t the M components were not n e g l i g i b l e compared t o the M components. 6 5 One i n t e r e s t i n g p o i n t o f the e x p e r i m e n t i s the e f f e c t on the l i n e w i d t h when i n c r e a s i n g the a n g l e between Keff and H 6. I t was noted i n the i n t r o d u c t i o n t h a t the l i n e w i d t h i n the r o t a t i n g frame s h o u l d narrow as the f u n c t i o n 3 c o s # - l becomes s m a l l e r i f the s p i n t e m p e r a t u r e h y p o t h e s i s i s v a l i d f o r the c a s e . The p r e d i c t i o n o f our t h e o r y i s not i m m e d i a t e l y a p p a r e n t but when a c o n s t a n t T i s used i n the e q u a t i o n s the -12-l i n e w i d t h s h o u l d be c o n s t a n t i n the r o t a t i n g frame. The r e l a t i o n o f the l i n e w i d t h a l o n g t o t h a t i n the l a b frame i s s i m p l e as l o n g as the l i n e w i d t h i s s m a l l compared t o H<?# . The r e l a t i o n f o l l o w s from e q u a t i o n 1.10. AWc* - K ^ l ^ . S o A / / w , A(H.^} (0, 30) So assuming the l i n e w i d t h i s a c o n s t a n t the dependence of the measured l i n e w i d t h i n the l a b frame s h o u l d be ^Ull . I t i s cos e u n f o r t u n a t e t h a t the h i g h e s t v a l u e s o f H, o b t a i n a b l e were o n l y around one gauss which i s not l a r g e enough t o t e s t t h i s p r e -d i c t i o n a c c u r a t e l y . F o r l a r g e angles, one gauss which i s not much l a r g e r t h a n the l i n e w i d t h . S i n c e the computer r e s u l t s matched the e x p e r i m e n t a l l i n e s f o r a l l c a s e s o b s e r v e d i t was r u n f o r a s e r i e s o f a n g l e s w i t h H^^^IO Hy. T h i s p l o t i s g i v e n a l o n g w i t h a p l o t of and AH-/X (3003*0-1) i n f i g u r e -IS. -13-3. APPARATUS The a p p a r a t u s c o n s i s t s of a h i g h power 34 GHz E l l i o t t y p e 8TFK2 k l y s t r o n c a p a b l e o f 10 w a t t s o u t p u t , an i s o l a t o r , l o a d , and a s s o c i a t e d microwave equipment i n c l u d i n g a TE104 r e t a n g u l a r c a v i t y , a b o l o m e t e r and H e w l e t t P a c k a r d Model 430C power meter, a f i e l d c o n t r o l l e d 9.5 i n c h Magnion magnet a l o n g w i t h f i e l d m o d u l a t i o n equipment, two m a r g i n a l o s c i l l a t o r s , a H e w l e t t P a c k a r d f r e q u e n c y c o u n t e r , a l o c k - i n d e t e c t o r . C h a r t r e c o r d e r and the a u x i l i a r y equipment needed f o r s t a b i l i z a t i o n . The arrangement i s shown i n the b l o c k diagram ( F i g u r e 2 ) . The k l y s t r o n i s water c o o l e d p r o v i d i n g enough s t a b i l i t y so t h a t i t c o u l d be r u n s a t i s f a c t o r i l y w i t h o u t an a . f . c . a f t e r an i n i t i a l warm up p e r i o d o f h a l f an hour. The k l y s t r o n c o u l d not be swept t h r o u g h an e n t i r e mode and d i s p l a y e d on the o s c i l l i s c o p e as t h i s i n v o l v e d m o d u l a t i n g the beam v o l t a g e : s u p p l y by about 150 v o l t s . I n p r a c t i c e then the a . f . c . was not used and the k l y s t r o n was a l l o w e d t o d r i f t s l i g h t l y . T h i s meant r e t u n i n g the k l y s t r o n t o the c a v i t y f r e q u e n t l y . T h i s was a c c o m p l i s h e d i n two s t e p s . F i r s t a rough s e t t i n g o f the f r e q u e n c y was o b t a i n e d t h r o u g h the c a l i b r a t e d wavemeter. Then the k l y s t r o n was f i n e tuned u n t i l the maximum s i g n a l s t r e n g t h was o b t a i n e d . The f r e q u e n c y c o u l d be found t o about t e n MHz w i t h the wavemeter. The c o u p l e r i s d e s c r i b e d i n d e t a i l i n 13 Enga's t h e s i s . W h i l e the s t r e n g t h o f the f i e l d can be c o n t r o l l e d t h r o u g h the c o u p l i n g t h e r e i s no c a l i b r a t i o n and the s e t t i n g s -14-a r e n o t r e p r o d u c i b l e . The c a v i t y used was the same as d e s c r i b e d i n Enga's t h e s i s and a l t h o u g h the s i l v e r p l a t i n g had d e t e r i o r a t e d the Q was not s e r i o u s l y a f f e c t e d . The c a v i t y was a l s o c o o l e d t h r o u g h an e x t e r n a l copper tube a f f i x e d t o the s i d e o f the c a v i t y . D e t a i l s o f the c a v i t y and t e f l o n sample h o l d e r s a r e shown i n f i g u r e 4. The sample c o i l s i z e depended on the f r e q u e n c y d e s i r e d but c o i l s made o f 100 t u r n s o f no. 46 enameled copper w i r e o s c i l l a t e d a t about t h r e e MHz. The o t h e r microwave equipment i s d e s c r i b e d i n d e t a i l i n Enga's t h e s i s . A s c h e m a t i c o f the k l y s t r o n power s u p p l y and i t s c o n n e c t i o n s t o the k l y s t r o n i s shown i n f i g u r e 7. The power s u p p l y i s n o i s y and d e t e c t i o n of e . s . r . by m o n i t o r i n g power ab s o r b e d from the microwave system would be d i f f i c u l t . An a d d i t i o n a l d i f f i c u l t y was t h a t the k l y s t r o n mode c o u l d not be d i s p l a y e d t h u s making i t d i f f i c u l t t o match the k l y s t r o n f r e q u e n c y t o the c a v i t y r e s o n a n t f r e q u e n c y . A H e w l e t t P a c k a r d f r e q u e n c y c o u n t e r w i t h a 50-100 MHz p l u g i n u n i t was used t o de t e r m i n e the e x a c t f r e q u e n c i e s f o r the e . s . r . resonance a l o n g lie$f and the p r o t o n resonance f r e q u e n c y i n H 0. U s i n g the p l u g i n u n i t f r e q u e n c i e s both above and below 50 MHz can be d e t e r m i n e d by c h a n g i n g the s e l e c t o r knob. T h i s a v o i d s the n e c e s s i t y o f two c o u n t e r s . The m a r g i n a l o s c i l l a t o r i s a s l i g h t l y m o d i f i e d form 14 of (the one d e s c r i b e d by Benedek and K u s h i d a and V o l k o f f 15 e t a l . The 6J6 tube o p e r a t e s as a p u s h - p u l l o s c i l l a t o r w i t h the r E s o n a n t f r e q u e n c y d e t e r m i n e d by the sample c o i l and -15-b u t t e r f l y t u n i n g c a p a c i t o r s . The g r i d - p l a t e feedback c a p a c -i t o r s a r e a d j u s t e d f o r m a r g i n a l o s c i l l a t i o n . A t r e s o n a n c e , e nergy i s absorbed from the sample c o i l c h a n g i n g i t s Q and hence c h a n g i n g the c u r r e n t d e l i v e r e d t o the c i r c u i t v i a the p l a t e s u p p l y . T h i s s i g n a l i s modulated by the H e l m h o l t z m o d u l a t i o n c o i l s a t w and i s f e d v i a a c o a x i a l c a b l e t o the l o c k - i n d e t e c t o r . T h i s i n t u r n i s c o n n e c t e d t o a c h a r t r e c o r d e r . A t r a n s i s t o r i z e d a u d i o a m p l i f i e r c o n n e c t e d t o the p l a t e c i r c u i t i s used t o m o n i t o r the f r e q u e n c y . I t p r o v i d e s enough g a i n t o d r i v e the H e w l e t t P a c k a r d model HP5245 L f r e q u e n c y c o u n t e r . T h i s amounts t o over .1 v o l t a t the f r e q u e n c i e s used. The d.c. meter on the f r o n t o f the m a r g i n a l o s c i l l a t o r e s s e n t i a l l y measures the v o l t a g e d e v e l o p e d by the g r i d c u r r e n t a c r o s s a 51 K r e s i s t o r on the p o s i t i v e h a l f o f the wave form. See f i g u r e 3. The s m a l l g r i d c a p a c i t o r s can be u sed t o a d j u s t the g r i d v o l t a g e and hence the d.c. f i e l d s t r e n g t h . The u n m o d i f i e d o s c i l l a t o r was c a p a b l e of o p e r a t i n g between about 5 and 60 MHz. F o r o p e r a t i o n a t lower f r e q u e n c i e s e x t r a c a p a c i t a n c e was added t o the g r i d c a p a c i t o r s . With a c o i l o f about 100 t u r n s and a c a p a c i t o r i n p a r a l l e l w i t h the g r i d trimmer of 47 pf a f r e q u e n c y of 3 t o 3.5 MHz c o u l d be o b t a i n e d . A p a r a l l e l c a p a c i t a n c e o f 100 pf would g i v e f r e q -u e n c i e s clown t o 2.3 MHz. However a t such f r e q u e n c i e s the t u n i n g range of the o s c i l l a t o r was l i m i t e d t o a few hundred k i l o c y c l e s . To o b t a i n a l l d e s i r e d f r e q u e n c i e s from 2.3 t o 4 MHz i t was n e c e s s a r y t o use two s p e c t r o m e t e r s , one m o d i f i e d -16-anci one u n m o d i f i e d , and two s p e c t r o m e t e r t o s i g n a l c o i l c a b l e l e n g t h s . When o p e r a t i n g p r o p e r l y the s i g n a l t o n o i s e r a t i o of t he o s c i l l a t o r was over 100 f o r the n=l resonance l i n e i n L i F . The p h y s i c a l c o n s t r u c t i o n o f the o s c i l l a t o r i s d i s c u s s e d i n Enga's t h e s i s , and a s c h e m a t i c diagram i s g i v e n i n f i g u r e 3. F i g u r e 6 c o n t a i n diagrams o f the c a v i t y c o u p l i n g mechanism and a TE011 mode c y l i n d r i c a l c a v i t y . T h i s c a v i t y was not used i n the p r e s e n t e x p e r i m e n t s but was used by Enga i n h i s o r i g i n a l e x p e r i m e n t s and so i s i n c l u d e d f o r c o m p l e t e n e s s . -17-B L O C K D I A G R A M O F T H E E X P E R I M E N T A L A P P A R A T U S COUNTER MARGINAL OSCILLATOR B A L A N C E D LINES ... L O C K - I N AMPLIFIER MARGINAL OSCILLATOR / / / MAGNION 9.5 MAGNET BOLOMETER. POWER M E T E R r. V CHART RECORDER 4 0 0 HZ OSCILLATOR AUDIO A M ? / 34 GHZ CAVITY T E ^ S A M P L E HOLDER : / 104 MODULATION COILS KLYSTRON POWER SUP. DIRECTIONAL COUPLER (lOdb) KLYSTRON AFC Vs --f\7L. XTAL ATT WAVEMETER F I G U R E 2 3 PORT CIRCULATOR -18-FIG. 3. -19-Coup!u-iCj Hols-.071 Diet. W/G Fiance. S o l d e r e d o n t o lev ion Sample HoltScr Spectrometer Coil Silver* Pjqttncj S f l m pie. W a t e r Cooliriq S I D E VIEW CUTAWAY S l o t f o r introa'ucinj sample. $urrevs\<lm^ metal rcac'iinecC t o .QQ'f ilr\i$!tnes$ TOP VIEW r .07} J l • ) — 1 "I I* T _rn_ "\nc - . 1 0 0 Oia. >.02S" OIO -#<S0 O n II LUC ITS PLASTIC FORM WHICH S A M P L E . B0 .B51N ( T c R O N ) C A V I T Y !VAS ELECTROPLATED FIG. V- t£ 0/'} iM0D£ CAVITY (RECTAWG-ULAR) cuTavwy VKW saowm ggass jtsms in' vn-zz vilo FIG 6. ' VARIABLE COUPLING DEVICE -22-BLOCK DIAGRAM OF KLYSTRON SUPPLY c u r r e n t o p e r a t e d r e l a y EIIT SUPPLY r n c a v i t y c o l l e c t o r c o n n e c t e d t o g h t e r i f a . f . c . not used h e a t e r FIGURE 7 -23-•4. EXPERIMENTAL PROCEDURE AND RESULTS The e x p e r i m e n t s were c a r r i e d o u t a t room t e m p e r a t u r e i n a l l c a s e s . The s t a t i c magnetic f i e l d H Dwas modulated a t 400 Hz. and a l o c k - i n d e t e c t i o n system was employed. The s i g n a l s were o b t a i n e d by s e t t i n g the k l y s t r o n a t a f i x e d f r e q u e n c y c o r r e s p o n d i n g t o t h e c a v i t y r esonance and s e t t i n g the m a r g i n a l o s c i l l a t o r f o r the f r e q u e n c y which gave the d e s i r e d ESif and then s l o w l y sweeping H ^ v i a the c a l i b r a t e d sweep power s u p p l y . T y p i c a l sweep speeds were o f the o r d e r o f a few gauss per minute. The d e r i v a t i v e s i g n a l s were d i s p l a y e d on a c h a r t r e c o r d e r . P r e l i m i n a r y r u n s were f i r s t made a t f r e q u e n c i e s g r e a t e r t h a n 6 MHz. F o r t h e s e r u n s l i n e s s i m i l a r t o t h o s e found by Enga were o b s e r v e d . F i g u r e 8 shows t y p i c a l t r a c e s t a k e n a t t h e s e f r e q u e n c i e s . FIG. 9 -24-F i g . 9 shows the q u a l i t a t i v e e f f e c t o f i n c r e a s i n g H,through i n c r e a s i n g the c o u p l i n g t o the c a v i t y . A l l t h e s e r u n s were a t 3.58 MHz and a p p r o x i m a t e l y 10 t o 14|iA of o s c i l l a t o r meter r e a d i n g w h i c h i n d i c a t e s the r f f i e l d s t r e n g t h . . H , i n c r e a s e s from (a) t h r o u g h (d) but no a c c u r a t e q u a n t i t a t i v e v a l u e s f o r H (were o b t a i n e d f o r t h e s e r u n s . -25-In fig.10 a l l the signals were obtained at a frequency 2.6 MHz. The current reading r e f l e c t s the r e l a t i v e strength of H ^ b u t H , i s not c a l i b r a t e d . Figure (b), however, has the largest value of H.. ( c ) 12 (iA FIGURE 10 ( d ) 2 0 jiA - 2 6 -The magnetic f i e l d was not c a l i b r a t e d i n t h e s e p r e l i m -i n a r y r u n s but from the c a l i b r a t i o n o b t a i n e d i n subsequent r u n s i t was a p p a r e n t t h a t a t t h e s e o s c i l l a t o r f r e q u e n c i e s the microwave f i e l d s t r e n g t h was s m a l l compared t o H e ^ . I n the u s u a l f i r s t t r e a t m e n t o f resonance the e q u a t i o n f o r the e v o l u t i o n o f the m a g n e t i z a t i o n , i s s o l v e d e x a c t l y by s e p a r a t i n g H i n t o the v e c t o r coiaponents: T h i s i m p l i e s t h a t the p e r p e n d i c u l a r r f f i e l d i s r o t a t i n g about H 0 . O b v i o u s l y a l i n e a r l y o s c i l l a t i n g f i e l d 2Kcos;jt can be decomposed i n t o two o p p o s i t e l y r o t a t i n g magnetic f i e l d s . HL = C V, COSCO< - 0 H, StJVb/t I f H ^ i s l a r g e compared t o K ( i t can be shown t h a t o n l y one r o t a t i n g component i s i m p o r t a n t toward e f f e c t i n g a r e s o n a n c e . Hence when H j i s s m a l l compared t o I I d a l i n e a r l y o s c i l l a t i n g r f f i e l d may be t r e a t e d as a r o t a t i n g f i e l d . D o i n g t h i s , o n l y one c o n d i t i o n f o r r esonance i s p o s s i b l e and t h i s o c c u r s when Uf = i ^ a n d 2 i r i J v f c o r r e s p o n d s t o the Larmor p r e c e s s i o n if f r e q u e n c y . In a c t u a l p r a c t i c e a l i n e a r l y o s c i l l a t i n g f i e l d i s a l m o s t always used. I n t h i s e x p e r i m e n t the l i n e a r l y o s c i l l a t i n g r f f i e l d was comparable i n s t r e n g t h t o f o r low f r e q u e n c i e s o f H r f and h i g h r f f i e l d s t r e n g t h s . In t h i s c a s e i t i s no l o n g e r p r o p e r t o n e g l e c t one o f the r o t a t i n g components of the f i e l d . I n f a c t an a n a l y s i s o f the f i e l d s -27-pi^esent i n the d o u b l y r o t a t i n g frame ( t h e one r o t a t i n g about the a x i s d e f i n e d by n^^5), shows t h a t t h e r e a r e o t h e r resonance 16 c o n d i t i o n s t h a n the one g i v e n above. Y / i n t e r ' s a n a l y s i s shows t h a t f o r a r f magnetic f i e l d w i t h non z e r o components i n a l l o f the c o o r d i n a t e d i r e c t i o n s r e s o n a n t c o n d i t i o n s e x i s t f o r U=WJ6 where G J 0 i s the Larmor p r e c e s s i o n f r e q u e n c y i n the r o t a t i n g frame about . The chances o f o b s e r v i n g t h e s e h i g h e r o r d e r r e s o n a n c e s i s g r e a t l y enhanced as the r f f i e l d s t r e n g t h becomes comparable t o t h e s t a t i c f i e l d . Thus t h e r o t a t i n g frame p o s s i b l y a f f o r d s the o n l y means t o s t u d y "these h i g h e r o r d e r r e s o n a n c e s f o r e l e c t r o n s p i n r e s o n a n c e . There a r e tv/o ways t o a c h i e v e an l a r g e r e l a t i v e t o H e ^ j , e i t h e r r e q u i r e t h a t t h e o s c i l l a t o r produce a l a r g e o s c i l l a t i n g f i e l d or make Heyy s m a l l by u s i n g a lo w e r r f f r e q u e n c y . The l a t t e r i s b e t t e r because t h e s e n s i t i v i t y o f the o s c i l l a t o r i s g r e a t e s t f o r low f i e l d s t r e n g t h s . F o r t h e s e r e a s o n s i t was d e c i d e d t o make ru n s a t as low as p r a c t i c a l r f f r e q u e n c i e s . A p e r t u r b a t i o n 17 c a l c u l a t i o n shows t h a t i f t h e e f f e c t i s s m a l l the n=2 reson a n c e s i g n a l s t r e n g t h s h o u l d be p r o p o r t i o n a l t o llri w h i l e t h e n=3 re s o n a n c e s h o u l d be p r o p o r t i o n a l t o H£. Of c o u r s e the n=l resonance s t r e n g t h i s p r o p o r t i o n a l t o . F i g u r e 11 shows the e f f e c t o f i n c r e a s i n g I I v j w h i l e the o t h e r p a r a m e t e r s a r e h o l d c o n s t a n t , the f r e q u e n c y b e i n g 2.6 MHz. F i g u r e 12 shows the same e x p e r i m e n t but w i t h f ^ = 3 . 0 MHz. I t i s o b v i o u s from th e s e r e s u l t s t h a t the s t r e n g t h of the h i g h e r o r d e r r e s o n a n c e s i s h i g h l y s e n s i t i v e t o the s t r e n g t h of II . -28-A l s o f o r the same r f f i e l d s t r e n g t h the s t r e n g t h of the h i g h e r o r d e r r e s o n a n c e s i n c r e a s e s as d e c r e a s e s . T h i s i s the same as d e c r e a s i n g H^^. . A s e m i - q u a n t i t a t i v e d e s c r i p t i o n o f the e f f e c t o f H T i s t r e n g t h on the r e l a t i v e s i g n a l s t r e n g t h s was made by n o t i n g t h a t f o r a L o r e n t z i a n l i n e shape the i n t e g r a t e d a r e a under the a b s o r p t i o n c u r v e i s p r o p o r t i o n a l t o the peak h e i g h t and the d e r i v a t i v e peak h e i g h t assuming t h a t the l i n e w i d t h i s c o n s t a n t . T h i s i s a good a p p r o x i m a t i o n f o r the ranges o f the magnetic f i e l d s used i n t h i s e x p e r i m e n t . F i g u r e 13 shows the r e s u l t s o f t h i s measurement f o r a t y p i c a l r u n a t 3.0 MHz. S i x t r a c e s were made f o r H y ^ f i e l d s t r e n g t h s o f from 18 7 t o 30 |iA as r e a d on the o s c i l l a t o r meter. F o r each v a l u e o f H Y ^ t h e r a t i o o f peak h e i g h t 2 - t o - l , 3 - t o - l , and 4 - t o - l was measured. Only a t the two h i g h e s t v a l u e s o f H t j c o u l d measurements be made f o r the r a t i o o f peak 4 t o peak 1. The peak h e i g h t s c o u l d not be compared between one v a l u e o f I-L , and. a n o t h e r because the s e n s i t i v i t y o f the o s c i l l a t o r i s some unknown f u n c t i o n o f the o s c i l l a t o r s i g n a l s t r e n g t h . From the s l o p e of the l o g l o g p l o t i t i s seen t h a t : i . W i t h i n the l i m i t s o f e r r o r t h e n , T h i s i s what the p e r t u r b a t i o n t h e o r y f o r low H v* would p r e d i c t , a l t h o u g h p e r t u r b a t i o n t h e o r y i s d e f i n i t e l y not a p p l i c a b l e as -3 (c) 30 FIGURE 12 -32-The n e x t t h r e e f i g u r e s (14-16J show n u m e r i c a l s o l u t i o n s f o r t he e q u a t i o n s d e v e l o p e d i n the t h e o r y . They a r e p l o t s o f "77?''' . F i g u r e 14 shows the e f f e c t of c h a n g i n g the r e l a x a t i o n o Ho time r . S h o r t e n i n g T has the e f f e c t o f b r o a d e n i n g the resonance l i n e s . However as w i l l be shown o t h e r parameters can have s i r a i l a r e f f e c t s and the b r o a d e n i n g due t o T and t h a t due t o o t h e r e f f e c t s must be d i s t i n g u i s h e d i f the p r o p e r r i s t o be found i n t h i s manner. I t w i l l be seen t h a t the b r o a d e n i n g due t o t h e f a c t t h a t we a r e o b s e r v i n g a r o t a t i n g frame r e s o n -ance i n the l a b i s i m p o r t a n t o n l y when cos 0 i s s i g n i f i c a n t l y d i f f e r e n t from u n i t y . T h e r e f o r e the s e a r c h f o r the c o r r e c t v a l u e o f T s h o u l d be l i m i t e d t o f r e q u e n c i e s h i g h e r t h a n 3.5 MHz and low f i e l d s t r e n g t h s . The r e s u l t i s t h a t the b e s t r was found -7 -7 t o be 1.5x10 seconds. T h i s i s comparable to 1.9x10 c a l c u l a t e d from the peak t o peak d e r i v a t i v e l i n e w i d t h . The d i s c r e p e n c y -7 c o u l d be due t o the f a c t t h a t the 1.9x10 second f i g u r e was d e r i v e d from the u n s a t u r a t e d resonance l i n e o b t a i n e d from a low power k l y s t r o n . A l l the s i g n a l s were o b t a i n e d i n the r o t a t i n g frame w i t h H r f a s the p e r t u r b i n g f i e l d . The r a t h e r h i g h f i e l d s t r e n g t h s o f H r f s h o u l d i n d u c e s i g n i f i c a n t s a t u r a t i o n . The k l y s t r o n power w i l l not i t s e l f produce s i g n i f i c a n t s a t u r a t i o n s i n c e a l l o f the s i g n a l s were o b t a i n e d a t l e a s t 2 MHz o r about. 3 h a l f w i d t h s from the k l y s t r o n r e s o n a n t f r e q u e n c y . 19 The h a l f w i d t h a t resonance i s g i v e n by: where ^/£is the u n s a t u r a t e d l i n e w i d t h . F o r the y o f the e l e c t r o n -7 and a T o f 1.5x10 s e c . and Hr^.of .5 gauss, ^ ' - j f£y -33-1.9 The o b s e r v e d d i f f e r e n c e , 1 . 5 g i v e s o n l y Ayz =/*2fA'/x , but the argument e x p l a i n s the e f f e c t q u a l i t a t i v e l y . F i g u r e 15 shows the e f f e c t of i n c r e a s i n g H , w h i l e the o t h e r p a r a m e t e r s a r e h e l d c o n s t a n t , and f i n a l l y , f i g u r e 16 shows t h e e f f e c t o f i n c r e a s i n g H ,. I n some r e s p e c t the e f f e c t s o f i n c r e a s i n g H , o r H r^ a r e q u i t e s i m i l a r and the d i f f e r e n c e s r a t h e r s u b t l e . I t i s q u i t e easy t o see why the e f f e c t s s h o u l d be s i m i l a r . Only the component o f Yirf p e r p e n d i c u l a r t o H i s e f f e c t i v e i n c a u s i n g r e s o n a n t a b s o r p t i o n . T h i s component i s H t f s i n 0 . From f i g u r e 1, sine=H, /H e , so the p e r t u r b i n g f i e l d i s a c t u a l l y H /H c# where H e ^ i s d e t e r m i n e d by urf which i s k e p t c o n s t a n t . Thus i f the o n l y component o f H ^ w e r e a l o n g t h e z a x i s the e f f e c t s on the peak h e i g h t due t o H, and II,. ^  would be i n d i s t i n g u i s h a b l e . E x p e r i m e n t a l l y however t h e r e i s alw a y s a component o f H ^ i n the x-y p l a n e and t h i s component p r o b a b l y a c c o u n t s f o r t h e d i f f e r e n c e s i n the l i n e s h a p e o b s e r v e d even when the p r o d u c t H r i H , i s k e p t c o n s t a n t . Of c o u r s e H , e f f e c t s the a n g l e e and thus the s e p a r a t i o n o f the a b s o r p t i v e and e m i s s i v e resonance l i n e s . H ^ h a s no such e f f e c t . F i g u r e s 17 and 18 show a co m p a r i s o n o f the t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s f o r some t y p i c a l r u n s . The parameters i n the t h e o r e t i c a l c u r v e s have been chosen t o g i v e the b e s t f i t t o the e x p e r i m e n t . F i g u r e 17 shows r u n s a t two f r e q u e n c i e s w h i l e the o t h e r parameters a r e h e l d c o n s t a n t . The main e f f e c t i s c l e a r l y a change i n r e l a t i v e peak h e i g h t s . The parameters THE T H E O R E T I C A L E F F E C T OF C H A N G I N G T 9 = 3.S rfc FIGURE 14 -39-FIGURE 16 (c) Hrf=1.0g used f o r the t h e o r e t i c a l p l o t s a r e g i v e n b e s i d e the f i g u r e s " . The e x p e r i m e n t a l parameters agree w i t h i n e x p e r i m e n t a l e r r o r . F i g u r e 18 shows t h r e e c u r v e s which were run a t a p p r o x i m a t e l y the same f r e q u e n c y (2.6MHz) w h i l e H ( and U n v a r i e d . A g a i n the e x p e r i m e n t a l v a l u e s agree w e l l w i t h the t h e o r e t i c a l v a l u e s . A s l i g h t d i s c r e p a n c y i s t h a t near the c e n t e r o f the t r a c e s , i.e.,where (H ~ ^ ) = 0 , the e x p e r i m e n t a l c u r v e s appear b r o a d e r and show l e s s d e t a i l than the t h e o r e t i c a l ones. A p o s s i b l e e x p l a n a t i o n i s t h a t the t r e a t m e n t of L i F c r y s t a l s w i t h n e u t r o n i r r a d i a t i o n has the e f f e c t o f p r o d u c i n g p l a t e l e t s 20 o f l i t h i u m m e t a l of about l j i i n d i a m e t e r . The s k i n depth a t 34GHz i s 1 m i c r o n a l s o . T h i s means t h a t the microwave f i e l d s t r e n g t h c o u l d v a r y by 30% over the l i t h i u m p a r t i c l e o f lp. i n d i a m e t e r . I f the above e s t i m a t e of s i z e i s wrong and the s i z e i s a c t u a l l y say . l>i i n d i a m e t e r , the f i e l d s t r e n g t h 8 v a r i e s by o n l y 1 p a r t i n 10 • A c c e p t i n g the v a l u e q u o t e d F I G U R E 1 7 THEORY EXPERIMENT FIGURE I S THEORY f=2.60MHz -6 X.=lxl0 -7 r = I . 5 x l 0 sec, Hi = .40g Kri =.83g gauss H i =.60g A- H r i=1.03g 4 H, =.75g gauss above o f l p may a c c o u n t f o r minor d i s c r e p a n c i e s o f our theox'y and e x p e r i m e n t a l c u r v e s because near H~u/y-0 a v a r i a t i o n i n Hj c a u s e s a p p r e c i a b l e s h i f t i n g o f a b s o r p t i o n peak p o s i t i o n s which would f o r an inhomogeneous H, a c r o s s t h e sample l e a d t o a b r o a d e n i n g o f t h e l i n e s r e l a t i v e t o tho s e o b t a i n e d by the t h e o r y assuming a c o n s t a n t I I , over the ' p a i - t i c l e s i z e . W h i l e i t i s t r u e t h a t t h e r e s h o u l d be l i n e b r o a d e n i n g near the p o i n t where t h e k l y s t r o n i s r e s o n a n t w i t h the e l e c t r o n Larmor f r e q u e n c y , H 0—~-0 t due t o the s a t u r a t i o n o f the s p i n f o system w i t h t h e k l y s t r o n power, t h i s e f f e c t s h o u l d be i n c o r -p o r a t e d i n o u r B l o c h e q u a t i o n f o r m u l a t i o n and l e a d t o no d i s c r e p a n c i e s 'between the t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s . F i g u r e 19 shows the n=l reso n a n c e l i n e r u n a t about MHz and f o r f o u r v a l u e s o f H . E x p e r i m e n t a l l y II ,could not be changed enough t o o b s e r v e the e f f e c t of the l i n e shape and peak p o s i t i o n on H,.- T h i s i s s i m p l y a c c o m p l i s h e d w i t h the computer program however. A t the f r e q u e n c y chosen i s about 4.5 gauss and v i s i n g v a l u e s of H,of l,2,3,and4 gauss, a n g l e s o f up t o CO de g r e e s can be o b t a i n e d . I t i s n o t i c e d t h a t the l i n e w i d t h and peak p o s i t i o n ( d e t e r m i n e d by the d e r i v a t i v e z e r o ) a r e bo t h s t r o n g l y dependent on Q . F i g u r e 20 shows f o u r c u r v e s . The c u r v e denoted "computer" i s the peak t o peak l i n e w i d t h as a f u n c t i o n of H , d e t e r m i n e d from the computer p l o t . The second shows the peak t o peak w i d t h e x p e c t e d on t h e b a s i s o f a c o n s t a n t 'width i n the r o t a t i n g frame. The t h i r d c u r v e i s f i t t e d t o n a t c h the second c u r v e . I t s 'formula i s : Y~5/8X~.£0 where Y i s the v a l u e f o r a p o i n t on c u r v e COMPARISON OF DIFFERENT PREDICTIONS OF THE LINE WIDTH AS MEASURED IN THE LAB-. FRAME 30 3o 10 SO €0 70 Fz&. 2 0 t h r e e and X i s a v a l u e on c u r v e two. The f o u r t h c u r v e i s a. 2 "'. . p l o t o f ( S c o s O ^ l } w h i c h shows the dependency w h i c h one c o s o w ould e x p e c t i f t h e R e d f i e l d t h e o r y were s t r i c t l y a p p l i c a b l e . W h i l e t h e v a l u e s o f H , used i n the e x p e r i m e n t s ' w e r e not l a r g e enough t o c o m p l e t e l y d e t e r m i n e t h i s dependency, f i g u r e 9 shows t h a t t h e l i n e w i d t h d i d i n c r e a s e r o u g h l y as the a s s u m p t i o n o f a c o n s t a n t w i d t h i n t h e r o t a t i o n frame i n d i c a t e d i t would. From f i g u r e 1 t h e s e p a r a t i o n of the two n-1 resonance * > '/z l i n e s i n t h e l a b frame s h o u l d be H ^ cose=(II ( ? J f-H, ). S i n c e t h i s i s t h e f o r m u l a used t o o b t a i n H , from the e x p e r i m e n t a l r e s u l t s i t i s i n t e r e s t i n g t o see i f i t h o l d s f o r t h e t h e o r e t i c a l p l o t s . The s e p a r a t i o n i s computed from f i g u r e 20 f o r cose= .975, .895, .745, and .460 and t h i s i s coiapared w i t h t h e s e p a r a t i o n a t 0=0 t i m e s c o s 6 . The s e p a r a t i o n o f t h e l i n e s on t h e t h e o r e t i c a l p l o t does not f o l l o w BcosG e x a c t l y where B- the s e p a r a t i o n a t 0-0. However f o r s m a l l v a l u e s o f G t h e agreement i s good and the c o n c l u s i o n i s t h a t e q u a t i o n 1.10 i s a v a l i d one t o c a l c u l a t e H ,. F i g u r e 21 shows .this c o m p a r i s o n . The l i n e s e p a r a t i o n i s a l s o a f f e c t e d by the B l o c h S i e g a r t e f f e c t . The e x p r e s s i o n f o r the p o s i t i o n of the r e s o n a n c e i n terms o f the p o s i t i o n where resonance would o c c u r n e g l e c t i n g t h e e f f e c t i s g i v e n by_. •---•( ' T a k i n g H =11 as a t y n i c a l example u ~ 7€ ^ ° T h i s i s r* I T a s h i f t o f a b o u t 6%. The u n c e r t a i n t i e s i n H rbetween one r u n and a n o t h e r a r e g r e a t e r t h a n t h i s so no e x p e r i m e n t a l v e r i f i -c a t i o n o f t h e B l o c h - S i e g a r t e f f e c t was made. 21 SEPARATION IN THE LAB. FRAME OF THE TWO N=l RESONANCE LINES AS A FUNCTION OF 0. (a) AS FOUND FROM THE THEORY DEVELOPED IN THE TEXT, AND (b) C039 FOR COMPARISON .? .g .? . 6 , i -47-5. T H E M E A S U R E M E N T O F T H E R E L E V A N T M A G N E T I C F I E L D S I t w a s n e c e s s a r y t o m e a s u r e a n d c a l i b r a t e t h e v a r i o u s m a g n e t i c f i e l d s u s e d i n t h e e x p e r i m e n t . T h e m e a s u r e m e n t o f ,H, i s d i f f i c u l t e x p e r i m e n t a l l y . I t c a n b e d e t e r m i n e d t o s o m e d e g r e e o f a c c u r a c y f r o m t h e e x p e r i m e n t a l c u r v e s a n d t h e e q u a t i o n 1.10. H o w e v e r i t w o u l d b e d e s i r e a b l e t o m a k e a n i n d e p e n d e n t d i r e c t m e a s u r e m e n t o f H,. T h e b e s t w a y t o d o t h i s i s t o m e a s u r e t h e i n c i d e n t p o w e r t o t h e c a v i t y , t h e r e f l e c t e d p o w e r f r o m t h e c a v i t y a n d t h e Q o f t h e c a v i t y . T h e n k n o w i n g t h e f i e l d d i s t r i b u t i o n i n t h e c a v i t y o n e c a n c a l c u l a t e t h e a p p r o x i m a t e f i e l d s t r e n g t h a t a n y g i v e n p o i n t . F r o m t h e d e f i n i t i o n o f Q a n e q u a t i o n r e l a t i n g t h e Q , t h e p o w e r d i s s i p a t e d i n t h e c a v i t y , a n d t h e m a g n e t i c f i e l d 22 d i s t r i b u t i o n i n t h e c a v i t y c a n b e f o u n d . T h e c r i t i c a l m e a s u r e m e n t i s t h u s a m e a s u r e o f t h e i n c i d e n t a n d r e f l e c t e d p o w e r t o t h e c a v i t y . T h e s e m e a s u r e m e n t s w e r e a t t e m p t e d w i t h a H e w l e t t P a c k a r d b o l o m e t e r a n d p o w e r m e a s u r e -m e n t m e t e r b u t e x p e r i m e n t a l d i f f i c u l t i e s m a d e t h i s i n a c c u r a t e . T h e m a i n d i f f i c u l t y i s t h a t t h e b o l o m e t e r c o u l d n o t b e s u b -j e c t e d t o o v e r a f e w m i l l i w a t t s w h i l e t h e k l y s t r o n p r o d u c e d a b o u t 10 w a t t s o f p o w e r . T h e r e f o r e t w o l O d b d i r e c t i o n a l c o u p l e r s w e r e u t i l i z e d t o r e d u c e t h e i n c i d e n t p o w e r b y a 4 f a c t o r o f 10. T h e r e f l e c t i o n s f r o m t h e c o u p l i n g s a n d - 4 8 -d i s c o n t i n u i t i e s i n the gui d e system made i t d i f f i c u l t i f not i m p o s s i b l e t o de t e r m i n e the d i f f e r e n c e o f the i n c i d e n t and r e f l e c t e d power t o the c a v i t y w i t h s u f f i c i e n t a c c u r a c y . A l s o i t was not p o s s i b l e t o l o c k the k l y s t r o n t o the c a v i t y and .as the k l y s t r o n d r i f t s s l i g h t l y from the c a v i t y r e s o n a n t f r e q u e n c y the f i e l d s t r e n g t h i n the c a v i t y can f l u x u a t e s i g n i f i c a n t l y . I t i s f o r t h e s e r e a s o n s t h a t no a c c u r a t e e x p e r i m e n t a l v a l u e can be g i v e n t o the H , f i e l d s w hich i s indepen d e n t o f the l i n e s e p a r a t i o n measurements. The r e s u l t s o f the power meter measurement was t h a t a maximum o f 1 t o 2 w a t t s o f power v/ere d i s s i p a t e d i n the c a v i t y . The l o a d e d Q was measured t o be 1600 and a c a l c u l a t i o n shows t h a t the maximum f i e l d s t r e n g t h i n the c a v i t y i s about 1 . 4 3 t i m e s the average f i e l d . These v a l u e s g i v e w i t h our e q u a t i o n above a maximum H = 3 . 2 gauss. T h i s i s i n agreement w i t h the v a l u e s o b t a i n e d by Enga but the l a r g e s t H f f i e l d s measured i n the p r e s e n t e x p e r i m e n t s were about one gauss. T h i s i s p r o b a b l y due t o the f a c t t h a t the sample g r e a t l y r e d u c e s the Q and most o f the power i s d i s s i p a t e d i n the sample and not . i n the c a v i t y . From e q u a t i o n 5.1 i t i s seen t h a t the average f i e l d s t r e n g t h i s p r o p o r t i o n a l t o t h e square r o o t of Q f o r a g i v e n power d i s s i p a t e d . A d e t e r m i n a t i o n o f the f i e l d s t r e n g t h H was made i n d i r e c t l y . In the s p e c t r o m e t e r c i r c u i t t h e r e a r e two 51 K r e s i s t o r s w hich a r e i n s e r i e s and p l a c e d i n p a r a l l e l w i t h the s p e c t r o m e t e r c o i l . Hence the v o l t a g e a c r o s s the r e s i s t o r s i n d i c a t e s the v o l t a g e a c r o s s the c o i l . I f an-attempt was made t o measure the v o l t a g e a c r o s s the c o i l d i r e c t l y w i t h the o s c i l l i s c o p e the o s c i l l a t o r was i n t e r r u p t e d and ceased t o o s c i l l a t e . However the v o l t a g e c o u l d be measured a c r o s s one o f the r e s i s t o r s on a f a s t r i s e t i me o s c i l l i s c o p e . The form seemed t o be a pure s i n e wave w i t h v e r y l i t t l e harmonic c o n t e n t o v e r the s t r e n g t h s and f r e q u e n c i e s used. Of p a r t i c -u l a r i n t e r e s t i s t h a t the v o l t a g e v a r i e d i n d i r e c t p r o p o r t i o n t o the g r i d c u r r e n t as r e a d on the o s c i l l a t o r meter. F i g u r e 22 i s a p l o t o f the i n d u c e d c o i l v o l t a g e vs the c u r r e n t r e a d on the o s c i l l a t o r meter. T h i s d a t a was f o r a 100 t u r n c o i l o p e r a t i n g a t a p p r o x i m a t e l y 3.5 MHz. « 1 3 H S~ FIG. VOLTAGE ACROSS COIL IN VOLTS - 5 0 -Assuming t h a t the sample i s c o n f i n e d t o a p o i n t on the a x i s o f the f i e l d produced by the c o i l and t h a t t h e r e a r e n e g l i g i b l e s h i e l d i n g e f f e c t s due t o the c a v i t y , the f i e l d s t r e n g t h a t the sample can be c a l c u l a t e d knowing the v o l t a g e i n d u c e d a c r o s s i t and the f r e q u e n c y o f the o s c i l l a t o r . A s k e t c h o f the c a l c u l a t i o n f o l l o w s . 6=£3 c o n . 2 where B i s i n Webers per meter a i s i n meters and I i s i n amps, z i s the d i s t a n c e of the sample from the c e n t e r o f the c o i l . Then Where B i s i n gauss E i s i n v o l t s and f i s i n megacycles. U s i n g t h i s c a l c u l a t i o n and f i g u r e 22 i t i s seen t h a t f o r a f r e q u e n c y o f 3.5 MHz, H v ^ c o u l d be v a r i e d from about .4 t o 1.3 gauss when the g r i d c u r r e n t r e a d i n g v a r i e d from 7 t o 30 |iA. T h i s c o r r e s p o n d s c l o s e l y w i t h the e x p e r i m e n t a l r e s u l t s as f i t t e d by the computer program t o our t h e o r y . The l a r g e magnetic f i e l d H nwas s e t by a f i e l d r e g u l a t e d power s u p p l y d r i v i n g the Magnion magnet. However the f i e l d d i a l e d i n on the power s u p p l y was found to be m i s c a l i b r a t e d by approx-i m a t e l y 100 gauss. A l s o the sweep r a t e q u o t e d on the power s u p p l y was found t o be i n e r r o r . In f a c t the sweep r a t e depended on whether the sweep was up or down. T h i s was r e a l i z e d from the w i d t h of the s i g n a l s . S i n c e i t was d e s i r e d t o know the s e p a r a t i o n of the r e s o n a n c e s a c c u r a t e l y the sweep had t o be c a l i b r a t e d . The way t h i s was done a c c u r a t e l y was t o use a s e p a r a t e m a r g i n a l o s c i l l a t o r m o n i t o r i n g the m a g n e t i z a t i o n i n a g l y c e r i n e ^ sample. T h i s o s c i l l a t o r was o p e r a t e d a t about 55 MHz f o r the p r o t o n resonance i n a 12 k i l o g a u s s f i e l d . The sweep r a t e c o u l d be found t o t h r e e s i g n i f i c a n t f i g u r e s t h i s way. At the 5 gauss p e r minute s e t t i n g the sweep up was 3.49 gauss per minute w h i l e the r a t e down was 5.04 gauss per minute. A c t u a l l y the r a t e was c a l i b r a t e d on the c h a r t paper so t h a t no e r r o r s owing t o a m i s c a l i b r a t i o n of the c h a r t speed c o u l d e n t e r . -52-APPENDIX T h i s a p pendix i n c l u d e s a complete s t a t e m e n t of the com-p u t e r program used t o s o l v e e q u a t i o n s ( 2 - 1 1 ) ~ ( 2 ~ 2 9 ) f o r Mf 1 and d/dt Mf"1.' Below i s a b r i e f d e s c r i p t i o n o f the symbols which appear i n the program t h a t f o l l o w s . I n l i n e one M r e f e r s t o the m a t r i x o f the c o e f f i c i e n t s o f the m a g n e t i z a t i o n . The d i m e n s i o n 50 i s w r i t t e n here as i t wasn't a p p a r e n t i n the b e g i n n i n g t h a t a 36x36 would be adequate. The m a j o r i t y o f t h e program i s the r e a d i n g i n o f t h e m a t r i x e l e m e n t s . A l l s t a t e -ments o f t h e form M(a,b) r e f e r t o s p e c i f i c e l e m e n t s . The numbers which must be s u p p l i e d f o r t h e e q u a t i o n s a r e : N, the d i m e n s i o n o f the m a t r i x , which i n t h i s case i s 36, A, the maximum v a l u e o f (E0-y ), B, the a n g u l a r f r e q u e n c y o f the r a d i o f r e q u e n c y f i e l d , C l , the s p i n l a t t i c e r e l a x a t i o n t i m e , D, the f i e l d s t r e n g t h of the microwave f i e l d R^ , G, the gyromagnetic r a t i o o f the e l e c t r o n , GM, the r a d i o f r e q u e n c y f i e l d s t r e n g t h , DA, the i n c r e m e n t i n ( H e - y) which was t a k e n t o be .1 gauss, Dend i s the end o f the sv/eep range o f (Ha~y) which was -2 gauss, GA i s the e l e c t i ' o n i c magnetic s u s c e p t i b i l i t y o f -6 l i t h i u m which I chose t o be 1x10 the P a u l i paramagnetism r e s u l t . E x p e r i m e n t a l e v i d e n c e i n d i c a t e s t h i s i s low, the 23 t r u e v a l u e b e i n g two t o t h r e e t i a e s as l a r g e . However as d i s c u s s e d e a r l i e r t h i s e n t e r s o n l y as a s c a l i n g f a c t o r and hence o n l y the r e l a t i v e v a l u e i s o f i n t e r e s t h e r e . The a c t u a l s o l u t i o n o f the m a t r i x i s c a r r i e d out i n a computer l i b r a r y s u b r o u t i n e . The d e r i v a t i v e i s o b t a i n e d by t a k i n g -53-d i f f e r e n c e s o f M f o r s u c c e s s i v e v a l u e s o f ( H 0 ~ u ), T h i s r has the e f f e c t of s h i f t i n g the c e n t e r o f the ;trace .05 .:, gauss t o the r i g h t . The program i n c l u d e s i n s t r u c t i o n s t o have the o u t p u t p l o t t e d , though i t does not i n c l u d e a s c a l i n g r o u t i n e and the axes must be a d j u s t e d t o f i t the p a r t i c u l a r o u t p u t . -54-''ftJTIME 10 . : $PAGE 50 $IBFIC MAIN DIMENSION M(50,50) , 0(50) CALL PLOTS REAL M NSOS =10 YMIN^2.5E-3 XMIN=-6. YL=10. XL=3. DY=.5E-3 READ 10, N, A, B, C I , D 10 FORMAT (15, 4D15.8) READ 20, G, GM, DA, DEND 20 FORMAT (4E15.8) READ 21, GA, GAI 21 FORMAT (2E15.5) PRINT 101,N,A,B,C1,D,G,GM,DA,DEND,GA,GAI DAD=A 101 F0RMAT(I5,5E16.8/5E16.8) NLL=(DEND-DAD J/DA+1.5 CALL GRID (NLL,DAD,DEND,.5,101,-20.,20.,4.} DO 113 11=1,5 CALL AXIS (0.,0.,6HX AXIS,-6,XL,0.,XMIH,DX) CALL AXIS (0.,0.,6HY AXIS,6,YL,90.,YMIN,DY) A=DAD 100 DO 30 1=1,N DO 30 J=1,N M(I,J)=0.0 30 CONTINUE DO 40 1=1,N M(I,I)=1.0 40 CONTINUE M(7,3)=-1.0*B*C1 M(9,10)=M(7,8) M(11,12)=M(7,8) M(13,14)=M(7,8)*2.0 M(15,16)=M(13,14) M(17,18)=M(13,14) M(19,20)=M(7 f8)*3.0 M(21,22)=M(19,20) M(23,24)=M(19,20) M(25,26)=M(7,8)*4.0 M(27,28)=M(25,26) M(29,30)=M(25,26) M(31,32)=M(7,S)*4.0 M(33,34)=M(31,32) M(35,36)=M(31,32) -55-FACT=-1.0*G*C1*A NCUE=0 NST=-1 70 DO 50 1=1,N NCUE=NCUE+1 IF(NCUE.LT.3) GOTO 51 IF(NCUE.LT.6) GOTO 50 NCUE=0 ' GOTO 50 51 J=I-i-2 M(I,J)=FACT 50 CONTINUE IF(NST.GT.O) GOTO 60 NST=10 NCUE=4 FACT=1.G*C1*G*D GOTO 70 60 DO 80 1=1,N DO 80 K=l,2 J=I+K 80 M ( J , I ) = M( I , J ) M(3,7}=2.0*C1*G*GM M(9,13)=M(3,7)/2.0 M(10,14)=M(9,13) M(15,19)=M(9,13) M(16,20)=M(9,13) M(22,26)=M(9,13> M(1,9)=-1.0*M(3,7) M(7,15)=M(l,9)/2.0 M(8,16)=M(7,15) M(13,21)=M(8,16) M (14,22) =41 (13, 21) M(19,27)=M(14,22) M(20,28)=M(19,27j M(7,3)=M(20,28) . M(8,4)=M(7,3) M(13,9)=M(8,4) M(14,10)=M(13,9) M(19,15)-M(14,10) M(20,16)=M(19,15) M(25,21)=M(20,16) M(26,22)=M(25,21) M(9,1)=~1.0*M(26,22) M(10,2)=M(9,1J M(15,7)=M(10,2) M(16,S)=M(15,7) M(21,13)=M(16,8) M(22,14)=M(21,13) M(27,19)=M(22,14) -56-M(28,20)=M(27,19) M(25,33)=M(7,15) M(26,34)=M(25,33) M(31,27)=M(26,34) M(32,28)=M(31,27) M(27,31)=-1.0*M(32,28) . M(28,32)=M(27,31) M(33,25)=M(28,32) M(34,26)=M(33,25) DO 90 1=1,N 90 C(I)==0.0 C(2)=GAI*D C(5)=1.2E-2 C(6)=GAI*A C(11)«GA*GM IF(NSOS.GT.O) GO TO 200 NS0S=10 CALL MATOUT (M,N,50) CALL VECOUT (C,N) 200 CONTINUE CALL SOLTN(H,C,N,50,DET) CC=(C(12)-CC)*10. PRINT 78,A,C(12),CC,DET 78 FORMAT (1P4E16.6) CALL POINT (A,CC*l.E-!-4,II) XX=(A-XMIN)/DX YY=(CC-YMIN)/DY CALL SYMBOL (XX,YY,.07,3,0.,-1) CC=C(12) A=A+DA. IF (A.GT.DEND) GO TO 112 GO TO 100 112 D=D+.25 CALL PLOT (.1. 5*XL, 0. , -3 ) 113 CONTINUE CALL PLOTND CALL OUTPUT 99 STOP END -57-BIBLIOGRAPHY 1. Pake, G. , P a r a m a g n e t i c Resonance W.A. Benjamin I n c . New York. 1962 2. R e d f i e l d , A., Phys. Rev. 98, 1787 (1955) 3. F r a n z , J . , D o c t o r a l T h e s i s , U. o f I l l i n o i s , 1965 ( u n p u b l i s h e d ) 4-5. Abragam, A., The P r i n c i p l e s o f N u c l e a r Magnetism ( O x f o r d U n i v e r s i t y P r e s s . London, 1961") ~~ 6-7. R e d f i e l d , A., Phys. Rev. 98, 1787 (1955) 8. Enga, E., M a s t e r s T h e s i s , U,B eC., 1966 ( u n p u b l i s h e d ) 9. R y t e r , Ch., Phys. Rev. L e t t e r s , 10, (1960) 10. G a r s t e n s , M.A. and K a p l a n , J . I . , Phys. Rev. 99, 459 (1955) 11. W h i t f i e l d , G., and R e d f i e l d , A.G., Phys. Rev. 106, 918 (1957) 12. Koss, T.A., A l e x a n d e r , S., and S c h w e r d t f e g e r , C.F. Can. J . Phys. (August 1968) 13. Enga, E., op. c i t . 14. Benedek, G.B. and K u s h i d a , T., Phys. Rev. 118, 46 (1960) 15. V o l k o f f , G., P e t c h , H., and S m e l l i e , D., Can. J . Phys. 30, 270 (1952) 16-17. W i n t e r , J.M., Ann. de Phys. 4, 745 (1959) 18. S c h w e r d t f e g e r , C.F. and Koss, T.A., Phys. Rev. 166, 259 (1968) 19. Abragam, A., op. c i t . 20. R y t e r , Ch., op. c i t . 21. Abragam A., op. c i t . 22. P o r t i s , A.M., Phys. Rev. 91, 1071 (1953) 23. Schumacher, R.T. and S l i c h t e r , C., Phys. Rev. 101, 58 (1956) 

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