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Electric quadrupole interaction in the nuclear magnetic resonance Smellie, Donald William 1951

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nr> $7 ELECTRIC QUADRUPOLE INTERACTION IN THE NUCLEAR MAGNETIC RESONANCE DONALD WILLIAM SMELLIE 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 the r e q u i r e m e n t s f o r the d e g r e e o f M a s t e r o f A p p l i e d S c i e n c e i n t h e Department o f E n g i n e e r i n g P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e s t a n d a r d r e q u i r e d f r o m c a n d i d a t e s f o r the d e g r e e o f M a s t e r of A p p l i e d S c i e n c e Members o f the Department 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 September 1951 ABSTRACT The g e n e r a l case o f the dependence on c r y s t a l o r i e n t a t i o n o f the f r e q u e n c y s p l i t t i n g o f the n u c l e a r magnetic resonance a b s o r p t i o n l i n e i n a s i n g l e c r y s t a l sample i n t o 21 components due t o the c o u p l i n g between the n u c l e a r e l e c t r i c quadrupole moment and the e l e c t r i c f i e l d g r a d i e n t i n the c r y s t a l a t the s i t e o f the n u c l e i i n q u e s t -i o n i s examined b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y . The t h e o r e t i c a l p a r t o f t h i s t h e s i s c o n s i s t s of a d e t a i l e d e x p o s i t i o n o f the f i r s t o r d e r p e r t u r b a t i o n t h e o r y a p p l i c a b l e t o c r y s t a l s w i t h a x i a l l y symmetric f i e l d s which was o u t l i n e d i n condensed form by P r o f . R. V. Pound, and of the e x t e n s i o n o f t h i s f i r s t o r d e r t h e o r y t o n o n - a x i a l l y symmetric cases suggested by P r o f . G. M. V o l k o f f . The e x p e r i m e n t a l p a r t o f t h i s t h e s i s c o n s i s t s o f the d e s c r i p t i o n o f an e x p l o r a t o r y e xperiment proposed by the a u t h o r , and performed by him, w i t h some a s s i s t a n c e from Mr. H. E. P e t c h , f o r the purpose o f o b t a i n i n g a p r e l i m i n a r y check on the t h e o r y , and o f d e m o n s t r a t i n g the f e a s i b i l i t y o f a l a t e r more c a r e f u l l y p e r f o r m e d e x p e r i m e n t . Such an experiment w i t h an improved c r y s t a l mount has s i n c e then been performed by Mr. P e t c h . A s i n g l e c r y s t a l o f spodumene was us e d , and the for Li? a n g u l a r dependence o f the f r e q u e n c y s p l i t t i n g ^ w a s measured as the c r y s t a l was r o t a t e d about the C - a x i s o f i t s m o n o c l i n i c s t r u c t u r e . The apparatus used was an o s c i l l a t i n g d e t e c t o r type o f n u c l e a r magnetic resonance s p e c t r o m e t e r p r e v i o u s l y d e s i g n e d and b u i l t by Dr. T. L. C o l l i n s . The a x i s o f r o t a t i o n o f the c r y s t a l was k e p t p e r p e n d i c u l a r t o the u n i f o r m magnetic f i e l d o f the s p e c t r o m e t e r . The d i r e c t i o n o f the fci-axis o f the c r y s t a l i n f e r r e d from t h i s r o t a t i o n was found t o c o i n c i d e w i t h i n e x p e r i m e n t a l e r r o r w i t h , i t s o r i e n t a t i o n as o b t a i n e d l a t e r by s t a n d a r d o p t i c a l methods. An experiment was d e s i g n e d t o i n d i c a t e the r e l a t i v e magnitudes o f the s p i n - l a t t i c e r e l a x a t i o n b y m a g n e t i c d i p o l e and e l e c t r i c quadrupole c o u p l i n g i s proposed. - i i -ACKNOWLEDGMENTS The a u t h o r i s i n d e b t e d t o Dr. G. M. V o l k o f f f o r h i s p a t i e n t guidance i n t h i s work as w e l l as o t h e r p roblems, and f o r the e l u c i d a t i o n o f the g e n e r a l t h e o r y o f the e l e c t r i c quadrupole i n t e r a c t i o n . Mr. H. E. P e t c h deserves most of the c r e d i t f o r the e x c e l l e n t f u n c t i o n i n g o f the apparatus w h i l e t h i s work was i n p r o g r e s s . He a l s o v e r y k i n d l y took some o f the e x p e r i m e n t a l d a t a . The a u t h o r i s a l s o i n d e b t e d t o the f o l l o w i n g members o f the Department o f Geology and Geography: Dr. R. M. Thompson, f o r p r o v i d i n g the spodumene c r y s t a l , Mr. J . A. Gower f o r e n l i g h t e n i n g d i s c u s s i o n s , Dr. L. D. Mantuani f o r k i n d l y l o c a t i n g the o p t i c a l a x i s o f the c r y s t a l , and Dr. K. C. MacTaggart f o r f u r t h e r o p t i c a l work on the c r y s t a l w h i c h l e d t o a c o r r e c t i d e n t i f i c a t i o n o f i t s axes. L a s t l y , I would l i k e t o thank my e a r l y t e a c h e r s , Dr. H. W. Harkness, Dr. J . A. "Gray, P r o f . J . K. Robertson; and r e s e a r c h s u p e r v i s o r s , Dr. H i n Lew and Dr. L. G. E l l i o t t , f o r t h e i r e a r l y i n s p i r a t i o n w h i c h u l t i m a t e l y l e d t o t h i s work. The s u p p o r t o f t h e N.R.C. i n t h e form o f a B u r s a r y t o t h e a u t h o r and a g r a n t - i n - a i d o f t h e p r o j e c t t o Dr. V o l k o f f i s g r a t e f u l l y acknowledged. - i i i -TABLE OP CONTENTS Page ABSTRACT i ACKNOWLEDGMENTS i i i INTRODUCTION 1 CHAPTER I : THE PHENOMENON OF NUCLEAR PARA-MAGNETIC RESONANCE 3 CHAPTER I I : THE APPARATUS 7 CHAPTER I I I : THE NUCLEAR ELECTRIC QUADRUPOLE INTERACTION IN CRYSTALS 10 T h e o r y 11 The Spodumene C r y s t a l 22-, E x p e r i m e n t a l 23 C o n c l u s i o n s 25 CHAPTER IV : SUGGESTIONS FOR FURTHER WORK .. 26 REFERENCES : L I S T OF ILLUSTRATIONS F i g . 1 - R e c o r d e d A b s o r p t i o n L i n e D e r i v a t i v e s , f a c i n g page 23 F i g . 2 - F r e q u e n c y s p l i t t i n g i n k i l o c y c l e s o f s a t e l l i t e a b s o r p t i o n l i n e s as a f u n c t -i o n o f a n g l e between b - a x i s and Ho, f o r the t h r e e - l i n e L i 7 n u c l e a r m a g n e t i c r e s o n a n c e a b s o r p t i o n s p e c t r u m i n spodumene f a c i n g page ELECTRIC QUADRUPOLE INTERACTION IN THE NUCLEAR MAGNETIC RESONANCE INTRODUCTION N u c l e a r p a r a m a g n e t i c r e s o n a n c e s a b s o r p t i o n (PTP) o r n u c l e a r i n d u c t i o n (BHP) p r o v i d e s a means o f o b s e r v i n g the i n t e r a c t i o n between a t o m i c n u c l e i and t h e i r s u r r o u n d i n g s . These i n t e r a c t i o n s may be e i t h e r o f a m a g n e t i c o r e l e c t r i c t y p e . F o r n u c l e i o f s p i n 1/2, t h e i n t e r a c t i o n s are o f n e c e s s i t y m a g n e t i c o n l y . N u c l e i o f s p i n I g r e a t e r t h a n 1/2 may p o s s e s s an e l e c t r i c q u a d r u p o l e moment w h i c h w i l l i n t e r a c t w i t h th e g r a d i e n t o f an e l e c t r i c f i e l d a t the n u c l e u s . I n l i q u i d s , where t h e e l e c t r i c f i e l d a t t h e n u c l e u s i s o s c i l l a t i n g r a p i d l y due t o random m o t i o n s o f n e i g h b o u r i n g i o n s , t h i s , e l e c t r i c q u a d r u p o l e i n t e r a c t i o n may w e l l be the d o m i n a t i n g mechanism f o r the t h e r m a l o r 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 ( P 5 0 ) . I n i o n i c c r y s t a l s , t h e o s c i l l a t i n g e l e c t r i c f i e l d s due t o l a t t i c e v i b r a t i o n s a l s o g i v e r i s e t o an e l e c t r i c q u a d r u p o l e i n t e r a c t i o n w h i c h w i l l be t h e d o m i n a t i n g r e l a x a t i o n mechanism i f p a r a m a g n e t i c i m p u r i t i e s 2 are absent (B48, P51). I n some c r y s t a l s , t h e r e w i l l be a s t e a d y component o f the e l e c t r i c f i e l d g r a d i e n t a t the s i t e o f the n u c l e u s under s t u d y , and the n u c l e a r magnetic resonance a b s o r p t i o n l i n e w i l l be broken up i n t o 21 components due to a p e r t u r b a t i o n o f the normal Zeeman l e v e l s o f the n u c l e a r s p i n system. The f r e q u e n c y s e p a r a t i o n between components depends on c r y s t a l o r i e n t a t i o n , s i n c e the l a t t e r d e t e r -mines the r e l a t i o n between the e x t e r n a l magnetic f i e l d and the p r i n c i p a l axes w i t h r e s p e c t t o which the 3X3 m a t r i x r e p r e s e n t i n g the r e c t a n g u l a r components o f the e l e c t r i c f i e l d g r a d i e n t t e n s o r i s d i a g o n a l i z e d . The a n g u l a r dependence o f the f r e q u e n c y s p l i t t i n g o f the s a t e l l i t e a b s o r p t i o n l i n e s f o r the L i 7 resonance i n c r y s t a l l i n e L i A l ( S i 0 3 ) 2 s e r v e s as an i l l u s t r a t i o n of the g e n e r a l case of the e l e c t r i c quadrupole i n t e r a c t i o n i n a c r y s t a l w i t h n o n - a x i a l symmetry. R. V. Pound, i n the p i o n e e r i n g work i n t h i s s u b j e c t (P50) has examined the e l e c t r i c quadrupole i n t e r -a c t i o n i n c r y s t a l s p o s s e s s i n g an a x i s o f symmetry. The p r e s e n t work i n v e s t i g a t e s the more g e n e r a l case o f c r y s t a l s w i t h n o n - a x i a l l y symmetric f i e l d s . I t i s f o u n d t h a t the a n a l y s i s o f the a n g u l a r dependence f o r t h r e e r o t a t i o n s about m u t u a l l y p e r p e n d i c u l a r axes y i e l d s the c o u p l i n g energy f o r the c r y s t a l , the e l e c t r i c f i e l d g r a d i e n t asymmetry, and the o r i e n t a t i o n o f the p r i n c i p a l axes of the e l e c t r i c f i e l d g r a d i e n t t e n s o r . 3 CHAPTER I THE PHENOMENON OF NUCLEAR PARAMAGNETIC RESONANCE Si n c e i t s c o n c e p t i o n i n 1946 by the groups at S t a n f o r d (BHP) and Har v a r d (PTP) the phenomenon o f n u c l e a r paramagnetic resonance a b s o r p t i o n o r n u c l e a r i n d u c t i o n has r e c e i v e d much a t t e n t i o n b o t h t h e o r e t i c a l l y and ex p e r i m e n t -a l l y . S e v e r a l e x c e l l e n t r e v i e w a r t i c l e s have been w r i t t e n , t o w h i c h the re'ader i s r e f e r r e d (Pu48, R48, Pa50). I n view of the l a r g e volume o f e x i s t i n g l i t e r a t u r e , o n l y a b r i e f d i s c u s s i o n w i l l be g i v e n . f i e l d Ho i t w i l l p r e c e s s about the f i e l d d i r e c t i o n w i t h a f r e q u e n c y c a l l e d the Larmor p r e c e s s i o n f r e q u e n c y which i s independent o f the angle between Ho and JU, the magnetic d i p o l e moment. A n u c l e u s o f s p i n I , magnetic moment JH may e x i s t i n a magnetic f i e l d i n one o f a d i s c r e t e s e t o f energy l e v e l s c a l l e d the Zeeman l e v e l s . These l e v e l s c o r r e s p o n d to d i f f e r e n t v a l u e s o f M, the p r o j e c t i o n o f the n u c l e a r a n g u l a r momentum o r s p i n v e c t o r I on the f i e l d d i r e c t i o n . The energy l e v e l s are such t h a t a t r a n s i t i o n between a d j a c -ent l e v e l s i n v o l v e s the e m i s s i o n o r a b s o r p t i o n o f a quantum of energy o f f r e q u e n c y e q u a l t o the Larmor p r e c e s s i o n frequency;. One e x p e r i m e n t a l t e c h n i q u e i n v o l v e s p l a c i n g a I f a.magnetfc d i p o l e i s p l a c e d i n a magnetic 4 s m a l l c o i l e x c i t e d by an r - f . v o l t a g e w i t h i t s a x i s a t r i g h t a n g l e s to the s t e a d y magnetic f i e l d Ho and d e t e c t i n g the a b s o r p t i o n o f r - f . energy by a sample p l a c e d w i t h i n the c o i l when the Larraor p r e c e s s i o n f r e q u e n c y i s re a c h e d . S i n c e the p r o b a b i l i t y f o r a b s o r p t i o n o f a photon by the n u c l e a r s p i n system i s the same as the p r o b a b i l i t y f o r in d u c e d e m i s s i o n , t h e r e w i l l be a n e t a b s o r p t i o n i n the sample o n l y i f the number o f n u c l e i i n l o w e r energy s t a t e s i s g r e a t e r than the number i n h i g h e r energy s t a t e s . T h i s i s r e a l i z e d because o f the Boltzmann f a c t o r . I f a sample c o n t a i n i n g n u c l e a r magnetic moments i s s u d d e n l y t h r u s t i n t o the gap o f an e l e c t r o m a g n e t p r o d u c i n g a h i g h f i e l d , the excess number o f n u c l e i i n lower energy s t a t e s w i l l i n i t i a l l y be z e r o and w i l l approach the e q u i l i b r i u m v a l u e d e f i n e d by the Boltzmann d i s t r i b u t i o n a c c o r d i n g t o the r e l a t i o n n = no |Ti _ exp ( - t / T i ) J where T i i s c a l l e d the s p i n - l a t t i c e o r t h e r m a l r e l a x a t i o n t i m e . T i i s det e r m i n e d by the i n t e r a c t i o n between the n u c l e a r s p i n system and f i e l d s o f f r e q u e n c y a t o r v e r y near the Larmor p r e c e s s i o n f r e q u e n c y caused by random motions i n the s u r r o u n d i n g l a t t i c e . I t i s a measure of the time r e q u i r e d f o r t h e r m a l e q u i l i b r i u m t o be e s t a b -l i s h e d i n the sample. T i a l s o determines the r a t e a t whi c h the s p i n system " r e l a x e s " a f t e r i t has absorbed r - f . energy, i n o t h e r words, t r a n s m i t s energy t o the l a t t i c e and r e t u r n s to i t s o r i g i n a l energy s t a t e . F or a g i v e n r - f . l e v e l i n the d r i v i n g c o i l , T]_ must be s u f f i c i e n t l y s h o r t o r the sample w i l l s a t u r a t e . The r e l a t i v e phases o f p r e c e s s i n g n u c l e a r s p i n s w i l l be d e s t r o y e d i n a time o f the o r d e r o f T2 , the s p i n -s p i n r e l a x a t i o n t i m e . T2 i s a measure o f the r e c i p r o c a l o f the l i n e w i d t h o f the n u c l e a r magnetic resonance absorp t i o n l i n e . T h i s l i n e has a f i n i t e w i d t h because o f the v a r i a t i o n i n l o c a l f i e l d a t d i f f e r e n t p o s i t i o n s w i t h i n the sample, g i v i n g r i s e to a s l i g h t spread i n the Larmor p r e c e s s i o n f r e q u e n c i e s . A n u c l e u s may i n t e r a c t so s t r o n g l y w i t h i t s s u r r o u n d i n g s t h a t , i n the absence o f s t r o n g f l u c t u a t i o n s i n f i e l d s s e t up by the s u r r o u n d i n g medium, an o t h e r w i s e s i n g l e a b s o r p t i o n l i n e may be broken up i n t o s e v e r a l component l i n e s . There are two s o u r c e s o f t h i s " f i n e s t r u c t u r e " . F i r s t l y , n e i g h b o u r i n g magnetic d i p o l e s may be s u f f i c i e n t l y c l o s e t h a t t h e i r m u tual magnetic i n t e r -a c t i o n p e r t u r b s the n o r m a l Zeeman l e v e l s i n a manner depending ;on the r e l a t i v e o r i e n t a t i o n s o f the two d i p o l e s (Pa48, GKPP). Thus t r a n s i t i o n s w i l l o c c u r a t f r e q u e n c i e s s l i g h t l y d i f f e r e n t from the normal Larmor p r e c e s s i o n f r e q u e n c y . S e c o n d l y , an e l e c t r i c quadrupole i n t e r a c t i o n may p e r t u r b the n o r m al Zeeman l e v e l s i n such a way t h a t the 21 p o s s i b l e t r a n s i t i o n s between a d j a c e n t l e v e l s o c c u r a t s l i g h t l y d i f f e r e n t f r e q u e n c i e s . 6 Other s o u r c e s o f l i n e s t r u c t u r e and l i n e b r o a d e n i n g w i l l n o t be d i s c u s s e d . They are u s e f u l f o r d e t e r m i n i n g s t r u c t u r e and I n t e r n a l m o t i o n I n s o l i d s (GP). 7 CHAPTER I I THE APPARATUS The a p p a r a t u s used f o r the p r e s e n t r e s e a r c h was the n u c l e a r magnetic resonance s p e c t r o m e t e r b u i l t by C o l l i n s (C50). I t c o n s i s t s o f a s t a b i l i z e d e l e c t r o m a g n e t to p r o v i d e the s t e a d y magnetic f i e l d Ho, t o g e t h e r w i t h an o s c i l l a t i n g d e t e c t o r , used t o d e t e c t the n u c l e a r magnetic resonance a b s o r p t i o n . Only a b r i e f d e s c r i p t i o n o f the a pparatus w i l l be g i v e n . The r e a d e r i s r e f e r r e d t o the t h e s i s o f i t s o r i g i n a t o r f o r d e t a i l s . The magnet c u r r e n t i s c o n t r o l l e d by a bank o f 19 t w i n t r i o d e 6AS7 t u b e s . D-c. s t a b i l i z a t i o n i s o b t a i n e d by comparing the v o l t a g e a c r o s s p a r t o f a 1 ohm p o t e n t i o -meter w i t h 3 v o l t s from a p a i r o f d r y c e l l s . The d i f f e r e n c e v o l t a g e i s a m p l i f i e d and a p p l i e d t o the g r i d s o f the 6AS7's. A-c. v o l t a g e s a p p e a r i n g a c r o s s the magnet c o i l are f e d i n t o the t h i r d s tage o f t h i s d-c. a m p l i f i e r and a l s o cause a compensating v o l t a g e to appear on the g r i d s o f the 6AS7«s. The magnetic f i e l d I s a l s o s t a b i l i z e d by means o f a p r o t o n resonance s i g n a l t o b e t t e r than 1 p a r t i n 100,000. An o s c i l l a t i n g d e t e c t o r o f a d e s i g n s i m i l a r t o the one used f o r s i g n a l d e t e c t i o n i s i n use. The sample b e i n g i n v e s t i g a t e d i s p l a c e d i n the 8 o s c i l l a t i n g d e t e c t o r ' s tank c o i l w h i c h i s mounted i n the magnetic f i e l d . The o s c i l l a t i n g d e t e c t o r i s a weakly o s c i l l a t i n g o s c i l l a t o r whose a m p l i t u d e I s v e r y s e n s i t i v e t o the a b s o r p t i o n o f energy from the t a n k c o i l . As the f r e q u e n c y o f the o s c i l l a t o r i s v a r i e d , the a m p l i t u d e drops s h a r p l y when a p p r e c i a b l e a b s o r p t i o n o f r - f . energy by the sample takes p l a c e . The f i e l d Ho i s modulated a t an audio f r e q u e n c y by means o f s m a l l "sweep c o i l s " mounted on e i t h e r s i d e o f the tank c o i l w i t h axes i n the Ho d i r e c t i o n . Thus when the o s c i l l a t o r f r e q u e n c y i s n e a r r e s o n a n c e , the t o t a l f i e l d (Ho + a u d i o ) sweeps o v e r p a r t o f the a b s o r p t i o n l i n e . We t h e n have a modulated r - f . o u t p u t from the o s c i l l a t o r w h i c h i s a m p l i f i e d and then d e t e c t e d . The r e s u l t i n g a u d i o wave I s a m p l i f i e d by s e v e r a l s t a g e s . A p h a s e - s e n s i t i v e o r " l o c k - i n " d e t e c t o r r e c t i f i e s s i g n a l s i n a narrow f r e q u e n c y band about, and n e a r l y i n phase w i t h , an a p p l i e d audio s i g n a l . T h i s s i g n a l i s o b t a i n e d from the same audio o s c i l l a t o r which s u p p l i e s the c o i l s f o r magnetic f i e l d m o d u l a t i o n , b u t i t passes t h r o u g h an a d j u s t a b l e phase s h i f t i n g network b e f o r e r e a c h i n g the p h a s e - s e n s i t i v e d e t e c t o r . The r e c t i f i e d audio wave i s smoothed u s i n g an a d j u s t a b l e time c o n s t a n t , passes I n t o a vacuum-tube v o l t m e t e r (VW) and the o u t p u t i s r e c o r d e d on an E s t e r l i n e - A n g u s R e c o r d i n g M i l l i a m e t e r . The e x p l o r a t o r y work on the n u c l e a r e l e c t r i c q uadrupole i n t e r a c t i o n d e s c r i b e d i n t h i s t h e s i s was performed u s i n g a s i n g l e c r y s t a l o f spodumene mounted i n a s l o t on the end o f a b a k e l i t e r o d ; and i n s e r t e d i n the tank c o i l o f the o s c i l l a t i n g d e t e c t o r . The method o f me a s u r i n g angles o f r o t a t i o n o f the c r y s t a l was r a t h e r crude and w i l l n o t be d e s c r i b e d . An improved mount has been c o n s t r u c t e d by Mr. H. E. P e t c h and w i l l be d e s c r i b e d i n a f o r t h c o m i n g paper (VPS). CHAPTER I I I THE NUCLEAR ELECTRIC QUADRUPOLE INTERACTION IN CRYSTALS I n a c r y s t a l i n w h i c h t h e symmetry i s s u c h t h a t t h e e l e c t r i c f i e l d g r a d i e n t does n o t v a n i s h a t the s i t e o f a n u c l e u s u n d e r s t u d y , t h e n u c l e a r m a g n e t i c r e s o n a n c e a b s o r p t i o n l i n e i s s p l i t up i n t o 21 components. I f t h e i n t e r a c t i o n i s n o t t o o s t r o n g , so t h a t f i r s t o r d e r p e r t u r -b a t i o n t h e o r y i s a d e q u a t e , i t i s f o u n d t h a t t h e s e p a r a t i o n i n f r e q u e n c y between t h e components i s p r o p o r t i o n a l t o t h e e l e c t r i c f i e l d g r a d i e n t i n t h e Ho d i r e c t i o n and t o t h e e l e c t r i c q u a d r u p o l e moment o f t h e n u c l e u s u n d e r s t u d y . 21 components o c c u r b e c a u s e the e l e c t r i c q u a d r u p o l e i n t e r a c t i o n p e r t u r b s t h e n o r m a l Zeeman l e v e l s o f t h e n u c l e a r s p i n s y s t e m I n s u c h a way t h a t the 21 a l l o w e d t r a n s i t i o n s between a d j a c e n t l e v e l s o c c u r a t s l i g h t l y d i f f e r e n t f r e q u e n c i e s . F o r example, f o r 1 - 3 / 2 , P e r t u r b e d l e v e l s N ormal l e v e l s Upper S a t e l l i t e L i n e C e n t r a l L i n e ( u n d i s p l a c e d ) Lower S a t e l l i t e L i n e The above d i a g r a m i s f o r a p o s i t i v e c o u p l i n g energy. F o r s m a l l i n t e r a c t i o n s , such, as the L i i n t e r a c t i o n i n spodumene, the c e n t r a l resonance l i n e I s u n d i s p l a c e d , the phenomenon b e i n g d e s c r i b e d u s i n g f i r s t o r d e r p e r t u r b a t i o n t h e o r y . Theory The t h e o r y p r e s e n t e d below i s a more d e t a i l e d e x p o s i t i o n o f the b r i e f s k e t c h g i v e n by Pound (P50) f o r a x i a l l y symmetric f i e l d s , and an e x t e n s i o n o f Pound's work t o n o n - a x i a l l y symmetric c r y s t a l s s u g g e sted by P r o f . V o l k o f f . The e l e c t r o s t a t i c energy o f i n t e r a c t i o n between a system o f n u c l e a r c h a r g e s I and an e x t e r n a l charge d i s t r i b u t i o n j i s g i v e n by Expa n d i n g i n Legendre p o l y n o m i a l s t h i s may be w r i t t e n The term c o r r e s p o n d i n g t o k = 0 r e p r e s e n t s the energy o f the t o t a l n u c l e a r charge c o n c e n t r a t e d a t the o r i g i n o f c o o r d i n a t e s i n a p o t e n t i a l d e t e r m i n e d by the e x t e r n a l charge d i s t r i b u t i o n . The k = 1 term r e p r e s e n t s the energy o f an e l e c t r i c d i p o l e i n the e l e c t r i c f i e l d o f the e x t e r n a l charge d i s t r i b u t i o n . However, s i n c e n u c l e i do not p o s s e s s e l e c t r i c d i p o l e moments, t h i s term w i l l be n o n - e x i s t e n t i n the n u c l e a r c a s e . The k - 2 term r e p r e s e n t s the energy o f the n u c l e a r e l e c t r i c q uadrupole i n the e l e c t r i c f i e l d g r a d i e n t at t h e o r i g i n o f c o o r d i n a t e s . By u s i n g c e r t a i n theorems p e r t a i n i n g to s p h e r i c a l h a r m o n i c s , we may e x p r e s s V as i , j i -k ( 3 ) i always r e f e r s t o the n u c l e a r charge c o n f i g u r a t i o n , j to the e x t e r n a l charge c o n f i g u r a t i o n . F o l l o w i n g Racah (R42), and Condon and S h o r t l e y ' s d e f i n i t i o n (CS) f o r the n o r m a l -i z e d A s s o c i a t e d Legendre P o l y n o m i a l s I s used, v i z . 1 The e l e c t r i c quadrupole i n t e r a c t i o n I s e x p r e s s e d by the k = 2 term i n ( 3 ) , (4) We may a l s o e x p r e s s the e l e c t r i c q uadrupole i n t e r a c t i o n as the d y a d i c (second rank t e n s o r ) s c a l a r p r o d u c t P - Q . V E - t^irfdtiy^ ( 5 ) 1—* where Qq : ^ H*" C ^ (6) i ( ? e L » £ c ! , i J ) (7) j ^ To o b t a i n $ and V E i n the above form, we may f i r s t o f a l l c o n s i d e r the r e c t a n g u l a r components o f V E and show how t o t r a n s f o r m t h e s e components i n t o the form ( 7 ) . The components o f the e l e c t r i c quadrupole moment o p e r a t o r t r a n s f o r m i n an analogous f a s h i o n . The 3X3 m a t r i x r e p r e s e n t i n g the r e c t a n g u l a r components o f V E , v i z . / 3 Ex 9 Ex 3 Ex \ 9 x 3 j 9 z 3Ey 3 x a y 3 z dEz dEz £Ez \ d x a y a z y may be s i m p l i f i e d when i t i s c o n s i d e r e d t h a t E = - \7 V, hence VXE = 0 , i . e . the t e n s o r i s symmetric, r e d u c i n g I t t o s i x Independent components. The r e q u i r e m e n t t h a t d i v E = 0 reduces i t t o f i v e independent components. . and 14 J R J 3 1*1 _ 3 *., yj 0 _ 3 xj yj ff \ The C's may be w r i t t e n I n the form c : - - ± r t e * - T r % ) r2 _ - p i _ * ^ - i _ L H . \ (9) (10) I t may be shown t h a t ( 7 E ) c , (17 E ) t | , (\7 E ) n bear the same r e l a t i o n t o the r e c t a n g u l a r t e n s o r components (9) as the above C's (10) do t o the s p h e r i c a l harmonics d e f i n e d i n Racah's n o t a t i o n ( 3 ) . In f a c t , dz 9 £ y (11) 9v By analogy, the e l e c t r i c quadrupole moment o p e r a t o r OlO-Ee^C'/O * F a c t o r o f 1/2 i n t h i s e x p r e s s i o n i s m i s s i n g i n Pound (£50) i s e x p r e s s i b l e i n components where Q i s the s c a l a r n u c l e a r e l e c t r i c quadrupole moment d e f i n e d i n the u s u a l manner ( C 3 6 , F L ) . The f i r s t o r d e r energy p e r t u r b a t i o n on the normal Zeeman l e v e l s o f the n u c l e a r s p i n system i s g i v e n by the m a t r i x element o f the quadrupole i n t e r a c t i o n term F d e f i n e d by ( 5 ) Thus i n the f i r s t o r d e r t h e o r y o n l y the component ( ^ E ) C o c c u r s . The o t h e r f o u r components o f V E need n ot be known u n l e s s second o r d e r c a l c u l a t i o n s are co n t e m p l a t e d . The remainder o f our problem i s now the d e t e r m i n a t i o n o f ( V ' E ) Q f o r the g e n e r a l case o f no a x i a l symmetry. The e l e c t r i c f i e l d g r a d i e n t t e n s o r may be t r a n s f o r m e d to a c o o r d i n a t e system i n w h i c h the 3 X 3 m a t r i x r e p r e s e n t i n g i t s r e c t a n g u l a r components I s d i a g o n a l i z e d . When t h i s i s done, we may choose our z - a x i s a l o n g the p r i n c i p a l a x i s h a v i n g l a r g e s t component, and — r ~ ~ > where e<^  = ^ — r » j 7 J ' . i (?E)tl ' O where 71 — 9 Ex* dx" (14) 3 y 9 €*.» Here; the (double p r i m e s ) r e f e r to the c o o r d i n a t e system i n w h i c h ( v'E) i s d i a g o n a l i z e d . The (unprimed) axes w i l l c o n t i n u e t o r e f e r t o a l a b o r a t o r y c o o r d i n a t e system i n w h i c h z i s i n the Ho d i r e c t i o n and x, y, z form a r i g h t -handed o r t h o g o n a l s e t . L e t the t r a n s f o r m a t i o n m a t r i x between the x y z system and the x" y" z" system be: x y z x " P l l P 1 2 P 1 3 y" P 2 1 P 22 P23 (15) z " P 3 1 P 3 2 P 33 The t e n s o r ( V E ) t r a n s f o r m s i n e x a c t l y the same way as the t e n s o r C ? and S u b s t i t u t i n g f o r x " 2 , y " 2 , z " 2 , x"y", x " z " , y " z " , . u s i n g A r e l a t i o n s (10) we* o b t a i n a r e l a t i o n f o r C 2 i n terms o f the C* ' s. Then, by analogy, and making use o f t h e f a c t t h a t S u b s t i t u t i n g r e l a t i o n s ( 1 4 ) , (16) I t remains now to o b t a i n e x p l i c i t e x p r e s s i o n s f o r P i 3 ( i = 1,2,3) i n terms o f c r y s t a l o r i e n t a t i o n . Let us i n t r o d u c e a s e t o f any t h r e e m u t u a l l y p e r p e n d i c u l a r axes f i x e d w i t h r e s p e c t t o the c r y s t a l which are e a s i l y r e c o g n i z a b l e e x t e r n a l l y , t o be r e f e r r e d t o as the ( s i n g l e primed) s e t . We may choose, I f we w i s h (though i t i s n o t e s s e n t i a l ) some o f these s i n g l e primed axes t o c o i n c i d e w i t h some o f the c r y s t a l l o g r a p h i c axes o f the c r y s t a l . Let the t r a n s f o r m a t i o n from the (primed) t o t h e (double-primed) systems o f axes be d e f i n e d by the n i n e d i r e c t i o n c o s i n e s : / ^ i ( i = 1,2,3), w h i c h d e f i n e the o r i e n t a t i o n o f the x' a x i s w i t h r e s p e c t to the ( d o u b l e -primed) s e t , a n d / ' i and ( f o r the y' and z1 axes r e s p e c t -i v e l y ) . F u r t h e r , l e t the (primed) system be r o t a t e d about an a x i s f i x e d w i t h r e s p e c t t o b o t h the x',y', z 1 and the x, y, z ( l a b ) systems. We have chosen z to c o i n c i d e w i t h Ho. We choose an a x i s o f r o t a t i o n p e r p e n d i c u l a r t o Ho, d e s i g n a t i n g i t y. We now c o n s i d e r r o t a t i n g the c r y s t a l i n t u r n about i t s x', y', z' axes, making the a x i s o f r o t a t i o n i n each case c o i n c i d e w i t h the y - a x i s i n the l a b system. I f we l a b e l the t h r e e r o t a t i o n s r e s p e c t i v e l y X, Y, Z and choose the i n i t i a l p o s i t i o n s o f the c r y s t a l so t h a t x', y», z', I n i t i a l l y c o i n c i d e w i t h y, z, x ( f o r the X r o t a t i o n ) ; x, y, z ( f o r the Y ) ; z, x, y ( f o r the Z ) , then the t h r e e t r a n s f o r m a t i o n m a t r i c e s are d e f i n e d by the f o l l o w i n g T a b l e : X X y z X' 0 1 0 y s i n e - x 0 cos x z ' 0 - s i n - 0 - X IS Y X y 2 COS &y 0 - s i n y 0 1 0 Z ' sin 0 COS Gy Z X y z sin &z 0 COS y cos -0^ 0 - s i n 2 ' 0 l 0 The p r o d u c t o f the t r a n s f o r m a t i o n from x n , y", z" t o x', y', z', and the t r a n s f o r m a t i o n from x', y 1 , z* to x, y, z then g i v e s : ^ ^ x 2 Z* 1 c o s ^ x - ^ I s i n % ( P l 3 ) y - Vi cosSy- sin-9-y ( 1=1,2,3) (17) (n . 3> Z : ^ i <soa&z-J/± s i n ^ - z where the c y c l i c p e r m u t a t i o n i s e v i d e n t . 4 3 S u b s t i t u t i o n o f these i n t o eq. (16) then y i e l d s \yE)a - ^ - ( ^ f b* C o 5 + c* 5 i K 1 (18) Two s i m i l a r s e t s f o r Y and Z r o t a t i o n s are o b t a i n e d by c y c l i c p e r m u t a t i o n . From eq. (13) i t f o l l o w s t h a t the f r e q u e n c y s p l i t t i n g between l i n e s c o r r e s p o n d i n g t o t r a n s i t i o n s ta-*~m - 1 and - (n - 1 ) * * - m i s g i v e n by: = / \ + 6 Co 2 & + C 5 i « 2 & (20) where A = 2 k a , B - 2 k b , C - 2 k c , fr - £ ^ a 3 ( I D M arO.r-0 P e r f o r m i n g the t h r e e r o t a t i o n s X,Y,Z i t appears t h a t one o b t a i n s n i n e r e l a t i o n s o f the form ( I , I I ) i n v o l v i n g n i n e e x p e r i m e n t a l l y d e t e r m i n e d A B C's. However, t h e s e are n o t a l l independent. Thus (7£ ) a = -~-^r i s t h e s a m e f o r = ° and "©k = ~ and i n b o t h cases e q u a l to — ^ - • — r — = -^-/cL + bx) = -^ -(«.-t.) S i m i l a r e x p r e s s i o n s h o l d f o r Y and Z axes, and L a p l a c e ' s e q u a t i o n then guarantees t h a t a x + ay + a z = 0 ( a l s o b x + b y + b z =0). Thus o n l y any one p a i r AB i s an independent p a i r , the o t h e r s b e i n g r e l a t e d to i t . Thus, e x p e r i m e n t a l l y we o b t a i n from t h r e e r o t a t i o n s o n l y f i v e independent c o n s t a n t s : the t h r e e C's and any two o f the f o l l o w i n g t h r e e ; « = Ay - 5Y - Az *• e>z * - ZA* O + 0 *• r - o) Since e x p e r i m e n t a l l y we cannot determine which o f the two l i n e s i s m -*-•• m - 1 and which -(m - 1)-*-— - m, we o b t a i n o n l y the r e l a t i v e s i g n s o f C^, Cy , C2 , / 5 , Y but not the a b s o l u t e s i g n . Of the n i n e d i r e c t i o n c o s i n e s o n l y t h r e e are independent ( b y v i r t u e o f s i x o r t h o n o r m a l i t y r e l a t i o n s ) , s i n c e o n l y t h r e e parameters are r e q u i r e d to d e s c r i b e the o r i e n t a t i o n o f one s e t o f axes w i t h r e s p e c t to the o t h e r . These t h r e e d i r e c t o n c o s i n e s p l u s the v a l u e s o f 7t and k form a s e t o f f i v e unknowns determined by our s e t o f f i v e independent c o n s t a n t s . The Spodumene Crystal A single crystal of spodumene, dimensions about 37 mm. x 8 mm. x 3 mm. was used in the experimental work. It originated in the Minas Geraes region of Brazil and was purchased from Minerals Unlimited by Dr. R. M. Thompson of the Department of Geology and Geography. The orient-ation of the crystallographic C-axis was determined by Dr. L.D. Mantuani, using a polarising microscope. The orientation of the b-axis was described by the supplier. The unit c e l l of monoclinic spodumene (WB) cont-ains four L i atoms, but these atoms have similar e lectr ica l surroundings. This equivalence of the arrangement of neighbouring atoms is due to the existence of two-fold rotation axes and screw axes in the b-direction within the structure. The symmetry is such that the electr ic f i e ld at two of the Li nuclei i s equal in magnitude but opposite in direction to that at the other two Li nuclei . The electr ic f i e ld gradient (v*E)„ , being of the second rank, w i l l have the same sign at a l l four sites. In other words, the orientation of the coordinate system in which \7 E is diagonalized is the same at a l l Li nuclei , thus giving rise to a unique set of experimental results. If this condition did not hold, there would be a super-position of two or more different frequency sp l i t t ing patterns at each crystal orientation. FIG.l• RECORDED ABSORPTION LINE DERIVATIVES The symmetry p r o p e r t i e s o f spodumene are a l s o such t h a t the b - a x l s c o i n c i d e s w i t h one o f the p r i n c i p a l axes o f the e l e c t r i c f i e l d g r a d i e n t t e n s o r . E x p e r i m e n t a l The L I 7 resonance was l o c a t e d at about 11.84 Mc. i n a f i e l d o f about 8000 gauss. The c r y s t a l was r o t a t e d about the G - a x i s i n t o p o s i t i o n s where the d i r e c t i o n w h i c h was b e l i e v e d to be the c r y s t a l l i n e b - a x i s made v a r i o u s a n g l e s w i t h the d i r e c t i o n o f the magnetic f i e l d Ho. T r a c e s o f the t h r e e - l i n e n u c l e a r magnetic resonance a b s o r p t i o n l i n e spectrum f o r t h e L i 7 resonance were o b t a i n e d a t a ngles d i f f e r i n g by 22.5°. A few i n t e r m e d i a t e r e a d i n g s were ta k e n as c h e c k s . T y p i c a l t r a c e s o f the a b s o r p t i o n l i n e d e r i v a t i v e as a f u n c t i o n o f f r e q u e n c y as r e c o r d e d by the E s t e r l i n e - A n g u s r e c o r d i n g m i l l i a m e t e r are shown i n f i g . 1. Frequency r e a d i n g s were taken a t v a r i o u s p o i n t s a l o n g the t r a c e w h i l e the f r e q u e n c y was v a r y i n g s l o w l y t h r o u g h the r e g i o n o f the t h r e e r e s o n a n c e s . F o r the w e l l -r e s o l v e d r e s o n a n c e s , the f r e q u e n c y d i f f e r e n c e between the two s a t e l l i t e a b s o r p t i o n l i n e s was measured d i r e c t l y f r o m the c h a r t . F o r u n r e s o l v e d r e s o n a n c e s , the r e c o r d e d d e r i v -a t i v e c u r v e s were i n t e g r a t e d u s i n g a p l a n i m e t e r and the s e p a r a t i o n o f the s a t e l l i t e s e s t i m a t e d from the i n t e g r a t e d a b s o r p t i o n c u r v e s . The a n g u l a r v a r i a t i o n o f the s p l i t t i n g Kc baxis 010 plane 4 b axis 010 plane F i g . 2 — Frequency s p l i t t i n g i n ki l o c y c l e s of s a t e l l i t e absorption !!«»•»*» * function of angle between +b axis and Ho, for the t h r * « 7 l i " \ L i l n ^ * ? r magnetic resonance absorption spectrum i n spodumene. Ci r c l e s represent experimental points. The s o l i d curve i s 2AV - 13.1 • 51.9 cos 2l»+V7 ). The c- axis of the c r y s t a l i s v e r t i c a l , i . e . perpendicular to H 0 * of the s a t e l l i t e l i n e s i s shown i n f i g . 2. C i r c l e s r e p r e s -ent experimental p o i n t s ; the s o l i d curve i s Z = |3-l 4-51-3 Cos Z(&+ 47°) which i s of the form (20). This r e l a t i o n s h i p , u s i n g the o r i g i n a l l y - s p e c i f i e d axes, does not agree with the known f a c t that one of the p r i n c i p a l axes of V E c o i n c i d e s w i t h the b- a x i s o f the. c r y s t a l . This i m p l i e s t h a t the curve be symmetric about the b - a x i s . Thus, an e r r o r i n the o r i g i n a l assignment o f axes as shown i n f i g . 2 was suspected, and i t appeared that the b - a x i s should be s h i f t e d e i t h e r to = 43° or •£)- = - 47°. Very r e c e n t l y , Dr.K. C. MacTaggart examined a t h i n s e c t i o n o f the c r y s t a l o p t i c a l l y and found t h a t the b-axis.• should be i n a ..position given approximately by •0" = 45° (see fig. " 2 ) . A l l o w i n g f o r an e r r o r of about 2° i n measurements, the curve i s symmetric about the true b-axis and i n terms of t h i s c o r r e c t d e s i g n a t i o n , 2 A V - 1 3 1 - 5 1 3 C o s Zf , where ? i s the angle between the b - a x i s and Ho. Or, since only r e l a t i v e signs f o r the s p l i t t i n g are d i s c e r n i b l e , This agrees w i t h i n a few percent w i t h subsequent measurements made f o r the same r o t a t i o n by Petch (PSV) and also by Pake (Pa51). C o n c l u s i o n s The a n g u l a r dependence o b t a i n e d above I s c o n s i s t -ent w i t h the t h e o r y o f the g e n e r a l case o f the a n g u l a r dependence o f the n u c l e a r e l e c t r i c quadrupole i n t e r a c t i o n I n c r y s t a l s w i t h asymmetric f i e l d s and i s the f i r s t e xper-i m e n t a l t r e a t m e n t o f i t . P r e v i o u s work (P50) has been concerned w i t h the a n g u l a r dependence about an a x i s o f symmetry. From subsequent r e s u l t s (PSV) o b t a i n e d f o r t h i s and two other, r o t a t i o n s by H. E. P e t c h , Q = 7 5 c kc./sfic . , 71 « 0 - 7 3 3 h f o r ^ p o y o j T i e n e . Other r e s u l t s , such as the o r i e n t a t i o n o f the p r i n c i p a l axes o f the f i e l d g r a d i e n t t e n s o r , are quoted i n the r e f e r e n c e . In o r d e r t o deduce a v a l u e o f the magnitude o f the n u c l e a r e l e c t r i c quadrupole moment o f L i 7 from t h e s e r e s u l t s , i t i s s t i l l n e c e s s a r y to know eq, f o r t h i s c r y s t a l . CHAPTER IV SUGGESTIONS FOR FURTHER WORK ' An exp e r i m e n t s i m i l a r t o the one d e s c r i b e d i n t h i s t h e s i s might w e l l be performed w i t h a s i n g l e c r y s t a l o f j a d e i t e , N a A l ( S I 0 3 ) 2 , u s i n g the N a ^ 3 resonance. The c r y s t a l s t r u c t u r e o f j a d e i t e i s almost I d e n t i c a l w i t h t h a t o f spodumene. An e s t i m a t e o f the quadrupole moment r a t i o s QNa23 / Q L I 7 m i g h t be o b t a i n e d . Work on the s p l i t t i n g o f the n u c l e a r magnetic resonance a b s o r p t i o n l i n e due to magnetic d i p o l e - d l p o l e i n t e r a c t i o n s between n e i g h b o u r i n g n u c l e i (Pa48) might be c a r r i e d o u t u s i n g the p r o t o n resonance i n v a r i o u s h y d r a t e d c r y s t a l s , s uch as the m i n e r a l s a n a l c i t e , NaAl(8103)2.HgO, g o e t h i t e , FegOs.HgO, and d i a s p o r e , AI2O3.H2O. I n c r y o l i t e , Na3AlF5 , the A l 2 7 resonance s h o u l d show a f i n e s t r u c t u r e due to magnetic i n t e r a c t i o n w i t h the l a r g e magnetic moments o f n e i g h b o u r i n g p l 9 n u c l e i , w h i c h are l o c a t e d a mere 1.8A 0 away (NS). I f the g r a d i e n t o f the e l e c t r i c f i e l d a t the A l s i t e s i n t h e c r y s t a l i s n o n - z e r o , an e l e c t r i c quadrupole i n t e r a c t i o n would a l s o t a k e p l a c e , s i n c e A l ' has a l a r g e quadrupole moment ( L 4 9 ) . T h i s would p r o v i d e a s t r i k i n g i l l u s t r a t i o n o f b o t h e f f e c t s . The p r e s e n t a p p a r a t u s i s p a r t i c u l a r l y adapted to the s e a r c h f o r unknown reson a n c e s and the measurement of l i n e s p l i t t i n g due t o e l e c t r i c quadrupole c o u p l i n g . F o r more a c c u r a t e work on l i n e s t r u c t u r e s and r e l a x a t i o n t i m e s , a b r i d g e type c i r c u i t w i t h v a r i a b l e r - f . i n p u t would be i n v a l u a b l e . The method o f p r o g r e s s i v e s a t u r a t i o n (BPP) c o u l d then be used f o r the d e t e r m i n a t i o n o f the s p i n - l a t t i c e r e l a x a t i o n time Ti . The f o l l o w i n g p r o posed experiment might shed some l i g h t on the r e l a t i v e magnitudes o f the r e l a x a t i o n by magnetic d i p o l e as compared t o e l e c t r i c quadrupole i n t e r a c t i o n . The r e l a x a t i o n time Ti f o r a s e r i e s o f n u c l e i w hich are r o u g h l y known to have p r o g r e s s i v e l y l a r g e r e l e c t r i c quadrupole moments c o u l d be measured. F o r example, one might compare the L i 7 , N a ^ and A l ^ 7 r e s o n -ances i n s o l u t i o n s o f the s u l p h a t e s o f l i t h i u m , sodium and aluminum, a l l at the same c o n c e n t r a t i o n s . Of c o u r s e , the s t r e n g t h o f the i n t e r a c t i o n would a l s o depend on the i o n i c r a d i i o f the atoms whose n u c l e i are b e i n g compared, and t h i s would have t o be a l l o w e d f o r . T^ c o u l d a l s o be measured f o r s e v e r a l n u c l e i o f s p i n 1/2 under c o n d i t i o n s s i m i l a r t o the above, and the r e s u l t s compared w i t h the r e s u l t s f o r L I , Na and A l , where the r e l a x a t i o n i s s u p p o s e d l y t h r o u g h the e l e c t r i c quadrupole mechanism. I n t h i s way, one m ight d i s c e r n whether o r not the e l e c t r i c quadrupole i n t e r a c t i o n i s a c t u a l l y the d o m i n a t i n g mechanism f o r t h e r m a l r e l a x a t i o n f o r n u c l e i o f s p i n I > 1/2. REFERENCES (B48) N. Bloembergen, T h e s i s , L e i d e n (1948) (BHP) B l o c h , Hansen, P a c k a r d , Phys. Rev. 70, 474 (1946) (BPP) Bloembergen, P u r c e l l , Pound, Phys. Rev. 73, 679 (1948) (C36) H.B.G. C a s i m i r , On the I n t e r a c t i o n between Atomic N u c l e i and E l e c t r o n s ( T e y l e r Tweede Genootschap, Haarlem, 1936) (C50) T.L. C o l l i n s , Ph.D. T h e s i s , B r i t i s h Columbia, 1950 (CS) E.U. Condon and G.H. S h o r t l e y , The Theory o f Atomic S p e c t r a (Cambridge U n i v e r s i t y P r e s s , London, 1935) (F L ) B. F e l d and W.E. Lamb, Phys. Rev. 67, 15 (1946) (GP) H.S. Gutowsky and G.E. Pake, J . Chem. Phys. 18 , 162 (1950) (GKPP) Gutowsky, K i s t i a k o w s k y , Pake, P u r c e l l , J . Chem. Phys. 17, 972 (1949) (L49) H i n Lew, Phys. Rev. 76, 1086 (1949) (NS) S t . v. Naray-Szabo and K. S a s v a r i , Z e i t s . f . K r i s t (A) 99, 27 (1938) (P50) R.V. Pound, Phys. Rev. 79, 685 (1950) (P51) R.V. Pound, Phys. Rev. 81, 156 (1951) (Pa48) G.E. Pake, J . Chem. Phys. 16, 327 (1948) (Pa50) G.E. Pake, Am. J o u r n . ^hys. 18, 438; 18, 473 (1950) (Pa51) G.E. Pake, p r i v a t e communication (Pu48) E.M. P u r c e l l , S c i e n c e 107, 433 (1948) (PSV) P e t c h , S m e l l i e , V o l k o f f , Phys. Rev. ( t o be p u b l i s h e d ) (PTP) P u r c e l l , T o r r e y , Pound, Phys. Rev. 69, 37 (1946) (R42) G. Racah, Phys. Rev. 62, 438 (1942) 29" (R48) B.V. R o l l i n , Rep. P r o g . P h y s . 12, 48 (1948) (VPS) V o l k o f f , P e t c h , S m e l l i e , Can. J o u r n . P hys. ( t o be p u b l i s h e d ) (VW) G.E. V a l l e y and H. Wallman, Vacuum Tube A m p l i f i e r s , p. 480, Nr. 18, M. I . T. R a d i a t i o n Lab. S e r i e s , M c G r a w - H i l l , N. Y., 1940 (WB) B.E. Warren and -J. B i s c o e , Z e i t s . f . K r i s t , 80 391 (1931) 

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