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UBC Theses and Dissertations

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UBC Theses and Dissertations

Nuclear magnetic resonance in a thallium single crystal Schratter, Jacob Jack 1968

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NUCLEAR MAGNETIC RESONANCE IN A THALLIUM SINGLE CRYSTAL JACOB JACK SCHRATTER DIPLOMAT UNIVERSITAR OP UNIVERSITY OP BUCHAREST, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the requi red standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1968 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r ai advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and s t u d y , I f u r t h e agree that permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s represen-t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date i i ABSTRACT Nuclear magnetic resonance s tudies i n s i n g l e c r y s t a l s of t h a l l i u m have "been performed f o r the f i r s t t ime. The resonance frequency, l i n e w i d t h and second moment were s t u d i e d as a f u n c t i o n of c r y s t a l o r i e n t a t i o n w i t h respect to the magnetic f i e l d . The Knight s h i f t parameters were determined a t l i q u i d helium and l i q u i d n i t r o g e n temperatures. The a n i s o t r o p i c Knight s h i f t r e s u l t s were i n disagreement w i t h the r e s u l t s ohtained on powder specimens hy other workers . i i i .TABLE OP CONTENTS C H A P T E R Page 1 I N T R O D U C T I O N 1 2 mUIPMENT 3 3 REVIEW O P THE THEORY (a) The I s o t r o p i c Knight S h i f t 6 (b) The A n i s o t r o p i c Knight S h i f t 11 (c) The Moments of the Resonance Line 12 4 D I P O L A R B R O A D E N I N G F O R A H E X / L G O H A L C L O S E D P A C K E D L A T T I C E 20 5 E X P E R I M E N T A L C O N S I D E R A T I O N S 25 6 T H E A N A L Y S I S OF T H E L I N E S H A P E 34 7 E X P E R I M E N T A L R E S U L T S A N D D I S C U S S I O N 39 A P P E N D I X 48 R E F E R E N C E S 51 i v LIST OF FIGURES FIGURE Page 1 B l o c k Diagram of the Spectrometer 5 2 The Angular Dependence of the D i p o l a r C o n t r i b u t i o n to the Second Moment ( T l 2 0 5 ) 2 4 3 Separat ion of Modes f o r a t y p i c a l Thal l ium Line 3 8 4 Experimental Resonance Line of T l 2 0 5 a t 1.2 °K 4 0 5 Experimental Resonance Line of T l 2 0 5 a t 77°K 4 1 6 Experimental Resonance Line of T l 2 0 ^ at 1.2°K 4 2 7 The Resonance Frequency i n T l 2 < ^5 as a Funct ion of C r y s t a l O r i e n t a t i o n at 77°K 4 3 8 The Resonance Frequency i n T l 2 ^ 5 as a Funct ion of C r y s t a l O r i e n t a t i o n a t 1.2°K 4 4 9 The Linewidth as a Funct ion of C r y s t a l O r i e n t a t i o n at 1.2°K 4 6 ACKNOWLEDGEMENT I wish to express my s incere g r a t i t u d e to Dr . D. Llewelyn W i l l i a m s f o r h i s guidance and help throughout the per iod of t h i s work, and f o r h i s support i n ob ta in ing f i n a n c i a l ass is tance f o r my s t u d i e s . I a lso wish to thank D r . Surrendra Sharma f o r h i s u s e f u l suggestions regarding the experimental technique and f o r a l l o w i n g me to use h i s PKW spec-trometer. I wish to acknowledge the f i n a n c i a l support provided hy the U n i v e r s i t y of B r i t i s h Columbia through the award of the U n i v e r s i t y Scholarship and the Summer Grant rece ived from the Dean's Committee on Research. C H A P T E R I I N T R O TJU C T I O M " The d e t a i l e d c h a r a c t e r i s t i c s of the nuclear magnetic resonance i n metals are s t r o n g l y i n f l u e n c e d hy the presence of the conduction e l e c t r o n s . They are responsible f o r the i s o t r o p i c and a n i s o t r o p i c Knight s h i f t of the resonance frequency, as w e l l as f o r the pseudodipolar and exchange i n t e r a t i o n s which i n -f luence the shape and width of the l i n e . The K M g h t s h i f t i s due to the mag-n e t i c f i e l d produced by the conduction e lec t rons at the s i t e of the nucleus through the hyperf ine i n t e r a t i o n . The i s o t r o p i c p a r t of the s h i f t i s generated by the Fermi contact term and i s p r o p o r t i o n a l to the s -character of the e l e c t -ron wavefunctions, whereas the a n i s o t r o p i c c o n t r i b u t i o n i s a r e s u l t of the d i p o l a r term which i s r e l a t e d to the anisotropy of the non s-character of the wavefunctions. The hyperf ine i n t e r a t i o n increases as a f u n c t i o n of atomic number, and consequently the i s o t r o p i c Knight s h i f t f o r t h a l l i u m i s r e l a t i v e l y large ( ( 1 . 6 % ) . By t h i s token, one expected i n experiments on n a t u r a l t h a l l i u m ( 2 9 . 5 % T I 2 0 3 , 7 0 . 5 % T l 2 ° 5 ) , the T I 2 0 3 l i n e to be narrower that the T l 2 ° 5 l i n e . The experiments revealed an opposite e f f e c t , and more than t h a t , the broad-ening was an order of magnitude l a r g e r than that p r e d i c t e d . The e f f e c t i s due to an i n d i r e c t exchange i n t e r a t i o n between n u c l e i ^ v i a the conduction e l e c t r o n s , and, since i t i s p r o p o r t i o n a l to the square of the hyperf ine i n t e r -a c t i o n , i t becomes the dominant c o n t r i b u t i o n f o r heavy metals . T h a l l i u m , a heavy meta l , having two isotopes of h igh abundance and a s trong KMR s i g n a l , i s thus a favourable m a t e r i a l f o r Knight s h i f t and exchange i n t e r a c t i o n s t u d i e s . However, a l l previous work was performed on powdered specimens. The reason f o r t h i s l i e s i n the smal l s i g n a l to noise r a t i o f u r n i s h e d by b u l k metal samples. The s k i n e f f e c t prevents the r . f . f i e l d from p e n e t r a t i n g i i n the c r y s t a l f o r more than a few microns . Moreover, the obtained s i g n a l i n of the complex nuclear s u s c e p t i b i l i t y , but r a ther t o ^ ' t _ ^ ' . By u s i n g r e l a t i v e l y large c r y s t a l s , a s e n s i t i v e Pound Knight Watkins spectrometer, and a technique of separa t ion of the absorpt ion mode from the d i s p e r s i o n mode, these d i f f i c u l t i e s can be o v e r -come, and f a i r l y good s i g n a l to noise r a t i o s obta ined. The experiments y i e l d e d Knight s h i f t parameters a t l i q u i d helium and n i t -rogen temperatures which were s i g n i f i c a n t l y d i f f e r e n t from those observed by Bloembergen and Rowland on powdered samples _1]. A study of l i n e w i d t h and second moment as a f u n c t i o n of c r y s t a l o r i e n -t a t i o n was performed. The second moments were found to be l a r g e r than those quoted by previous workers, presumably due to the b e t t e r s i g n a l to noise r a t i o s obtained i n t h i s vrork. A d e t a i l e d c a l c u l a t i o n of the angular dependence of the d i p o l a r c o n t r i b -u t i o n to the second moment, f o r a c losed packed hexagonal l a t t i c e i n g e n e r a l , and the t h a l l i u m l a t t i c e i n p a r t i c u l a r , i s descr ibed i n chapter 4* A semi-t h e o r e t i c a l a n a l y s i s of a L o r e n t z i a n l i n e shape i s performed i n chapter 6. The experimental considerat ions mentioned i n chapter 5 m a y be of p r a c t i c a l use to students us ing the steady state technique on s i n g l e c r y s t a l s . CIIAPTER I I EQUIPMENT Figure 1 shows a b l o c k diagram of the equipment used i n the experiments. The sample, w i t h a copper wire c o i l wound around i t , and r i g i d l y fastened i n a p l a s t i c h o l d e r , was kept i n a copper bomb. A s t a i n l e s s s t e e l c o a x i a l l i n e , 3/8" i n diameter, l e a d to an o s c i l l a t i n g detector tha t , except f o r minor changes, was of the Pound-Knight-Watkins type £ l o ] . The frequency of the o s c i l l a t o r was cont inuously monitored w i t h a Hewlett -Packard e l e c t r o n i c counter , model 5245^. The output of the PKW o s c i l l a t o r was f e d through a narrow band a m p l i f i e r and phase-sens i t ive detector i n t o a V a r i a n recorder , model C-11A. An " a u d i o " o s -c i l l a t o r (Hewlet t -Packard, model 202D), whose output was f e d through an a t t e n -uator and power a m p l i f i e r to the f i e l d c o i l s , provided the audio f i e l d modul-a t i o n . I t s output was a l so used as the reference input to the phase-sens i t ive detector and, through a phase s h i f t e r , as the h o r i z o n t a l d r i v e of the CRO I and I I . The probe measuring the magnetic f i e l d c o n s i s t s of a g lass v i a l , ^> mm xn diameter, f i l l e d w i t h g l y c e r o l and having a copper c o i l wound around i t . A few of such probes were a v a i l a b l e f o r f i e l d measurements v a r y i n g from 5 to 11 k i l o -gauss. A f i e l d monitor ing o s c i l l a t o r p i cks up the proton s i g n a l and d i s p l a y s i t . on CRO I I . I t s frequency i s d i sp layed on*an other Hewlett-Packard e l e c t r o n i c counter (model 525A) and p r i n t e d out . Simultaneously w i t h the p r i n t out a pulse i s f e d i n t o the V a r i a n recorder , a c t i v a t i n g an i n d i c a t o r pen on the recording c h a r t . The output of the phase-sens i t ive detector i s d i sp layed on CRO I . I t pro— ' v ides a v i s u a l check of the b a l a n c i n g , as w e l l as the overloading of the detec tor . 3 The f i e l ' d was s w e p t "by means o f a m o t o r c o u p l e d t o t h e c u r r e n t c o n t r o l o f t h e m a gnet. The PKW o s c i l l a t i n g d e t e c t o r was " b u i l t b y D r . S. S harma [ l 0 _ . I t h a d a g o o d r e s p o n s e f o r f r e q u e n c i e s v a r y i n g b e t w e e n 8 a n d 20 M c / s e c . The f r e q u e n c y s t a b i l i t y was b e t t e r t h a n 1 i n 1 o5 o v e r t i m e i n t e r v a l s o f t h e o r d e r o f 10 m i n -u t e s ; t h e r e was no n e e d f o r b e t t e r s t a b i l i t y i n t h e p e r f o r m e d e x p e r i m e n t s . The n a r r o w b a n d a m p l i f i e r h a d a v a i l a b l e f i l t e r s o f t h e Y J h i t e t w i n - t e e t y p e , v a r y i n g b e t w e e n 15 a n d 400 c p s a n d h a v i n g a b a n d w i d t h o f 5%« T h e r e were two p h a s e -s e n s i t i v e d e t e c t o r s a v a i l a b l e , a n d b o t h w e r e u s e d i n d i f f e r e n t e x p e r i m e n t s . One o f them was' o f S c h u s t e r ' s t y p e 1.11 j , t h e o t h e r one was a c o m m e r c i a l l o c k - i n a m p l i f i e r (PAR, m o d e l JB4). The f i e l d was s u p p l i e d b y a V a r i a n r o t a t i n g mag-n e t w i t h 1 2 " p o l e f a c e s a n d 2_' g a p . The h o m o g e n e i t y o f t h e f i e l d b e t w e e n t h e p o l e s was o f t h e o r d e r o f 1 i n io5 , w e l l w i t h i n t h e n e c e s s a r y r e q u i r e m e n t s o f t h i s w o r k . The maximum a v a i l a b l e f i e l d was 11 k i l o g a u s s . E a c h o f t h e m o d u l a t i o n c o i l s h a d 60 t u r n s o f Bo. 18 c o p p e r w i r e wound o n b a k e l i t e f o r m s , w h i c h , i n t u r n , were mounted o n t h e magnet p o l e c a p s . The f i e l d m o n i t o r i n g o s c i l l a t o r was r i g i d l y m o u n t e d o n t h e magnet, t h u s r o t a t i n g w i t h i t i n a n g u l a r d e p e n d e n c e e x -p e r i m e n t s . A r i g i d c o a x i a l l i n e l e a d t o t h e g l y c e r o l p r o b e , w h i c h was s i t u a t e d c l o s e t o t h e s a m p l e . The r e l a t i v e l y h i g h h o m o g e n e i t y o f t h e f i e l d made a n y s p e c i a l p r e c a u t i o n i n p o s i t i o n i n g t h e p r o b e u n n e c e s s a r y . N e v e r t h e l e s s , t h e r e l a t i v e p o s i t i o n b e t w e e n p r o b e a n d s a m p l e was k e p t c o n s t a n t t h r o u g h o u t a l l t h e e x p e r i m e n t s . A v a r i a b l e c a p a c i t o r o f 2—8 p f , i n p a r a l l e l w i t h a No. 116 v a r -i c a p , was u s e d i n t h e r e s o n a n c e c i r c u i t o f t h e f i e l d m o n i t o r i n g o s c i l l a t o r . A 90 v o l t b a t t e r y o v e r a 1 0 0 k H e l i p o t p r o v i d e d t h e v a r i a b l e v o l t a g e f o r t h e v a r i c a p . T h u s , a v e r y f i n e c o n t r o l o f t h e f r e q u e n c y c o u l d be a c h i e v e d . The c i r c u i t h a d a f r e q u e n c y s t a b i l i t y o f 1 i n 10-^  a n d h i g h e nough s i g n a l t o n o i s e r a t i o t o d i s p l a y t h e p r o t o n s i g n a l c l e a r l y o n t h e s c o p e . P K W Detector f l o d - U Coil a n on Q F f e U Pro_e 21 1 De i _c|W Audio 0_ci /lator F i g u r e 1 B l o c k D i a g r a m of t h e S p e c t r o m e t e r CHAPTER I I I REVIEW OF THE THEORY (a) The I s o t r o p i c Knight S h i f t . The f o l l o w i n g c a l c u l a t i o n i s based on the assumption that the i s o t r o p i c Knight s h i f t i s due to the f i e l d experienced by the nucleus as a r e s u l t of the s -s ta te hyperf ine coupl ing w i t h the spins of the conduction e l e c t r o n s . The i n t e r a c t i o n f o r any g iven nucleus i n v o l v e s many conduction e lec t rons and i s therefore an average e f f e c t . I n the presence of an e x t e r n a l magnetic f i e l d Ho, the e lec t rons are p a r t l y p o l a r i z e d , thus g i v i n g a nonvanishing c o u p l i n g . The s -s ta te hyperf ine coupl ing i s g iven by the w e l l known express ion where J-.lt i s the magnetic moment assoc iated w i t h the e l e c t r o n spin ,y ! ( . , i s that assoc ia ted w i t h the nuclear s p i n , and R i s the p o s i t i o n vector of the nucleus. S i n c e ~ } £ h 3 a - n d / t l ^ - ^ h l , the coupl ing over a l l n u c l e i and e lec t rons i s of the form — 2. r \ \ y^KxJ^lS.MV^ (,3) Because of the r e l a t i v e weakness of the hyperf ine coupl ing the problem can be t reated by p e r t u r b a t i o n theory. The unperturbed Hamil tonian i s of the f orm , i . e . there i s no i n t e r a c t i o n between the nuclear and e l e c t r o n i c system and the unperturbed wavefunction can be taken as a product of the e l e c -t r o n i c and nuclear wavefunctionst^ z Kj^lf^ • F i r s t order per turbat ion theory e v i d e n t l y y i e l d s , } l e n = _ Tt^^X (3-3) where, out of convenience, the nuclear observable s t i l l appears as an op-erator and d^indicates i n t e g r a t i o n over s p a c i a l and s p i n e l e c t r o n coordinates . 6 7 The c o n t r i h u t i o n from the j - t h nucleus i s where we have taken f o r convenience ^ e n o w have to s p e c i f y the m u l t i e -l e c t r o n w a v e f u n c t i o n ^ . Prom a r igorous viewpoint t h i s i s a d i f f i c u l t task because the e lec trons are very s t r o n g l y coupled to each other by the Coulomb i n t e r a c t i o n . However, Bohm and Pines _7_ have shown by means of a canonica l t ransformat ion that the p r i n c i p a l e f f e c t of the Coulomb i n t e r a c t i o n i s to g ive r i s e to a set of c o l l e c t i v e modes of o s c i l l a t i o n , the plasma modes. I t turns o u t . t h a t f o t low energy processes , that do not exc i te the plasma modes, the e lec t rons can be t rea ted as weakly i n t e r a c t i n g . Vie can then assume that S^T i s a product of one e l e c t r o n wavefunctions. For the i n d i v i d u a l f i u i c t i o n s vie take • / rl % V X Is (5.3) where U^[Y)JI S ^ are B loch f u n c t i o n s and are s p i n f u n c t i o n s . Since the operator acts on one p a r t i c l e o n l y , we do not have to worry about ant i - symmetr iza t ion and (4.3) becomes We assume the e lec t rons quantized along the Z - d i r e c t i o n by H 0 . Then, -*• ^ the only c o n t r i b u t i o n from X _ S { i s X ; 3 - S - - . We a lso change from 2 to__ ; when we do t h i s we e v i d e n t l y must m u l t i p l y the expression by a f a c t o r p ( k , s ) that i s 1 i f the state (k , s ) i s occupied by an e l e c t r o n and zero otherwise . F i n -a l l y , s ince i> 2 f^~rrssf~ 1 "the i n t e g r a l w i l l f u r n i s h I U^(c}| jr)s , where ms=|r or — | r . Hence (X'J.=¥ x^l^i^U (6.3) I f we average t h i s expression over a set of occupations p ( k , s ) that are representat ive of the temperature of the e l e c t r o n s , we can replace p ( k , s ) by 8 the Fermi f u n c t i o n f ( k , s ) o r , which i s the same, by f l f ) - ~ y 7 ^ r ; E * E p * E s p ? n where Ej , i s the sum of the k i n e t i c and p o t e n t i a l energy of an e l e c t r o n of wave— vector k , c a l l e d the t r a n s l a t i o n a l energy, and E g p _ n i s the energy assoc ia ted w i t h the s p i n o r i e n t a t i o n . Hence a t y p i c a l term of the sum (6.3) w i l l he of the Since—__fS_>-fit » ^ n e quant i ty i n the bracket i s minus the average c o n t r i b u t i o n of state k to the z—component of the e l e c t r o n magnet izat ion of the sample. C a l l i t _ „ _ , , (8.3) The t o t a l z-magnet izat ion of the e lec t rons e v i d e n t l y i s Let us now define the t o t a l e l e c t r o n s p i n s u s c e p t i b i l i t y ^ b y ^ ^ ^ H , , and the quant i ty ^ by 77 v * U * (10.3) Then (9-3) i s equivalent to X* s _ - Xf a n < ^ "the e f f e c t i v e i n t e r a c t i o n f o r the j - t h ? k « s p i n becomes — .1 $ ( } t j r ^ m z i ^ U i ( 1 1 . 3 ) We s h a l l assume that the f u n c t i o n IU£(0)1 v a r i e s s lowly enough i n k-space to j u s t i f y the use of a dens i ty f u n c t i o n that describes the number of al lowed k -values i n any r e g i o n . We therefore def ine j(EpA)^^h as the number of al lowed -A k-values l y i n g i n the volume whose bottom and top surfaces dA are a t E£ and Ej_ + dE£, r e s p e c t i v e l y . We a lso define r ( E c ) = j ^ f , A ) i A ( l 2" 3 ) Under these circumstances the summation i n (11.3) can he replaced hy an i n t e g r a l / I P t ® \ % - - ) H ^ W ) J f t U (13.3) By (8.3) and (10.3), depends on the Fermi f u n c t i o n s f („,-§-) and f ( k , - | - ) , i . e . on hoth the energy Eg. and the d i f f e r e n c e i n energy between s p i n up and s p i n down ( f o r a s p i n i n s ta te K ) . I t i s reasonable to assume that \f i s the same k f o r any s tates k having the same value of t r a n s l a t i o n a l energy Ej_ ( i . e . to assume that the d i f f e r e n c e i n energy between s p i n up and s p i n down i s the same f o r two s tates k and k i f E£ = Ej_'). Then )_f -?f(5f)and (13.3) becomes. 1 4 1 - • . Define the average value of the f u n c t i o n It^Wj over the surface E_ = const < | U C » ) | I ^ ^ p N 1 3 ( E t , A ) J A k ' K £f-.cor>si k VJith t h i s d e f i n i t i o n (14«3) becomes I f l u(«i'Xj ' [ x K E r K i u c ^ f i ^ J E j (15.3) Since f o r both s p i n s tates are 100% populated and f o r n e i t h e r s p i n state i s occupied, /^5(E^)is zero f o r a l l values of Ej£ that are not near the Fermi energy E p . Therefore, the i n t e g r a n t of (15.3) i s zero except f o r a narrow r e g i o n centered a t Ep and of l i n e w i d t h KT. I f we assume that v a r i e s s l o w l y i n that r e g i o n , i t can be taken out of the i n t e g r a l and (11.3) becomes , , _ _ - 3 M -y •• , fc_ YJe see that t h i s express ion has the form of an i n t e r a c t i o n of the j - t h nuclear s p i n w i t h a magnetic f i e l d 1Q Note that the f r a c t i o n a l s h i f t i s independent of H Q and increases w i t h the atomic number Z . F o r , the atoms w i t h the l a r g e r Z w i l l have a l a r g e r value of j , corresponding to the p u l l i n g i n of t h e i r wavefunctions by the l a r g e r nuc lear charge. The above c a l c u l a t i o n considers only the s - s ta te hyperf ine c o u p l i n g w i t h the spins of the conduction e l e c t r o n s . There are two other c o n t r i b u t i o n s to the i s o t r o p i c p a r t of the Knight s h i f t that have to be taken i n c o n s i d e r a t i o n , namely o r b i t a l paramagnetism and core p o l a r i z a t i o n . The l a t t e r i s due to the f a c t that the unpaired s -s ta te e lec t rons at the Fermi surface i n t e r a c t w i t h the core e l e c t r o n spins through the exchange i n t e r a c t i o n . The exchange i n t e r -a c t i o n comes about through an i n t e r p l a y of the Coulomb forces and the P a u l i p r i n c i p l e . For ins tance , the ferromagnetic s tate i n which a p a r a l l e l a l l i g n -ment of e l e c t r o n spins i s e n e r g e t i c a l l y favourable can be expla ined through the f a c t that the P a u l i p r i n c i p l e keeps e lec t rons w i t h p a r a l l e l spins apart of each other , thus reducing the e f f e c t of the r e p u l s i v e Coulomb f o r c e s . A s i m i l a r phenomenon i s responsible f o r the f a c t that any unpaired e l e c t r o n of a g iven s p i n o r i e n t a t i o n w i l l modify the inner s h e l l o r b i t s d i f f e r e n t l y f o r d i f f e r e n t s p i n o r i e n t a t i o n s , thus d i s t u r b i n g the balance between the con-t r i b u t i o n s to the Knight s h i f t from the core e l e c t r o n s . I n other words, due to the r e n o r m a l i z a t i o n of the wavefunctions, the nucleus experiences a net mag-n e t i c f i e l d from the core e l e c t r o n s , which i s p r o p o r t i o n a l to the p o l a r i z a t i o n of the conduction e l e c t r o n s . These two c o n t r i b u t i o n s coming from the conduction e lec t rons are i n d i s t i n g u i s h a b l e . The o r b i t a l paramagnetic c o n t r i b u t i o n to the Knight s h i f t i s due to the o r b i t a l angular momentum, t h a t , even though quenched i n i t s ground s t a t e , may contr ibute to o r b i t a l paramagnetism through a second order mechanism, i n v/hich the o r b i t a l angular momentum operator mixes unoccupied e x c i t e d states i n t o 11 occupied-ground s t a t e s . I t turns out that c o n t r i b u t i o n s come only from matr ix elements between l e v e l s i n the same p a r t i a l l y f i l l e d band. This c o n t r i b u t i o n i s important f o r t r a n s i t i o n metals , s ince they have a p a r t i a l l y f i l l e d d-band. (b) The A n i s o t r o p i c Knight S h i f t i s due to the d i p o l a r coupl ing between the nuclear s p i n and the e l e c t r o n s p i n f o r non s - s t a t e s . Since these states correspond to zero p r o b a b i l i t y of the e l e c t r o n being a t the nuc leus , i t i s a good approximation to use the Hamil tonian of a p a i r of magnetic d i p o l e s , i . e . I f we assume the e l e c t r o n spins quantized e i t h e r p a r a l l e l or a n t i p a r a l l e l to _> _ • the d . c . f i e l d E Q , and c a l l c l the angle between r and H 0 , we have To f i n d the corresponding Knight s h i f t a c a l c u l a t i o n s i m i l a r to the one done i n s e c t i o n (a) must be performed, i . e . we have to c a l c u l a t e the expectat ion value of the i n t e r a c t i o n w i t h respect to the e l e c t r o n i c wavefunctions. Bloembergen _1 ] has made t h i s c a l c u l a t i o n f o r the case of a x i a l symmetry and p-type wavefunctions and has obtained (17.3) or (18.3) 0 where V Q i s the atomic volume, N(Ey) i s the densi ty of s tates at the Fermi s u r f a c e , 9 i s the angle made by the d . c . f i e l d H Q w i t h the ax i s of symmetry,, chosen as the z - a x i s , and o 12 a n d t h e t o t a l K n i g h t s h i f t c a n t h e r e f o r e be e x p r e s s e d a s ( c ) The Moments o f t h e R e s o n a n c e L i n e . F o r a r e s o n a n c e c u r v e d e s c r i b e d b y a s h a p e f u n c t i o n J(~) , t h e n - t h moment w i t h r e s p e c t t o t h e p o i n t O 0 i s d e f i n e d b y n 1 r- - -to (20.3) N o t e t h a t i f j(^>) i s n o r m a l i z e d , t h e d e n o m i n a t o r i s u n i t y , a n d i f i t i s sym-m e t r i c a l w i t h r e s p e c t t o , a l l o d d moments v a n i s h . The i m p o r t a n c e o f t h e moments l i e s i n t h e f a c t t h a t t h e y c a n be c a l c u l a t e d f r o m f i r s t p r i n c i p l e s w i t h -o u t h a v i n g t o f i n d t h e e i g e n s t a t e s o f t h e t o t a l H a m i l t o n ! a n d e s c r i b i n g t h e l i n e . When m a k i n g t h e o r e t i c a l c a l c u l a t i o n s o f t h e moments one must k e e p a p a r t t h e b r o a d e n i n g due t o i n t e r a c t i o n s b e t w e e n s p i n s f r o m t h o s e due t o a s p r e a d i n L a r m o r f r e q u e n c i e s , c a u s e d b y e x t e r n a l f a c t o r s , l i k e t h e i n h o m o g e n e i t y o f t h e d. c . f i e l d , t h e i n s t a b i l i t y o f t h e r . f . o s c i l l a t o r , i m p e r f e c t i o n o f t h e c r y s t a l s , e t c . The l a t t e r g r o u p e d i n t h e c l a s s c a l l e d i n h o m o g e n e o u s b r o a d e n i n g , a r e o f random t y p e , a n d t h e r e f o r e n o t i n t e r e s t i n g f r o m a t h e o r e t i c a l p o i n t o f v i e w . I n r e s o n a n c e e x p e r i m e n t s one t r i e s t o r e d u c e t h e i n h o m o g e n e o u s b r o a d e n i n g t o a minimum. The b r o a d e n i n g due t o c o u p l i n g s b e t w e e n t h e s p i n s o f t h e s y s t e m m a i n l y i n c l u d e s d i p o l e - d i p o l e , p s e u d o d i p o l a r a n d e x c h a n g e i n t e r a c t i o n s . We w i l l demon-s t r a t e t h e m e t h o d o f moments b y g i v i n g a s h o r t r e v i e w o f t h e t h e o r y o f t h e d i -p o l a r l i n e w i d t h i n a r i g i d l a t t i c e „ ] • The c l a s s i c a l i n t e r a c t i o n b e t w e e n two m o m e n t s ^ a n d y u 2 i s g i v e n b y t h e e x p r e s s i o n (17.3): w_ /LA - 4^{M± .13 The Hamil tonian f o r the system of spins i s therefore g iven by Since a l l spins are coupled w i t h each other , the s t a t i s t i c a l d e s c r i p t i o n of the N - s p i n system i s g iven by a (2l+l) x (21+1) dens i ty m a t r i x , corresponding to the Gibbs approach. Vie consider the system i n a l i n e a r regime, i . e . a l i n e a r l y p o l a r i z e d r . f . f i e l d H-tH,COStot (22.3) along an a x i s ox creates under s teady-state condi t ions a magnetizat ion Mx w i t h a component along the same a x i s , g iven by n ^ H ^ X ' M ^ w t + X " ( ^ ) 5 ' ^ t ] (23.3) where the s u s c e p t i b i l i t i e s ^ and y^ ' are independent of H-j. I f we denote by ^Xthe quantum mechanical operator f o r the macroscopic magnet-i z a t i o n M of the sample, we e v i d e n t l y have ft~-<J),>^T<{fJi\ (24.3) w h e r e p i s the dens i ty matr ix of the system. The equation of motion forj> i n the presence of the r . f . f i e l d (22.3) i s where'k^u i s the Hamil tonian of the system i n the absence of the r . f . f i e l d and JL i s the magnetizat ion operator per u n i t volume (V = volume of sample). Assuming that the r . f . f i e l d was turned on when the sample was i n thermal e q u i -l i b r i u m , descr ibed by j ^ - 0 0 ) ^ » ~kn e s o l u t i o n of the equation (25.3), i n f i r s t order i n H l f i s T __#t' ' iMi" F u r t h e r , assuming that before the a p p l i c a t i o n of the r . f . f i e l d there was magnetization along the x - a x i s , i . e . Tr(£l»\-nx(-~).o no 14 we obta in- immediate ly , by comparison w i t h (23.3) and (24.3) o' S t a t i s t i c a l mechanics f u r n i s h e s f o r the dens i ty matr ix of a system i n thermal e q u i l i b r i u m a t temperature T the express ion n~ KT £ W W ] and c o n s i d e r i n g the f a c t that the temperatures are almost always s u f f i c i e n t l y high to a l l o w a l i n e a r expansion, we can w r i t e where*£ i s the u n i t operator . Since the trace i s i n v a r i a n t w i t h respect to r e p -r e s e n t a t i o n , we are permit ted to use the representa t ion We then have -Utt' t-jct'- ~ h __. . -tact ' m\ -Q*t'-- m~ -act' i^ t ' , LV s i h u t ' i l < M J i > ' > f (Eh.-E,.Ui I n t e g r a t i o n by parts immediately gives 1 5 0 The f i r s t o f t h e t e r r a s i s z e r o , b e c a u s e f o r t =o, £i» <Ol - O f a n d f o r t - ^ o 0 , ex/)(-1(Fnt' =0 , so t h a t we a r e l e f t w i t h 0 I n t r o d u c e t h e H e i s e r i b e r g o p e r a t o r w here U ( t ) i s t h e e v o l u t i o n o p e r a t o r o f t h e s y s t e m . S i n c e ~}l i s n o t d e p e n d e n t e x p l i c i t e l y o n t i m e , whence D e f i n e t h e c o r r e l a t i o n f u n c t i o n o f t h e m a g n e t i z a t i o n o f t h e s y s t e m a s 6(t)'Tr^>U.l (27"3) M a k i n g a c a l c u l a t i o n s i m i l a r t o t h a t o f t h e p r e v i o u s page we o b t a i n ^ f o - M t ^ , , , , v l 1 (28.3) whence y V ) _ - ¥ ^ L o 5 ^ ' ^ t')it' (29 .3) T h i s e x p r e s s i o n i s o f u t m o s t i m p o r t a n c e , s i n c e i t r e p r e s e n t s t h e l i n k b e t w e e n t h e t h e o r e t i c a l c a l c u l a t i o n o f t h e moments, e x p r e s s e d i n t h e r e l a t i o n s (27 .3) a n d (28 .3) d e f i n i n g [l), a n d t h e e x p e r i m e n t a l s t e a d y - s t a t e a b s o r p t i o n l i n e ^-(w) t h a t i s p r o p o r t i o n a l t o S i n c e we hav e c h o s e n t o w o r k i n t h e r e p r e s e n t a t i o n o f t h e H a m i l -t o n i a n i n t h e a b s e n c e o f t h e r . f . f i e l d , we n e e d t o d e s c r i b e i t i n more d e t a i l . We c a n w r i t e fi (^2 0 +}v l\ where 16 kK,~-~)f^0T~li ( 3 0 . 3 ) i s the Zeeman Hamil tonian of the spins and ^ ^ , i s the perturbing: Hamil tonian g iven by ( 2 1 . 3 ) , and responsib le f o r the d i p o l a r broadening of the l i n e . The l a t t e r can be w r i t t e n as a sum of s i x terms, out of which f o u r can be shown to be respons ib le f o r s a t e l l i t e l i n e s of double and t r i p l e the resonance f r e -quency. Since such l i n e s would add a large and undesired c o n t r i b u t i o n to the moments of a g i v e n frequency (see ( 2 0 . 3 ) ) , these terms must be l e f t out , and one obtains .2,1 1 t r JK <1K where and a BA -i( i l i v 2 - t - ' W - w y . - tdi i . j . i.)(i - 3 c«'e i k) where 0;, i s the angle between the p o s i t i o n vec tor j o i n i n g the two spins and the d . c . f i e l d H 0 , I t can be e a s i l y shovm that , commutes wi th Since the MB. l i n e i s narrow, we can neglect the v a r i a t i o n of u)over the width of the l i n e and assume that the shape of the l i n e i s descr ibed by A.{U?)/U> as w e l l as by • Then, i f we c a l l j- { t h e normualized shape f u n c t i o n , ( 2 9 . 3 ) can be w r i t t e n as 1 where vA, i s a n o r m a l i z a t i o n constant . By tak ing the F o u r i e r transform, and us ing the f a c t that .-u) , w e ob ta in _ ( f j = ^ f('-0)cosi4 J.to ( 3 1 . 3 ) F u r t h e r , s ince commutes w i t h i t , , , we can w r i t e 17 t - t - t . t so t h a t , u s i n g ( 2 6 . 3 ) , (27-3) , and ( 3 0 . 3 ) , we o b t a i n where By u s i n g the f a c t that the operator £Xp[iu?0 represents a r o t a t i o n of angle W.t about the z - a x i s , one obtains $W -^Ji)coStoJ,where i s c a l l e d the reduced a u t o c o r r e l a t i o n f u n c t i o n . This f u n c t i o n can be e a s i l y shown to s a t i s f y the r e l a t i o n v p fi'wi e4 1 t - 0 -MT>[[ti .[tf. ,L-~, !XiUiU-Ul>i ( 3 2 - 2 > On the other hand, us ing (31.3) and the d e f i n i t i o n (20.3) of the moments, the f o l l o w i n g express ion i s obtained. i n $,(<>) \ d i / t - . 0 I t has the advantage that i t cons i s t s of traces of operators and i s there-fore independent of the representa t ion chosen f o r t h e i r computation. Thus^ne chooses the most convenient representa t ion , namely the one i n which the e i g e n -values m«' of the i n d i v i d u a l s p i n operators 1^  are good quantum numbers. For example, i n the case of the second moment of a sample w i t h i d e n t i c a l spins one obtains from (32.3) by somewhat ted ious , but s t r a i g h t f o r w a r d c a l c u l a t i o n s ( 3 4 - 3 ) A c a l c u l a t i o n of the d i p o l a r broadening f o r u n l i k e spins shows that the c o n t r i b u t i o n of the f o r e i g n spins to the second moment i s smaller by a f a c t o r of 4 / 9 . The pseudodipolar and exchange broadening come about through the i n d i r e c t c o u p l i n g of the nuclear spins v i a the conduction e l e c t r o n s . This i s a second order process , and v/as a p p l i e d to s o l i d s by Bloembergen and Rowland [ 2 ] and Ruderman and K i t t e l [ 8 ] . A short o u t l i n e of the comptitations i s g iven below. Consider the e l e c t r o n nuclear coupl ing f o r two n u c l e i where i s g iven by (1.3) and (I8.3): <R z X (contact) + ^ ( d i p o l a r ) , or where r^ = r — R__. The change i n energy due to t h i s i n t e r a c t i o n i s g iven by where the s tates |(?cO and\W.> are product s ta tes between e l e c t r o n and nuclear s t a t e s . From here on one proceeds i n a s i m i l a r manner as i n the previous s e c t i o n s ; an e f f e c t i v e coupl ing that does not depend e x p l i c i t e l y on the nuclear wavef unc t i ons i s i n t r o d u c e d . The conduction e l e c t r o n wavefunctions are chosen as products b e -tween B loch func t ions and s p i n f u n c t i o n s . The sum over quantum numbers i s r e -placed by the sum over occupied s tates ! k , $ / S a n d the occupation f a c t o r i s r e -placed by the Fermi f u n c t i o n . The expression (35*3) w i l l e v i d e n t l y y i e l d three type of terms. A quadrat ic contact term, a quadrat ic d i p o l a r term and cross terms. The f i r s t one gives the so c a l l e d exchange i n t e r a c t i o n and has a s c a l a r form: X ^ u ^ r i * (36.3) When computing moments t h i s term must be added to the Hamil tonian 1A\ i n (34•3)• Since commutes wi th 7 L x , t h i s term has zero c o n t r i b u t i o n to the second moment i n the case of l i k e s p i n s . The c o n t r i b u t i o n to the f o u r t h moment, however, does not v a n i s h , so that the exchange term i s e f f e c t i v e l y narrowing the l i n e . I n the case of u n l i k e spins the s i t u a t i o n i s somewhat d i f f e r e n t be-cause both the second moment and f o u r t h moment c o n t r i b u t i o n do not v a n i s h . 19 T h e . q u a d r a t i c d i p o l a r t e r m w i l l u s u a l l y he s m a l l e n o u g h t o be n e g l e c t e d . The c r o s s t e r m s y i e l d a p s e u d o d i p o l a r c o u p l i n g o f t h e f o r m ( 3 7 - 3 ) where B - y d e p e n d s o n t h e c h a r a c t e r o f t h e e l e c t r o n w a v e f u n c t i o n s a n d g o e s as f o r l a r g e r ^ -. N o t e t h a t t h e p s e u d o d i p o l a r t e r m h a s t h e same a n g u l a r d e -p e n d e n c e a s t h e d i p o l a r H a m i l t o n i a n ( 2 1 . 3 ) • CHAPTER IV DIPOLAR BROADENING FOR A CLOSED PACKED HEXAGONAL LATTICE The dipolar broadening i s expressed by (34-3) where the sum i s taken over a l l neighbouring sites and O^j i s the angle between the vector r ^ (from spin I to neighbour spin 1^) and the s t a t i c f i e l d H 0. Let us calculate the sum. Consider the Cartezian system of coordinates X C } Y c Z c having 0Z O p a r a l l e l to the pri n c i p a l axis of the c r y s t a l , so that X cOY c rep-resents the basal plane. Vie choose 0XC such as to have reflexion symmetry with respect to the plane ZC0XG5 this i s always possible f o r a closed packed hex-agonal l a t t i c e . Consider the d.c. f i e l d H Q having the polar angles , and an arbitrary l a t t i c e point r° having the polar coordinates (v^ ) dj ^ j . C a l l A - A C the angle of r j with respect to H 0. We f i r s t f i n d the dependence of the sum i the addition theorem for spherical harmonics (see Appendix A). We have We use 20 21 Applying (4-A) i n this case ( 8 ,6^0; , ^ r ^ f ) lf = ( f ! ) ' w e h a v e '6 »v ^ ^uep4zr y;<e,T) y / ( « and Using the f a c t that }^ = H ) ^ , we have where 4 7 fU + *>) T Hence Denote and Then SZ'fi~ /- 5Z. J w. v^' . Further, we evidently have f p j r a = fW and • ( V A + , V ) ^ ' ^(v*+^'>^f - t U W ) f - l ( » W ) < f 2 2 The second,term of the l a s t expression gives zero contribution, because for ^ -0 i t i s zero, and f o r ^ ' ^ O there exists i n the j-expansion for each j-terrn ^ " " j and %'> ~ - f ; • Hence and we are l e f t with C0_ a <f£cos2<p' + ^ U C 0 S 3 f J c o 5 3 f n < J . < j ^ - 4<f] Consider the term coi^f cosU?. Because of the three-fold sym-6 . 1 metry of the l a t t i c e , there exists i n the j-expansion, for each j-term, other two terms j'and £ having 9^ = 9j" = 8 I and ^ = -f-HO" ; f ' „ = LfC r ?Jl 0°>. Consequently there w i l l be groups of the form A [ c w ? j t ^ s ( f • + HO°) + c o . ( f - t 2 ^0) ] = 0 Similarly, the terms containing co$_*|f and cosh <j?• w i l l be zero. Consider now the term Because of the symmetry with respect to the basal plane, there exists i n the j-expansion, for each j-term, an other term j', having *f•, - \f- and Q:> - Tf~ Ol . Hence there w i l l be groups of the form But P™ has the parity ( - l ) 1 - m . Therefore and the above term i s zero. Yfe are l e f t with Q 23 or 5 " i M M . F>«e'4)T[P>Oi0)]V2(i,f[p>o50])]l[RV«e)]V Using the expressions ^(u) = 3 ( l ~ n ) ; P2(u) = 3uYl-u , P2(u) = -J-(3u -1) and denoting )f- -Si>i*$: we o b t a i n I t i s a general formula , v a l i d f o r any hexagonal c losed packed l a t t i c e . Note the independence of . 205 Let us now apply t h i s to a c r y s t a l of pure TI . For the 1-st s h e l l we have f&h &•= .7216, whence x-; = .3426, r ^ = 3.4104 and t h i s y i e l d s J j j S i m i l a r c a l c u l a t i o n s were made f o r the next 28 s h e l l s , comprising t o t a l l y 268 nearest neighbours. The r e s u l t i s = 3^x10* (l.ll ^ B-k.lksinl0 f 2.0$) 203 Let us now take i n t o c o n s i d e r a t i o n TI . The r a t i o between the two c o n t r i b u t i o n s w i l l be h oV°> .JLI ,203 Using the f a c t that the f r a c t i o n of T 1 ^ " J i s 29.5%, we obta in ( A v ' ) - n ^ = 0.7^(3.S1 5 / ^ 0 - ^ 7 ^ 5 i h 2 8 a , o ^ ) ( K c / 5 e c ) 2 S i m i l a r l y one obtains ( A 7 ) n 2 0 i = 0.550(,,^ W^'k.lh S,V0 t ? , < ) ( K c A f , ) " Figure 2 shows as a f u n c t i o n of the angle u between the p r i n c i p a l a x i s and the d . c . f i e l d 24 CHAPTER V EXPERIMENTAL CONSIDERATIONS r^ i e sample was a single c r y s t a l of natural thallium, i . e . 70.5% T l 2 0 ^ and 29.5% T l 2 ( ^ . The thallium crystal has a hexagonal closed packed l a t t i c e . I t i s a soft metal (similar to lead), and must therefore he handled with great care, especially i n the case of single crystal work, where a l l the signal comes from a thin surface layer. I t was stored on foam rubber, and the surface reserved for the winding of the c o i l was touched as l i t t l e as possible. I f s u p e r f i c i a l scratch-ing occurs, i t can be removed by etching. Thallium, when exposed to a i r , oxidizes quite rapidly, loosing i t s metallic l u s tre. Special care was taken to keep the metallic surface free of oxide. The reasons for this are the following: (1) The resonance of thallium oxide distorts the line shape. (2) The thallium oxide layer influences detrimentally the f i l l i n g factor. (3) I t i s d i f f i c u l t to obtain perfect single crystals of thallium because i t has a phase transition just below the melting point. The sample had i n the corners small c r y s t a l l i t e s whose orientation was different from the one of the main monocrystal. When the metallic surface i s clean, these c r y s t a l l i t e s are easily noticed with the naked eye and can be l e f t out when winding the c o i l . (4) When x-raying the c r y s t a l , the metallic surface must be free of oxide, i f a clear picture i s to be obtained. I t was found that even very thin layers of oxide blurred the f i l m . The etching was done with a solution of 80% acetic acid and 20% hydrogen peroxide. Extreme care has to be taken i n this process, since i t i s an exother- • mic reaction that becomes very violent as soon as the solution heats up. I f the 25 2.6 beaker becomes hot the c r y s t a l must be removed immediately, otherwise i t d i s -olves i n a matter of seconds. A u s e f u l precaut ion i s to perform the e t c h i n g i n a wide v e s s e l of f a i r l y large volume, to r e t a r d the heat ing process. The s t o r i n g s o l u t i o n that proved most s a t i s f a c t o r y f o r t h a l l i u m was b o i l e d water. Not only does the c r y s t a l mainta in per fec t m e t a l l i c l u s t r e when kept t h e r e i n , but i t was found that t h i n l a y e r s of oxide disappeared w i t h i n a few hours . Tape wound around the sample, a f t e r the a p p l i c a t i o n of the c o i l proved u s e f u l i n keeping the c o i l i n p l a c e , as w e l l as i n the prevent ion of o x i d a t i o n . When performing experiments, the c r y s t a l was kept as much as p o s s i b l e i n vacuum or i n a helium atmosphere. This was e a s i l y accomplished, s ince a l l experiments were performed i n dewers that were part of a vacuum system. Two c r y s t a l s , produced by Semi-Elements, I n c . , v;ere a v a i l a b l e f o r t h i s work. One of them was c y l i n d r i c a l i n shape, 3/4" i n length and l / 4 " i n diameter. Only p r e l i m i n a r y work was performed w i t h t h i s sample. The other c r y s t a l had a p u r i t y of 9 9 . 9 9 % a n d - was l a r g e r : a p a r a l l e l i p i p e d w i t h the dimensions 1" x 3/8" x 3/8". The large surface a v a i l a b l e f o r winding the c o i l proved bene-f i c i a r y i n improving the s i g n a l - t o - n o i s e r a t i o . The manufacturer claimed that the p r i n c i p a l a x i s 0 0 0 1 of the c r y s t a l was perpendicular to one of the p a r a l l e l -i p i p e d f a c e s . T h i s , as w e l l as a monocharacter of the c r y s t a l was checked by x - r a y i n g the four d i f f e r e n t faces of the sample. The r e s u l t s were i n agreement w i t h the quoted c h a r a c t e r i s t i c s . A molybdenum tube proved optimum f o r t h a l l i u m x - r a y i n g . A v a r i e t y of c o i l s was t r i e d out f o r the d i f f e r e n t experiments. When wind-i n g c o i l s one i s faced w i t h the choice of the f o l l o w i n g parameters: (1) The gauge of the w i r e . ( 2 ) The f i l l i n g f a c t o r (determined by the number of mylar l a y e r s ) . (3) The t o t a l number of t u r n s . ( 4 ) The number of turns per cm. 27 When performing an experiment, the magnitude of the d.c. f i e l d i s usually pre-determined within certain l i m i t s . This immediately determines the desired o s c i l l a t i n g frequency of the resonance c i r c u i t . Since the capacitance of the c i r c u i t located i n the P K W detector varies over a r e l a t i v e l y small range, and should he kept at a minimum to reduce the losses, i t i s the inductance of the c o i l that has to he chosen such as to provide the desired frequency. Since a l l four of the above mentioned parameters determine the inductance of the c i r c u i t , one i s faced with a multitude of choices. From the accumulated experience the following optimizing procedure became apparent. The f i l l i n g factor (determined by the number of layers of mylar) was chosen such as to render the c o i l with high enough q u a l i t y factor to o s c i l l a t e at the desired temperature, but not above i t (the Q of the c o i l decreases with increasing temperature). This brought the system very close to the ideal conditions of a marginal o s c i l l a t o r and seemed to y i e l d the best performance of the detection apparatus. The wire was chosen of f a i r l y high gauge (38-4-0), especially i n the cases with l i t t l e mylar, to prevent damaging of the surface when winding the c o i l . The number of turns and their spread was then chosen such as to y i e l d the right frequency and have the c o i l cover the whole usable surface of the sample. This can always be accom-plished, since the inductance increases with the number of turns and decreases with the spread between them. When working with the maximum f i e l d s available, i t i s useful to keep i n mind that the o s c i l l a t i o n frequency of the resonant c i r c u i t usually increases from l i q u i d nitrogen to l i q u i d helium temperatures by 6 to 8fo. Experiments were performed at l i q u i d helium, l i q u i d nitrogen, dry i c e , and room temperatures. The signal-to-noise ratio at dry ice and room temperature was not high enough to y i e l d concludent results. However, these experiments w i l l be repeated, possibly with a d i g i t a l integrator. The low temperature system was of conventional design. An outer dewer was kept f u l l with l i q u i d nitrogen, and 28 l i q u i d helium was pumped i n t o an inner dewer, that was part of a vacuum system, so that experiments could he performed at both temperatures, 1.2°K (when pumping) and 4-2°K (at atmospheric pressure). I t was found that, i n many of the experiments, the l o s s of s i g n a l due to the change i n temperature from 1.2°K to 4.2°K was com-pensated by the b e t t e r damping, provided by the l i q u i d helium when not s u p e r f l u i d . Audio-frequency noise, due to the free v i b r a t i o n s of the sample at the end of the coax, could not be f i l t e r e d out very s u c c e s s f u l l y at the modulation frequency of 20 cps. The inner dewer had a capacity of 1.75 l i t e r s , and, the system was w e l l enough i n s u l a t e d to y i e l d 20 hour helium runs without r e f i l l i n g . The o r i e n t a t i o n of the c r y s t a l was f i r s t performed v i s u a l l y . In t h i s way the p o s i t i o n of the 0001 a x i s can be determined w i t h i n i5°. I n the case of ex-periments with good signal-to-noise r a t i o t h i s can be improved to an accuracy of ±2° or b e t t e r , by using the r e s u l t s of the anisotropic Knight s h i f t . During l i q u i d helium experiments, a very accurate method of determining the o r i e n t a t i o n of the p r i n c i p a l axis was found. I t i s known that at these temperatures the magnetoresistance pick-up becomes very strong. Since i t i s i n phase with the s i g n a l , i t cannot be f i l t e r e d out by e i t h e r the narrow band a m p l i f i e r , nor the phase-sensitive detector. I t was found that the e f f e c t i s strongly a n i s o t r o p i c and has a minimum when the s t a t i c f i e l d i s p a r a l l e l with the p r i n c i p a l a x i s . By applying a high modulation amplitude ( 20Kcps proton), and by watching on CR0 I the overloading produced by the pick-up, the o r i e n t a t i o n of the p r i n c i p a l axis could be determined w i t h i n 1/3 of a degree, by changing the p o s i t i o n of the magnet such as to produce minimum overloading of the phase-sensitive detector;, Even higher accuracy can be obtained, i f the v i s u a l check on the scope i s r e -placed by the measurement of the zero p o s i t i o n of the baseline as a function of o r i e n t a t i o n . E v i dently, these measurements must be performed outside the resonance l i n e . Before s t a r t i n g the experiments on thallium the system was optimized on 29 a single c r y s t a l of aluminum. Different techniques were t r i e d . One of them was to replace the phase-sensitive detector hy a waveform eductor (100 storing channels). I t was also t r i e d to feed the output of the phase-sensitive detector, used with a low time constant into the waveform eductor, thus using the i n t e -gration capacities of both devices. The results of both methods were i n f e r i o r to those obtained by the phase-sensitive detector alone. The main reason for this l i e s i n the storing time l i m i t a t i o n of the eductor; p r a c t i c a l l y , the time i n t e r -val available f o r the apparatus to gather information about the l i n e i s limited to about 10 minutes. The technique using the phase-sensitive detector and recorder, when used with a time constant of 30 sec and corresponding slow sweep through the l i n e , provides longer time int e r v a l s . I t i s clear that a d i g i t a l integrator, having p r a c t i c a l l y no storing time li m i t a t i o n , would have great advantages over the eductor. Second harmonic detection was also t r i e d , but the results were not satisfactory. The apparatus was i n i t i a l l y designed for sweeping the frequency. This was accomplished by applying a linear, sawtooth voltage (0-100 volts) from a modified Tektronix waveform generator to a voltage sensitive diode capacitor (varicap PC116), i n p a r a l l e l with the capacitance of the resonant ci r c u i t y linear sweeps from 1 msec to several hours were available. The method i s satisfactory f or t i n and other materials having r e l a t i v e l y narrow l i n e s . I t was used i n a l l the pre-liminary experiments on thallium. However, for an exact analysis of the lineshape, where the baseline d r i f t has to be reduced to a minimum, the sweeping of the f i e l d proved more satisfactory. The reason for this l i e s i n the f a c t that i n the case of thallium one has to sweep 200-300 Kc/sec, and therefore use a relative large portion of the varicap range. This causes a change i n the l e v e l of o s c i l l a t i o n which, i n turn causes a baseline d r i f t . The modulation frequency used by Bloembergen and Rowland [2] was 280c/sec. In this work different frequencies between 15 and 400 cps were t r i e d . The PKW 30 'spectrometer seemed to give the best signal-to-noise r a t i o for 20 cps and, there-fore this frequency was used i n most of the performed experiments. The only d i s -advantage caused hy this low value was the f a c t that the commercial lock-in amplifier could not he used under optimum conditions; the manufacturer recom-mends, frequencies of 100 cps or higher, for best results. The modulation amplitude was chosen on an experimental basis. The most sen-s i t i v e feature of the line-shape to modulation i s i t s linewidth^ (Me w i l l use throughout this work the peak to peak linewidth of the derivative curve) therefore the safe modulation amplitude, that i s , the amplitude that does not d i s t o r t the li n e appreciably, can be obtained by making a series of recordings of the same l i n e , while using decreasing modulation amplitudes. When the amplitude became smaller than l / 5 of the linewidth the d i s t o r t i o n was negligible. In this manner i t was also found that, i n the case of the thallium c r y s t a l , the modulation am-plitude, even when having a strong influence on the linewidth, had a r e l a t i v e l y l i t t l e e f f e c t (5-10% of the linewidth) on the resultant resonant frequency (obtained by the method described i n the next chapter). Higher modulation am-plitudes than those given by 1/5 of the linewidth were used i n experiments with poor signal-to-noise r a t i o . When recording resonance lines with the purpose of measuring their second moments, large modulation amplitudes can be used, i n order to obtain a good s i g -nal strength f a r out i n the wings. A simple correction to the second moment for this case i s given by Andrew [9]. A further analysis of the influence of the equipment on the shape and resonant frequency of the line was carried out by using the following method. The output of the PKVJ spectrometer was fed simultaneously through a twin B1JC connector into two separate systems, each consisting of a narrow band amplifier, phase-sensitive detector, and recorder. Thus, one could vary different para-meters, ( l i k e attenuation, the time constant, etc.) independently and observe 31 'their e f f e c t on the l i n e shape. I t was found i n this manner that the time constant of the phase-sensitive detector had a stronger effect on the shape and resonant frequency of the li n e than was, at f i r s t , expected. Keeping the time constant l / l O of the sweeping time through the linewidth was found as unsatisfactory. A rati o of 1/20 i s rather recommendable. A safe procedure i s to sweep i n both dir e c -tions; i f the two lin e s thus obtained have the same resonant frequency and l i n e -width the sweeping speed can be considered adequate. I t i s interesting to note that, i n the case of experiments with single crystals, the time constant does not only affect the resonance frequency of the l i n e , but also i t s linewidth and ap-parent mixing of the absorption and dispersion mode ( i t was found that the r a t i o of the peaks was reduced when sweeping too f a s t ) . This effect i s due to the as-symmetry of the l i n e . I t i s clear that, i n the case of a line having the shape shown below, a f a s t sweeping rate w i l l displace the lower peak by a smaller distance than the upper one, ^ ^ thus creating a decrease i n \ ^ ~" linewidth. Note that the same \ effect w i l l increase the l i n e -sweefi'nfy \ / width i f the sweeping direction diftcf-'ior) T^---> i s reversed. The overloading of the phase-sensitive detector also proved to have a strong effect on the l i n e . When working at l i q u i d helium temperatures, where the magnet-oresistance pick-up i s very strong, one may be inclined to work with a f a i r l y over-loaded phase-sensitive detector. Under such circumstances i t was found that the mixing of the dispersion and absorption mode, determined by the rat i o of the peaks, was inconclusive. A safeguard against this type of error i s to sweep through the line twice, while using opposite p o l a r i t i e s on the modulation c o i l s , and check i f the same rat i o of the peaks i s obtained. The monitoring of the f i e l d was performed i n the following manner. The proton 32 S i g n a l g iven by the g l y c e r o l probe was d i s p l a y e d on CRO I I . The width of the h o r i z o n t a l sweep on the scope was a measure of the modulation amplitude, and t h e proton s i g n a l , when at the c e n t e r , gave an accurate measure of the f i e l d . As the l a t t e r was swept by means of a motor, the proton s i g n a l appeared on the scope t r a v e l l i n g from, say, l e f t to r i g h t . When pass ing through the c e n t e r , the corresponding proton frequency was punched out . The t i p of the proton s i g n a l _ was not wider than 1 mm. The width of the h o r i z o n t a l l i n e on the scope was 6 cm. The sweep was slow enough to permit accurate e s t i m a t i o n of the t i p p o s i t i o n o n the scope ( w i t h i n 1 mm). Therefore, f o r , say, a modulation amplitude of 5Kc/sec p r o t o n , the f i e l d could be measured w i t h an accuracy o f t ,2Kc/sec proton. The s i g n a l , i f d e s i r e d , c o u l d be brought back on the scope w i t h i n seconds by means of the v a r i c a p c o n t r o l of the f i e l d monitor ing o s c i l l a t o r . T h i s , however, turned out as unnecessary, s ince the sweep of the f i e l d was l i n e a r enough to provide accurate r e s u l t s by i n t e r p o l a t i o n , even when the proton frequencies were punched out every 5-10 minutes. The modulation amplitude was measured by r e c o r d i n g the proton frequency a t the two ends of the h o r i z o n t a l sweep on the scope. I f the. f i e l d modulation i s not p e r f e c t l y s i n u s o i d a l there can appear an e r r o r i n the measurement of the f i e l d . This can be corrected by t a k i n g the average of the two values obtained w i t h opposite p o l a r i t i e s on the modulation c o i l s . F i n a l l y , whenever necessary, a c o r r e c t i o n f o r the i n s t a b i l i t y of the PKW spectrometer was c a r r i e d out . The momentary frequency of the o s c i l l a t o r was recorded every 5 minutes on the recording c h a r t . I f i t happened to vary by ap-r p r e c i a b l e amounts; the f i e l d values were normalized to the frequency recorded at the center of the l i n e . 205 20^ The referrence resonance frequency TI ^ and TI was obtained as f o l l o w s . A saturated s o l u t i o n of t h a l l i u m acetate was placed i n a g lass v i a l l / 2 " i n diameter and a small c o i l of 2-3 turns of No.24 wire was wound around i t . The 33 c r y s t a l w a s replaced, by the referrence sample and a resonance experiment was performed at room temperature. The signal was strong enough to be seen c l e a r l y on the scope. Due to the narrow l i n e , a low modulation amplitude had to be used (,3Kc/sec proton). The results were (for 10 kilogauss)? (Tl 2°5) = 24,572.0 +.2 Kc/sec (Tl 2°3) = 24,332.8 +.2 Kc/sec CHAPTER VI THE ANALYSIS OP THE LINESHAPE The Lorentzian absorption and dispersion modes are given hy V - A 1 — - • V - A T^(^-^) where A i s a normalization constant. I n singl e c r y s t a l work vie are i n t e r e s t e d i n the mixed mode W h A U 1 - A L - ^ ^ A 'I Our experimental r e s u l t i s the de r i v a t i v e of t h i s mode 2 The general shape of such a l i n e i s shown below — absorption mode I t has two zeros V, a n d a n d three extrema at Va ,VC . The l a s t extremum at Vc i s u s u a l l y very shallow and therefore of l i t t l e i n t e r e s t . To f i n d the r e -maining four points, l e t us denote and AT2=B. Then 34 35 ^ - ' B ^ T W - 12.6) The "two zeros are evidently given by solving bx2-2x-b=0, i . e . The extreraa are found by solving y'(x)=0, i . e . The solutions are (4.6) y - 1 , ^ V ^ l r 0 ^ • y I 2</PTT ' y j , 2\/bNT f f , ^ " b T ~ > X a b f — b C 0 $ ^ ) > X ^ t —V~~ 5 \ i ^ 2 k 0 ) where £05 <j> (l^  I)2)~ ' / l . Note that, since 0$h<\ , we haveO^f <^5"° and | <y^ s so that X^X^XC» I n terms of frequencies: A 'V-iATr^m;""" 0"'^ The linewidth evidently i s given by The int e r v a l between the apparent resonance frequency V, and the true resonance frequency V0 i s given by A'-t-^r- ~^YiJ~- (6.6) In the case of pure absorption b=05and (3.6) becomes 3x^-1=0, giving x=±1 / 3 AUD ( A v ) a U ' TF^yf (7.6) I t i s important to note that the r a t i o of the peaks R depends on b only. Indeed, R=-Y(xa)/Y(xb) i s , by ( 2 . 6 ) , a function of b, x a, and x^, and, since x a and are functions of b only, i t results that R i s a function of b only. The dependence can be eas i l y calculated from the above formulae and i s shown below i n the form of a graph. 36 These expressions provide a method of obtaining the resonant frequency ~V0 and the linewidth AV by f i t t i n g Lorentzian experimental l i n e s . The thallium resonance line turned out Lorentzian i n the center over an interval of about three linewidths. (A Gaussian f i t was.also t r i e d , but proved unsatisfactory). The f i t t i n g was accomplished through a semiempirical method of successive ap-proximations that converged quite rapidly to satisfactory results. The procedure i s as follows! (1) Choose a baseline. (2) Measure the ratio of peaks R. (3) Use the graph to determine b from R. (4) Measure the experimental linewidth (5) Determine T 2 from (A V')exp and b. (6) Determine A from T^ and b. (7) Determine^ from A and{V , )e i i? • At this point, as a f i r s t check of the f i t , was compared to (\\-V0) and (Vo'Va), respectively, calculated from (4.6). I f this comparison did not turn out satisfactory i t could usually be improved by chang-ing the baseline and (A v)n|>. In the case of experiments where % and A v i s not needed to high accuracy one can stop here. However, for better f i t s , the following 37 extra steps were taken: (8) Using the f a c t that the experimental trace i s a sura of a symmetric l i n e (the dispersion mode) and an assymmetric line (the absorption mode) about^03 the two contributions were separated by respectively adding and subtracting, point by point, the two sides of the experimental l i n e . (9) Prom the absorption mode thus constructed, one determines (^"v^aLs* which, i n turn, furnishes T 2 through (7.6). This value of T 2 should correspond to the one obtained i n step (5). I f the comparison i s not satisfactory i t i s recommendable to repeat the previous steps (from (6) onwards) with the new T 2. (10) Construct the three modes (absorption, dispersion, and mixed) using the above obtained parameters and see i f they f i t the experimental modes. The above procedure may seem cumbersome at f i r s t sight, but i f repeated with a few l i n e s , i t provides-the operator with practice i n choosing the base-li n e and the linewidth, and this makes the f i t t i n g quickly convergent. We add a few useful hints i n shortening the process. Prom (2.6) we see that the absorp-tion and dispersion' mode are given respectively by A simple calculation of the rat i o of the peaks for the dispersion mode furnishes' R<3j_sp=l/8. Therefore i t i s the separation of the dispersion mode that provides the best information regarding the choice of baseline. I t was also observed that the absorption mode provided the most r e l i a b l e T 2. This i s understandable, since T 2 i s the scaling factor of x and therefore of strong influence i n f i t t i n g a l i n e with pronounced peaks as those of the absorption mode. Figure 3 shows the experimental and f i t t e d modes for a typical l i n e . Figure 3- Separation of Modes., f o r a t y p i c a l Thallium Line. CHAPTER VII EXPERIMENTAL RESULTS AND DISCUSSION In Figure 4| 5, a n d 6 are shovm three of the typical lines yielded hy the equipment. The f i r s t of the three i s a T l 2 0 ^ l i n e at l i q u i d helium temperature, with a modulation amptitude of 5Kc/sec proton (Ho=7800 oersted). The second line i s at l i q u i d nitrogen temperatures and has therefore a higher modulation amp-litu d e , to improve the signal-to-noise ratio (Ho=7600 oersted). F i n a l l y the l a s t of the three i s a Tl 2 <^3 resonance l i n e obtained at 1.2*K under optimum working conditions, at an orientation where the angle between the principal axis of the cr y s t a l and the f i e l d had a minimum, and therefore corresponded to minimum magneto-resistance pick-up. The modulation amplitude was 20Kc/sec proton and Ho=7900 oersted. The almost straight lines on the three figures represent the measurement of the monitored d.c. magnetic f i e l d and give the p o s s i b i l i t y of a point by point analysis of the resonance. Figure 7 and 8 show the angular dependence of the resonance frequency at l i q u i d nitrogen and l i q u i d helium temperatures, respectively. The experimental results are compared to a line of the functional form (see (19.3)). AH K , - f W e - O • C1-?) and furnish K i s o = (160.7+.8) x 10™4 K' = -(8.7±.8) x I O - 4 J at 77 K (2.7) K i g o = (163.0*.4) x 10 4 at 1.2°K (3.7) K' = -(9.8+.4) x 10-4 J The errors were calculated with the assumption that the resonance frequency was known with an accuracy of±2Kc/sec at 77°K and ±1Kc/sec at 1.2°K; this i s a very 39 p - p Modulation I 1 H = 7800 oersted Figure 4. Experimental Resonance Line of T l 2 0 5 at 1.2°K. Figure 5. Experimental Resonance Line of T l 2 0 5 at. 77°K (arbitrary u n i t s ) . Figure 6. Experimental Resonance Line, of T l 2 0 ^ at 1.2°K (arbitrary u n i t s ) . Figure 7. The Resonance Frequency i n T l 2 0 5 as a Function of Crystal Orientation at 77°K. Figure 8. The Resonance Frequency i n TI 5 as a Function of Crystal Orientation at 1.2°K. '45 "^cautious assumption when regarded i n the l i g h t of the high s t a t i s t i c a l information obtained from the many experimental points f i t t i n g the functional form (1.7). Figure 9 shows the angular dependence of the linewidth at 1.2°K. The ac-curacy i s +1.5 Kc/sec. Table 1 gives the results of an experimental second moment calculation f or f i v e of the better l i n e s . These lines were starting to cut off faster than the » Table 1 corresponding Lorentzian f i t t e d line at a distance of the 0 M2 (Kc/sec)^ ~ ~ YT_3 ~" order of two line widths from the centre. 25 1107 (?) ?m 35 587 T l l e T l ^ - 5 l i n e had a resonance frequency of 40 694 ' , 55 g83 = 24,701 ± 5 Kc/sec along the principal axis. This gave the 70 733 p o s s i b i l i t y to check i f the Knight s h i f t parameters determined f o r T l 2 0 5 f i t t e d the Tl 2 ° 3 sample at this orientation. Using (1.7) and (3.7) and the referrence frequency obtained for T I 2 0 3 , we obtain for 0= 0,"i) = 24705 Kc/sec, in f a i r agreement with the above experimental value. The nuclear magnetic resonance i n thallium has been studied previously i n powdered specimens by Bloembergen and Rowland £l ,2 ] , and Karirnov and Shchegolev [3] . The l a t t e r quote the following value for the resonance frequency i n powder at l i q u i d helium temperature. V i s o ( T l 2 0 5 ) = 24,975 ± 3"Kc/sec; S)^ B 0(T1 2 <^3) = 24723 ± 3 Kc/sec. Since this frequency corresponds to \) (powder) = V0(1+KJ_ s o) , i t can be calculated from our results. One obtains V i E o ( T l 2 ° 5 ) = 24,972.5 ± 1 Kc/sec, V i s 0 ( T l 2 0 3 ) = 24730 ± 5 Kc/sec, i n f a i r agreement with the above res u l t s . The results obtained for the linewidth (25-32 Kc/sec), also have the right order of magnitude, when compared to the 33 Kc/sec linewidth obtained by Bloembergen and Rowland. On the other hand, the Knight s h i f t and second moment results disagree with the previous measurements. Bloembergen and Rowland [2] obtained K i s o = ( x54 - 5) x 1 0 ~ ^ a n d K ' = " x 6 ' 4 x 10~4 at 77"K. These values are Figure 9. The Linewidth as a Function of Crystal Orientation at 1.2°K. 47 --significantly d i f f e r e n t from the values obtained i n this work ( K i s o = (160.7 ± .8) x 10-4; K' = - (8 .7 ± .8) x 10~4), and this discrepancy seems to r e f l e c t the d i f f i c u l t y of determining accurate values of the Knight s h i f t parameters from powder results. Also, the preliminary measurements of the second moment, given i n Table 1 are larger than those measured by previous workers [2, 3 ] . This does not seem surprising when comparing the resonance lines of this work with the previously published experimental results, which had smaller signal-to-noise r a t i o s , and therefore were not measuring the signal f a r enough out i n the wings. So f a r we have performed Knight s h i f t measurements f o r only three tem-peratures, 1.2, 4«2^ and 77°K. No significant change between 1.2° and 4.2° was observed. Between 1.2° and 77° an increase of 1.5$ i n Kj_ s 0 and 12$ i n K' was obtained. This i s barely s i g n i f i c a n t when compared to the experimental error. Suggestions f o r further experiments: Work i s continued with the purpose of obtaining the experimental value of the exchange constant^. J i j i n (36.3) as well as the quantities Bj_j i n the expression (37»3) for the pseudodipolar con-t r i b u t i o n to the second moment. This could hopefully be accomplished by study-ing the overlapping of the two isotope lines and by trying to sort out the different contributions to the line i n angular dependence measurements of the second moment. Experiments at higher temperatures are planned with the purpose of estab-l i s h i n g the temperature dependence of the Knight s h i f t and to compare i t with the results obtained by Sharma and Williams Cl2] on cadmium which has the same cryst a l structure and valency as thallium and has exhibited a r e l a t i v e l y strong temperature dependence of both the isotropic and anisotropic part of the Knight s h i f t . APPENDIX THE ADDITION THEOREM FOR SPHERICAL HARMONICS The simultaneous angular momentum eigenfunctions L 2 and Lz> are In particular, the eigenfunction corresponding to the eigenvalue m=0 Is . The addition theorem i s the formula expressing the particular eigenfunction (m = 0) f^(co5§)of angular momentum about a z'-axis | of L z . Such an expansion must s a t i s f y the following requirements: (a) Since P-^  i s also an eigenfunction of L 2 , we must have L 2P^ = l ( l + l ) P ^ . Therefore only spherical harmonics with the same subscript 1 can appear i n the expansion. (b) An interchange of 9 with/3 and^ with<l must leave the expansion unchanged. For such an interchange corresponds to the interchange of r with oz' , i . e . leaves their r e l a t i v e position unchanged and therefore also L z/ (c) In a r i g i d rotation of the figure about oz,<^  and ^ "change by equal amounts, and 6' remains constant. Hence, P^l^u) must be a function of { f~<i) . 49 A l l these requirement^ can he s a t i s f i e d i f Indeed, the coef f i c i e n t s of the expansion are Cm/| (^j^) • f i i ' s t requirement i s c l e a r l y s a t i s f i e d , (h) i s s a t i s f i e d because, by making the change of summation index m =• -m', we obtain <-^~ \ / " , r ' ' s \F"^' and this sum i s the same as (2.A), providing C_„m = Cm. F i n a l l y , (c) i s sat-i s f i e d since thed, and ^ dependence i n (2.A) i s , by (1.A), of the form . The coefficients Cm can be determined by using the evident condition L.£ (coiQ)~Q • This gives C rfiil - ~C.t^  whence £ m £ 0 . To de-termine C0 consider the special case , Then, by (2.A) But, by (1.A) Y 7 n \ \ ?"(\) Since f jptO^u.we have "^Jo,*) ^ jj2^1)/^ ] *; But X / ^ f K ^ O A u P ^ (CDS9) . H e n c e Co^*/(2ft-0 and, f i n a l l y ^ where we have used' " ( - I ^ Y j REFERENCES 1. N. Bloembergen and T. J . Rowland: Acta Metalurgica 1_, 731 (1953). 2. N. Bloembergen and T. J . Rowland: Phys. Rev. I679 (1955). 3 . Yu. S. Karimov and I. F. Shchegolev: J.E.T.P. l/j_, 772 (1962). 4 . J . H. Van Vleck: Phys. Rev. JJ., 1168 (1948). 5. A. Abragam: Principles of Nuclear Magnetism, Oxford U. Press, I 9 6 I . 6. C. P. S l i c h t e r : Principles of Magnetic Resonance, Harper & Row.Publishing Co., N. Y. (1963). 7. D. Pines: S o l i d State Physics Vol. 1, p. 38 (1955). 8. M. A. Ruderman and C. K i t t e l : Phys. Rev. 9_6, 99 (1954). 9. E. R. Andrew: Phys. Rev. 9_1 , 425 (1953). 10. S. N. Sharma: Ph.D. Thesis, University of B. C. (1967). 11. N. A. Shuster: RSI 22, 254 ( l 9 5 l ) . 12. S. N. Sharma and D. Llewelyn Williams: Magnetic Resonance and Relaxation, Editor, R. Blinc (North Holland) I967, page 480. 51 

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