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Some NMR studies of NbSe₂ Abdolall, Khaled 1974

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SOME NMR STUDIES OF NbSe 2 by KHALED ABDOLALL B . S c , U n i v e r s i t y of Waterloo, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF 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 re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1974 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an 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 study. I f u r t h e r agree that permission f o r extensive 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 r e p r e s e n 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 permission. 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 ABSTRACT A s e n s i t i v e n u c l e a r magnetic resonance spectrometer has been constructed using F i e l d E f f e c t T r a n s i s t o r s i n a Robinson c o n f i g u r a t i o n . The spectrometer has been used to study the anomalous n u c l e a r magnetic resonance spectra o f s i n g l e c r y s t a l s of NbSe^. An a n a l y s i s of the f i e l d dependence of the l i n e width i n the low temperature phase has demonstrated that t h i s r e s u l t s from a d i s t r i b u t i o n of Knight S h i f t s . Such a d i s t r i b u t i o n i s not c o n s i s t e n t with a s t r u c t u r a l transformation i n v o l v i n g only two nonequivalent s i t e s as proposed by Ehrenfreund et a l . In a d d i t i o n accurate measurements of the Knight S h i f t and e l e c t r i c f i e l d gradient tensor have been made i n the high temperature phase at 77K and 300K. The Knight S h i f t has a very large a n i s o t r o p i c component but an almost zero i s o t r o p i c component which i s i n d i c a t i v e o f n e g l i g i b l e s - e l e c t r o n character at the: Fermi surface. TABLE OF CONTENTS Abstract Table of Contents L i s t of Tables L i s t of I l l u s t r a t i o n s ' . Acknowledgements CHAPTER I: I n t r o d u c t i o n CHAPTER I I : T h e o r e t i c a l Considerations The Quadrupole Hamiltonian Energy Levels i n s i n g l e cryst. a) C r y s t a l s with c y l i n d r i c a l symmetry: Magnetic f i e l d p a r a l l e l t o c r y s t a l a x i s . b) C r y s t a l s w i t h l e s s than c y l i n d r i c a l symmetry: a r b i t r a r y o r i e n t a t i o n of the magnetic f i e l d Theory of the Knight S h i f t I s o t r o p i c Knight S h i f t A n i s o t r o p i c Knight S h i f t CHAPTER I I I : Experimental A b r i e f i n t r o d u c t i o n to steady s t a t e d e t e c t i o n of NMR Apparatus 1 . The spectrometer I. a) The r . f . a m p l i f i e r I. b) Noise f i g u r e of the r . f . a m p l i f i e r I I . The l i m i t e r I I I . The detector IV. The audio a m p l i f i e r V. P r i n c i p l e of operation V I . C o n s t r u c t i o n design of the spectrometer V I I . Performance 2. Sample and c r y s t a l holder 3 i O r i e n t a t i o n of the c r y s t a l 4) Temperature c o n t r o l CHAPTER IV: Re s u l t s Temperature dependence of the J9/2, l/2^> —i>J9/2, l i n e .2 Determination of and the Knight S h i f t s D i s c u s s i o n Conclusion References CHAPTER V: Appendix V LIST OF TABLES TABLE (I) : Noise f i g u r e o f the r . f . a m p l i f i e r f o r 3 d i f f e r e n t source impedances. TABLE ( I I ) : The quadrupole i n t e r a c t i o n and the Knight S h i f t s at room temperature and 77 K. v i LIST OF ILLUSTRATIONS Page F i g . (1): 93 Energy l e v e l diagrams f o r Nb nucleus of s p i n 9/2 i n NbSe 2 s i n g l e c r y s t a l w i t h the magnetic f i e l d p a r a l l e l to the high symmetry a x i s of the c r y s t a l (the c a x i s ) . 5 F i g . (.2): O r i e n t a t i o n of the magnetic f i e l d H w i t h respect to 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 gradient tensor. 6 F i g . C3): Schematic arrangement f o r steady s t a t e NMR absorption 18 experiments F i g . (4): The r . f . a m p l i f i e r 19 F i g . (5) : A cascode a m p l i f i e r 20 F i g . C6): The l i m i t e r 21 F i g . C7): The d e t e c t o r 22 F i g . (8): Schematic diagram,of the d e t e c t i o n process using the n o n l i n e a r r e l a t i o n between the gate b i a s and d r a i n current of a FET. 23 F i g . C 9 ) : The Audio a m p l i f i e r F i g . (.10): Frequency response of the audio a m p l i f i e r 25 F i g . d i ) : The attenuator 26 F i g . (12): A block diagram of the spectrometer 27 F i g . (13): Resonance curve of a tuned LC C i r c u i t . I t shows how a noise v o l t a g e AV i s introduced by a small change +Af where f i s the o s c i l l a t i o n frequency d i f f e r e n t from that of the resonance frequency f of the tank. F i g . (.14): C o n s t r u c t i o n layout of the spectrometer 29 F i g . (15): Low temperature c r y o s t a t and c r y s t a l holder 31 v i i Page F i g . (16): Angular dependance of the 1 9 / 2 , l/2> -> j 9/2, - l/2> resonance 32 F i g . (17): Nb i n NbSe 2 ]9/2, l/2> |9/2, - l/2> l i n e shape at 5.9 K and 12.22 MHz with S p a r a l l e l to c. 35 Q 3 F i g . (18): Nb i n NbSe 2 |9/2, l/2> -> |9/2, - l/2> l i n e shape at 21.3 K -> -> and 12.22 MHz with H p a r a l l e l t o c. 36 Q 3 F i g . (19): Nb i n NbSe 2 19/2, l/2> -> 19/2, -l/2> l i n e shape at 29 K -> and 12.22 MHz with H p a r a l l e l to c. 37 93 I i F i g . (20): Nb i n NbSe 2 j 9/2, l/2> J 9/2, - l/2> l i n e shape at 6.6 K and 18.00 MHz w i t h H p a r a l l e l t o c. 38 F i g . (21): N b 9 3 i n NbSe 2 |9/2, l/2> •+ |9/2, - l/2> l i n e shape at 12.6 K -> and 18.00 MHz w i t h H p a r a l l e l to c. 39 93 i i F i g . C22): Nb i n NbSe 2 19/2, l/2> 19/2, - l/2> l i n e shape at 20.6 K and 18.00 MHz w i t h H p a r a l l e l t o c. 40 F i g . (23): Temperature and f i e l d dependence of the l i n e width f o r the |9/2, l/2> -> |9/2, - l/2> l i n e w i t h H p a r a l l e l t o c. 41 F i g . (24): A sample c a l c u l a t i o n of K . The c a l c u l a t e d resonance " x : frequency (E_-jy 2 - E i / 2 ^ / ^ * s p l o t t e d as a f u n c t i o n of a p p l i e d f i e l d . The observed resonance frequency v e x p I s due to an e f f e c t i v e f i e l d H at the s i t e of the Nb e f f nucleus which i s lower than the a p p l i e d f i e l d H . 42 * r exp F i g . (25): The v a r i a t i o n of the |9/2, l/2> + |9/2, - l/2> frequency as a f u n c t i o n of n. 45 F i g . (26): F i e l d dependence of the l i n e width of the J9/2. l/2> |9/2, - l/2> l i n e w e l l below the t r a n s i t i o n . 46 ACKNOWLEDGEMENTS I would l i k e to express my s i n c e r e g r a t i t u d e to Dr.DL'Ll. W i l l i a m s f o r h i s v a l u a b l e i n s t r u c t i o n throughout the work. I would l i k e to express my g r a t i t u d e to Dr. M.I. V a l i c f o r h i s many va l u a b l e suggestions and h i s help i n the experiments. I am very g r a t e f u l to Dr. Walter N. Hardy f o r h i s great help i n e x p l a i n i n g the c i r c u i t diagrams of the spectrometer, i t s theory of o p e r a t i o n , and h i s v a l u a b l e suggestions regarding i t s c o n s t r u c t i o n . I am a l s o indebted to him f o r reviewing the s e c t i o n on the spectrometer. -1-CHAPTER I INTRODUCTION Niobium d i s e l e n i d e i s a layered t r a n s i t i o n metal dichalcogenide which e x i s t s i n three s t a c k i n g polytypes. Wilson and Yoffe (1969) have published a thorough d e s c r i p t i o n of these layered s t r u c t u r e s and t h e i r p r o p e r t i e s . At room temperature the 2H polytype has a layered hexagonal s t r u c t u r e t h a t has the symmetry o f the space group V6^/vmc. This m a t e r i a l showed an anomalous s i g n r e v e r s a l of the H a l l c o e f f i c i e n t at 26K and a maximum i n the magnetic s u s c e p t i b i l i t y near 40K (Lee et a l (1969)), which were taken to be i n d i c a t i v e of a phase t r a n s f o r m a t i o n . NMR stu d i e s on powders of 2H-NbSe2 were made by Ehrenfreund et a l (1971) . They conclude t h a t the anomalous behaviour of NbSe2 at low temperatures i s due to a s t r u c t u r a l t r a n s f o r m a t i o n and an associated conduction e l e c t r o n r e d i s -t r i b u t i o n . In t h e i r model, they suggest that at and below 20K there are two in e q u i v a l e n t niobium s i t e s w i t h d i f f e r e n t e l e c t r i c f i e l d g r a d i e n t s . T h e i r a n a l y s i s i s based on the informat i o n e x t r a c t e d from the powder p a t t e r n l i n e shape, which i s assumed to be the convol u t i o n of a Gaussian shape f u n c t i o n w i t h the frequency d i s t r i b u t i o n expected from a randomly o r i e n t e d m i c r o c r y s t a l i t e sample i n the presence of quadrupole e f f e c t s and an i n i s o t r o p i c Knight s h i f t . The purpose of t h i s work i s to f u r t h e r study the p r o p e r t i e s of NbSe2 by NMR methods usi n g s i n g l e c r y s t a l s . The advantages of using s i n g l e c r y s t a l s (as was demonstrated i n t h i s l a b o r a t o r y i n previous work on Gallium and Aluminium) are: (a) a d i r e c t study of the anisotropy i n the Knight s h i f t due to c r y s t a l symmetry, (b) e l i m i n a t i o n of l i n e broadening caused by the a n i s o t r o p i c Knight s h i f t , and (c) consequently more accurate measurements of the Knight s h i f t s and the e l e c t r i c f i e l d gradient parameters. - 2 -CHAPTER I I THEORETICAL CONSIDERATIONS The Quadrupolar Hamiltonian: The Hamiltonian operator f o r the i n t e r a c t i o n of the nuclear quadrupole moment w i t h the e l e c t r i c f i e l d gradient e x i s t i n g at the nucleus i s given by [4] where I + = I x , +.iY y l; I x , , I Y I and I z , are'the components o f the nuclear s p i n I; eQC i s the s c a l a r quadrupole moment and n i s an asymmetry parameter defined as: qX'X' " qY*Y' n = where q X ' X " qY'Y' a n d qZ*Z' = q are defined by the e l e c t r i c f i e l d gradient tensor; 2 ( 1 - n ) qX'X< 0 \ 0 0 1 J O^qy'Y' 0 0 0 "'^Z'Z1 \ and X', Y', Z* are the p r i n c i p a l axes of q. Energy l e v e l s i n s i n g l e c r y s t a l s : In the presence of an ap p l i e d magnetic f i e l d S the Hamiltonian f o r a Spin I w i t h a quadrupole moment i s given by (Abragam P.232). 3 = " + tf% i - i ) <3Iz< " 1 + J (I+ + :-)] (a) C r y s t a l s w i t h c y l i n d r i c a l symmetry; magnetic f i e l d p a r a l l e l to c r y s t a l a x i s , - 3 -For c r y s t a l s w i t h a x i a l symmetry, which i s the case of NbSe^ at temperatures above 24 K, n vanishes because qX'X' = qY'Y' I f H i s chosen p a r a l l e l to the symmetry a x i s of the c r y s t a l (the Z' a x i s ) , then the Hamiltonian takes the form £ = - T h H I z , + 4U2I-1) l31h " 1 so that i n the | l , M > r e p r e s e n t a t i o n where I , i s d i a g o n a l , the energies of t h e d i f f e r e n t l e v e l s are given by: 2 ' E m = - yhHm + 4 ^ , ^ [3m?- I OD] o r , u sing the n o t a t i o n s Y JJ _ 3e 2qQ V L 2TT ' VQ h 2 I ( 2 I - l ) The corresponding frequencies of the allowed t r a n s i t i o n s are E - E m-1 m v , = m-l,m n i = V L + ^2 C 1 _ 2 m ) For I = ^ of the Nb 9 3 nucleus the energy l e v e l diagram i s given i n F i g . ( l ) . Thus i n a nuclear magnetic resonance experiment on Nb 9 3 i n NbSe2 where the magnetic f i e l d i s along the symmetry a x i s o f the c r y s t a l one would expect to see nine l i n e s corresponding t o the frequencies: v r = V L - r vQ' r = °' 1 ' 2 ' 3 ' 4 w i t h a c e n t r a l l i n e corresponding to the | j , y > I §"» ~ ^  t r a n s i t i o n and f o u r s a t e l l i t e s on each s i d e e q u a l l y spaced by V Q the quadrupole frequency o f the Nb 9 3 nucleus. (b) C r y s t a l s w i t h l e s s than c y l i n d r i c a l symmetry; a r b i t r a r y o r i e n t a t i o n s o f the magnetic f i e l d . - A -For the more general problem of an a r b i t r a r y o r i e n t a t i o n of the magnetic f i e l d H the Hamiltonian takes the form , T T r s i n '6 'cos <f> , T T s i s i n 6 s i r i <J> , T T . Q T - i H = - yhH { = - (I +1 ) = ( I - I )+ cos 6 I 7 } + ^ fsQ r 3 I2 _ I f I + n + t r i2 + I 2-) i 41(21-1) L Z» + 2 1 + - J-l where the angles 6 , <j> d e f i n e the o r i e n t a t i o n of H w i t h respect t o the p r i n c i p a l axes X', Y', Z' as shown i n F i g . (2). Making the s u b s t i t u t i o n 2 V Y , t 3e qQ v 6 L v) = - — H \> = Y = ,L 2-n VQ h2I(2I-l)» v Q gives H = Y { S i n ? C O S • (I + I ) - 1 S I N I S i n * (I - I ) + cos 6 I ,} + 31*, - K I + l ) + j ( I * + I J ] The appendix i n c l u d e s an e x p l i c i t form o f the 10 x 10 matrix of H f o r a 9 s p i n 1 = 2 " a n d a computer programme f o r d i a g o n a l i z i n g such a matrix and f i n d i n g the exact eigenvalues and the corresponding normalized eigenvectors. - 5 -9/2 A 9/2 J - 7 / ^ C D i — C L ) C L U zero |9/2 3 - '/2> 9/2, >/£> I 9 / 2 . , 9/2>5/2> F i g . CI): Energy l e v e l diagrams f o r N b 9 3 nucleus o f s p i n 9/2 i n NbSe ? s i n g l e c r y s t a l w i t h the magnetic f i e l d p a r a l l e l to the high symmetry a x i s of the c r y s t a l (the c a x i s ) . F i g , ( 2 ) : O r i e n t a t i o n of the magnetic f i e l d H w i t h respect to 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 gradient tensor. THEORY OF THE KNIGHT SHIFT For a given f i x e d a p p l i e d f i e l d the nuclear magnetic resonance frequency f o r a nucleus i n a m e t a l l i c substance i s d i f f e r e n t or ' s h i f t e d ' from t h a t of the same nucleus i n a non m e t a l l i c reference compound f o r the same a p p l i e d f i e l d . This s h i f t , known as the Knight S h i f t , i s due to the i n t e r a c t i o n of the nuclear spins w i t h the conduction e l e c t r o n s i n the metal. The Hamiltonian operator f o r the i n t e r a c t i o n of the conduction e l e c t r o n s w i t h a nuclear s p i n I i s given by [1 ] ; I J o D t,t V / ^  ^ 3 r ( S . r \ 8TT - ± , , - » - . . , H. = 2$YhI'> { — r - —T + 3 + r r S Sfr)} ( l j e where r and S are the p o s i t i o n and s p i n of an e l e c t r o n e and I i s the o r b i t a l mo-mentum. The summation i s taken over a l l the e l e c t r o n s . I f the e l e c t r o n i c o r b i t a l momentum i s completely quenched then i n the one e l e c t r o n d e s c r i p t i o n o f the conduction e l e c t r o n s the expectation value of t h i s operator i s : + 3(r\ .% ) r . S. 0 _ -yh I.I<k|26 V> I* k - + | l ? k 6 (?)}| k > k k k = - yh ?•! \'% (2) -y 3?, r, s\ „ + where T = <k|2g {—^ - ~V i 1 « |k> k r k r k 3 k i s a tensor w i t h components depending on the s t a t e s |k>. The summation i s over u n f i l l e d s t a t e s |k> because e l e c t r o n s i n f i l l e d s t a t e s do not c o n t r i b u t e t o t h i s averaged i n t e r a c t i o n because t h e i r t o t a l s p i n moments are equal to zero. I f has approximately the same value f o r a l l u n f i l l e d s t a t e s near the top of the Fermi d i s t r i b u t i o n then yh I-l t • S, = yh I'T-S (3) where S = I s. k -> - » - - » • I f the tensors x^ are not i d e n t i c a l then T i s an average of x^ over a l l the s t a t e s |k> near the Fermi surface. In the presence of an ap p l i e d f i e l d H q we have: 2g A A p H o 3 = - L X X - - H where Xp i s the magnetic s u s c e p t i b i l i t y per u n i t volume and i t becomes a tensor i f the o r b i t a l momentum i s not completely quenched and V i s the volume. S u b s t i t u t i n g f o r S i n (3) gives - hYH X P ( 4 ) That i s the nuclear s p i n I sees an a d d i t i o n a l f i e l d Vx^ x*H o/23superimposed on H Q causing a s h i f t i n the nuclear resonance frequency. In the one e l e c t r o n d e s c r i p t i o n the s p a t i a l c o r r e l a t i o n between the e l e c t r o n s i s not taken i n t o account. In f a c t i t can be shown usin g the d e n s i t y matrix formalism [1] that r e l a t i o n (4) i s independent of the one e l e c t r o n d e s c r i p t i o n . I s o t r o p i c Knight S h i f t : I f the symmetry of the e l e c t r o n i c environment of the nuclear s p i n i s cubic or higher then only the s c a l a r part of the tensor i s d i f f e r e n t from zero, t h a t i s , only S e l e c t r o n s w i l l c o n t r i b u t e to the s h i f t . In t h i s case (4) reduces to - Vyh CI-H) X p ' y- < I KO) | 2 >F where ip(0) i s the value of the wave f u n c t i o n o f an e l e c t r o n at the nucleus and the average < >p i s made over a l l s t a t e s at the top of the Fermi d i s t r i b u t i o n . The t o t a l Hamiltonian becomes .yh f . 3 o CI + V X p- y- < U CO) | 2 >p) - 9 This corresponds to a p o s i t i v e frequency s h i f t K given by AH . 8ir i s o H A n i s o t r o p i c Knight S h i f t I f the symmetry of the riuclear environment i s l e s s than cubic the tensor x i s not a s c a l a r and the Knight s h i f t w i l l depend on the o r i e n t a t i o n of the ap p l i e d magnetic f i e l d w i t h respect to the c r y s t a l a x i s . Therefore we must add to the i s o t r o p i c s h i f t an a n i s o t r o p i c s h i f t given by: yh i ' K ' t i yh ( I Y 1 , I Y I , I 7 , ) K XX 0 K' 0 o \ YY 0 0 K' zz H, H„ where I , K' and H are now defined w i t h respect to the p r i n c i p a l axes X', Y', Z ' of the tensor t. Since the a n i s o t r o p i c i n t e r n a l f i e l d i s u s u a l l y much smaller than the a p p l i e d f i e l d H Q [9], the only e f f e c t i v e component i n s h i f t i n g the resonance frequency i s that along H q which can be shown to be [1] H (K' c o s 2 0 + K' s i n 2 6 c o s 2 <j> + K' s i n 2 9 s i n 2 a)) O LtLt A A I I where 6, <f> s p e c i f y the o r i e n t a t i o n of H q w i t h respect to X' ,Y! ,11. For a x i a l symmetry we have K'' (K 1 + K' ) from the t r a c e l e s s property of K' ZZ XX YY the relat-ive- frequency,,, s h i f t r becomes l„ ; _^ = K. + I K ! C3 c o s 2 6 - 1) H i s o " 2 || = K. + k K . (3 c o s 2 0 - 1 ) i s o 2 aniso - 1 0 -where K . = K1 i s a measure of the anisotropy i n the charge d i s t r i b u t i o n aniso || r } 6 and i s r e l a t e d to the conduction e l e c t r o n wave f u n c t i o n by [ 6 ] I , 1 2 ( 3 c o s 2 6 - 1 ) , 3 which i s a p o s i t i v e q u a n t i t y i f the charge d e n s i t y i s greatest i n the d i r e c t i o n of the Z a x i s . For the extreme cases we have K - = K. + K. . f o r 9 = 0 II i s o aniso K = K. - \ K . f o r 0 = \ j_ i s o 2 aniso 2 This gives K. = f CK„ + 2 K, ) i s o 3 V || \ _ K - CK„ - K J aniso 3 ^ || JJ - 11 -CHAPTER I I I EXPERIMENTAL A b r i e f i n t r o d u c t i o n to steady s t a t e d e t e c t i o n of NMR: Steady s t a t e d e t e c t i o n of nuclear magnetic resonance absorption c o n s i s t s of observing the response of the nuclear s p i n system to a conti n u o u s l y a p p l i e d r a d i o frequency f i e l d . The specimen i s placed i n s i d e an r . f . c o i l which forms part of a tuned c i r c u i t as shown i n F i g . (3). At resonance the nuclear s p i n system absorbs energy from the r . f . f i e l d H^ i n s i d e the c o i l . This resonance e f f e c t i s detected by i t s r e a c t i o n on the c i r c u i t supplying the r . f . f i e l d . The nuclear resonance absorption by the spins w i t h i n the c o i l changes the q u a l i t y f a c t o r , Q, of the c o i l by [5]: s CqD = 4 IT -n x M where n i s the f i l l i n g f a c t o r and x" i s the imaginary part of the nuclear s u s c e p t i b i l i t y . I f i n the absence of s i g n a l the r . f . v o l t a g e across the c o i l i s then near resonance a change by 6 (—) w i l l r e s u l t i n 6 V1 = V1 Q <5 (i) Thus d e t e c t i o n of nuclear magnetic resonance absorption i s reduced to d e t e c t i n g the change i n the r . f . l e v e l across the c o i l c o n t a i n i n g the sample. A v a r i e t y o f c i r c u i t s f o r d e t e c t i n g nuclear magnetic resonance absorption have been used. The two most s u c c e s s f u l of the s i n g l e c o i l types are the one by Robinson [5], and the one by Pound, Knight and Watkins [ 8 ] . APPARATUS 1 : The Spectrometer: For t h i s work a t r a n s i s t o r i z e d v e r s i o n of the Robinson c i r c u i t u s i n g F i e l d E f f e c t T r a n s i s t o r (FETS) was b u i l t . The c i r c u i t was designed by Profess o r V. Frank - 12 -at the Tec h n i c a l U n i v e r s i t y of Denmark. The f o l l o w i n g i s a d e s c r i p t i o n of such a c i r c u i t , i t s p r i n c i p l e of op e r a t i o n , c o n s t r u c t i o n design and an evalu-a t i o n of i t s performance. I. a) The r . f . A m p l i f i e r : The r . f . a m p l i f i e r i s wide band and c o n s i s t s o f an input stage and fou r a m p l i f i e r stages that are connected i n cascade w i t h the input stage. F i g . (4) i s the c i r c u i t diagram. The FETS , Q , and form two cascodes i n p a r a l l e l . A cascode a m p l i f i e r c o n f i g u r a t i o n i s obtained u s i n g a common source and a common gate FET connected as shown i n F i g . (5). This c o n f i g u r a t i o n o f f e r s good s t a b i l i t y , low noise a m p l i f i c a t i o n , and lar g e power g a i n . The s t a b i l i t y i s due to the reduced M i l l e r e f f e c t i n the common source stage and the small d r a i n source capacitance of the common gate stage. Both f a c t o r s c o n t r i b u t e t o a very small reverse t r a n s f e r admittance f o r the combination. I t s low noise a m p l i f i c a t i o n i s due to the f a c t t h a t the v o l t a g e gain of the f i r s t stage i s u n i t y and t h e r e f o r e does not c o n t r i b u t e to the no i s e . While the second stage allows a m p l i f i c a t i o n without a d d i t i o n i n noise [3]. Thus the noise f i g u r e of the combination corresponds to the noise f i g u r e of the f i r s t stage. The l a r g e power gain of the cascode i s a r e s u l t of the decreased output admittance. The p a r a l l e l combination of the two cascodes reduces the noise v o l t a g e of the input stage by a f a c t o r o f two, g i v i n g f u r t h e r improvement i n low noise a m p l i f i c a t i o n when the source impedance i s l e s s than the optimum v a l u e . Q 5 and are i n the emitter f o l l o w e r c o n f i g u r a t i o n . The p a r a l l e l arrangement provides low output impedance. - 13 - 12 The remaining four a m p l i f i e r stages each c o n s i s t of a common source c o n f i g u r a t i o n followed by an e m i t t e r f o l l o w e r c o n f i g u r a t i o n . The emitter f o l l o w e r serves as a power a m p l i f i e r as w e l l as an impedance matching device. The t o t a l v o l t a g e gain of the r . f . a m p l i f i e r can be v a r i e d i n steps from 4.6 to 1430. I. b) Noise f i g u r e of the r . f . a m p l i f i e r : The noise f i g u r e of the r . f . a m p l i f i e r was measured f o r d i f f e r e n t values of source impedance at a frequency of 10 MHz. The r e s u l t s are shown i n Table (I) below. SOURCE IMPEDANCE NOISE FIGURE 244 fi 3.44 .984 K 1.44 7.3 K 1,46 Table (.1) : Noise f i g u r e of the r . f . a m p l i f i e r f o r 3 d i f f e r e n t source impedances. I I : The L i m i t e r : The c i r c u i t i s shown i n F i g . (6). The source f o l l o w e r Qg makes use of the high dynamic r e s i s t a n c e i n the source c i r c u i t provided by the constant current supply Q-^Q- This improves the l i n e a r i t y and performance of the source f o l l o w e r . The input s i g n a l f o r the l i m i t e r i s the output of the r . f . a m p l i f i e r . The output of the source f o l l o w e r i s a p p l i e d to a p a i r of diodes and T)^ c a l l e d ' c l i p p e r s ' . The output waveform i s l i m i t e d to the v o l t a g e s set by the reverse biases of D^  and D 2. Diodes D^, D^, D^ and D^ provide the necessary b i a s v o l t a g e s . The l i m i t i n g a c t i o n can be adjusted by a l t e r i n g the reverse - 14 -bi a s v o l t a g e s of and by means of the swit c h i n g arrangement shown. The arrangement of and forms a source coupled 0°/180° phase i n v e r t e r . Since each a m p l i f i e r stage changes the phase by 180°, the phase i n v e r t e r i s necessary to ensure t h a t the t o t a l phase s h i f t i s zero when the number of a m p l i f i e r stages i s odd. I l l : The Detector: Detection i s the process of reco v e r i n g from a modulated r . f . c a r r i e r a s i g n a l that v a r i e s i n accordance w i t h the modulation present on the c a r r i e r . Several methods of d e t e c t i o n are a v a i l a b l e . The method used i n t h i s spectrometer makes use of the n o n - l i n e a r r e l a t i o n (approximately quadratic) between the gate b i a s and d r a i n current of a FET. F i g . (7) i s a c i r c u i t diagram of the d e t e c t o r . i s biased n e a r l y to pinch o f f and t h e r e f o r e i s operating i n the no n - l i n e a r r e g i o n . When a modulated r . f . c a r r i e r i s a p p l i e d to the gate the averaged output s i g n a l w i l l vary approximately as the modulation envelope.. This i s i l l u s t r a t e d i n F i g . (8). The D.C. component o f the r e c t i f i e d s i g n a l i s a m p l i f i e d by the D.C. a m p l i f i e r Q ^. The r . f . l e v e l i s then monitored u s i n g a D.C. ammeter. The v a r i a b l e r e s i s t a n c e R q serves as a D.C. o f f - s e t f o r the D.C. a m p l i f i e r . IV: The Audio A m p l i f i e r : The output o f the detector i s a m p l i f i e d by the low noise audio a m p l i f i e r shown i n F i g . (9). The low frequency components o f the detector output are blocked by the high pass f i l t e r at the input of the a m p l i f i e r . Q^j. and are i n common source c o n f i g u r a t i o n and i s i n a source f o l l o w e r c o n f i g u r a t i o n . The three stages are connected i n cascade and a gain of about 3200 i s achieved. The frequency response i s shown i n F i g . (10). - 15 -The h i g h frequency r o l l o f f of the a m p l i f i e r i s at 1.1 kHz. A higher frequency r o l l o f f at 12 kHz can be obtained by means of the 5 nF c a p a c i t o r shown i n the diagram. V: P r i n c i p l e of Operation: A block diagram of the spectrometer i s shown i n F i g . (12). O s c i l l a t i o n s i n the tank c i r c u i t are sustained by the p o s i t i v e feed-back provided by the l i m i t e r through the attenuator. I f the phase s h i f t between the input of the r . f . a m p l i f i e r and the output of the attenuator (as the r . f . s i g n a l goes through the r . f . a m p l i f i e r , l i m i t e r , and attenuator) i s p r e c i s e l y zero, o s c i l l a t i o n s w i l l occur at the centre of the resonance curve of the tuned c i r c u i t . Under such c o n d i t i o n s the o s c i l l a t i o n amplitude i s l e a s t s e n s i t i v e to small changes i n frequency. However, i f phase s h i f t s are introduced by some elements of the c i r c u i t such as s t r a y capacitances, e t c . o s c i l l a t i o n s * w i l l s t i l l occur, but i n order to maintain a zero phase s h i f t , the frequency of o s c i l l a t i o n w i l l be s h i f t e d from the centre of the resonance curve. In t h i s case a small change i n frequency can cause a l a r g e change i n o s c i l l a t i o n amplitude and the spectrometer w i l l not be operating under optimum c o n d i t i o n s ; This i s i l l u s t r a t e d i n F i g . (13). The r . f . l e v e l across the c o i l i s determined by the amount of feed-back. Lower r . f . l e v e l s are obtained by decreasing the amount of feed-back and a d j u s t i n g the gain of the r . f . a m p l i f i e r such that i t s output i s s u f f i c i e n t to d r i v e the l i m i t e r . Nuclear resonance a b s o r p t i o n changes the q u a l i t y f a c t o r of the c o i l and thus changes the r . f . l e v e l across i t . In the presence of an audiomodulated magnetic f i e l d the s p i n system w i l l go i n and out.of resonance p e r i o d i c a l l y w i t h a frequency equal to that of the modulation frequency. Consequently the r . f . l e v e l across the c o i l w i l l be amplitude modulated. This i s a m p l i f i e d by the r . f . a m p l i f i e r . a n d then fed i n t o the d e t e c t o r . The output of the d e t e c t o r i s - 1,6 -approximately p r o p o r t i o n a l to the modulation envelope and i s f u r t h e r a m p l i f i e d by the audio stage. The output i s then e i t h e r d i s p l a y e d on a scope o r , i n the case of weak s i g n a l s , i s fed i n t o a phase s e n s i t i v e d e t e c t o r to improve the s i g n a l t o noise r a t i o . By the nature of phase s e n s i t i v e d e t e c t i o n one obtains the d e r i v a t i v e of the absorption s i g n a l r a t h e r than the s i g n a l i t s e l f . VI: C o n s t r u c t i o n Design of the Spectrometer: F i g . (14) i s a layout of the spectrometer. Each stage i s enclosed i n a separate compartment to s h i e l d i t from other stages and outside i n t e r f e r e n c e . The r . f . a m p l i f i e r stages, l i m i t e r and attenuator are arranged i n the way shown to e l i m i n a t e the un d e s i r a b l e use of a c o a x i a l l i n e i n the feedback loop. Feed through c a p a c i t o r s were used to provide the supply v o l t a g e as w e l l as r i g i d supports f o r mounting the FET's. Ground lugs were soldered d i r e c t l y to the c h a s s i s . Whenever i t was p r a c t i c a l o n ly one ground p o i n t was used f o r each stage. This has the e f f e c t of reducing s e l f o s c i l l a t i o n s and ground loops. The leads were kept as short as p o s s i b l e i n order to reduce s t r a y p i c k up and microphonics. Double pole double throw switches JMT 223 were used f o r connecting r . f . a m p l i f i e r stages i n cascade. In one p o s i t i o n of the switch the a m p l i f i e r stage i s connected,in the other i t i s bypassed. In the l i m i t e r a double pole f i v e p o s i t i o n r o t a r y switch A l c o s w i t c h MRA-25 i s used. I t provides 5 l i m i t i n g o p t i o n s . A l l the sides o f the spectrometer were made removable except the f r o n t panel where a l l the c o n t r o l knobs are mounted. This provides easy access to a l l the component s. - 17 -V I I : Performance: The c i r c u i t performed w i t h good s e n s i t i v i t y throughout the frequency range 4 - 4 0 MHz. T y p i c a l absorption s p e c t r a are shown i n F i g s . (20), (21), and (22). No attempt was made t o compare the s e n s i t i v i t y of t h i s c i r c u i t w i t h that of the marginal o s c i l l a t o r (Pound Knight box) p r e s e n t l y used i n t h i s l a b . However, according to Brofessor V. Frank the c i r c u i t i s capable of performing w i t h a s e n s i t i v i t y about a f a c t o r of 2 b e t t e r than that of the marginal o s c i l l a t o r . A d d i t i o n a l advantages of t h i s c i r c u i t over the marginal o s c i l l a t o r c i r c u i t are [5],(a) i t can be adjusted t o very low r . f . l e v e l s ; (b) i t can be used w i t h c i r c u i t s of very low L/C r a t i o . I f the shunt impedance of the tuned c i r c u i t i s low, not only i s the noise f i g u r e of a marginal o s c i l l a t o r impaired, but al s o the c i r c u i t may f a i l t o o s c i l l a t e at a l l . In the case of a metal sample the c o i l can be wound d i r e c t l y on the sample when u s i n g a Robinson c i r c u i t . But f o r the marginal o s c i l l a t o r i t i s o f t e n necessary t o wrap a piece of mylar around the sample before winding the c o i l i n order f o r the c i r c u i t to o s c i l l a t e . This reduces the f i l l i n g f a c t o r and thus a f f e c t s the s e n s i t i v i t y . In c o n c l u s i o n t h i s makes the Robinson c i r c u i t more a t t r a c t i v e f o r NMR s t u d i e s of metal s i n g l e c r y s t a l s . K Circuit supp-& I lying the r-f field Detector F i g . (3): Schematic arrangement f o r steady s t a t e NMR absorption experiments. + 30-35" V 4.7 K 100 — JL/w\ 10k IZK 5 6 Ii-I OIL T 4 -301/ /OO ion 2X 8F/84 2.X U(994 u, ST T 1^ /OOMF ALL FBSISi TORS ARE 1/2 WATT (/A/L£SS OT/tEXW/S£ SPEC/F/ED F i g . (4): The r . f . a m p l i f i e r F i g , C.5): A cascode a m p l i f i e r 5nF_ 400 JJF — 1 ^ -22.0SL _AAAA I80IL W W + 3 0 V R£S/STOKS 4Z£ V2 WATT UNLE5S Or//£ZW/SE SPEZI/ttD . F i g . ( 6 ): The l i m i t e r - 2 2 -I UI994 180PF\ .WVVv- | 0 O K 13 UI9S4 V4 OUTPUT TC ft.p. AMPLIFIER 8-2 K - 3 0 1 / /JZ/! RESISTOAS AR£ V2 WATT UA/LESS 0T#£RW/S£ SP£C/F/£1> F i g - C7).: The detector - 23 -F i g . (8). '< Schematic diagrams of the d e t e c t i o n process using the n o n l i n e a r r e l a t i o n between the gate b i a s and d r a i n current of a FET. - 24 -4-301/ ALL RESISTORS f\R£ l/2 WATT UNLESS OTHERWISE SPECIFIES F i g . (9): The Audio A m p l i f i e r 10 100 Frequency (HZ ) 1000 F i g . (10): Frequency response of the audio a m p l i f i e r F"/?OM TANK 22 PF ,1 \ 2-2 PF 11 2. < '3 . 2-2 P F 11 4 . 2.2 PF 11 5 - 3.3 Pf= h y\ \t / II 11 -2.2PF -II - 2.2 PF -11 - 2.2 PF z 11 -2-2PF -F i g . (11): The Attenuator r-f amplifier • Detector A-F amplifier to out F i g . (12): A block diagram of the spectrometer - 2 8 -F i g - (13): Resonance curve of a tuned LC C i r c u i t . I t shows how a noise voltage AV i s introduced by a small change +Af where f i s the o s c i l l a t i o n frequency d i f f e r e n t from that of the resonance frequency f of the tank. Xy ,pU+ s+acje []) 5 r / | 4 ^ k O U T phase I i mi ier Biasing netujo r K5 F i g . (14): Construction layout of the spectrometer - 30 -2. Sample and C r y s t a l Holder: A s i n g l e c r y s t a l of NbSe2 was obtained from Dr. R. F r i n d t of Simon Fraser U n i v e r s i t y , grown i n the form of a t h i n sheet (about .1 mm t h i c k ) w i t h the c a x i s p e r p e n d i c u l a r to the plane of the sheet. To minimize damage to the c r y s t a l , the sample was placed between two t h i n sheets of mylar before the r . f . c o i l of #40 copper wire was wound on i t . The whole sample assembly was glued w i t h v a r n i s h onto a t h i n microscope s l i d e which was then glued to the copper block as shown i n F i g . (15). 3 . . O r i e n t a t i o n o f the. C r y s t a l :•  ••/: il: I n i t i a l o r i e n t a t i o n of the c r y s t a l c - a x i s was done v i s u a l l y . The f i n a l o r i e n t a t i o n was then determined from the angular dependence of the | § - j > + 9 i I y - 2 > resonance. By r o t a t i n g the magnet and f i n d i n g the resonance f i e l d the graph of F i g . (16) was obtained. From t h i s graph the o r i e n t a t i o n of the c a x i s was determined to w i t h i n h a l f a degree. 4 . Temperatur.e.,Corit>r6T:; I . Regulation o f temperature between 4.2 K and 50 K i s achieved by e l e c t r i c a l heating i n the low temperature cryostat,shown i n F i g . (15). Desired temperatures were obtained by a manual c o n t r o l of the heating current and the amount of exchange gas i n s i d e the c r y o s t a t . For temperatures above 20°K i t was necessary to pump out a l l the exchange gas i n order to reduce the heater current and thus minimize Helium b o i l - o f f . As a r e s u l t the time r e q u i r e d f o r the Cu block and sample to reach thermal e q u i l i b i r u m was much longer (about 30 minutes). For measuring the temperature a gold + .03% at Fe Vs chromel thermo-couple was used. The reference j u n c t i o n was kept at 4.2 K by ensuring that i t was i n contact with the brass block at the bottom of the c r y o s t a t which was i n t u r n i n d i r e c t contact w i t h the helium bath. The thermocouple v o l t a g e was measured w i t h a K e i t h l e y 148 nanovolt meter. - 31 -BNC connector feed through electrical connections L vto pump thin wall stainless steel heater sample thermocouple brass copper block thin wall stainles£ steel tubing (heat leak) lead seal F i g . (15): Low temperature c r y o s t a t and c r y s t a l holder - 32 -F i g . (16): Angular dependance of the 19/2, l/2> -> j 9/2, - l/2> resonance. - 33 -CHAPTER IV RESULTS The p r e l i m i n a r y data upon the sample was taken by Dr. M.I. V a l i c . 19 1 19 1 This data showed th a t the Ij, |y, T-—> l i n e observed w i t h the magnetic f i e l d p a r a l l e l t o the c ax i s o f the c r y s t a l was unchanged as a f u n c t i o n of temperature f o r temperatures above 30K. At lower temperatures the l i n e broadened and a t r a n s i t i o n was observed near 24K. At s t i l l lower temperatures (10K and lower) an a d d i t i o n a l broad base l i n e f e a t u r e appeared. In contrast 19 1 the l i n e s a f f e c t e d by the quadrupole i n t e r a c t i o n , f o r example the |y, -j> ->-19 3 12", 2"> l i n e , broadened as the temperature was decreased below 40K and were unobservably broad at 24K. To f u r t h e r study t h i s problem, a new set of data was taken at a higher f i e l d and both sets of data were analysed. For the purpose of t h i s t h e s i s 19 1 we w i l l r e s t r i c t our a n a l y s i s to the temperature dependence of the \-^ , y> -»-• 9 i 12", l i n e and the d e t e r m i n a t i o n o f the quadrupole i n t e r a c t i o n and the Knight s h i f t s at room temperature and 77K. 19 1 19 1 Temperature dependence of the ^ + |<f> l i n e Line shapes were obtained at temperatures between 5.2K and 50K f o r f i x e d frequencies of 101.443 MHZ, 12.217 MHZ, 15.822 MHZ and 18.000 MHZ. At temperatures below 5.OK the sample was a superconductor. At lower f i e l d s the l i n e s broadened as the temperature was lowered below 30K and a t r a n s i t i o n was observed near 24K. The high f i e l d data showed a t r a n s i t i o n near 18K and no broad base l i n e f e a t u r e . F i g s . 17-22 show the l i n e shape above, near, and below the t r a n s i t i o n . The v a r i a t i o n of the l i n e width as a f u n c t i o n of temperature i s shown i n F i g . 23. Determination of e 2q^- and the Knight S h i f t s The quadropole frequency and the Knight S h i f t s K^ and K^ were c a l c u l a t e d from the sp e c t r a obtained at room temperature and 77K. The - 34 -i 9 1 19 ' 1 •9 1 , 9 3 separation between the \^ -, j> -* and \-^-, 2~>"~K 12' "2* •*-^nes S a v e d i r e c t measurement of v^. A sample c a l c u l a t i o n of i s s h o w n i n ..p F i g . 24... The c a l c u l a t e d values of e 2q^- = 24v_., K. and K . are l i s t s n h Q' i s o aniso i n Table ( I I ) below. Room temperature 77 K I S O aniso e 2qQ h 0.41 + .04% -0.12 +_ .01% 0.06 +_ .02% 0.35 +_ .03% 59853.6 +60.0 kHz 0.39 +_ .01% -0.19 + .02% 0.00 +_ .02% 0.39 + .02% 61864.8 + 32.0 kHz Table ( I I ) : The quadrupole i n t e r a c t i o n and the Knight S h i f t s at room temperature and 77 K. 10 gauss F i g . (17): Nb i n NbSe 0 j 9/2, l/2> -> ]9/2, - l/2> l i n e shape at 5.9'K and 12.22 MHz w i t h H p a r a l l e l t o c - 37 -1 ( F i g . 23): Temperature and f i e l d dependence of the l i n e width f o r the |9/2, l/2> + ]9/2, - l/2> l i n e with H p a r a l l e l to c. The l i n e width s p e c i f i e d i s the peak-to-peak of the d e r i v a t i v e spectrum. - 42 -(24): A sample c a l c u l a t i o n of K x . The c a l c u l a t e d resonance frequency ( E _ l / 2 - E 1 ^ 2 ) / h i s p l o t t e d as a f u n c t i o n of a p p l i e d f i e l d . The observed resonance frequency v i s due to an e f f e c t i v e f i e l d H at the s i t e of 6 X p G i t the Nb nucleus which i s lower than the a p p l i e d f i e l d H - 43 -Discussio n I f the t r a n s i t i o n observed at low temperatures i s a r e s u l t of having two i n e q u i v a l e n t Nb s i t e s with d i f f e r e n t e l e c t r i c f i e l d gradients (EFG) as explained by Ehrenfreund and Gossard [ 2 ] , then i t should be p o s s i b l e to f i t the l i n e shape obtained at temperatures below the t r a n s i t i o n w i t h two l i n e s corresponding t o the two d i f f e r e n t s i t e s . This was t r i e d and could not be done. Moreover, the observed t r a n s i t i o n could not be duei to a low temperature s t r u c t u r a l d i s t o r t i o n alone. Fig.(24)shows the dependence of the c a l c u l a t e d 19 1 19 1 frequency of the j> -> \j, -j> l i n e as a f u n c t i o n o f the asymmetry parameter f). I t i s c l e a r that n must be much l a r g e r than i. 05, as quoted by Gossard, i n order t o have any e f f e c t compatible with our experimental r e s u l t s . The c a l c u l a t e d frequency s h i f t f o r n = .05 i s 0.5 KHZ and i s independent of f i e l d i n the range between 13 and 17.5 KG. The e f f e c t of t a k i n g n i n t o account i s not only n e g l i g i b l e but a l s o y i e l d s the wrong f i e l d dependence. However i f the s p l i t t i n g below the t r a n s i t i o n i s p l o t t e d as a f u n c t i o n of a p p l i e d f i e l d , as shown i n F i g . (26), i t can be seen that the zero f i e l d s p l i t t i n g i s j u s t the l i n e width obtained above the t r a n s i t i o n s . This i n d i c a t e s that the l i n e shape i s not a s u p e r p o s i t i o n of j u s t two l i n e s but i s a r e s u l t of a d i s t r i b u t i o n of Knight S h i f t s . In other words the l i n e shape below the t r a n s i t i o n i s a convolution of some d i s t r i b u t i o n f u n c t i o n whose breadth i s l i n e a r i n f i e l d , and the l i n e shape f u n c t i o n above the t r a n s i t i o n . The change i n the t r a n s i t i o n temperature from 24K to 18K at high f i e l d i s an i n d i c a t i o n t h a t the NMR p r o p e r t i e s of the sample are dependent upon i t s mechanical and thermal h i s t o r y . A p r e l i m i n a r y experiment on a specimen with 1% excess Nb showed no t r a n s i t i o n at a l l . F i n a l l y we note that the temperature v a r i a t i o n of the Knight S h i f t - 44 -and e 2q^-between 77K and room temperature i s i n d i c a t i v e o f some change, i n the e l e c t r o n wavefunctions w i t h temperature. I t i s p a r t i c u l a r l y i n t e r -e s t i n g that K. i s very small and i n f a c t e s s e n t i a l l y zero at 77K. This b I S O i n d i c a t e s that the S-electron character of the e l e c t r o n wavefunctions at the Fermi surface i s very s m a l l . In contrast the very large anisotropy of the Knight S h i f t — as f a r as we know, the l a r g e s t yet observed — i s strong evidence f o r a large P^ . component i n the e l e c t r o n wavefunctions (where Z i s p a r a l l e l to c ) . This strong d i r e c t i o n a l e f f e c t must be r e l a t e d to the s t r u c t u r e of the Fermi surface of NbSe 0. (25): The v a r i a t i o n of the j9/2, l/2> + 19/2, - l/2> frequency as a f u n c t i o n of n. I I I I 1 I I I I 1 I I I I I I I I I 1 5 10 15 20 Field ( expressed in MHz of Nb frequency) F i g . (26): F i e l d dependence of the l i n e width of the 19/2, l/2> -> 19/2, - l/2> l i n e w e l l below the t r a n s i t i o n . - 47 -CONCLUSION In t h i s t h e s i s we have s u c c e s s f u l l y constructed an FET Robinson spectrometer which we have used i n an i n i t i a l study of NbSe2. The r e s u l t s obtained demonstrate that below a c e r t a i n t r a n s i t i o n temperature the Nb n u c l e i experience a d i s t r i b u t i o n of Knight s h i f t s which i m p l i e s that some long range feature i s ass o c i a t e d w i t h the low temperature phase which r e s u l t s i n many non-equivalent Nb s i t e s . I t i s c l e a r that to r e s o l v e t h i s f e a t u r e r e q u i r e s a d e t a i l e d study of NbSe^ as a f u n c t i o n of sample p u r i t y , magnetic f i e l d , and temperature. In a d d i t i o n the study of the temperature v a r i a t i o n o f the Knight s h i f t and e l e c t r i c f i e l d gradient parameters i n the hexagonal phase should give i n t e r e s t i n g i nformation on the e l e c t r o n wavefunctions. In a l l of the above, c l e a r l y the use of s i n g l e c r y s t a l specimens i s e s s e n t i a l . - 48 -REFERENCES A. Abragam, The P r i n c i p l e s of Nuclear Magnetism. Oxford U n i v e r s i t y Press (1961) , p. 200-205. E. Ehrenfreund, A.C. Gossard, F.R. Gamble and T.H. Gebale, J . of A p l . Phys. 42, 4 (1971). Walter N. Hardy, unpublished l e c t u r e notes, U n i v e r s i t y o f B r i t i s h Columbia (.1972) . Charles P. Poole, J r . , and Horacio A. Farach, The Theory of Magnetic  Resonance . John Wiley and Sons (1972), p.208 F. N.H. Robinson, J o u r n a l of S c i e n t i f i c Instruments, V o l ; 3_6_ (1959). T.J. Rowland, Progress i n M a t e r i a l s Science, V o l . 9_, No. 1 (1961), p . l . J.A. Wilson and A.D. Y a f f e , Advances i n Physics 18^ , 73 (1969). G. D. Watkins, Thesis, Harvard U n i v e r s i t y . J . Winter, Magnetic Resonance i n M e t a l s ; Oxford U n i v e r s i t y Press (1971)^ U n l i s t e d references: M.I. V a l i c , T h e s is, U n i v e r s i t y of B r i t i s h Columbia. A. V a n d e r z e i l , Noise Sources, C h a r a c t e r i z a t i o n and Measurement.. P r e n t i c e H a l l (1970). A. V a n d e r z e i l , Noise i n S o l i d State Devices and Lasers, I.E.E.E. P r o c , 58, 8 (August, 1970). Richard S.C. Golbold, Theory and A p p l i c a t i o n of Fets. John Wiley and Sons (1970). - 4 9 -APPENDIX A 9 E x p l i c i t form f o r the matrix of the Hamiltonians f o r a s p i n a r b i t r a r y o r i e n t a t i o n of the magnetic f i e l d . For a r b i t a r y 6, <}> the t o t a l Hamiltonian (Zeeman plus quadrupolar) i s given, i n the no t a t i o n s o f Chapter I I , by: •H = [- Y { s i n e c o s f _ i s i i . e s i i u r ( I + - I J + c o s 6 2 - 2 Z' + 3I|, - 1(1 + 1) + \ ( I 2 + I 2 ) ] For b r e v i t y l e t v sinGcos* 2 = a v sin6sin<J> _ 2 = 3 'Y cose = 5. Then H h v Q 6 Or i n dimensionless u n i t s , [- a ( I + + I_) + i B ( I + - 1 ^ + J (I 2 . + I 2 ) ] j ^ - f ' H ) = [- a ( I + + I_) + i B ( I + - I_) + j ( I 2 + I 2 ) ] The matrix r e p r e s e n t a t i v e f o r t H ) i s given by - 5 1 -APPENDIX B C A COMPUTER PROGRAMME FOR DIAGONALIZING THE TOTAL C HAMILTONIAN(ZEEM AN+QU ADROPOL AR) FOR A SPIN 9/2 FOR AN C ARBITRARY ORIENTATION OF THE MAGNETIC FIELD. C IT CALCULATES THE EXACT EIGENVALUES AND THE CORRESPONDING C NORMALIZED EIGENVECTORS. C THE INPUT PARAMETRS ARE: C BZERO=APPLIED MAGNATIC FIELD C ZMDQ=THE QUDRU POLE FREQUENCY C THETA,PHI SPECIFY THE ORIENTATION OF THE MAGNETIC FIELD C WITH RESPECT TO THE PRINCPAL AXES OF THE ELECTRIC FIELD C GRADIENT TENSOR. C ETA= ASYMMETRY PARAMETER. IMPLICIT SEAL*8(A-H,0-Z) REAL*8DSQRT DIMENSION AH(10,10) ,AI ( 1 0 , 10) , ER (10) , EI(10) , VR (10,10) , VI (10,10) 10 READ(5,2,END=999)BZERO,ZMUQ,THETA,PHI,IT A PRINT 503, BZERO,ZMUQ,THETA, PHI, ETA 503 FORMAT('1',5X,T15,'BZERO',T3 0,'ZMUQ*,T50,*THETA*, 1T69, ' P H I ' , ^ , »ETA»//6X, 2F15. 3, 3F20.6//) XMUL=1,0407D0*BZERO Y=6. DO *XMUL/ZMUQ WRITE (6,19) Y 19 FORMAT (/, ' Y=',F15.5) ALFA=Y*DSIN (THETA) *DCOS (PHI) /2. DO BETA=Y*DSIN(THETA)*DSIN (PHI)/2.DO DELTA=Y* DCOS (THETA) ^ 2 FORMAT(5F15.6) DO 3 J=1,10 DO 3 1=1,10 AR ( I , J) =0. DO 3 AI(I,J)=0.D0 AR (1 , 1) =36. DO-4. 5D0*DELTA AR(2,2) =12.D0-3.5D0*DELTA AR (3, 3) =-6. DO-2. 5DO*DELTA AR(4,4) =-18.DO-1 .5DO*DELTA AR (5,5) = -24. D0-0.5D0*DELTA AR(6,6) =-24.D0+0.5D0*DELTA AR (7,7) =-18. DO + 1. 5D0*DELTA AR (8,8) =-6.D0+2. 5DO*DELTA AR (9 ,9) = 12. DO+3. 5DO*DELTA AR(10,10) =36.D0+4.5D0*DELTA AR (1 ,2)=-3. DO*ALFA AI{1,2) =3.D0*BETA AR (2,3) =-4. DO*ALFA AI(2,3)=4.D0*BETA AR (3,4) =-DSQRT (2 1. DO) *ALFA AI (3,4) =DSQRT (21 .DO) *BETA AR (4,5)=-2. D0*DSQRT (6. DO) *ALFA - 6 - 2 -AI (4,5)=2.D0*DSQBT (6.D0)*BETA AR (5,6) =-5, DO*ALFA AI (5,6)=5.D0*BETA AR (6 ,7) =-2. DO*DSQRT (6. DO) * ALFA AI (6,7) =2.D0*DSQRT (6 » DO) *BETA AS (7,8) =-DSQRT (21. DO) * ALFA AI (7, 8) =DSQRT (21 . DO) *BET A AB (8,9)=-4.D0*ALFA AI (8,9) =4.D0*BETA AR (9 , 10) =-3. DO*ALFA AI (9,10)=3.D0*BETA DO 100 M=1,9 DO 100 N=2,10 AR (N , M) = A R (M, N) 100 AI (N,M) =-1.D0*AI (M,N) AR (1,3) =6.D0*ETA AR (2,U) =2.D0*DSQBT (21 .DO) *ETA AR (3,5)=3.D0*DSQBT (14. DO) *ET A AR(U,6) =5.D0*DSQRT (6. DO)* ETA AR (5,7) =5.D0*DSQRT (6. DO) *ETA AR (6,8) =3.D0*DSQRT (14. DO) *ETA AB (7,9) =2. DO*DSQBT (21. DO) *ETA AB (8,10)=6.D0*ETA DO 200 MM=1,8 DO 200 NN=3,10 200 AR (NN,MM)=AB (MM, NN) CALL DCEIGN(AB , AI, 10 ,10,EB,EI,VB,VI,IEBBOB, 1,0) N1 = 10 IF (IEBBOB.GT.O) GO TO 20 DO 500 IB0W=1,10 SUM=0.D0 DO 501 IC0L=1,10 501 SUM=SUM + VB(IROW,ICOL) **2 + VI (IBOW,ICOL)**2 SUM=DSQBT (SUM) DO 502 ICOL=1,10 VR (IBOW,ICOL)=VR (IBOW,ICOL)/SUM 502 VI (IROW,ICOL)=VI (IROW,ICOL)/SUM 500 CONTINUE PBINT 4 4 FORMAT(«0',T5,«EIGENVALUES•,T50,'EIGENVECTORS'//) DO 5 1=1,10 5 PBINT 6 , EB (I) , (V B (I, J) , J= 1, 10) 6 FOBHAT('0«,F11.5 #lX f10F9.5) 20 N1=IEBROB-1 DO 8 KK=1,10 DO 8 KKK=1,10 A R (KK , KKK) = 0 . D0 8 AI (KK,KKK)=0.DO DO 9 KJ=1,10 9 ER(KJ)=0.D0 GO TO 10 999 PBINT 11 11 FOBMAT ( * 1 ') STOP END 

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