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The direct detection of double quantum coherence Legros, Mark Anthony 1981

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THE DIRECT DETECTION OF DOUBLE QUANTUM COHERENCE by MARK ANTHONY LEGROS B.Sc.,Massey U n i v e r s i t y , 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of P h y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1981 © Mark Anthony LeGros, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t 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 r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n 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 or her 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 p e r m i s s i o n . Department of P h ysics The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 8, 1984 i i A b s t r a c t When a spi n - 1 nucleus i s p l a c e d i n a l a r g e magnetic f i e l d and an e l e c t r i c f i e l d g r a d i e n t , the energy l e v e l diagram f o r the s t a t e s d e s c r i b i n g i t ' s o r i e n t a t i o n c o n s i s t of three unequally spaced l e v e l s . I f the hami l t o n i a n (which i s the sum of a Zeeman term and a quadrupolar term) i s t r u n c a t e d to i n c l u d e only terms which commute with the Zeeman term, magnetic d i p o l e t r a n s i t i o n s between l e v e l s with m=1 have f i n i t e t r a n s i t i o n p r o b a b i l i t i e s . When the nonsecular terms, those which do not commute with l z , are taken i n t o account another magnetic d i p o l e t r a n s i t i o n can occur; a t r a n s i t i o n i n which m=2. The theory and experimental techniques a s s o c i a t e d with the ob s e r v a t i o n of m=2 t r a n s i t i o n s i s presented. The i n t e n t i o n i s to provide i n s i g h t s and examples of when the spectroscopy of such t r a n s i t i o n s i s a va l u a b l e procedure. The r e s u l t s show that the spectroscopy of t h i s f o r b idden t r a n s i t i o n i s p r a c t i c a l for spin - 1 n u c l e i i n systems with l a r g e (>500Khz) quadrupole s p l i t t i n g s , and long s p i n - s p i n r e l a x a t i o n times. The inf o r m a t i o n a v a i l a b l e by doing the m=2 experiments i s ob t a i n a b l e from other resonance techniques and the major fe a t u r e which prevents the experiments from being r e l e g a t e d to the catagory of a " S c i e n t i f i c c u r i o s i t y " i s t h e i r s i m p l i c i t y . i i i T able of Contents A b s t r a c t i i L i s t of Tables i v L i s t of F i g u r e s v Acknowledgement v i Chapter I INTRODUCTION 1 Chapter II BACKGROUND 3 A. THE FIRST ORDER QUADRUPOLE SPECTRUM 5 B. NONSECULAR EFFECTS 17 Chapter III HAMILTONIANS ; 18 A. PERTURBATION THEORY AND THE FICTITIOUS SPIN 1/2 21 B. FICTITIOUS SPIN-1/2 24 C. PREPARATION OF DOUBLE QUANTUM COHERENCE 30 D. TWO HARD 90 DEGREE PULSES 30 E. THE SOFT DOUBLE QUANTUM PULSE 31 F. THE EVOLUTION OF DOUBLE QUANTUM COHERENCE 37 G. THE SECOND ORDER SHIFT 4 1 Chapter IV EXPERIMENTAL 44 A. PROBE 4 5 B. COILS 45 C. SAMPLES 47 D. COMPARISON BETWEEN THE EXCITATION METHODS 56 E. APPLICATIONS 58 F. FURTHER APPLICATIONS 59 BIBLIOGRAPHY 61 APPENDIX A - ANGULAR MOMENTUM 62 i v L i s t of Tables I. B a s i s Tensors and Axioms of L i o u v i l l e Space 11 I I . Angular momentum changes d u r i n g a |1> to |-1> t r a n s i t i o n 67 V L i s t of F i g u r e s 1. The f i r s t order quadrupole spectrum 6 2. Powder p a t t e r n f o r a spin-1 nucleus 8 3. R e l a t i o n s h i p between the primed and the unprimed frames 28 4. E i g e n s t a t e s i n the r o t a t i n g frame 32 5. T r a n s i t i o n p r o b a b i l i t i e s 35 6. The second order s h i f t powder p a t t e r n 43 7. C o i l s ...46 8. S a t e l l i t e l i n e s 49 9. Transverse e x c i t a t i o n 51 10. L o n g i t u d i n a l e x c i t a t i o n 53 11. R o t a t i o n p a t t e r n 54 v i Acknowledgement I would l i k e to thank my s u p e r v i s o r , P r o f . Myer Bloom f o r i n v e n t i n g the c e n t r a l idea f o r the t h e s i s , and I wish to thank the c o h a b i t a n t s of Myer's l a b , "Room 100" f o r p r o v i d i n g an enjoyable work p l a c e . I must a l s o g i v e s p e c i a l mention to Dr.Tom Pratum ,Dr. Alexander Mackay and Edward S t e r n i n f o r h e l p i n g me with the t e c h n i c a l aspects of my work i n a p a t i e n t f r i e n d l y f a s h i o n . 1 I. INTRODUCTION T h i s t h e s i s r e p o r t s a novel magnetic resonance technique, r e l e v a n t to n u c l e i which experience an e l e c t r i c quadrupole i n t e r a c t i o n with l o c a l e l e c t r i c f i e l d g r a d i e n t s . The method i s a c o n t r i b u t i o n to a c l a s s of experiments c o l l e c t i v e l y termed, " S o l i d s t a t e high r e s o l u t i o n t e c h n i q u e s " . The aim of these techniques i s the o b s e r v a t i o n of a small p a r t of a sp i n h a m i l t o n i a n which i s normally masked by a much l a r g e r i n t e r a c t i o n . The experimental procedures are r a d i c a l l y d i f f e r e n t from those normally employed i n s o l i d s t a t e NMR, as the heart of the method i n v o l v e s t r a n s i t i o n s which are only p a r t i a l l y a l l o w e d . The ideas are s i m i l a r to the usual methods, i n that the experiment i n v o l v e s the p r e p a r a t i o n of a s p e c i a l i n i t i a l s t a t e , which i s u n a f f e c t e d by the troublesome p a r t of the h a m i l t o n i a n . Before beginning a d e t a i l e d d e s c r i p t i o n of the " D i r e c t d e t e c t i o n of double quantum coherence" i t may be wise to c l a r i f y a few n o t a t i o n a l p o i n t s f o r the b e n e f i t of those "not i n the business'.' The resonance experiments r e p o r t e d here are done u s i n g a l a r g e e x t e r n a l magnetic f i e l d , and the d i r e c t i o n p a r a l l e l to t h i s f i e l d i s termed l o n g i t u d i n a l and des i g n a t e d the l a b o r a t o r y z a x i s . The plane which l i e s p e r p e n d i c u l a r to the f i e l d i s c a l l e d , the t r a n s v e r s e , or x, y, plane. For the experiments on nitrogen-14, I have desi g n a t e d the doublet which a r i s e s i n normal F o u r i e r transform NMR as s a t e l l i t e l i n e s . The term s a t e l l i t e i s used to d i s t i n g u s h these l i n e s , which occur at the fr e q u e n c i e s Wo+Wq and Wo-Wq, from the Am=2 t r a n s i t i o n at Wo i n a 2 s p i n - 1 s y s t e m . 3 I I . BACKGROUND The i n t e r a c t i o n between the e l e c t r i c quadrupole moment of a nucleus and the e l e c t r i c f i e l d g r a d i e n t at the s i t e of the nucleus i s governed by a tensor i n t e r a c t i o n of rank two. In the absence of any a p p l i e d f i e l d s , the hami l t o n i a n d e s c r i b i n g the nuclear e l e c t r i c quadrupole i n t e r a c t i o n i s r o t a t i o n a l l y i n v a r i a n t w.r.t r o t a t i o n of the molecule as a whole. I f , as i n the case of high f i e l d NMR , a l a r g e e x t e r n a l magnetic f i e l d i s present the h a m i l t o n i a n i s no longer i n v a r i a n t to molecular r o t a t i o n s , and the spectrum c o n t a i n s i n f o r m a t i o n on the r e l a t i v e o r i e n t a t i o n s of the e l e c t r i c f i e l d g r a d i e n t and the a p p l i e d f i e l d . T h i s f a c t i s the main reason why NMR of n u c l e i p o s s e s s i n g e l e c t r i c quadrupole moments has become a p r e c i s e t o o l for the i n v e s t i g a t i o n of s p a t i a l l y ordered systems and a n i s o t r o p i c molecular p r o c e s s e s . H i s t o r i c a l l y the f i r s t major use of the quadrupole i n t e r a c t i o n i n s o l i d s was the study of s i n g l e c r y s t a l s , g i v i n g i n f o r m a t i o n on s t r u c t u r e and dynamics. T h i s i n i t i a l c h o i c e of system was d i c t a t e d by the e x i s t i n g technology and the requirement that simple systems be understood f i r s t . The subsequent advances i n e l e c t r o n i c s and computing, with the i n t r o d u c t i o n of pul s e d NMR, v a s t l y i n c r e a s e d the range and nature of systems which c o u l d be s t u d i e d . In recent years much a t t e n t i o n has been p a i d to systems c o n s i s t i n g of c r y s t a l l i t e s having a d i s t r i b u t i o n of o r i e n t a t i o n s . The most common d i s t r i b u t i o n i s that of an i s o t r o p i c powder, which cannot g i v e i n f o r m a t i o n on the o r i e n t a t i o n of e l e c t r i c f i e l d s , but the r e s u l t i n g powder p a t t e r n 4 s p e c t r u m h a s f e a t u r e s w h i c h e n a b l e t h e d e t e r m i n a t i o n o f t h e m a g n i t u d e a n d symmetry p r o p e r t i e s o f t h e i n t e r a c t i o n . The most f r u i t f u l s t u d i e s have been p e r f o r m e d on s p i n - 1 n u c l e i , p r o b a b l y b e c a u s e t h e r e s u l t i n g s p e c t r a a r e t h e s i m p l e s t t o i n t e r p r e t . T h r e e common s p i n - 1 i s o t o p e s a r e d e u t e r i u m , n i t r o g e n - 1 4 , a n d l i t h i u m - 6 , o f w h i c h d e u t e r i u m i s by f a r t h e e a s i e s t t o s t u d y . A c o n c i s e s t a t e m e n t o f what makes a n u c l e u s e a s y t o s t u d y i s n e c e s s a r y , a s t h i s t h e s i s i s c o n c e r n e d w i t h a t e c h n i q u e t o w i d e n t h e r a n g e o f s y s t e m s a c c e s s i b l e t o h i g h f i e l d NMR. 1) The n u c l e u s must p o s s e s s a l a r g e m a g n e t i c d i p o l e moment. 2) The q u a d r u p o l e h a m i l t o n i a n must h a v e a s m a l l c o u p l i n g c o n s t a n t . T h e s e two f e a t u r e s a r e e s s e n t i a l t o o b t a i n i n g g ood s i g n a l t o n o i s e a n d u n i f o r m i n i t i a l c o n d i t i o n s o v e r t h e w h o l e s p e c t r u m . A s i m p l e i n i t i a l c o n d i t i o n c a n be p r o d u c e d when t h e r a t i o o f t h e Zeeman i n t e r a c t i o n t o t h e q u a d r u p o l a r i n t e r a c t i o n i s l a r g e , t h i s means t h e e f f e c t o f t h e q u a d r u p o l e i n t e r a c t i o n on t h e e v o l u t i o n o f t h e s y s t e m i s n e g l i g i b l e d u r i n g t h e p r e p a r a t i o n o f a n o n e q u i l i b r i u m s t a t e . A g o o d e x a m p l e i s t h e q u a d r u p o l e e c h o s e q u e n c e ; i f t h e power s p e c t r u m o f t h e e x c i t a t i o n v a r i e s a p p r e c i a b l y o v e r t h e f r e q u e n c i e s t o be e x c i t e d , t h e f o u r i e r t r a n s f o r m o f t h e e c h o e x h i b i t s f r e q u e n c y d e p e n d e n t d i s t o r t i o n . F o r w i d e b a n d w i d t h NMR e x p e r i m e n t s , s u c h a s t h e q u a d r u p o l a r e c h o , t h e most i m p o r t a n t p a r a m e t e r i s t h e d u r a t i o n o f t h e 90 d e g r e e p u l s e w h i c h p r e p a r e s t h e s y s t e m i n a s t a t e o f t r a n s v e r s e m a g n e t i s a t i o n . The l e n g t h o f t h i s p u l s e i s t h e s t a n d a r d w i t h w h i c h t h e l a r g e s t q u a d r u p o l e f r e q u e n c y must be c o m p a r e d . L i m i t a t i o n s on t h e 90 d e g r e e p u l s e 5 degree pulse time are of an experimental nature and w i l l be d i s c u s s e d i n the experimental s e c t i o n . For deuterium a two microsecond p u l s e i s r e a d i l y a t t a i n a b l e , and t h i s allows s p e c t r a 300 KHZ wide to be recorded with l i t t l e d i s t o r t i o n . The width of p r a c t i c a l l y a l l deuterium quadrupole s p e c t r a f a l l w i t h i n t h i s range, and most i f not a l l deuterium compounds can be examined by c o n v e n t i o n a l NMR methods. Nitrogen-14 i s much l e s s s u i t a b l e for s o l i d s t a t e NMR s t u d i e s as i t s range of quadrupole c o u p l i n g c o n s t a n t s extends over many megahertz, and i t s gyromagnetic r a t i o i s approximately h a l f that of deuterium. The best 90 degree p u l s e time one can achieve with n i t r o g e n i s approximately 5 microseconds, t h i s combined with the l a r g e quadrupole c o u p l i n g c o n s t a n t s makes n i t r o g e n compounds d i f f i c u l t to study using c o n v e n t i o n a l NMR; u n l e s s s i n g l e c r y s t a l s are a v a i l a b l e . A. THE FIRST ORDER QUADRUPOLE SPECTRUM The f i r s t order quadrupole h a m i l t o n i a n i s a t r u n c a t e d v e r s i o n of the t r u e high f i e l d h a m i l t o n i a n i n which only terms commuting with Iz have been r e t a i n e d , these terms a f f e c t the energy l e v e l s i n f i r s t order as shown i n f i g . ( l ) . as [H,lz]=0 the t r u n c a t e d h a m i l t o n i a n i s i n v a r i a n t to r o t a t i o n s about the main magnetic f i e l d , and the e i g e n s t a t e s w i l l transform i r r e d u c i b l y under the 2-dimensional r o t a t i o n group. Because of t h i s the t o t a l angular momentum, and the z p r o j e c t i o n of angular momentum, of the s p i n system w i l l be good quantum numbers for the e i g e n s t a t e s . S e l e c t i o n r u l e s f o r magnetic d i p o l e t r a n s i t i o n s between the e i g e n s t a t e s , can be d e r i v e d by noting /\ y\ A /V that I can be expanded i n terms of, Iz,I+,I-,which transform 6 F i g u r e 1 - The f i r s t order quadrupole spectrum F i r s t order quadrupole spectrum f o r a spin - 1 nucleus i n an a x i a l l y symmetric e l e c t r i c f i e l d g r a d i e n t . -> < |0> 1 1 1 t . —- " i < ^ , -—• i Wo - C J a i Do (JJO 2 . L O G L (Jo A A H= -Dolz. + (J<aTz.z. 7 i r r e d u c i b l y under the symmetry group as, L=1 m=0,L=1 m=1, L=1 m=-1, r e s p e c t i v e l y . R e f l e c t i o n i n the t r a n s v e r s e plane i s a l s o a symmetry o p e r a t i o n but i t commutes with the 2-dimensional r o t a t i o n group and adds nothing u s e f u l to l a b e l l i n g the e i g e n s t a t e s . The s e l e c t i o n r u l e a l s o i m p l i e s that the system can possess a net magnetisation only i f there i s phase coherence between l e v e l s d i f f e r i n g by |m|=1,0. The s e c u l a r h a m i l t o n i a n has a p e r i o d i c i t y of 2 0 about a r o t a t i o n a x i s p e r p e n d i c u l a r to the z d i r e c t i o n , consequently the m=1 s t a t e s are s h i f t e d by the same amount from the pure Zeeman s p l i t t i n g . The f i r s t order spectrum i s a d i r e c t r e s u l t of an o r i e n t a t i o n dependent energy of the m=0 s t a t e . If the sample i s a powder then each o r i e n t a t i o n has equal p r o b a b i l i t y , but as the f i r s t order spectrum depends s o l e l y on the angle between the a p p l i e d f i e l d and the z a x i s of the e l e c t r i c f i e l d g r a d i e n t , the f i r s t order powder p a t t e r n i s weighted by SiNfpi^i) oj^ ^ f i g ( 2 ) . If a d i r e c t way to monitor the energy d i f f e r e n c e between the m=1 and m=-1 s t a t e s e x i s t e d , systems i n which the powder p a t t e r n s p e c t r a are too wide f o r t r a d i t i o n a l spectroscopy c o u l d be s t u d i e d . The m=1 to m=-1 t r a n s i t i o n i s u n a f f e c t e d by the f i r s t order quadrupole i n t e r a c t i o n and f o r cases i n which t h i s approximation i s v a l i d the powder p a t t e r n spectrum i s j u s t a s i n g l e l i n e . Some f e a t u r e s of the s p e c t r a i n f i g . ( 2 ) need to be ex p l a i n e d f o r those not f a m i l i a r with NMR. The standard experimental proceedure i s to prepare the system i n a s t a t e of t r a n s v e r s e magnetisation and monitor the decay of t h i s 8 F i g u r e 2 - Powder p a t t e r n f o r a s p i n - 1 n u c l e u s 77=0 powder pattern - 2 0 0 0 200 Frequency (kHz) 9 m a g n e t i s a t i o n . The r e s u l t i s the a u t o c o r r e l a t i o n f u n c t i o n f o r the t r a n s v e r s e m a g n e t i s a t i o n . The F o u r i e r transform of t h i s a u t o c o r r e l a t i o n f u n c t i o n i s e q u i v a l e n t to the t r a n s v e r s e components of the high temperature magnetic s u s c e p t i b i l i t y t e n s o r . The s p e c t r a obtained v i a the F o u r i e r transform are o f t e n represented i n a reduced frequency scheme i n which the e x c i t a t i o n frequency, Wo, i s s u b t r a c t e d from each l i n e i n the spectrum. T h i s r e l a t e s the observed spectrum to the method of phase s e n s i t i v e d e t e c t i o n , and to the common t h e o r e t i c a l d e s c r i p t i o n of s e c u l a r h a m i l t o n i a n s ; that of an i n t e r a c t i o n frame r o t a t i n g at Wo about the z a x i s . A most u s e f u l f o r m a l i s i m to d e s c r i b e magnetic resonance i s the L i o u v i l l e f o r m a l i s i m of quantum mechanics, because i t enables the e s s e n t i a l p h y s i c s to be d e a l t with i n a unique economical f a s h i o n . L i o u v i l l e f o r m a l i s i m and the d e n s i t y matrix are reviewed i n many a r t i c l e s , so I w i l l c o n c e n t r a t e on the f e a t u r e s of the theory s p e c i f i c to the d e s c r i p t i o n of a s p i n - 1 n u c l e u s . The f i r s t d e c i s i o n i s the choice of r e l e v a n t v a r i a b l e s . For an i s o l a t e d s p i n system these c o n s i s t of a l l sums of products of the fundamental observables I x , I y , I z . For s p i n - 1 n u c l e i there are nine independent o p e r a t o r s which can be p a r t i t i o n e d i n t o three t e n s o r s ; each tensor transforms i r r e d u c i b l y under the three dimensional r o t a t i o n group, P\3 .We o b t a i n an i n v a r i a n t t ensor, a second rank antisymmetric tensor, and a t r a c e l e s s second rank symmetric t e n s o r . The ch o i c e of b a s i s w i t h i n each of these t e n s o r s can be made by r e q u i r i n g the components i n each b a s i s transform i r r e d u c i b l y under a subgroup 10 of P\3 , such as C o o or C o o v • T n e subgroup chosen to define the basis i s taylored to the symmetry of the hamiltonian to be dealt with. During the r . f . pulses the quadrupole interaction can be ignored and the evolution operator i s e s s e n t i a l l y a rotation, hence states belonging to d i f f e r e n t irreducible representations of a r e n o t mixed. During evolution under the secular quadrupole hamiltonian the p r e v a i l i n g symmetry is C o o o r Coov a n < 3 states belonging to d i f f e r e n t representations of these groups are not coupled. Table ( 1 ) shows the form of the two basis systems in terms of the, fundumental observables, and symmetry groups used to label the basis states. The presentation of invariant subspaces under p , ^ is straightforward, but catagorisation of the components by their transformation properties under C ^ and C < * v require some explanation. For an iso l a t e d spin system interacting with an external magnetic f i e l d , (defined to be in the z direction) the symmetry i s C o o r which is an albelian group and has 1 -dimensional representations. The appearance of more than one tensor component beside each representation l a b e l , merely indicates that they transform, in the same way under the group operations. The group C o o y i s the symmetry group of our spin as described in a resonant rotating frame, so that there is no remnant of the Zeeman interaction ( r e f l e c t i o n symmetry i s broken i f the hamiltonian contains terms linear in the angular momentum). This group i s nonabelian and has 2-dimensional representations, the partners for a p a r t i c u l a r representation are enclosed in brackets. 11 Table I - B a s i s Tensors and Axioms of L i o u v i l l e Space B a s i s Tensors grouped i n terms of i r r e d u c i b l e r e p r e s e n t a t i o n s of R 3 Sprier ica l A A 2 T « = 1 ( I + ) A A A A A Txi = - l ( I * + fxO= 1 ( 3 1 2 f V l - l ( ± z ± - - r ± - i E ) Ta-A = J_ I -2 Car tes ian ^ , A A A /V Txy = l ( I x l y + I y l x ) Txz = 1. ( I z l * + I x l . f 2 Z = J L ( 3 i z - A ) tyz: = J_ ( l ^ I y + T y I z ) TV-y* = 1 ( I x + I y ) A A A A I* A A A I-= J x - Z X y I x A A Xz. 1 2 Table-1 c o n t i n u e d Tensors grouped i n terms of i r r e d u c i b l e r e p r e s e n t a t i o n s of  subgroups of R3. ' o o v Tao T»-i I- ( I x . I y ) ( T y 2 , T x 2 ) ( T x y , T x ' Y ) T* 13 T a b l e-1 c o n t i n u e d A x i o m s o f L i o u v i l l e s p a c e (1) L i o u v i l l e s p a c e i s a l i n e a r v e c t o r s p a c e s p a n n e d by a s e t o f o p e r a t o r s { |A i> }. ( 2 ) A q u a n t u m m e c h a n i c a l s t a t e i s r e p r e s e n t e d by a l i n e a r (3) A s c a l a r p r o d u c t i s d e f i n e d a s < A i | A j > = T r a ( A i A j ) . (4) L i n e a r o p e r a t o r s a c t i n g on t h e s p a c e f o r m a n a l g e b r a s p a n n e d by a l l sums a n d p r o d u c t s o f o p e r a t o r s o f t h e f o r m A i * A j ; w h e r e A . A A . A . A . A , A . A . A i * A j | A k > = | A i A j A k > , w i t h A i a n d A ] b e i n g a n y o p e r a t o r s f r o m L i o u v i l l e s p a c e . (5) The e v o l u t i o n p r o b l e m i s s p e c i f i e d by a n i n i t i a l c o n d i t i o n I £Ho>>= £cuco) iAi> , a n d t h e d y n a m i c a l e q u a t i o n : s u p p o s i t i o n o f o p e r a t o r s t w h e r e H i s t h e h a m i l t o n i a n o f t h e s y s t e m . 1 4 As an example l e t us now s o l v e the f o l l o w i n g i n i t i a l v a lue A problem,; If the s t a t e at time t=0 i s Ix, what i s the s t a t e at time t>0 i f the system i n t e r a c t s with a l a r g e Zeeman h a m i l t o n i a n and a s e c u l a r quadrupole h a m i l t o n i a n . The problem can be s i m p l i f i e d by t r a n s f e r r i n g to a frame r o t a t i n g at the larmor frequency, and d e f i n e the r o t a t i n g frame so that the x a x i s of t h i s frame c o i n c i d e s with the l a b o r a t o r y x a x i s at t=0. Formally the problem i s expressed as: A more t r a c t a b l e s o l u t i o n can be found i f one notes that e i g e n f u n c t i o n s of the operator L must transform i r r e d u c i b l y under the symmetry group of L, i e . Coov ' a n ^ that matrix elements of L v a n i s h unless i t i s a matrix element between o p e r a t o r s which transform under the same row of the same i r r e d u c i b l e r e p r e s e n t a t i o n ; hence <A|L|lx> = 0 u n l e s s 7\=Tyz. We know that the e i g e n f u n c t i o n s of L w i l l be of the form alx+bTyz, and some a l g e b r a shows the e i g e n f u n c t i o n s and c o r r e s p o n d i n g e i g e n v a l u e s a r e : and lp-[Hp-pH] H=u)aT. I x + i T y a Our s o l u t i o n can be w r i t t e n : ( 0 ( t ) = a ( I ^ + i T y z ) e + b ( I x - i T \ , z ) e - i .U )a+ ( 2 ) 1 5 The s t a t e o f t h e s y s t e m a t a n y t i m e c a n be g i v e n by l i s t i n g t h e c o m p o n e n t s o f a v e c t o r i n a n 8 - d i m e n s i o n a l s p a c e . I f t h e h a m i l t o n i a n h a s a h i g h d e g r e e o f s y m m e t r y t h e e v o l u t i o n p r o c e s s p a r t i t i o n s i n t o d i s j o i n t s u b s p a c e s , a n d i f t h e i n i t i a l c o n d i t i o n s a r e s u f f i c i e n t l y s i m p l e , t h e d e n s i t y o p e r a t o r c a n be e x p r e s s e d i n t e r m s o f a s m a l l number o f b a s i s o p e r a t o r s . The s e c u l a r q u a d r u p o l e h a m i l t o n i a n i s b i l i n e a r i n t h e s p i n a n g u l a r momenta a n d c o u p l e s t h e i n i t i a l s t a t e o f v e c t o r m a g n e t i s a t i o n t o a s t a t e o f s e c o n d r a n k t e n s o r p o l a r i z a t i o n . We h a v e s e e n f r o m e q u a t i o n ( 2 ) t h a t a s y s t e m i n i t i a l l y i n t h e s t a t e I x p r e c e s s e s i n t o a s t a t e T y z a t a f r e q u e n c y d e t e r m i n e d by t h e . q u a d r u p o l e c o u p l i n g c o n s t a n t . The W i g n e r t h e o r e m f o r t e n s o r o p e r a t o r s t e l l s u s t h a t i f t h e n u c l e a r s t a t e s a r e s u c h t h a t L a n d m a r e g o o d q u a n t u m n u m b e r s , t h e m a t r i x e l e m e n t o f a n y o p e r a t o r a c t i n g on t h e n u c l e a r c o o r d i n a t e s , a n d t r a n s f o r m i n g a c c o r d i n g t o t h e same row o f t h e same i r r e d u c i b l e r e p r e s e n t a t i o n o f P\ 3 a s T y z , w i l l be p r o p o r t i o n a l t o t h o s e o f /\ T y z b e t w e e n s t a t e s l a b e l l e d w i t h t h e same L a n d m. An o p e r a t o r w i t h t h e s e c h a r a c t e r i s t i c s i s t h e y - z c o m p o n e n t o f t h e n u c l e a r e l e c t r i c q u a d r u p o l e moment o p e r a t o r . The e x p e c t a t i o n v a l u e f o r t h e o p e r a t o r f o r a g i v e n n u c l e a r s t a t e i s a m e a s u r e o f t h e e l e c t r i c q u a d r u p o l e moment o f t h e n u c l e a r c h a r g e d i s t r i b u t i o n . The e v o l u t i o n p r o b l e m c a n h a v e t h e f o l l o w i n g v i s u a l i z a t i o n ; i n i t i a l l y t h e e n s e m b l e o f n u c l e i i s i n a s t a t e " i x , a n d t h e n u c l e a r d a n c e i s s u c h t h a t a n e t m a g n e t i c d i p o l e moment e x i s t s . The e l e c t r i c f i e l d g r a d i e n t a t t h e n u c l e u s p r o d u c e s a t o r q u e d u e t o t h e q u a d r u p o l e i n t e r a c t i o n , a n d t h e n u c l e a r d a n c e c h a n g e s s o 16 that the ensemble develops an e l e c t r i c quadrupole moment ; and so the o s c i l l a t i o n c o n t i n u e s . We can couple to an o s c i l l a t i n g d i p o l e moment by wrapping a c o i l around the sample and monitoring the v o l t a g e induced i n the c o i l , s i m i l a r l y i f the sample i s put i n t o the c e n t e r of a s u i t a b l y o r i e n t e d quadrupole c a p a c i t o r the o s c i l l a t i n g quadrupole moment w i l l induce a vo l t a g e a c r o s s the c a p a c i t o r . Comparison between the nuclear magnetic d i p o l e moment and the e l e c t r i c quadrupole moment shows that the c o u p l i n g to the c a p a c i t o r i s unobservably s m a l l . The other second rank s t a t e s correspond to the quadrupole charge d i s t r i b u t i o n having d i f f e r e n t o r i e n t a t i o n s w.r.t. the l a b frame. Tzz i s a s t a t i c e l e c t r i c quadrupole d i s t r i b u t i o n , and A A Txy precesses i n t o Txx-yy at twice the Larmor frequency. The p r e c i s e nature of the nuclear charge d i s t r i b u t i o n i s not u s u a l l y of i n t e r e s t to the s p e c t r o s c o p i s t , but the fr e q u e n c i e s and l i f e t i m e s of the o s c i l l a t i n g nonequi1ibrium s t a t e s a r e . We have seen how to observe the o s c i l l a t i o n of the Ix i n t o the s t a t e Tyz, and from t h i s we can deduce the A A f r e q u e n c i e s a s s o c i a t e d with Txy and Txx-yy, but as remarked e a r l i e r i f f o r some reason the t r a n s v e r s e s i g n a l cannot be seen we must f i n d another way of o b s e r v i n g the m=2 energy d i f f e r e n c e between the m=1 and m=-1 s t a t e s . The f i r s t method used to observe t h i s t r a n s i t i o n was a continuous wave method ( Y a t s i v e t . a l (1)) where the s t r e n g t h of the r . f . f i e l d i s i n c r e a s e d u n t i l an a b s o r p t i o n peak c o r r e s p o n d i n g to a two photon process c o u p l i n g m=1 to m=-1 appears. The theory of l i n e s h a p e f o r such processes i s complicated because the exact s t r e n g t h of the r . f . 17 f i e l d i s an important parameter in the spectrum, and the system i s no longer i n the l i n e a r response regime. More r e c e n t l y techniques of m u l t i p l e quantum spectroscopy have been developed, and although the p h i l o s o p h i e s and experimental methods are w e l l s u i t e d t o the problem of monitoring m u l t i p l e quantum t r a n s i t i o n s , they g e n e r a l l y r e q u i r e that the system can be manipulated by a t r a n s v e r s e f i e l d at the Larmor frequency. B. NONSECULAR EFFECTS The d i s c u s s i o n up to t h i s p o i n t has been concerned p r i n c i p a l l y with s e c u l a r h a m i l t o n i a n s , and as w i l l be shown in the theory s e c t i o n the quadrupole h a m i l t o n i a n w i l l u s u a l l y have some nonsecular terms. The approximation of a s e c u l a r h a m i l t o n i a n to d e s c r i b e s p i n dynamics in most NMR experiments i s very reasonable as the spi n system i s manipulated as a t r a n s v e r s e magnetisation o s c i l l a t i n g at the larmor frequency. I n c l u s i o n of nonsecular terms in these experiments would l e a d to a great d e a l of c o m p l i c a t i o n i n the theory of s p i n dynamics f o r l i t t l e e x perimental reward. If the nonsecular e f f e c t s are i n c l u d e d and one does experiments at f r e q u e n c i e s other than the Larmor frequency, and or d e t e c t s l o n g i t u d i n a l l y p o l a r i z e d m a g n e t i s a t i o n , very observable e f f e c t s occur. The nonsecular e f f e c t s examined in t h i s t h e s i s r e s u l t i n an o s c i l l a t i n g m a gnetisation from an i n i t i a l s t a t e of double quantum coherence. The p r i n c i p a l f e a t u r e s of t h i s m agnetisation are that i t o s c i l l a t e s at twice the Larmor frequency, and i s p o l a r i z e d i n the z d i r e c t i o n . 18 I I I . HAMILTONIANS The e l e c t r i c quadrupole i n t e r a c t i o n i s a second rank tensor i n t e r a c t i o n between the e l e c t r i c f i e l d g r a d i e n t at the nucleus, and the e l e c t r i c quadrupole moment of the nucleus. The i n t e r a c t i o n may be expressed i n terms of s p h e r i c a l t e n s ors as: Wa = 2 j A x B x Q m--2 where the pCx are spin o p e r a t o r s d e f i n e d to be: A° = eo. ( 5 ± » - ± a > i ) -• - - A ti= e a ^ ( L L t i t t ) 1(2.1-1) 1(21-1)4 rY* = eqf6 x* - (D 1(21-1)4 The §>* are o p e r a t o r s which a c t on e l e c t r o n s surrounding the nucleus and they are given by: B a ^ l f O ^ t i v V ) , B * = A V z z . • - fi~a = l (X/xx-Vyy t a iVxy ) -<_> V i j are operat o r s which y i e l d second d e r i v a t i v e s of the e l e c t r o s t a t i c p o t e n t i a l when a p p l i e d to the e l e c t r o n wave f u n c t i o n . The h a m i l t o n i a n given i n eq.(5) i s an approximation to the true quadrupole h a m i l t o n i a n , and j u s t i f i c a t i o n f o r being able to express the i n t e r a c t i o n i n t h i s way can be found i n Abragam ( 2 ) . If we choose the p r i n c i p a l a x i s system of the e l e c t r i c f i e l d g r a d i e n t (e.f.g.) to d e s c r i b e both s p i n c o o r d i n a t e s and the e . f . g . , (1) becomes: fta = e z c| Q c^fl - i ( i - n ) - i - i n ( T * + T - n V £ I T I = \ A i - v ¥ y (f) 4 1(21-1)1 A J 19 I n t h e p r e s e n c e o f a l a r g e e x t e r n a l m a g n e t i c f i e l d , H , t h e t o t a l h a m i l t o n i a n b e c o m e s : .# = - y * H - I + & a We now w i s h t o t r a n s f o r m t h e s p i n c o o r d i n a t e s y s t e m s o t h a t t h e z a x i s o f t h e s y s t e m i s p a r a l l e l t o t h e e x t e r n a l m a g n e t i c f i e l d . ( w h i c h i s d e f i n e d t o l i e a l o n g t h e l a b o r a t o r y z a x i s ) . I f t h e p r i n c i p a l a x i s s y s t e m i s o b t a i n e d f r o m t h e l a b s y s t e m v i a a r o t a t i o n d e s c r i b e d by t h e E u l e r a n g l e s <X , |3 , V ( 3 ) ; t h e n (8) g i v e n i n t e r m s o f s p i n c o o r d i n a t e s r e f e r e d t o t h e l a b s y s t e m i s : • V + |-3e"'*snvi(p)Cos (Bi+ne^siNtp^i+cosiBije*1 + ne*siN(B)|cosip)-ij e>llfjf»i e* 1*-n e i w6 I N(B)/± J - C O S(B ) \ e X l * \ f i - i + p e siNcpjcos(f3)-ne s i N ( f i ) | c o s j B ) - i | e e s i N ( B ) | i + c o s ( B j e j + j 3 e 1' * s i w z( B) + n e~* (ii^scg)) 2 e 1 , r+tie' l l c < ^ i-cos ( B)jV'' j f i a The p u r p o s e o f t h i s t r a n s f o r m a t i o n i s t o d i a g o n a l i z e t h e Zeeman i n t e r a c t i o n f r o m a g e n e r a l m i s a l i g n m e n t o f t h e l a b a n d m o l e c u l a r f r a m e s . O n l y two E u l e r r o t a t i o n s a r e r e q u i r e d t o a l i g n t h e z a x e s o f t h e c o o r d i n a t e s y s t e m s . The f i n a l r o t a t i o n , w h i c h i s a r o u n d t h e z a x i s by a n a n g l e c< , a l i g n s t h e m o l e c u l a r x a n d y a x i s w i t h t h e l a b x a n d y a x e s . T h i s r o t a t i o n i s n o t a s y m m e t r y o p e r a t i o n o f t h e h a m i l t o n i a n b u t i t d o e s n o t c h a n g e t h e e n e r g y l e v e l s p a c i n g a n d f o r t h e p u r p o s e o f s i m p l i c i t y I w i l l s e t a = 0 . The v a l u e o f e*. i s i m p o r t a n t f o r some e x p e r i m e n t s a n d w i l l be d i s c u s s e d i n a l a t e r s e c t i o n . The h a m i l t o n i a n i s now 2 0 reexpressed i n terms of two d i f f e r e n t s p i n operator b a s i s systems as each b a s i s has merit f o r c e r t a i n types of c a l c u l a t i o n s : = - Yf.H l a + U)a I j J T (3COS A(B) -1 + TI SlN^tB) COS*( X)) Tzi. + (3 -C5T Sil\J(B) C05(B)-f5:n 5iN((3)C05(p.)C05CA^) T X H -iJTTl 5IIV/(|3) 5|I\J(A*) T « , z + j 3 il3C 5 1 NAB) t T l A ^ l - C O S ( B ) j A C Q 5 ( A y ) + X n ^ - t - C O S ( f i ) j 2 C 0 6 ( X y ) j "fV-yl + |VXn | l + C05(p)j a6lMqY)-^njJ.-COS(|3)j ZSl^(SLnjTxy j (±o) ^ - - YhH I2 -r WQ r j J J 3O0S*(B)-1) +_L5iW*p) COS( A 2 0 j ( 3 l i - I ( I t l ) + |3€lN(B) C05(6)-r_5lN((3)C05((3 )CO5CA!r)j ( ± 2 ( ± + + X - ) +(1+ + X - ) X z ) + i n si*/rp) siM^y) ( ± z ( i * -X-) + (X* - x - ) x « ) + 17L 51 W( R V) / 5 1 ) - C O S 4 " ^ ) j f±I - ±-) ) © 2 1 A . PERTURBATION THEORY AND THE F I C T I T I O U S S P I N 1 / 2 The m o s t e f f i c i e n t way t o s e e e f f e c t s due t o t h e n o n s e c u l a r t e r m s , i s t o d o a p e r t u r b a t i o n c a l c u l a t i o n o f t h e e i g e n s t a t e s , a n d t o d e t e r m i n e m a t r i x e l e m e n t s g o v e r n i n g m a g n e t i c d i p o l e t r a n s i t i o n s . To do t h e c a l c u l a t i o n I w i l l u s e t h e s i m p l e s t f o r m o f t h e h a m i l t o n i a n w h i c h g i v e s t h e d e s i r e d e f f e c t s : # a = - U ) o ± 2 + _ k ( 3 ± * - 1 ( 1 + 1 ) + A ( I z ( T + + ± - ) + ( X - + - X + ) ± , ; + fcCEi-rii) £j«=3e*qo. F O R S p i N - l A = O V smfft)coses) 4^ * Ye = i Ja SlW^p) (_?) 4 U / R i T l N G W - H o + H ' H o = - u ) o i z + 6 J a ( 3 ± ^ - I ( X + l ) ) a n d t h e z e r o o r d e r e i g e n f u n c t i o n s a r e e i g e n f u n c t i o n s o f Ho g i v e n by | + > , |0> , |— >• U s i n g s t a n d a r d n o n d e g e n e r a t e p e r t u r b a t i o n t h e o r y t h e e i g e n f u n c t i o n s t o f i r s t o r d e r i n H ' a r e g i v e n b y : i > ' - l m ) + £ _ _ H K _ _ J K > kpm HK-m = < k l H ' l o i ) k. Em~E\K. a n d t h e Em a r e e i g e n v a l u e s o f H o . The r e s u l t i n g e i g e n f u n c t i o n s a r e : l + ) ' - - l + > + 1 _ A I O ) + ^ e j ~ > (CL)«+ 3 bJa) 0)o 10>'= 10> - A f X A I +>' +- _ _ _ A _ J - > (yh (k>° + 3 U » ) (LJo-30Ja) I->'= l-> ~ A I T A I0> — I +> ( 3 0 J a - C J o ) U> A l t h o u g h t h e e n e r g y d i f f e r e n c e b e t w e e n |+>' a n d |~~>' i s u n c h a n g e d i n f i r s t o r d e r , t h e l e v e l s a c q u i r e a n o r i e n t a t i o n a l l y d e p e n d e n t e n e r g y d i f f e r e n c e i n s e c o n d o r d e r . The p e r t u r b a t i o n 22 expansion f o r the eigenvalues i s : + Yt i H ^ m l * |c^m (f§) The r e s u l t of the second order c a l c u l a t i o n of the energy d i f f e r e n c e between the |+>' and |—>' l e v e l s i s : EC ~ E- = A cJo -i-U)* SlN*(p) (CQ5*(6) + JL SlN*(ft)) (l6) * U)o 4 where ^3 i s the angle between the symmetry a x i s of the e . f . g . and the z d i r e c t i o n . Let us now c a l c u l a t e the matrix elements f o r magnetic d i p o l e r a d i a t i o n at a frequency 2Wo, as t h i s i s resonant with the |+>' |—>' t r a n s i t i o n : <+LX«l-/= - SW*(p) _ ZDo (g) <+IIx|->'=JA)aS»IVl(p) COS(B) As the matrix elements c o r r e s p o n d i n g to t h i s t r a n s i t i o n do not vanish, i t i s p o s s i b l e to induce t r a n s i t i o n s between the s t a t e s |+>' and |—>' using r a d i a t i o n p o l a r i z e d i n e i t h e r the x or z d i r e c t i o n , and o s c i l l a t i n g at 2Wo. By the same reasoning i f the system i s i n a s t a t e which has a net phase coherence between |+>' and I—>', then an o s c i l l a t i n g m a g n etisation would be d e t e c t e d p o l a r i z e d i n the x and z d i r e c t i o n . Let us c o n c e n t r a t e on the case of l o n g i t u d i n a l p o l a r i z a t i o n , as e v o l u t i o n of Ix from a g e n e r a l s t a t e of |+>' and |—>' coherence i s c o m p l i c a t e d ; u n l e s s the i n i t i a l s t a t e has been prepared using a t r a n s v e r s e f i e l d at 2Wo. When c o n s i d e r i n g the induced l o n g i t u d i n a l m a g n e t i s a t i o n we can ignore the 23 presence of the middle l e v e l |0>* as i t does not c o n t r i b u t e to a s i g n a l and i s not coupled to the other l e v e l s by t h i s form of r . f . f i e l d . T h i s approximation has been known f o r many years and was f i r s t a p p l i e d to magnetic resonance by Bloom e t . a l . ( 4 ) . ( see a l s o Feynman e t . a l . ( 5 ) ). The common name f o r t h i s e f f e c t i v e two l e v e l system i s the " F i c t i t i o u s spin-1/2 approximation". The net r e s u l t i s that the system may be co n s i d e r e d as a two l e v e l system with three fundamental observables, which can be chosen to have angular momentum commutation r e l a t i o n s h i p s . The s t a t e of the system can be d e s c r i b e d as the r o t a t i o n of a v e c t o r i n a three dimensional space, and the mathematics i s e q u i v a l e n t to a spin-1/2 with an a n i s o t r o p i c gyromagnetic r a t i o , e x p e r i e n c i n g some combination of e f f e c t i v e a p p l i e d f i e l d s . I w i l l now s y s t e m a t i c a l l y develop the F i c t i t i o u s spin-1/2 approximation f o r our system. 24 B. FICTITIOUS SPIN-1/2 By assuming that the molecular frame and the l a b o r a t o r y frame, d i f f e r only by a r o t a t i o n of 90 degrees about the x a x i s , and that the e . f . g . i s a x i a l l y symmetric, the h a m i l t o n i a n takes the form: 6 a = - U ) o I * + U ) a [ ( - i ( 3 i i - ± ( I + l ) j +3.(±l+±-} ] @) 3 * ' 4 which has the f o l l o w i n g matrix r e p r e s e n t a t i o n i n the |+>, |0>, | —> b a s i s : fro.--Do • ( J a 3 (Ja 4 3 Lk 4 •Do _TL 3 LJ« J l /I 1 1. 1 @ 25 The h a m i l t o n i a n can be d i a g o n a l i z e d by a s i m i l a r i t y t r a n s f o r m a t i o n : T = K S 1 a b [\2 XL* 8=f 1+ 1 ( U . -h-Joo + J1X")' IT XL XL 2. If we a l s o apply a l o n g i t u d i n a l f i e l d o s c i l l a t i n g at 2Wo, the t o t a l h a m i l t o n i a n i n the primed b a s i s i s : -•wTTsiF + ( J i C O S ( ZLdot) e Y e - K S - K S ( ( J o A > V i L * ' )((Jb--Jub*+ JL 7 ) (A!) The terms e and d are small compared with /\jL0o+ s\?~ and can be ignored i n what f o l l o w s . I f the s t a t e s of i n t e r e s t o n l y i n v o l v e coherence between the hig h e s t and lowest energy e i g e n v e c t o r s , the d e n s i t y matrix may be expressed as a combination of the f o l l o w i n g m a t r i c e s : X* =.1 A o I x = l A -r l i = I A i I 26 We can remove components of the h a m i l t o n i a n matrix which commute with the above m a t r i c e s , as these components do not c o n t r i b u t e to the e v o l u t i o n of the system. The h a m i l t o n i a n can be w r i t t e n in terms of the primed o p e r a t o r s as: As the primed 'operators have angular momentum commutation r e l a t i o n s h i p s the system may be v i s u a l i s e d as a f i c t i c i o u s s p i n -1/2, experieYicing a s t a t i c magnetic f i e l d , and a t r a n s v e r s e o s c i l l a t i n g f i e l d of s t r e n g t h W1. The s o l u t i o n may be w r i t t e n down by i n s p e c t i o n f o r an i n i t i a l c o n d i t i o n of the form: (pco)= 1+ ^ d i l i i - % , y , « ar\d (?<t>= X-r ( a* COS( 2 0 , C l y SlW( AU. t)) l a i + Qx( Ix CO'bCXUo-L) -r Xi SlW^U.'-fc)) +-( Oy COS(xU y at)5lW ( A . U . t ) ) x(iy cosdu't) + i x siiMUcJo't)). a '={ijf+i? @ If the i n i t i a l s t a t e of the system i s taken to be the thermal e q u i l i b r i u m s t a t e i n the absence of time v a r y i n g f i e l d s we may w r i t e : Trae FOR B U » « 1 Trae- 6 *-O i ^ f l B A l J o (A?) The primed frame o p e r a t o r s can be expressed i n terms of l a b o r a t o r y angular momentum o p e r a t o r s by a p p l y i n g the 27 transformat i o n : T I j T , T l y T = - l T V y T L T = L)° X* +• IL T V - y A or A ej; T I x T = OJo TV-y* - II. Xz A U ' A O , ' The r e l a t i o n s h i p between the two systems can be c o n v e n i e n t l y represented as i n - f i g . ( 3 ) . The s o l u t i o n of the e v o l u t i o n problem a f t e r the l o n g i t u d i n a l p u l s e i s represented as a v e c t o r A p r e c e s s i n g about the Iz' a x i s . The p r o j e c t i o n of t h i s motion on the Iz a x i s r e s u l t s i n a l i n e a r l y p o l a r i z e d l o n g i t u d i n a l m agnetisation o s c i l l a t i n g at 2Wo'. E v o l u t i o n d u r i n g the pulse can be r e presented i n an i n t e r a c t i o n frame which r o t a t e s at the frequency of the a p p l i e d r . f . I t should be noted that the A r o t a t i n g frame Iy' i s not p r o p o r t i o n a l to the l a b o r a t o r y frame pure double quantum coherence, and only f o r s p e c i a l values of r . f . f i e l d can the system be transformed from a s t a t e of I z ' to pure double quantum coherence. More s u c c i n c t l y , the primed frame i n t e r a c t i o n p i c t u r e t r a n s f o r m a t i o n i s not a r o t a t i o n when represented in the l a b o r a t o r y frame. We can now put i n some t y p i c a l v a l u e s to get an idea of the s i g n a l s expected. The i n t e r a c t i o n p i c t u r e d e n s i t y matrix a f t e r a l o n g i t u d i n a l pulse of l e n g t h T i s : (?<:T) = fi( I - A j x _ . ( C 0 5 ( Z U ) . teJ) l a + S l N U U i t f e T ) ±y)) @ > 28 F i g u r e 3 - R e l a t i o n s h i p between the primed and the unprimed frames The primed frame can be obtained from the l a b o r a t o r y frame by r e f l e c t i o n s i n the Ix, Tx-y, plane f o l l o w e d by a r o t a t i o n about Txy by: g'TfliyPViL \ 29 The maximum p r o j e c t i o n of o s c i l l a t i n g z magnetisation i s : i i imox) = zn SIM ( 3 . ( J i X e T ) £ iq ) I f Da = 3e*qQ- = 3.TT A 3, x IO* r o d s"1 and (Jo = A T T A / 6-q x 10* rod s"X , the l e n g t h of a 90 degree pulse i s 80 microseconds f o r e x p e r i m e n t a l l y a t t a i n a b l e values of W1. The magnitude of the magnetisation, compared to the t r a n s v e r s e magnetisation induced, a f t e r a 90 degree p u l s e at the Larmor frequency, i s reduced by the f a c t o r %(do~15o" ' F o r a r b i t r a r y o r i e n t a t i o n of the p r i n c i p a l axes: SI = 5Sll\A(3) e'gq 4 4 1 ( 7 1 - 1 ) and the h a m i l t o n i a n w i l l c o n t a i n o p e r a t o r s which do not admix the |0> s t a t e i n t o |+>' and |~>', but these admixtures a f f e c t the magnitude of the matrix element of i n t e r e s t v i a n o r m a l i z a t i o n of the perturbed wave f u n c t i o n , and the change amounts to a few percent which can be d i s r e g a r d e d . I f the c a l c u l a t i o n i s done f o r the most g e n e r a l h a m i l t o n i a n , equation ( 9 ) , the r e s u l t s are e s s e n t i a l l y the same but the r e l a t i o n s h i p between the primed and l a b o r a t o r y frames i s a more gen e r a l r o t a t i o n . Because the t r a n s f o r m a t i o n between the two frames pr e s e r v e s the phase r e l a t i o n s h i p between the e x c i t a t i o n and the response, the ..result "of any experiment i s i n v a r i a n t to r o t a t i o n s of the c r y s t a l about the l a b z a x i s . 30 C. PREPARATION OF DOUBLE QUANTUM COHERENCE P r e p a r a t i o n of a s t a t e of double quantum coherence can be achieved i n three ways. (1) A l o n g i t u d i n a l pulse at 2Wo. (2) Two hard 90 degree p u l s e s . (3) A s o f t double quantum p u l s e . Method (1) has been d i s c u s s e d i n the pre v i o u s s e c t i o n and I w i l l now d e s c r i b e the other methods i n d e t a i l (see Pines e t . a l . ( 6 ) ) . D. TWO HARD 90 DEGREE PULSES In t h i s case the e v o l u t i o n i s examined i n the hard t r a n s v e r s e p u l s e regime, which means that the e x c i t a t i o n i s a p p l i e d i n the t r a n s v e r s e plane at the larmor frequency, and the st r e n g t h of the r . f . f i e l d i s much l a r g e r than the quadrupole s p l i t t i n g . I t i s l e g i t i m a t e to ignore e v o l u t i o n due to the quadrupole i n t e r a c t i o n d u r i n g the p u l s e s , and a p u l s e may be t r e a t e d as an instantaneous r o t a t i o n about an a x i s s p e c i f i e d by the phase of the e x c i t a t i o n . What we seek i s a s o l u t i o n to the L i o u v i l l e equation with an i n i t i a l c o n d i t i o n of the form, <o(o)^(I-&(JoIi) , and a h a m i l t o n i a n c o n s i s t i n g of a 90 degree p u l s e about the x a x i s , f o l l o w e d by e v o l u t i o n f o r time ,T, under a s e c u l a r quadrupole h a m i l t o n i a n , and then another 90 degree p u l s e along the +x a x i s . The pu l s e s are a p p l i e d at the Larmor frequency and we can sol v e the problem i n an i n t e r a c t i o n frame which r o t a t e s at Wo around the z a x i s . In t h i s r o t a t i n g frame the p u l s e s l o s e t h e i r time dependence and the equations of motion can be simply i n t e g r a t e d : 31 = fl(l+UW COS(<J aT)± a +SlN(UaT) Txy)) @ ) Kt So the system can be taken from a s t a t e of Zeeman order to a s t a t e of pure double quantum coherence; p r o v i d e d c J a T _ - ^ - . - The e v o l u t i o n of the system a f t e r the p r e p a r a t i o n of double quantum coherence i s e s s e n t i a l l y the same as when the system has been prepared v i a a l o n g i t u d i n a l p u l s e ; there are important d i f f e r e n c e s which are d i s c u s s e d i n the s e c t i o n on powders. E. THE SOFT DOUBLE QUANTUM PULSE P r e p a r a t i o n by a s o f t double quantum pu l s e i s not q u i t e so s t r a i g t f o r w a r d , but under the c o n d i t i o n s l i s t e d below the dynamics i s e x a c t l y s o l u b l e as the ham i l t o n i a n can be d i a g o n a l i s e d a n a l y t i c a l l y f o r a r b i t r a r y values of r . f . f i e l d and quadrupole frequency. The problem i s so l v e d under the assumptions that the pulse i s a p p l i e d i n the t r a n s v e r s e plane at the Larmor frequency, and that the quadrupole ha m i l t o n i a n i s s e c u l a r : 6 - - - W o I » + fcfcL(3±i--l(±+l))-XCJACOS(0,t)±x @ A f t e r making the r o t a t i n g wave approximation and t r a n s f e r r i n g to a r o t a t i n g frame, the ha m i l t o n i a n becomes time independent .and i s given by: 6i = U a ( 3 ± £ - I ( ± + l ) ) - U i l * -(34) E i g e n s t a t e s f o r the i n t e r a c t i o n p i c t u r e h a m i l t o n i a n i n both the presence, and absence of the r . f . f i e l d are given i n f i g ( 4 ) . 32 Fi g u r e 4 - E i g e n s t a t e s i n the r o t a t i n g frame A) Shows the e i g e n s t a t e s of the r o t a t i n g frame h a m i l t o n i a n i n the absence of an a p p l i e d r . f . f i e l d . B) Shows the energy l e v e l diagram i n the presence of an r . f . f i e l d . (See page 31) l + > l " > ~J— (x3a 3 a. 3 _ I 0 > 3 6Jo . -h S 6 3> 6= (cJa + ecoi") 1. A A #=10(3 T z z A 33 The i n t e r a c t i o n p i c t u r e e i g e n s t a t e s are expressed i n terms of angular momentum s t a t e s q u a n t i z e d along the z a x i s as: i i>=A( i+>-n -> + e i o > ) e l ' , iz> =_i_(i +>—i-))q 1 fx 3 > = B ( n - > + i - > - ^ i o>)e l < J 3' t 6= \/ (Jo" +9 0)1*" -H U)a , 3= (Ja. - y/ bJo + A = (*-t-eA)~* , E>=(x+J31]^ Ux - - (Ja -t-1 / iJa+9tJ* f-6Jz»(Ja , Ui = - u)g - 1 V (Jg + 0 U& The dynamical problem i s now s o l v e d , and e v o l u t i o n from any i n i t i a l s t a t e can be determined by a T=0 expansion of the i n i t i a l s t a t e i n terms of |1>, |2>, |3>. The e i g e n s t a t e s i n the absence of the a p p l i e d f i e l d are | +>, |0>, |—> and i t i s convenient to have these s t a t e s expanded i n terms of |1>, |2>, |3>: I+> = AI±> + 1_IX>+BI3> , l-? = flll>-J_+ B|3> ft fX lO> = Aei±> + B : B l 3 > (36) From an i n i t i a l s t a t e Iz=\+><+1-><— | the system i s d e s c r i b e d at time T by: QLt.)- I a (ft C05[ U,~Uz)t + QX COS ( - U)3) € J + fiT I y (9 A*S I W( UJa- lj,)-t 1- 3 B*Si M ( U i - U $ + "CC fxy ( 2 f t 5 " 5 l N ( U 3 - U , ) t -t-2.3ZSIN((Ji-[Ji)t)-t-^ fx Z(eA 2COS((J J.-(J 1)t + 5&:iCOS(6Ji-iJv^ — ® — Passage from the i n i t i a l s t a t e to eqn. (37) i s simple but tedi o u s a l g e b r a ; expansion of ^(+) i n c a r t e s i a n o p e r a t o r s d e f i n e d on page(9) can be performed by expanding the fundamental 34 o b s e r v a b l e s i n t h e |+>, |0>, |—> b a s i s . The s o l u t i o n ( 3 7 ) i s f a i r l y c o m p l i c a t e d f o r a r b i t r a r y W1 a n d Wq, b u t s i m p l i f i e s c o n s i d e r a b l y when W1<<Wq o r W1>>Wq. F o r W1>>Wq: a n d t h i s i s n o t h i n g m o r e t h a n t h e h a r d p u l s e a p p r o x i m a t i o n w h e r e e v o l u t i o n u n d e r t h e r . f f i e l d i s t r e a t e d a s a r o t a t i o n . When W 1 « W q : a n d t h e s y s t e m p r e c e s s e s f r o m a s t a t e o f I z t o a s t a t e o f d o u b l e q u a n t u m c o h e r e n c e , t h e a m p l i t u d e s o f t h e s t a t e s i n v o l v i n g c o h e r e n c e b e t w e e n |0> a n d |+> o r |0> a n d |—> v a n i s h e s w i t h W1/Wq. To g e t a b e t e r u n d e r s t a n d i n g o f t h e a b o v e l i m i t s we c o n s i d e r t h e e v o l u t i o n o f a s y s t e m p r e p a r e d i n a s t a t e o f p ( o ) = | + > < + | a n d e x a m i n e t h e p r o b a b i l i t i e s t o be i n o t h e r s t a t e s a s a f u n c t i o n o f t i m e f i g ( 5 ) . F o r t h e h a r d p u l s e c a s e we s e e t h e p r o b a b i l i t y t o be i n t h e s t a t e |0> d e v e l o p s much m o r e q u i c k l y a n d t o a l a r g e r v a l u e , t h a n t h e p r o b a b i l i t y t o be i n t h e s t a t e |—>. The p r e s e n c e o f d o u b l e q u a n t u m c o h e r e n c e i s d u e t o t h e f a c t t h a t t h e i n i t i a l s t a t e h a s some s e c o n d r a n k c o m p o n e n t s , b u t t h e r e i s no c o n v e r s i o n o f s e c o n d r a n k t e n s o r s t o f i r s t r a n k t e n s o r s d u r i n g t h e e v o l u t i o n . I n t h e s o f t p u l s e l i m i t t h e p r o b a b i l i t y t o be i n t h e s t a t e |0> r e m a i n s s m a l l e v e n t h o u g h t h e l o w e s t o r d e r t r a n s i t i o n p r o b a b i l i t y f r o m |0> t o |+> i s much g r e a t e r t h a n f r o m |+> t o | - > ( i n f a c t o n e h a s t o go t o t h e n e x t o r d e r i n t i m e t o s e e a t r a n s i t i o n f r o m |+> t o |—> a s s u c h a 35 F i g u r e 5 - T r a n s i t i o n p r o b a b i l i t i e s T h e s e e q u a t i o n s d e s c r i b e t h e e v o l u t i o n o f t h e p r o b a b i l i t i e s f o r a s p i n - 1 i n i t i a l l y i n a s t a t e |+>, t o be i n t h e s t a t e s |+>, |0>, hr>. \f> = £ C i ft) I i > i= t, o , -i I C * < T ) I * = 3 + ± cos((j.t) + icos(au.+) e x 8 ICo(t) l*= 1 ( 1 - C 0 S ( A ( J . + ) ) 4 IC-W I = 3 - l C O S ( U i t ) + J-C0S(2(J.+) a . a e |C+(t)l = 0-46 +0-44COS(0-5 t ) + 0 0 3 C 0 5 ( 3 + ) + O - 0 7 C O 5 ( 3 - 5 i ) ICo't) ix= a i l ( l - C O S ( 3 t ) ) |C-(+) l*= 0-4-6-0-44COS(0-5t) +• 0 - 0 3 C 0 S ( 3 t ) - O-O ? C O S (3-5 t ) 36 F i g u r e 5 co n t i n u e d Graphs of the t r a n s i t i o n p r o b a b i l i t i e s d e s c r i b e d by the equations on the pr e v i o u s page. A) Shows the case of no quadrupole i n t e r a c t i o n or a l a r g e r . f . Zeeman i n t e r a c t i o n compared to the quadrupole i n t e r a c t i o n . B) Shows the case where the r . f . Zeeman i n t e r a c t i o n i s comparable to the quadrupole i n t e r a c t i o n . a= lC*W|l, b=lc.o(0|\ c = \ C - ( + ) r * A 0.0 1.2 2.4 - 3.6 4.8 6.0 Time 37 process i s a two photon p r o c e s s ) . The amplitude to be i n |0> never develops because the i n t e r a c t i o n s c o n s i d e r e d are h i g h l y n o n l i n e a r and one must add c o n t r i b u t i o n s from many multiphoton processes to compute amplitudes to be i n any given s t a t e at any time. The frequency of the a p p l i e d r . f . i s not commensurate with e i t h e r the |0> to |+> or the 10> to |-r> t r a n s i t i o n and the phase of the zero l e v e l changes r e l a t i v e to the r . f . , thus the sum of processes c o n t r i b u t i n g to the amplitude of the |0> s t a t e i n t e r f e r e d e s t r u c t i v e l y . For the two photon process connecting |+> to |—> the c o n t r i b u t i o n s add i n phase, even though such processes are mediated by the amplitude to be i n the |+> s t a t e . The phase d i f f e r e n c e i n going from |+> to |—> i s e x a c t l y c a n c e l l e d by passage from 10> to |-r>. F. THE EVOLUTION OF DOUBLE QUANTUM COHERENCE Using the d e n s i t y matrix f o r m a l i s i m I w i l l now look at the development of l o n g i t u d i n a l magnetisation from a system prepared in a s t a t e of double quantum coherence. As we are c o n s i d e r i n g l o n g i t u d i n a l magnetisation at 2Wo, p a r t s of the h a m i l t o n i a n which couple s t a t e s d i f f e r i n g by m=1 or m=-1 are ignored. T r u n c a t i o n of the h a m i l t o n i a n i n t h i s way does l i t t l e to e f f e c t matrix elements of l z between |+>' and |-> '. The h a m i l t o n i a n i s a s p e c i a l i z a t i o n of eqn.(10) to an a x i a l l y symmetric e . f . g . , but the c a l c u l a t i o n d e s c r i b e s the essence of the non a x i a l l y symmetric case as w e l l : £ = - y H i t + U a J J J O C O S ^ p J - l j f z z + J S i N Z f B ) ( C O S ( 2 « x ) f x ' V + SlW(2<x)f xy) " (j(o)oc I + a f xy (£d) 3 8 S u c c e s s i v e commutators of H with the i n i t i a l c o n d i t i o n shows which L i o u v i l l e subspace w i l l s u f f i c e to d e s c r i b e the dynamics. I t i s convenient to do the c a l c u l a t i o n s i n a r e f e r e n c e frame i n which the s p i n c o o r d i n a t e s are r o t a t e d by an angle of. about the A A A. z a x i s . The dynamical subspace l z , Txy, Tx-y, i s l e f t i n v a r i a n t under t h i s t r a n s f o r m a t i o n and the ha m i l t o n i a n assumes the form: /\+- -A A y\ ft /Kit 3_ (3 C O S ( 6 ) - l ) T ^ z + _ 3 _ T x y LN 2. f2" ^lo) = O i ( o ) I z + C U ( o ) T x 1 - y 1 +-Qs(o)Txy •<Q> The g e n e r a l s o l u t i o n of the L i o u v i l l e equation i s : (>(t)'= Q, (+) l z + C\x(t) f xy + Q3(t)f x'-f' QM) = Tra(X* (3(t))- ( J o ( UoX-Q^Q)-tSlQUo)) + f Q, (o)/1 - U)o l c^S ( .1uU) + JL-Qii^ SiM(2(Jot) V OJ^iL1" a ( J o W - f l - L I ((Jo ' i-iL 1)/ t J o X + i I 1 J VcJoSit1-cuc t ) = a»co) cos(ii^'t)-VuL±JL-(a'W i - (J° . \ - .a ( J ° C K M swdu'-t) - - ( 4 . 5 1 ) — • If we l i m i t the i n i t i a l c o n d i t i o n to the form: (D (O) = Qxlo) TV-y1- + C h ( o ) Txy A A and c o n s i d e r only the e x p e c t a t i o n value of l z " =Iz we f i n d : 39 - Q. (t)^ Qi(o)Jl (Jo fl-C0SUa't)) + XL CU(o) SiW(iU't) (u)oVii z) v VLJO +xil D u r i n g the t r a n s v e r s e e x c i t a t i o n of a powder sample, a l l c r y s t a l l i t e s a r e t r e a t e d as h a v i n g s e c u l a r h a m i l t o n i a n s . The A i i n i t i a l s t a t e of the i t h c r y s t a l l i t e a f t e r p r e p a r a t i o n i s Txy, where the s p i n c o o r d i n a t e system i s the same f o r a l l i . The h a m i l t o n i a n f o r a c r y s t a l l i t e w i t h an o r i e n t a t i o n w . r . t . the l a b frame d e f i n e d by the E u l e r a n g l e c< (we a r e o n l y concerned w i t h the f i n a l r o t a t i o n about the l a b z a x i s , pg. (I *?)) i s g i v e n i n terms of the i t h s p i n c o o r d i n a t e s a s : 6-Z = r\*(<*0|^-yH±k + - L J a ( ( 3 C O S Z ( p ) - l ) T i z + 3. SIf\J2((3) Txy ) " f t z f r f i ) @ A. and the e x p e c t a t i o n v a l u e of I z f o r the whole powder i s g i v e n by: I f we assume the d i s t r i b u t i o n t o be c o n t i n u o u s : < I* >- J Tra ( e R i W T*yR*(«)e FUU) I a ) p<*) do*. (g) p ( ( * ) i s the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n f o r o<. , and i f p(e*0 i s symmetric about z e r o : IT 0=J(Ra(-<) T x y fta(«0) pc*)cU (g) -Tr We see t h a t i n the case of a powder eqn.(47) h o l d s and the s i g n a l w i l l v a n i s h f o r e x c i t a t i o n by e i t h e r a s o f t double 40 quantum pulse or two hard 90 degree pulses. If longitudinal e x c i t a t i o n i s used then the phase factor i s of no consequence, as the i n i t i a l condition for each c r y s t a l l i t e depends on the c r y s t a l l i t e orientation in such a way as to maintain the, excitation-response, phase rela t i o n s h i p constant over the whole powder. 41 G. THE SECOND ORDER SHIFT I f the sample i s a p e r f e c t c r y s t a l with m a g n e t i c a l l y e q u i v a l e n t n u c l e i , the spectrum of f r e q u e n c i e s a s s o c i a t e d with the m=2 t r a n s i t i o n i s a s i n g l e l i n e . The width of t h i s l i n e can be a t t r i b u t e d to a wide v a r i e t y of s p i n - s p i n r e l a x a t i o n processes, some of which w i l l be d e s c r i b e d i n the experimental s e c t i o n . For the case of an i s o t r o p i c powder with one quadrupole nucleus per u n i t c e l l , which e x p e r i e n c e s an a x i a l e . f . g . , the spectrum i s inhomogeneously broadened by the o r i e n t a t i o n a l l y dependent second order s h i f t . The i n t e n s i t y of the powder p a t t e r n i s dependent on the e f f e c t i v e gyromagnetic r a t i o , the number of s p i n s with a given frequency and the d u r a t i o n of the e x c i t a t i o n . The most convenient way to express the i n t e n s i t y i s as a f u n c t i o n of |3 , the angle between the e . f . g . symmetry a x i s and the s t a t i c f i e l d : . The degree of e x c i t a t i o n a c r y s t a l l i t e r e c e i v e s i s a f u n c t i o n not only of the a p p l i e d r . f . f i e l d but a l s o o r i e n t a t i o n , hence a uniform i n i t i a l c o n d i t i o n f o r a l l c r y s t a l l i t e s i s impossible to a c h i e v e . T h i s i s due to the angular dependence of the of the spectrum a l s o has a small dependence on the s t r e n g t h of the e x t e r n a l s t a t i c f i e l d , and the s t r e n g t h d u r a t i o n product, W1T, of the e x c i t a t i o n . The i m p o s s i b i l i t y of a c h i e v i n g a o e f f e c t i v e gyromagnetic r a t i o Ye • For a given sample the shape 42 uniform i n i t i a l condition over the whole spectrum also has some important consequences as i t means the Fourier transform of the free induction decay i s no longer equivalent to the high temperature magnetic s u s c e p t i b i l i t y . The th e o r e t i c a l powder pattern i s given in f i g ( 6 ) ; the main point about the spectrum i s that i t has sharp edges which, due to experimental considerations, are the only features which can be measured accurately. 43 F i g u r e 6 - The second order s h i f t powder p a t t e r n The frequency i s normalized to the second order s h i f t at 90 degrees to remove the f i e l d dependence of the spectrum. For t h i s diagram Ti=0, the n o n a x i a l l y symmetric spectrum has i n t e n s i t y spread over a wider range of frequency. 1.300 1.114 -0.928 -'175 0.742 C Q) 0.557 0.571 -0.185 0.000 0.00 0.42 0.84 1.26 Normalized frequency 44 IV. EXPERIMENTAL The equipment r e q u i r e d to do the experiments i s a standard m u l t i n u c l e a r spectrometer. The gyromagnetic r a t i o of the l o n g i t u d i n a l magnetisation i s small and the spectrometer must have, low n o i s e d e t e c t i o n c i r c u i t r y , and a t r a n s m i t t e r capable of d e l i v e r i n g at l e a s t 1kw of r . f . power. If the double quantum coherence i s e x c i t e d by a t r a n s v e r s e r . f . f i e l d of frequency Wo, and the s i g n a l d e t e c t e d at 2Wo , the spectrometer must pr o v i d e both of these f r e q u e n c i e s . The s i g n a l s are small and have to be c o h e r e n t l y averaged to get good s i g n a l to noise s p e c t r a . T h i s means that the phase r e l a t i o n s h i p s between, Wo, and 2Wo, must be s t a b l e . The only way to achieve the r e q u i r e d s t a b i l i t y i s to c r e a t e the 2Wo frequency by d o u b l i n g the Wo frequency. The output of the doubler must be f i l t e r e d to remove any components at Wo, because any coherent s i g n a l at t h i s frequency present at the , r e f e r e n c e input of the phase s e n s i t i v e d e t e c t o r , can l e a d to d e t e c t i o n of s i n g l e quantum processes at the Larmor frequency. The experiments should be done at the h i g h e s t value of e x t e r n a l f i e l d , Bo, because although the e f f e c t i v e gyromagnetic r a t i o of the t r a n s i t i o n magnetic moment i s i n v e r s e l y p r o p o r t i o n a l to Bo, both the p o p u l a t i o n d i f f e r e n c e between the l e v e l s and the v o l t a g e induced in the d e t e c t i o n c i r c u i t , i n c r e a s e l i n e a r l y with Bo. 45 A. PROBE The probe used f o r experiments r e p o r t e d here i s u n s o p h i s t i c a t e d by modern NMR standards. The c i r c u i t i s a s e r i e s tuned LC c i r c u i t with no matching c a p a c i t o r , matching being performed by a d j u s t i n g the c o i l geometry. In order to study the angular dependence of the s i g n a l s the probe has a goniometer, which allows r o t a t i o n about an a x i s p e r p e n d i c u l a r to the a p p l i e d f i e l d with a r e s o l u t i o n of 1 degree. B. COILS The c o i l geometry ( f i g u r e 7) used f o r the experiments i s one of two b a s i c types depending on the nature of the e x c i t a t i o n . For l o n g i t u d i n a l e x c i t a t i o n a saddle c o i l with i t s t r a n s v e r s e a x i s a l i g n e d along the e x t e r n a l f i e l d was used f o r both e x c i t a t i o n and d e t e c t i o n . A saddle c o i l i s i n f e r i o r to a s o l e n o i d a l c o i l i n a l l f e a t u r e s except f i l l i n g f a c t o r , and the reason i t was used was because i t allowed easy r o t a t i o n of the sample about an a x i s p e r p e n d i c u l a r to the e x t e r n a l f i e l d . For t r a n s v e r s e e x c i t a t i o n and l o n g i t u d i n a l d e t e c t i o n a c r o s s e d c o i l arrangement was used. The t r a n s v e r s e e x c i t a t i o n c o i l was s o l e n o i d a l , and the d e t e c t i o n c o i l was a saddle c o i l mounted c o a x i a l l y with the s o l e n o i d . The saddle c o i l should be mounted on the i n s i d e of the s o l e n o i d to improve the d e t e c t i o n c i r c u i t f i l l i n g f a c t o r and decrease the c a p a c i t i v e c o u p l i n g to noise sources. To prevent a r c i n g d u r i n g the intense t r a n s v e r s e p u l s e , the c o i l s had to be separated by at l e a s t 2mm of t e f l o n . The probe c i r c u i t r y a s s o c i a t e d with the d e t e c t i o n i s almost 46 F i g u r e 7 - C o i l s A ) Shows t h e t w o c o i l a r r a n g e m e n t w i t h t h e s a d d l e c o i l s e p a r a t e d f r o m t h e s o l e n o i d by a t e f l o n s h r o u d . B) Shows t h e o r i e n t a t i o n o f t h e l o n g i t u d i n a l e x c i t a t i o n - d e t e c t i o n c o i l w . r . t t h e e x t e r n a l f i e l d . B) 4 7 as simple as that a s s o c i a t e d with the e x c i t a t i o n , with the i n c l u s i o n of a matching c a p a c i t o r . I t i s very d i f f i c u l t to b u i l d a high Q d e t e c t i o n c i r c u i t as the c a p a c i t i v e c o u p l i n g between the c o i l s causes c o n s i d e r a b l e l o s s e s , even though the f i e l d s produced by the two are o r t h o g o n a l . These p o i n t s p r e j u d i c e any d e c i s i o n to use a c r o s s c o i l c o n f i g u r a t i o n e s p e c i a l l y when account i s taken of the i n t r i n s i c a l l y poor s i g n a l to n o i s e . The nature of the experiment i s such that the high power components are i n v a r i a b l y pushed past t h e i r l i m i t and these components should be easy to r e p l a c e . C. SAMPLES I t i s u s e f u l to review p o i n t s c o n s i d e r e d i n chosing a sample to demonstrate the theory, as these g i v e a good i n d i c a t i o n of the types of systems amenable to t h i s technique. The s i g n a l of i n t e r e s t i s s m a l l as the , e f f e c t i v e gyromagnetic r a t i o i s only a s m a l l f r a c t i o n of the i n t r i n s i c nuclear gyromagnetic r a t i o . T h i s f r a c t i o n i n c r e a s e s l i n e a r l y with the quadrupole c o u p l i n g constant and we need a sample with a l a r g e quadrupole i n t e r a c t i o n . A s m a l l s i g n a l means we have to achieve a h i g h d e n s i t y of n u c l e i , and the need to s i g n a l average r e q u i r e s that the s p i n l a t t i c e r e l a x a t i o n time be s h o r t . Phase coherence w i t h i n the ensemble double quantum coherence should be maintained as long as p o s s i b l e , to allow narrow banding of the l o n g i t u d i n a l s i g n a l and to accommodate the narrow power spectrum of the e x c i t a t i o n . The major broadening mechanisims are h e t e r o n u c l e a r d i p o l a r c o u p l i n g , ( i n those systems c o n t a i n i n g p r o t o n s ) , and the second order quadrupole s h i f t i n imperfect 48 c r y s t a l s . The magnetic i n t e r a c t i o n s , such as d i p o l e - d i p o l e c o u p l i n g , are e s p e c i a l l y damaging as the double quantum coherence dephases at twice the r a t e of the s i n g l e quantum coherence under an inhomogeneous Zeeman term. Nitrogen-14 was the nucleus of c h o i c e because of a high n a t u r a l abundance and a l a r g e quadrupole i n t e r a c t i o n . The best r e s u l t s have been obtained on sodium n i t r i t e , . NaN02, which i s an i o n i c s o l i d a v a i l a b l e as a l a r g e s i n g l e c r y s t a l . The c r y s t a l has one n i t r i t e per u n i t c e l l and the n i t r o g e n nucleus experiences a quadrupole c o u p l i n g constant G QQ- =5.8flhz, and an asymmetry n parameter n=0.4. The s p i n l a t t i c e r e l a x a t i o n time i s 1.5 seconds at room temperature. Sodium N i t r i t e i s a f e r r o e l e c t r i c c r y s t a l , and the o s c i l l a t i n g e l e c t r i c f i e l d produced by the high power e x c i t a t i o n induces p i e z o e l e c t r i c r i n g i n g , which o b l i t e r a t e s the s i g n a l f o r 300 microseconds a f t e r the p u l s e . P i e z o e l e c t r i c r i n g i n g i n NaN02 has been s t u d i e d by Hughes(7) and v a r i o u s methods of overcoming the problem suggested. I used an echo technique to observe the s i g n a l , the pulse spacing being chosen to ensure the echo formed o u t s i d e the r i n g i n g . The n i t r o g e n s a t e l l i t e l i n e s were s t u d i e d and a rough d e t e r m i n a t i o n of the o r i e n t a t i o n of the e . f . g . with respect t o the c r y s t a l h a b i t made ( F i g . ( 8 ) ) . An a c c u r a t e d e t e r m i n a t i o n of the r o t a t i o n p a t t e r n s was not attempted as the goniometer d i d not possess the r e q u i r e d r e p r o d u c i b i l i t y or r e s o l u t i o n . The s a t e l l i t e l i n e s were inhomogeneously broadened by an imperfect c r y s t a l s t r u c t u r e , the broadening was o r i e n t a t i o n a l l y dependent and ranged from 4 to 50 k i l o h e r t z . 49 A 30 us i 1 B 50 khz F i g u r e 8 - S a t e l l i t e l i n e s A) Shows the f . i . d . of a s a t e l l i t e l i n e u s i n g a Hahn echo sequence. B) The F o u r i e r transform of the above echo with the zero of time taken to be the echo peak. 50 Although complete r o t a t i o n diagrams were not measured the o r i e n t a t i o n of the z a x i s of the e . f . g . was determined by f i n d i n g the p o s i t i o n which gave the maximum s e p a r a t i o n of the s a t e l l i t e l i n e s , c o n s i s t a n t with the known valu e s f o r Q C\Q-h and Ti. The s t r e n g t h ^ of the t r a n s v e r s e r . f . . f i e l d was determined by f i n d i n g the time r e q u i r e d f o r a 90 degree r o t a t i o n of the magnetisation a s s o c i a t e d with a s a t e l l i t e l i n e . With the dual c o i l arrangement the s h o r t e s t time I c o u l d achieve was 8.5 microseconds. A f a c t o r of 1.4 was gained from the l a r g e r e f f e c t i v e gyromagnetic r a t i o of a s a t e l l i t e examined in i s o l a t i o n . To e s t a b l i s h the decay time of the s i g n a l due to homogeneous (non-refocusable) processes a normal Hahn echo was performed on the s a t e l l i t e , the r e l a x a t i o n time of the echo amplitude was 1.5 m i l l i s e c o n d s , although there was some angular dependence of t h i s r e l a x a t i o n time. A f t e r the p r e l i m i n a r y i n v e s t i g a t i o n s were f i n i s h e d the c r y s t a l was mounted a few degrees o f f the magic angle, where the s e p a r a t i o n of the quadrupole doublet was 400 k i l o h e r t z . The system was then i r r a d i a t e d with a t r a n s v e r s e f i e l d with ^ n p -=25KHZ, and frequency Wo. The l e n g t h of pulse which t r a n s f e r s the maximum amount of Zeeman order i n t o double quantum coherence can be computed using the s o f t double pulse approximation, and i s given by T= =80us. The r e s u l t i n g l o n g i t u d i n a l s i g n a l , f i g ( 9 ) , was averaged using an add s u b t r a c t r o u t i n e , i n which the phase of s u c c e s s i v e p u l s e s i s s h i f t e d by 90 degrees, and the s i g n a l a l t e r n a t e l y added and s u b t r a c t e d from the a c q u i s i t i o n memory. The double quantum coherence precesses at twice the 51 F i g u r e 9 - Transverse e x c i t a t i o n A) Shows the l o n g i t u d i n a l f . i . d . f o l l o w i n g a t r a n s v e r s e double quantum p u l s e . 52 i n t e r a c t i o n frequency of the Zeeman h a m i l t o n i a n , hence a change of phase by 90 degrees at the Larmor frequency changes the phase of the double quantum coherence and l o n g i t u d i n a l m a g n i t i s a t i o n by 180 degrees. The l o n g i t u d i n a l p u l s e was a p p l i e d using a saddle c o i l tuned to 33.6fthz, and the value of r . f . f i e l d produced was t e s l a ( f i g . ( 1 0 ) ) . The resonance frequency can be found by scanning over the second order s h i f t frequency range. T h i s range i s narrow enough, approximately 200khz, to r e q u i r e a minimum r e t u n i n g of the probe and the spectrometer c h a r a c t e r i s t i c s remain unchanged over t h i s range. When sear c h i n g f o r a resonance i t must be remembered that the c r y s t a l may be o r i e n t e d at an angle where the matrix element of 'iz va n i s h e s . I f the quadrupole c o u p l i n g constant i s known i t i s b e t t e r to set the spectrometer frequency to a value where the corresponding gyromagnetic r a t i o , i s l a r g e , and then search f o r the resonance by r o t a t i n g the c r y s t a l . The second order quadrupole s h i f t r o t a t i o n p a t t e r n s have been p l o t t e d f o r two orthogonal o r i e n t a t i o n s of the c r y s t a l f i g ( 1 1 ) . One has the z a x i s of the e . f . g . p e r p e n d i c u l a r to the r o t a t i o n a x i s , the other has the z a x i s p a r a l l e l . E x p r e s s i o n s f o r the resonance frequency and the e f f e c t i v e gyromagnetic r a t i o when r\.^0 are given by: " 3 . S I N A ( B ) ( 4 - C O S ^ ) -f±) + T l C O S J ( K ) 5 l N \ 6 ) ( 4 C O S i ( ( 3 ) - l ) Do - 3 U)o + Uo. 3.4- LJo 3 + n 1 (z - 4-cos\xp)(cos\$) - i f j J3Siw a(:B)+ncosttir)(s^ 53 F i g u r e 1 0 - L o n g i t u d i n a l e x c i t a t i o n The l o n g i t u d i n a l s i g n a l has been e x c i t e d and d e t e c t e d using a saddle c o i l with the o r i e n t a t i o n given i n f i g - 7 b on page 44. 5 4 34.015 33.860 0 30 60 90 Rotation angle y (°) F i g u r e 11 - R o t a t i o n p a t t e r n R o t a t i o n p a t t e r n w i t h t h e z a x i s of t h e e . f . g . o r i e n t e d p e r p e n d i c u l a r t o t h e r o t a t i o n a x i s . 5 5 33.875 33.850 N O c CD 3 cr o 33.825 33.800 33.775 33.750 1 1 J 1 1 1 J « -0 - 4 0 80 120 160 Rotation angle (3 (°) F i g u r e 11 c o n t i n u e d R o t a t i o n p a t t e r n w i t h t h e z a x i s o f t h e e . f . g . p a r a l l e l t o t h e r o t a t i o n a x i s 56 The magnitude of tfe appears as i n g e n e r a l , #e =<1|IZ|^1>, w i l l be a complex number. The phase f a c t o r i s a f u n c t i o n of the E u l e r angle and i t has no e f f e c t i n the l o n g i t u d i n a l e x c i t a t i o n experiment. When n f O the e x p r e s s i o n s f o r the resonance frequency and the e f f e c t i v e gyromagnetic r a t i o are c o m p l i c a t e d , but i n p r i n c i p a l (1) can be used to f i n d a l l f i v e components of the e . f . g tensor i f r o t a t i o n p a t t e r n s about three orthogonal axes are measured. Important p o i n t s concerning (2) are that Ye does not v a n i s h f o r any o r i e n t a t i o n when and that a s i g n a l can be observed even when z a x i s of the e . f . g . i s p a r a l l e l to the e x t e r n a l f i e l d . D. COMPARISON BETWEEN THE EXCITATION METHODS For powder samples the only e x c i t a t i o n method which r e s u l t s in an observable s i g n a l i s the l o n g i t u d i n a l p u l s e . L i m i t a t i o n s on systems which can be s t u d i e d are determined by a comparison of the d u r a t i o n of the e x c i t a t i o n with the T a of the induced double quantum coherence. The d u r a t i o n of the pulse r e q u i r e d to t r a n s f e r an a p p r e c i a b l e amount of Zeeman order i n t o double quantum coherence i s i n v e r s e l y p r o p o r t i o n a l to the quadrupole c o u p l i n g c o n s t a n t , and l i n e a r i n the Zeeman i n t e r a c t i o n , hence i t i s the r a t i o of Wq to Wo which i s the important parameter. The optimal experimental arrangement i s a s o l e n o i d a l c o i l o r i e n t e d along the z d i r e c t i o n , as the s o l e n o i d o f f e r s the best performance and the sample does not r e q u i r e r o t a t i o n . For s i n g l e c r y s t a l s i t i s more d i f f i c u l t to c h a r a c t e r i z e which of the three d i f f e r e n t e x c i t a t i o n methods should be used, as there are many f a c t o r s which i n v o l v e spectrometer 57 c a p a b i l i t i e s and sample p r o p e r t i e s . The best approach i s to e x p l a i n the s t r e n g t h s and weaknesses of each method and make comparisons only of a general nature. The hard pulse e x c i t a t i o n method i s good because the t y p i c a l time f o r the e x c i t a t i o n sequence can be 10-20 microseconds, and the s p i n system can be sub j e c t e d to a wide range of pul s e sequences. The disadvantage i s that the f i r s t order quadrupole s p l i t t i n g has to be w e l l w i t h i n the power spectrum of the e x c i t a t i o n , l i m i t i n g t h i s method to systems with small quadrupole i n t e r a c t i o n s or angles c l o s e to the magic angle. The s o f t double quantum pulse i s u s e f u l f o r an inte r m e d i a t e range of quadrupole s p l i t t i n g s . If the s p l i t t i n g i s too small the s o f t p u l s e approximation breaks down and i f the s p l i t t i n g becomes too l a r g e , >1Mhz, the l o n g i t u d i n a l pulse becomes more e f f e c t i v e . The main disadvantage of the s o f t pulse i s t h a t i t can not be used f o r angles c l o s e to the magic angle and, as i n the case of the hard pulse method, r e q u i r e s a two c o i l system. The l o n g i t u d i n a l p u l s e becomes p r a c t i c a l at quadrupole c o u p l i n g c o n s t a n t s >500khz; i t works over a much l a r g e r range of c r y s t a l o r i e n t a t i o n s and r e q u i r e s only a s i n g l e c o i l . A s i n g l e c o i l system has tremendous s i g n a l to nois e advantages as there i s no t r a n s v e r s e c o i l c o u p l i n g i n noise and the sample f i l l i n g f a c t o r can be op t i m i s e d . 58 E. APPLICATIONS I must now make some statements on the u s e f u l n e s s of these experiments, compared to other resonance methods which can y i e l d the same i n f o r m a t i o n . The primary m o t i v a t i o n f o r these experiments was an attempt to view systems i n the absence of the f i r s t order quadrupole i n t e r a c t i o n . The c e n t r a l aim was to observe f e a t u r e s normally obscured by the f i r s t order spectrum, such as d i p o l e - d i p o l e i n t e r a c t i o n s . The d i s c u s s i o n again d i v i d e s i n t o powders and s i n g l e c r y s t a l s . For powders the only method of e x c i t a t i o n i s the l o n g i t u d i n a l , p u l s e , and t h i s l i m i t s the samples to those with quadrupole c o u p l i n g c o n s t a n t s >500khz. For t h i s s i z e of i n t e r a c t i o n the second order quadrupole i n t e r a c t i o n w i l l dominate the magnetic i n t e r a c t i o n s . The values of r . f . f i e l d a t t a i n a b l e make i t impossible f o r the power spectrum of a 90 degree pulse to cover the whole spectrum, thus i t i s only p o s s i b l e to observe a p o r t i o n of t h i s spectrum in one experiment. A l l i s not l o s t as the powder p a t t e r n spectrum does possess d i s t i n c t i v e edges and f i n d i n g these can l e a d to a d e t e r m i n a t i o n of the quadrupole c o u p l i n g constant and the asymmetry parameter. I t must be remarked that these q u a n t i t i e s are more e a s i l y and more a c c u r a t e l y a t t a i n e d by NQR. As a s p e c t r o s c o p i c technique the l o n g i t u d i n a l ' pulse f a r e s p o o r l y , because the lineshape i s determined by the power spectrum of the p u l s e , but i t may be used to get some l i m i t e d i n f o r m a t i o n on the s p i n l a t t i c e r e l a x a t i o n r a t e . For s i n g l e c r y s t a l s the prospects are a l i t t l e b r i g h t e r , as 59 c o n v e n t i o n a l NMR has to focus on the quadrupole doublet l i n e s i n i s o l a t i o n when the s p l i t t i n g i s l a r g e . The angular dependence of the s a t e l l i t e t r a n s i t i o n f r e q u e n c i e s v a r i e s over many times the spectrometer bandwidth, and to produce r o t a t i o n p a t t e r n s the spectrometer needs frequent r e t u n i n g . The l o n g i t u d i n a l e x c i t a t i o n and d e t e c t i o n method makes i t unnecessary to know the frequency of the s a t e l l i t e l i n e s and mapping out the second order s h i f t p a t t e r n i s very easy. If the sample has a w e l l c h a r a c t e r i s e d , quadrupole c o u p l i n g constant and asymmetry parameter, i n f o r m a t i o n on magnetic i n t e r a c t i o n s such as a n i s o t r o p i c chemical s h i f t s can be measured. In the case of an imperfect c r y s t a l i t i s o f t e n d i f f i c u l t to determine the exact frequency of a s a t e l l i t e l i n e , whereas the Am=2 t r a n s i t i o n i s broadened only i n second order and the c h a r a c t e r of s m a l l magnetic i n t e r a c t i o n s can be found. In summary I do not t h i n k the technique i s capable of g i v i n g i n f o r m a t i o n which i s not a l r e a d y a v a i l a b l e by c o n v e n t i o n a l NMR or NQR, but the experiments d e s c r i b e d i n t h i s t h e s i s are easy to do f o r samples with l a r g e quadrupole i n t e r a c t i o n s , and f o r the l o n g i t u d i n a l pulse method, they r e q u i r e l i t t l e more than t u r n i n g the c o i l of a c o n v e n t i o n a l probe to l i e p a r a l l e l to the z d i r e c t i o n . F. FURTHER APPLICATIONS There are some extensions of the ideas presented i n t h i s t h e s i s which deserve mention. Most n i t r o g e n samples have protons near to the n i t r o g e n nucleus and h e t e r o n u c l e a r d i p o l e broadening can be p r o h i b i t i v e . The problem can be overcome by d e c o u p l i n g the protons by i r r a d i a t i n g them with a resonant r . f . 60 f i e l d many times the s t r e n g t h of the c o u p l i n g . The best setup would be a t r a n s v e r s e s o l e n o i d a l proton c o i l mounted on the i n s i d e of a saddle c o i l which i s tuned to twice the n i t r o g e n larmor frequency. The ideas are a l s o a p p l i c a b l e to n u c l e i of higher s p i n , and i t may be f r u i t f u l to search f o r unconventional resonances i n sp i n 3/2-nuclei such as L i 7 , C135, Na23. The admixture terms are l i m i t e d to those due to a second rank nonsecular hamiltonian and f o r m>2 the e f f e c t i v e gyromagnetic r a t i o s a s s o c i a t e d with the p a r t i a l l y f o r b idden t r a n s i t i o n s s c a l e as VkJ&)n. , where n i s an i n t e g e r g r e a t e r than 1. t h i s means the samples would have to have very l a r g e values of Wq to warrant study by these methods. 61 BIBLIOGRAPHY 1) S . Y a t s i v , Phys. Rev. n_3 , 1522. (1959). 2) A.Abragam, The P r i n c i p l e s of Nuclear Magnetism. Oxford Univ. Press, London. (1961) 3) G o l d s t e i n , C l a s s i c a l Mechanics (page 107). Addison Wesley Pub. (1959) 4) M.Bloom, L.B.Robinson< and G.M.Volkoff, Can.J.Phys 36 1286 (1958). 5) R.P.Feynman, F.L.Vernon, and R.W.Hellworth, J.Appl.Phys 28 , 49 (1957). 6) S.Vega, and A.Pines, J.Chem.Phys., 66 5624 (1977). 7) D.G.Hughes and Lakshman Pandy, J.Magnetic Resonance 5_6 428 (1984). 62 APPENDIX A ~ ANGULAR MOMENTUM Whenever a complex many bodied system i s s t u d i e d from the po i n t of view of quantum mechanics, one must i n v a r i a b l y make approximations concerning the i n i t i a l s t a t e and the h a m i l t o n i a n . For a completely i s o l a t e d system the h a m i l t o n i a n w i l l be i n v a r i a n t to a r b i t r a r y r o t a t i o n s and time t r a n s l a t i o n s . These i n v a r i a n c e s r e s u l t i n the c o n s e r v a t i o n of the magnitude of the angular momentum, the z p r o j e c t i o n of the angular momentum w.r.t. any a x i s , and the i n t e r n a l energy. The symmetry groups and c o n s e r v a t i o n laws n a t u r a l l y p a r t i t i o n i n t o two s e t s , one set co n t a i n s o p e r a t i o n s on v a r i a b l e s of the c l o s e d system, the other d e a l s s t r i c t l y with the world o u t s i d e the system. The i n t r o d u c t i o n of a c o u p l i n g between the system and the r e s t of the world upsets t h i s p a r t i t i o n , and some of the o p e r a t i o n s of the o r i g i n a l set are no longer symmetry o p e r a t i o n s unless they are performed as part of a simultaneous t r a n s f o r m a t i o n of the system and the o u t s i d e world. If the i n t r o d u c t i o n of v a r i a b l e s from the r e s t of the world complicates the problem u n n e c e s s a r i l y , they can be excluded and r e p l a c e d by t h e i r e x p e c t a t i o n value over some p r e s c r i b e d s t a t e . T h i s replacement of v a r i a b l e s i n the h a m i l t o n i a n by a number means a l o s s of c o n s e r v a t i o n laws, f o r example the very presence of a c o u p l i n g with the o u t s i d e world means that the energy of the system w i l l not be a constant f o r a l l p o s s i b l e motions. The accuracy of an approximation which excludes a f u l l dynamical d e s c r i p t i o n depends on the c o u p l i n g being small compared to the i n t e r n a l 63 h a m i l t o n i a n , and on the r e s t of the world being i n a s t a t e which i s not measurably a f f e c t e d by the motions of the system of i n t e r e s t . With these statements i n mind, l e t us co n s i d e r the c o n s e r v a t i o n of momentum and energy as a p p l i e d to our s p i n system, which experiences a nonsecular quadrupole ha m i l t o n i a n and a l o n g i t u d i n a l o s c i l l a t i n g f i e l d . If we imagine an experiment i n which the system, assumed to be composed of the nucle a r s p i n s and the r . f . f i e l d , i s taken from an i n i t i a l A A. s t a t e Iz to a s t a t e - I z ; we see that the z component of angular momentum i s not conserved (a l o n g i t u d i n a l f i e l d can c a r r y no angular momentum w.r.t. the z a x i s ) . T h i s i s the most s t r i k i n g d i f f e r e n c e from normal NMR, where the system c o n s i s t s of the spi n s and a t r a n s v e r s e f i e l d at Wo. The ha m i l t o n i a n i s i n v a r i a n t to a r o t a t i o n of both the s p i n c o o r d i n a t e s and f i e l d c o o r d i n a t e s about the z d i r e c t i o n (conserving the p r o j e c t i o n of angular momentum on t h i s a x i s ) . To re g a i n t h i s c o n s e r v a t i o n law for the nonsecular case enough of the r e s t of the world must be co n s i d e r e d u n t i l the ha m i l t o n i a n regains a x i a l symmetry. We begin with the f o l l o w i n g h a m i l t o n i a n : H ( x t a l ) i s an approximation which t r e a t s the c r y s t a l as an ob j e c t with s i x degrees of freedom. T h i s approximation i s v a l i d because the motions i n which we w i l l be i n t e r e s t e d i n v o l v e very small energy changes, and the f r e q u e n c i e s of the other degrees of freedom are r a p i d enough to f o l l o w the changes, m a i n t a i n i n g an e q u i l i b r i u m . If we ignore t r a n s l a t i o n and c o n s i d e r the na 64 A c r y s t a l to be u n r e s t r a i n e d the e i g e n s t a t e s of H ( x t a l ) w i l l be f u n c t i o n s c h a r a c t e r i s e d by good quantum numbers w.r.t. angular momentum. The c o o r d i n a t e system f o r H ( x t a l ) has i t s z a x i s p a r a l l e l to the main magnetic f i e l d . The energy l e v e l diagram of H ( x t a l ) c o n s i s t s of very c l o s e l y spaced l e v e l s with energy: 3"(r+i) - o~=o,i,--- © I i s , the moment of i n e r t i a of the c r y s t a l about the z a x i s ; i^.«(Jo w e c a n a s s u m e the c r y s t a l to be s p h e r i c a l . The A A energy l e v e l s of Wolz+WqTzz are once again s t a t e s of good angular momentum i f the q u a n t i z a t i o n a x i s i s chosen to l i e along the main magnetic f i e l d . The energy l e v e l spacing of the s p i n degrees of freedom i s very much l a r g e r than the r o t a t i o n a l energy l e v e l s p a c i n g . Once again we are only i n t e r e s t e d i n l o n g i t u d i n a l e x c i t a t i o n at 2Wo and I w i l l l i m i t the d i s c u s s i o n to l e v e l s coupled by such a p e r t u r b a t i o n . In the absence of spi n l a t t i c e c o u p l i n g the e i g e n f u n c t i o n s are tensor products of A A A e i g e n f u n c t i o n s of, Wolz+WqTzz, and H ( x t a l ) . The p e r t u r b a t i o n x -x —X x iX has the form, C ( R B -t A 6 ) , where g>" are s p i n o p e r a t o r s 1 , o - \ xx r and the f\ are o p e r a t o r s a c t i n g on the o r b i t a l angular momentum s t a t e s and transform as second rank s p h e r i c a l t e n s o r s . The t r u e p e r t u r b a t i o n i s one i n v o l v i n g a' p a r t i c u l a r nucleus and the e l e c t r o n s i n i t ' s immediate v i c i n i t y , however we are assuming that the exact H ( x t a l ) has r o t a t i o n a l i n v a r i a n c e , ( d u e to the c o u p l i n g between degrees of freedom the r o t a t i o n must be a simultaneous t r a n s f o r m a t i o n of a l l degrees of freedom except spin) and we may use the Wigner theorem to 65 r e p l a c e the e l e c t r o n operator with an operator p o s s e s s i n g the same r o t a t i o n a l p r o p e r t i e s . A p p l y i n g p e r t u r b a t i o n theory using the product f u n c t i o n s as zero order f u n c t i o n s the f o l l o w i n g p o i n t s a r i s e . ( 1 ) The p e r t u r b a t i o n can not couple degenerate l e v e l s . ( 2 ) The energy d i f f e r e n c e between s t a t e s which are coupled i s to a very good approximation, equal to the energy d i f f e r e n c e of the spin degrees of freedom alon e . The zero order f u n c t i o n s of i n t e r e s t are given by: l l > - | + > ) ^ m I-l>=|-> f « which to f i r s t order i n C becomes: I 1>'= l+>/m + C |-> ( Q f i m A + b/'mVz + C ^ L l + 6 f n A + ) EL+ - E-i-i> = i->/£ - c i+> (-f fk-x + g y>i*-i + h fLx + i + j j _ E-The c o e f f i c i e n t s a-j a r i s e from c a l c u l a t i n g the r e s u l t s f o r s p e c i f i c L and m and may be zero i n some cases. A t r a n s i t i o n from a s t a t e | 1 > ' to a s t a t e [ - 1 > ' , with L=L ' + 2 and m=m '-2, i s induced by a l o n g i t u d i n a l f i e l d . Table ( 2 ) shows the changes i n angular momentum which occur. As the s p i n system i s d r i v e n by the r . f . f i e l d a torque i s exerted on the l a t t i c e . The change in z p r o j e c t i o n of angular momentum of the s p i n system i s accompanied by a compensating change i n the angular momentum of the l a t t i c e . For a r e a l experiment the c r y s t a l has a well d e f i n e d o r i e n t a t i o n , hence the s t a t e of the l a t t i c e c o n s i s t s of 66 I n c l u s i o n of the l a t t i c e s t a t e s does l i t t l e to a f f e c t the matrix elements of a spi n o perator and such a comprehensive treatment i s unnecessary except f o r angular momentum c o n s i d e r a t i o n s . 67 Table II - Angular momentum changes d u r i n g a |-1> to |1> t r a n s i t i o n IL>= l + > VV-A + C |-l>^ Q|//m- -rbfn' + Cy - f - C l J / m ' + e J^m' j i-> f i - g g ( f f t : * 9 f z * y z > * « ^ K £ ) l-l> m' + C(m'- 2) - i - t - C E*-E-m-2. -+ Con' 1 - C E*-E- E*-E-

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