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UBC Theses and Dissertations

Nuclear magnetic resonance in inhomogeneous magnetic fields Norwood, Timothy John 1985

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NUCLEAR MAGNETIC RESONANCE IN INHOMOGENEOUS MAGNETIC FIELDS by TIMOTHY JOHN NORWOOD B.Sc. (Hons.), U n i v e r s i t y of London, 1983 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Ch e m i s t r y ) We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1985 © Timothy John Norwood, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C H E M I S T R Y The University of B r i t i s h Columbia 1956 Main Mall Van couve r, Canada V6T 1Y3 D a t e 19 SEPTEMBER 1785 i i A B S T R A C T The work d e s c r i b e d i n t h i s t h e s i s was i n i t i a t e d i n an attempt t o overcome the l i m i t a t i o n s imposed upon NMR s p e c t r o s c o p y by magnetic f i e l d inhomogeneity i n two s p e c i f i c a r e a s : h i g h r e s o l u t i o n s p e c t r o s c o p y i n i s o t r o p i c l i q u i d s , and c h e m i c a l s h i f t r e s o l v e d NMR imaging i n i s o t r o p i c l i q u i d s . In b o th c a s e s magnetic f i e l d inhomogeneity may degrade the r e s o l u t i o n of s p e c t r a t o such an e x t e n t t h a t no u s e f u l i n f o r m a t i o n can be o b t a i n e d from them. In h i g h r e s o l u t i o n NMR s p e c t r o s c o p y i t i s n e c e s s a r y t o be a b l e t o e x t r a c t a c c u r a t e l y the parameters p r e s e n t w i t h i n the spectrum such as c h e m i c a l s h i f t s , c o u p l i n g c o n s t a n t s and peak a r e a s . In c h e m i c a l s h i f t r e s o l v e d imaging experiments the r e q u i r e m e n t s are l e s s s t r i n g e n t ; and i t i s o n l y n e c e s s a r y t h a t the resonances of d i f f e r e n t c h e m i c a l s p e c i e s be r e s o l v e d . However, even the l e s s s t r i n g e n t r e q u i r e m e n t s of NMR imaging are o f t e n d i f f i c u l t t o meet as the sample volumes r e q u i r e d a r e o f t e n s e v e r a l o r d e r s of magnitude l a r g e r than those r e q u i r e d i n c o n v e n t i o n a l h i g h r e s o l u t i o n NMR s p e c t r o s c o p y . The use of zero-quantum coherence has been i n v e s t i g a t e d as a p o t e n t i a l s o l u t i o n t o the magnetic f i e l d inhomogeneity problem i n both of these a r e a s . Zero-quantum coherences a r e independent of magnetic f i e l d inhomogeneity and c o n t a i n the parameters d e s i r e d i n both c a s e s , though they a r e d i s p l a y e d i n a way which d i f f e r s from c o n v e n t i o n a l NMR s p e c t r a . In t h i s t h e s i s , e x i s t i n g zero-quantum coherence e x p e r i m e n t s have been e v a l u a t e d f o r use w i t h inhomogeneous magnetic f i e l d s , and, where n e c e s s a r y , adapted f o r t h i s purpose. S e v e r a l c o m p l e t e l y new e x p e r i m e n t s have been developed f o r p r o d u c i n g broad-band d e c o u p l e d zero-quantum coherence s p e c t r a and a l s o f o r p r e s e n t i n g c o u p l i n g c o n s t a n t s and c h e m i c a l s h i f t s i n a manner which i s as c l o s e t o c o n v e n t i o n a l NMR s p e c t r a as p o s s i b l e , hence f a c i l i t a t i n g ease of use. Zero-quantum coherence has been e v a l u a t e d as a t o o l f o r i d e n t i f y i n g unknown compounds and a l s o f o r i d e n t i f y i n g the components of complex m i x t u r e s by " s i g n a t u r e " r e c o g n i t i o n . Both d e c o u p l e d and non-decoupled zero-quantum coherence e x p e r i m e n t s are adapted t o p r o v i d e imaging e x p e r i m e n t s which a l l o w the s e p a r a t i o n of the images of d i f f e r e n t c h e m i c a l s p e c i e s i n inhomogeneous magnetic f i e l d s . The t w o - d i m e n s i o n a l J - r e s o l v e d experiment i s a l s o adapted f o r t h i s purpose. i v T A B L E O F C O N T E N T S Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i CHAPTER I - INTRODUCTION 1 1.1 - Magnetic F i e l d Inhomogeneity 2 1.2 - M u l t i p l e - Q u a n t u m Coherence 4 1.3 - The Produc t O p e r a t o r F o r m a l i s m 10 - R e f e r e n c e s 15 CHAPTER I I - NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY IN AN INHOMOGENEOUS MAGNETIC FIELD 17 2.1 - Zero-Quantum S p e c t r o s c o p y i n an Inhomogeneous Ma g n e t i c F i e l d 18 2.2 - E d i t i n g Zero-Quantum Coherence S p e c t r a 30 2.3 - N o n - S e l e c t i v e E x c i t a t i o n and D e t e c t i o n of Zero-Quantum Coherence i n an Inhomogeneous M a g n e t i c F i e l d 31 2.4 - Homonuclear Broad-Band Decoupled Zero-Quantum Coherence S p e c t r o s c o p y 36 2.5 - Single-Quantum J - R e s o l v e d Broad Band Decoupled Zero-Quantum S p e c t r o s c o p y 48 V 2.6 - The R e c o n s t r u c t i o n of Single-Quantum S p e c t r a i n an Inhomogeneous Magnetic F i e l d 61 2.7 - The Assignment of Zero-Quantum Coherence S p e c t r a i n Inhomogeneous Magnetic F i e l d s 66 2.8 - The A n a l y s i s of M i x t u r e s i n an Inhomogeneous Magnetic F i e l d by the R e c o g n i t i o n of the Zero-Quantum Coherence S i g n a t u r e s of T h e i r C o n s t i t u e n t s 74 2.9 - E x p e r i m e n t a l 82 - R e f e r e n c e s 84 CHAPTER I I I -NUCLEAR MAGNETIC RESONANCE IMAGING IN AN INHOMOGENEOUS MAGNETIC FIELD 87 3.1 - N u c l e a r Magnetic Resonance Imaging 88 3.2 - Chemical S h i f t R e s o l v e d Imaging 92 3.3 - Zero-Quantum Coherence R e s o l v e d Imaging 95 3.4 - Broad-Band Decoupled Zero-Quantum Coherence R e s o l v e d Imaging 105 3.5 - J - R e s o l v e d Imaging 114 3.6 - E x p e r i m e n t a l 121 - R e f e r e n c e s 122 CHAPTER IV - CONCLUSION 126 v i APPENDIX I - PRODUCT OPERATOR EVOLUTION 130 A1.1 - I n t r o d u c t i o n t o the E v o l u t i o n of Product O p e r a t o r s 131 A1.2 - Chemical S h i f t E v o l u t i o n 132 A1.3 - S p i n - S p i n C o u p l i n g E v o l u t i o n 133 A1.4 - Radio-Frequency P u l s e E v o l u t i o n 133 A1.5 - Zero-Quantum Coherence E v o l u t i o n 134 - R e f e r e n c e s 135 APPENDIX I I - ZERO-QUANTUM SPECTRA OF AMINO ACIDS 137 V I 1 LIST OF TABLES TABLE I Phase c y c l i n g schemes f o r m u l t i p l e - q u a n t u m coherence o r d e r s e l e c t i o n TABLE I I E f f e c t s of the p u l s e a=90° on the pr o d u c t o p e r a t o r s p r e s e n t a t the end of the e v o l u t i o n time t ^ of the broad-band de c o u p l e d zero-quantum experiment f o r an AX 2 s p i n - s y s t e m . TABLE I I I E f f e c t s of an a r b i t r a r y p u l s e a on the p r o d u c t o p e r a t o r s p r e s e n t a t the end of the e v o l u t i o n time trj of the broad-band de c o u p l e d zero-quantum experiment f o r an AX 2 s p i n - s y s t e m . v i i i LIST OF FIGURES Page CHAPTER I F i g u r e 1.1. S p i n v e c t o r and r o t a t i n g frame r e p r e s e n t a t i o n s of an ensemble of s p i n s a t t h e r m a l e q u i l i b r i u m and as si n g l e - q u a n t u m coherence. 6 F i g u r e 1.2. Energy diagram f o r an AX s p i n - s y s t e m showing the phase coherences which a r e p o s s i b l e . 8 F i g u r e 1.3. Pr o d u c t o p e r a t o r s and t h e i r c o r r e s p o n d i n g r o t a t i n g frame r e p r e s e n t a t i o n s . 13 CHAPTER I I F i g u r e 2.1. B a s i c zero-quantum ex p e r i m e n t . 19 F i g u r e 2.2. B a s i c zero-quantum experiment w i t h magnetic f i e l d g r a d i e n t o r d e r s e l e c t i o n . 19 F i g u r e 2.3. R e f o c u s s e d zero-quantum e x p e r i m e n t . 23 F i g u r e 2.4. Zero-quantum s p e c t r a o f , A. L - a l a n i n e , B. L - t h r e o n i n e , C. L - v a l i n e , D. L - a s p a r a g i n e , i n d i v i d u a l l y ' i n a homogeneous magnetic f i e l d , and E. t o g e t h e r i n an inhomogeneous magnetic f i e l d . 28 F i g u r e 2.5. Single-quantum s p e c t r a of a m i x t u r e of L - a l a n i n e , L - t h r e o n i n e , L - v a l i n e and L - a s p a r a g i n e , A. i n homogeneous and B. inhomogeneous magnetic f i e l d s . 29 F i g u r e 2.6. Zero-quantum s p e c t r a of L - t h r e o n i n e o b t a i n e d w i t h d i f f e r e n t l e n g t h s of p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s w i t h the r e f o c u s s e d zero-quantum e x p e r i m e n t . - 32 F i g u r e 2.7. R e f o c u s s e d zero-quantum experiment w i t h A. an a c c o r d i o n p r e p a r a t i o n p e r i o d , and B. w i t h a c c o r d i o n p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s . 34 i x F i g u r e 2.8. Zero-quantum s p e c t r a of L - t h r e o n i n e o b t a i n e d w i t h the r e f o c u s s e d zero-quantum experiment w i t h A. an a c c o r d i o n p r e p a r a t i o n p e r i o d , and B. w i t h a c c o r d i o n p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s . " 35 F i g u r e 2.9. Broad-band d e c o u p l e d zero-quantum e x p e r i m e n t . 38 F i g u r e 2.10. Zero-quantum s p e c t r a of a m i x t u r e of e t h a n o l and 2 - p r o p a n o l , A. undecoupled, B-D. d e c o u p l e d , w i t h a=90°. 43 F i g u r e 2.11. Graph of zero-quantum coherence i n t e n s i t y v s . l e n g t h of e v o l u t i o n time f o r e t h a n o l and 2-propanol w i t h the broad-band d e c o u p l e d zero-quantum exp e r i m e n t . 45 F i g u r e 2.12. Broad-band d e c o u p l e d zero-quantum s p e c t r a of e t h a n o l and 2-propanol w i t h a=45°. 48 F i g u r e 2.13. Single-quantum J - r e s o l v e d broad-band d e c o u p l e d zero-quantum e x p e r i m e n t . 55 F i g u r e 2.14. Zero-quantum s p e c t r a of L - a l a n i n e , A. undecoupled, B. d e c o u p l e d , C. o b t a i n e d w i t h the si n g l e - q u a n t u m J - r e s o l v e d broad-band d e c o u p l e d zero-quantum e x p e r i m e n t . 58 F i g u r e 2.15 Single-quantum J - r e s o l v e d broad band d e c o u p l e d zero-quantum spectrum of e t h a n o l . 58 F i g u r e 2.16. Single-quantum J - r e s o l v e d broad-band d e c o u p l e d zero-quantum s p e c t r a of 2 - p r o p a n o l , A. w i t h a=45°, B. w i t h a=22.5°. 59 F i g u r e 2.17. S p e c t r a of L - t h r e o n i n e , A. s i n g l e - q u a n t u m J - r e s o l v e d broad-band decoupled spectrum, B. d i a g r a m a t i c r e p r e s e n t a t i o n of the method of r e c o n s t r u c t i o n of s i n g l e - q u a n t u m s p e c t r a from t h o s e o b t a i n e d i n the s i n g l e - q u a n t u m J - r e s o l v e d broad-band decoupled zero-quantum e x p e r i m e n t . C. R e c o n s t r u c t e d s i n g l e - q u a n t u m s p e c t r a of L - t h r e o n i n e from the spectrum i n p a r t A. D. Normal s i n g l e - q u a n t u m spectrum of L - t h r e o n i n e . 64 X F i g u r e 2.18. S p e c t r a of a m i x t u r e of L - v a l i n e , L - l e u c i n e , L - a s p a r a g i n e , L - a s p a r j t i c a c i d , and L - i s o l e u c i n e . A. Single-quantum spectrum. B. Zero-quantum spectrum w i t h T,T'=60 msec. C. zero-quantum spectrum w i t h T,T'=140 msec. 77 F i g u r e 2.19. Broad-band d e c o u p l e d zero-quantum s p e c t r a o f : A. L - v a l i n e , L - l e u c i n e , L - a s p a r a g i n e , L - a s p a r t i c a c i d and L - i s o l e u c i n e , B. L - v a l i n e , C. L - l e u c i n e , D. L - a s p a r a g i n e , E. L - a s p a r t i c a c i d , F, L - i s o l e u c i n e . 80 CHAPTER I I I F i g u r e 3.1. A. One g r a d i e n t p r o j e c t i o n imaging e x p e r i m e n t . B. R e p r e s e n t a t i o n of a water phantom. C. R e p r e s e n t a t i o n of a p r o j e c t i o n of the phantom (B) o b t a i n a b l e w i t h the one g r a d i e n t p r o j e c t i o n e x p e r i m e n t . 90 F i g u r e 3.2. Two-dimensional s p i n - d e n s i t y imaging experiment 93 F i g u r e 3.3. C h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t . 93 F i g u r e 3.4. Zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t . 98 F i g u r e 3.5. A. Phantom of 2 - p r o p a n o l , e t h a n o l and water. B. Zero-quantum coherence r e s o l v e d image of the phantom ( A ) . C. Zero-quantum spectrum of the phantom ( A ) . D. Single-quantum spectrum of the phantom ( A ) . 101 F i g u r e 3.6. D a t a s e t s o f : A. c h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t , B. Zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t . 104 F i g u r e 3.7. Broad-band d e c o u p l e d zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t . 1 07 F i g u r e 3.8. Images of a phantom of X 1 e t h a n o l , 2 - p r o p a n o l , and water: A. undecoupled, B-D. d e c o u p l e d . E. Single-quantum s p e c t r a of phantom. 109 F i g u r e 3.9. Zero-quantum s p e c t r a c o r r e s p o n d i n g t o the images i n F i g u r e 3.8. p a r t s A-D. 112 F i g u r e 3.10. A. R e p r e s e n t a t i o n of the phantom used t o o b t a i n the images i n F i g u r e 3.8. B-I s l i c e s taken t h r o u g h the images i n F i g u r e 3.8 A-D. 113 F i g u r e 3.11. A. Homonuclear t w o - d i m e n s i o n a l J - r e s o l v e d e x p e r i m e n t . B. Homonuclear t w o - d i m e n s i o n a l J - r e s o l v e d imaging e x p e r i m e n t . 116 F i g u r e 3.12. 3 1 P s p e c t r a : A. J-spectrum of ATP. B. J- s p e c t r u m of ADP. C. Single-quantum spectrum of ATP. D. Single-quantum spectrum of ADP. 117 F i g u r e 3.13. 3 1 P J - r e s o l v e d image of ADP and ATP. 120 APPENDIX I I F i g u r e A2.1. Zero-quantum s p e c t r a of L - g l u t a m i n e . 139 F i g u r e A2.2. Zero-quantum s p e c t r a of L - m e t h i o n i n e . 139 F i g u r e A2.3. Zero-quantum s p e c t r a of L - g l u t a m i c a c i d . 140 F i g u r e A2.4. Zero-quantum s p e c t r a of L - a s p a r a g i n e . 140 F i g u r e A2.5. Zero-quantum s p e c t r a of L-0-phenyl a l a n i n e . 141 F i g u r e A2.6 Zero-quantum s p e c t r a of L - l e u c i n e . 141 F i g u r e A2.7. Zero-quantum s p e c t r a of L - t h r e o n i n e . 142 F i g u r e A2.8. Zero-quantum s p e c t r a of L - i s o l e u c i n e . 142 x i i F i g u r e A2.9. Zero-quantum spectrum of L - v a l i n e . 143 F i g u r e A2.10. Zero-quantum s p e c t r a of L - p r o l i n e . 143 F i g u r e A2.11. Zero-quantum spectrum of L - c y s t e i n e . 143 F i g u r e A2.12. Zero-quantum s p e c t r a of L - t r y p t o p h a n . 144 F i g u r e A2.13. Zero-quantum s p e c t r a of L - s e r i n e . 144 F i g u r e A2.14. Zero-quantum s p e c t r a of L - a r g i n i n e . 145 F i g u r e A2.15. Zero-quantum spectrum of L - a l a n i n e . 145 F i g u r e A2.16. Zero-quantum spectrum of L - a s p a r t i c a c i d . 145 XI 1 1 ACKNOWLEDGMENT I would l i k e t o acknowledge my deep g r a t i t u d e t o Dr. L.D. H a l l f o r the c o n s t a n t h e l p and guidance he has g i v e n me w h i l e u n d e r t a k i n g the work h e r e i n . I would a l s o l i k e t o acknowledge my g r a t i t u d e t o Mr. Jacques B r i a n d f o r proof r e a d i n g t h i s t h e s i s w i t h g r e a t speed and a t s h o r t n o t i c e and a l s o f o r h e l p f u l and e n l i g h t e n i n g d i s c u s s i o n s . I would a l s o l i k e t o thank Mr S t a n l e y Luck and Mr L a l i t h T a l a g a l a f o r h e l p f u l d i s c u s s i o n s and a s s i s t a n c e . 1 CHAPTER I * INTRODUCTION 2 1.1 Mag n e t i c F i e l d Inhomogeneity M a g n e t i c f i e l d inhomogeneity has always been a major problem i n NMR. O r i g i n a l l y the problem was c o n f i n e d t o a c u b i c c e n t i m e t e r or so of sample deep w i t h i n the bowels of a magnet, a l t h o u g h i n r e c e n t y e a r s some a p p l i c a t i o n s have n e c e s s i t a t e d i n c r e a s i n g the sample volume by s e v e r a l o r d e r s of magnitude. The degree of magnetic f i e l d homogeneity a t t a i n a b l e w i t h i n the magnet of a c o n v e n t i o n a l h i g h r e s o l u t i o n NMR s p e c t r o m e t e r has i n c r e a s e d c o n s i d e r a b l y i n r e c e n t t i m e s due t o both the improvement of magnet t e c h n o l o g y and the i m p l e m e n t a t i o n of s o p h i s t i c a t e d shimming t e c h n i q u e s . However, magnetic f i e l d inhomogeneity s t i l l remains a problem, p a r t i c u l a r l y a t h i g h f i e l d s . For example, a magnetic f i e l d i nhomogeneity over the volume of i n t e r e s t of 1 p a r t i n 10 s w i l l r e s u l t i n a l i n e w i d t h of 0.1 Hz a t 10 MHz, but a t 100 MHz the same l e v e l of homogeneity w i l l r e s u l t i n a l i n e w i d t h of 1 Hz, and a t 1,000 MHz the l i n e w i d t h w i l l be 10 Hz. C l e a r l y , the h i g h e r the magnetic f i e l d the g r e a t e r the l e v e l of homogeneity needed t o o b t a i n the same r e s o l u t i o n . T h i s i s , p e r h a p s , the major problem c o n f r o n t e d as one goes t o h i g h e r magnetic f i e l d s , c o u n t e r b a l a n c i n g t h e i r advantages of i n c r e a s e d s p e c t r a l d i s p e r s i o n and b e t t e r s i g n a l - t o - n o i s e . The o t h e r area of NMR where magnetic f i e l d inhomogeneity i s a problem, and here even a t low f i e l d s t r e n g t h s , i s NMR imaging. Here the problem eminates from the i n h e r e n t l y l a r g e volume of some of the o b j e c t s which i t i s d e s i r e d t o image, 3 such as human b e i n g s , and i s p a r t i c u l a r l y c r i t i c a l i n two c a t e g o r i e s of e x p e r i m e n t s : those which seek t o o b t a i n s p e c t r a from w i t h i n a s p e c i f i c s m a l l volume w i t h i n the sample, and those imaging experiments which a l l o w one t o e x t r a c t the image of a s p e c i f i c resonance. In both t y p e s of experiment i t i s s p e c t r o s c o p i c r a t h e r than s p a t i a l i n f o r m a t i o n which i s o f t e n f a t a l l y s u s c e p t i b l e t o magnetic f i e l d inhomogeneity. In b oth c o n v e n t i o n a l h i g h r e s o l u t i o n NMR s p e c t r o s c o p y , where one would l i k e t o go t o h i g h e r f i e l d s t r e n g t h s t o i n c r e a s e s p e c t r a l d i s p e r s i o n , and i n NMR imaging, where one would l i k e t o r e s o l v e r e l a t i v e l y w e l l s e p a r a t e d resonances a t c o m p a r a t i v e l y low f i e l d s t r e n g t h s , magnetic f i e l d inhomogeneity i s a problem, a problem which cannot c u r r e n t l y be met due t o e i t h e r t e c h n i c a l or p r a c t i c a l l i m i t a t i o n s . G i v e n t h i s r e a l i t y can the l i m i t a t i o n s i t imposes be overcome? The b a s i c r equirement f o r a s o l u t i o n t o t h i s problem would seem t o be t o make the r e l e v a n t s p e c t r o s c o p i c p a r a m e t e r s , c h e m i c a l s h i f t s and s c a l a r c o u p l i n g s , independent o f , or a t l e a s t l e s s dependent upon, magnetic f i e l d i nhomogeneity. T h i s may appear t o be w i s h f u l t h i n k i n g , but i t i s n o t ! I t has been known f o r some time t h a t among the c l a s s of phenomena known as m u l t i p l e - q u a n t u m coherence [1-4] one p a r t i c u l a r o r d e r of cohere n c e , zero-quantum coherence, w h i l e c o n t a i n i n g c h e m i c a l s h i f t and s c a l a r c o u p l i n g i n f o r m a t i o n , i s a l s o independent of magnetic f i e l d inhomogeneity. 4 The o b j e c t of t h i s t h e s i s i s t o e x p l o r e ways of overcoming the l i m i t a t i o n s imposed upon NMR by magnetic f i e l d i nhomogeneity. For the most p a r t i t i s the p o t e n t i a l i t i e s of zero-quantum coherence which are of i n t e r e s t i n t h i s r e s p e c t , b o th i n the c o n t e x t of s p e c t r o s c o p y (Chapter I I ) , and a l s o i n the c o n t e x t of imaging f o r o b t a i n i n g the s e p a r a t e d s p i n - d e n s i t y images of d i f f e r e n t c h e m i c a l s p e c i e s . ( C h a p t e r I I I ) . In both cases the scope and l i m i t a t i o n s of e x i s t i n g e x p e r i m e n t s are d i s c u s s e d , and new ones a r e d e v e l o p e d . In the l i g h t of t h e s e o b j e c t i v e s and t h e means by which i t w i l l be attempted t o meet them a q u e s t i o n must be answered: what i s m u l t i p l e - q u a n t u m coherence? 1.2 M u l t i p l e - Q u a n t u m Coherence M u l t i p l e - q u a n t u m c o h e r e n c e s are those c o h e r e n c e s which do not obey the w e l l known t r a n s i t i o n r u l e Am= ± 1. T h i s d e f i n i t i o n by i t s e l f i s not n e c e s s a r i l y v e r y e n l i g h t e n i n g as i t does not convey any i d e a as t o what, p h y s i c a l l y , a m u l t i p l e - q u a n t u m coherence i s . T h i s problem has been e x a c e r b a t e d by the l a c k of e x p l a n a t i o n p r o v i d e d by the w i d e l y used c l a s s i c a l and s e m i c l a s s i c a l v e c t o r models.-One of the e a s i e s t ways t o a t t a i n an u n d e r s t a n d i n g of the n a t u r e of m u l t i p l e - q u a n t u m coherence i s t o do so as an e x t e n s i o n of an u n d e r s t a n d i n g of the n o r m a l l y observed s i n g l e - q u a n t u m c o h e r e n c e . T h i s , d e s p i t e w i d e l y h e l d b e l i e f s t o the c o n t r a r y , i s i t s e l f o f t e n m i s u n d e r s t o o d . M i s c o n c e p t i o n s of the n a t u r e of s i n g l e - q u a n t u m cphererice u s u a l l y a r i s e from a l a c k of a p p r e c i a t i o n of the 5 l i m i t a t i o n s of the models b e i n g used t o d e s c r i b e i t , t h a t i s , where i t i s t h a t the model d e p a r t s from p e r c e i v e d r e a l i t y . F i g u r e 1.1A i s a r e p r e s e n t a t i o n of the p r e c e s s i o n of an ensemble of v e c t o r s w i t h random phase w i t h r e s p e c t t o the xy - p l a n e i n a magnetic f i e l d B Q. Each v e c t o r r e p r e s e n t s the magnetic d i p o l e of a n u c l e u s . T h i s how one might expect t o f i n d an ensemble of s p i n s a t e q u i l i b r i u m i n a magnetic f i e l d , w i t h s l i g h t l y more s p i n s a l i g n e d w i t h the f i e l d than a g a i n s t i t . F i g u r e 1.1B d e p i c t s the c o r r e s p o n d i n g r o t a t i n g frame r e p r e s e n t a t i o n . The v e c t o r r e p r e s e n t s the net mac r o s c o p i c m a g n e t i z a t i o n of the ensemble. When a 90° r a d i o - f r e q u e n c y p u l s e i s a p p l i e d t o the ensemble i t has the e f f e c t , i n the r o t a t i n g frame model, of t i p p i n g t he net m a g n e t i z a t i o n v e c t o r t h r o u g h 90° onto the y - a x i s . A common m i s c o n c e p t i o n i s t h a t a l l the i n d i v i d u a l s p i n v e c t o r s , r e p r e s e n t i n g the n u c l e a r magnetic d i p o l e moments a r e now a l l a l i g n e d a l o n g the y - a x i s . T h i s i s not the cas e . The r o t a t i n g frame model's v e c t o r a l o n g the y - a x i s a c t u a l l y r e p r e s e n t s a phase coherence between the two s t a t e s of the s p i n , w i t h and a g a i n s t the f i e l d , a and |3, i n the x y - p l a n e as can be seen from F i g u r e 1.1D. The 90° p u l s e has the e f f e c t of e q u a l i z i n g the p o p u l a t i o n s of the two s p i n s t a t e s a and j3 and c r e a t i n g a phase coherence between them w i t h r e s p e c t t o the x y - p l a n e , but w i t h more s p i n s i n phase a l o n g the +y-a x i s than the - y - a x i s . 6 B X Z -1 Y D X • Y F i g u r e 1.1. A. P r e c e s s i o n i n a magnetic f i e l d B c of an ensemble of v e c t o r s , r e p r e s e n t i n g the magnetic d i p o l e s of s p i n - 1 / 2 n u c l e i , w i t h random phase i n the x y - p l a n e , and w i t h net m a g n e t i s a t i o n o n l y i n the z - d i r e c t i o n , as one might expect t o f i n d a t t h e r m a l e q u i l i b r i u m . B. C o r r e s p o n d i n g r o t a t i n g frame r e p r e s e n t a t i o n of p a r t A, the v e c t o r r e p r e s e n t s the net m a c r o s c o p i c m a g n e t i s a t i o n . C. Phase-coherence of an ensemble of s p i n s i n the x y - p l a n e a l o n g the y - a x i s , as would be p r e s e n t a f t e r the a p p l i c a t i o n of a 90$ r a d i o - f r e q u e n c y p u l s e t o the ensemble at e q u i l i b r i u m ( p a r t A ) . D. C o r r e s p o n d i n g r o t a t i n g frame r e p r e s e n t a t i o n of p a r t C. 7 Whereas si n g l e - q u a n t u m coherences a re phase coherences between the two s t a t e s of one s p i n i n the x y - p l a n e , m u l t i p l e - q u a n t u m coherences c o n s i s t of phase c o h e r e n c e s between the a and j5 s t a t e s of a number of d i f f e r e n t c o u p l e d s p i n s i n the x y - p l a n e . U n l i k e s i n g l e - q u a n t u m c o h e r e n c e s , m u l t i p l e - q u a n t u m coherences cannot be c r e a t e d d i r e c t l y by a 90° p u l s e or d e t e c t e d d i r e c t l y [ 5 ] , as they do not co u p l e w i t h the r e c e i v e r c o i l ; t h e i r c r e a t i o n and d e t e c t i o n w i l l be d i s c u s s e d l a t e r ( s e c t i o n 2.1). The s i m p l e s t m u l t i p l e - q u a n t u m coherences o c c u r f o r an AX s p i n system, which may g i v e r i s e t o z e r o - and d o u b l e - as w e l l as s i n g l e - q u a n t u m c o h e r e n c e s , F i g u r e 1.2. From the F i g u r e i t can be seen t h a t Am=0 f o r the zero-quantum cohe r e n c e . A l t h o u g h f o r the A - s p i n AmA=1 between energy l e v e l s 2 and 3, f o r the X - s p i n Amx=-1, g i v i n g , o v e r a l l , Am=0. L i k e w i s e i t can be seen t h a t f o r the double-quantum t r a n s i t i o n Am=2. The a c t u a l p r e c e s s i o n a l f r e q u e n c y of a s p i n i n the r o t a t i n g frame may be e x p r e s s e d as co' =CJ +7AB(x,y,z) (1.1) where CJ i s the p r e c e s s i o n a l f r e q u e n c y of the s p i n i n the absence of magnetic f i e l d inhomogeneity and AB(x,y,z) i s the inhomogeneity e x p e r i e n c e d by a s p i n a t the s p a t i a l c o o r d i n a t e s x, y, and z. 8 4 j8 AjS x i 01^ SQC = Single-quantum coherence ZQC = Zero-quantum coherence DQC = Double-quantum coherence F i g u r e 1 . 2 . Energy diagram f o r an AX s p i n - s y s t e m showing the the phase coherences which a re p o s s i b l e . 9 The g e n e r a l e x p r e s s i o n f o r the c h e m i c a l s h i f t of a m u l t i p l e - q u a n t u m coherence i s g i v e n by c o p Q C = £ A m k ( t o k +7AB(x,y,z)) (1.2) where Am=±1 depending on the change of magnetic quantum number of s p i n k f o r a P-order m u l t i p l e - q u a n t u m c o h e r e n c e , where P i s g i v e n by: P = £Am k (1.3) From e q u a t i o n 1.2 i t can be seen t h a t a double-quantum coherence between two c o u p l e d s p i n s A and B w i l l p r e c e s s a t a fre q u e n c y g i v e n by: "DQCT W A + a JB + 2 7 A B ( x , y , z ) (1.4) C l e a r l y double-quantum coherence i s t w i c e as s u s c e p t i b l e t o magnetic f i e l d inhomogeneity as s i n g l e - q u a n t u m c o h e r e n c e . S i m i l a r l y the p r e c e s s i o n a l f r e q u e n c y of zero-quantum coherence may be e x p r e s s e d as UZQC= "A ~UB ( K 5 ) which i s the d i f f e r e n c e i n f r e q u e n c i e s of s p i n s A and B. Co n s e q u e n t l y as AB(x,y,z) i s e f f e c t i v e l y the same f o r s p i n s w i t h i n the same mol e c u l e the term c a n c e l s out making zero-quantum coherences independent of magnetic f i e l d 10 inhomogeneity. M u l t i p l e - q u a n t u m coherences o n l y e x h i b i t c o u p l i n g s t o s p i n s not a c t i v e w i t h i n the coheren c e , t h i s w i l l be d i s c u s s e d i n more d e t a i l l a t e r ( s e c t i o n s 2.5, A1.5). 1.3 The P r o d u c t Operator Formalism The development of m u l t i p u l s e NMR experiments i n r e c e n t y e a r s has been accompanied by the development of models w i t h which t o d e s c r i b e them. These models can g e n e r a l l y be c a t e g o r i s e d as be i n g of one of two t y p e s . One approach has been t o use c l a s s i c a l or s e m i c l a s s i c a l v e c t o r models. A l t h o u g h adequate f o r d e s c r i b i n g e x p e r i m e n t s such as s p i n - e c h o e s [ 6 ] , slow c h e m i c a l exchange [7,8] and s p i n imaging [9] t h e s e models a r e inadequate f o r more s o p h i s t i c a t e d experiments such as thos e i n v o l v i n g m u l t i p l e - q u a n t u m coherence. The o t h e r approach i s d e n s i t y o p e r a t o r t h e o r y [ 1 0 - 1 2 ] , T h i s can cope w i t h a r b i t r a r i l y complex p u l s e s e q u e n c i e s a l t h o u g h t h i s i s o f t e n a t the expense of p h y s i c a l i n t u i t i o n . The p r o d u c t o p e r a t o r approach i n t r o d u c e d by Sorensen, E i c h , L e v i t t , Bodenhausen and E r n s t i s e s s e n t i a l l y a m i d d l e road between the two b a s i c approaches r e f e r r e d t o above. A l t h o u g h o r i g i n a t i n g i n d e n s i t y o p e r a t o r t h e o r y i t r e t a i n s the i n t u i t i v e c o n c e p t s and p h y s i c a l i n s i g h t of the c l a s s i c a l or s e m i c l a s s i c a l v e c t o r models. P r o d u c t o p e r a t o r s a r e d e f i n i t i v e l y i n t r o d u c e d i n r e f e r e n c e [13] which the r e a d e r of t h i s t h e s i s i s s t r o n g l y recommended t o read. As can be seen 11 from the r e f e r e n c e , an attempt t o i n t r o d u c e p r o d u c t o p e r a t o r s a d e q u a t e l y would take a c o n s i d e r a b l e amount of space, more than i s a v a i l a b l e h e r e , t h e r e f o r e o n l y a summary of the c o n t e n t s of r e f e r e n c e [13] w i l l be g i v e n h e r e i n . ' In t h i s s e c t i o n the n a t u r e of p r o d u c t o p e r a t o r s and t h e i r o r i g i n s w i l l be d i s c u s s e d , and the r u l e s f o r t h e i r use are g i v e n i n Appendix I . A d e n s i t y o p e r a t o r o can be e x p r e s s e d as a l i n e a r c o m b i n a t i o n of base o p e r a t o r s B : These base o p e r a t o r s may, f o r example, be e x p r e s s e d as i r r e d u c i b l e t e n s o r o p e r a t o r s [14,15] o r , as w i l l be done h e r e i n , as p r o d u c t o p e r a t o r s : where N = t o t a l number of s p i n - 1 / 2 n u c l e i i n the s p i n system, k=index of n u c l e u s , v=x, y or z, q=number of s i n g l e s p i n o p e r a t o r s i n the p r o d u c t , a=1 f o r q n u c l e i and a=0 f o r N-q n u c l e i . A l t h o u g h p r o d u c t o p e r a t o r s f o r spin-1/2 n u c l e i a r e o r t h o g o n a l w i t h r e s p e c t t o t r a c e f o r m a t i o n they a r e not n o r m a l i s e d , i . e . a ( t ) = p s ( t ) B s (1.6) B (1.7) Tr{B ,B } =5 r s r , s (1.8) 1 2 The s e t of pr o d u c t o p e r a t o r s f o r a two s p i n system can be g e n e r a t e d u s i n g e q u a t i o n 1.7: q=0 (1/2)E (E=unity o p e r a t o r ) q = 1 T k x ' : k y ' 1kz' 1lx' 1ly' 1 l z *= 2 2 I k x : i x ' 2 I k x J l y ' 2 I k x I l z ' 2 I k y : i x ' 2 I k y Z l y ' 2 I k y I l z ' 2 I k z J l x ' 2 I k z I l y ' 2 I k z : i z Any d e n s i t y o p e r a t o r may be e x p r e s s e d as a l i n e a r c o m b i n a t i o n of such a s e t . Product o p e r a t o r s g r e a t l y s i m p l i f y the c a l c u l a t i o n s of p u l s e e x p e r i m e n t s a p p l i e d t o weakly c o u p l e d systems because the f a t e of i n d i v i d u a l o p e r a t o r terms can be f o l l o w e d and a s s o c i a t e d w i t h c l e a r p h y s i c a l meaning. S i n g l e - s p i n p r o d u c t o p e r a t o r s may be d e s c r i b e d as f o l l o w s : J k z J k x J k y l o n g i t u d i n a l m a g n e t i z a t i o n of s p i n k in-phase x - m a g n e t i z a t i o n of s p i n k, in-phase y - m a g n e t i z a t i o n of s p i n k, Ij^ r e p r e s e n t s m a g n e t i z a t i o n a l o n g the z - a x i s of the r o t a t i n g frame, F i g u r e 1.3.A, as one might f i n d when the system i s a t th e r m a l e q u i l i b r i u m . and 1^ c o r r e s p o n d t o the components of s p i n k which a r e i n phase a l o n g the x - a x i s and y - a x i s , F i g u r e 1.3.B, of the r o t a t i n g frame. For example, a 90° r a d i o - f r e q u e n c y p u l s e a c t s upon the s p i n k a t e q u i l i b r i u m , 1^ t o g i v e I k . 13 F i g u r e 1.3. P r o d u c t o p e r a t o r s and t h e i r c o r r e s p o n d i n g r o t a t i n g frame r e p r e s e n t a t i o n s : A. L o n g i t u d i n a l m a g n e t i z a t i o n of s p i n k. B. In-phase y - m a g n e t i z a t i o n of s p i n k. C. x - m a g n e t i z a t i o n of s p i n k a n t i p h a s e w i t h r e s p e c t t o s p i n 1. 14 Two-spin p r o d u c t o p e r a t o r s may be d e s c r i b e d as f o l l o w s : 2 I k x J l z : a n t i P n a s e x - m a g n e t i z a t i o n of s p i n k w i t h r e s p e c t t o s p i n 1, 2 I k y I l z : a n t i P n a s e y - m a g n e t i z a t i o n of s p i n k w i t h r e s p e c t to s p i n 1, 2 I k x : i x ' 2 I * y l l y > 2 1 k x T l y a n d " k y h x ' t w o - s p i n coherences of s p i n s k and 1, 2 I k z I l z : l° n9i t u c^i n al t w o - s p i n o r d e r of s p i n s k and 1. A n t i p h a s e m a g n e t i z a t i o n , 2 I ^ x ^ z f o r example, F i g u r e 1.3.C, r e p r e s e n t s two components of a m u l t i p l e t which have o p p o s i t e phases. As i t s i n t e g r a t e d i n t e n s i t y i s z e r o i t does not c o u p l e w i t h the r e c e i v e r . Two-spin coherences r e p r e s e n t a s u p e r p o s i t i o n of z e r o - and double-quantum coherence. L o n g i t u d i n a l t w o - s p i n o r d e r c o r r e s p o n d s t o s p i n - c o r r e l a t e d p o p u l a t i o n of energy l e v e l s w i t h o u t net p o l a r i z a t i o n and w i t h o u t o b s e r v a b l e m a g n e t i z a t i o n . In l a r g e r s p i n systems t h r e e s p i n terms appear: ^ I k x I l z I m z : x - m a g n e t i z a t i o n of s p i n k a n t i p h a s e w i t h r e s p e c t t o s p i n s 1 and m, ^ I k x I l x I m z : t w o s P * n coherence of s p i n s k a n d . l a n t i p h a s e w i t h r e s p e c t t o s p i n m, 41, I , I : t h r e e - s p i n coherence, kx l x mx r 4 I k z I l z I m z : l o n g i t u d i n a l t h r e e - s p i n o r d e r . A n t i p h a s e t w o - s p i n coherence c o n s i s t s of z e r o - and 1 5 double-quantum coherence w i t h m u l t i p l e t components t h a t have o p p o s i t e phases depending upon t h e p o l a r i z a t i o n of the " p a s s i v e " s p i n m. T h r e e - s p i n c o h e r e n c e c o n s i s t s of a s u p e r p o s i t i o n of si n g l e - q u a n t u m c oherence ( c o m b i n a t i o n l i n e s ) and t r i p l e - q u a n t u m coherence. The e f f e c t s of c h e m i c a l s h i f t and s c a l a r c o u p l i n g e v o l u t i o n and r a d i o - f r e q u e n c y p u l s e s can be d e s c r i b e d by a s e r i e s of t r a n s f o r m a t i o n s of the t y p e : e x p { - i * B r } B s e x p { i * B r } = £b f c s(r,$)B f c ( 1 . 9 ) where 4>Br can tak e the form ( ^ ) l T ) I j c z f ° r c h e m i c a l s h i f t p r e c e s s i o n , ( nJ^± ) 21 ^  I f o r weak s c a l a r c o u p l i n g e v o l u t i o n and 01^ f o r a r a d i o f r e q u e n c y p u l s e about the v - a x i s a p p l i e d t o the n u c l e u s k'. How one p u t s t h i s i n t o p r a c t i c e , t o c a l c u l a t e the e f f e c t s of the p u l s e s and d e l a y s of a p u l s e sequence on a s p i n system, i s d i s c u s s e d i n d e t a i l i n Appendix I . Ref e r e n c e s 1. Anderson, W.A., Phys. Rev. ( 1 9 5 6 ) , 1 0 4 , 8 5 0 . 2 . Anderson, W.A., Freeman, R., and R e i l l e y , C.A., J . Chem. Phys. ( 1 9 6 3 ) , 3 9 , 1518 3 . Y a t s i v , S . , Phys. Rev. ( 1 9 5 9 ) , n _ 3 , 1522 4 . Bax, A., Two-Dimensional N u c l e a r Magnetic Resonance i n L i q u i d s , D e l f U n i v . P r e s s , M i j n b o u w p l e i n , ( 1 9 8 2 ) , pp. 139 5 . Wokaun, A., and E r n s t , R.R., Chem. Phys. L e t t . ( 1 9 7 7 ) , 5 2 , 16 407 6. C a r r , H.Y., and P u r c e l l , E.M., Phys. Rev. (1954), 94, 630 7. J e e n e r , J . , M e i e r , B.H., Bachmann, P., and E r n s t , R.R., J . Chem. Phys. (1979), 7J_, 4546 8. Bodenhausen, G., and E r n s t , R.R., J . Am. Chem. Soc (1982), 104, 1304 9. M a n s f i e l d , P., and M o r r i s , P.G., NMR Imaging i n B i o m e d i c i n e , Academic P r e s s , New York, (1982) 10. Fano, U., Rev. Mod. Phys. (1957), 29, 74 11. S l i c h t e r , C P . , P r i n c i p l e s of Magnetic Resonance, (2nd e d i t i o n ) , S p r i n g e r , B e r l i n , (1978) 12. Blum, K., D e n s i t y M a t r i x Theory and A p p l i c a t i o n s , Plenum P r e s s , New York, (1981 ) 13. Sorensen, O.W., E i c h , G.W., L e v i t t , M.H., Bodenhausen, G., and E r n s t , R.R., P r o g r . N u c l . Magn. Reson. S p e c t r o s c . ( 1 981 ) , J_4, 1 37 14. B a i n , A.D., and B r o w s t e i n , J . Magn. Reson. (1982), 47, 409 15. S a n c t u a r y , B.C., J . Chem. Phys. (1976), 64, 4352 17 CHAPTER I I NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY IN AN INHOMOGENEOUS MAGNETIC FIELD 18 2.1 Zero-Quantum S p e c t r o s c o p y i n an Inhomogeneous Magnetic  F i e l d Zero-quantum c o h e r e n c e s , u n l i k e s i n g l e - q u a n t u m c o h e r e n c e s , cannot be p r e p a r e d d i r e c t l y by the a p p l i c a t i o n of a s i n g l e 90° p u l s e t o a s p i n system at e q u i l i b r i u m , and n e i t h e r can they be observed d i r e c t l y . C o n s e q u e n t l y a more s o p h i s t i c a t e d p u l s e sequence i s needed which can g e n e r a l l y be broken down i n t o t h r e e p a r t s [ 1 ] . F i r s t l y i t i s n e c e s s a r y t o p r e p a r e the zero-quantum coherence, s e c o n d l y i t must be a l l o w e d t o e v o l v e , and t h i r d l y i t must be c o n v e r t e d back i n t o s i n g l e - q u a n t u m c o h e r e n c e , which can be d i r e c t l y o b served. T h i s must be done i n such a way t h a t the o b s e r v e d s i n g l e - q u a n t u m coherence i s modulated as a f u n c t i o n of the e x t e n t of zero-quantum coherence e v o l u t i o n . The most common p u l s e sequence used t o do t h i s i s g i v e n i n F i g u r e 2.1. The f i r s t 90° p u l s e c r e a t e s in-phase s i n g l e - q u a n t u m coherence i n the r o t a t i n g frame which e v o l v e s d u r i n g the p r e p a r a t i o n p e r i o d T under the i n f l u e n c e of c h e m i c a l s h i f t and s c a l a r c o u p l i n g s i n t o a n t i p h a s e s i n g l e - q u a n t u m coherence. The a n t i p h a s e s i n g l e - q u a n t u m coherence i s c o n v e r t e d i n t o m u l t i p l e - q u a n t u m coherences by the second 90° p u l s e , and these e v o l v e d u r i n g t , . Next the t h i r d 90° p u l s e p a r t i a l l y c o n v e r t s them back i n t o a n t i p h a s e s i n g l e - q u a n t u m c o h e r e n c e s , the a m p l i t u d e s of which a r e modulated as a f u n c t i o n of the e x t e n t of m u l t i p l e - q u a n t u m coherence e v o l u t i o n . The i n v i s i b l e 19 90£ 90|, t i Preparation Zero-quantum evolution Acquisition F i g u r e 2.1 B a s i c zero-quantum e x p e r i m e n t . T a b l e I . Phase c y c l i n g schemes f o r m u l t i p l e - q u a n t u m coherence o r d e r s e l e c t i o n . P u l s e phase $ 0° 90° 180° 270° Orders s e l e c t e d R e c e i v e r 0° 0 1 2 3 4 5 6 7 8 phase 0° 0° 0 2 4 6 8 0° 180° 1 3 5 7 0° 0° 0° 0° 0 4 8 0° 180° 0° 180° 2 6 90J 905 90S Gradient Z T t i \ f\(\\r I) Preparation Zero-quantum evolution Acquisition F i g u r e 2.2 B a s i c zero-quantum experiment! w i t h magnetic f i e l d g r a d i e n t o r d e r s e l e c t i o n . 20 a n t i p h a s e s i n g l e - q u a n t u m c o h e r e n c e s rephase i n t o v i s i b l e s i n g l e - q u a n t u m coherences which a r e d e t e c t e d . I f n e x p e r i m e n t s are performed i n c r e m e n t i n g t , by a c o n s t a n t amount, A t , , each time a d a t a m a t r i x S ( t 1 f t 2 ) w i l l be o b t a i n e d . F o u r i e r t r a n s f o r m a t i o n w i t h r e s p e c t t o t 2 , the rows of the m a t r i x , w i l l y i e l d the s i n g l e - q u a n t u m spectrum c o r r e s p o n d i n g t o each v a l u e of t , . F o u r i e r t r a n s f o r m a t i o n w i t h r e s p e c t t o t 1 f the columns of the m a t r i x , w i l l y i e l d the m u l t i p l e - q u a n t u m spectrum w i t h a sweep w i d t h g i v e n by l / 2 A t , Hz. There a r e two p r a c t i c a l methods of s e p a r a t i n g out the v a r i o u s o r d e r s of m u l t i p l e - q u a n t u m c o h e r e n c e , by phase c y c l i n g or by u s i n g a p u l s e d magnetic f i e l d g r a d i e n t . Phase c y c l i n g methods make use of the d i f f e r i n g s e n s i t i v i t i e s of d i f f e r e n t o r d e r s of m u l t i p l e - q u a n t u m coherence t o a phase change of a r a d i o f r e q u e n c y p u l s e [ l - 3 ] . I f a p u l s e i s phase s h i f t e d by an a n g l e <p, then a p-order m u l t i p l e - q u a n t u m coherence w i l l e x p e r i e n c e a phase s h i f t of p</>. There a r e s e v e r a l d i f f e r e n t methods of s e p a r a t i o n based on phase p r o p e r t i e s [ 2 , 4 - 7 ] , In the most w i d e l y used of t h e s e [2] f o r each v a l u e of t , s e v e r a l e x p e r i m e n t s are performed w i t h d i f f e r e n t e x c i t a t i o n or d e t e c t i o n p u l s e phases and a l i n e a r c o m b i n a t i o n of the Free I n d u c t i o n Decays (FIDs) t a k e n . T h i s r e s u l t s i n the c a n c e l l a t i o n of s i g n a l due t o unwanted o r d e r s . For the b a s i c m u l t i p l e - q u a n t u m e x p e r i m e n t g i v e n i n f i g u r e 2.1 the phase c y c l i n g scheme used t o s e l e c t zero-quantum coherence i s g i v e n i n T a b l e I . T h i s assumes o n l y the a b i l i t y t o phase 21 s h i f t by m u l t i p l e s of 90° (an u n a v o i d a b l e c o n s t r a i n t on most NMR s p e c t r o m e t e r s ) and c o n s e q u e n t l y a l s o l e t s t h r o u g h f o u r t h and e i g h t h o r d e r m u l t i p l e - q u a n t u m coherence, though i n i s o t r o p i c s o l u t i o n the r e l a t i v e i n t e n s i t i e s of th e s e o r d e r s ar e u s u a l l y s m a l l . To be most e f f e c t i v e the m a g n e t i z a t i o n must be a l l o w e d t o r e t u r n t o e q u i l i b r i u m a f t e r each experiment (>5T,), and i t a l s o n e c e s s i t a t e s the. p e r f o r m i n g a minimum number of f o u r e x p e r i m e n t s f o r each v a l u e of t , . The p u l s e d f i e l d g r a d i e n t method of s e p a r a t i n g out the d i f f e r e n t o r d e r s of m u l t i p l e - q u a n t u m coherence [9] makes use of the d i f f e r i n g s e n s i t i v i t i e s of d i f f e r e n t o r d e r s of m u l t i p l e - q u a n t u m coherence t o magnetic f i e l d inhomogeneity [ 2 , 8 ] . T h i s has the same t h e o r e t i c a l b a s i s as phase s e n s i t i v i t y d i s c u s s e d above. An o f f s e t Aw caused by magnetic f i e l d inhomogeneity i s e x p e r i e n c e d by a p-order m u l t i p l e - q u a n t u m coherence as pAa>. Con s e q u e n t l y i f a magnetic f i e l d g r a d i e n t i s a p p l i e d f o r a time T d u r i n g t , w i t h i n the b a s i c m u l t i p l e - q u a n t u m experiment dephasing p r o p o r t i o n a l t o pT w i l l o c c u r . A f t e r b e i n g c o n v e r t e d back i n t o s i n g l e - q u a n t u m coherence m a g n e t i z a t i o n r e s u l t i n g from p-order m u l t i p l e - q u a n t u m coherence can be rephased by a p p l y i n g the same magnetic f i e l d g r a d i e n t f o r p times as l o n g . T h i s i s because s i n g l e - q u a n t u m coherence w i l l o n l y dephase or rephase at 1/p the r a t e of p-order m u l t i p l e - q u a n t u m coherence. Other o r d e r s w i l l remain dephased. To s e l e c t zero-quantum coherence which i s u n a f f e c t e d by magnetic f i e l d inhomogeneity i t i s o n l y n e c e s s a r y t o a p p l y the magnetic f i e l d g r a d i e n t once, d u r i n g t . 22 t o dephase a l l o t h e r o r d e r s p r e s e n t , F i g u r e 2.2. Due t o i t s g r e a t e r f l e x i b i l i t y and e f f i c i e n c y the p u l s e d f i e l d g r a d i e n t method of o r d e r s e l e c t i o n i s used h e r e i n u n l e s s o t h e r w i s e s t a t e d . In an inhomogeneous magnetic f i e l d the p u l s e sequence g i v e n i n F i g u r e 2.2 does not work f o r two r e a s o n s . F i r s t l y m a g n e t i z a t i o n w i l l dephase d u r i n g the p r e p a r a t i o n p e r i o d , and s e c o n d l y i n v i s i b l e a n t i p h a s e single-quanum coherence c r e a t e d by the t h i r d 90° p u l s e w i l l dephase b e f o r e i t can e v o l v e i n t o v i s i b l e in-phase m a g n e t i z a t i o n . T h i s problem can be s o l v e d by the i n t r o d u c t i o n of two 180° p u l s e s i n t o the p u l s e sequence. These r e f o c u s dephasing due t o magnetic f i e l d i n homogeneity, one i n the c e n t r e of the p r e p a r a t i o n p e r i o d and one i n the c e n t r e of an a d d i t i o n a l r e f o c u s s i n g p e r i o d a f t e r the t h i r d 90° p u l s e , F i g u r e 2.3, t o form the r e f o c u s s e d zero-quantum e x p e r i m e n t . T h i s sequence i s s i m i l a r t o one i n t r o d u c e d by Sorensen, L e v i t t and E r n s t i n the c o n t e x t of m u l t i p l e - q u a n t u m f i l t e r i n g [ 1 0 ] . The e f f e c t s of the r e f o c u s s e d zero-quantum experiment on a s p i n system can be d e s c r i b e d i n terms of the p r o d u c t o p e r a t o r f o r m a l i s m [ 1 1 ] , For an AX 2 s p i n system, n e g l e c t i n g t r a n s v e r s e r e l a x a t i o n , t h i s i s as f o l l o w s : The f i r s t 90° p u l s e produces m a g n e t i z a t i o n a l o n g the -y a x i s of the r o t a t i n g frame, - ( I . + 2 I V ). T h i s s u b s e q u e n t l y A y A y e v o l v e s d u r i n g the p r e p a r a t i o n p e r i o d r due t o the H a m i l t o n i a n Gradient Z 9 C 8 0 o V V2 9 0 o t l 9 0 8 0 o Y IP1' Preparation Zero-quantum i evolution Refocussing Acquisition F i g u r e 2.3 Refocussed zero-quantum experiment w i t h magnetic f i e l d g r a d i e n t order s e l e c t i o n . The 180° p u l s e s i n the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s a r e e s s e n t i a l i n an inhomogeneous magnetic f i e l d t o r e v e r s e d e p h a s i n g due t o magnetic f i e l d i n h o m o g e n i e t i e s . 24 2 7 r J A V r I . I v , c h e m i c a l s h i f t e v o l u t i o n b e i n g c a n c e l e d out by AA AZ AZ the 180° p u l s e at T / 2 . - ( l A y + 2 I X y ) 2 ' J A Z T l A , I Z z ; - I A y c o s M , J A X r ) . ^ W x z ^ ^ A X ^ ^ ^ A X ^ + 2 W x z s i n ( 7 r J A X T ) c 0 s ( 7 r J A X T ) ^ ^ y ^ z ^ z 5 1 " 2 ^ ^ ^ - 2 I X y c o s ( 7 r J A X T ) ^ W A Z ^ ^ A X ^ ( 2 ' 1 ) The second 90° p u l s e c o n v e r t s the f i r s t and f i f t h terms of e q u a t i o n 2.1 i n t o 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 I A z and 2 I X z . T h i s w i l l be c o n v e r t e d back i n t o I . and 2 I V t r by the t h i r d 90° Ay Ay p u l s e and w i l l c o n t r i b u t e t o the peak a t 0.0 Hz i n the F1 dim e n s i o n of the d a t a s e t as i t i s not modulated w i t h r e s p e c t t o t , . The f o u r t h term i s c o n v e r t e d i n t o the term 4 I A z I X y I X y , zero-quantum coherence between the two X s p i n s a n t i p h a s e w i t h r e s p e c t t o the p a s s i v e A s p i n . T h i s w i l l not e v o l v e due t o s c a l a r c o u p l i n g s or c h e m i c a l s h i f t d u r i n g t 1 f and hence when c o n v e r t e d back i n t o s i n g l e - q u a n t u m coherence w i l l o n l y c o n t r i b u t e t o the peak a t 0.0 Hz i n F1. The second, t h i r d , and s i x t h terms of the e q u a t i o n a r e c o n v e r t e d i n t o t w o - s p i n c o h e r e n c e s , 2 I A x I X y » 2 I A x J X y ' a n d 2 I X x I A y r e s P e c t i v e l Y which c o n t a i n b o t h z e r o - and double-quantum coherence [ 1 1 ] . Pure zero-quantum coherence i s g i v e n by: ( 1 / 2 ) ( 2 I A X I X X + 2 I A y I X y ) = { Z Q T } x ( 2 ' 2 ) and ^^Vxx -2IAxV= f ZQT} y (2.3) T h e r e f o r e the a m p l i t u d e of zero-quantum coherence p r e s e n t 25 a f t e r the second 90° p u l s e i s g i v e n by: X 2 s i n ( 7 r J A X r ) -2cos ( 7 T J A X T ) s i n (7 r d" A XT ) = {ZQT} y. (2.4) Non-zero m u l t i p l e - q u a n t u m coherences w i l l be dephased by the p u l s e d magnetic f i e l d g r a d i e n t . Zero-quantum coherences w i l l e v o l v e d u r i n g t , under the i n f l u e n c e of both c h e m i c a l s h i f t and s c a l a r c o u p l i n g s t o p a s s i v e s p i n s [ 1 1 ] , the e f f e c t i v e c h e m i c a l s h i f t i s g i v e n by: n e f f = n A - n x (2.5) and the e f f e c t i v e s c a l a r c o u p l i n g by: J e f f = J A X " JXX ( 2 * 6 ) The e x t e n t of e v o l u t i o n d u r i n g t t i s g i v e n by: {ZQT} 7 r J A X t 1 2 I A z I X z . R A t 1 I A z . Q X t 1 I X z . « y > { Z Q T } y c o s ( 7 r J e f f t , ) c o s ( Q e f f t 1 ) - 2 I X z { Z Q T } x s i n ( 7 r J e f f t , ) c o s f i e f f t , ) - { Z Q T } x c o s ( j r J e f f t , ) s i n ( O e f f t , ) - 2 I X z ( Z Q T } y s i n ( 7 r J e f f t , ) s i n ( Q e f f t , ) (2.7) A f t e r t 1 a t h i r d 90° p u l s e i s a p p l i e d : X {ZQT} y ( 7 r / 2 ) ( I A x + I X x \ ( 1/2) ( 2 I A z I X x + 2 I A x I X z ) ( 2 ' 8 ) 2 I V {ZQT} ( T T / 2 ) ( I A X - H X X ) ( , / 2 ) < 4 I X y W x x + " x y W x z ' ( 2 - 9 ) ( Z Q T ) X U / 2 > ( I A x + I X x ' ; < 1 / 2 ) < 2 I A x I X x + ' A z ' x z ' 2 6 ( 2 . 1 0 ) 21 v { ZQT} ( T T / 2 ) ( I A X - H X X ) ^ / ^ " x y W x x " 4 I Z y I A x I X z ) ( 2 ' 1 l ) The terms on the r i g h t s i d e of e q u a t i o n 2.8 are a n t i p h a s e single-quantum coherences which w i l l be observed upon e v o l v i n g i n t o in-phase s i n g l e - q u a n t u m coherence. The terms, on the r i g h t s i d e of e q u a t i o n 2.9 a r e t h r e e - s p i n c o h e r e n c e , which i s not d i r e c t l y o b s e r v a b l e , and a n t i p h a s e s i n g l e - q u a n t u m coherence. The l a t t e r does not become o b s e r v a b l e e i t h e r as ^ X X = 0 . The terms on the r i g h t of e q u a t i o n 2 . 1 0 a r e u n a f f e c t e d zero-quantum coherence and l o n g i t u d i n a l t w o - s p i n o r d e r r e s p e c t i v e l y . The terms' on the r i g h t of e q u a t i o n 2 . 1 1 a r e a n t i p h a s e zero-quantum coherence. D u r i n g the r e f o c u s s i n g p e r i o d T' the a n t i p h a s e single-quantum coherence formed by the t h i r d 9 0 ° p u l s e (on the r i g h t of e q u a t i o n 2 . 8 ) w i l l e v o l v e under the i n f l u e n c e of s c a l a r c o u p l i n g s , c h e m i c a l s h i f t e v o l u t i o n b e i n g c a n c e l e d out by the 1 8 0 ° p u l s e a t T ' / 2 , t h u s : < 1 / 2 ) < 2 I A « I X x - 2 I A x I X z > ' ^ A z ^ z , ( l / 2 ) [ 2 I A z I X x + C O s ( 7 r J A X T ' ) + I X y s i n U j A X ^ ) - 2 I A x : X z C O s 2 ( , r J A X r , ) - ^ A y W x z ^ ^ A X 7 ' > s i n ( 7 T J A X r ' ) - I A y s i n ( 7 r J A X T ' ) c o s ( 7 r J A X r ' ) + I A x I X z s i n 2 ( * J A X r , ) 1 ( 2 ' 1 2 ) C o n s e q u e n t l y t h e a m p l i t u d e of i n - p l i a s e , and hence v i s i b l e , 27 single-quantum coherence p r e s e n t a t time T' o r i g i n a t i n g from the zero-quantum coherence c o n s i s t i n g of the s p i n s A and X i s g i v e n by: I X y + I A y = ( 1/2) t s i n ( 7 r J A X T ' . ) - s i n ( T T J ^ T ' ) cos ( T T J ^ T ' ) ] (2.13) I t s h o u l d be noted t h a t t h i s has the same form as e q u a t i o n 2.4. The r e f o c u s s e d zero-quantum e x p e r i m e n t ( f i g u r e 2.3) was used t o o b t a i n the zero-quantum s p e c t r a of the amino a c i d s L - a l a n i n e , L - v a l i n e , L - t h r e o n i n e , and L - a s p a r a g i n e i n a homogeneous magnetic f i e l d , F i g u r e 2.4 A-D. The homogeneity of the magnetic f i e l d was then degraded, F i g u r e 2.5, and the zero-quantum spectrum of a m i x t u r e of the amino a c i d s o b t a i n e d w i t h the same p u l s e sequence and the same para m e t e r s , F i g u r e 2.4 E. The zero-quantum coherences of t h e i n d i v i d u a l components of the m i x t u r e a r e c l e a r l y d i s c e r n i b l e whereas the c o r r e s p o n d i n g s i n g l e - q u a n t u m spectrum i s of l i t t l e use. Consequently the d a t a s e t , s ( t 1 , t 2 ) , was o n l y F o u r i e r t r a n s f o r m e d w i t h r e s p e c t t o t , . As t h e FID i n t 2 decayed v e r y r a p i d l y as a consequence of the magnetic f i e l d inhomogeneity o n l y the t , FIDs w i t h i n the f i r s t 2-3 m i l l i s e c o n d s of t 2 c o u l d be co-added t o b u i l d up s i g n a l - t o - n o i s e (S/N). In the case of extreme inhomogeneity o n l y the f i r s t p o i n t of the t 2 FID, c o l l e c t e d a t the t o p of the s p i n - e c h o , may be of use. Each zero-quantum spectrum took under 15 m i n u t e s t o a c q u i r e , which i s by no means p r o h i b i t i v e when one c o n s i d e r s the l a c k of a l t e r n a t i v e s i n an i n h e r e n t l y inhomogeneous magnetic D ~1 I I I I I I 1 I 1 1 " 1 1 1 I 1 ' X~ 800 600 400 200 0.0 -200 -400 -600 Hz to CD F i g u r e 2.4 Zero-quantum s p e c t r a o b t a i n e d a t 270 MHz w i t h the r e f o c u s s e d zero-quantum experiment. In a homogeneous magnetic f i e l d : A. L - a l a n i n e , B. L - t h r e o n i n e , C. L - v a l i n e , D. L - a s p a r a g i n e . In an inhomogeneous magnetic f i e l d : E. A m i x t u r e of the above. A l l c o n c e n t r a t i o n s were 0.1 molar i n D 20; T,T'=60 msec, At,=600 /usee, t o t a l a c q u i s i t i o n time was 15 minutes i n each ca s e . i i 1 1 1 1 1 1 1 1 1 1 1 1 r 600 400 200 0.0 -200 -400 -600 Hz F i g u r e 2.5 S i n g l e quantum s p e c t r a a t 270 MHz of a s o l u t i o n i n D 20 of L - a l a n i n e , L - t h r e o n i n e , L - v a l i n e and L - a s p a r a g i n e , a l l c o n c e n t r a t i o n s 0.1 molar. A. In a homogeneous magnetic f i e l d , B. In an inhomogeneous magnetic f i e l d c o r r e s p o n d i n g t o F i g u r e 2.4 E. 30 environment such as one might f i n d i n the c o n t e x t of NMR imaging. 2.2 E d i t i n g Zero-Quantum Coherence S p e c t r a Up u n t i l the p r e s e n t time most p r a c t i t i o n e r s of zero-quantum NMR have been more concerned w i t h e n s u r i n g t h a t they o b t a i n the whole of the zero-quantum spectrum [12] r a t h e r than o n l y p a r t of i t , a l t h o u g h t h e r e have been q u a l i f i e d e x c e p t i o n s [ 1 3 ] . From e q u a t i o n 2.4 i t can be seen t h a t the e f f i c i e n c y of e x c i t a t i o n of zero-quantum coherence i s dependent upon the s c a l a r c o u p l i n g s of the s p i n s and the l e n g t h of the p r e p a r a t i o n p e r i o d T . Consequently the a b s o l u t e and r e l a t i v e i n t e n s i t i e s of the zero-quantum coherences o b s e r v e d w i l l v a r y f o r a g i v e n s p i n system from one v a l u e of T t o a n o t h e r . At c e r t a i n v a l u e s , whenever e q u a t i o n 2.4 i s e q u a l t o z e r o i n the case of an AX 2 s p i n system, the zero-quantum coherence w i l l be absent from the spectrum e n t i r e l y . The dependency upon s c a l a r c o u p l i n g s and r w i l l v a r y from one s p i n system t o a n o t h e r , a l t h o u g h i t w i l l be g e n e r a l l y easy t o c a l c u l a t e [ 1 1 ] . As has been noted above, magnetic f i e l d inhomogeneity may l i m i t the e f f e c t i v e d a t a a c q u i s i t i o n time t o 2-3 m i l l i s e c o n d s d u r i n g which time the s p i n system w i l l undergo n e g l i g i b l e e v o l u t i o n due t o s c a l a r c o u p l i n g s , assuming the c o u p l i n g c o n s t a n t s concerned a r e not l a r g e . C o n s e q u e n t l y the i n t e n s i t y of m a g n e t i z a t i o n d e t e c t e d o r i g i n a t i n g from zero-quantum 31 c o h e r e n c e , and hence the i n t e n s i t y of t h a t zero-quantum c o h e r e n c e observed i n the r e s u l t a n t spectrum, w i l l be dependent upon the r e l e v a n t s c a l a r c o u p l i n g s and the l e n g t h of t h e r e f o c u s s i n g p e r i o d r ' . T h i s i s analogous t o the dependency of zero-quantum e x c i t a t i o n upon s c a l a r c o u p l i n g s and r d i s c u s s e d above. For an AX 2 s p i n system t h i s g i v e n i n e q u a t i o n 2.13, and can be e a s i l y c a l c u l a t e d f o r o t h e r s p i n systems [ 1 1 ] . Both p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s p r o v i d e an o p p o r t u n i t y t o e d i t the observed s i g n a l . The importance of t h i s a b i l i t y , p a r t i c u l a r l y a t lower f i e l d s , i s r e a d i l y a p p r e c i a t e d when one c o n s i d e r s t h a t i n zero-quantum space the p r o b l e m of resonance o v e r l a p i n a spectrum i s o f t e n worse than f o r i t s s i n g l e - q u a n t u m c o u n t e r p a r t . T h i s i s because zero-quantum coherences o n l y occur a t the d i f f e r e n c e i n r e sonance f r e q u e n c i e s of two c o u p l e d s p i n s and t h e r e f o r e u s u a l l y occur w i t h i n a narrower f r e q u e n c y range. The e d i t i n g e f f e c t s of v a r y i n g the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s upon a zero-quantum spectrum are demonstrated i n F i g u r e 2.6 f o r L - t h r e o n i n e . 2.3 N o n - S e l e c t i v e E x c i t a t i o n and D e t e c t i o n of Zero-Quantum  Coherence i n an Inhomogeneous Ma g n e t i c F i e l d G i v e n the c o n s t r a i n t s imposed upon the d e t e c t i o n of zero-quantum coherence i n inhomogeneous magnetic f i e l d s by p r e p a r a t i o n and r e f o c u s s i n g t i m e s , how can one be c e r t a i n of a r e a s o n a b l y u n i f o r m e x c i t a t i o n and d e t e c t i o n of zero-quantum 32 A I i i i r—|—i—i—i—i—|—i—1-71—1—j—1—1—1—1 [ 1—1—1—1—j—1—1—1—1—j 300 200 100 0.0 -100 -200 Hz F i g u r e 2.6 Zero-quantum s p e c t r a o b t a i n e d a t 80 MHz i n an inhomogeneous magnetic f i e l d w i t h t he r e f o c u s s e d zero-quantum experiment of 0.1 molar L - t h r e o n i n e i n D 20 w i t h : A. T,T'=60 msec, B. T,T'=100 msec, C. T,T'=140 msec, D. T=100 msec, T'=60 msec. In each case At,=1.67 msec, 256 b l o c k s were c o l l e c t e d w i t h 4 a c q u i s i t i o n s per b l o c k . A c q u i s i t i o n t i m e s 15-16 min u t e s . c o h e r e n c e s ? 33 One p o s s i b i l i t y would be t o re p e a t the experiment f o r s e v e r a l d i f f e r e n t v a l u e s of T and r' and add the r e s u l t i n g s p e c t r a t o g e t h e r [ 1 3 ] . One might, f o r example, add t o g e t h e r the s p e c t r a i n F i g u r e 2.6 A and C t o o b t a i n complete zero-quantum spectrum of L - t h r e o n i n e . The major d i s a d v a n t a g e of t h i s method i s the e x t r a time i t i n v o l v e s . A l t e r n a t i v e l y , E r n s t and co-workers [12] have proposed the use of an " a c c o r d i o n " p r e p a r a t i o n p e r i o d i n which the p r e p a r a t i o n p e r i o d i s i ncremented by nAt, becoming r+nt,. U s u a l l y n i s s e t t o SO.25 so t h a t the J - e n c o d i n g i t w i l l cause t o be superimposed on t o p of the zero-quantum spectrum w i l l be s c a l e d down s u f f i c i e n t l y so as not t o d i s t o r t i t s i g n i f i c a n t l y . I t works because the l e n g t h of the p r e p a r a t i o n p e r i o d w i l l c o r r e s p o n d t o t h a t needed t o e x c i t e each zero-quantum coherence f o r a t l e a s t p a r t of the ex p e r i m e n t , assuming t h a t the time range i s adequate. In F i g u r e 2.7A the p r e p a r a t i o n sub-sequence of the r e f o c u s s e d zero-quantum experiment has been r e p l a c e d w i t h the " a c c o r d i o n " p r e p a r a t i o n p e r i o d . D e s p i t e e n s u r i n g e x c i t a t i o n of a l l zero-quantum coherence, t h i s sequence does not ensure t h a t a l l w i l l be d e t e c t e d . T h i s i s because i t assumes the e f f e c t i v e a c q u i s i t i o n time t o be l o n g enough f o r a l l components of i n v i s i b l e a n t i p h a s e m a g n e t i z a t i o n t o e v o l v e t h r o u g h v i s i b l e i n -phase m a g n e t i z a t i o n and hence not t o be e d i t e d out of the spectrum by c h o o s i n g a c e r t a i n v a l u e of the r e f o c u s s i n g time.. T h i s would indeed be the case i n a homogeneous f i e l d where the e f f e c t i v e a c q u i s i t i o n time i s r e l a t i v e l y l o n g , but not i n an Gz T+nt, III 2 l l T+nt, 2 I U T 2 1 1 1 1 Preparation j Zero-quantum ; evolution i Refocussing Acq. B G 2 T+nt, T+nt, ti T+nt, T+nt, 2 2 2 2 ..„ CO i i i l i i 1 1 Preparation j i i i i i i i 1 j Refocussing i i Acq. F i g u r e 2.7 A. Refocussed zero-quantum e x p e r i m e n t w i t h " a c c o r d i o n " p r e p a r a t i o n p e r i o d . B. R e f o c u s s e d zero-quantum experiment w i t h " a c c o r d i o n " p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s . 35 A I J V J B I i i i — i — | — i — i — i — i — | — i — i — i — i — j — i — i — i — i — | — i — i — i — i — j — i — i — i — i | 3 0 0 2 0 0 1 0 0 0 . 0 - 1 0 0 - 2 0 0 H z F i g u r e 2.8 Zero-quantum s p e c t r a a t 80 MHz of 0.1 molar L - t h r e o n i n e i n D 20 w i t h : A. The r e f o c u s s e d zero-quantum experiment w i t h an " a c c o r d i o n " p r e p a r a t i o n p e r i o d ( F i g u r e 2.7A), B. The r e f o c u s s e d zero-quantum experiment w i t h " a c c o r d i o n " p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s . In both c a s e s T=70 msec, A t , = 1.67 msec, n=.12 . 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . A c q u i s i t i o n t i m e s 35-37 m i n u t e s . 36 inhomogeneous f i e l d where the e f f e c t i v e a c q u i s i t i o n time i s o n l y a c o u p l e of m i l l i s e c o n d s . C o n s e q u e n t l y d e s p i t e the " a c c o r d i o n " p r e p a r a t i o n p e r i o d the r e f o c u s s i n g p e r i o d w i l l s t i l l cause e d i t i n g , F i g u r e 2.8A. T h i s problem was overcome by u s i n g an " a c c o r d i o n " r e f o c u s s i n g as w e l l as p r e p a r a t i o n p e r i o d , F i g u r e 2.IB. T h i s p u l s e sequence was used t o o b t a i n the spectrum i n F i g u r e 2.8B which i s c l e a r l y more u n i f o r m than i t s c o u n t e r p a r t o b t a i n e d w i t h a c o n s t a n t r e f o c u s s i n g p e r i o d i n F i g u r e 2.8A. The parameters used were the same i n both c a s e s . Both sequences were found t o produce s i g n i f i c a n t l y l e s s i n t e n s e peaks, r e f l e c t e d i n the p o o r e r S/N compared t o the r e f o c u s s e d zero-quantum experiment ( F i g u r e 2.6) a l t h o u g h t w i c e as many a c q u i s i t i o n s per b l o c k were c o l l e c t e d . Hence the t o t a l a c q u i s i t i o n time was t w i c e as l o n g as t h a t used f o r the r e f o c u s s e d zero-quantum experiment i n F i g u r e 2.5. T h i s i s i n h e r e n t i n a c c o r d i o n t e c h n i q u e s . V a r y i n g the l e n g t h of the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s r e s u l t s i n the c r e a t i o n and d e t e c t i o n of a g i v e n zero-quantum coherence b e i n g a t an optimum f o r o n l y p a r t of the experiment whereas the f i x e d p r e p a r a t i o n and r e f o c u s s i n g i n t e r v a l experiment r e s u l t s i n the zero-quantum coherence chosen b e i n g o p t i m i s e d throughout the whole of the e x p e r i m e n t . 2.4 Homonuclear Broad-Band Decoupled Zero-Quantum Coherence  S p e c t r o s c o p y Zero-quantum s p e c t r a have s e v e r a l major d i s a d v a n t a g e s over t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s . Among thes e a r e the 37 time they t a k e t o a c q u i r e (each spectrum i s a t w o - d i m e n s i o n a l experiment) and the narrow frequency range r e l a t i v e to the c o r r e s p o n d i n g single-quantum s p e c t r a over which they o c c u r . The l a t t e r f e a t u r e i s a d i r e c t r e s u l t of the p r o p e r t y of zero-quantum coherences which occur a t the d i f f e r e n c e i n resonance f r e q u e n c i e s of two c o u p l e d s p i n s . O f t e n t h i s l e a d s t o o v e r l a p p i n g , a more crowded and hence l e s s e a s i l y i n t e r p r e t a b l e spectrum. T h i s problem can be overcome by m a n i p u l a t i n g the p r e p a r a t i o n and r e f o c u s s i n g t i m e s t o e d i t the spectrum ( s e c t i o n 2.1). Time consuming t r i a l and e r r o r i s r e q u i r e d i f the type of s p i n system and i t s c o u p l i n g c o n s t a n t s are not known. A l t e r n a t i v e l y the r e s o l u t i o n of the spectrum can be i n c r e a s e d to t r y and r e s o l v e crowded and o v e r l a p p i n g resonances by i n c r e a s i n g the number of t , i n c r e m e n t s used; t h i s a l s o t a k e s time. A t h i r d a l t e r n a t i v e which might a l l e v i a t e b o t h of the problems r e f e r r e d t o .above would be t o decouple the e n t i r e zero-quantum spectrum. T h i s would l e a d t o a much s i m p l e r spectrum, p o s s i b l y r e q u i r i n g lower r e s o l u t i o n and hence l e s s a c q u i s i t i o n t i m e . I f a l l the i n t e n s i t i e s of a former m u l t i p l e t now went i n t o one peak fewer a c q u i s i t i o n s would be r e q u i r e d t o b u i l d up comparable S/N. A p u l s e sequence was d e s i g n e d t o a c c o m p l i s h the broad-band d e c o u p l i n g of zero-quantum s p e c t r a which i s shown i n F i g u r e "2.9. I t has been known f o r some time how t o produce homonuclear broad-band d e c o u p l e d s i n g l e - q u a n t u m s p e c t r a . T h i s Gradient Z 9C »x T/2 180 o T/2 9C >x 1 t,/2 80 o Y t d - t x / 2 a o T/2 180 o 1 1 1 1 i 1 Preparation | Zero-quantum evolution i i i i i Refocussing i Acquisition Figure 2.9 Broad band decoupled zero-quantum experiment. 39 was o r i g i n a l l y a c h i e v e d by t a k i n g a p r o j e c t i o n t h r o u g h a t w o - d i m e n s i o n a l J - r e s o l v e d spectrum [ 1 4 ] . L a t e r a p u l s e sequence was d e s i g n e d s p e c i f i c a l l y t o o b t a i n homonuclear broad-band de c o u p l e d s p e c t r a [15] which was s u b s e q u e n t l y adapted t o o b t a i n a J e e n e r spectrum decoupled i n one dimension [ 1 6 ] . The u n d e r l y i n g p r i n c i p l e s of d e c o u p l i n g s i n g l e - q u a n t u m and zero-quantum s p e c t r a a r e f u n d a m e n t a l l y the same. I t i s most e a s i l y u n d e r s t o o d by c o n s i d e r i n g what i t t a k e s t o put i n f o r m a t i o n i n t o the F1 dimension of a t w o - d i m e n s i o n a l spectrum. For each v a l u e of t , the p r o p e r t y of i n t e r e s t must v a r y i n such a way t h a t the a c q u i r e d s i g n a l i s modulated as a f u n c t i o n of t , due t o t h i s p r o p e r t y . For example, i n a zero-quantum experiment such as t h a t i n F i g u r e 2.3, a t each v a l u e of t , a g i v e n zero-quantum coherence w i l l have e v o l v e d t o a d i f f e r e n t e x t e n t due t o c h e m i c a l s h i f t , and t h i s modulates the a c q u i r e d s i g n a l ( e q u a t i o n s 2.7-2.13). L i k e w i s e a c o r r e s p o n d i n g e v o l u t i o n due t o s c a l a r c o u p l i n g s r e s u l t s i n t h e i r m o d u l a t i n g the a c q u i r e d s i g n a l as w e l l . T h i s g i v e s r i s e t o a zero-quantum spectrum complete w i t h m u l t i p l e t s t r u c t u r e when F o u r i e r t r a n s f o r m e d w i t h r e s p e c t t o t , . T h e r e f o r e , t o d ecouple a zero-quantum spectrum a l l one has t o do i s t o ensure t h a t w h i l e the e x t e n t of c h e m i c a l s h i f t e v o l u t i o n v a r i e s w i t h t , , the e x t e n t of s c a l a r c o u p l i n g e v o l u t i o n does n o t . I t has been w e l l documented [17,18] t h a t a 180° p u l s e a p p l i e d t o a homonuclear s p i n system, as i n a s p i n - e c h o e x p e r i m e n t , has the e f f e c t of r e v e r s i n g the 1 e v o l u t i o n of the 40 s p i n system due t o c h e m i c a l s h i f t and magnetic f i e l d i n h o m o g e n e i t i e s but l e a v e s e v o l u t i o n due t o s c a l a r c o u p l i n g s u n a f f e c t e d . T h i s a p p l i e s e q u a l l y t o z e r o - and si n g l e - q u a n t u m c o h e r e n c e s . In the deco u p l e d zero-quantum e x p e r i m e n t , F i g u r e 2.9, the zero-quantum e v o l u t i o n t i m e , t ^ , i s c o n s t a n t . T h e r e f o r e the e x t e n t of e v o l u t i o n due t o s c a l a r c o u p l i n g s of a zero-quantum coherence c r e a t e d by the second 90° p u l s e w i l l be c o n s t a n t , b e i n g u n a f f e c t e d by the p o s i t i o n of the 180° p u l s e w i t h i n i t . Thus the a c q u i r e d s i g n a l w i l l not be modulated due t o s c a l a r c o u p l i n g s i n the t , dimension of the data s et S ( t 1 r t 2 ) . The e x t e n t of c h e m i c a l s h i f t e v o l u t i o n on the o t h e r hand w i l l v a r y from a maximum a t t,/2=0 t o z e r o when t 1 / 2 = t c j / 2 , c h e m i c a l s h i f t e v o l u t i o n a t t h i s p o i n t i n the experiment b e i n g c o m p l e t e l y r e v e r s e d , t o a maximum a g a i n a t t,/2=t^. T h i s • r e s u l t s i n the a c q u i r e d s i g n a l b e i n g modulated w i t h r e s p e c t t o t , due t o c h e m i c a l s h i f t e v o l u t i o n . F o u r i e r t r a n s f o r m a t i o n w i t h r e s p e c t t o t , w i l l t h e r e f o r e y i e l d the deco u p l e d zero-quantum spectrum. For an AX 2 s p i n system the e v o l u t i o n of a zero-quantum coherence between s p i n s A and X d u r i n g t ^ can be d e s c r i b e d t h u s : {2QT} y "WV^Wxz, W 2 ) l A 2 j W 2 ) I X z : '^Ax + W,  y JAX ( td- t1 / 2 ) 2 IAz IXz, 0 A ( y V 2 ) V W V 2 ) I X Z ; 41 {ZQT} yCOS[fi e f f ( t , - t d ) ]cOS(7TJef f t f l ) *+{ZQT} xsin [ n e f f ( t , - t d ) ] c o s ( 7 r J e f f t d ) - 2 I X z ( Z Q T } x c o s [ n e f f ( t 1 - t d ) ] s i n U J e f f t d ) + 2 I X z { Z Q T } y s i n [ J 2 e f f ( t , - t d ) ] s i n ( i r J e f f t d ) (2.14) C l e a r l y a l l s c a l a r c o u p l i n g e v o l u t i o n i s independent of t , . An example of zero-quantum d e c o u p l i n g i s g i v e n i n F i g u r e 2.10 B f o r a s o l u t i o n of e t h a n o l and 2-propanol i n D 20, and the c o r r e s p o n d i n g undecoupled zero-quantum spectrum i n F i g u r e 2.10 A. From an e x a m i n a t i o n of e q u a t i o n 2.14 i t can be seen t h a t whether a p a r t i c u l a r zero-quantum coherence i s p r e s e n t as the in-phase terms 2 I A x I X x , 2 I A y I X y , 2 I A x I X y , and 2 I A y I X x , or as a n t i p h a s e terms such as 4 I ^ y I X y I X z a t t* i e e n d °^ fcd * s dependent o n l y upon the c o u p l i n g c o n s t a n t s concerned and t d . I t f o l l o w s t h a t a coherence may be c o m p l e t e l y in-phase or c o m p l e t e l y a n t i p h a s e , o r , more n o r m a l l y , p a r t i a l l y both at the end of t d f o r a l l v a l u e s of t , . T h i s has i m p o r t a n t consequences as can be seen from T a b l e I I . The t a b l e shows the e f f e c t s of the p u l s e a=90° at the end of t d on the product o p e r a t o r s p r e s e n t a t t h a t time f o r an AX 2 s p i n system. 42 Table I I . E f f e c t s of the p u l s e a=90° on the p r o d u c t o p e r a t o r s p r e s e n t a t the end of the e v o l u t i o n time t ^ of the broad-band decoupled zero-quantum experiment f o r an AX 2 s p i n system. Product o p e r a t o r s P r o d u c t o p e r a t o r s p r e s e n t p r e s e n t b e f o r e a f t e r p u l s e a=90° p u l s e a ( v i s i b l e terms i n b o l d type) 2 I A y J X x 2 I A z J X x " ^ A x ^ y ' " A x 1 ! ! ^ A x ^ x 2 I A x : X x 2 I A y J X y 2 I A z I X z ~ 4 I A x I X x I X z 41 I I * Ax Xx Xy " ^ A y V x z 4 I A z I X z I X y 4 I A y I X x I X z ~ 4 I A z I X x I X y ~ 4 I A x I X y I X z 4 I A x I X z I X y C l e a r l y o n l y the in-phase terms 2 I A X I X y a n d 2 I A y I X x a r e c o n v e r t e d i n t o a n t i p h a s e s i n g l e - q u a n t u m coherence which w i l l be v i s i b l e a f t e r r e p h a s i n g . In o t h e r words, o n l y t h a t p a r t of a zero-quantum coherence which i s in-phase a t the end of t ^ w i l l be o b s e r v e d . T h e r e f o r e t ^ can be chosen such t h a t a l l of a zero-quantum coherence i s o n l y p r e s e n t as a n t i p h a s e m a g n e t i z a t i o n a t the end of the p e r i o d and hence w i l l be e d i t e d out of the observed zero-quantum spectrum. T h i s i s demonstrated f o r a s o l u t i o n of e t h a n o l and 2-propanol i n D 20 i n F i g u r e 2.10 C and D where 2-propanol and e t h a n o l r e s p e c t i v e l y have been e d i t e d out of the spectrum. The graph i n F i g u r e 2.11 shows how the i n t e n s i t i e s of the components of F i g u r e 2.10. Zero-quantum s p e c t r a of e t h a n o l and 2 - p r o p a n o l , 1:1 i n D 20. A. C o n v e n t i o n a l zero-quantum s p e c t r a o b t a i n e d w i t h the r e f o c u s s e d zero-quantum e x p e r i m e n t ; T,T'=60 msec, At,=1.67 msec. 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . B-D. Broad band decoupled zero-quantum s p e c t r a : B. ^=443 msec, C. td=413 msec, D. td=493 msec. In B-D T=60 msec, At,=1.67 msec, a=90°. 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . Peak as s i g n m e n t s : e t h a n o l ±197 Hz, 2-propanol ±225 Hz. 44 the above s o l u t i o n were found t o v a r y w i t h t ^ f o r a=90°. These i n t e n s i t i e s can be e a s i l y p r e d i c t e d by c o n s t r u c t i n g the a p p r o p r i a t e p r o d u c t o p e r a t o r s . The i n t e n s i t y of the zero-quantum coherence of an AX n s p i n system was found t o be p r o p o r t i o n a l t o c o s n 1( T3 J e£f f f)> a n d t h i s dependency i s r e f l e c t e d by 2-propanol i n F i g u r e 2.11. For s p i n systems w i t h more than two d i f f e r e n t c o u p l e d s p i n s the s i t u a t i o n i s somewhat d i f f e r e n t . Even when a=90° some a n t i p h a s e terms of a zero-quantum coherence w i l l be c o n v e r t e d i n t o a n t i p h a s e s i n g l e - q u a n t u m coherence by the p u l s e a which w i l l be c a p a b l e of r e f o c u s s i n g and hence w i l l be observed t h u s t h w a r t i n g a t t e m p t s a t e d i t i n g . For example, f o r an AMX s p i n system a zero-quantum coherence between s p i n s A and M w i l l e v o l v e d u r i n g t ^ i n t o a n t i p h a s e terms i n c l u d i n g 4 I A y I M y I X z " A P u l s e w i H c o n v e r t t h i s term i n t o 4 I A z I M z I X y ' a n t i p h a s e m a g n e t i z a t i o n of the p a s s i v e s p i n X w i t h r e s p e c t t o the s p i n s A and M. As n e i t h e r J X A nor J X M i s e q u a l t o z e r o i t w i l l e v o l v e i n t o o b s e r v a b l e in-phase s i n g l e - q u a n t u m coherence. Hence the zero-quantum coherence between s p i n s A and M w i l l be observed even i f t ^ i s s e t so t h a t a l l of i t i s a n t i p h a s e a t the end of the e v o l u t i o n p e r i o d . A c o r r e s p o n d i n g a n t i p h a s e zero-quantum coherence f o r an AX n s p i n system might be 4 I ^ y I X y I X z * 0 n a P P l i c a t i ° n o f a 90° p u l s e t h i s term would become 4 I ^ z I X z I X y ' a n t i P n a s e m a g n e t i z a t i o n of the p a s s i v e s p i n X w i t h r e s p e c t t o the s p i n s A and X. As ^ X x = ^ w o u l d never rephase i n t o v i s i b l e inphase s i n g l e - q u a n t u m coherence and hence e d i t i n g w i l l be e f f e c t i v e . Peak intensity l — i — i — i — i — i — i — i — r — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i 273 313 353 393 433 473 513 t^ (msec) F i g u r e 2 . 1 1 . Graph of peak i n t e n s i t y v s . t ^ f o r the broad band decoupled zero-quantum experiment f o r a 1:1 s o l u t i o n of e t h a n o l and 2 - p r o p a n o l i n D 2 0 . 46 Table I I I . E f f e c t s of a p u l s e a on the p r o d u c t o p e r a t o r s p r e s e n t a t the end of the e v o l u t i o n time t ^ of t h e broad-band d e c o u p l e d zero-quantum experiment f o r an AX 2 s p i n system. Produc t o p e r a t o r s P r o d u c t o p e r a t o r s p r e s e n t a f t e r p u l s e a p r e s e n t b e f o r e ( v i s i b l e terms i n b o l d t y p e ) p u l s e a 2Vxx 2 I A y I X x c o s a 2 I A z I X x s i n a ' " A x 1 * * _ 2 I A x I X y c o s a - 2 I A K X X z s i n a 2 I A x I X x 2 I A x : X x 21. I v Ay Xy 2 I A y I X y c o s 2 a 2 I A Y I X z c o s a s i n a 2 I A z I X y s i n a c O S a 2 l A z I X z s i n 2 a _ 4 I A x I X x I X z - " A X W X Z 0 0 8 " 4 I A x I X x I X y s i n a - ^ ^ y ^ y ^ z 0 0 8 ' 1 1 4 I A y I X y I X y c o s 2 a s i n a " 4 I A y I X z I X z C O s 2 a s i n a  _ 4 I A z I X y I X z c o s 2 a s i n a  4 I A y I X z I X y C O S a s i n 2 a  4 I A z I X y I X y C O S a s i n 2 a -UAz1Xz1XzCOSasin2a 4 I A z I X z I X y s i n 3 a 4 I A y I X x I X z 4 I A y I X x I X z C O s 2 a - * I A y I X x I X y c o s a s i n a 4 I A z I X x I X z C O S a s i n a " 4 I A z I X x I X y s i n 2 a _ 4 I A x I X y I X z " 4 I A x I X y I X z c o s 2 a 4 I A x I X y I X y C O S a s i n a - 4 I A x I X z I X z C O S a s i n a  4 I A x I X z I X y s i n 2 a One problem i s how t o ensure t h a t one o b t a i n s the complete d e c o u p l e d zero-quantum spectrum. There a r e two p o s s i b l e s o l u t i o n s : f i r s t l y , the s p e c t r a o b t a i n e d w i t h s e v e r a l 47 d i f f e r e n t v a l u e s of t ^ can be co-added, o r , a l t e r n a t i v e l y , the p u l s e a can be v a r i e d from 90°. The e f f e c t s of an a r b i t r a r y p u l s e a on the pr o d u c t o p e r a t o r s p r e s e n t f o r an AX 2 s p i n system a f t e r t ^ can be e v a l u a t e d from Table I I I . Some of the a n t i p h a s e zero-quantum coherence terms, none of which become v i s i b l e when a=90° or an i n t e g e r m u l t i p l e of i t , g i v e r i s e t o terms which w i l l become v i s i b l e a f t e r r e p h a s i n g when a does not meet t h i s c o n d i t i o n . Terms which w i l l be obser v e d a f t e r r e p h a s i n g a r e i n b o l d t ype i n Ta b l e I I I . S e t t i n g a=45° w i l l r e s u l t i n a n t i p h a s e zero-quantum coherence b e i n g c o n v e r t e d i n p o t e n t i a l l y o b s e r v a b l e terms w i t h maximum e f f i c i e n c y . G e n e r a l l y , the more n u c l e i t h a t a re i n v o l v e d i n an a n t i p h a s e zero-quantum coherence p r o d u c t o p e r a t o r the l e s s e f f i c i e n t w i l l be i t s c o n v e r s i o n i n t o p o t e n t i a l l y o b s e r v a b l e terms [ 1 2 ] . The e x p e r i m e n t s i n F i g u r e 2.10 B-D were r e p e a t e d w i t h a=45° and t h e s e r e s u l t s a r e shown i n F i g u r e 2.12. In each case both c o h e r e n c e s a re v i s i b l e , and 2-propanol g i v e s r i s e t o the most i n t e n s e resonance even when the v a l u e of t ^ was used which had p r e v i o u s l y , when a=90°, e d i t e d i t o u t . T h i s r e f l e c t s t h r e e f a c t o r s . F i r s t l y , i t i s dependent upon the p r o p o r t i o n of a n t i p h a s e zero-qantum coherence terms f o r each s p i n system which w i l l y i e l d p o t e n t i a l l y o b s e r v a b l e terms when a c t e d upon by the p u l s e a. S e c o n d l y , i t r e f l e c t s the r e l a t i v e s i z e of th e s e terms, and t h i r d l y , the e f f i c i e n c y w i t h which they a re c o n v e r t e d i n t o p o t e n t i a l l y o b s e r v a b l e terms. For an AX 2 s p i n system the s i z e of the s e a n t i p h a s e zero-quantum coherence terms can be e v a l u a t e d from e q u a t i o n 2.14, and t h e i r 48 PI F i g u r e 2.12. Broad band d e c o u p l e d zero-quantum s p e c t r a of a 1:1 s o l u t i o n of e t h a n o l and 2-propanol i n D 20 w i t h a=45°: A. t ( j = 4 l 3 msec, B. t<3=443 msec, C. td=493 msec. T = 60 msec, At!=1.67 msec, 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . Peak a s s i g n m e n t s : e t h a n o l ±197 Hz, 2-propanol ±225 Hz. 49 c o n v e r s i o n e f f i c i e n c y can be e v a l u a t e d from Table I I I . I t s h o u l d be n o t e d t h a t the coherence i n t e n s i t i e s of F i g u r e 2.11, u n l i k e F i g u r e 2.10, a r e no l o n g e r s y m m e t r i c a l about 0.0Hz. T h i s i s due t o the p r e f e r e n t i a l t r a n s f e r of the p o s i t i v e zero-quantum fre q u e n c y t o one of the p a r t i c i p a t i n g s p i n s and t h a t of the n e g a t i v e f r e q u e n c y t o the o t h e r induced by v a r y i n g a from 90°. T h i s phenomenon w i l l be d i s c u s s e d i n d e t a i l l a t e r [ 12,19]. R e s o l u t i o n i n the F1 spectrum i s dependent upon the number of t , i n c r e m e n t s , n, used and hence upon t ^ as t ^ n A t , . T h i s i s p o t e n t i a l l y a l i m i t a t i o n f o r m o l e c u l e s where s p i n - s p i n r e l a x a t i o n t i m e s a r e s h o r t , p a r t i c u l a r l y when one c o n s i d e r s t h a t up t o 200 m i l l i s e c o n d s more w i l l be taken up by the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s . T h i s would seem t o r u l e out most in vivo a p p l i c a t i o n s of t h i s e x p e r i m e n t . I t has p r e v i o u s l y been demonstrated t h a t two d i m e n s i o n a l zero-quantum s p e c t r a can be used t o map s p i n - s p i n c o u p l i n g networks i n a homogeneous magnetic f i e l d [ 2 0 , 2 1 ] , most r e c e n t l y i n an e x p e r i m e n t which g i v e s r i s e t o a spectrum w i t h a s i m i l a r format t o t h a t of the SECSCY experiment [ 2 1 - 2 3 ] . The u s e f u l n e s s of t h e s e e x p e r i m e n t s c o u l d be enhanced by d e c o u p l i n g the zero-quantum c o h e r e n c e s , i m p r o v i n g r e s o l u t i o n and hence u s e f u l n e s s . 2.5 Single-Quantum J - R e s o l v e d Broad-Band Decoupled Zero-quantum S p e c t r o s c o p y 50 One of the major problems f o r d e t e r m i n i n g c o n n e c t i v i t y between zero-quantum c o h e r e n c e s , and hence i n d e t e r m i n i n g the s t r u c t u r e of c h e m i c a l s p e c i e s s o l e l y from such c o h e r e n c e s , l i e s i n t h e i r m u l t i p l e t s t r u c t u r e . The c o u p l i n g c o n s t a n t s and a l s o the m u l t i p l e t s t r u c t u r e of zero-quantum coherences are n o t , on the whole, the same as t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s . In g e n e r a l the e f f e c t i v e c o u p l i n g c o n s t a n t of a m u l t i p l e - q u a n t u m coherence c o n s i s t i n g of s p i n s k c o u p l i n g t o a p a s s i v e s p i n m i s g i v e n by: J e f f = K A m k J k m ( 2 ' 1 5 ) where Am^=±1, the change i n the magnetic quantum number of s p i n k. C o n s e q u e n t l y not o n l y a r e the e f f e c t i v e c o u p l i n g c o n s t a n t s d i f f e r e n t from those found on s i n g l e - q u a n t u m c o h e r e n c e s , but a s p i n which can p a r t i c i p a t e i n more than one zero-quantum coherence may e x h i b i t d i f f e r e n t c o u p l i n g c o n s t a n t s i n each c a s e . C l e a r l y t h i s makes the e l u c i d a t i o n of s p i n - s p i n c o u p l i n g networks, and hence i d e n t i f y i n g c h e m i c a l s p e c i e s , d i f f i c u l t . I t would seem t h e r e f o r e t h a t i f zero-quantum coherences e x h i b i t e d s i n g l e - q u a n t u m m u l t i p l e t s t r u c t u r e and c o u p l i n g c o n s t a n t s t h a t t h e i r use would be g r e a t l y enhanced i n t h i s r e s p e c t . B u t, i t s h o u l d be no t e d , the problem i s not q u i t e so s i m p l e t o overcome. Each zero-quantum coherence i s a coherence between two s p i n s and hence i f one were somehow t o r e p l a c e the zero-quantum m u l t i p l e t s t r u c t u r e one would be d o i n g so w i t h not one but two s i n g l e - q u a n t u m ones. T h i s , i t would seem, would not l e a v e one any b e t t e r o f f . 51 T h e r e f o r e not o n l y does one have t o r e p l a c e the z e r o - w i t h s i n g l e - q u a n t u m m u l t i p l e t s t r u c t u r e but one has t o do i t s e l e c t i v l y . The s ingle-quantum m u l t i p l e t s t r u c t u r e of one p a r t i c i p a t i n g s p i n must be found o n l y a t the p o s i t i v e zero-quantum frequency and t h a t of the o t h e r o n l y a t the n e g a t i v e zero-quantum f r e q u e n c y . Given these r e q u i r e m e n t s , the s o l u t i o n would seem t o p r e s e n t i t s e l f as f o l l o w s . F i r s t l y decouple the zero-quantum spectrum. Secondly t r a n s f e r the p o s i t i v e zero-quantum coherence f r e q u e n c y encoding t o one p a r t i c i p a t i n g s p i n and the n e g a t i v e t o the o t h e r . T h i r d l y J-encode the r e s u l t i n g s i n g l e - q u a n t u m coherences b e f o r e a c q u i s i t i o n . The d e c o u p l i n g of zero-quantum coherences has been d i s c u s s e d p r e v i o u s l y ( s e c t i o n 2.4). I t was noted t h a t i f a i s v a r i e d from 90° i n the d e c o u p l e d zero-quantum experiment p r e f e r e n t i a l t r a n s f e r of coherence w i l l occur [12,19,21], The p o s i t i v e zero-quantum frequency w i l l be p r e f e r e n t i a l l y t r a n s f e r e d t o one p a r t i c i p a t i n g s p i n and the n e g a t i v e zero-quantum fre q u e n c y t o the o t h e r . T h i s i s p o s s i b l e because v a r y i n g a from 90° r e s u l t s i n phase i n s t e a d of a m p l i t u d e m o d u l a t i o n of s i n g l e - q u a n t u m coherence o r i g i n a t i n g from zero-quantum coherence. T h i s makes p o s s i b l e quadrature-phase d e t e c t i o n w i t h r e s p e c t t o t , . With phase m o d u l a t i o n , u n l i k e a m p l i t u d e m o d u l a t i o n , p o s i t i v e and n e g a t i v e f r e q u e n c i e s encoded can be d i s t i n g u i s h e d . 52 From T a b l e I I and e q u a t i o n s 2.2, 2.3 and 2.14 i t can be seen t h a t f o r an AX 2 s p i n system the p o t e n t i a l l y o b s e r v a b l e a n t i p h a s e s i n g l e - q u a n t u m coherence p r e s e n t a f t e r the p u l s e a-90° o r i g i n a t i n g from zero-quantum coherence i n the broad-band d e c o u p l i n g experiment i s g i v e n by: o P ( r + t d + ) = ( 2 IAz XXx - 2 I A x I X z ) c o s I ( 0 A - ° X ) ( t i - t d ) ] c o s ( i r J e f f t d ) ( 2 ' 1 5 ) C l e a r l y the a n t i p h a s e s i n g l e - q u a n t u m coherence i n e q u a t i o n 2.15 i s o n l y a m p l i t u d e modulated. T h e r e f o r e p o s i t i v e and n e g a t i v e zero-quantum f r e q u e n c i e s a re i n d i s t i n g u i s h a b l e and w i l l both be c a r r i e d by bo t h p a r t i c i p a t i n g s p i n s . From T a b l e I I I and e q u a t i o n s 2.2, 2.3 and 2.14 i t can be seen t h a t f o r an AX 2 s p i n system the g e n e r a l e x p r e s s i o n f o r p o t e n t i a l l y o b s e r v a b l e a n t i p h a s e s i n g l e - q u a n t u m coherence c r e a t e d from zero-quantum coherence by an a r b i t r a r y p u l s e a f o r an X s p i n i s g i v e n by: ° X ( ' + t d + ) = 2 I A z I X x c o s [ ( 0 A - f l x ) ( t i - t d ) ] c o s U J e f f t d ) s i n a + 2 I A z I X y s i n [ ( ^ A " f i x ) ( t , - t ^ ) ] c o s ( r r J e f f t ^ J s i n a c o s a (2.16) and f o r an A s p i n : - 2 I A x I X z c o s [ ( n A " ^ x ) ( t i ~ t d ) ] c o s ( 7 r J e f f t d ) s i n a + 2 I A y I X z s i n [ ( f i A _ f i x ) ( t , - t d ) ] c o s ( 7 r J e f f t d ) s i n a c o s a 53 •41. I v I v s i n [ (S2 -fl„) ( t , - t , ) ]sin ( 7 r J , , t J s i n a c o s a Ax Xz Xz A X a e r r a ~ 4 I A y I X z I X z c o s ^ ( n A ~ f l X ) ( t l _ t d ) ] s i n ( i r J e f f t d ) s i n a c o s 2 a (2.17) Both s p i n s now have two o r t h o g o n a l components of a n t i p h a s e s i n g l e - q u a n t u m coherence o r i g i n a t i n g from in-phase zero-quantum coherence (the f i r s t two terms i n e q u a t i o n s 2.16 and 2.17). One i s modulated by c o s [ ( f i A - f l x ) ( t , - t ^ ) ] and the o t h e r by s i n [ ( f l A " f l x ) ( t , - t ^ ) ] i n each c a s e . Hence each s p i n w i l l be phase modulated as a f u n c t i o n of the e x t e n t of c h e m i c a l s h i f t e v o l u t i o n of the p a r e n t zero-quantum coherence making p o s s i b l e q u a d r a t u r e - p h a s e d e t e c t i o n w i t h r e s p e c t t o t , . There i s an i m p o r t a n t d i f f e r e n c e between e q u a t i o n s 2.16 and •2.17, i n t h a t the l a t t e r has a n e g a t i v e term. T h i s r e s u l t s i n the sense of phase m o d u l a t i o n of the two s p i n s b e i n g d i f f e r e n t ; one s p i n w i l l have a p o s i t i v e phase m o d u l a t i o n w i t h r e s p e c t t o t , and the o t h e r w i l l have a c o r r e s p o n d i n g n e g a t i v e phase m o d u l a t i o n . The d i r e c t r e s u l t of t h i s i s t h a t when F o u r i e r t r a n s f o r m e d w i t h r e s p e c t t o t , the s p i n w i t h a p o s i t i v e encoded phase w i l l g i v e r i s e t o the p o s i t i v e zero-quantum peak, and the o t h e r , w i t h a n e g a t i v e encoded phase, the n e g a t i v e zero-quantum peak. Hence p r e f e r e n t i a l t r a n s f e r of coherence i s a c h i e v e d as d e s i r e d . U n f o r t u n a t e l y t h e r e a r e c o m p l i c a t i o n s . The f i r s t two terms of e q u a t i o n s 2.16 and 2.17 show d i f f e r e n t dependencies upon the p u l s e a n g l e a, s i n a and s i n a c o s a r e s p e c t i v e l y . I f s i n a = s i n a c o s a then both terms w i l l have e q u a l i n t e n s i t i e s and pure phase m o d u l a t i o n w i l l r e s u l t , but i f they are not- e q u a l t h e i r m o d u l a t i o n can be 54 broken down i n t o two components. One component i s phase modulated and the o t h e r a m p l i t u d e modulated. The phase modulated component r e s u l t s i n p r e f e r e n t i a l t r a n s f e r of coherence as d e s c r i b e d above, but the a m p l i t u d e modulated m a g n e t i z a t i o n w i l l r e s u l t i n n o n - p r e f e r e n t i a l t r a n s f e r , g i v i n g r i s e t o both p o s i t i v e and n e g a t i v e zero-quantum f r e q u e n c i e s upon F o u r i e r t r a n s f o r m a t i o n . T h i s reduces the o v e r a l l e f f i c i e n c y of the p r e f e r e n t i a l t r a n s f e r p r o c e s s . As a tends t o z e r o s i n a tends t o s i n a c o s a and the e f f i c i e n c y of p r e f e r e n t i a l t r a n s f e r w i l l i n c r e a s e a l t h o u g h the o v e r a l l s i g n a l i n t e n s i t y w i l l d e c r e a s e . I t has been shown elsewhere t h a t the i n t e n s i t y r a t i o of the two peaks a r i s i n g from the F o u r i e r t r a n s f o r m a t i o n of the m o d u l a t i o n of one k i n d s p i n i s p r o p o r t i o n a l t o t a n 2 ( a / 2 ) [ 2 1 ] . The t h i r d and f o u r t h terms of e q u a t i o n 2.17 o r i g i n a t e from a n t i p h a s e zero-quantum coherence and, as w i t h t h e f i r s t two terms, c o n s i s t of phase and a m p l i t u d e modulated components. T h i s w i l l g i v e r i s e t o p r e f e r e n t i a l t r a n s f e r w i t h the same e f f i c i e n c y as the f i r s t two terms, a l t h o u g h w i t h a d i f f e r e n c e i n phase of 90°. In more complex s p i n systems coherence may be t r a n s f e r r e d t o s p i n s of a k i n d not i n v o l v e d i n the zero-quantum coherence concerned (page 4 4 ) . The e f f i c i e n c y of t h i s p r o c e s s i s s i g n i f i c a n t l y reduced when a i s reduced from 90° t o 45° and c o n s e q u e n t l y i s not a s i g n i f i c a n t problem. 90; 180°, 90S 180°. OiJ 180° Gradient Z 1 T/2 T/2 t:/2 t d - t i / 2 n t , 2 1 n t , 1 Xniiiii,,.. cn 1 1 Preparation ] Zero-quantum evolution 1 1 1 1 1 Refocussing cn Acquisition I I F i g u r e 2.13. Single-quantum J - r e s o l v e d broad band d e c o u p l e d zero-quantum experiment. 56 T a k i n g the above i n t o account i t can be seen t h a t the c h o i c e of a w i l l be a compromise between p r e f e r e n t i a l t r a n s f e r , which becomes more e f f i c i e n t as a t e n d s t o z e r o , and o v e r a l l s i g n a l i n t e n s i t y , which d e c r e a s e s as a tends t o z e r o . The t h i r d requirement i s t o J-encode t h e m a g n e t i z a t i o n , as i s done i n the c o n v e n t i o n a l J - r e s o l v e d t w o - d i m e n s i o n a l experiment [24,25]. T h i s i s a c h e i v e d by r e p l a c i n g the p u r e l y r e f o c u s s i n g subsequence which f o l l o w s the p u l s e a by one which r e f o c u s s e s and J-encodes. T h i s i s done by r e p l a c i n g the r e f o c u s s i n g time r w i t h n t , , and r e t a i n i n g t h e 180° p u l s e at the c e n t r e of t h i s p e r i o d , F i g u r e 2.13. T h i s subsequence f u n c t i o n s i n the same way as a normal J - r e s o l v e d t w o - d i m e n s i o n a l experiment by i n t r o d u c i n g phase m o d u l a t i o n due to s c a l a r c o u p l i n g e v o l u t i o n as a f u n c t i o n of t , [ 2 4 , 2 5 ] , n i s a J - s c a l i n g f a c t o r : the s i n g l e - q u a n t u m m u l t i p l e t superimposed on t o p of the decoupled zero-quantum coherence w i l l be s c a l e d - u p r e l a t i v e t o the r e s t of the spectrum by a f a c t o r of n. A v a l u e of n>1 would i n c r e a s e the r e s o l u t i o n of the s i n g l e - q u a n t u m m u l t i p l e t s , a l t h o u g h a l s o p o t e n t i a l l y i n c r e a s i n g the o v e r l a p of a d j a c e n t c o h e r e n c e s . A v a l u e of n<1 would decrease the r e s o l u t i o n of the s i n g l e - q u a n t u m m u l t i p l e t s t r u c t u r e , u s e f u l where t h e r e i s o v e r l a p p i n g , a l t h o u g h i t s h o u l d be noted t h a t i n s u f f i c i e n t r e s o l u t i o n may l e a d t o c a n c e l l a t i o n of peaks. The e v o l u t i o n of i n v i s i b l e a n t i p h a s e s i n g l e - q u a n t u m coherence i n t o v i s i b l e in-phase s i n g l e - q u a n t u m coherence and 57 J - e n c o d i n g both depend upon e v o l u t i o n of the m a g n e t i z a t i o n due t o s c a l a r c o u p l i n g s . The c e n t r a l peak of an odd numbered m u l t i p l e t does not e v o l v e d u r i n g the J - e n c o d i n g subsequence. C o n s e q u e n t l y the c e n t r a l peaks of odd numbered m u l t i p l e t s superimposed upon de c o u p l e d zero-quantum coherences w i l l not be o b s e r v e d i n t h i s e x p e r i m e n t . T h i s i s because a l l s i n g l e - q u a n t u m coherence modulated as a f u n c t i o n of the e x t e n t of zero-quantum coherence e v o l u t i o n i s i n i t i a l l y i n v i s i b l e a n t i p h a s e m a g n e t i s a t i o n a f t e r the p u l s e a. C o n s e q u e n t l y as they do not e v o l v e due t o s c a l a r c o u p l i n g s the c e n t r a l peaks of odd numbered m u l t i p l e t s w i l l remain a n t i p h a s e and hence i n v i s i b l e . However, t h i s r e s u l t s i n o n l y minor i n c o n v e n i e n c e as the s i n g l e - q u a n t u m m u l t i p l e t s t r u c t u r e of both s p i n s a c t i v e i n the zero-quantum coherence w i l l be p r e s e n t i n the r e s u l t i n g spectrum. As the two m u l t i p l e t s must share a c o u p l i n g c o n s t a n t and as t h e r e must be a s u f f i c i e n t number of each k i n d of s p i n t o cause the observed s p l i t t i n g of the o t h e r the absence of a peak can be r e a d i l y deduced. Any r e m a i n i n g a m b i g u i t y may be removed by r e f e r e n c e t o the c o n v e n t i o n a l zero-quantum spectrum. The experiment i s demonstrated f o r a s o l u t i o n of 0.1 molar L - a l a n i n e i n D 20 i n F i g u r e 2.14 C, w i t h the J - s c a l i n g f a c t o r n s e t t o 2 t o improve r e s o l u t i o n of the m u l t i p l e t s . A d o u b l e t and a q u a r t e t a r e p r e s e n t as e x p e c t e d and e x h i b i t one c o u p l i n g c o n s t a n t as would be e x p e c t e d f o r an AX 3 s p i n system. The c o n v e n t i o n a l and broad-band d e c o u p l e d zero-quantum s p e c t r a a r e g i v e n i n F i g u r e s 2.14 A and B r e s p e c t i v e l y . 58 B 300 F i g u r e 2.14. S p e c t r a of 0.1 molar L - a l a n i n e i n D 20. A. C o n v e n t i o n a l zero-quantum spectrum o b t a i n e d w i t h the r e f o c u s s e d zero-quantum e x p e r i m e n t . T , T ' = 6 0 msec, At,=1.67 msec. 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . B. Broad band d e c o u p l e d zero-quantum spectrum. T=60 msec, t (j=443 msec, At, = 1.67 msec. 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . C. Single-quantum J - r e s o l v e d broad band d e c o u p l e d zero-quantum spectrum. a=22.5°, T=60 msec, t^=443 msec, At,=1.67 msec, n=2. 256 b l o c k s c o l l e c t e d , 8 a c q i s i t i o n s per b l o c k . ~\—r 1 I r 200 - i — i — i — r 300 100 T 0.0 ~i—r ~i—r T ~i—i—i—r T ~ i — r -100 -200 1 Hz F i g u r e 2.15. Single-quantum J - r e s o l v e d broad band d e c o u p l e d zero-quantum spectrum of e t h a n o l i n D 20 ( 2 : 1 ) . a=45°, T=60 msec, t ( j = 4 l 3 msec, At , = 1.67 msec, n=2. 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . 59 300 200 100 0.0 -100 -200 Hz F i g u r e 2.16. Single-quantum J - r e s o l v e d broad band d e c o u p l e d zero-quantum s p e c t r a of 2-propanol i n D 20 ( 3 : 1 ) . A. a=45°. B. a=22.5°. In both c a s e s T=60 msec, tqj=493 msec, At,=1.67 msec, n=2. 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . 60 The c o r r e s p o n d i n g spectrum of e t h a n o l i s g i v e n i n F i g u r e 2.15. The c e n t r a l peak of the t r i p l e t i s m i s s i n g though as • s t a t e d above t h i s i s o b v i o u s g i v e n the n e c e s s i t y of a common c o u p l i n g c o n s t a n t . In t h i s case i t can be seen t h a t n o n - p r e f e r e n t i a l t r a n s f e r of coherence i s not s i g n i f i c a n t . Due t o the o v e r l a p of resonances i t i s not p o s s i b l e t o a s c e r t a i n the e x t e n t of n o n - p r e f e r e n t i a l t r a n s f e r of coherence f o r L - a l a n i n e . In F i g u r e 2.16 A i s g i v e n the c o r r e s p o n d i n g spectrum f o r 2 - p r o p a n o l . A l t h o u g h the d o u b l e t i s p r e s e n t as e x p e c t e d the s e p t e t , minus i t s c e n t r a l peak, has an e x t r a d o u b l e t s t i c k i n g out of i t due t o n o n - p r e f e r e n t i a l t r a n s f e r of c o h e r e n c e . As was d i s c u s s e d above, the e f f i c i e n c y of p r e f e r e n t i a l t r a n s f e r i n c r e a s e s as a d e c r e a s e s ( p r o p o r t i o n a l t o t a n 2 ( a / 2 ) ) . T h e r e f o r e by r e d u c i n g a from 45° t o 22.5° t h i s problem can be g r e a t l y a l l e v i a t e d , a l t h o u g h a t the c o s t of r e d u c i n g o v e r a l l s i g n a l i n t e n s i t y , F i g u r e 2.16 B. I n t e r f e r e n c e due t o coherence t r a n s f e r t o s p i n s not of a k i n d i n v o l v e d i n a g i v e n zero-quantum coherence w i l l a l s o be reduced by r e d u c i n g the a n g l e a. Coherence t r a n s f e r t o such s p i n s can be f u r t h e r reduced by c h o o s i n g a v a l u e of t ^ f o r which the zero-quantum terms t h a t g i v e r i s e t o i t are e i t h e r absent or r e l a t i v e l y s m a l l . I f the optimum v a l u e of t ^ i s not known the experiment can be p e r f o r m e d w i t h s e v e r a l d i f f e r e n t v a l u e s and the r e s u l t s co-added t o o b t a i n the complete spectrum. L i k e w i s e t o ensure t h a t t h e i n i t i a l e x c i t a t i o n of zero-quantum coherences i s complete and r e s o n a b l y u n i f o r m i t may be n e c e s s a r y t o use s e v e r a l v a l u e s of T . However, the spectrum o b t a i n e d from one 61 experiment w i l l a l l o w one t o d etermine whether or not a l l the zero-quantum coherences of a r e p r e s e n t e d s p i n system are p r e s e n t from the number of d i f f e r e n t s c a l a r c o u p l i n g s e x h i b i t e d by the m u l t i p l e t s p r e s e n t . 2.6 The R e c o n s t r u c t i o n of Single-Quantum S p e c t r a i n  Inhomogeneous Magnetic F i e l d s . N u c l e a r Magnetic Resonance as a t o o l i n a n a l y t i c a l c h e m i s t r y i s c e n t r e d upon the a b i l i t y t o i n t e r p r e t s i n g l e - q u a n t u m s p e c t r a , the a r e a of peaks, t h e i r m u l t i p l e t s t r u c t u r e s , c o u p l i n g c o n s t a n t s , and c h e m i c a l s h i f t s [ 2 6 - 2 8 ] , C o n v e n t i o n a l zero-quantum s p e c t r a g i v e r i s e t o none of t h i s i n f o r m a t i o n d i r e c t l y [ 1 2 , 2 0 ] , T h e i r peak a r e a s a r e complex f u n c t i o n s of c o u p l i n g c o n s t a n t s and p r e p a r a t i o n t i m e . Thesir m u l t i p l e t s t r u c t u r e s are d i f f e r e n t . T h e i r e f f e c t i v e c o u p l i n g c o n s t a n t s are d i f f e r e n t . A l s o , o n l y the magnitude of the d i f f e r e n c e of the c h e m i c a l s h i f t s of two c o u p l e d s p i n s i s r e v e a l e d . A l t h o u g h i t i s c o n c e i v a b l e t h a t new a n a l y t i c a l methods might be d e v e l o p e d to a l l o w ready e x t r a c t i o n of i n f o r m a t i o n from zero-quantum s p e c t r a as from t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s , the c o m p l e x i t y of such a t a s k , g i v e n the complex r e l a t i o n s h i p s between the c o u p l i n g c o n s t a n t s , peak a r e a s , c h e m i c a l s h i f t s e t c . of the two would seem t o make such a development u n l i k e l y . 62 As NMR a n a l y s i s has developed around and c e n t e r e d upon si n g l e - q u a n t u m s p e c t r a the most u s e f u l methods of o b t a i n i n g s p e c t r a l i n f o r m a t i o n i n inhomogeneous magnetic f i e l d s a r e l i k e l y t o be those which p r o v i d e s i n gle-quantum parameters i n the most d i r e c t l y a c c e s s i b l e way. The single-quantum J - r e s o l v e d broad-band decoupled zero-quantum experiment comes c o n s i d e r a b l y n e a r e r t o t h i s g o a l than the c o n v e n t i o n a l zero-quantum e x p e r i m e n t . T h i s i s because i t r e p l a c e s the e f f e c t i v e zero-quantum c o u p l i n g c o n s t a n t s and m u l t i p l e t s t r u c t u r e s w i t h the single-quantum c o u p l i n g c o n s t a n t s and m u l t i p l e t s t r u c t u r e s of those s p i n s a c t i v e i n a g i v e n zero-quantum coherence. B e s i d e s making a c c e s s i b l e the i n f o r m a t i o n i n h e r e n t w i t h i n these parameters i t a l s o means t h a t s c a l a r c o u p l i n g networks can be e a s i l y t r a c e d . T h i s i s because a s p i n a c t i v e i n two d i f f e r e n t zero-quantum coherences w i l l g i v e r i s e t o the same m u l t i p l e t s t r u c t u r e on one peak f o r each coherence. T h i s i s demonstrated f o r L - t h r e o n i n e i n F i g u r e 2.17A. The presence of the o c t e t , though i n c o m p l e t e l y r e s o l v e d , on one peak of each zero-quantum coherence i n d i c a t e s the c o n n e c t i v i t y . Hence the s p i n system may be r e a d i l y deduced. The two d o u b l e t s have d i f f e r e n t c o u p l i n g c o n s t a n t s and so do not r e s u l t from the same s p i n . A l a r g e p a r t of the a n a l y s i s of single-quantum s p e c t r a i s based upon c h e m i c a l s h i f t s ; methylene p r o t o n s r e s o n a t e i n one r e g i o n of the spectrum, a r o m a t i c p r o t o n s i n a n o t h e r , e t c . T h i s i n f o r m a t i o n would seem t o be l a c k i n g i n a zero-quantum spectrum which o n l y p r e s e n t s one w i t h the d i f f e r e n c e i n 63 c h e m i c a l s h i f t s . A zero-quantum f r e q u e n c y of a g i v e n magnitude may c o r r e s p o n d t o a c o u p l i n g between two s p i n s s e p a r a t e d by t h a t f r e q u e n c y i n any number of r e g i o n s i n the s i n g l e - q u a n t u m spectrum. Hence one i s g i v e n a much l e s s c e r t a i n i d e a as t o t h e i r i d e n t i t y . I f o n l y one c o u l d t e l l from the s i n g l e - q u a n t u m J - r e s o l v e d broad-band d e c o u p l e d zero-quantum spectrum which of the two m u l t i p l e t s on a g i v e n coherence was, say, d o w n f i e l d of the o t h e r i t would a l l o w one t o a c c u r a t e l y reassemble from t h i s experiment the complete single-quantum spectrum except f o r s i n g l e t s and a r e f e r e n c e such as TMS. From e q u a t i o n s 2.16 and 2.17 i t can be seen t h a t a f t e r the p u l s e a the a n t i p h a s e s i n g l e quantum coherence of each s p i n c o n s i s t s of two t y p e s of o r t h o g o n a l term. One i s •modulated by c o s [ ( & A ~ & X ) ( t i - t ^ ) ] , and the o t h e r by s i n [ ( n A ~ n x ) ( t , - t ^ ) ] . The f i r s t of t h e s e i s o n l y dependent upon the magnitude of (^ A~^ x) ' ^ u t fc^e s e c o n c ^ ' t n e s i n e modulated term, i s a l s o dependent upon the s i g n . T h e r e f o r e i t can be seen t h a t i f n A > & x the m o d u l a t i o n of s p i n X w i l l have a n e g a t i v e phase m o d u l a t i o n w i t h r e s p e c t t o t , and s p i n A a p o s i t i v e phase modulated f r e q u e n c y . I f & A < f l x the r e v e r s e w i l l be t r u e : s p i n X w i l l have a . p o s i t i v e phase m o d u l a t i o n and s p i n A a n e g a t i v e phase m o d u l a t i o n . Consequently the s i n g l e - q u a n t u m m u l t i p l e t a t the n e g a t i v e zero-quantum frequency w i l l always be the most u p f i e l d of the two s p i n s and v i c e v e r s a . The presence of a 180° p u l s e b e f o r e a c q u i s i t i o n w i l l r e v e r s e the s i g n s of a l l zero-quantum f r e q u e n c i e s . 64 l i i i i j — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | 300 200 100 0.0 -100 -200 Hz B Zero-quantum coherence connectivity Reconstructed single-quantum spectrum 11 65 F i g u r e 2.17. A. Single-quantum J - r e s o l v e d broad band d e c o u p l e d zero-quantum spectrum of 0.5 molar L - t h r e o n i n e i n D 20 r e s u l t i n g from the a d d i t o n of two s p e c t r a o b t a i n e d i n e x p e r i m e n t s i n which r was s e t t o 60 msec and 140 msec. A l l o t h e r p a r a m e t e r s were the same: ^=431 msec, At,=1.67 msec, n=2. 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . B. R e p r e s e n t a t i o n of the method of r e c o n s t r u c t i o n of a s i n g l e - q u a n t u m spectrum from the spectrum o b t a i n e d i n the s i n g l e - q u a n t u m J - r e s o l v e d broad band d e c o u p l e d zero-quantum experiment f o r L - t h r e o n i n e . F i r s t l y i t i s n e c e s s a r y t o r e c o g n i s e t h e zero-quantum c o n n e c t i v i t i e s which a r e r e v e a l e d by the o c c u r a n c e of the same s i n g l e - q u a n t u m m u l t i p l e t on d i f f e r e n t zero-quantum c o h e r e n c e s . Connected zero-quantum coherences a r e then r e a r r a n g e d so t h a t t h e i r common m u l t i p l e t s a r e superimposed upon one a n o t h e r . T h i s g i v e s r i s e t o the s i n g l e - q u a n t u m spectrum except f o r s i n g l e t s which do not p a r t i c i p a t e i n zero-quantum c o h e r e n c e s . C. Single-quantum spectrum of L - t h r e o n i n e r e c o n s t r u c t e d from the spectrum i n p a r t A a c c o r d i n g t o the method d e s c r i b e d above D. C o n v e n t i o n a l s i n g l e - q u a n t u m spectrum of L - t h r e o n i n e o b t a i n e d under the same c o n d i t i o n s . 66 C o n s e q u e n t l y , w i t h t h i s experiment i t i s p o s s i b l e t o e a s i l y reassemble the s i n g l e - q u a n t u m spectrum even i n an inhomogeneous magnetic f i e l d where o b t a i n i n g the normal single-quantum s p e c t r a i s o t h e r w i s e i m p o s s i b l e , e x c e p t . The e x c e p t i o n s t o t h i s a re s i n g l e s p i n s (which do not form zero-quantum coherences) and a r e f e r e n c e such as TMS. T h i s i s demonstrated i n F i g u r e 2.17 f o r L - t h r e o n i n e . One advantage t h i s t e c h n i q u e does have over a c o n v e n t i o n a l s i n g l e - q u a n t u m spectrum b e s i d e s independence of magnetic f i e l d i n h o m o g e n e i t i e s i s t h a t any m u l t i p l e t hidden underneath a s i n g l e t such as HDO w i l l now be r e v e a l e d . T h i s experiment has t h r e e major d i s a d v a n t a g e s . F i r s t l y , peak a r e a s are complex f u n c t i o n s of c o u p l i n g c o n s t a n t and the l e n g t h of the p r e p a r a t i o n p e r i o d T . S e c o n d l y , the s m a l l v a l u e of the p u l s e a n e c e s s a r y t o a c h i e v e e f f i c i e n t p r e f e r e n t i a l coherence t r a n s f e r r e s u l t s i n the a c q u i r e d s i g n a l h a v i n g a r e l a t i v e l y low i n t e n s i t y which may be r e f l e c t e d i n poor S/N. T h i r d l y , the l e n g t h of time i n v o l v e d : each experiment t a k e s a minimum time of ( r + t ^ ) which may e a s i l y amount t o 400-500 msec, and i n a d d i t i o n , the Nth b l o c k of the experiment w i l l c o n t a i n an a d d i t i o n a l d e l a y due t o J-encoding of ( N n A t , ) . T h i s i s p o t e n t i a l l y a major problem i f the c h e m i c a l s p e c i e s of i n t e r e s t have s h o r t s p i n - s p i n r e l a x a t i o n t i m e s . 2.7 The Assignment of Zero-Quantum Coherence S p e c t r a i n  Inhomogeneous Magnetic F i e l d s 67 In t h i s s e c t i o n the assignment of zero-quantum coherence s p e c t r a w i l l be d i s c u s s e d w i t h r e f e r e n c e t o a number of r e p r e s e n t a t i v e amino a c i d s f o r wh i c h the zero-quantum s p e c t r a o b t a i n e d under a s t a n d a r d s e t of c o n d i t i o n s a re g i v e n i n Appendi x 11 . The assignment of the zero-quantum spectrum of a known compound i s p o t e n t i a l l y t r i v i a l i f a f u l l y a s s i g n e d s i n g l e - q u a n t u m spectrum i s a v a i l a b l e . I t i s o n l y n e c e s s a r y to c a l c u l a t e the d i f f e r e n c e i n resonance f r e q u e n c i e s of each p a i r of c o u p l e d s p i n s and t o lo o k f o r a peak at t h a t f r e q u e n c y i n the zero-quantum spectrum. For example, the s i n g l e - q u a n t u m spectrum of L - t h r e o n i n e i n D 20 has t h r e e resonances [ 2 9 ] , at 288 Hz, 347 Hz, and 105 Hz, c o r r e s p o n d i n g t o the a, 0, and 7 p r o t o n s . C o n s e q u e n t l y , assuming a l l the p r o t o n s have m u t u a l l y r e s o l v e d s c a l a r c o u p l i n g s , one would expect t o f i n d the zero-quantum coherences a-/3, £-7, and a-7 a t 59 Hz, 242 Hz, and 183 Hz r e s p e c t i v e l y . The former two coherences o c c u r as p r e d i c t e d ( F i g u r e A2.7). A l t h o u g h i t does occur the l a t t e r i s of r e l a t i v e l y low i n t e n s i t y as t h e c o u p l i n g c o n s t a n t i s s m a l l . S m a l l v a l u e s of f and r' t e n d t o emphasise zero-quantum coherences between s p i n s w i t h a l a r g e mutual c o u p l i n g c o n s t a n t whereas zero-quantum coherences between more weakly c o u p l e d s p i n s w i l l be emphasised by l a r g e r v a l u e s of T and T ' . T h i s i s because the zero-quantum coherence of two s p i n s i s g e n e r a t e d from those components of t h e i r s i n g l e - q u a n t u m coherence which are a n t i p h a s e w i t h r e s p e c t t o each o t h e r . The r a t e a t which in-phase s i n g l e - q u a n t u m coherence w i l l e v o l v e i n t o a n t i p h a s e 68 si n g l e - q u a n t u m coherence w i t h r e s p e c t t o a g i v e n s p i n i s dependent upon t h e i r mutual c o u p l i n g c o n s t a n t . C onsequently i f the c o u p l i n g c o n s t a n t i s l a r g e the d e s i r e d a n t i p h a s e s i n g l e - q u a n t u m coherence w i l l e v o l v e q u i c k l y , and i f i t i s s m a l l s l o w l y . I f two or more zero-quantum coherences o v e r l a p a knowledge of t h e i r f r e q u e n c i e s may be i n s u f f i c i e n t t o determine which are p r e s e n t . In t h i s case t h e r e a re t h r e e a l t e r n a t i v e c o u r s e s of a c t i o n . F i r s t l y , the zero-quantum coherences may have d i f f e r e n t m u l t i p l e t s t r u c t u r e s , i n which c a s e , i f r e s o l u t i o n i s s u f f i c i e n t , the coherences p r e s e n t can be de t e r e m i n e d . S e c o n d l y , by u s i n g d i f f e r e n t v a l u e s of r and r ' , the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s , i t may be p o s s i b l e t o e d i t out some of the o v e r l a p p i n g coherences t o s i m p l i f y the spectrum. For example, the zero-quantum spectrum of L - i s o l e u c i n e o b t a i n e d w i t h T and T'=60 msec c o n t a i n s a number of o v e r l a p p i n g c o h e r e n c e s , F i g u r e A2.8.A. By changing T and r' t o 100 msec, F i g u r e A2.8.B, or 140 msec, F i g u r e A2.8.C, the spectrum i s g r e a t l y s i m p l i f i e d and the o v e r l a p of coherences reduced. T h i r d l y , the zero-quantum spectrum can be broad-band de c o u p l e d t o t r y and r e s o l v e coherences whose m u l t i p l e t s o v e r l a p . For example, from an e x a m i n a t i o n of the si n g l e - q u a n t u m spectrum of L - v a l i n e [30] one might expect 69 t h r e e s t r o n g zero-quantum c o h e r e n c e s : a -0, 0~7, and 0-7' a t 107 Hz, 99 Hz, and 94 Hz r e s p e c t i v e l y . The o v e r l a p of m u l t i p l e t s i n the zero-quantum spectrum makes i t i m p o s s i b l e t o determine which of the s e coherences a r e p r e s e n t ( F i g u r e A2.9). Upon broad-band d e c o u p l i n g the zero-quantum spectrum, F i g u r e 2.19.B, i t can be seen t h a t o n l y the a-0 and 0-7 coherences a r e p r e s e n t , a l t h o u g h the peaks are i n c o m p l e t e l y r e s o l v e d . What does one do i f an a s s i g n e d s i n g l e - q u a n t u m spectrum i s not a v a i l a b l e ? In si n g l e - q u a n t u m NMR s p e c t r o s c o p y i t i s o f t e n p o s s i b l e t o a s s i g n a spectrum, p a r t i a l l y or w h o l l y , by the use of fo u r types of par a m e t e r s : c h e m i c a l s h i f t s , m u l t i p l e t s t r u c t u r e s , c o u p l i n g c o n s t a n t s , and peak a r e a s . C h emical s h i f t s t e l l one about the environment of a s p i n w i t h i n a m o l e c u l e , whether i t i s a r o m a t i c , a d j a c e n t t o a h e t e r o n u c l e u s e t c . The m u l t i p l e t s t r u c t u r e of a resonance p r o v i d e s i n f o r m a t i o n on n e i g h b o u r i n g s p i n s ; how many s p i n s i t i s c o u p l e d t o and whether they a r e i d e n t i c a l or d i f f e r e n t and hence what k i n d of s p i n system i t i s p a r t o f . C o u p l i n g c o n s t a n t s t e l l one about the r e l a t i v e l o c a t i o n s of c o u p l e d s p i n s , by t h e i r magnitude whether a c o u p l i n g i s th r o u g h two, t h r e e , or f i v e bonds, and they may a l s o p r o v i d e c o n f o r m a t i o n a l i n f o r m a t i o n . Peak a r e a s t e l l one how many s p i n s g i v e r i s e t o a p a r t i c u l a r resonance and are p a r t i c u l a r l y u s e f u l when a number 70 of resonances o v e r l a p , and f a c i l i t a t e the i d e n t i f i c a t i o n of s p i n systems. To what e x t e n t i s the i n f o r m a t i o n summarized above a c c e s s i b l e w i t h i n a zero-quantum coherence spectrum? Zero-quantum coherences occur a t the d i f f e r e n c e i n f r e q u e n c i e s of two c o u p l e d s p i n s and a r e " t h e r e f o r e found w i t h i n a narrower frequency range than t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s . Zero-quantum f r e q u e n c i e s a r e r e l a t i v e r a t h e r than a b s o l u t e and c o n s e q u e n t l y the i n f o r m a t i o n which can be e x t r a c t e d from them i s a l s o r e l a t i v e . One cannot t e l l , f o r example, whether a p r o t o n a c t i v e i n a zero-quantum coherence i s l i k e l y t o be a r o m a t i c or a d j a c e n t t o a h e t e r o n u c l e u s - f r o m i t s p o s i t i o n i n the spectrum. One can o n l y t e l l whether i t a r i s e s from two s p i n s i n a s i m i l a r e nvironment, hence h a v i n g s i m i l a r c h e m i c a l s h i f t s and g i v i n g r i s e t o a s m a l l zero-quantum f r e q u e n c y , or between two s p i n s i n d i f f e r e n t e nvironments and hence g i v i n g r i s e t o a l a r g e zero-quantum f r e q u e n c y . The m u l t i p l e t s t r u c t u r e of a zero-quantum coherence s t i l l p r o v i d e s i n f o r m a t i o n on the s p i n system of which the s p i n s a c t i v e i n the coherence are p a r t , but i n a d i f f e r e n t way from s i n g l e - q u a n t u m c o h e r e n c e s . Zero-quantum coherences o n l y e x h i b i t c o u p l i n g s t o those s p i n s not a c t i v e i n the coherence and hence g i v e r i s e t o d i f f e r e n t s p l i t t i n g p a t t e r n s . 71 The c o u p l i n g c o n s t a n t s e x h i b i t e d by zero-quantum coherences o f t e n c o r r e s p o n d t o the d i f f e r e n c e of two s i n g l e - q u a n t u m c o u p l i n g c o n s t a n t s ( e q u a t i o n 2.15) making them more d i f f i c u l t t o i n t e r p r e t . A s m a l l zero-quantum c o u p l i n g c o n s t a n t may c o r r e s p o n d t o the d i f f e r e n c e of two s m a l l or two l a r g e s i n g l e - q u a n t u m c o u p l i n g c o n s t a n t s . I f the r e l e v a n t s i n g l e - q u a n t u m c o u p l i n g c o n s t a n t s a r e v e r y s i m i l a r the c o r r e s p o n d i n g zero-quantum c o u p l i n g c o n s t a n t may be u n r e s o l v e d and may even l e a d t o the mutual c a n c e l l a t i o n of u n r e s o l v e d peaks. The peak a r e a s of zero-quantum coherences a r e complex f u n c t i o n s of s i n g l e - q u a n t u m c o u p l i n g c o n s t a n t s and the l e n g t h s of the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s of the zero-quantum e x p e r i m e n t . C o n s e q u e n t l y they cannot e a s i l y be used t o r e v e a l the number of s p i n s g i v i n g r i s e t o a zero-quantum coherence. T a k i n g t h e s e f a c t o r s i n t o account the i n f o r m a t i o n p r o v i d e d by zero-quantum coherences would seem t o be more ambiguous and t o have a lower c o n t e n t of u s e f u l i n f o r m a t i o n than t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s . From the amino a c i d s s u r v e y e d i t can be seen t h a t the assignment of the zero-quantum s p e c t r a of some w i l l be t r i v i a l ; t hose which g i v e r i s e t o o n l y one zero-quantum coherence such as L - c y s t e i n e ( F i g u r e A2.11), L - a l a n i n e ( F i g u r e A2.15), and L - a s p a r t i c a c i d ( F i g u r e A2.16). These are a l l compounds which have o n l y two t y p e s of c o u p l e d s p i n and hence produce o n l y one zero-quantum c o h e r e n c e , and a r e r a r e i n s t a n c e s of zero-quantum s p e c t r a 72 b e i n g e a s i e r t o a s s i g n than s i n g l e - q u a n t u m ones. Most compounds of i n t e r e s t however g i v e r i s e t o more than one zero-quantum coherence. For a known compound i t i s p o s s i b l e t o c a l c u l a t e the approximate s i n g l e - q u a n t u m resonance f r e q u e n c i e s of i t s s p i n s by the use of c h e m i c a l s h i f t t a b l e s , and hence t o c a l c u l a t e the f r e q u e n c i e s of zero-quantum c o h e r e n c e s from t h e s e . U n f o r t u n a t e l y c h e m i c a l s h i f t t a b l e s have l i m i t e d a c c u r a c y which i s degraded s t i l l f u r t h e r when a zero-quantum fr e q u e n c y i s c a l c u l a t e d . In one t y p i c a l c h e m i c a l s h i f t t a b l e [31] an a c c u r a c y of ±0.2 ppm i s q u o t e d . For a zero-quantum fr e q u e n c y thus c a l c u l a t e d the a c c u r a c y w i l l be reduced t o ±0.28 ppm, or 22.5 Hz a t 80.3 MHz. A l t h o u g h t h i s may be adequate t o a s s i g n s i m p l e r compounds w i t h a few w i d e l y d i s p e r s e d resonances such as L - t h r e o n i n e , f o r compounds which a r e o n l y a l i t t l e more complex i t p r o v e s t o be i n a d e q u a t e . Of the amino a c i d s surveyed over h a l f have cohe r e n c e s which a r e w i t h i n 22.5 Hz (0.28 ppm) of each o t h e r . For example, L - m e t h i o n i n e w i t h i n the range 30-50 Hz ( F i g u r e A2.2.C), L - g l u t a m i c a c i d w i t h i n the range 10-40 Hz ( F i g u r e A2.3.B) and L - i s o l e u c i n e w i t h i n the range 0-100 Hz ( F i g u r e A2.8.A). To an e x t e n t such a m b i g u i t i e s may be removed i f the c a l c u l a t e d m u l t i p l e t s t r u c t u r e s of t h e s e c o h e r e n c e s a r e d i f f e r e n t . T h i s has the p r e r e q u i s i t e t h a t the r e s o l u t i o n of the zero-quantum spectrum be h i g h enough t o r e s o l v e the s c a l a r c o u p l i n g s concerned. T h i s i s p o t e n t i a l l y a problem as the c o u p l i n g c o n s t a n t s of zero-quantum c o h e r e n c e s may be c o n s i d e r a b l y s m a l l e r than t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s thus l e a d i n g t o 73 a m b i g u i t y as t o t h e i r p e r c e i v e d m u l t i p l i c i t y . T h i s problem may be reduced by i n c r e a s i n g the r e s o l u t i o n of the spectrum, but t h i s would r e q u i r e more tim e , b oth i n i n c r e a s i n g the number of t , increments used and a l s o i n i n c r e a s i n g the number of a c q u i s i t i o n s per increment needed t o a t t a i n the same S/N. As the e v o l u t i o n t i m e , t , , i n c r e a s e s the e x t e n t t o which the m a g n e t i z a t i o n w i l l have r e l a x e d b e f o r e a c q u i s i t i o n a l s o i n c r e a s e s hence d e c r e a s i n g S/N. T h i s i s l i a b l e t o be p r o b l e m a t i c , p a r t i c u l a r l y f o r compounds w i t h s h o r t s p i n - s p i n r e l a x a t i o n t i m e s . In a d d i t i o n t o the e v o l u t i o n t i m e , s e v e r a l hundred m i l l i s e c o n d s (120-280 msec f o r the amino a c i d s p e c t r a g i v e n i n appendix 2) may be spent i n the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s of the e xperiment. O v e r l a p may a l s o p r e v e n t one from d e t e r m i n i n g the m u l t i p l i c i t y of a coherence a l t h o u g h t h i s problem may be a l l e v i a t e d by c h a n g i n g the l e n g t h s of the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s as d e s c r i b e d p r e v i o u s l y ( s e c t i o n 2.2). In the l i g h t of t h e s e c o n s i d e r a t i o n s i t would seem t h a t the assignment of zero-quantum s p e c t r a , i n the absence of any o t h e r NMR s p e c t r o s c o p i c i n f o r m a t i o n , i s not a r e a l i s t i c o b j e c t i v e , except f o r the v e r y s i m p l e s t compounds, w i t h c o n v e n t i o n a l zero-quantum e x p e r i m e n t s . However, t h e r e i s an a l t e r n a t i v e which has been developed h e r e i n : the s i n g l e - q u a n t u m J - r e s o l v e d broad-band d e c o u p l e d zero-quantum e x p e r i m e n t . T h i s experiment produces s p e c t r a w i t h many of the f e a t u r e s of s i n g l e - q u a n t u m s p e c t r a o f t e n r e g a r d e d as e s s e n t i a l f o r the assignment of c o h e r e n c e s . Each zero-quantum coherence 74 e x h i b i t s the sin g l e - q u a n t u m m u l t i p l e t s of the two c o u p l e d s p i n s which g i v e r i s e t o i t - r e v e a l i n g a s i n g l e - q u a n t u m s p i n - s p i n c o n n e c t i v i t y . A l s o , d i f f e r e n t zero-quantum c o h e r e n c e s which have a s p i n i n common have a m u l t i p l e t i n common. T h i s a l l o w s one t o extend s p i n - s p i n c o u p l i n g networks t h r o u g h d i f f e r e n t zero-quantum coherences and hence t o deter m i n e the s p e c i f i c l o c a t i o n w i t h i n a s p i n system of the s p i n s a c t i v e i n a p a r t i c u l a r coherence. With t h i s experiment i t may a l s o be p o s s i b l e t o i d e n t i f y unknown compounds, or a t l e a s t the s p i n systems they c o n t a i n , by v i r t u e of the f a c t t h a t t h e i r s i n g l e - q u a n t u m s p e c t r a can be l a r g e l y reassembled from the one produced i n t h i s e x p e r i m e n t . A l t h o u g h the s p e c t r a l w i d t h of the experiment i s r e l a t i v e l y narrow t h i s can be compensated f o r . A h i g h e r magnetic f i e l d can be used, or the spectrum can be e d i t e d by v a r y i n g the l e n g t h of the p r e p a r a t i o n p e r i o d t o reduce crowding or o v e r l a p p i n g of co h e r e n c e s . A l t e r n a t i v e l y the J - s c a l i n g f a c t o r n can be reduced. I t s h o u l d not be f o r g o t t e n t h a t t h i s experiment does have s i g n i f i c a n t d i s a d v a n t a g e s which were d i s c u s s e d above ( s e c t i o n 2.6). 2.8 The A n a l y s i s of M i x t u r e s i n an Inhomogeneous Magnetic  F i e l d by the R e c o g n i t i o n of the Zero-Quantum Coherence  S i g n a t u r e s of T h e i r C o n s t i t u e n t s B e s i d e s u s i n g the s p e c t r o s c o p i c parameters c o n t a i n e d w i t h i n the zero-quantum spectrum i t s e l f t o deduce the i d e n t i t y of the compound g i v i n g r i s e t o i t i t i s p o s s i b l e t o use 75 zero-quantum s p e c t r a i n a q u i t e d i f f e r e n t way. T h i s i s done not so much by u s i n g the parameters which can be e x t r a c t e d from them but by u s i n g the p a t t e r n of a spectrum as a whole as i t a r i s e s under a g i v e n s e t of c o n d i t i o n s as a " s i g n a t u r e " of the compound p r o d u c i n g i t . T h e r e f o r e use i s made not o n l y the coherence f r e q u e n c i e s and m u l t i p l e t s t r u c t u r e s , as one might do i n an attempt t o a s s i g n the spectrum, but a l s o of the r e l a t i v e i n t e n s i t i e s of coherences as p a r t of the s i g n a t u r e which i s not p r a c t i c a l when a t t e m p t i n g t o a s s i g n the spectrum. To make use of zero-quantum coherence s p e c t r a i n t h i s way i t i s n e c e s s a r y t o a c q u i r e a c a t a l o g u e of s p e c t r a f o r the compounds one w i l l come a c r o s s , or a t l e a s t f o r those compounds which one i s s p e c i f i c a l l y l o o k i n g f o r , i n a s i m i l a r manner t o those p r e v i o u s l y c o m p i l e d of s i n g l e - q u a n t u m s p e c t r a [ 3 2 ] . The essence of t h i s method has been demonstrated above i n F i g u r e 2.4 where the components of a m i x t u r e of amino a c i d s were d e t e r m i n e d by r e f e r e n c e t o the p r e v i o u s l y c o l l e c t e d zero-quantum s p e c t r a of i t s i n d i v i d u a l components t o c o n s i s t of L - a l a n i n e , L - t h r e o n i n e , L - v a l i n e , and L - a s p a r a g i n e . To d e t e r m i n e the presence of a compound i n a m i x t u r e by i t zero-quantum coherence " s i g n a t u r e " u n l i k e a s s i g n i n g a zero-quantum spectrum does not r e q u i r e t h a t a l l of i t s coherences be c o m p l e t e l y r e s o l v e d from those produced by the o t h e r components, but merely t h a t enough are d i s c e r n i b l e so as to make the d e t e r m i n a t i o n of i t s presence or absence 76 unambiguous. When u s i n g the r e f o c u s s e d zero-quantum experiment the v a l u e s of r and T ' , the l e n g t h s of the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s , can be v a r i e d both t o e d i t out coherences and a l s o t o determine the r e l a t i v e i n t e n s i t i e s of those which a r e l e f t . In the c o n t e x t of s i g n a t u r e r e c o g n i t i o n t h i s has two u s e s ; t o reduce the number of coherences i n an overcrowded spectrum hence f a c i l i t a t i n g the r e c o g n i t i o n of s i g n a t u r e s among the r e m a i n i n g c o h e r e n c e s . A l s o two or more p o s s i b l e components of a m i x t u r e produce i n d i s t i n g u i s h a b l e coherences a t the same fr e q u e n c y i t may be p o s s i b l e t o d i s t i n g u i s h between them by the way i n which the i n t e n s i t i e s of the peaks v a r y w i t h r and T ' . The broad-band d e c o u p l e d zero-quantum experiment can be used i n a s i m i l a r manner. I t produces a spectrum which i s l e s s prone t o o v e r l a p p i n g as each coherence i s o n l y p r e s e n t as a s i n g l e t . T h i s makes the d e t e r m i n a t i o n of which peaks a r e p r e s e n t l e s s ambiguous, but i t a l s o reduces the amount of i n f o r m a t i o n a v a i l a b l e i n t h a t m u l t i p l e t s t r u c t u r e , a u s e f u l p a r t of a s i g n a t u r e , i s removed. By way of compensation t h e r e are a d d i t i o n a l o p p o r t u n i t i e s f o r e d i t i n g the spectrum v i a the l e n g t h of the e v o l u t i o n time t ^ . T h i s can be used i n a s i m i l a r manner t o T and r ' , t o s u p r e s s unwanted co h e r e n c e s and t o determine the r e l a t i v e i n t e n s i t i e s of those which remain. A d d i t i o n a l i n t e n s i t y i n f o r m a t i o n can be encoded i n the spectrum by v a r y i n g the a n g l e of the p u l s e a from 90°. T h i s g i v e s r i s e t o p r e f e r e n t i a l t r a n s f e r of coherence and hence a spectrum whose i n t e n s i t i e s a r e no l o n g e r 77 A I—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—| 1000 800 600 400 200 0 -200 -400 -600 -800 H z B C I i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—| 300 200 100 0.0 -100 -200 H z F i g u r e 2.18. S p e c t r a of a s o l u t i o n of L - v a l i n e , L - l e u c i n e , L - a s p a r a g i n e , L - a s p a r t i c a c i d and L - i s o l e u c i n e i n D 20, a l l c o n c e n t r a t i o n s 0.1 molar. A. Single-quantum spectrum. Zero-quantum s p e c t r a o b t a i n e d w i t h the r e f o c u s s e d zero-quantum e x p e r i m e n t ; B. T , T ' = 6 0 msec, C. T , T ' = 1 4 0 msec. For p a r t s B and C At,=1.67 msec, 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . 78 s y m m e t r i c a l about 0.0 Hz. T h i s i s p o t e n t i a l l y a f u r t h e r way of d i s t i n g u i s h i n g between compounds whose r e s o l v e d coherences a r e o t h e r w i s e the same. To determine the e f f e c t i v e n e s s of the a p p l i c a t i o n of the t e c h n i q u e s d e s c r i b e d above t o s i g n a t u r e r e c o g n i t i o n a m i x t u r e was made up c o n s i s t i n g of s e v e r a l amino a c i d s . The zero-quantum s p e c t r a of t h e s e amino a c i d s are c a t a l o g u e d i n Appendix I I . Due t o m agnetic f i e l d inhomogeneity, F i g u r e 2.18.A, the c o n v e n t i o n a l s i n g l e - q u a n t u m spectrum c o n t a i n e d no u s e f u l i n f o r m a t i o n . The c o n v e n t i o n a l zero-quantum spectrum of the m i x t u r e o b t a i n e d w i t h r,r'=60 msec, F i g u r e 2.18.B, does not s u f f e r from t h i s drawback. By comparison w i t h those s p e c t r a c a t a l o g u e d i n Appendix I I i t i s p o s s i b l e t o r u l e out the p r e s e n c e of many of the amino a c i d s w i t h i n the m i x t u r e . With the e x c e p t i o n of t h e s m a l l peak a t ±230 Hz t h e r e a r e no coherences v i s i b l e above the n o i s e l e v e l i n the range ±120 t o ±300 Hz. T h i s r u l e s out the presence of L - a l a n i n e , L - a r g i n i n e , L - p r o l i n e , L - g l u t a m i n e , L - m e t h i o n i n e , L - g l u t a m i c a c i d and L - t h r e o n i n e . Remaining p o s s i b l e c o n s t i t u e n t s a r e L - a s p a r t i c a c i d , L-/3-phenyl a l a n i n e , L - l e u c i n e , L - v a l i n e , L - c y s t e i n e , L - a s p a r a g i n e , L - i s o l e u c i n e and L - s e r i n e . Of t h e s e t h e r e i s e v i d e n c e t o support L - l e u c i n e , as the peak a t ±230 Hz c o r r e s p o n d s t o one i n t h e c a t a l o g u e d spectrum of L - l e u c i n e ( f i g u r e A2.6.A), and L - v a l i n e due t o the peaks i n the range ±80 t o ±120 Hz which resemble those of the L - v a l i n e spectrum ( F i g u r e A2.9) a l t h o u g h t h e r e i s c l e a r l y o v e r l a p w i t h the o t h e r components of the m i x t u r e . The r e m a i n i n g c oherences are of 79 l i t t l e use due t o the e x t e n s i v e degree of o v e r l a p p i n g found t h e r e i n . The c o r r e s p o n d i n g experiment w i t h T and T ' s e t t o 140 msec g i v e s r i s e t o a spectrum w i t h fewer coherences and g r e a t l y reduced o v e r l a p , F i g u r e 2.18.C. The absence of peaks i n the a p p r o p r i a t e r e g i o n s of t h i s spectrum r u l e out the presence of L - c y s t e i n e ( F i g u r e A2.11), L - 0 - p h e n y l a l a n i n e ( F i g u r e A2.5.C) and L - s e r i n e ( F i g u r e A2.13.B). The peak at ±230 Hz has a l l but d i s a p p e a r e d as would be e x p e c t e d i f i t i s due t o L - l e u c i n e . Many of the peaks i n the r e g i o n ±80 t o ±120 Hz have a l s o d i s a p p e a r e d as one might expect i f they were due t o L - v a l i n e ( F i g u r e A2 .9) . The r e m a i n i n g peaks i n t h i s r e g i o n may be due t o the o v e r l a p p i n g d o u b l e t s of L - a s p a r a g i n e ( F i g u r e A2.4.C) and L - a s p a r t i c a c i d ( F i g u r e A2.16). The presence of L - a s p a r a g i n e i s f u r t h e r suggested by the presence of a peak at ±67 Hz. T h i s l e a v e s as p o s s i b l e components of the m i x t u r e L - v a l i n e , L - a s p a r a g i n e , L - a s p a r t i c a c i d , L - l e u c i n e and L - i s o l e u c i n e . Of t h e s e , f u l l y r e s o l v e d peaks are o b s e r v a b l e f o r L - a s p a r a g i n e and L - l e u c i n e , and, of the remainder, p o s s i b l e p a r t i a l l y r e s o l v e d peaks a r e observed f o r L - v a l i n e and L - a s p a r t i c a c i d a l t h o u g h t h i s by i t s e l f i s i n s u f f i c i e n t t o determine t h e i r p r e s e n c e w i t h c e r t a i n t y . F i g u r e 2.19. Broad band decoupled zero-quantum s p e c t r a of 0.1 molar s o l u t i o n s i n D 20 o f : A. L - v a l i n e , L - l e u c i n e , L - a s p a r a g i n e , L - a s p a r t i c a c i d and L - i s o l e u c i n e . B. L - v a l i n e . C. L - l e u c i n e . D. L - a s p a r a g i n e . E. L - a s p a r t i c a c i d . F. L - i s o l e u c i n e . In each case t(3=364 msec, T=60 msec, A t ^ l . 6 7 msec, 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . 80 34 150 100 T 1 1 1 1 1 1 1 1 1 1 1 1 1 50 0.0 ~i—i—i—|—i—i—i—i—|—i—r -50 -100 -150 Hz L-valine (1) L-asparagine (3) L-aspartic acid (4) L-isoleucine (5) | i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — r 150 100 50 0.0 -50 ~ i — i — | — i — i — i — i — | -100 -150 Hz 81 As the main problem here e n c o u n t e r e d i s one of o v e r l a p of coherences the broad-band d e c o u p l e d zero-quantum s p e c t r a was o b t a i n e d f o r both the m i x t u r e as a whole, F i g u r e 2.19.A, and a l s o of i t s p o s s i b l e components, F i g u r e 2.19 B-F. In both c a s e s a was s e t t o 45° t o take advantage of the r e s u l t i n g u n s y m m e t r i c a l i n t e n s i t y d i s t r i b u t i o n about 0.0 Hz. From i n s p e c t i o n of the s p e c t r a i t can i m m e d i a t e l y be seen t h a t L - v a l i n e , L - l e u c i n e and L - i s o l e u c i n e a r e p r e s e n t . L - i s o l e u c i n e went u n d e t e c t e d i n the c o n v e n t i o n a l zero-quantum spectrum. Whether L - a s p a r t i c a c i d as w e l l as L - a s p a r a g i n e i s p r e s e n t i s l e s s i m m e d i a t e l y o b v i o u s . The f o u r o u t e r coherences of the L - a s p a r a g i n e spectrum ( F i g u r e 2.19.D) can be i d e n t i f i e d i n the spectrum of the m i x t u r e ( F i g u r e 2.19.A), but the f r e q u e n c y of one of these c o i n c i d e s w i t h t h a t of the L - a s p a r t i c a c i d c o herence. T h i s makes the presence of the l a t t e r ambiguous. However, t h e r e i s e v i d e n c e which suggest t h a t L - a s p a r t i c a c i d i s p r e s e n t . The most i n t e n s e peak a s s i g n a b l e t o L - a s p a r a g i n e on the l e f t hand s i d e of the spectrum i s bordered on both s i d e s by two l e s s i n t e n s e peaks. A l t h o u g h these peaks can a l l be d i s c e r n e d i n F i g u r e 2.19.A, t h e i r r e l a t i v e i n t e n s i t i e s a re not what one would expect from the spectrum of the amino a c i d . The c e n t r a l of the t h r e e peaks has a g r e a t e r r e l a t i v e i n t e n s i t y than a n t i c i p a t e d s u g g e s t i n g t h a t another peak i s p r e s e n t a t t h i s f r e q u e n c y . T h i s would indeed be the case i f L - a s p a r t i c a c i d were p r e s e n t as can be seen from the spectrum of L - a s p a r t i c a c i d , F i g u r e 2.19.E. S e c o n d l y , the r e l a t i v e i n t e n s i t y of the peak at -90 Hz i s g r e a t e r than t h a t a t +90 Hz 82 i n the spectrum of L - a s p a r a g i n e whereas i n the m i x t u r e t h i s s i t u a t i o n has been r e v e r s e d , a g a i n s u g g e s t i n g the presence of a n o t h e r peak a t +90 Hz as w e l l as t h a t due t o L - a s p a r a g i n e . A g a i n t h i s i n d i c a t e s the presence of L - a s p a r t i c a c i d as can be seen from i t s broad-band d e c o u p l e d zero-quantum spectrum. I f a had been s e t t o 90° i t would be i m p o s s i b l e t o t e l l which was p r e s e n t from the zero-quantum spectrum. The peaks on e i t h e r s i d e of those a t ±90 Hz would be b u r i e d beneath those of L - v a l i n e and L - i s o l e u c i n e and hence u n a v a i l a b l e f o r i n t e n s i t y c omparisons, and the i n t e n s i t y of c o h e r e n c e s about 0.0 Hz would be s y m m e t r i c a l making any comparison of t h e i r r e l a t i v e i n t e n s i t i e s p o i n t l e s s . Thus, w i t h the use of c o n v e n t i o n a l and d e c o u p l e d zero-quantum s p e c t r a , making use of r e l a t i v e i n t e n s i t i e s and p r e f e r e n t i a l t r a n s f e r of coherence as w e l l as coherence f r e q u e n c i e s and m u l t i p l e t s t r u c t u r e s i t i s p o s s i b l e t o i d e n t i f y a l l the components of a complex m i x t u r e i n an inhomogeneous magnetic f i e l d by comparison of s p e c t r a of the m i x t u r e as a whole w i t h t h o s e of i n d i v i d u a l p o s s i b l e components o b t a i n e d w i t h the same e x p e r i m e n t a l parameters. .2 . 9 Exper i m e n t a l A l l s p e c t r a , w i t h the e x c e p t i o n of those i n F i g u r e s 2.4 and 2.5, were r e c o r d e d a t room temperature u s i n g a s p e c t r o m e t e r equipped w i t h an O x f o r d Research Systems 1.89 T, 30 cm h o r i z o n t a l bore magnet and a N i c o l e t NT-300 c o n s o l e c o n t r o l l e d by a N i c o l e t 1280 computer and a 293C 83 pulse-programmer o p e r a t i n g a t 80.3 MHz f o r 1H. The samples used were p r e p a r e d a n a l y t i c a l l y , by a c c u r a t e w e i g h i n g where the m o l a r i t y i s g i v e n , and by volume w i t h a p i p e t t e where the r a t i o of the components i s g i v e n . The magnetic f i e l d g r a d i e n t s were produced by' the shim c o i l s u s i n g d r i v i n g v o l t a g e s g e n e r a t e d by d i g i t a l - t o - a n a l o g u e c o n v e r t e r s i n the 293C u n i t . Samples were p l a c e d w i t h i n a 7 cm a x i a l r e s o n a t o r probe. P u l s e sequences were r e a d i l y implementable w i t h the s o f t w a r e p r o v i d e d e xcept f o r those i n v o l v i n g d e c r e m e n t a l d e l a y s . The broad-band decoupled zero-quantum experiment and the s i n g l e - q u a n t u m J - r e s o l v e d broad-band decoupled zero-quantum experiment which both i n c o r p o r a t e the d e c r e m e n t a l d e l a y ( t d - t , / 2 ) . The s o f t w a r e d i d not p r o v i d e f o r d e c r e m e n t a l d e l a y s and c o n s e q u e n t l y a d e l a y l i s t had t o be used. T h i s was c a p a b l e of c o n t a i n i n g up t o 32 v a l u e s , and was used i n c o n j u n c t i o n w i t h a n o t h e r f i x e d d e l a y . One v a l u e from the d e l a y l i s t was used f o r each b l o c k of the e x p e r i m e n t , and when the l i s t was used up the experiment was stopped a u t o m a t i c a l l y . Each time the d e l a y l i s t was f i n i s h e d the f i x e d d e l a y was decremented s u c c e s s i v e l y through c o n s e c u t i v e i n t e g e r m u l t i p l e s of the l a r g e s t d e l a y v a l u e i n the l i s t . T h i s was done so t h a t when the experiment was run a g a i n the v a l u e s of the d e c r e m e n t a l time i n t e r v a l would f o l l o w on from those p r e v i o u s l y used i n s t e a d of r e p e a t i n g them. To a l l o w the experiment t o r e a c h e q u i l i b r i u m a g a i n a f t e r i t was r e s t a r t e d a f t e r c h a n g i n g the a p p r o p r i a t e d e l a y v a l u e s , the f i r s t 6 a c q u i s i t i o n s of each b l o c k were d i s c a r d e d . In a l l e x p e r i m e n t s a s i n e a p o d i z a t i o n 84 f u n c t i o n was used on the FIDs which were then z e r o - f i l l e d to 2K. For s y m m e t r i c a l s p e c t r a (about 0.0 Hz) the s p e c t r a were r e v e r s e d and added t o themselves t o improve S/N i f t h i s was l a c k i n g i n the o r i g i n a l spectrum. Only s p e c t r a w i t h i n the f i r s t 2-3 msec of t 2 , the a c q u i s i t i o n t i m e , were co-added t o b u i l d up S/N due t o r a p i d c o l l a p s e of the FID r e s u l t i n g from magnetic f i e l d inhomogeneity. Those experiments r e p o r t e d a t 270 MHz ( F i g u r e s 2.4 and 2.5 o n l y ) were re c o r d e d on a h o m e - b u i l t NMR s p e c t r o m e t e r based upon an O x f o r d Instruments s u p e r c o n d u c t i n g 6.35 T magnet, a Bruker WP-60 c o n s o l e and a N i c o l e t 1180 computer and a 293B pulse-programmer. The samples used were p r e p a r e d a n a l y t i c a l l y by a c c u r a t e w e i g h i n g and the s o l v e n t added by p i p e t t e and p l a c e d w i t h i n 5 mm diameter NMR t u b e s . R e f e r e n c e s 1. Aue, W.P., B a r t h o l d i , E., and E r n s t , R.R., J . Chem. Phys. (1976), 64, 2229 2. Wokaun, A., and E r n s t , R.R., Chem. Phys. L e t t . (1977), 52, 407 3. Bax, A., Two-Dimensional N u c l e a r Magnetic Resonance i n L i q u i d s , D e l f U n i v . P r e s s , D e l f , (1982), pp. 139 4. P i n e s , A., Weemer, D.E., Tang, J . , and S i n t o n , S., B u l l . Am. Phys. Soc. (1978), 23, 21 5. Bodenhausen, G., V o i d , R.L., and V o i d , R.R., J . Magn. Reson. (1980), 37, 93 6. Bax, A., Freeman, R., F r e n k i e l , T.A., and L e v i t t , M.H., J . Magn. Reson. (1981), 43, 478 7. F r e n k i e l , T.A., L e v i t t , M.H., and Freeman, R., J . Magn. Reson. (1981), 44, 409 8. Maudsley, A.A., Wokaun, A., E r n s t , R.R., Chem. phys. L e t t . ' (1978), 55, 9 9. Bax, A., De Jong, P.G., Mehlkopf, A.F., and Smidt, J . , Chem. Phys. L e t t . (1980), 69, 567 10. Sorensen, O.W., L e v i t t , M.H., and E r n s t , R.R., J . Magn. Reson. (1983), 55, 104 11. Sorensen, O.W., E i c h , G.W., L e v i t t , M.H., Bodenhausen, G., and E r n s t , R.R., P r o g r . N u c l . Magn. Reson. S p e c t r o s c . (1983) , J_6, 163 12. B r a u n s c h w e i l e r , L., Bodenhausen, G., and E r n s t , R.R. , Molec. Phys. (1983), 48, 535 13. Ranee. M., Sorensen, O.W., L e u p i n , W., K o g l e r , H., W u t h r i c h , K., and E r n s t , R.R., J . Magn. Reson. (1985), 61, 67 14. Aue, W.P., Karhan, J . , and E r n s t , R.R., J . Chem. Phys. (1976), 64, 4226 15. Bax, A., Mehlkopf, A.F., and Smidt, J . , J . Magn. Reson. (1979), 35, 167 16. Bax, A., Two-Dimensional N u c l e a r Magnetic Resonance i n L i q u i d s , D e l f U n i v . P r e s s , D e l f , (1982), pp. 87 17. Hahn, E.L., Phys. Rev. (1950), 80, 580 18. Benn, R., and Gunther, H., Angew. Chem. I n t . Ed. E n g l . (1983), 22, 350 19. M a r e c i , T.H., and Freeman, R., J . Magn. Reson. (1983), 51, 86 531 • 2 0 . P o u z a r d , G., Sukumar, S., and H a l l , L.D., J . Am. Chem. Soc. (1981), 103, 4209 21. M u l l e r , L., J . Magn. Reson. (1984), 59, 326 22. Nagayama, K., W u t h r i c h , K., and E r n s t , R.R., Biochem. B i o p h y s . Res. Comm. (1979), 90, 305 23. Nagayama, K., Kumar, A., W u t h r i c h , K., and E r n s t , R.R., J . Magn. Reson. (1980), 40, 321 24. M u l l e r , L., Kumar, A., and E r n s t , R.R., J . Chem. Phys. (1975) , 63, 5496 25. Aue, W.P., Karhan, J . , and E r n s t , R.R., J . Chem. Phys. (1976) , 64, 4226 26. M a t h i e s o n , D.W., N u c l e a r Magnetic Resonance f o r Organi c C h e m i s t s , Academic P r e s s , London, (1967) 27. K a r p l u s , M. , J . Chem. Phys. (1959), 3_0, 11 28. K a r p l u s , M., J . Am Chem. Soc. (1963), 85, 2870 29. S t a n d a r d NMR S p e c t r a C o l l e c t i o n , S a d t l e r R e s e a r c h L a b o r a t o r i e s , r e s e a r c h e r s , e d i t o r s , and p u b l i s h e r s , U.S.A. (1980), 11978 30. I b i d . 15096 31. W i l l i a m s , D.H., and F l e m i n g , I . , S p e c t r o s c o p i c Methods i n Or g a n i c C h e m i s t r y , 2nd ed. M c G r a w - H i l l ( U . K . ) , Maidenhead, (1973), pp. 132 32. S t a n d a r d NMR S p e c t r a C o l l e c t i o n , S a d t l e r R e s e a r c h L a b o r a t o r i e s , r e s e a r c h e r s , e d i t o r s , and p u b l i s h e r s , U.S.A. (1980) 87 CHAPTER I I I NUCLEAR MAGNETIC RESONANCE IMAGING IN AN INHOMOGENEOUS MAGNETIC FIELD 88 3.1 N u c l e a r Magnetic Resonance Imaging Over the l a s t twenty y e a r s NMR has advanced a t an almost e x p l o s i v e r a t e , and f o r over h a l f of t h i s time NMR imaging has caused not a l i t t l e of the e x c i t e m e n t . The f i r s t account of NMR imaging was p u b l i s h e d i n 1973 by L a u t e r b u r [1] f o l l o w e d s h o r t l y by a number of o t h e r s [ 2 - 4 ] , j u s t two y e a r s a f t e r Jeener had i n t r o d u c e d the concept of the t w o - d i m e n s i o n a l F o u r i e r t r a n s f o r m a t i o n [ 5 ] . I t has o n l y been more r e c e n t l y , a f t e r the f i r s t waves of the 2-D r e v o l u t i o n had d i e d down, t h a t NMR imaging has r e a l l y taken o f f as a f i e l d of r e s e a r c h i n i t s own r i g h t . A l a r g e p a r t of the i n t e r e s t i n NMR imaging has been i n r e l a t i o n t o b i o l o g i c a l e x p e r i m e n t s , though i t s h o u l d be noted t h a t t h i s i s not a new phenomenon. T r a d i t i o n has i t t h a t 30 y e a r s ago Edward P u r c e l l put h i s head i n a magnet i n an attempt t o observe the e f f e c t s of i n t e n s i v e t h i n k i n g on the NMR s i g n a l [ 6 ] . The fundamental i d e a b e h i n d NMR imaging i s t o encode the o b s e r v e d s i g n a l w i t h s p a t i a l i n f o r m a t i o n . T h i s i s done by making the magnetic f i e l d s t r e n g t h e x p e r i e n c e d by a n u c l e u s , and hence i t s p r e c e s s i o n a l f r e q u e n c y , s p a t i a l l y dependent. I f a l i n e a r magnetic f i e l d g r a d i e n t i n the x d i r e c t i o n , G , i s superimposed upon the s t a t i c magnetic f i e l d B , and ,assuming t h a t the magnitude of the magnetic f i e l d c r e a t e d by the g r a d i e n t i s much l e s s than B 0 [ 7 ] , the Larmor f r e q u e n c y , 89 CJ, of the n u c l e u s becomes: CJ= 7 B Q + T x G x (3.1) On F o u r i e r t r a n s f o r m a t i o n t h i s would g i v e a f r e q u e n c y spectrum which i s a o n e - d i m e n s i o n a l p r o j e c t i o n of the o b j e c t of i n t e r e s t t a k e n a l o n g the x d i r e c t i o n , F i g u r e 3.1. C l e a r l y one p r o j e c t i o n of the o b j e c t i s i n s u f f i c i e n t t o d e f i n e c o m p l e t e l y the shape of an o b j e c t and i t s s p a t i a l l o c a t i o n . In h i s f i r s t e x p e r i m e n t s L a u t e r b u r , [ 1 ] , overcame t h i s problem by r o t a t i n g the o b j e c t about an a x i s p e r p e n d i c u l a r t o the g r a d i e n t d i r e c t i o n . T h i s gave a number of p r o j e c t i o n s from which the image c o u l d be r e c o n s t r u c t e d by the use of back p r o j e c t i o n t e c h n i q u e s [ 8 - 1 2 ] . More p r a c t i c a l l y , i n s t e a d of r o t a t i n g the o b j e c t , two g r a d i e n t s can be v a r i e d such t h a t : 7 ( x G x + y G y ) = c j k (3.2) where a>k i s a c o n s t a n t f r e q u e n c y , t o c r e a t e an a r b i t r a r y g r a d i e n t of c o n s t a n t magnitude i n any d i r e c t i o n i n the p l a n e d e f i n e d by t h e two g r a d i e n t s . The use of p r o j e c t i o n r e c o n s t r u c t i o n t e c h n i q u e s has been superseded by F o u r i e r zeugmatography [ 4 ] . T h i s t e c h n i q u e makes use of m u l t i d i m e n s i o n a l F o u r i e r t r a n s f o r m a t i o n s , i n s t e a d of back p r o j e c t i o n , t o r e c o n s t r u c t the image of ah o b j e c t . E x p e r i m e n t s u t i l i s i n g t h i s t e c h n i q u e g e n e r a l l y i n v o l v e the phase e n c o d i n g of a l l but one of t h e dimensions of i n t e r e s t , such as x,y, and z, p r i o r t o a c q u i s i t i o n and the f r e q u e n c y 90 90* Gradient X 1 lllw 1 A c q . B Y A o o o o o X CO F i g u r e 3.1. A. One-pulse experiment w i t h a magnetic f i e l d g r a d i e n t i n the x - d i r e c t i o n on d u r i n g a c q u i s i t i o n t o frequency encode the x - c o o r d i n a t e s of s p i n s c o n t r i b u t i n g t o the FID. B. R e p r e s e n t a t i o n of a phantom c o n s i s t i n g of v i a l s of H 20. C. R e p r e s e n t a t i o n of the spectrum o b t a i n e d when the p u l s e sequence i n p a r t A i s used on the phantom i n p a r t B. 91 e n c o d i n g of the r e m a i n i n g d i m e n s i o n d u r i n g a c q u i s i t i o n . Phase enco d i n g may be a c h i e v e d by i n c r e m e n t i n g the a p p l i c a t i o n time of a g r a d i e n t of c o n s t a n t magnitude, or by i n c r e m e n t i n g the a m p l i t u d e of a g r a d i e n t a p p l i e d f o r a f i x e d t i m e . The l a t t e r method i s used h e r e i n and w i l l be d e s c r i b e d below. I f a magnetic f i e l d g r a d i e n t G y i s a p p l i e d t o a sample f o r a time t , , at the end of t h a t time each s p i n w i l l have p r e c e s s e d through an a n g l e g i v e n by: A"=7B 0t, +7G yyt 1 (3.3) hence the phase change i n c u r r e d by a s p i n d u r i n g t , f A C J , w i l l be dependent upon i t s y - c o o r d i n a t e . I f the experiment i s r e p e a t e d n t i m e s i n c r e m e n t i n g G y by a c o n s t a n t amount, A t , , each t i m e , then the a c q u i r e d s i g n a l w i l l be dependent upon G y g i v i n g r i s e t o a d a t a m a t r i x S(G , t 2 ) . F o u r i e r t r a n s f o r m a t i o n w i t h r e s p e c t t o t 2 , the rows of the m a t r i x , w i l l g i v e r i s e t o the f r e q u e n c y s p e c t r a c o r r e s p o n d i n g t o each v a l u e of G y. F o u r i e r t r a n s f o r m a t i o n w i t h r e s p e c t t o G y, the columns of the m a t r i x , w i l l g i v e r i s e t o the f r e q u e n c y s p e c t r a of any o b s e r v a b l e s which caused m o d u l a t i o n of the m a g n e t i z a t i o n as a f u n c t i o n of G y. As can be seen from e q u a t i o n 3.3 i n t r i n s i c c h e m i c a l s h i f t , the f i r s t term i n the r i g h t h a l f of the e q u a t i o n , i s independent of G y and c o n s e q u e n t l y w i l l not be encoded i n t h i s d i m e n s i o n . The phase change o c c u r r i n g d u r i n g t , due t o a s p i n y - c o o r d i n a t e on the o t h e r hand i s c l e a r l y dependent upon G y. C o n s e q u e n t l y t h i s d imension of the data s e t w i l l , when F o u r i e r t r a n s f o r m e d , c o r r e s p o n d t o the s p a t i a l 92 c o o r d i n a t e y and hence the s p e c t r a o b t a i n e d w i l l be r e p r e s e n t a t i v e of s p i n d e n s i t y of the sample i n the y -d i r e c t i o n . I f i n a d d i t i o n t o a phase encoding g r a d i e n t , a s t a t i c g r a d i e n t i s a p p l i e d d u r i n g a c q u i s i t i o n , l e a d i n g t o f r e q u e n c y encoding of the s p a t i a l dimension c o r r e s p o n d i n g t o the d i r e c t i o n of t h a t g r a d i e n t ( e q u a t i o n 3.1), the r e s u l t a n t t w o - d i m e n s i o n a l spectrum a f t e r F o u r i e r t r a n s f o r m a t i o n w i l l be a s p i n d e n s i t y map. The phase encoded s p a t i a l c o o r d i n a t e w i l l be a l o n g F1 and the f r e q u e n c y encoded s p a t i a l dimension a l o n g F2. The p u l s e sequence f o r t h i s experiment i s g i v e n i n F i g u r e 3.2. I t can be seen from e q u a t i o n 3.1 t h a t c h e m i c a l s h i f t as w e l l as the s p a t i a l l y dependent freq u e n c y v a r i a t i o n i n d uced by the s t a t i c magnetic f i e l d g r a d i e n t w i l l be encoded d u r i n g a c q u i s i t i o n . C o n s e q u e n t l y , i f the image i s not t o be d i s t o r t e d , the s t a t i c g r a d i e n t must be s u f f i c i e n t l y l a r g e t o make c h e m i c a l s h i f t f r e q u e n c y d i f f e r e n c e s n e g l i g i b l e compared w i t h the g r a d i e n t induced s p a t i a l l y - d e p e n d e n t f r e q u e n c y - v a r i a t i o n . 3.2 C hemical S h i f t R e s o l v e d Imaging C h e m i c a l s h i f t r e s o l v e d imaging experiments are those imaging e x p e r i m e n t s which b e s i d e s c o n t a i n i n g s p a t i a l d i m ensions a l s o have an i n t r i n s i c c h e m i c a l s h i f t d imension [13-17]. 29x i i Gradient Y i i Gradient X 1 1 ' A c q . F i g u r e 3.2. Two-dimensional s p i n - d e n s i t y imaging e x p e r i m e n t . The y-dimension i s phase encoded p r i o r t o a c q u i s i t i o n and the x-dimension i s frequency encoded d u r i n g a c q u i s i t i o n . Gradient X Gradient Y 9 0 x 180° Phase encoding Acquisition F i g u r e 3.3. C h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t . Two s p a t i a l d i m e n s i o n s , x and y, a r e phase encoded p r i o r t o a c q u i s i t i o n , and the c h e m i c a l s h i f t spectrum i s a c q u i r e d d i r e c t l y . 94 A l t h o u g h l o c a l i s e d h i g h r e s o l u t i o n s p e c t r a can be o b t a i n e d w i t h such t e c h n i q u e s as those u t i l i s i n g s u r f a c e c o i l s [ 1 8 - 2 2 ] , f i e l d p r o f i l i n g [ 2 3 ] , and the s e n s i t i v e p o i n t method [24] t h e s e methods are i m p r a c t i c a l f o r o b t a i n i n g c h e m i c a l s h i f t r e s o l v e d images e i t h e r due t o the time r e q u i r e d , low s e n s i t i v i t y , or o t h e r t e c h n i c a l l i m i t a t i o n s . The f i r s t c h e m i c a l s h i f t r e s o l v e d imaging experiment proposed by L a u t e r b u r [13] was based upon p r o j e c t i o n r e c o n s t r u c t i o n , though the c u r r e n t l y most w i d e l y used methods ar e v a r i a t i o n s on the F o u r i e r zeugmatography ex p e r i m e n t . T h i s e x p e r i m e n t , i n t r o d u c e d by E r n s t [ 4 ] , uses m u l t i d i m e n s i o n a l F o u r i e r t r a n s f o r m a t i o n s t o r e c o n s t r u c t an image. In one of the more common of i t s forms, t h i s experiment uses two or t h r e e phase e n c o d i n g g r a d i e n t s t o encode the s p a t i a l d i mensions of the d a t a s e t . D u r i n g a c q u i s i t i o n a l l g r a d i e n t s a r e s w i t c h e d o f f and the c h e m i c a l s h i f t spectrum i s a c q u i r e d , F i g u r e 3.3. A 180° p u l s e i s i n c l u d e d t o r e f o c u s dephasing due t o magnetic f i e l d i n h o m o g e n e i t i e s . The image i s o b t a i n e d by F o u r i e r t r a n s f o r m i n g a l l d i m e n s i o n s . The c h e m i c a l s h i f t dimension w i l l be F2. Due t o the s u b s t a n t i a l a c q u i s i t i o n and p r o c e s s i n g time i n v o l v e d i n m u l t i d i m e n s i o n a l e x p e r i m e n t s a t o t a l of t h r e e d i m e n s i o n s , x,y, and c h e m i c a l s h i f t , a r e u s u a l l y a c q u i r e d . S l i c e s e l e c t i o n t e c h n i q u e s [25-26] a r e used t o s e l e c t a s l i c e t h r o u g h the r e m a i n i n g d i m e n s i o n . They do t h i s by e n s u r i n g t h a t o n l y t h o s e s p i n s w i t h i n a c e r t a i n narrow range a l o n g the r e m a i n i n g d i m e n s i o n a r e e x c i t e d . A s l i c e t aken through the d a t a s e t o b t a i n e d o r t h o g o n a l t o the c h e m i c a l s h i f t a x i s a t the 95 resonance frequency of a c h e m i c a l s p e c i e s w i l l y i e l d the image of t h a t s p e c i e s . U n f o r t u n a t e l y the degree of magnetic f i e l d homogeneity r e q u i r e d t o o b t a i n h i g h r e s o l u t i o n c h e m i c a l s h i f t s p e c t r a , which i s of the o r d e r of 1 p a r t i n 10 7 i s c u r r e n t l y o n l y a t t a i n a b l e i n volumes of a p p r o x i m a t l y 10 cm i n d i a m e t e r . The o b j e c t s which i t i s d e s i r e d t o image, such as human b e i n g s , a r e o f t e n somewhat l a r g e r . C onsequently the r e s o l u t i o n o b t a i n e d i n c h e m i c a l s h i f t r e s o l v e d images i s t y p i c a l l y as poor as 2 ppm, adequate t o d i s t i n g u i s h f a t from water but l i t t l e e l s e . A p a r t i a l s o l u t i o n t o t h i s problem i s t o map the magnetic f i e l d d i s t r i b u t i o n . Assuming t h a t the d i s t r i b u t i o n i s s t a t i c , t h i s i n f o r m a t i o n can be used t o g e n e r a t e a c u r v e d i n s t e a d of p l a n a r s l i c e through the c h e m i c a l s h i f t a x i s thus compensating f o r i r r e g u l a r i t i e s i n magnetic f i e l d d i s t r i b u t i o n [ 2 7 ] . 3.3 Zero-Quantum Coherence R e s o l v e d Imaging The major drawback t o c h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t s , as d i s c u s s e d above, i s the t e c h n i c a l l y demanding requirement f o r B 0 ~ f i e l d homogeneity t o exceed 1 p a r t i n 10 7 over the volume of i n t e r e s t . I t does not seem r e a l i s t i c a t the p r e s e n t time t o expect t h i s requirement t o become t e c h n i c a l l y f e a s i b l e on a r o u t i n e b a s i s i n the near f u t u r e . C o n s e q u e n t l y a l t e r n a t i v e s o l u t i o n s a r e needed, ex p e r i m e n t s which w h i l e s t i l l a l l o w i n g the s e p a r a t i o n of the s p i n d e n s i t y images of 96 d i f f e r e n t s p i n systems a r e not f a t a l l y s u s c e p t i b l e t o magnetic f i e l d i n h o m o g e n e i t i e s as a r e a l l c u r r e n t l y used c h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t s . The b a s i c requirement of such an ex p e r i m e n t would be t h a t i t produce a data s e t which encodes i n one dimensi o n some p r o p e r t y which i s a f u n c t i o n of the s p i n system p r o d u c i n g i t . I t must a l s o be independent of magnetic f i e l d inhomogeneity. An o b v i o u s c a n d i d a t e would seem t o be zero-quantum coherence as i t f u l f i l l s both of t h e s e r e q u i r e m e n t s . Zero-quantum coherence cannot be d i r e c t l y o bserved [ 2 8 ] . Consequently i t cannot be encoded d u r i n g a c q u i s i t i o n as i s the case w i t h c h e m i c a l s h i f t i n many c h e m i c a l s h i f t r e s o l v e d imaging experiments [ 1 3 - 1 7 ] . T h e r e f o r e i t would seem l o g i c a l t o f requency encode one of the s p a t i a l d i m e n s i o n s d u r i n g a c q u i s i t i o n and t o modulate the m a g n e t i z a t i o n p r i o r t o a c q u i s i t i o n w i t h r e s p e c t t o the r e m a i n i n g s p a t i a l d i m e n s i o n s and zero-quantum c o h e r e n c e . The s p a t i a l d i m e n s i o n s may be encoded as a f u n c t i o n of an i n c r e m e n t a l g r a d i e n t , and zero-quantum coherence may be encoded as a f u n c t i o n of an i n c r e m e n t a l time t , . An i n c r e m e n t a l g r a d i e n t encodes a s p a t i a l c o o r d i n a t e by phase e n c o d i n g ( e q u a t i o n 3.3), and t o c o n v e r t zero-quantum coherence i n t o s i n g l e - q u a n t u m coherence most e f f i c i e n t l y r e q u i r e s a 90° p u l s e which i m p l i e s a m p l i t u d e m o d u l a t i o n ( e q u a t i o n 2.15). C o n s e q u e n t l y i t does not make sense t o phase encode the m a g n e t i z a t i o n ( s p a t i a l encoding) b e f o r e a m p l i t u d e e n c o d i n g (zero-quantum coherence encoding) as 97 the l a t t e r d e s t r o y s the sense of phase (±) of the former. T h e r e f o r e the m a g n e t i z a t i o n m u s t . f i r s t be a m p l i t u d e modulated t o encode zero-quantum coherence d u r i n g an i n c r e m e n t a l time t , . Then i t must be phase encoded w i t h i n c r e m e n t a l g r a d i e n t s t o encode a l l but one of the s p a t i a l d i m e n s i o n s , and l a s t l y f r e q u e n c y encoded i n the r e m a i n i n g s p a t i a l d imension by a s t a t i c g r a d i e n t d u r i n g a c q u i s i t i o n . The a n t i p h a s e single-quantum coherence o r i g i n a t i n g from zero-quantum coherence must be a l l o w e d t o rephase b e f o r e i t can be d e t e c t e d so i t would seem t o make sense t o phase encode the m a g n e t i z a t i o n w h i l e t h i s i s o c c u r r i n g and t o i n c l u d e , as i n the r e f o c u s s e d zero-quantum e x p e r i m e n t , a r e f o c u s s i n g p u l s e t o c a n c e l out dephasing due t o magnetic f i e l d i n h o m o g e n e i t i e s . I t would a l s o be d e s i r a b l e t o c o l l e c t the whole echo i n s t e a d of j u s t s t a r t i n g t o a c q u i r e at the t o p of the echo as t h i s w i l l improve S/N and r e s o l u t i o n . The p u l s e sequence t o which th e s e c o n s i d e r a t i o n s would l o g i c a l l y seem t o l e a d one, i s g i v e n i n F i g u r e 3.4 . T h i s experiment w i l l g i v e r i s e t o a d a t a s e t S ( t 1 f G , t 2 ) . Only the t 2 dimension of the d a t a s e t w i l l be s u b j e c t t o d i s t o r t i o n s by magnetic f i e l d i n h o m o g e n e i t i e s . The phase e n c o d i n g g r a d i e n t i s a p p l i e d f o r a c o n s t a n t time ( o n l y the a m p l i t u d e i s incremented) and c o n s e q u e n t l y t h e r e w i l l be no m o d u l a t i o n of the G y dimension due t o i n h o m o g e n e i t i e s . Zero-quantum coherences a r e u n a f f e c t e d by them. Gradient X Gradient Y -Gradient Z Preparation Zero-quantum evolution Spatial encoding Relaxation delay F i g u r e 3.4. Zero-quantum coherence r e s o l v e d imaging experiment. 99 I f t h r e e s p a t i a l d i m e n s i o n s a re r e q u i r e d , the z - g r a d i e n t may a l s o be used f o r phase encoding a t the same time t h a t the y - g r a d i e n t i s . I f t h i s i s done they must be incremented i n d e p e n d e n t l y , and z e r o - o r d e r m u l t i p l e - q u a n t u m coherence s e l e c t e d by phase c y c l i n g [ 2 8 ] . A t w o - d i m e n s i o n a l s l i c e image at a p a r t i c u l a r v a l u e of the t h i r d unencoded s p a t i a l dimension can be o b t a i n e d by the use of e x i s t i n g s l i c e s e l e c t i o n t e c h n i q u e s [ 2 5 , 2 6 ] . These work by e n s u r i n g t h a t o n l y the s p i n s i n a s l i c e a few m i l l i m e t e r s t h i c k a t the d e s i r e d c o o r d i n a t e i n the unencoded dimens i o n are e x c i t e d by a g i v e n r a d i o f r e q u e n c y p u l s e , and would r e q u i r e o n l y minor a l t e r a t i o n s t o t he experiment. The f i r s t normal ( i . e . hard) 90° p u l s e i s r e p l a c e d by a s o f t p u l s e which w i l l e x c i t e a much narrower freq u e n c y range than the normal hard p u l s e would. D u r i n g the s o f t p u l s e a s l i c e s e l e c t i o n g r a d i e n t c o r r e s p o n d i n g t o the unencoded dimension i s s w i t c h e d on t o make the resonance f r e q u e n c i e s of the s p i n s s p a t i a l l y dependent. C o n s e q u e n t l y the narrow range of f r e q u e n c i e s e x c i t e d by the s o f t p u l s e c o r r e s p o n d s t o a range i n the unencoded d i m e n s i o n , the s p a t i a l c o o r d i n a t e of which i s dependent upon the t r a n s m i t t e r f r e q u e n c y . The t h i c k n e s s of the s l i c e i s dependent upon the magnitude of the g r a d i e n t and the c h a r a c t e r i s t i c s of the s o f t p u l s e . The s l i c e s e l e c t i o n g r a d i e n t must be s u f f i c i e n t l y l a r g e t h a t the frequ e n c y d i f f e r e n c e s of the s p i n s due t o c h e m i c a l s h i f t s and magnetic f i e l d i n h o m o g e n e i t i e s a r e n e g l i g i b l e r e l a t i v e t o those i n d u c e d by the g r a d i e n t ( e q u a t i o n 3.1). 100 I t i s p o s i b l e t o reduce the t h r e e d i m e n s i o n a l experiment t o two dim e n s i o n s by i n c r e m e n t i n g the phase e n c o d i n g g r a d i e n t s i m u l t a n e o u s l y w i t h t 1 f i n s t e a d of i n d e p e n d e n t l y . T h i s g i v e s r i s e t o a t w o - d i m e n s i o n a l data set i n which the phase encoded s p a t i a l dimension and the zero-quantum coherence d i m e n s i o n become p a r a l l e l a l o n g one a x i s , and the frequency encoded s p a t i a l dimension i s a l o n g the o t h e r . E f f e c t i v e l y t h i s r e s u l t s i n t he image of a s p e c i e s b e i n g superimposed on t o p of i t s zero-quantum coherence. Due t o the l i m i t e d a b i l i t y of the computer used to handle m u l t i d i m e n s i o n a l data t h i s v e r s i o n of the experiment was performed. The experiment was demonstrated w i t h a phantom c o m p r i s i n g seven v i a l s c o n t a i n i n g p r o p i o n i c a c i d , 2 - p r o p a n o l , and water, F i g u r e 3.5.A. The zero-quantum coherence r e s o l v e d image of the phantom i s g i v e n i n F i g u r e 3.5'B. The images of 2-propanol and p r o p i o n i c a c i d are c o m p l e t e l y r e s o l v e d and a r e superimposed on t o p of t h e i r zero-quantum c o h e r e n c e s . M a g n e t i z a t i o n which was l o n g i t u d i n a l d u r i n g t 1 f and hence was not modulated w i t h r e s p e c t t o i t , produced a c o n v e n t i o n a l image c e n t r e d a t 0.0 Hz i n F 1. T h i s m a g n e t i z a t i o n o r i g i n a t e d from s i n g l e s p i n s such as w a t e r , 2-propanol and p r o p i o n i c a c i d which was not c o n v e r t e d i n t o zero-quantum c o h e r e n c e , and s p i n - l a t t i c e r e l a x a t i o n of zero-quantum coherence. A s i m i l a r complete r e s o l u t i o n of t h e s e c h e m i c a l s p e c i e s would have been i m p o s s i b l e w i t h the t r a d i t i o n a l c h e m i c a l s h i f t r e s o l v e d imaging experiment g i v e n the magnetic f i e l d inhomogeneity e v i d e n t i n the c o r r e s p o n d i n g s i n g l e - q u a n t u m spectrum of the phantom, F i g u r e 3.5.D. 101 2-Propanol Propionic acid Magnetisation longitudinal during t i | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — p — i — i — i — i — | — i — i — i — i — | — i — r — i — i — | 300 240 180 120 60 0 H z 102 F i g u r e 3.5. A. The s t r u c t u r e and c o m p o s i t i o n of the phantom used. B. the t w o - d i m e n s i o n a l zero-quantum coherence r e s o l v e d image o b t a i n e d w i t h the phantom i n p a r t A, t , and G v were incremented s i m u l t a n i o u s l y . T=60 msec, At,=1.67 msec, AGy=1.84x10~ 3 Gem - 1 ( a p p l i c a t i o n time=l0 msec), G x=0.41 Gem - 1, 255 b l o c k s c o l l e c t e d , 16 a c q u i s i t i o n s per b l o c k . T o t a l a c q u i s i t i o n t i m e 75 m i n u t e s . C. Zero-quantum spectrum of the phantom o b t a i n e d w i t h the r e f o c u s s e d zero-quantum e x p e r i m e n t . T=60 msec, At,=1.67 msec, 256 b l o c k s c o l l e c t e d , 8 a c q u i s i t i o n s per b l o c k . T o t a l a c q u i s i t i o n time 35 m i n u t e s . D. Single-quantum spectrum of the phantom. 103 I t s h o u l d be noted t h a t i n the t w o - d i m e n s i o n a l v e r s i o n of the e x periment demonstrated here t h e r e i s d i s t o r t i o n of the image i n F1 r e s u l t i n g from the m u l t i p l e t s t r u c t u r e o f f the zero-quantum c o h e r e n c e s . T h i s s t r u c t u r e i s r e a d i l y d i s c e r n i b l e , p a r t i c u l a r l y i n the case of p r o p i o n i c a c i d i n F i g u r e 3 . 5 . B . In the t h r e e and f o u r d i m e n s i o n a l v e r s i o n s of the experiment m u l t i p l e t s t r u c t u r e i s c o n f i n e d t o the zero-quantum coherence d i m e n s i o n . A s l i c e taken through the d a t a s e t o r t h o g o n a l t o the zero-quantum coherence a x i s a t the frequency of a c e r t a i n zero-quantum coherence, w i l l y i e l d the s p i n d e n s i t y image of the c h e m i c a l s p e c i e s p r o d u c i n g i t . In t he c o n v e n t i o n a l c h e m i c a l s h i f t r e s o l v e d imaging experiment a s l i c e taken through the c h e m i c a l s h i f t d i m e n s i o n at a s p e c i f i c resonance may be non-planar due t o s p a t i a l B Q inhomogeneity, F i g u r e 3.6.A. A l t h o u g h t h i s may be compensated f o r by d a t a m a n i p u l a t i o n [27] the problem does not a r i s e w i t h zero-quantum coherence r e s o l v e d imaging, F i g u r e 3.6.B, as i t i s independent of magnetic f i e l d inhomogeneity. As d e s c r i b e d p r e v i o u s l y ( s e c t i o n 2.2) the zero-quantum coherence d i m e n s i o n can be e d i t e d , which i s p a r t i c u l a r l y u s e f u l i n p r e v e n t i n g o v e r l a p p i n g of images i n the t w o - d i m e n s i o n a l experiment. T h i s i s because the e f f i c i e n c y of e x c i t a t i o n of zero-quantum coherence i s dependent upon the l e n g t h of the p r e p a r a t i o n p e r i o d T and the s c a l a r c o u p l i n g c o n s t a n t s i n v o l v e d . E d i t i n g may a l s o be u s e f u l i n the t h r e e and f o u r d i m e n s i o n a l 104 B F i g u r e 3.6. A. D a t a s e t f o r a c h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t . S l i c e s taken t h r o u g h the the F2 ( c h e m i c a l s h i f t ) d i m e n s i o n may have t o be n o n - p l a n a r t o compensate f o r a s p a t i a l l y inhomogeneous magnetic f i e l d . B. D a t a s e t f o r a zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t . S l i c e s t a k e n t h r o u g h the F1 (zero-quantum coherence) d i m e n s i o n a r e p l a n a r , independent t o magnetic f i e l d i nhomogeneity, and r e q u i r e no s p e c i a l m a n i p u l a t i o n . 105 e x p e r i m e n t s i n cases where coherences o v e r l a p which may become a c o n s i d e r a b l e problem a t lower B c f i e l d s . A l t e r n a t i v e l y the use of " a c c o r d i o n " [29] p r e p a r a t i o n and r e f o c u s s i n g sequences ( s e c t i o n 2.3) w i l l ensure a f a i r l y even e x c i t a t i o n of a l l zero-quantum coherences a l t h o u g h a t the c o s t of reduced s i g n a l i n t e n s i t y and hence S/N ( s e c t i o n 2.3). 3.4 Broad-Band Decoupled Zero-Quantum Coherence R e s o l v e d  Imaging The zero-quantum coherence d i m e n s i o n of a zero-quantum coherence r e s o l v e d image has i n common w i t h c o n v e n t i o n a l zero-quantum s p e c t r a a number of d i s a d v a n t a g e s r e l a t i v e t o t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s . O c c u r r i n g as they do o n l y a t the d i f f e r e n c e i n f r e q u e n c i e s of two c o u p l e d s p i n s they g e n e r a l l y occupy a narrower f r e q u e n c y range. Hence they a r e more i n c l i n e d t o be crowded and t o o v e r l a p . T h i s may l e a d t o problems when t r y i n g t o a s s i g n s p e c t r a , when t r y i n g t o s e p a r a t e the s p i n d e n s i t y images of two o v e r l a p p i n g c o h e r e n c e s , and o v e r l a p p i n g may l e a d t o mutual c a n c e l l a t i o n of c o h e r e n c e s . A p o s s i b l e s o l u t i o n t o a l l of t h e s e problems would be t o broad-band d e c o u p l e the zero-quantum coherence dimension of the e x p e r i m e n t . As d i s c u s s e d p r e v i o u s l y ( s e c t i o n 2.4), r e d u c i n g the zero-quantum coherence m u l t i p l e t s t o s i n g l e t s would d e c r e a s e o v e r l a p , reduce mutual c a n c e l l a t i o n of peaks and may r e q u i r e lower r e s o l u t i o n and hence l e s s a c q u i s i t i o n t i m e . I f a l l the i n t e n s i t y of a former m u l t i p l e t now goes i n t o 1 06 one peak, fewer a c q u i s i t i o n s per b l o c k may be r e q u i r e d t o b u i l d up an a c c e p t a b l e l e v e l of S/N. A broad-band decoupled zero-quantum coherence r e s o l v e d imaging experiment can be c o n s t r u c t e d i n a s i m i l a r manner t o the undecoupled zero-quantum coherence r e s o l v e d imaging experiment d i s c u s s e d p r e v i o u s l y . T h i s i s done by the a d d i t i o n t o the e x i s t i n g s p e c t r o s c o p i c experiment of g r a d i e n t s d u r i n g the r e f o c u s s i n g and a c q u i s i t i o n t i m e s t o encode the d e s i r e d s p a t i a l d i m e n s i o n s . A l s o , i n a n a l o g y t o the undecoupled zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t , the whole s p i n echo i s a c q u i r e d i n s t e a d of j u s t the second h a l f s t a r t i n g a t the t o p of the echo. T h i s improves the S/N and r e s o l u t i o n of the image. The r e s u l t i n g imaging experiment i s g i v e n i n F i g u r e 3.7. As w i t h the undecoupled experiment i t can be used t o produce a f o u r , t h r e e , or t w o - d i m e n s i o n a l d a t a s e t . In the f o u r d i m e n s i o n a l experiment one encodes broad-band d e c o u p l e d zero-quantum coherence and a l s o the t h r e e s p a t i a l d i m e n s i o n s , x, y, and z, u s i n g two phase and one frequency encoding g r a d i e n t s . In the t h r e e - d i m e n s i o n a l experiment o n l y two s p a t i a l d i m e n s i o n s are encoded, one by a f r e q u e n c y and one by a phase e n c o d i n g g r a d i e n t . I f d e s i r e d , a s l i c e may be s e l e c t e d t h r ough the t h i r d unencoded s p a t i a l d imension by r e p l a c i n g the f i r s t h a r d 90° p u l s e w i t h a s o f t p u l s e and a s l i c e s e l e c t i o n g r a d i e n t as d e s c r i b e d p r e v i o u s l y ( s e c t i o n 3.3). In the t w o - d i m e n s i o n a l experiment the a c q u i r e d s i g n a l i s s t i l l encoded w i t h two s p a t i a l d i m e n s i o n s and broad-band d e c o u p l e d zero-quantum coherence but the i n c r e m e n t a l d e l a y t , , 90S I80 Y 901 I 8 0 Y Wx '80° Gradient X Gradient Y Gradient Z T/2 T/2 t,/2 t d - 1 , /2 T/2 ffl • L • Preparation i i Zero-quantum evolution [ Spatial encoding i i i i I • F i g u r e 3.7. Broad band d e c o u p l e d zero-quantum coherence r e s o l v e d imaging experiment. 108 which l e a d s t o zero-quantum coherence e n c o d i n g , and the phase e n c o d i n g g r a d i e n t are incremented s i m u l t a n e o u s l y i n s t e a d of i n d e p e n d e n t l y . T h i s g i v e s r i s e t o a t w o - d i m e n s i o n a l data s e t w i t h one dimension modulated due t o the f r e q u e n c y encoding and the o t h e r modulated due t o both broad-band d e c o u p l e d zero-quantum coherence and the phase encoded s p a t i a l d i m e n s i o n . When F o u r i e r t r a n s f o r m e d i n both d i m e n s i o n s , t h i s d a t a s e t g i v e s r i s e t o an image i n which the image of a c h e m i c a l s p e c i e s w i l l be superimposed on top of i t s zero-quantum c o h e r e n c e s . I t i s i n the t w o - d i m e n s i o n a l v e r s i o n of the experiment t h a t the a b i l i t y t o broad-band decouple zero-quantum coherence i s most s i g n i f i c a n t . In the absence of d e c o u p l i n g d i s t o r t i o n s a r e i n t r o d u c e d i n t o the image by the presence of the zero-quantum coherence m u l t i p l e t s t r u c t u r e . T h i s d i s t o r t i o n can c l e a r l y be seen i n F i g u r e 3.8.A which c o n s i s t s of the zero-quantum coherence r e s o l v e d image of a phantom of s i x v i a l s of 1:1 2-propanol and e t h a n o l i n D 20 a r r a n g e d i n a hexagon. 2-Propanol and e t h a n o l produce zero-quantum coherences a t s i m i l a r f r e q u e n c i e s . C o n s e q u e n t l y when t h e i r r e s p e c t i v e images a r e superimposed on t o p of t h e i r zero-quantum coherences i t r e s u l t s i n t h e i r images o v e r l a p p i n g . The a d d i t i o n a l d i s t o r t i o n s i n t r o d u c e d by the zero-quantum coherence m u l t i p l e t s t r u c t u r e makes i t almost i m p o s s i b l e t o d i s t i n g u i s h the shape of the phantom and i t s c o n s i s t e n t v i a l s and l e a d s t o c o n s i d e r a b l e merging of peaks. The broad-band d e c o u p l e d experiment was used t o o b t a i n the c o r r e s p o n d i n g d e c o u p l e d image of the phantom, F i g u r e 3.8.B. In 109 1 10 T — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i r~ 200 100 0 -100 -200 Hz F i g u r e 3.8. Images and s p e c t r a of a phantom c o n s i s t i n g of 6 v i a l s , d i a m e t e r 6 mm, c o n t a i n i n g 2-propanol and e t h a n o l , 1:1 i n D 20, a r r a n g e d i n a hexagon w i t h s i d e s of l e n g t h 9mm. A. Zero-quantum coherence r e s o l v e d image of the phantom; r=60 msec, At,=1.67 msec, A G v = 1 . 5 X 1 0 ~ 3 Gem - 1 ( a p p l i e d f o r 10 msec), G x= 0.34 Gem - 1, 256 b l o c k s c o l l e c t e d , 16 a c q u i s i t i o n s per b l o c k . B-D, broad band d e c o u p l e d zero-quantum coherence r e s o l v e d images of the phantom: B. t (j=443 msec ( b o t h e t h a n o l and 2-propanol p r e s e n t ) , C. t ( j = 4 l 3 msec ( 2 - p r o p a n o l e d i t e d o u t ) , D. t (j=493 msec ( e t h a n o l e d i t e d o u t ) . For B-D r=60 msec, A t , = 1.67 msec, AGy=5.55x10 - 4 Gem - 1 ( a p p l i e d f o r 27 msec), G x=0.34 Gem - 1, 25b b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . E. Single-quantum spectrum of phantom. Assignments: -100 Hz, 100 Hz - o v e r l a p p i n g peaks of b o t h e t h a n o l and 2 - p r o p a n o l , 200 Hz -HDO. 111 shape of the phantom i s much more r e a d i l y d e t e r m i n e d as a r e the shapes of i t s c o n s t i t u e n t v i a l s and the merging of peaks has been g r e a t l y reduced. D e s p i t e t h i s improvement i n r e s o l u t i o n the images of the two s p i n systems s t i l l o v e r l a p . T h i s i s one of the b i g g e s t d i s a d v a n t a g e s of the t w o - d i m e n s i o n a l experiment. As was d i s c u s s e d p r e v i o u s l y ( s e c t i o n 2.4 ) the broad-band decoupled experiment i s more f l e x i b l e w i t h r e g a r d s t o s p e c t r a l e d i t i n g than the c o n v e n t i o n a l zero-quantum e x p e r i m e n t , and t h i s i s e q u a l l y an a t t r i b u t e of the imaging e x p e r i m e n t . By a l t e r i n g the v a l u e of t ^ i t i s p o s s i b l e t o e d i t out unwanted c o h e r e n c e s , and hence the images a s s o c i a t e d w i t h them. T h i s i s demonstrated i n F i g u r e 3.8 C and D where 2-propanol and e t h a n o l r e s p e c t i v e l y have been e d i t e d out l e a v i n g the u n d i s t o r t e d image of the r e m a i n i n g coherence. The c o r r e s p o n d i n g zero-quantum s p e c t r a a r e g i v e n i n F i g u r e 3.9. In a d d i t i o n t o d i s t o r t i n g the shape of the image the presence of m u l t i p l e t s t r u c t u r e was a l s o found t o d i s t o r t i t s i n t e n s i t y i n the t w o - d i m e n s i o n a l e x p e r i m e n t . I t can be seen from the c r o s s s e c t i o n s taken t h r o u g h the image i n F i g u r e 3.7.A which a re g i v e n i n F i g u r e 3.8 B and C t h a t t h i s d i s t o r t i o n makes the experiment p r a c t i c a l l y u s e l e s s f o r d e t e r m i n i n g the r e l a t i v e i n t e n s i t i e s , and hence s p i n d e n s i t i e s , of a p a r t i c u l a r s p e c i e s . The broad-band d e c o u p l e d experiment however does not e x p e r i e n c e t h i s problem as can be seen from taken from the images i n F i g u r e 3.8 B-D g i v e n i n 1 12 F i g u r e 3.9. S p e c t r a c o r r e s p o n d i n g t o the images i n F i g u r e 3.8 A-D of e t h a n o l and 2 - p r o p a n o l , 1:1 i n D 20. A. Zero-quantum spectrum o b t a i n e d w i t h the r e f o c u s s e d zero-quantum e x p e r i e m e n t ; T=60 msec, A=1.67 msec, 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . B-D. Broad band d e c o u p l e d zero-quantum s p e c t r a : B. t (j=443 msec ( b o t h e t h a n o l and 2-propanol p r e s e n t ) , C. tj3=4l3 msec ( 2 - p r o p a n o l e d i t e d o u t ) , D. t^=493 msec ( e t h a n o l e d i t e d o u t ) . For B-D T=60 msec, At,=1.67 msec, 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k . Peak a s s i g n m e n t s : e t h a n o l ±197 Hz, 2-propanol ±225 Hz. 113 D H u n 800 260 200 ' I ' 160 100 60 T 1 " 0.0 T-,-1 | I I I . | . Hi SOO 250 T 1 -200 F l 160 100 F l 60 o.o Hi Figure 3.10. A. Representation of the phantom used to obtain the images in Figure 3.8 ind i c a t i n g s l i c e s taken through them which are given in parts B-I. B and C: s l i c e s taken through Figure 3.8.A. D and E: s l i c e s taken through Figure 3.8.B. F and G: s l i c e s taken through Figure 3.8.C. H and I: s l i c e s taken through Figure 3.8.D. 1 1 4 F i g u r e 3.10 D-I. The r e l a t i v e s p i n d e n s i t i e s of d i f f e r e n t c h e m i c a l s p e c i e s are even more d i f f i c u l t t o determine w i t h the broad-band d e c o u p l e d zero-quantum experiment than f o r i t s undecoupled c o u n t e r p a r t . In the l a t t e r case the i n t e n s i t y of a c o h e r e n c e i s a f u n c t i o n of the r e l e v a n t c o u p l i n g c o n s t a n t s and t h e p r e p a r a t i o n and r e f o c u s s i n g t i m e s . In the former case i t i s a d d i t i o n a l l y dependent upon t ^ . The b i g g e s t d i s a d v a n t a g e of the broad-band d e c o u p l e d zero-quantum coherence r e s o l v e d imaging experiment w h i c h a p p l i e s t o a s i g n i f i c a n t but l e s s e r e x t e n t t o the undecoupled experiment i s the time i n v o l v e d . In both e x p e r i m e n t s up t o 200 msec may be taken up by the p r e p a r a t i o n and r e f o c u s s i n g p e r i o d s a l o n e . With the the broad-band decoupled e x p e r i m e n t every experiment i s l o n g e r by a t l e a s t nAt, ( as t ^ n A t , ) where n i s the number of t , increments used. These c o n s i d e r a t i o n s would seem t o make in vivo a p p l i c a t i o n s of t h e s e e x p e r i m e n t s i m p r a c t i c a l , g i v e n the s h o r t s p i n - s p i n r e l a x a t i o n t i m e s encountered t h e r e i n . 3.5 J - R e s o l v e d Imaging The homonuclear t w o - d i m e n s i o n a l J - r e s o l v e d e x p e r i m e n t [30,31] s h a r e s one important a t t r i b u t e w i t h homonuclear zero-quantum coherence e x p e r i m e n t s , independence of magnetic f i e l d inhomogeneity i n the F1 d i m e n s i o n . 115 Why i s t h i s so? L i k e any o t h e r p r o p e r t y , i f i t i s t o be encoded i n the t , dimension of a t w o - d i m e n s i o n a l experiment the e x t e n t t o which m a g n e t i z a t i o n e v o l v e s due t o magnetic f i e l d inhomogeneity must change i n each s u c c e s s i v e experiment as a f u n c t i o n of an i n c r e m e n t a l d e l a y or some o t h e r p r o p e r t y b e i n g s y s t e m a t i c a l l y i n c r e m e n t e d . T h i s produces a t w o - d i m e n s i o n a l d a t a m a t r i x i n which the m a g n e t i z a t i o n i n the dimension c o r r e s p o n d i n g t o the s y s t e m a t i c a l l y i n c r e m e n t e d d e l a y or o t h e r v a r i a b l e i s modulated by the p r o p e r t y one d e s i r e s t o encode. In the b a s i c homonuclear two-dimension J - r e s o l v e d e x p e r i m e n t , F i g u r e 3.11.A, a t the end of the e v o l u t i o n time t , the system has o n l y e v o l v e d due t o i t s s c a l a r c o u p l i n g s . C h emical s h i f t and magnetic f i e l d inhomogeneity a r e c a n c e l e d out by the 180° p u l s e i n the m i d d l e of t , . C o n s e q u e n t l y the t , dimensio n of the da t a s e t w i l l o n l y be phase modulated as a f u n c t i o n of the e x t e n t of e v o l u t i o n due t o s c a l a r c o u p l i n g s . T h i s i s demonstrated f o r the 3 1 P s p e c t r a of a d e n o s i n e t r i p h o s p h a t e and adenosine d i p h o s p h a t e i n F i g u r e 3.12 A and B which a r e c l e a r l y u n a f f e c t e d by magnetic f i e l d inhomogeneity whereas the c o r r e s p o n d i n g s i n g l e - q u a n t u m coherence s p e c t r a , F i g u r e 3.12 C and D, a r e , t h e i r s c a l a r c o u p l i n g s b e i n g u n r e s o l v e d . Only the t , di m e n s i o n of the da t a s e t a c q u i r e d , S ( t , , t 2 ) , has been F o u r i e r t r a n s f o r m e d , g i v i n g S ( F 1 , t 2 ) . In the case of extreme inhomogeneity F2 w i l l c o n t a i n no u s e f u l c h e m i c a l s h i f t or s c a l a r c o u p l i n g i n f o r m a t i o n . 1 16 A 2 2 Phase encoding Acquisition B 90x Gradient X Gradient Y Gradient Z Phase encoding Acquisition F i g u r e 3.11. A. Homonuclear t w o - d i m e n s i o n a l J - r e s o l v e d e x p e r i m e n t . B. Homonuclear J - r e s o l v e d imaging experiment w i t h two phase encoded and one fre q u e n c y encoded s p a t i a l d i m e n s i o n s . 1 1 7 T—i—i—|-1—i—i—r—i—i—i | i—i—i—|—i—i—i—[• — i i—i—|—i—i—i—|—i—i—i—|—i—i—i—|—i—i—i—r 200 0 -200 H z 200 0 -200 H z F i g u r e 3.12. 3 1 P s p e c t r a o b t a i n e d a t 32.5 MHz. A. J-spectrum of ATP, 0.5 molar i n H 20 o b t a i n e d w i t h the homonuclear t w o - d i m e n s i o n a l J - r e s o l v e d e x p e r i m e n t . B. J-spectrum of ADP, 0.5 molar i n H 20, o b t a i n e d w i t h the homonuclear t w o - d i m e n s i o n a l J - r e s o l v e d e x p e r i m e n t . For p a r t s A and B At , = 11.1 msec, 64 b l o c k s c o l l e c t e d , 8 a c q i s i t i o n s per b l o c k . C. Single-quantum spectrum of ATP. D. Single-quantum spectrum of ADP. 118 I n c o r p o r a t e d i n t o an imaging experiment i n s t e a d of a c h e m i c a l s h i f t r e s o l v e d dimension J - e n c o d i n g would t h e r e f o r e a l l o w one t o o b t a i n the image of a c h e m i c a l s p e c i e s where the magnetic f i e l d i s t o o inhomogeneous f o r a c h e m i c a l s h i f t r e s o l v e d image t o be u s e f u l . I t was found t o be i n s u f f i c i e n t merely t o add frequ e n c y and phase enc o d i n g magnetic f i e l d g r a d i e n t s t o encode the d e s i r e d s p a t i a l d i m e n s i o n s t o the e x i s t i n g t w o - d i m e n s i o n a l J - r e s o l v e d e x p e r i m e n t . T h i s was because the r a p i d decay of the FID under the i n f l u e n c e s of the fr e q u e n c y encoding g r a d i e n t and magnetic f i e l d inhomogeneity r e s u l t e d i n inadequate r e s o l u t i o n and S/N. T h i s problem was overcome by d e s i g n i n g a p u l s e sequence i n which the whole echo i s a c q u i r e d , i n s t e a d of j u s t the second h a l f , and which y e t r e t a i n s independence of magnetic f i e l d i n h o m ogeneity w i t h r e s p e c t t o t , . The r e s u l t i n g experiment i s g i v e n i n F i g u r e 3.11.B. B e g i n i n g d a t a a c q u i s i t i o n w i t h i n the i n c r e m e n t a l time t , would r e s u l t i n magnetic f i e l d i n homogeneity and c h e m i c a l s h i f t b e i n g encoded w i t h r e s p e c t t o t , a l o n g w i t h s c a l a r c o u p l i n g e v o l u t i o n . T h e r e f o r e an a d d i t i o n t i m e p e r i o d r/2 was i n t r o d u c e d b e f o r e the 180° p u l s e and T- s e t e q u a l t o t h e a c q u i s i t i o n t i m e . T h i s r e s u l t s i n the a c q u i s i t i o n of the whole s p i n echo which r e a c h e s i t s maximum a t the c e n t r e of the FID g i v i n g r i s e t o maximum r e s o l u t i o n and S/N w i t h o u t i n t r o d u c i n g magnetic f i e l d i nhomogeneity i n t o the F1 dimension of the image. 119 T h i s sequence was i n i t i a l l y e v a l u a t e d w i t h a phantom c o n t a i n i n g e t h a n o l , p r o p i o n i c a c i d and 2-propanol i n water. Only one s p a t i a l dimension was encoded, w i t h a fr e q u e n c y e n c o d i n g g r a d i e n t d u r i n g a c q u i s i t i o n . The o t h e r d i m e n s i o n was J-encoded. The experiment was found t o be i n a d e q u a t e . S h o r t T 2's t o g e t h e r w i t h the r e l a t i v e l y s m a l l s i z e s of most p r o t o n - p r o t o n c o u p l i n g c o n s t a n t s l i m i t e d the r e s o l u t i o n o b t a i n a b l e i n F1. The n e c e s s i t y of u s i n g the a b s o l u t e v a l u e mode of d i s p l a y due t o phase d i s t o r t i o n s which made a u t o m a t i c phase c o r r e c t i o n i m p o s s i b l e , a l s o h e l p e d t o make i t u n f e a s i b l e t o s e p a r a t e out the d i f f e r e n t c o n s t i t u e n t s (by t a k i n g s l i c e s t h r o ugh the F1 dimension of the image). T h i s problem may be e x a c e r b a t e d by the l a r g e number of m u l t i p l e t s a s s o c i a t e d w i t h many compounds which may l e a d t o s i g n i f i c a n t o v e r l a p p i n g of peaks i n the F1 di m e n s i o n . A phantom c o n s i s t i n g of a row of s i x v i a l s c o n t a i n i n g a d e nosine t r i p h o s p h a t e and adenosine d i p h o s p h a t e was c o n s t r u c t e d , and a l i g n e d p a r a l l e l t o the d i r e c t i o n of the x - g r a d i e n t . O b s e r v i n g 3 1 P , and a g a i n u s i n g o n l y a frequ e n c y e ncoding g r a d i e n t the experiment was r e p e a t e d . Phosphorus-phosphorus c o u p l i n g c o n s t a n t s a r e c o n s i d e r a b l y l a r g e r than t h e i r p r o t o n c o u n t e r p a r t s . C o n s e q u e n t l y they c o u l d be e x p e c t e d t o be b e t t e r r e s o l v e d . As o n l y ATP and ADP of those phosphorus compounds found in vivo a r e o t h e r than s i n g l e t s , t h e J - r e s o l v e d dimension of the experiment may be ex p e c t e d t o be l e s s s u s c e p t i b l e t o peak o v e r l a p . The r e s u l t i n g image i s g i v e n i n F i g u r e 3.13. U n f o r t u n a t e l y the T 2's of ADP 120 F 2 Cx+6) F i g u r e 3.13. 3 1 P homonuclear J - r e s o l v e d image of a phantom c o n s i s t i n g of a row of 6 e q u a l l y spaced v i a l s p a r a l l e l t o the d i r e c t i o n of the x - g r a d i e n t c o n t a i n i n g a l t e r n a t l y ATP and ADP, 0.5 molar i n H 20. Diameter of v i a l s 6 mm, d i s t a n c e between c e n t r e s of a d j a c a n t v i a l s 7mm. Only a frequ e n c y e n c o d i n g g r a d i e n t , G x, was used. I n f o r m a t i o n encoded i n image d i m e n s i o n s : F1=J, F2=x+5. G x=0.27 Gem" 1, At!=20 msec, 64 b l o c k s c o l l e c t e d , 200 a c q u i s i t i o n s per b l o c k , t o t a l a c q u i s i t i o n time=7.4 h o u r s . 121 were found t o be s h o r t , and t h i s , combined w i t h the s i n e a p o d i s a t i o n f u n c t i o n used i n the F1 d i m e n s i o n of the data set r e s u l t e d i n i t s absence from the image. The s h o r t n e s s of the s p i n - s p i n r e l a x a t i o n time of ADP r e l a t i v e t o ATP i s r e f l e c t e d i n the r e l a t i v e l y poor S/N of ADP t o ATP i n F i g u r e 3.12 A and B. The s e p a r a t i o n of the d o u b l e t and t r i p l e t i n the F2 d i m e n s i o n i s due t o the d i f f e r e n c e i n c h e m i c a l s h i f t s between two of the phosphorus peaks of ATP (20ppm) which the magnitude of the frequency e n c o d i n g g r a d i e n t used was i n s u f f i c i e n t t o overcome. C l e a r l y , a l t h o u g h the experiment works i t s u f f e r s from severe l i m i t a t i o n s i n terms of the r e s o l u t i o n r e q u i r e d and t h a t a c t u a l l y o b t a i n a b l e , and, p a r t i c u l a r l y w i t h p r o t o n s , s i g n i f i c a n t o v e r l a p of m u l t i p l e t s of d i f f e r e n t c h e m i c a l s p e c i e s . In c a s e s where i t does work, such as phosphorus, the c h e m i c a l s h i f t s e n c o u n t e r e d are l a r g e enough t o make c h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t s adequate i n a l l but cases of extreme magnetic f i e l d inhomogeneity. Consequently t h i s experiment would seem t o be redundant. 3.6 E x p e r i m e n t a l The s p e c t r o m e t e r used was t h a t based upon a 1.98 T magnet d e s c r i b e d p r e v i o u s l y i n s e c t i o n 2.9. The s o f t w a r e used t o generate the i n c r e m e n t a l g r a d i e n t s f o r the zero-quantum coherence r e s o l v e d imaging experiment was found t o be inadequate f o r the broad band decoupled 1 22 zero-quantum coherence r e s o l v e d imaging experiment as i t d i d not a l l o w one t o choose the i n i t i a l v a l u e of the incremented g r a d i e n t . T h i s i s e s s e n t i a l f o r t h i s experiment as i t has t o be a c q u i r e d i n s e t s of 32 b l o c k s (see s e c t i o n 2.9) on t h i s s p e c t r o m e t e r . C o n s e q u e n t l y , every time the experiment i s r e s t a r t e d the i n c r e m e n t a l g r a d i e n t s t a r t e d at the same i n i t i a l v a l u e , r e p e a t i n g i t s e l f , i n s t e a d of c a r r y i n g on from where i t had l e f t o f f a t the end of the p r e v i o u s s e t of 32 b l o c k s . T h i s problem was overcome by c o n s t r u c t i n g an i n c r e m e n t a l g r a d i e n t w i t h a time d e l a y of c o n s t a n t l e n g t h which i t s e l f c o n s i s t e d of two d e l a y s . One was d e c r e m e n t a l , and one was i n c r e m e n t a l . Both were changed from b l o c k t o b l o c k by the same amount, hence ke e p i n g the o v e r a l l time of the two c o n s t a n t . D u r i n g the i n c r e m e n t a l d e l a y a p o s i t i v e g r a d i e n t was s w i t c h e d on, and d u r i n g the decremental d e l a y the same g r a d i e n t was i n v e r t e d , though i t s magnitude was kept c o n s t a n t . T h i s has the same e f f e c t as u s i n g an i n c r e m e n t a l g r a d i e n t , p r o d u c i n g an "increment e q u i v a l e n t " dependent upon the magnitude of the g r a d i e n t and the s i z e of the increment/decrement of the two d e l a y s used. The d e l a y s used were the same as t h o s e used t o e f f e c t the i n c r e m e n t a l and d e c r e m e n t a l d e l a y s w i t h i n the e v o l u t i o n time t ^ of the experiment ( s e c t i o n 2.9) and c o n s e q u e n t l y r e q u i r e d no a d d i t i o n a l m a n i p u l a t i o n of parameters between s e t s of 32 b l o c k s t o ensure c o n t i n u i t y . R e f e r e n c e s 1. L a u t e r b u r , P . C , N a t u r e , (1973), 242, 190 1 23 2. M a n s f i e l d , P., and G r a n n e l l , P.K., J . Phys. (1973), C 6, L422 3. Hinshaw, W.S., Phys. L e t t . ( 1 974), 48A, 87 4. Kumar, A., W e l t i , D., and E r n s t , R.R., J . Magn. Reson. (1975), J_8, 69 5. J e e n e r , J . , Ampere I n t e r n a t i o n a l Summer S c h o o l , Basko, P o l j i , Y u g o s l a v i a (1971) 6. M a n s f i e l d , P., and M o r r i s , P.G., NMR Imaging i n B i o m e d i c i n e , Academic P r e s s , New York, (1983), pp. 2 7. I b i d . pp. 33 8. B r a c e w e l l , R.N., and R i d d l e , A.C., A s t r o p h y s . J . (1967), 1 50, 427 9. V a i n s h t e i n , B.K., S o v i e t P h y s i c s - C r y s t a l l o g r a p h y , (1971), J_5, 781 10. Ramachandran, G.N., and Lakshminarayan, A.V., P r o c . U.S. Nat. Acad. S c i . (1971), 68, 2236 11. Gordon, R., and Herman, G.T., Comm. Assoc . Comput. Mach. (1 971 ) , _1_4, 759 12. K l u g , A., and Crowt h e r , R.A., N a t u r e , (1972), 238, 435 13. L a u t e r b u r , P . C , Karmer, D.M., House J r , W.V., C h i n g - N i e n , C , J . Am. Chem. Soc. (1975), 97, 6866 14. B e n d e l , P., L a i , C-M. , L a u t e r b u r , P . C , J . Magn. Reson. ( 1 980) , 3_8, 342 15. Cox, S . J . , and S t y l e s , P., J . Magn. Reson. (1980), 40, 209 16. Brown, T.R., K i n c a i d , B.M., and U g u r b i l , K., P r o c . N a t l . Acad. S c i . USA. (1982), 79, 3523 17. H a l l , L.D., and Sukumar, S., J . Magn. Reson. (1984), 56, 124 314 18. Ackerman, J.J.H., Grove, T.H., Wong, G.G., Gaddian, D.G., and Radda, G.K., N a t u r e , (1980), 19. B e n d a l l , M.R., and Gordan, R.E., J , Magn. Reson. (1983), 53, 365 20. Shaka, A . J . , and Freeman, R., J . Magn. Reson. (1984), 59, 169 21. Tycko, R., and P i n e s , A., J . Magn. Reson. (1984), 60, 156 22. Shaka, A . J . , and Freeman, R. , J . Magn. Reson. (1985), 6_2, 340 23. Gordan, R., Hanley, P., Shaw, D., Gaddian, D., Radda, G.K., S t y l e s , P., Bore, P., and Chan, L., N a t u r e , (1980), 287, 736 24. Hinshaw, W.S., J . A p p l . Phys. (1976), 47, 3709 25. Garroway, A.N., G r a n n e l l , P.K., and M a n s f i e l d , P., J . Phys. C: (1974), 7, L457 26. S u t h e r l a n d , R.J., and H u t c h i s o n , J.M.S., J . Phys. E. ( 1978) , JM, 79 27. Maudsley, A.A., H i l a l , S.K., and Permar, W.H., J . Magn. Reson. (1983), 5J_, 147 28. Wokaun, A., and E r n s t , R.R., Chem. Phys. L e t t . (1977), 5_2, 407 29. B r a u n s c h w e i l e r , L., Bodenhausen, G., and E r n s t , R.R., Molec. Phys. (1983), 48, 535 30. M u l l e r , L., Kumar, A., and E r n s t , R.R., J . Chem. Phys. (1975), 63, 5496 31. Aue, W.P., Karhan, J . , and E r n s t , R.R., J . Chem. Phys. (1976), 64, 4226 126 CHAPTER IV CONCLUSION 1 27 C o n c l u s i o n At the s t a r t of these s t u d i e s (September 1984) i t was a l r e a d y known t h a t zero-quantum coherences c o u l d be measured and t h a t t h e i r l i n e w i d t h s were independent of magnetic f i e l d i n homogeneity. F u r t h e r m o r e , a r i c h l e g a c y of s p i n p h y s i c s was a v a i l a b l e t h a t i n c l u d e d s e v e r a l e x t r e m e l y p o w e r f u l c o n c e p t u a l t o o l s . For example, the use of a 180° r a d i o f r e q u e n c y p u l s e e i t h e r t o r e f o c u s the phase d i s p e r s i o n i n t r o d u c e d by magnetic f i e l d inhomogeneity, or t o remove s c a l a r c o u p l i n g s . The i n i t i a l m o t i v a t i o n of the s t u d i e s summarized i n t h i s t h e s i s was t o e x p l o r e the p o t e n t i a l of t h e s e and o t h e r methods to f a c i l i t a t e the measurement of c h e m i c a l l y u s e f u l i n f o r m a t i o n i n magnetic f i e l d s of l i m i t e d homogeneity (1 p a r t i n 10 5 r a t h e r than 1 p a r t i n 10 9 which i s u s u a l l y u s e d ) . The f i r s t group of e x p e r i m e n t s ( s e c t i o n s 2.1-2.3) soon demonstrated t h a t zero-quantum s p e c t r a c o u l d be used t o i d e n t i f y the components of a m i x t u r e by i d e n t i f y i n g the c h a r a c t e r i s t i c zero-quantum coherences of it's components. T h i s was demonstrated under c o n d i t i o n s of low magnetic f i e l d homogeneity where c o n v e n t i o n a l s p e c t r a c o n t a i n no u s e f u l l i n f o r m a t i o n . Zero-quantum s p e c t r a t e n d t o o v e r l a p more than t h e i r s i n g l e - q u a n t u m c o u n t e r p a r t s . To s o l v e t h i s problem e x i s t i n g e d i t i n g t e c h n i q u e s were e v a l u a t e d and, where n e c e s s a r y , adapted f o r use i n an inhomogeneous magnetic f i e l d . In a d d i t i o n , a new experiment was d e v e l o p e d t o broad-band d e c o u p l e zero-quantum s p e c t r a ( s e c t i o n 2.4). T h i s experiment 128 was found t o g r e a t l y reduce the problem of peak o v e r l a p , and t o have a d d i t i o n a l e d i t i n g p o s s i b i l i t i e s . A u s e f u l e x t e n s i o n of t h i s t e c h n i q u e would be t o use broad-band d e c o u p l i n g i n m u l t i p l e - q u a n t u m coherence c o n n e c t i v i t y e x p e r i m e n t s performed i n homogeneous magnetic f i e l d s t o improve the r e s o l u t i o n of peaks. A l t h o u g h zero-quantum s p e c t r a can r e a d i l y be used f o r " s i g n a t u r e " r e c o g n i t i o n t hey are of l i t t l e use f o r i d e n t i f y i n g unknown compounds. T h i s i s l a r g e l y because of the way i n which they p r e s e n t c h e m i c a l s h i f t and s c a l a r c o u p l i n g i n f o r m a t i o n . T h e r e f o r e a new experiment was d e s i g n e d ( s e c t i o n 2.5-6) from which the c o n v e n t i o n a l s i n g l e - q u a n t u m spectrum can be l a r g e l y r e c o n s t r u c t e d . Single-quantum s p e c t r a are much more r e a d i l y i n t e r p r e t a b l e i n t h i s c o n t e x t than t h e i r zero-quantum c o u n t e r p a r t s . A c o m b i n a t i o n of c o n v e n t i o n a l zero-quantum s p e c t r a and broad-band decoupled zero-quantum s p e c t r a have been shown t o be p o w e r f u l t o o l s f o r a n a l y s i n g complex m i x t u r e s ( s e c t i o n 2.8). For zero-quantum s p e c t r a t o become w i d e l y used f o r t h i s purpose i t would be n e c e s s a r y t o assemble a c a t a l o g u e of s p e c t r a o b t a i n e d under a s e t of s t a n d a r d c o n d i t i o n s f o r those compounds which are of i n t e r e s t . C h e m i c a l s h i f t r e s o l v e d imaging e x p e r i m e n t s a r e f a t a l l y s u s c e p t i b l e t o magnetic f i e l d inhomogeneity. Three new e x p e r i m e n t s were d e v e l o p e d t o t r y and overcome t h i s 1 29 p r o b l e m ( s e c t i o n s 3.3-3.5). A zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t , a broad-band d e c o u p l e d zero-quantum coherence r e s o l v e d imaging e x p e r i m e n t , and a J - r e s o l v e d imaging e x p e r i m e n t . A l t h o u g h the l a t t e r proved t o be of l i t t l e use t h e e x p e r i m e n t s u t i l i z i n g zero-quantum coherence were demo n s t r a t e d t o be e f f e c t i v e . The l e n g t h of these e x p e r i m e n t s makes f u t u r e in vivo a p p l i c a t i o n s of them u n l i k e l y , g i v e n the s h o r t s p i n - s p i n r e l a x a t i o n t i m e s e n c o u n t e r e d t h e r e i n . I t i s t o o e a r l y t o p r o v i d e a mature overview of the f u t u r e i m p l i c a t i o n s of t h e s e s t u d i e s , both s p e c t r o s c o p i c and im a g i n g . However, s e v e r a l p o i n t s a r e c l e a r . In t h e i r p r e s e n t form t h e d a t a a c q u i s i t i o n t i m e s a r e too l o n g f o r t h e s e measurements t o be a p p l i e d t o s t u d i e s of c l i n i c a l p roblems. However, t h e r e are many o t h e r systems f o r which time i s a l e s s i m p o r t a n t c r i t e r i o n . I t i s towards t h e s e a r e a s t h a t the next emphasis s h o u l d be d i r e c t e d . For example, i t i s c o n c e i v a b l e t h a t t h e s e p r o c e d u r e s w i l l f i n d a p p l i c a t i o n i n the mapping of the o r g a n i c p r o d u c t s produced by b i o - r e a c t o r s . 130 APPENDIX I PRODUCT OPERATOR EVOLUTION 131 A1.1 I n t r o d u c t i o n t o the E v o l u t i o n of Product O p e r a t o r s T h i s appendix c o n s i s t s of a d i s c u s s i o n of the time e v o l u t i o n of p r o d u c t o p e r a t o r s which i s a summary of the d e f i n i t i v e p u b l i c a t i o n on the s u b j e c t which the r e a d e r of t h i s t h e s i s i s s t r o n g l y recommended t o read [ 1 ] . The d i s c u s s i o n here w i l l be c o n f i n e d t o weakly c o u p l e d s p i n systems, s i n c e s t r o n g c o u p l i n g c o m p l i c a t e s the s i t u a t i o n w i t h o u t p r o v i d i n g a d d i t i o n a l i n s i g h t i n t o an e x p e r i m e n t . The e v o l u t i o n of the d e n s i t y m a t r i x caused by the H a m i l t o n i a n can be a b b r e v i a t e d t h u s : o ( t ) H 1 r 1 ^ a ( t + r , ) H 2 r 2 ^ a ( t + T , + T 2 ) (A1.1) I f a f t e r the time i n t e r v a l such as t , t h e r e i s a r a d i o f r e q u e n c y p u l s e the d e n s i t y o p e r a t o r o ( t ^ _ ) d e s c r i b e s the system p r i o r t o the p u l s e , and o ( t 1 + ) the system a f t e r the p u l s e , i n d i c a t e s t h a t the e x p r e s s i o n of the d e n s i t y o p e r a t o r i s i n c o m p l e t e , and the a d d i t i o n of a s u b s c r i p t such as A t o g i v e i n d i c a t i n g t h a t the e x p r e s s i o n c o n s i s t s of the p a r t i a l d e n s i t y m a t r i x of a s p i n denoted A. The e v o l u t i o n of the d e n s i t y o p e r a t o r under the u n p e r t u r b e d weakly c o u p l e d H a m i l t o n i a n i s g i v e n by: H = Pk ( I kz ) + W J k l < 2 I k z I l z ) ( A U 2 ) w r i t t e n i n terms of p r o d u c t o p e r a t o r s . The c h e m i c a l s h i f t f r e q u e n c i e s , fi, c o r r e s p o n d t o r o t a t i n g frame f r e q u e n c i e s . 132 P o s i t i v e r o t a t i o n s a r e d e f i n e d i n the r i g h t hand sense, i . e . a p o s i t i v e r o t a t i o n about the z - a x i s w i l l l e a d from x t o y t o -x to -y. S i n c e a l l terms i n e q u a t i o n A1.2 commute, the e v o l u t i o n caused by the i n d i v i d u a l terms may be computed s e p a r a t e l y i n a r b i t r a r y o r d e r , s y m b o l i c a l l y : 0,7-1. Q~TI~„ a 1 1z^ 2 Zz^ .... ^ ^ i z ^ z , , ^ n ^ l z ^ z , .... o ( t + r) (A1.3) A1.2 Ch e m i c a l S h i f t E v o l u t i o n The e f f e c t s of a freq u e n c y s h i f t flk a r e d e s c r i b e d by I l t x n k T l k z ; i I k x c o s n R T + I k y s i n Q k r (A1.4) flkTlkz^ I k y c o s n k r - I ^ s i n J ^ r (A1.5) For a.product o p e r a t o r of more than one s p i n the p r o d u c t of the e f f e c t s of e v o l u t i o n on each s p i n computed s e p a r a t e l y i s taken 2 I k x I l x V i l z , . 2 l ( I k x c o s V + I k y s i n V ) ( I l x C O S f i l T + I i y s i n f l i T ) (A1.6) The e f f e c t s of s c a l a r c o u p l i n g e v o l u t i o n and r a d i o - f r e q u e n c y p u l s e s on m u l t i - s p i n p r o d u c t o p e r a t o r s a r e d e a l t w i t h i n the same way. 133 A1.3 S p i n - S p i n C o u p l i n g E v o l u t i o n The r u l e s f o r s c a l a r c o u p l i n g e v o l u t i o n a r e I k x ^ k l ^ k z h z , I k x c o s U J k l r ) + 2 I k y I l z s i n ( 7 r J k l r ) (A1.7) I. 7 r J k l r 2 I k z I l z I. c o s d r J . .. T) -21. I. s i n ( j r J . , T ) (A1.8) ky > ky k l kx l z k l The c o r r e s p o n d i n g r u l e s f o r the e v o l u t i o n of a n t i p h a s e m a g n e t i z a t i o n i n t o in-phase m a g n e t i z a t i o n a re 2 I k x I l 2 ^ " k z h z , 2 I k x I l z c o s U J k l r ) + I k y s i n ( 7 r J k l r ) (A1.9) 21. I , ffJklT2IkzIlz. 21. I , COS(7TJ,,T) ky l z > ky l z k l - I k x s i n ( i r J K L T ) (Al .10) A1.4 Radio-Frequency P u l s e E v o l u t i o n The e f f e c t of a r a d i o - f r e q u e n c y p u l s e of f l i p a n g l e /3 about the x - a x i s f o r one s i n g l e - s p i n o p e r a t o r i s g i v e n by I k z ^ X k x 3 I k z c o s | 3 - I k y s i n / 3 (A1.11) I. ^ X k x I, cos/3 +1. sin/3 (A1.12) ky > ky ^ kz K and about the y - a x i s 134 I k x I k x c o s / 3 - I k z s i n ^ ( A L U ) For o p e r a t o r s of more than one s p i n the e f f e c t s on each may be c a l c u l a t e d s e p a r a t e l y and the p r o d u c t of the r e s u l t s t a k e n . E f f e c t i v e l y a summation of the t r a n s f o r m a t i o n over a l l s p i n s a f f e c t e d by the p u l s e i s c a r r i e d o u t , i . e . c ( t _ ) P I k v 3 ^ l v ? ^ mv^... o ( t + ) (A1.15) A1 .5 Zero-quantum coherence e v o l u t i o n The t w o - s p i n p r o d u c t o p e r a t o r 2 I k x I l x , 2 1 k y I l y , 2 I k x I l y and 2 I k y I ^ x each c o n t a i n b oth z e r o - and double-quantum co h e r e n c e . Pure zero-quantum coherence i s g i v e n by ' / ^ ^ k x ^ x +2IkyV = { Z Q C } x ( A 1 - 1 6 ) 1 / 2 ( 2 I k y I l x -2IkxIly) = { Z Q C } y < A K 1 7 ) and pure double-quantum coherence by 1 / 2 ( 2 I k x I l x ""kyV = { D Q C } x ( A 1 ' l 8 ) 1 / 2 ( 2 I k x I l y +2IkyIlx) = { D Q C } y ( A K 1 9 ) The p r e c e s s i o n of zero-quantum coherence due t o c h e m i c a l s h i f t may be d e s c r i b e d i n a manner analogous t o si n g l e - q u a n t u m coherence e x c e p t t h a t i t p r e c e s s e s a t the d i f f e r e n c e i n 1 35 f r e q u e n c i e s of i t s two s p i n s , f o r example, i n ana l o g y t o e q u a t i o n A1.4, {ZQC} ( f l k X k z + Q 1 I 1 Z ) t . {ZQC} c o s ( f l , - f l , ) r j£ ^ X K X + { Z Q C } y s i n ( n k - 0 1 ) T (A1.20) M u l t i p l e - q u a n t u m coherences o n l y e x h i b i t s c a l a r c o u p l i n g s w i t h those s p i n s not a c t i v e w i t h i n t h a t coherence. The e f f e c t i v e c o u p l i n g c o n s t a n t of a s p i n m t o a m u l t i p l e quantum coherence c o n s i s t i n g of s p i n s k may be g i v e n by J e f f = £ A m k J k m ( A 1 ' 2 1 ) where the o r d e r , P, of the coherence i s g i v e n by P=AM =£Am k (A1.22) and s c a l a r c o u p l i n g e v o l u t i o n , l i k e c h e m i c a l s h i f t e v o l u t i o n , i s analogous t o si n g l e - q u a n t u m c o h e r e n c e , f o r example, i n analogy t o e q u a t i o n A1.7, {zqc} ^ 7 r J k m r 2 I k z I m z . {ZQC} COSTTJ f f r + 2 I m z { Z Q C } y s i n 7 r J e f f r (A1.23) R e f e r e n c e s 1. Sorensen, O.W., E i c h , G.W., L e v i t t , M.H., Bodenhausen, G., and E r n s t , R.R., P r o g r . N u c l . Magn. Reson. S p e c t r o s c . 1 36 (1981), 14, 137 1 3 7 APPENDIX I I ZERO-QUANTUM SPECTRA OF AMINO ACIDS 1 38 Appendix I I T h i s appendix c o n s i s t s of a c a t a l o g u e of the zero-quantum coherence s p e c t r a of 16 amino a c i d s o b t a i n e d a t 80.3 MHz w i t h t h r e e s e t s of s t a n d a r d p a r a m e t e r s . Where d i f f e r e n t s e t s of parameters produce d u p l i c a t e r e s u l t s o n l y one of the s p e c t r a i s g i v e n . I f no coherences were v i s i b l e i n a spectrum then t h i s t o o i s not g i v e n . S o l u t i o n s of the amino a c i d s i n D 20 were used w i t h c o n c e n t r a t i o n s of e i t h e r 0.1 molar or t h e i r maximum s o l u b i l i t y , whichever was l e s s . The t h r e e s e t s of parameters used d i f f e r o n l y i n the v a l u e s g i v e n t o T and T'. The v a l u e s of b o t h of the s e time i n t e r v a l s were e q u a l i n each c a s e . The v a l u e s chosen f o r T and T 1 were: 60 msec, 100 msec, and 140 msec. A l l o t h e r parameters were the same: A t ^ l . 6 7 msec, 256 b l o c k s c o l l e c t e d , 4 a c q u i s i t i o n s per b l o c k , t o t a l a c q u i s i t i o n time 14-17 minutes. An e x c e p t i o n was made f o r L - t r y p t o p h a n f o r whi c h , as i t i s o n l y s p a r i n g l y s o l u b l e i n water, 32 a c q u i s i t i o n s per b l o c k were c o l l e c t e d . 139 300 F i g u r e A2.1. Zero-quantum s p e c t r a of L - g l u t a m i n e : A. T,T'=60 msec, B. T,T'=100 msec, C. T,T'=140 msec. T—i—i—i—i—i—i—i—i—I—r 300 200 100 T—I—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—r 0.0 -100 -200 Hz F i g u r e A2.2 Zero-quantum s p e c t r a of L - m e t h i o n i n e : A. T,T'=60 msec, B. T,T'=100 msec, C. T,T'=140 msec. 1 40 A I—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—| 300 200 100 0.0 -100 -200 Hz F i g u r e A2.4. Zero-quantum s p e c t r a of L - a s p a r a g i n e : A. T , T ' = 6 0 msec, B. T , T * = 1 0 0 msec, C. r,r'=140 msec. A I i i i i — | i i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—| 300 200 100 0.0 -100 -200 Hz F i g u r e A2.6. Zero-quantum s p e c t r a of L - l e u c i n e : A. T,T'=60 msec, B. T , T ' = 1 0 0 msec, C. r,r'=140 msec. 142 A I—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i i—i | i i i i | i i i i | 300 200 100 0.0 -100 -200 Hz F i g u r e A2.7. Zero-quantum s p e c t r a of L - t h r e o n i n e : A. T,T'=60 msec, B. T , T ' = 1 0 0 msec, C. T , T ' = 1 4 0 msec. A 300 200 100 0.0 -100 -200 Hz F i g u r e A2.8. Zero-quantum s p e c t r a of L - i s o l e u c i n e : A. T,T'=60 msec, B. T , T * = 1 0 0 msec, C. T , T ' = 1 4 0 msec. 143 I — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — n — i — | — i — i — i — i — | — i — i — i — i — | 300 200 100 0.0 -100 -200 Hz F i g u r e A2.9. Zero-quantum spectrum of L - v a l i n e f o r T , T ' = 6 0 msec. (The same f o r T , T ' = 1 0 0 msec, no cohere n c e s observed f o r T , T * = 1 4 0 msec) B " i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — I — I — i — i — i — I — i — i — i — r 300 200 100 0.0 •100 -200 Hz F i g u r e A2.10. Zero-quantum s p e c t r a f o r L - p r o l i n e : A. T , T ' = 6 0 msec, B. T , T ' = 1 4 0 msec. (No coherences o b s e r v e d f o r T , T ' = 1 0 0 msec) ~t—i—r " i — i — i — i — I — i — i — i — r 300 200 100 ( ~ i — i — i — i — | — i — i — i — i — | — i — i — i — i — | o!o -100 -200 Hz F i g u r e A2.11. Zero-quantum spectrum of L - c y s t e i n e f o r T , T ' = 1 0 0 msec. (For T , T ' = 6 0 msec and 140 msec s p e c t r a t h e same.) 144 | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i i i i | i i ' ' | 300 200 100 0.0 -100 -200 Hz F i g u r e A2.12. Zero-quantum s p e c t r a of L - t r y p t o p h a n : A. T , T ' = 6 0 msec, B. T , T ' = l 0 0 m s e c , C. T , T'=140 msec. | — i — i — i — i — | — i — i — i — i — | — r — i — i — i — | — i — i — i — i — | i i i i f - ! 1 1 1 | 300 200 100 0.0 -100 -200 Hz F i g u r e A2.13. Zero-quantum s p e c t r a of L - s e r i n e : A. T , T ' = 6 0 msec, B. T , T ' = 1 4 0 msec. (For T , T ' = 1 0 0 msec spectrum the same as p a r t B.) 145 A I—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—j 300 200 100 0.0 -100 -200 Hz F i g u r e A2.14. Zero-quantum s p e c t r a of L - a r g i n i n e : A. T , T ' = 1 0 0 msec, B. T , T ' = 1 4 0 msec. (For T , T ' = 6 0 msec no c o h e r e n c e s i n spectrum.) | i i i i | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | 300 200 100 0.0 -100 -200 Hz F i g u r e A2.15. Zero-quantum spectrum of L - a l a n i n e f o r T T'=100 msec. (For T , T ' = 6 0 msec spectrum the same, f o r T ,T'=140msec no coherences i n spectrum.) | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | 300 200 100 0.0 -100 -200 Hz F i g u r e A2.16. Zero-quantum spectrum of L - a s p a r t i c a c i d f o r T , T ' = 100 msec. (For T , T ' = 60 msec and 140 msec s p e c t r a the same.) 

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