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UBC Theses and Dissertations

A novel phase compensated NMR system Hayden, Michael Edward 1986

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A NOVEL PHASE COMPENSATED NMR SYSTEM by MICHAEL EDWARD HAYDEN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES E n g i n e e r i n g P h y s i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Fe b r u a r y 1986 © M i c h a e l Edward Hayden, 1986 7 8 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the The U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . E n g i n e e r i n g P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: F e b r u a r y 1986 A b s t r a c t A novel continuous-wave (cw) Nuclear Magnetic Resonance (NMR) Q-meter which u t i l i z e s phase compensation to f o r c e the resonant frequency of the tuned c i r c u i t to t r a c k the input r f frequency has been developed. T h i s technique e l i m i n a t e s the d i s t o r t i o n s i n the NMR l i n e s h a p e which c h a r a c t e r i z e c o n v e n t i o n a l systems. A p r o t o t y p e system based upon t h i s p r i n c i p l e was c o n s t r u c t e d and used to measure the enhanced nuclear alignment of a p o l a r i z e d deuteron t a r g e t . A g e n e r a l t h e o r e t i c a l d e s c r i p t i o n of the performance of t h i s type of system was developed. The adequacy of t h i s d e s c r i p t i o n was t e s t e d and v e r i f i e d i n the prototype system. D e v i a t i o n s from the t h e o r e t i c a l p r e d i c t i o n s are d i s c u s s e d and the important f a c t o r s which need be c o n s i d e r e d i n the design of such a system are presented. Deuteron thermal e q u i l i b r i u m NMR l i n e s h a p e s were measured with a p r e c i s i o n of ± 5% using the prototype system. R e s u l t s have been shown to be r e p r o d u c i b l e f o r p e r i o d s of at l e a s t three weeks without the need f o r any parameter adjustment i n the phase compensation system. In a d d i t i o n the tensor p o l a r i z a t i o n r e s u l t s o b t ained from the NMR data are shown to be i n agreement with e x p e c t a t i o n s based upon n u c l e a r s c a t t e r i n g d a t a . i i T a b l e of C o n t e n t s A b s t r a c t i i L i s t of F i g u r e s . ....y Acknowledgements v i i i I . I n t r o d u c t i o n 1 I I . C hapter 1 5 A. N u c l e a r S p i n P o l a r i z a t i o n Theory 5 B. The A b s o r p t i o n of Radio Frequency Energy by a Resonant N u c l e a r S p i n System .10 C. N u c l e a r R e l a x a t i o n P r o c e s s e s 18 D. The Shape F u n c t i o n f ( u ) and the B l o c h E q u a t i o n s 19 E. Measurement of Sample P o l a r i z a t i o n 23 F. Dynamic N u c l e a r P o l a r i z a t i o n 25 G. F r o z e n S p i n T a r g e t s 27 H. Quadrupole I n t e r a c t i o n s and the DMR L i n e s h a p e ..29 I I I . C hapter 2 34 A. The P r a c t i c a l Measurement of x(w) 34 B. F a c t o r s Which D i s t o r t the N u c l e a r S p i n Induced M o d u l a t i o n s 43 1. C o a x i a l C a b l e 43 2. D i s t o r t i o n s I n h e r e n t i n the C o n s t a n t C u r r e n t S e r i e s Tuned NMR C i r c u i t 47 C. Other Design C o n s i d e r a t i o n s 52 IV. Chapter 3 55 A. P r i n c i p l e of O p e r a t i o n 56 B. D e s i g n of a Phase Compensated NMR System 59 C. Time Response and S t a b i l i t y 82 D. The System as a Whole 99 V. Chapter 4 " 102 A. The TRIUMF DMR E l e c t r o n i c s Package 103 B. P o s t NMR E l e c t r o n i c s P r o c e s s i n g 117 i i i C. F a c t o r s Which Cause D e v i a t i o n s From I d e a l B e h a v i o u r 121 1. The I n f l u e n c e of the Non - I d e a l V a r a c t o r ....121 2. The I n f l u e n c e of the S t a t i c M a g n e t i c F i e l d .124 3. T h e r m a l , M e c h a n i c a l and E l e c t r o m a g n e t i c N o i s e 127 D. E x p e r i m e n t a l Performance of the TRIUMF P r o t o t y p e 132 E. C o n c l u d i n g Remarks 138 V I . Appendix A 146 A. V a r a c t o r D e v i c e s 146 B. The Simple V a r a c t o r 147 C The MOSFET as a V a r a c t o r 14fl BIBLIOGRAPHY l 4 * i v L i s t of F i g u r e s 1.1 The R o t a t i n g F i e l d 11 1.2 L i n e Broadening 12 1.3 The D i f f e r e n t S p i n P o p u l a t i o n s 14 1.4 The Complex N u c l e a r M a g n e t i c S u s c e p t i b i l i t y 23 1.5 Quadrupole I n t e r a c t i o n s 31 2.1 The r f C o i l and Sample 35 2.2 The C o i l Impedance 37 2.3 The C o i l Impedance Near x(w) 38 2.4 C a n c e l l a t i o n of the R e a c t i v e Impedance 40 2.5 S e r i e s Resonance 42 2.6 S e r i e s Resonance Near x ( w ) 43 2.7 The Imaged Impedance Z' 45 2.8 The C o n s tant C u r r e n t , S e r i e s Tuned C i r c u i t 47 2.9 A d d i t i o n of a Q S p o i l i n g R e s i s t o r 54 3.1 C a n c e l l a t i o n of Net Reactance w i t h C v .56 3.2 Phase A n g l e s f o r S e r i e s Resonance 58 3.3 Phase Compensated NMR Q-Meter 59 3.4 P o t e n t i a l A c r o s s the Resonant C i r c u i t 61 3.5 V a r i a t i o n of the Feed P o t e n t i a l 62 3.6 C a p a c i t a n c e of a V a r a c t o r 64 3.7 (S>p as an A r b i t r a r y F u n c t i o n of \jj 67 3.8 M v s . \p w i t h V f a s e t such t h a t <//=0 a t f = f c 71 3.9 v s . i// as a F u n c i o n of Frequency 73 3.10 as a F u n c t i o n of \p and Frequency 7-^-3.11 R a i s i n g a>c by L o w e r i n g V^ 76 3.12 L o w e r i n g CJ by R a i s i n g V, 77 v 3.13 ^ v s . iji as G Approaches » 78 3.14 v!/ F o r c e d t o 0 as G i s I n c r e a s e d 80 3.15 Rate of Response t o an E r r o r S i g n a l 81 3.16 RC Time C o n s t a n t due t o V a r a c t o r C a p a c i t a n c e 83 3.17 F i l t e r i n g and C a p a c i t i v e L o a d i n g 84 3.18 Frequency Dependent A m p l i f i c a t i o n 84 3.19 I n t e g r a t i o n F o l l o w i n g Phase D e t e c t i o n 85 3.20 The i t h Low Pass F i l t e r 86 3.21 Example Root Locus Diagram 94 3.22 O s c i l l a t o r y B e h a v i o u r of \p 97 3.23 E f f e c t of I n c r e a s i n g A m p l i f i e r G a i n 99 4.1 The TRIUMF NMR E l e c t r o n i c s 105 4.2 E a r l y Phase D e t e c t i o n System 110 4.3 Output Waveform Showing Phase V a r i a t i o n Without Compensation 114 4.4 Output Waveform Showing Phase V a r i a t i o n W i t h Compensation 115 4.5 The Uncompensated Q Curve 116 4.6 The Compensated "Q Curve" 116 4.7 The E l e c t r o n i c s A s s o c i a t e d w i t h the NMR System ....118 4.8 Power D i s s i p a t i o n i n C v 122 4.9 Example TE S i g n a l 134 4.10 Example Enhanced S i g n a l 137 A.1 pn J u n c t i o n Doping 144 A. 2 The MOSFET 1 46 A.3 MOSFET i n A c c u m u l a t i o n 147 A.4 MOSFET i n A c c u m u l a t i o n 146 A.5 MOSFET i n D e p l e t i o n 148 v i A.6 MOSFET i n D e p l e t i o n 149 A.7 MOSFET i n I n v e r s i o n 1 50 A.8 MOSFET i n I n v e r s i o n 1 5.V A.9 Reverse C u r r e n t A c r o s s M e t a l l u r g i c a l J u n c t i o n 1 %\ A.10 Reverse C u r r e n t A c r o s s F i e l d Induced J u n c t i o n 15a A. 11 Reverse C u r r e n t Due t o S u r f a c e G e n e r a t i o n 152.' A.12 T o t a l Reverse C u r r e n t 153 v i i Acknowledgement The r e s u l t s p r e s e n t e d i n t h i s t h e s i s r e p r e s e n t f a r more time and e f f o r t than one p e r s o n c o u l d devote t o such a p r o j e c t i n so s h o r t a t i m e . I am i n d e b t e d t o s e v e r a l people f o r t h e i r a s s i s t a n c e i n many a s p e c t s of t h i s t h e s i s . My e x p e r i m e n t a l c o l l a b o r a t o r s c o n s i s t of the members of the TRIUMF t a r g e t s group. In p a r t i c u l a r , Dr. Gary W a i t , Dr. Dennis Healey and Dr. P a u l D e l h e i j were v e r y c l o s e l y i n v o l v e d w i t h the p r o j e c t . T h i s m a n u s c r i p t has undergone s e v e r a l r e v i s i o n s s i n c e the f i r s t d r a f t . I would l i k e t o thank my t h e s i s s u p e r v i s o r Dr. G a r t h J o n e s , Dr. Gary W a i t , Graeme Luke, and my mother and f a t h e r f o r t h e i r many h e l p f u l s u g g e s t i o n s and t h e i r a s s i s t a n c e i n p r o o f r e a d i n g the v a r i o u s d r a f t s . F i n a l l y , but f a r from t h e l e a s t I would l i k e t o thank Jean H o l t f o r drawing most of the diagrams which appear throughout t h e t h e s i s . v i i i I . INTRODUCTION N u c l e a r M a g n e t i c Resonance (NMR) l i n e s h a p e s can be measured by c o u p l i n g the magnetic moments of the sample n u c l e i t o a r a d i o f r e q u e n c y ( r f ) s i g n a l which i s a p p l i e d t o the c o i l of a s e r i e s or p a r a l l e l r e s o n a n t c i r c u i t . T h i s c o u p l i n g r e s u l t s i n a t r a n s f e r of energy between the r f source and the s p i n system which can be ob s e r v e d as a m o d i f i c a t i o n of the impedance of the r e s o n a n t c i r c u i t a t f r e q u e n c i e s near t h e Larmor p r e c e s s i o n f r e q u e n c y of the magnetic momments. Such CW-NMR t e c h n i q u e s can be used t o measure the n u c l e a r v e c t o r p o l a r i z a t i o n of a p o l a r i z e d t a r g e t used f o r n u c l e a r s c a t t e r i n g or a b s o r p t i o n e x p e r i m e n t s . In t h i s type of work, however, i t i s n e c e s s a r y t o view the c o i l impedance th r o u g h a l e n g t h of c o a x i a l c a b l e as the t a r g e t i s l o c a t e d i n s i d e a c r y o s t a t or r e f r i g e r a t o r . The imaged c o i l impedance i s f r e q u e n c y dependent. Thus, d i s t o r t i o n s t o the NMR l i n e s h a p e a r e en c o u n t e r e d due t o the r e a c t i v e n a t u r e of the resonant c i r c u i t and the c a b l e . The g o a l of t h i s t h e s i s was t o d e v e l o p a n o v e l "phase compensated NMR system" which i s c a p a b l e of a v o i d i n g d i s t o r t i o n s t o the NMR l i n e s h a p e by f o r c i n g the resonant c i r c u i t t o s t a y a t resonance a t any fr e q u e n c y w i t h i n a s p e c i f i e d range and hence e l i m i n a t i n g the net r e a c t a n c e of the c i r c u i t . T h i s d e v i c e i s e s p e c i a l l y s u i t e d t o a p p l i c a t i o n i n p o l a r i z e d t a r g e t s where the need t o d e t e c t a v e r y s m a l l energy exchange between the t a r g e t n u c l e i and the re s o n a n t 1 2 c i r c u i t l e a v e s no room f o r masking the d a t a w i t h v a r i o u s n o n l i n e a r i t i e s . In a d d i t i o n , as the t e c h n i q u e l e a d s t o a f r e q u e n c y independent impedance f o r the r e s o n a n t c i r c u i t , t he dynamic range of the r f s i g n a l i s d r a s t i c a l l y reduced by c omparison w i t h the uncompensated c a s e . In t h i s manner, phase compensation becomes a p r a c t i c a l a l t e r n a t i v e t o the c o n v e n t i o n a l t e c h n i q u e s 1 f o r a v o i d i n g a m p l i f i e r s a t u r a t i o n . The f i r s t c h a p t e r of t h i s t h e s i s d e s c r i b e s the b a s i c t h e o r y of n u c l e a r s p i n p o l a r i z a t i o n as a p p l i e d t o p o l a r i z e d t a r g e t work. W h i l e not c e n t r a l t o the development of the phase compensation i d e a , t h i s m a t e r i a l i s u s e f u l from the p o i n t of view of u n d e r s t a n d i n g the p h y s i c a l s i g n i f i c a n c e of the p o l a r i z a t i o n measurement. T h i s c h a p t e r i s i n t e n d e d t o be a c o l l e c t i o n of p e r t i n e n t i n f o r m a t i o n f o r the non-NMR s p e c i a l i s t . F a r more complete t r e a t m e n t s of t h i s m a t e r i a l can be found i n the r e f e r e n c e s by Abragam and S l i c h t e r . The second c h a p t e r i s concerned w i t h an i n v e s t i g a t i o n of the v a r i o u s n o n l i n e a r i t i e s which a r e e n c o u n t e r e d i n the measurement of NMR l i n e s h a p e s i n a p o l a r i z e d t a r g e t e n vironment. In a d d i t i o n , s e v e r a l n o i s e r e l a t e d f a c t o r s which i n f l u e n c e the p o l a r i z a t i o n measurement a r e c o n s i d e r e d . The phase compensation concept i s i n t r o d u c e d i n the t h i r d c h a p t e r . The t h e o r e t i c a l p r i n c i p l e s of o p e r a t i o n of the d e v i c e and the v a r i o u s d e s i g n c r i t e r i a which need t o be c o n s i d e r e d a r e d i s c u s s e d . S e v e r a l g e n e r a l i z e d e q u a t i o n s which c h a r a c t e r i z e the b e h a v i o r of the system a r e d e v e l o p e d . 1H. K l e v e r and M. S c h l a a k , " A Simple C i r c u i t f o r cw NMR Measurements," Rev. S c i . I n s t r u m . 44 ( 1 ) , 25-27 (1973). 3 The f i n a l c h a p t e r d e s c r i b e s an o p e r a t i o n a l " p r o t o t y p e " phase compensated NMR system which has been b u i l t and implemented i n a p o l a r i z e d t a r g e t environment. Rather than p r e s e n t a d e t a i l e d a n a l y s i s of t h i s system, t h i s c h a p t e r p r e s e n t s a d i s c u s s i o n of the v a r i o u s d e v i a t i o n s from i d e a l b e h a v i o r which can be e x p e c t e d i n such a system and r e f e r s t o the p r o t o t y p e system as an example. An e v a l u a t i o n of the performance of the p r o t o t y p e and the v a l i d i t y of the t h e o r e t i c a l developments of c h a p t e r t h r e e i s based upon v a r i o u s e x p e r i m e n t a l r e s u l t s i n c l u d i n g an independant measurement of the t a r g e t p o l a r i z a t i o n . The work d e s c r i b e d i n t h i s t h e s i s a r o s e out of the need f o r a system t o measure the p o l a r i z a t i o n of a p o l a r i z e d d e u t e r o n t a r g e t a t the TRIUMF n u c l e a r r e s e a r c h f a c i l i t y i n Vancouver, Canada. The o r i g i n a l d e s i g n of the system and the development of the t h e o r e t i c a l p r i n c i p l e s of o p e r a t i o n of t h i s d e v i c e were c o n t r i b u t e d by the a u t h o r . V a r i o u s m o d i f i c a t i o n s and improvements t o the system a r e the r e s u l t of a j o i n t e f f o r t of G. W a i t 2 and the a u t h o r . W h i l e the work r e l a t e d t o the development of t h i s system was done i n d e p e n d e n t l y , i t has r e c e n t l y been d i s c o v e r e d t h a t s i m i l a r systems have been c o n c e i v e d of and b u i l t i n the p a s t , w i t h l i m i t e d s u c c e s s . The o r i g i n a l p r o p o s a l of the b a s i c concept s h o u l d be a t t r i b u t e d t o V. P e t f i c e k 3 w i t h f u r t h e r development work c a r r i e d out a t the 2Permanent a d d r e s s : TRIUMF, 4004 Wesbrook M a l l , Vancouver B.C. Canada V6T 2A3. 3 V. P e t f i c e k , "A L i n e a r i z e d Q-Meter C i r c u i t f o r Measurement of H i g h P r o t o n P o a r i z a t i o n i n a T a r g e t , " N u c l . I n s t r u m . and 4 R u t h e r f o r d H i g h Energy L a b o r a t o r y . " Mcont'd) Methods 58, 111-116 (1968). " A. R. Cash, R u t h e r f o r d L a b o r a t o r i e s Report No. RL-78-103, 1978. 11 . CHAPTER 1 T h i s c h a p t e r w i l l d e v e l o p the c o n c e p t s of n u c l e a r s p i n p o l a r i z a t i o n , the c o u p l i n g of the n u c l e a r s p i n system t o an e x t e r n a l s o u r c e of r f energy and v a r i o u s o t h e r t o p i c s which a r e p e r t i n e n t t o the development of an NMR system i n a p o l a r i z e d t a r g e t environment. The work p r e s e n t e d here i s not o r i g i n a l and can be found i n many books on the t o p i c of N u c l e a r Magnetic Resonance. T h i s i s s i m p l y an attempt t o summarize the r e l e v a n t i n f o r m a t i o n . A. NUCLEAR SPIN POLARIZATION THEORY C o n s i d e r a system of n u c l e a r s p i n s I h a v i n g a magnetic d i p o l e moment where y i s the gyromagnetic r a t i o and h7 = h/2ir. When p l a c e d i n a magnetic f i e l d H, the H a m i l t o n i a n f o r a p a r t i c u l a r s p i n i s ( 1 . 1 ) H = - M - H " ( 1 . 2 ) = - T K H I Z . Note t h a t H = \U\ and t h a t we have assumed t h a t H" i s 5 6 p a r a l l e l t o z. T h i s can a l s o be w r i t t e n as H = }icj I (1.3) L i z where a>L i s the Larmor f r e q u e n c y . E i g e n v a l u e s of t h i s H a m i l t o n i a n a r e of the form mh'wL where m t a k e s on 21+1 v a l u e s between - I and I . The energy d i f f e r e n c e between a d j a c e n t energy l e v e l s i s g i v e n by AE = 7KH . (1.4) The m a g n e t i z a t i o n of a c o l l e c t i o n of n u c l e a r s p i n s can be d e f i n e d as R = ££. (1.5) where t h e a r e the i n d i v i d u a l d i p o l e moments. In a s t a t i c magnetic f i e l d H"0 / M" can be r e l a t e d t o H"0 t h r o u g h the s t a t i c n u c l e a r s u s c e p t i b i l i t y Xo of the m a t e r i a l M 0 = Xo^o • (1 . 6 ) S i n c e t h e t h e r m a l e q u i l i b r i u m p o p u l a t i o n s P of the energy l e v e l s a r e p r o p o r t i o n a l t o -E m/kT iKmHo/kT = e ( i . e . a Boltzmann d i s t r i b u t i o n ) e m = e (1.7) M can be w r i t t e n i n the form 7h/mH0/kT M = NTffZme (1.8) 7hmH 0/kT Ze where the summations a r e over a l l m (-1 t o I ) . However, as 7h/mH0/kT i s almost always s m a l l i n ex p e r i m e n t s d e a l i n g w i t h n u c l e a r magnetism the e x p o n e n t i a l s can be expanded w i t h o n l y the l e a d i n g terms b e i n g r e t a i n e d . M = N7 2K 2H 0Zm 2 (1.9) kT(2I+1) = N7 2K 2I(I+1)H 0 3kT 8 where Xo i s the s t a t i c n u c l e a r s u s c e p t i b i l i t y Xo = N 7 2 M 2 I ( 1 + 1 ) . ( 1 . 1 0 ) 3 k T A n u c l e a r v e c t o r p o l a r i z a t i o n can a l s o be d e f i n e d as P = <I >/I . ( 1 . 1 1 ) For a s p i n 1/2 system such as a p r o t o n t h i s can be reduced t o P 1 / 2 = n t - n . ( 1 . 1 2 ) n + +n. = t a n h ( - A E / 2 k T ) where n + and n. r e f e r t o the s p i n p o p u l a t i o n s i n the two p o s s i b l e o r i e n t a t i o n s r e l a t i v e t o the magnetic f i e l d . For a s p i n 1 system (such as a de u t e r o n ) which p o s s e s s e s t h r e e s p i n s t a t e s (+, 0 , and -) t h i s r e l a t i o n becomes 9 P = ru ~ n. (1.13) n + + n 0 + n. = 4+tanh(-AE/2kT) . 3+tanh(-AE/2kT) At a p a r t i c u l a r magnetic f i e l d and t e m p e r a t u r e ( u s u a l l y such t h a t AE/2kT<<1) the n a t u r a l or t h e r m a l e q u i l i b r i u m p o l a r i z a t i o n of the s p i n s i s g i v e n by P 1 / 2 * MH/2kT (1.14) P, * (2/3)MH/kT . (1.15) For example, a t 1K and 2.5 T e s l a P ^ 2 = 0. 1277% P 1 = 0.05227% where th e magnetic d i p o l e moments M = 2.793 and M = .8574 n u c l e a r magnetons were used f o r the p r o t o n and the d e u t e r o n r e s p e c t i v e l y . W h i l e the two s p i n s t a t e s of the s p i n 1/2 system can be d e s c r i b e d a d e q u a t e l y by knowing the t o t a l number of s p i n s 10 and the r e l a t i v e p o p u l a t i o n of the two l e v e l s ( v e c t o r p o l a r i z a t i o n ) , a t h i r d r e l a t i o n s h i p i n v o l v i n g h 0 i s r e q u i r e d t o c o m p l e t e l y d e s c r i b e the d e u t e r o n system. The t e n s o r p o l a r i z a t i o n or a l i g n m e n t i s d e f i n e d t o be where p 0 = n 0 / ( n + + n 0 + n . ) . Note t h a t the a l i g n m e n t i s d e f i n e d such t h a t A = 0 f o r e q u i l i b r i u m where p 0 = 1/3. B. THE ABSORPTION OF RADIO FREQUENCY ENERGY BY A RESONANT  NUCLEAR SPIN SYSTEM C o n s i d e r now the case where a s m a l l r o t a t i n g f i e l d R* i s added t o a l a r g e dc magnetic f i e l d H"0 ( | H", | << | i ? 0 | ) . A = 3 1 2 - K l + 1) (1.16) z = 1 - 3p 0 H x = H , C O S C J t (1.17) H = H T s i n c j t U s i n g a p e r t u r b a t i o n t e c h n i q u e 5 the u n p e r t u r b e d H a m i l t o n i a n 5See f o r example the r e f e r e n c e s by Abragam or S l i c h t e r . 11 H , F i g u r e 1.1 The R o t a t i n g F i e l d . Note t h a t 6=cjt modulus 2ir can be w r i t t e n as H 0 = * J L I z (1.18) w h i l e the p e r t u r b i n g H a m i l t o n i a n i s d e s c r i b e d by H, = - 7 K ( H x I x + H y I y ) (1.19) = - ( l / 2 ) 7 K H 1 ( I + e " j c J t + I . e j c J t ) = - ( l / 2 ) K w i ( I + e " j w t + I . e j w t ) where u 1 = yH, and I + and I . a r e r a i s i n g and l o w e r i n g 12 o p e r a t o r s which o n l y p e r m i t non-zero m a t r i x elements between s t a t e s f o r which Am = ± 1 . The time-dependent p e r t u r b a t i o n a t f r e q u e n c y o w i l l o n l y i n duce t r a n s i t i o n s between s t a t e s s e p a r a t e d by an energy i n t e r v a l AE = tfa>L i f u> i s near co^. In p r a c t i c e the energy l e v e l s have a f i n i t e w i d t h d e s c r i b e d by some f u n c t i o n f(a>) such t h a t ff(o>)da> = 1 (1 . 2 0 ) or p(AE) such t h a t /p(AE)d(AE) = 1 ( 1 . 2 1 ) f (di) L F i g u r e 1 .2 L i n e B r o a d e n i n g Note t h a t t h i s i m p l i e s t h a t f(a>) = Kp(Kcj) • 1 3 U s i n g a f i r s t o r d e r p e r t u r b a t i o n t h e o r y ( i . e . F e r m i ' s Golden Rule) the t r a n s i t i o n p r o b a b i l i t y per u n i t time between s t a t e s \a> and |/3> can be w r i t t e n as Wa/3 = (2TT/K)P(AE) |<j3|H, | a> | 2 (1.22) so t h a t Wm,m-1 = (2 - r /K)(7KH 1/2 ) 2 |<m|l + jm-1>| 2 ( f (w L)/K) (1.23) = (ir/2)w, 2 (I+m) (l-m+1 )f (w_ ) . Next c o n s i d e r the case where the s a t u r a t i o n i s s m a l l so t h a t the d i f f e r e n c e s i n p o p u l a t i o n s of the l e v e l s can be w r i t t e n KwT/kT 1 (e L -1) (21+1) - 1 yio>T . • (21+1) k T b 1 4 I— < _l CL o Q_ m - i m MAGNETIC SUBLEVEL F i g u r e 1.3 The D i f f e r e n t S p i n P o p u l a t i o n s The r a t e of energy a b s o r p t i o n per u n i t volume, Pwr(cj), i s g i v e n by Pwr(w) = Z N ( A E ) ( P m .~P m)W (1.25) m _ i m m,m-i where t h e summation over m ranges from -1+1 t o I and N denotes the number of s p i n s per u n i t volume. U s i n g e q u a t i o n s 1.23 and 1.24 t h i s e x p r e s s i o n becomes Pwr(w) = Nho) }iu 7TCJ i f (OJ) Z | <m 11 ^  |m-1> | 2 (21+1)'kT 2 (1.26) 15 = 7rK272H2a>a)r -1(1 + 1 )Nf (CJ) . 3kT b By s u b s t i t u t i n g f o r the s t a t i c s u s c e p t i b i l i t y Xo from e q u a t i o n 1.10 we o b t a i n Pwr(w) = 7rcjo>LH2f (CJ )XO • (1.27) There of c o u r s e e x i s t s some type of r e l a x a t i o n p r o c e s s which p r o v i d e s a mechanism which a l l o w s the s p i n s t o s t a y a t the l a t t i c e t e mperature ( t h e r m a l e q u i l i b r i u m ) . That i s , the s p i n p o p u l a t i o n s P m remain a t t h e i r Boltzmann v a l u e s . I t i s i n s t r u c t i v e t o d e s c r i b e the r f power a b s o r p t i o n i n a n o t h e r way. Pwr = -H-dH (1.28) dt Assume t h a t the r o t a t i n g f i e l d H", i s produced by the l i n e a r l y p o l a r i z e d f i e l d H = 2H,coswt . (1.29) H i s a c t u a l l y the sum of H", and a c o u n t e r r o t a t i n g f i e l d 1 6 w i t h a n g u l a r f r e q u e n c y -CJ. I f the bandwidth of the NMR s i g n a l i s s m a l l compared t o CJ t then the c o u n t e r r o t a t i n g f i e l d i s a t a f r e q u e n c y of the o r d e r of 2 C J l away from the Larmor f r e q u e n c y and hence can be assumed t o have n e g l i g i b l e e f f e c t . That i s , the l i n e a r l y p o l a r i z e d f i e l d i s e q u i v a l e n t t o H", from the p o i n t of view of the n u c l e a r s p i n s . I f the r f e x c i t a t i o n i s s m a l l , the response of the s p i n s U i s p r o p o r t i o n a l t o H", where x(w) = x'~ jx" i s the complex s u s c e p t i b i l i t y of the s p i n s ; x' b e i n g the d i s p e r s i v e component and x" the a b s o r p t i v e component. These components a r e r e l a t e d by the w e l l known K r a m e r s - K r o n i g r e l a t i o n s 6 f o r the r e a l and i m a g i n a r y p a r t s of the response of a l i n e a r system t o a s i n u s o i d a l e x c i t a t i o n . i . e . R = xH"i (1 . 3 0 ) = (x'~ jx")H CO ( 1 . 3 1 ) 6 S e e f o r example the r e f e r e n c e s by J a c k s o n or S l i c h t e r . 17 CO X"(u) = "I P.fx' (u^-X^du' co CJ- e oo where Pfhju' )dco' means //m(J* + J) U s i n g t h i s f o r m a l i s m we can w r i t e ] C J t H v = 2H1J?e{e } (1 .32) jcot M = 2H,J?e{xe } ( 1 .33) = 2H 1 (x'coswt + x"sino;t) Thus T Pwr(w) = /M dH dt (1.34) T 0 X d t X = 2H?x w • The meaning of the term " a b s o r p t i v e component" of the complex s u s c e p t i b i l i t y i s o b v i o u s from t h i s s t a t e m e n t . 18 T h i s r e s u l t i s of paramount importance t o t h i s t h e s i s as i t e x p r e s s e s the p r o p o r t i o n a l i t y between the s u s c e p t i b i l i t y and the power a b s o r p t i o n by the s p i n system, which i s a measurable q u a n t i t y . Comparing the two e x p r e s s i o n s f o r the power a b s o r p t i o n we o b t a i n the u s e f u l e x p r e s s i o n C. NUCLEAR RELAXATION PROCESSES As demonstrated above, th e unequal p o p u l a t i o n s of the e x t e r n a l r f energy g r e a t e r than i n d u c e d e m i s s i o n f o r p o s i t i v e p o l a r i z a t i o n . The r e v e r s e i s t r u e f o r a n e g a t i v e v e c t o r p o l a r i z a t i o n . A s t e a d y s t a t e NMR experiment i s the r e s u l t of two competing p r o c e s s e s ; the a b s o r p t i o n of r f energy (from an e x t e r n a l g e n e r a t o r ) which tends t o d e s t r o y the Boltzmann i n e q u a l i t i e s and v a r i o u s r e l a x a t i o n p r o c e s s e s which t r y t o r e - e s t a b l i s h t h e s e i n e q u a l i t i e s . One form of r e l a x a t i o n p r o c e s s i n v o l v e s the t h e r m a l c o u p l i n g of the n u c l e a r s p i n s t o the l a t t i c e . A l t h o u g h t h i s " s p i n - l a t t i c e r e l a x a t i o n " can not be d e s c r i b e d a d e q u a t e l y by a s i n g l e time c o n s t a n t , i n most c a s e s i t i s c o n v e n t i o n a l t o i n t r o d u c e a s p i n - l a t t i c e r e l a x a t i o n time T, which a t l e a s t g i v e s the time s c a l e of the p r o c e s s . As a r e s u l t of l a r g e n u c l e a r c o n c e n t r a t i o n s and s m a l l d i s t a n c e s between n u c l e a r s p i n s i n b u l k m a t t e r , s p i n - s p i n X"(w) = ( l / 2 ) 7 r x o u L f (") • (1 . 3 5 ) magnetic energy l e v e l s , E ~e makes a b s o r p t i o n of 19 d i p o l e - i n t e r a c t i o n s a l s o p l a y a r o l e i n NMR work. The f i r s t e f f e c t i s the b r o a d e n i n g of resonance l i n e s due t o v a r i a t i o n s i n the l o c a l f i e l d a c r o s s the sample (a r e s u l t of c l o s e n e i g h b o u r i n g s p i n s ) . I n a d d i t i o n , t i g h t s p i n - s p i n c o u p l i n g a l l o w s f o r r a p i d energy t r a n s f e r from one s p i n t o a n o t h e r . T h i s a l l o w s the n u c l e a r s p i n system t o r e a c h i t s own t h e r m a l e q u i l i b r i u m a t a temperature d i f f e r e n t from t h a t of the l a t t i c e and i n much l e s s t i m e . The s p i n - s p i n r e l a x a t i o n time i s denoted by T 2 ( u s u a l l y T 2 « T , ) . D. THE SHAPE FUNCTION F (u>) AND THE BLOCH EQUATIONS In g e n e r a l , the shape f u n c t i o n f(w) i s a narrow b e l l shaped c u r v e w i t h i t s maximum a t the Larmor f r e q u e n c y CJ T LI ( r e f e r t o f i g u r e 1.2). These shape f u n c t i o n s a re c h a r a c t e r i s t i c of many s p e c t r o s c o p i c l i n e s . In o r d e r t o demonstrate how such a l i n e s h a p e i s m a n i f e s t e d a phe n o m e n o l o g i c a l d e s c r i p t i o n due t o F e l i x B l o c h ( 1 9 4 6 ) 7 i s p r e s e n t e d . When p l a c e d i n a f i e l d R, a magnetic moment M e x p e r i e n c e s a t o r q u e r = MxH" (1 .36) e q u a l t o t h e r a t e of change tfdl/dt of i t s a n g u l a r momentum. Thus f o r a n u c l e a r s p i n R = ytfl we have the e q u a t i o n of 7 A. Abragam, The P r i n c i p l e s of N u c l e a r Magnetism, ( O x f o r d U n i v e r s i t y P r e s s , New York, 1983), p. 44. 20 motion dR = T R X H . (1.37) dt In a l a r g e dc magnetic f i e l d H 0 which i s p a r a l l e l t o z, we can o f t e n d e s c r i b e the t r e n d of the m a g n e t i z a t i o n towards i t s e q u i l i b r i u m v a l u e M = M 0 = XoH 0 by the e x p r e s s i o n z dM = - M z M ° . (1.38) -z d t " T In a d d i t i o n , i f a s m a l l r f s i g n a l i s a p p l i e d a t r i g h t a n g l e s t o z, the n u c l e a r s p i n s w i l l respond by p i c k i n g up a component of m a g n e t i z a t i o n p e r p e n d i c u l a r t o H"0. T h i s t r a n s v e r s e m a g n e t i z a t i o n decays a p p r o x i m a t e l y as dM = -M dM = -M . (1.39) d t x T* d t y T^ Note t h a t the t r a n s v e r s e m a g n e t i z a t i o n can decay by the f a s t e r s p i n - s p i n r e l a x a t i o n p r o c e s s as t h i s decay c o n s e r v e s energy i n the s t a t i c f i e l d . 8 Thus i n the presence of a l a r g e dc f i e l d and a s m a l l o r t h o g o n a l r f f i e l d the e q u a t i o n of 8C. P. S l i c h t e r , P r i n c i p l e s of Magnetic Resonance, 2nd ed. ( S p r i n g e r - V e r l a g , New York, 1978), p. 33. 21 motion i s assumed to be of the form dM = yUxU dt M x + M y (M - M 0)z x yJ - z _ (1.40) T 2 T, Th i s i s known as Bloch's phenomenological equation f o r paramagnetic resonance. With v a r i o u s approximations or assumptions, t h i s equation may be s o l v e d f o r the components of the nuc l e a r m a g n e t i z a t i o n . For example, i n the case of the l i n e a r l y p o l a r i z e d r f f i e l d at frequency u> d i s c u s s e d e a r l i e r (the counter r o t a t i n g component of the f i e l d being n e g l e c t e d ) , we o b t a i n the steady s t a t e s o l u t i o n by s e t t i n g dM = dM = dM = 0 . d t x d t Y d t z (1.41) The s o l u t i o n can be w r i t t e n (CJ-CJT )7H 1T|coscot-7H 1T 2sincJt (1.42) M 1 + T| (u-a>T ) 2 + 7 2H 2T,T 2 Li X (CJ-COT ) 7 H 1 T l s i n o j t - 7 H 1 T 2 c o s c j t M. 1 + T\ (u)-u>L) 2 + 7 Z H 2 T 1 T 2 y 22 1 + T| (w-co r ) 2 M = z 1 + T l (u-u, ) 2 + 7 2 H 2 T 1 T 2 M 0 From t h e d e f i n i t i o n o f t h e r f s u s c e p t i b i l i t i e s ( s e e e q u a t i o n 1.30) we can w r i t e - ( 1 / 2 ) U L ( U - U L ) T ! ( 1 . 4 3 ) X = 1 + T 2. (co-co L) 2 + 7 2 H 2 T 1 T 2 X o ( l / 2 ) w r T 2 X 1 + T 1 ( C O - O J L ) 2 + 7 2 H Z T 1 T 2 X O B o t h o f t h e s e f u n c t i o n s a r e p r o p o r t i o n a l t o t h e n o r m a l i z e d L o r e n t z shape f u n c t i o n L ( u ) = T 2 1 ( 1 . 4 4 ) 1 + [ T 2 ( G J - C J T ) ] 2 where T 2 = T 2 [ 1 + 7 2 H 2 T , T 2 ] 1 ^ 2 23 F i g u r e 1.4 The Complex N u c l e a r M a g n e t i c S u s c e p t i b i l i t y x=x'-jx" E. MEASUREMENT OF SAMPLE POLARIZATION Hav i n g r e l a t e d x" t o a measurable q u a n t i t y ( i . e . power a b s o r p t i o n of r f energy from an e x t e r n a l s o u r c e ) , i t i s easy t o r e c o g n i z e the p r o p o r t i o n a l i t y between the sample p o l a r i z a t i o n and the i n t e g r a l 24 JV' ( & > ) d u . (1 .45) 0 By measuring the r f power a b s o r p t i o n of the s p i n s a t a l l f r e q u e n c i e s and hence e v a l u a t i n g t h i s i n t e g r a l we a r e a b l e t o compare r e l a t i v e t a r g e t p o l a r i z a t i o n s ( g i v e n the p h y s i c a l parameters d e s c r i b i n g the s t a t e ; the tem p e r a t u r e and the magnetic f i e l d s t r e n g t h ) . As w i l l be d i s c u s s e d s h o r t l y , i t i s p o s s i b l e t o r a i s e the p o l a r i z a t i o n of a sample above i t s t h e r m a l e q u i l i b r i u m v a l u e by the t e c h n i q u e of dynamic p o l a r i z a t i o n . By comparing the v a l u e of the above i n t e g r a l f o r the s p i n system i n th e r m a l e q u i l i b r i u m t o the same system a f t e r b e i n g d y n a m i c a l l y p o l a r i z e d the r e l a t i v e i n c r e a s e i n p o l a r i z a t i o n may be c a l c u l a t e d . i . e . PDyn = = PTE ( 1 ' 4 6 ) / X " ( w ) d c o 0 T E When used i n c o n j u n c t i o n w i t h e q u a t i o n 1.14 or e q u a t i o n 1.15 (which a r e used t o c a l c u l a t e the a b s o l u t e t h e r m a l e q u i l i b r i u m p o l a r i z a t i o n ) , e q u a t i o n 1.46 can be used t o c a l c u l a t e the a b s o l u t e dynamic p o l a r i z a t i o n . 25 F. DYNAMIC NUCLEAR POLARIZATION The t e c h n i q u e of d y n a m i c a l n u c l e a r p o l a r i z a t i o n makes i t p o s s i b l e t o enhance the n u c l e a r p o l a r i z a t i o n above the t h e r m a l e q u a l i b r i u m l e v e l by u t i l i z i n g the e l e c t r o n i c s p i n resonance and the c o u p l i n g between the e l e c t r o n i c and the n u c l e a r s p i n systems. W h i l e t h e r e a r e s e v e r a l mechanisms which can be u t i l i z e d t o produce t h i s e f f e c t , the S o l i d S t a t e e f f e c t i s the most i m p o r t a n t w i t h r e g a r d s t o the c o n s t r u c t i o n of p r a c t i c a l p o l a r i z e d t a r g e t s used i n n u c l e a r s c a t t e r i n g e x p e r i m e n t s . T h i s e f f e c t i s d e s c r i b e d below. C o n s i d e r a system of n u c l e a r s p i n s I and l o c a l i z e d e l e c t r o n i c s p i n s S" (due t o the i n t r o d u c t i o n of paramagnetic i m p u r i t i e s i n the sample). Denote the p o p u l a t i o n s of the e l e c t r o n i c and n u c l e a r s p i n s t a t e s (assumed t o be s p i n 1/2 f o r s i m p l i c i t y ) by N +,N.,n + and n. r e s p e c t i v e l y . In a d d i t i o n denote the v a r i o u s t r a n s i t i o n p r o b a b i l i t i e s i n a f a s h i o n s i m i l a r t o W „ „ which i n d i c a t e s a s i m u l t a n e o u s f l i p of the a/3-6 e c e l e c t r o n i c s p i n (a t o 6) and the n u c l e a r s p i n (S t o e ) . C o n s i d e r f i r s t a case where the o n l y a l l o w e d t r a n s i t i o n s a r e W „ c and WE „. Under stea d y s t a t e c o n d i t i o n s we have the r e q u i r e m e n t t h a t N n„W p , = N £n Ws „ . (1 .47) T h i s statement assumes t h a t t h e r e e x i s t s some i n f i n i t e r e s e r v o i r t o absorb or s u p p l y any n e c e s s a r y quanta of 26 energy. T h i s might be a l a t t i c e w i t h i n f i n i t e s p e c i f i c heat (or more' r i g o r o u s l y a l a t t i c e i n t h e r m a l e q u i l i b r i u m w i t h i t s e l f ) or an e x t e r n a l s o u r c e of r f energy. A d d i t i o n a l terms can be added as n e c e s s a r y t o d e s c r i b e o t h e r t r a n s i t i o n p r o b a b i l i t i e s . Next assume t h a t we have a d i f f e r e n t system i n which an e x t e r n a l source of r f energy a t the f r e q u e n c i e s = t J g + £ J I or 0 = " g - ^ j (1.48) where = "~7gH0 ( r e f e r t o e q u a t i o n s 1.2 and 1.3) " i = ~ 7 i H ° a>g>>o)j i n d u c e s the t r a n s i t i o n s W+. . + or W + + ... Note t h a t t h e s e i n v e r s e t r a n s i t i o n p r o b a b i l i t i e s a r e of c o u r s e e q u a l . In a d d i t i o n they a r e not the o n l y a l l o w e d t r a n s i t i o n p r o b a b i l i t i e s . Now i f the a v a i l a b l e power i s g r e a t enough t h a t the r a t e of t r a n s i t i o n s i s much f a s t e r than the n u c l e a r r e l a x a t i o n r a t e ( l / T , ^ but a t the same time much s l o w e r than the e l e c t r o n i c r e l a x a t i o n r a t e ( 1 / T ^ g we can o b t a i n a s i t u a t i o n i n which the e l e c t r o n i c p o p u l a t i o n d i s t r i b u t i o n i s e s s e n t i a l l y u n a f f e c t e d . That i s , they keep t h e i r Boltzmann 27 e q u i l i b r i u m v a l u e s ( N + ) 0 and ( N . ) 0 . The n u c l e a r p o p u l a t i o n s under t h i s new steady s t a t e c o n d i t i o n a r e g i v e n by ru = N. n. N. = e - h a ; s / k T i f = co s - ct (1 .49) ( i . e . W+. . +) and n^ = n. N + = e i f fl = cjg + cjj (1 .50 ) ( i . e . W+ + ..) The Boltzmann f a c t o r i n the p o p u l a t i o n r a t i o ( N + / N . ) 0 comes from the f a c t t h a t the " l a t t i c e " ( w i t h which the e l e c t r o n s a r e i n t h e r m a l e q u i l i b r i u m a t t e m p e r a t u r e T) i s r e c e i v i n g the net energy yito^ f o r the e l e c t r o n i c t r a n s i t i o n E +-E.. As mentioned e a r l i e r , the " l a t t i c e " has i n f i n i t e s p e c i f i c heat and can a b s o r b any energy g i v e n t o i t by the e l e c t r o n s . These e x p r e s s i o n s demonstrate the p o s s i b i l i t y of enhancing the n u c l e a r s p i n p o l a r i z a t i o n i n e i t h e r sense ( p o s i t i v e or n e g a t i v e ) by c h o o s i n g the a p p r o p r i a t e t r a n s i t i o n mechanism. Note the p o p u l a t i o n i n v e r s i o n i n the n e g a t i v e p o l a r i z a t i o n s t a t e ( i . e . n. > n + ) . G . FROZEN SPIN TARGETS In 1965 Schmugge and J e f f r i e s 9 s u g g e s t e d t h a t once enhanced, t a r g e t p o l a r i z a t i o n c o u l d be m a i n t a i n e d w i t h o u t an 9 T. J . Schmugge and C. D. J e f f r i e s , "High Dynamic P o l a r i z a t i o n of P r o t o n s , " Phys. Rev. J_38 ( 6 A ) , 1 785-1801 (1965). 28 e x t e r n a l microwave sou r c e s u p p l y i n g the energy #a>g ± • T h e i r p r e d i c t i o n was based upon t h e i r o b s e r v a t i o n t h a t n u c l e a r s p i n - l a t t i c e r e l a x a t i o n t i m e s f o r the p r o t o n s i n the water s of h y d r a t i o n , i n Lanthanum Magnesium N i t r a t e ( L a 2 M g 3 ( N 0 3 ) ! 2 • 2 4 H 2 0 ) e x t r a p o l a t e d t o v e r y l o n g t i m e s ( s e v e r a l hours) below 1 K a t f i e l d s of a few T e s l a . T h i s would t e n d t o reduce c o n s t r a i n t s upon the homogeneity and s t r e n g t h of the magnetic f i e l d , removing the need t o p l a c e the t a r g e t between heavy magnet p o l e s . That i s , a f t e r p o l a r i z a t i o n , a weaker " h o l d i n g " f i e l d c o u l d be used t o r e t a i n the p o l a r i z a t i o n . T h i s would of c o u r s e be i d e a l f o r s c a t t e r i n g e x p e r i m e n t s where l a r g e s o l i d a n g l e a c c e s s i s d e s i r a b l e and l a r g e f i e l d s a r e u n d e s i r a b l e because of t h e i r e f f e c t s on the t r a j e c t o r i e s of charged p a r t i c l e s . In the p a s t two decades s e v e r a l "Frozen S p i n T a r g e t s " as they have come t o be known, have been b u i l t a t v a r i o u s f a c i l i t i e s t h roughout the w o r l d . The "modern" f r o z e n s p i n t a r g e t u s u a l l y c o n s i s t s of a d i l u t i o n r e f r i g e r a t i o n system used t o o b t a i n the t e m p e r a t u r e s near 50 degrees m i l l i K e l v i n n e c e s s a r y f o r a d e q u a t e l y l o n g r e l a x a t i o n t i m e s ; a magnet or magnets used f o r the p o l a r i z a t i o n , p o l a r i z a t i o n r e v e r s a l and h o l d i n g f u n c t i o n s ; a microwave source t o p o l a r i z e the sample; and an NMR system f o r measurement of the n u c l e a r s p i n p o l a r i z a t i o n . T h i s l a s t i t e m i s the p r i m a r y s u b j e c t of the remainder of t h i s t h e s i s . In p a r t i c u l a r the t h e s i s i s concerned w i t h the d e s i g n , c o n s t r u c t i o n and o p e r a t i o n of a n o v e l type of NMR system 29 which i s i n t e n d e d t o c i r c u m v e n t some ve r y g e n e r a l problems c h a r a c t e r i s t i c of NMR systems used i n p o l a r i z e d s p i n t a r g e t work. A l t h o u g h the d e s i g n i s s p e c i f i c as t o i t s a p p l i c a t i o n , the c o n c e p t s i n t r o d u c e d have p o t e n t i a l f o r w i d e s p r e a d use i n NMR work. The a c t u a l system t o be d i s c u s s e d i s the P o l a r i z e d Deuteron T a r g e t b u i l t a t the TRIUMF n u c l e a r r e s e a r c h f a c i l i t y i n Vancouver Canada. Because i t i s a p o l a r i z e d d e u t e r o n t a r g e t , i t i s n e c e s s a r y t o b r i e f l y l o o k a t the q u a d r u p o l e resonances c h a r a c t e r i s t i c of d e u t e r i u m NMR (DMR). H. QUADRUPOLE INTERACTIONS AND THE DMR LINESHAPE To t h i s p o i n t we have n e g l e c t e d the q u a d r u p o l e i n t e r a c t i o n s i n the s t r u c t u r e of the magnetic s u b l e v e l s of the d e u t e r o n s p i n system. That i s , we have o n l y d i s c u s s e d the t h r e e e q u a l l y spaced l e v e l s c o r r e s p o n d i n g t o m = -1,0,1 ( f i g u r e 1.5). When we c o n s i d e r t h e s p a t i a l o r i e n t a t i o n of bonds (between d e u t e r o n s and some o t h e r atom, say f o r example th e carbon atoms found i n d e u t e r a t e d b u t a n o l ) w i t h r e s p e c t t o the magnetic f i e l d , we f i n d t h a t t h e s e energy d i f f e r e n c e s are no l o n g e r e q u a l . The q u a d r u p o l e moment Q of the d e u t e r o n s (Q = 0 . 0 2 X 1 0 " 2 4 cm 2) i n t e r a c t s w i t h the e l e c t r i c f i e l d g r a d i e n t a t the p o s i t i o n of the d e u t e r o n w i t h an energy E = ( 1 / 8 ) e q Q ( 3 c o s 2 0 - 1 ) ( 3 m 2 - I ( 1 + 1 ) ) (1.51) 30 where q = 9 2V i s the f i e l d g r a d i e n t a l o n g the p r i n c i p l e a x i s of the f i e l d g r a d i e n t t e n s o r (£ c o i n c i d e s w i t h a bond). A x i a l symmetry w i t h r e s p e c t t o the f i e l d i s assumed. 6 i s the a n g l e between the bond a x i s and H"0. The a n g u l a r dependence of t h i s i n t e r a c t i o n combined w i t h the s i n f l f a c t o r e n c o u n t e r e d i n the a n g u l a r average r e s u l t s i n the magnetic s u b l e v e l s b e i n g s h i f t e d . The s h i f t s a r e shown i n f i g u r e 1.5 f o r the extreme c a s e s where 0 = 0 and 6 = ir/2. The s i n 0 w e i g h t i n g y i e l d s the c h a r a c t e r i s t i c q u a d r u p o l a r spectrum f o r the two t r a n s i t i o n s . The two i n d i v i d u a l t r a n s i t i o n s have been shown as d o t t e d l i n e s w h i l e some b r o a d e n i n g has been added t o the r e s u l t a n t spectrum. .The e q u a t i o n s f o r the spectrum w i t h o u t b r o a d e n i n g a r e p r o p o r t i o n a l t o (1 .52) f o r A - 68 < ha> < A - 35 + b tfco - A + 3 5 35 n 1/2 f o r A - 35 < ho; < A + 35 31 m = - l /Is rr u z UJ A - 3 S A+ 6 8 m = 0 2cT T 3 8 A+ 3 8 T A - 6 8 3 8 EQUALLY SPACED TT MAGNETIC 0 = / 2 S U B L E V E L S T ©=0 SHIFTS IN THE ENERGY OF THE MAGNETIC S U B L E V E L S S = '/s eq Q H, C-D BOND AXIS AT AN ANGLE TO MAGNETIC FIELD A - 3 8 A A + 3 8 A + 6 8 QUADRUPOLAR SPECTRUM a i F i g u r e 1.5 Quadrupolar I n t e r a c t i o n s i n D e u t e r a t e d B u t a n o l 32 f(w) = b 36 - A + 3 6 n 1/2 f o r A + 36 < < A + 65 where A = K" L, 6 = eqQ/8 and R = a/b i s the r a t i o of the i n t e n s i t i e s of the t r a n s i t i o n s m = 1-0 t o m = 0--1. I f we assume a s p i n t emperature T D w i t h i n the d e u t e r o n s p i n system, then the p o p u l a t i o n s of the magnetic s u b l e v e l s a r e g i v e n by 2A/kT D \t = e where the q u a d r u p o l a r s h i f t s have been n e g l e c t e d w i t h r e s p e c t t o A. T h i s r e s u l t s i n R = n„ - n 0 = exp(A/kT ) . (1.54) n 0 - n. Hi n, A/kT = e D (1.53) U s i n g e q u a t i o n 1.13, P = n„. - n. n + + n 0 + n. (1.55) 33 and e q u a t i o n 1.53 i t i s p o s s i b l e t o w r i t e P = R 2 - 1 (1.56) R* + R + 1 which a l l o w s us t o c a l c u l a t e the v e c t o r p o l a r i z a t i o n from the l i n e s h a p e asymmetry. W i t h the p r e v i o u s assumption of a s p i n t e m p e r a t u r e T D 1 0 we can r e l a t e the v e c t o r p o l a r i z a t i o n t o the a l i g n m e n t T h i s a l l o w s the a l i g n m e n t t o be c a l c u l a t e d from the measured v e c t o r p o l a r i z a t i o n . 1 0M. B o r g h i n i and K. S c h e f f l e r , "A B u t a n o l P o l a r i z e d Deuteron T a r g e t , " N u c l . I n s t r u m . and Methods 95, 93-98 (1971). A 2 - 4A + 3 P 2 = 0 (1.57) I l l . CHAPTER 2 In t he p r e v i o u s c h a p t e r i t was shown t h a t a measurement of the a b s o r p t i v e l i n e s h a p e x " ( " ) of a c o l l e c t i o n of n u c l e a r s p i n s can p r o v i d e i n f o r m a t i o n r e g a r d i n g the p o l a r i z a t i o n ( v e c t o r or t e n s o r ) of those s p i n s . T h i s measurement f i n d s a p p l i c a t i o n i n P o l a r i z e d T a r g e t work where the knowledge of the n u c l e a r s p i n p o l a r i z a t i o n i s v a l u a b l e f o r s c a t t e r i n g e x p e r i m e n t s . T h i s c h a p t e r d e a l s w i t h the p r a c t i c a l measurement of the a b s o r p t i v e l i n e s h a p e u s i n g a t y p i c a l P o l a r i z e d T a r g e t NMR i n s t r u m e n t a t i o n scheme. As i t w i l l be shown s h o r t l y , the d e t e c t i o n of the NMR s i g n a l i t s e l f i s f a i r l y s i m p l e , w h i l e i n t e r p r e t a t i o n of the s i g n a l can be v e r y d i f f i c u l t . S e v e r a l reasons f o r t h i s d i f f i c u l t y w i l l then be d i s c u s s e d . L a t e r c h a p t e r s d i s c u s s a t e c h n i q u e of which a v o i d s t h e s e d i f f i c u l t i e s . A. THE PRACTICAL MEASUREMENT OF x(u) The n u c l e a r magnetic s u s c e p t i b i l i t y x ( ^ ) was i n t r o d u c e d w h i l e c o n s i d e r i n g the e f f e c t of a s m a l l r o t a t i n g or l i n e a r l y p o l a r i z e d f i e l d H~1f o r t h o g o n a l t o a l a r g e s t a t i c f i e l d R0. For a l a r g e f i e l d H"0 (say a few T e s l a ) most n u c l e a r Larmor f r e q u e n c i e s a r e i n the r a d i o f r e q u e n c y range of a few MHz t o a few hundred MHz. T h i s i m m e d i a t e l y s u g g e s t s the use of an r f c o i l t o s u p p l y the f i e l d H", . Now c o n s i d e r a c o l l e c t i o n of n u c l e a r s p i n s p l a c e d i n s i d e such an r f c o i l w i t h an i n d u c t a n c e L, s u p p l i e d w i t h a 34 c u r r e n t lRe{e }. Assume t h a t the f i e l d H 0 i s a p p l i e d p e r p e n d i c u l a r l y t o the a x i s of the c o i l such t h a t the n u c l e a r Larmor f r e q u e n c y w L i s near CJ . 35 I R e ^ ' SAMPLE MATERIAL / /// \ \ \ \ \ \ \ \ I IrU \\\\\ / NET MAGNETIZATION F i g u r e 2 . 1 The r f C o i l and Sample I n s i d e the c o i l , the magnetic i n d u c t i o n B i s g i v e n by I = H^+M ( 2 . 1 ) = 0+x)H, where the l a r g e s t a t i c component of the f i e l d has been n e g l e c t e d f o r the sake of s i m p l i c i t y . The magnetic f l u x <j> of the r f f i e l d i n s i d e the c o i l i s g i v e n by 36 0 = LReile jcot } (2.2) Combining t h e s e , i t i s c l e a r t h a t as f a r as any e x t e r n a l c i r c u i t r y i s concerned, the e f f e c t i v e i n d u c t a n c e of the c o i l i s g i v e n by That i s , the e f f e c t of t h e n u c l e a r s p i n s i s s i m p l y a m o d i f i c a t i o n of the c o i l i n d u c t a n c e near o>L. I f f a c t o r s such as the c o u p l i n g between H-, and the sample s p i n s were c o n s i d e r e d , a more c o r r e c t e x p r e s s i o n would be where i? i s a f i l l i n g f a c t o r g i v e n a p p r o x i m a t e l y by L(w) = L(1+X(W)) (2.3) L(u) = L(1+T?X(W)) (2.4) volume of sample (2.5) e f f e c t i v e volume of c o i l The t o t a l impedance of the c o i l i n the absence of a sample i s 37 Z = R 0 + jcoL (2.6) •AAV -TYYTY 1 F i g u r e 2.2 The C o i l Impedance Z(co) i n the Absence of any N u c l e a r S p i n s I where R 0 i s the e f f e c t i v e r e s i s t a n c e of the c o i l a t the fr e q u e n c y u t a k i n g the s k i n d epth una 1/2 (2.7) where M = c o i l p e r m e a b i l i t y a = c o i l c o n d u c t i v i t y i n t o a c c o u n t . Thus, i n the p r e s e n c e of a c o l l e c t i o n of s p i n s I, the t o t a l impedance becomes Z(CJ) =' [ R 0 + T?CJLX"(W)] + JCJL[1 + 7?x' (") 3 . (2.8) 38 R0 L — n n n n — . XM F i g u r e 2.3 The C o i l Impedance Z(CJ) i n the Presen c e of a C o l l e c t i o n of N u c l e a r S p i n s T T h i s e x p r e s s i o n shows the m o d i f i c a t i o n of the c o i l impedance e x p l i c i t l y . The r e s i s t i v e impedance R 0 i s modulated by the a b s o r p t i v e component of the s u s c e p t i b i l i t y x" (CJ) w h i l e the i n d u c t i v e impedance i s modulated by the d i s p e r s i v e component x' (co). Of th e s e f o u r "components" of the c o i l impedance, we are i n t e r e s t e d i n the a b s o r p t i v e m o d u l a t i o n 7 7CJLX"(W). U n f o r t u n a t e l y , b oth of the m o d u l a t i o n s t o the c o i l impedance due t o the n u c l e a r s p i n s a re u s u a l l y s m a l l . C o n s i d e r a t y p i c a l experiment w i t h a) = oj r = 2ir x 10MHz and L = 1 0 M H . W i t h r e f e r e n c e t o e q u a t i o n 1.43, the r e s o n a n t r i s e i n x" a t resonance i s r o u g h l y x"(^o) - <^SL t 2 X O f o r H , « H 0 . Thus, 2 the e f f e c t i v e r e s i s t i v e impedance of a c o l l e c t i o n of n u c l e a r s p i n s a t the Larmor f r e q u e n c y can be w r i t t e n as 39 R „ = ^cjgT 2Lxo • (2.9) X 2 For a common paramagnetic s u b s t a n c e Xo - 10" 6 and T 2 - 10" 5 seconds. I f we assume t h a t TJ = 1 , then the e f f e c t i v e r e s i s t a n c e of these s p i n s i s about 0.2 Ohms. T h i s can be compared t o a t y p i c a l r e s i s t i v e impedance R 0 of a few te n s of Ohms and an i n d u c t i v e impedance of coL = (27rx1 0x1 0 6Hz) ( 1 0x1 0" 6H) * 600 Ohms. O b v i o u s l y the impedance m o d u l a t i o n due t o a TE s i g n a l i s s m a l l by comparison t o t h e s e v a l u e s . On the o t h e r hand, w i t h a system of d y n a m i c a l l y p o l a r i z e d s p i n s , t h e p o l a r i z a t i o n may be i n c r e a s e d by a f a c t o r of s e v e r a l hundred, b r i n g i n g us i n t o the regime where the m o d u l a t i o n i s of the same r e l a t i v e magnitude as R 0. A d d i t i o n a l problems which a r i s e as a r e s u l t of t h i s l a r g e m o d u l a t i o n w i l l be d i s c u s s e d i n c h a p t e r f o u r . D e t e c t i o n i s s i m p l i f i e d w i t h the use of a c a p a c i t i v e r e a c t a n c e 1/jwC i n s e r i e s w i t h the c o i l such t h a t the s e r i e s 1 /2 resonant f r e q u e n c y co0 = 1/(LC) e q u a l s t h e Larmor f r e q u e n c y CJ^. T h i s a d d i t i o n a l r e a c t a n c e c a n c e l s the i n d u c t i v e r e a c t a n c e of the c o i l l e a v i n g o n l y the r e s i s t i v e impedance R 0 upon which the m o d u l a t i o n due t o x(^) i s 40 superimposed as |H 0| i s v a r i e d . I f the c u r r e n t s u p p l y t o t h i s c i r c u i t i s kept c o n s t a n t i n magnitude, the cha n g i n g impedance w i l l be r e f l e c t e d as a v a r y i n g p o t e n t i a l In the absence of a n u c l e a r s p i n i n d u c e d m o d u l a t i o n , V i s known as the c a r r i e r wave. F i g u r e 2.4 C a n c e l l a t i o n of the R e a c t i v e Impedance a t the Larmor Frequency W h i l e t h i s p r o p o s a l f o r c a n c e l l i n g the r e a c t i v e impedance of the re s o n a n t c i r c u i t a t C J 0 i s p e r f e c t l y r e s p e c t a b l e f o r many pu r p o s e s , s e v e r a l problems a r i s e when i t i s a p p l i e d t o p o l a r i z e d t a r g e t work. The major problem i s t h a t the l a r g e magnetic f i e l d i s u s u a l l y r e q u i r e d t o be c o n s t a n t d u r i n g the n u c l e a r s c a t t e r i n g e x p e r i m e n t . T h i s p r e c l u d e s measuring the n u c l e a r s p i n p o l a r i z a t i o n w h i l e t a k i n g d a t a f o r the s c a t t e r i n g e x p e r i m e n t . F u r t h e r m o r e , i t i s u s u a l l y d e s i r a b l e t o observe t h e p o l a r i z a t i o n p r o c e s s . A change i n |Ho|, however, r e s u l t s i n a change i n the jcjt V = ZReile } (2.10) 41 e l e c t r o n i c Larmor f r e q u e n c y . U n l e s s the microwave frequency can be l o c k e d t o the magnetic f i e l d , the p o l a r i z a t i o n p r o c e s s cannot be o b s e r v e d w i t h a f i e l d swept NMR system. These problems te n d t o r u l e out f i e l d swept a p p a r a t i f o r t h i s t y pe of work. In g e n e r a l , one has a g r e a t e r degree of c o n t r o l over the f r e q u e n c y of the r f c u r r e n t which i s i n p u t t o the r e s o n a n t c i r c u i t . U n f o r t u n a t e l y , as the c i r c u i t i s r e s o n a n t , complete c a n c e l l a t i o n of the r e a c t i v e impedance of the 1 /2 c i r c u i t o c c u r s o n l y a t resonance ( i . e . a>0 = 1/(LC) ). For a l l o t h e r f r e q u e n c i e s the r e a c t i v e impedance i s o n l y p a r t i a l l y c a n c e l l e d and we a r e f o r c e d t o d e a l w i t h a l l f o u r s i g n a l components (two e l e c t r o n i c s i g n a l s i n q u a d r a t u r e and the two components of the induced s i g n a l ) . T h i s s i t u a t i o n i s not q u i t e as bad f o r a low Q c i r c u i t where the magnitude and phase of the c a r r i e r s i g n a l do not change as r a p i d l y w i t h f r e q u e n c y ( i . e . a s i t u a t i o n i n which the s i g n a l bandwidth i s s m a l l compared t o the bandwidth of the r e s o n a n t c i r c u i t Aw = 3/2 1 /2 "o/Q = Ro/("L ' C ' ) ). Even i n t h i s c a s e , however, i n t e r p r e t a t i o n of the s i g n a l can be a d i f f i c u l t problem. The impedance of a s e r i e s r e s o n a n t c i r c u i t and i t s components a r e shown i n f i g u r e s 2.5 and 2.6, w i t h and w i t h o u t an i n d u c e d m o d u l a t i o n due to n u c l e a r s p i n s . Note t h a t the r e s i s t i v e impedance R 0 i s assumed t o be a c o n s t a n t and t h a t the magnitudes of the induced s i g n a l s i n f i g u r e 2.6 a r e e x a g g e r a t e d . 42 F i g u r e 2.5 Impedance of a S e r i e s Resonant C i r c u i t The problem e n c o u n t e r e d i n s i g n a l d e t e c t i o n l i e s i n the d e c o m p o s i t i o n of the v a r i o u s s i g n a l components. V a r i o u s f r e q u e n c y dependent n o n l i n e a r i t i e s such as the impedance of the c o a x i a l c a b l e used t o b r i n g the s i g n a l out of the c r y o s t a t t e n d t o mix the s i g n a l components. T h i s " m i x i n g " of terms becomes more s e r i o u s as the Q of the e l e c t r o n i c c i r c u i t i n c r e a s e s . Note t h a t t h i s i m p l i e s t h a t the c a r r i e r f o r t h e m o d u l a t i o n has an a m p l i t u d e and a phase which a r e s t r o n g f u n c t i o n s of f r e q u e n c y . On the o t h e r hand, r e d u c i n g the Q tends t o mask the v e r y s m a l l impedance m o d u l a t i o n . In the f o l l o w i n g s e c t i o n s the v a r i o u s n o n l i n e a r i t i e s which a r e 4 3 commonly e n c o u n t e r e d i n P o l a r i z e d t a r g e t NMR work a r e d i s c u s s e d . B. FACTORS WHICH DISTORT THE NUCLEAR SPIN INDUCED MODULATIONS 1. COAXIAL CABLE D i r e c t a c c e s s t o the impedance Z of the resonant c i r c u i t i s i n g e n e r a l not p o s s i b l e i n P o l a r i z e d T a r g e t work. The low t e m p e r a t u r e s n e c e s s a r y t o o b t a i n u s e f u l n u c l e a r s p i n - l a t t i c e r e l a x a t i o n t i m e s r e q u i r e the use of d i l u t i o n 4 4 r e f r i g e r a t i o n systems. Access t o the c o i l must be made v i a a l e n g t h of c o a x i a l c a b l e o f t e n s e v e r a l metres l o n g ( i . e . not n e g l i g i b l e w i t h r e s p e c t t o the wavelength X ) . There i s no s t a n d a r d t e c h n i q u e f o r p l a c i n g the c a p a c i t a n c e i n s e r i e s w i t h the c o i l . The c a p a c i t o r has a l t e r n a t e l y been p l a c e d i n s i d e 1 1 or o u t s i d e 1 2 of the c r y o s t a t depending upon the a p p a r a t u s (a t h i r d a l t e r n a t i v e i s t o r e s o n a t e the c o i l w i t h the c a b l e 1 3 ) . There a r e advantages and d i s a d v a n t a g e s t o each t e c h n i q u e . However, f o r the purposes of t h i s t h e s i s the o n l y case which w i l l be c o n s i d e r e d i s t h a t i n which the c a p a c i t o r i s p h y s i c a l l y c o n n e c t e d t o the c o i l . Thus i t becomes n e c e s s a r y t o c o n s i d e r the e f f e c t of the c a b l e impedance when l o o k i n g a t the impedance of the resonant c i r c u i t . The imaged impedance seen from o u t s i d e the c r y o s t a t tends t o d i s t o r t the impedance m o d u l a t i o n due t o the p r e s e nce of n u c l e a r s p i n s i n the f i e l d H 0. I n t e r p r e t a t i o n of t h i s d i s t o r t e d s i g n a l has a s t r o n g b e a r i n g upon the a c c u r a c y of the f i n a l p o l a r i z a t i o n measurement. At t h i s p o i n t i t s h o u l d be noted t h a t use of a nX/2 . c a b l e i n a f i x e d f r e q u e n c y , f i e l d swept measurement, p e r m i t s 1 1 D. W. Alderman, " L i q u i d H e l i u m Temperature cw NMR S/N Improvement U s i n g a MOSFET r f A m p l i f i e r , " Rev. S c i . I n s t r u m . 41 ( 2 ) , 192-197 (1970). "^G. C o u r t , i n P r o c e e d i n g s of the Second Workshop on  P o l a r i s e d T a r g e t M a t e r i a l s , e d i t e d by G. R. C o u r t , S. F. J . Cox, D. A. Cragg and T. 0. N i i n i k o s k i ( R u t h e r f o r d and A p p l e t o n L a b o r a t o r i e s , C h i l t o n , D i d c o t , Oxon, 1980), pp. 76-80. 1 3 L . Friedman,, i n Low Temperature T e c h n i q u e s , e d i t e d by Bob R i c h a r d s o n ( M a t e r i a l s S c i e n c e C e n t e r a t C o r n e l l U n i v e r s i t y , I t h i c a , New York, 1982), pp. 247-258. 45 one t o n e g l e c t the e f f e c t s of the c a b l e . That i s , i n such a case one has d i r e c t a c c e s s t o the resonant c i r c u i t . The imaged impedance Z' of the res o n a n t c i r c u i t impedance Z as seen t h r o u g h t h e c a b l e i s g i v e n by Z' = Z. Z + Z ctanh7/ Z c + Z t a n h 7 / (2.11) R 0 L l) LC^^RC.GC )—vv\—nnnn— u ^ ^ rh F i g u r e 2.7 The Imaged Impedance Z' where the c h a r a c t e r i s t i c impedance Z^ of the c a b l e i s g i v e n by Z c " R + TO)L c J c 1/2 (2.12) G c + j c C c 46 « Z, 1 i R 1 + c and Z 0 = ( L c / C c ) 1/2 The p r o p a g a t i o n c o n s t a n t y i s g i v e n by 7 = (R + jwL )(G + jwC ) c c c c 1/2 (2.13) The d i s t r i b u t e d c a b l e parameters R c , L c , C c and G c r e s p e c t i v e l y account f o r power l o s s i n the c a b l e c o n d u c t o r s , energy s t o r a g e i n the magnetic f i e l d s , energy s t o r a g e i n the e l e c t r i c f i e l d s and shunt power l o s s i n the d i e l e c t r i c . Each parameter i s d e f i n e d per u n i t l e n g t h . 1 4 L o s s e s have been i n c l u d e d i n t h e s e e x p r e s s i o n s as c r y o g e n i c c o a x i a l c a b l e tends t o be l o s s y . That i s , i n an attempt t o m i n i m i z e heat c o n d u c t i o n i n t o the c r y o s t a t , m a t e r i a l s w i t h poor t h e r m a l c o n d u c t i v i t y a r e used t o make the c a b l e s . 1 4 R. E. C o l l i n , i n F o u n d a t i o n s f o r Microwave E n g i n e e r i n g , ( M c G r a w - H i l l , T o r o n t o , Ont., 1966), pp.64-143. 47 2. DISTORTIONS INHERENT IN THE CONSTANT CURRENT SERIES TUNED  NMR CIRCUIT The s e r i e s tuned c i r c u i t f e d by a c o n s t a n t c u r r e n t w hich has been d e s c r i b e d t o t h i s p o i n t i s p r o b a b l y the s i m p l e s t p o s s i b l e scheme f o r the f r o n t end i n s t r u m e n t a t i o n of an NMR system. Such a c i r c u i t i s i l l u s t r a t e d d i a g r a m m a t i c a l l y i n f i g u r e 2.8. F i g u r e 2.8 The C o n s t a n t C u r r e n t , S e r i e s Tuned C i r c u i t The r e s i s t a n c e R >> |Z| s u p p l i e s a c o n s t a n t c u r r e n t t o the s e r i e s r e sonant c i r c u i t impedance Z (or Z' as the case may b e ) . The output of t h e a m p l i f i e r c o n s i s t s of two s i g n a l s w hich a r e i n q u a d r a t u r e . hRe{v.} = Avo Re{Z} + X[Re{Z] + 7m{Z}] (2.14) 1 R '[1 +. XReiz})2 + X 2/m{z}] 48 and y o i m\ L, s  R [1 + X t f e { z } ] s + X 2/m{z}] Im{Z] (2.15) where X = J_ + _1_ R. R I d e a l l y X = 0 and the r e a l p a r t of the s i g n a l i s a v o l t a g e p r o p o r t i o n a l t o the l o s s e s i n the c i r c u i t ( i . e . Re{Z} or Re{Z'}). In p r a c t i c e however t h i s may not always be p o s s i b l e . For a non-zero v a l u e of X, a second m i x i n g of the two q u a d r a t u r e components of t h e s i g n a l o c c u r s . These v a r i o u s e f f e c t s l e a d t o d i s t o r t i o n s which u s u a l l y cannot be i g n o r e d . F u r t h e r m a t h e m a t i c a l a n a l y s i s i s o f t e n n e c e s s a r y . I t i s p o s s i b l e t o expand the imaged impedance i n a s e r i e s of the form where x(<*>) = x'(w) ~ jx"(") and the a^(u) a r e g i v e n by the app r o x i m a t e e x p r e s s i o n s (some s m a l l terms have been dropped) Z ' = a0(cj) + a 1(cj)r?x(w) (2.16) + a 2 (CJ)T J 2 X 2 (W) + 4 9 a 0(w) = R + j(wL - 1/CJC) (2.17) + njr Z 0 L ZQ + (R + JCJL - j/coC) 1/(2Q C) + j 2 n + 1 ( A 0 + A) a, (CJ) = jcoL 1-2njr{R + jcoL - j/coC} { 1 / ( 2 Q n ) + j 2° + 1(A 0+A)} Z 0 C a 2(w) = (jcoL) 2njr Z 0 1/(2Q C) + j 2 n + 1 ( A 0 + A) where Q c = C J L C / R c , n i s the number of h a l f wavelengths i n the c a b l e and A 0 = ( C J 0 _ t o ) A = (cu - C J 0 ) (2.18) CJ0 a r e the r e l a t i v e f r e q u e n c y o f f s e t s of the c a b l e resonance a>c 1 / o and the Larmor f r e q u e n c y CJ^ = CJ0 = 1/(LC) / from the s i g n a l f r e q u e n c y co. V a r i o u s t e c h n i q u e s f o r m i s t u n i n g the c a b l e a r e p o s s i b l e , c a n c e l l i n g c e r t a i n terms i n the above e x p r e s s i o n s . I t i s not p o s s i b l e , however, t o o b t a i n an output s i g n a l p r o p o r t i o n a l t o x(<*>) a l o n e . I f the e x p r e s s i o n f o r Z' i s s u b s t i t u t e d i n t o t h e a m p l i f i e r o u t p u t e x p r e s s i o n hRe(v.) we o b t a i n hRe(v.) = Ay_o Re(Z') + X[Re~(Z') + IrhjZ') ]  1 R '[1 + X J J e ( z ' ) ] 2 + X J / A ( Z ' ) ] « A v , Re(Z') + X[-Ri(Z') + / m ( Z ' ) ] + X 2 { • • • } + ( 2 . 1 9 ) w h e r e Re{Z') a n d Im(Z') a r e a p p r o x i m a t e l y Re(Z') = R + n7r 2 Q C Z 0 Z | + R 2 - (wL - 1 ) 2 CJC -( 2 . 2 0 ) + n j r ( - l ) + 1 ( A 0 + A ) 2 R ( C J L - J _ ) Z 0 CJC + CJLX" I _ L_(1 " 2 n 7 r R / Z 0 ) L 2 Q „ + ( 2 n 7 r / Z 0 ) ( o j L - J _ ) ( - l ) n ( A 0 + A ) ] CJC J + u L x ' I n?r (wL - j _ ) + 2 n 7 r R ( - 1 ) n ( A 0 + A ) l L Z 0 C i CJC ZO J 51 - 2(wL) 2x'x"n7r(-1 ) n ( A 0 + A) - (uL) 2 (x' 2~x" 2 ) rur Z 0 2Z 0Q r Im{Z') = ( u L - l / u C ) + nrr Z 0 (-1) 1 Z§ + R 2 - (wL - 1 ) 2 wC (A 0 + A) (2.21) + R (wL - J_) Q_ coC + OJLX | 1 ~ rnr Z 0 R + 2 ( - 1 ) n ( w L - 1 ) ( A 0 + A) + cjLx" 2n7r[ 1 (wL - j _ ) + ( - l ) n 2 R ( A 0 + A)] Z 0 L Q c wC J + U)U2x'x"n7r + j 2 U ( w L ) 2 ( x Q„ Zo x" 2)n7r(A 0 + A) O b v i o u s l y r e d u c t i o n of t h e s e e q u a t i o n s i s not a t r i v i a l m a t t e r . A s i m p l e NMR i n s t r u m e n t a t i o n scheme as d e s c r i b e d above cannot a v o i d d i s t o r t i o n s i n the l i n e s h a p e . At be s t one can o n l y hope t o reduce d i s t o r t i o n s t o a few p e r c e n t . For some t y p e s of work t h i s i s q u i t e s a t i s f a c t o r y . U n f o r t u n a t e l y , when the s i g n a l bandwidth becomes comparable t o the bandwidth of the re s o n a n t c i r c u i t (or when x(") f a i l s 52 t o go t o z e r o q u i c k l y ) the d i s t o r t i o n s on the s i g n a l wings make s i g n i f i c a n t c o n t r i b u t i o n s t o the l i n e s h a p e . D i s t o r t i o n s such as t h o s e d e s c r i b e d above a r i s e i n p o l a r i z e d t a r g e t work when wo r k i n g w i t h the r e l a t i v e l y wide q u a d r u p o l a r s p l i t t i n g of the de u t e r o n NMR l i n e s h a p e . T h i s i s one of the main m o t i v a t i o n s f o r d e s i g n i n g a NMR system which bypasses or a v o i d s the m i x i n g of terms and r e s u l t a n t l o s s of i n f o r m a t i o n . C. OTHER DESIGN CONSIDERATIONS A s i d e from the problems a s s o c i a t e d w i t h the d i s t o r t i o n s which were d e s c r i b e d i n the p r e v i o u s s e c t i o n t h e r e a r e s e v e r a l o t h e r m o t i v a t i o n s f o r s e e k i n g t o improve the s i m p l i s t i c d e s i g n d e s c r i b e d e a r l i e r . Many e x p e r i m e n t e r s have been f o r c e d t o c o n s i d e r the problem of matching the impedance of the resonant c i r c u i t t o .the c o a x i a l c a b l e which b r i n g s the NMR s i g n a l out of the c r y o s t a t . 1 5 I s o l a t i n g the re s o n a n t c i r c u i t from the c a b l e would p r o v i d e a p a r t i a l s o l u t i o n t o the problems e n c o u n t e r e d i n the p r e v i o u s s e c t i o n s . Use of a low temp e r a t u r e a m p l i f i e r has the p o t e n t i a l t o i s o l a t e the resonant c i r c u i t and t o p r o v i d e low n o i s e a m p l i f i c a t i o n t o d r i v e the l o s s y c o a x i a l c a b l e o f t e n used i n c r y o g e n i c w o r k . 1 6 1 5 G. C o u r t , i n P r o c e e d i n g s of the Second Workshop on  P o l a r i s e d T a r g e t M a t e r i a l s , e d i t e d by G. R. C o u r t , S. F. J . Cox, D. A. Cragg and T. 0. N i i n i k o s k i ( R u t h e r f o r d and A p p l e t o n L a b o r a t o r i e s , C h i l t o n , D i d c o t , Oxon, 1980), pp. 76-80. 1 6 D. S. M i y o s h i and R. M. C o t t s , "Helium C o o l e d , Radio Frequency P r e a m p l i f i e r f o r Use i n NMR," Rev. S c i . I n s t r u m . 39 ( 1 2 ) , 1881-1884 (1968). 53 R e g a r d l e s s of the a m p l i f i c a t i o n used, a v e r y s e r i o u s d e s i g n c o n c e r n i n v o l v e s the dynamic range of the a m p l i f i e r . 1 7 Wideband s i g n a l s i n t r o d u c e l a r g e v a r i a t i o n s i n the c a r r i e r l e v e l due t o the e l e c t r o n i c Q c u r v e . These v a r i a t i o n s may be many tim e s the s i z e of the TE s i g n a l (see comparisons of r e l a t i v e impedance magnitudes f o r TE s i g n a l s and the r e a c t i v e impedances of the c i r c u i t g i v e n i n s e c t i o n A of t h i s c h a p t e r ) . S i m i l a r problems a r e e n c o u n t e r e d f o r h i g h d y n a m i c a l l y p o l a r i z e d s i g n a l s where the s i g n a l c o m p r i s e s a v e r y l a r g e m o d u l a t i o n of the c a r r i e r ( o f t e n t h i s s i g n a l i s of the same r e l a t i v e s i z e as the Q c u r v e " w i n g s " ) . Note t h a t i n the case of n e g a t i v e p o l a r i z a t i o n , the s p i n system i s a c t u a l l y c o n t r i b u t i n g energy t o the e l e c t r o n i c c i r c u i t , and i s thus m a n i f e s t e d as a n e g a t i v e r e s i s t a n c e . I f t h i s r e s i s t a n c e becomes much l a r g e r than R 0 an e f f e c t s i m i l a r t o o v e r m o d u l a t i o n i s o b s e r v e d . 1 8 T h i s u s u a l l y n e c e s s i t a t e s the i n t r o d u c t i o n of a Q s p o i l i n g r e s i s t o r i n s e r i e s w i t h the resonant c i r c u i t . Some of the more fundamental problems a r e r e l a t e d t o n o i s e s i g n a l s of v a r i o u s d e s c r i p t i o n s . A s i d e from Johnson n o i s e , i t must be remembered t h a t the e x p e r i m e n t a l environment i s u s u a l l y p o l l u t e d w i t h n o i s e of almost e v e r y f r e q u e n c y . In p a r t i c u l a r , m e c h a n i c a l v i b r a t i o n s , 1 9 temperature v a r i a t i o n s and h i g h power l i n e f r e q u e n c y s i g n a l s 1 7 H. K l e v e r and M. S c h l a a k , "A Simple C i r c u i t f o r cw NMR Measurements," Rev. S c i . I n s t r u m . 4_4 ( 1 ) , 25-27 ( 1973). 1 8 T h i s problem w i l l be d i s c u s s e d f u r t h e r i n c h a p t e r 4. 1 9 E . P o l t u r a k , i n Low Temperature T e c h n i q u e s , e d i t e d by Bob R i c h a r d s o n ( M a t e r i a l s S c i e n c e C e n t e r a t C o r n e l l U n i v e r s i t y , I t h i c a , New York, 1982), pp. 39-44. 54 u ' ^ F i g u r e 2.9 A d d i t i o n of a Q - S p o i l i n g R e s i s t o r must be c o n s i d e r e d i n a d d i t i o n t o r f n o i s e . These s i g n a l s can be c o u p l e d i n t o the NMR system v i a many mechanisms. The l o n g l e n g t h s of c o a x i a l c a b l e c a r r y i n g low l e v e l s i g n a l s , however, a r e e s p e c i a l l y s u s c e p t i b l e . 2 0 M e c h a n i c a l v i b r a t i o n s t r a n s m i t t e d t h r ough the c r y o s t a t , pumping l i n e s or even the h e l i u m r e f r i g e r a n t can wreak havoc upon the resonant c i r c u i t . V i b r a t i o n s can a l t e r the resonant f r e q u e n c y by amounts which a r e s i g n i f i c a n t i n s e n s i t i v e measurements. I s o l a t i o n of m e c h a n i c a l v i b r a t i o n s i s a c o n c e r n of g r e a t importance i n t h i s t ype of work. Even the i n f l u e n c e of temp e r a t u r e v a r i a t i o n s must be c o n s i d e r e d . W h i l e the e f f e c t of c h a n g i n g the temperature does not have much e f f e c t on the re s o n a n t c i r c u i t i t s e l f , any a c t i v e d e v i c e s p r e s e n t i n the c r y o s t a t a r e p o t e n t i a l l y v e r y s e n s i t i v e t o these changes. 2 0 " N o i s e i n A m p l i f i e r s , " P r i n c e t o n A p p l i e d R esearch C o r p e r a t i o n t e c h n i c a l note 243, 1976. IV. CHAPTER 3 We have a l r e a d y encountered the c o u p l i n g between the nuclear s p i n s and our "measuring d e v i c e " , the i n d u c t o r , v i a the s m a l l f i e l d H",. Information about the n u c l e a r sp i n s i s r e f l e c t e d as a frequency dependent (or f i e l d dependent) modulation of the impedance of the i n d u c t o r . In f i e l d swept a p p a r a t i the nuclear s p i n dependent modulation i s the only impedance modulation encountered. U n f o r t u n a t e l y , as the impedance of a resonant c i r c u i t i s h i g h l y frequency dependent near resonance, we must d e a l with t h i s second frequency dependent modulation i n frequency swept measurements. The mixing of these two modulations i s complex as was shown i n chapter two. I t i s n o n - t r i v i a l to attempt to r e s o l v e them. Information about the n u c l e a r s p i n s i s thus l i m i t e d by t h i s "mixing" of s i g n a l s , e s p e c i a l l y f o r f r e q u e n c i e s f a r from the Larmor frequency where the NMR s i g n a l i s small and the impedance of the resonant c i r c u i t i s 1 /2 l a r g e (Note that a>L has been set equal to co0 = 1/(LC) ). The obvious s o l u t i o n and the b a s i c premise of the design c o n s i d e r e d here i s to couple the n u c l e a r s p i n s to a device whose impedance i s frequency independent. While no p a s s i v e d e v i c e meeting t h i s c r i t e r i o n e x i s t s ( i . e . an i n d u c t i v e d e v i c e must be employed s i n c e the c o u p l i n g occurs through the magnetic f i e l d H-,), i t i s p o s s i b l e to d e v i s e a c t i v e systems having t h i s c h a r a c t e r i s t i c , at l e a s t as f a r as an e x t e r n a l observer i s concerned. 55 A. PRINCIPLE OF OPERATION The fr e q u e n c y dependence of the impedance of the resonant c i r c u i t i s due t o the presence of r e a c t i v e impedances. In o r d e r t o a c h i e v e the g o a l of a f r e q u e n c y independent c i r c u i t , i t i s n e c e s s a r y t o c a n c e l any r e a c t i v e component of the impedance f o r a l l f r e q u e n c i e s w i t h i n the o p e r a t i n g r e g i o n of the d e v i c e . T h i s r e q u i r e s the use of a v a r i a b l e r e a c t a n c e whose impedance can be c o n t r o l l e d by some e x t e r n a l i n p u t . The o b v i o u s c h o i c e i s some type of v a r a c t o r d e v i c e . By cha n g i n g i t s impedance i n some p r e d e t e r m i n e d f a s h i o n or i n a c c o r d w i t h some s e n s i n g d e v i c e , the v a r a c t o r can p o t e n t i a l l y c a n c e l any net r e a c t a n c e a t a g i v e n f r e q u e n c y . Ro AW L rrrr\ r h C v = OJ F i g u r e 3.1 C a n c e l l a t i o n of Net Reactance w i t h C A " s e l f a d j u s t i n g " system i s o b v i o u s l y f a r more v e r s a t i l e than an impedance programmed t o change a c c o r d i n g t o some p r e d e t e r m i n e d a l g o r i t h m . A programmed system i s not c a p a b l e of r e a c t i n g t o changes i n component v a l u e s or a b l e t o respond t o e x t e r n a l i n f l u e n c e s ( f o r example temperature 57 f l u c t u a t i o n s ) . In a d d i t i o n the a p p r o p r i a t e programming of such a d e v i c e would i n i t s e l f be a v e r y time consuming o p e r a t i o n . The s e l f a d j u s t i n g system i s h i g h l y d e s i r a b l e from a p r a c t i c a l p o i n t of view as i t i s o f t e n c o n v e n i e n t t o change the o p e r a t i n g f r e q u e n c y range or even the a c t u a l c o i l b e i n g used. The n a t u r e of a " r e s o n a n t " c i r c u i t whose r e a c t i v e impedance can be z e r o e d over the f r e q u e n c y range of i n t e r e s t w i l l now be d e s c r i b e d . B a s i c a l l y , the r e q u i r e m e n t i s a r e s o n a n t c i r c u i t whose r e s o n a n t f r e q u e n c y can be a d j u s t e d t o c o i n c i d e w i t h the i n p u t f r e q u e n c y . That i s , the c i r c u i t must be a b l e t o r e s o n a t e a t any f r e q u e n c y w i t h i n i t s o p e r a t i n g range. I t s impedance would be p u r e l y r e s i s t i v e and c o n s i s t of the sum of the r e s i s t a n c e s due t o any Q s p o i l i n g r e s i s t o r s , the R 0 of the c o i l and any l e a k a g e t h r o u g h the c a p a c i t o r s . Upon such a r e s i s t i v e impedance l e v e l i s superimposed the NMR l i n e s h a p e . The c o n t r i b u t i o n t o the t o t a l r e s i s t a n c e due t o a l l s o u r c e s o t h e r than the n u c l e a r s p i n s w i l l h e r e a f t e r be r e f e r r e d t o as the background impedance l e v e l or s i m p l y the background. The NMR l i n e s h a p e c o n s i s t s of two components added i n q u a d r a t u r e , the d i s p e r s i v e component x'(^) m a n i f e s t s i t s e l f as a net r e a c t a n c e which the v a r a c t o r w i l l s t r i v e t o c a n c e l . The a b s o r p t i v e component x" (") on the o t h e r hand i s m a n i f e s t e d as a r e s i s t i v e impedance which i s s i m p l y added t o the e x i s t i n g r e s i s t i v e impedance background i n a p u r e l y s c a l a r f a s h i o n . T h i s i s the b a s i s of the NMR system 58 d e s c r i b e d i n t h i s c h a p t e r . The o n l y o t h e r t o p i c which s h o u l d be mentioned a t t h i s p o i n t i s the t e c h n i q u e by which the resonance c o n d i t i o n i s a c t u a l l y "sensed". The resonance c o n d i t i o n i s c h a r a c t e r i z e d by a z e r o r e l a t i v e phase between the c u r r e n t t h r o u g h and the p o t e n t i a l a c r o s s the resonant c i r c u i t . Below resonance the c i r c u i t i s c a p a c i t i v e w i t h the c u r r e n t l e a d i n g the v o l t a g e s i g n a l . Above resonance the c i r c u i t i s i n d u c t i v e and the c u r r e n t l a g s b e h i n d the v o l t a g e . Measurement of t h i s r e l a t i v e phase p r o v i d e s not o n l y a measure of the d e v i a t i o n from resonance but a l s o a t r i b u t e s a s i g n t o t h e e r r o r . F i g u r e 3.2 Phase <t> of V o l t a g e S i g n a l w i t h Respect t o C u r r e n t ( S e r i e s Resonance) The remainder of t h i s c h a p t e r i s d e v o t e d t o the d e s c r i p t i o n of the d e s i g n of an NMR system based upon th e s e p r i n c i p l e s . 59 B. DESIGN OF A PHASE COMPENSATED NMR SYSTEM A phase compensated NMR system i s i l l u s t r a t e d d i a g r a m m a t i c a l l y i n f i g u r e 3 . 3 . CONSTANT VOLTAGE VARIABLE FREQUENCY r f SUPPLY > f vb F i g u r e 3.3 Phase Compensated NMR Q-Meter Upon i n s p e c t i o n of t h i s c i r c u i t one can e a s i l y r e c o g n i z e the c o n s t a n t c u r r e n t f e d , s e r i e s tuned NMR scheme. The resonant c i r c u i t c o n s i s t s of the i n d u c t o r L ( w i t h s e r i e s r e s i s t a n c e R 0) i n s e r i e s w i t h the p a r a l l e l c o m b i n a t i o n of C and the v a r a c t o r c a p a c i t y C^. The b l o c k i n g c a p a c i t o r C^ which i s used t o b i a s 60 C v has a r e a c t a n c e which i s n e g l i g i b l e a t the f r e q u e n c i e s of i n t e r e s t . I n a d d i t i o n the b i a s i n g r e s i s t o r R^ i s l a r g e t o pr e v e n t the c a p a c i t a n c e of the re s o n a n t c i r c u i t from b e i n g " s h o r t e d o u t " by the low output impedance of the a m p l i f i e r . The resonant c i r c u i t i s f e d by the c o n s t a n t c u r r e n t p r o v i d e d by the l a r g e r e s i s t a n c e R c c (whose impedance i s l a r g e compared t o the impedance of the resonant c i r c u i t ) . The p o t e n t i a l a c r o s s the resonant c i r c u i t i s a m p l i f i e d by the a m p l i f i e r A. T h i s s i g n a l i s passed t o the r f arm of the phase d e t e c t o r and i s a l s o f u r t h e r a m p l i f i e d by the a m p l i f i e r A'. The output of A' i s the s i g n a l which i s e v e n t u a l l y a n a l y z e d i n o r d e r t o measure the NMR l i n e s h a p e . At f r e q u e n c i e s below the r e s o n a n t f r e q u e n c y co0 = 1  •L(C + C y) (3.1) the v o l t a g e s i g n a l passed t o the r f arm of the phase d e t e c t o r l a g s b e h i n d the r e f e r e n c e c u r r e n t i D t h r o u g h the resonant c i r c u i t (due t o the net c a p a c i t i v e r e a c t a n c e of the c i r c u i t ) . At f r e q u e n c i e s above C J 0 t h i s v o l t a g e s i g n a l l e a d s the r e f e r e n c e c u r r e n t i R . At resonance the net impedance of the c i r c u i t i s p u r e l y r e s i s t i v e and as a r e s u l t the p o t e n t i a l a c r o s s the c i r c u i t i s i n phase w i t h i R . These r e l a t i o n s h i p s a r e i n d i c a t e d i n f i g u r e 3.4. The phase a n g l e <j> i s g i v e n by 61 t F i g u r e 3.4 Phasor Diagram of P o t e n t i a l A c r o s s Resonant C i r c u i t ( r e f e r e n c e phase i s i D ) * = t a n " 1 I" u T j ~ 1 / " ( C + C v ) l ( 3 2 )  L R 0 J In the i d e a l case R c c i s so much l a r g e r than the impedance of the resonant c i r c u i t t h a t i t i s e s s e n t i a l l y the o n l y impedance seen by the i n p u t r f p o t e n t i a l . As a r e s u l t , the i n p u t v o l t a g e s i g n a l i s a l w a y s i n phase w i t h the c u r r e n t t h r ough the r e s o n a n t c i r c u i t i R . T h i s i n p u t v o l t a g e s i g n a l i s c o n s e q u e n t l y f e d i n t o t h e l o c a l o s c i l l a t o r b ranch of the phase d e t e c t o r . Resonance i s t h u s d e t e r m i n e d by the r e l a t i v e phase of the c u r r e n t t h r o u g h and the p o t e n t i a l a c r o s s the r e s o n a n t c i r c u i t . T h i s phase d i f f e r e n c e i s z e r o a t r e s o n a n c e , as mentioned e a r l i e r . 62 In the n o n - i d e a l case the f e e d p o t e n t i a l i s o n l y i n phase w i t h i R a t resonance. At a l l o t h e r f r e q u e n c i e s the non-zero r e a c t i v e component of the resonant c i r c u i t impedance i n t r o d u c e s a phase s h i f t between t h e s e s i g n a l s . The v a r i a t i o n of the f e e d p o t e n t i a l w i t h r e s p e c t t o the r e f e r e n c e c u r r e n t i _ i s shown i n f i g u r e 3 . 5 . Im > O ZD > O < F i g u r e 3.5 Phasor Diagram of Feed P o t e n t i a l and Phase D e t e c t o r R e f e r e n c e ( r e f e r e n c e phase i s i p ) ( 3 . 3 ) G U < GL) 0 The r e s u l t i n g phase \p which i s measured i s l e s s than the a c t u a l phase <t> by the amount 8. 63 i . e . 4/ = <t> - 6 (3.4) Note that i n the l i m i t co - C J 0 we have the r e s u l t that _ (p _ o. That i s , at resonance the measured phase i s equal to the t r u e phase d i f f e r e n c e between the c u r r e n t and the v o l t a g e waveforms (which i s z e r o ) . Thus the resonance c o n d i t i o n i s s t i l l u n i q u e l y d e f i n e d . The e r r o r s i g n a l , however, i s p r o p o r t i o n a l to \p and not to <t> and as a r e s u l t i s s m a l l e r than the t r u e e r r o r . The s i g n of the e r r o r i s of course p r e s e r v e d . The d e t e c t e d phase d i f f e r e n c e $ r e s u l t s i n a p o t e n t i a l V c = Ki<// + being a p p l i e d to the v a r a c t o r C v, where K i = G K , and K , d e s c r i b e s the response of the phase d e t e c t o r . The b i a s i n g p o t e n t i a l V " c governs the value of C y . A t y p i c a l response f o r C i s of the f o r m 2 1 over most of the o p e r a t i n g range of the d e v i c e , n i s i n the range of 1/3 to 1/2 depending upon g e o m e t r i c a l and doping c o n s i d e r a t i o n s and V ^ ^ i s a b u i l t i n p o t e n t i a l ( t y p i c a l l y a f r a c t i o n of a v o l t ) . Above and below the r e g i o n i n which the 2 1 P. Horowitz and W. H i l l , The A r t of E l e c t r o n i c s , (Cambridge U n i v e r s i t y Press, New York, 1984), p. 556. (3.5) ( V c " V b i ) n 64 CAPACITANCE A (vc-vbi) . _ n V c - V b i F i g u r e 3.6 C a p a c i t a n c e of a V a r a c t o r c a p a c i t a n c e i s s e n s i t i v e t o V , the c a p a c i t a n c e of the d e v i c e i s e s s e n t i a l l y a c o n s t a n t (as i n d i c a t e d i n f i g u r e 3.6). The b e h a v i o u r of v a r a c t o r s i s d i s c u s s e d f u r t h e r i n appendix A. A change i n the c a p a c i t a n c e C v r e s u l t s i n a change i n the r e s o n a n t f r e q u e n c y co0 ( e q u a t i o n 3.1). T h i s i n t u r n r e s u l t s i n a change i n the phases <t> and 8 and hence r//. Some form of s e l f c o n s i s t e n c y must be found such t h a t t h e phase s h i f t i n t r o d u c e d by the d i f f e r e n c e between co and C J 0 i s the same as the phase s h i f t which s e t s the re s o n a n t f r e q u e n c y a t C J 0 . I t w i l l be shown t h a t an a d d i t i o n a l c o n d i t i o n must be imposed i n o r d e r t o ensure t h a t g i v e n time w i l l go t o z e r o and o>0 w i l l e q u a l w. That i s , under t h e s e c o n d i t i o n s w 0 w i l l i n d e e d t r a c k the i n p u t f r e q u e n c y u>. 65 The phase comensated NMR Q-meter i s a n e g a t i v e feedback system i n which the f r e q u e n c y u> i s the i n p u t and the output i s the r e s o n a n t f r e q u e n c y w 0 . By i n c r e a s i n g the l o o p g a i n co0 can be f o r c e d t o approach a r b i t r a r i l y c l o s e t o co. When the time r e s p o n s e s of the v a r i o u s elements of the feedback l o o p ar e c o n s i d e r e d , l i m i t s can be s e t f o r the maximum a l l o w a b l e g a i n i n o r d e r t o m a i n t a i n s t a b i l i t y . The time response and the s t a b i l i t y of t h i s system a r e the t o p i c s of the second s e c t i o n of t h i s c h a p t e r . The e q u a t i o n s d e s c r i b i n g the b e h a v i o u r of t h i s system ar e not i m m e d i a t e l y t r a n s p a r e n t . For t h i s reason a r a t h e r u n c o n v e n t i o n a l but more e n l i g h t e n i n g approach t o t h i s n e g a t i v e feedback problem i s t a k e n . A more c o n v e n t i o n a l approach i s used i n the s e c t i o n on s t a b i l i t y . L e t us i n t r o d u c e a f a c t o r /3(co,V^^-V^^) , such t h a t i f the measured phase s h i f t a t any time i s \p, the e f f e c t of the c l o s e d l o o p i s t o d r i v e the system towards a s t a t e such t h a t the measured phase s h i f t i s |3iiV. A s e l f c o n s i s t e n t s o l u t i o n i s one i n which /3 (co, v c ~ v j j j _ ) = From e q u a t i o n s 3.2, 3.3, 3.4 and 3.5 we can w r i t e an e x p r e s s i o n f o r the phase s h i f t B\jj which the system seeks t o o b t a i n , g i v e n an i n i t i a l phase i . e . /3i// = 4> - 6 66 or Pxjj = tan' Ro C + K-( K ^ + V B - V M ) n ( 3 . 6 ) - t a n - 1 R 0+R w(R 0+R ) cc cc C + K-(K;* + v b - v b i ) t a n " 1 wL -CJ C + K-( K ^ + V B - V b.) n R c c R 0 ( R 0 + R c c ) 1 + CJL -C + K-( K ^ + V F A - V B I ) ' R 0 ( R 0 + R _ _ ) cc which i s of the form M = t a n " 1 AX 1 +X' ( 3 . 7 ) where A = cc v/R 0(R 0+R ) cc ( 3 . 8 ) and X = coL -C + K-( K ^ + V F A V b i > n v / R 0 ( R o + R c c ) ( 3 . 9 ) 67 Giv e n a p a r t i c u l a r <//, u>, and V , the r e s u l t i n g phase P\p g i v e s an i n d i c a t i o n of which d i r e c t i o n and how "hard" \p i s f o r c e d t o change. The s o l u t i o n of t h i s t r a n s c e n d e n t a l e q u a t i o n f o r /3 = 1 y i e l d s the s e l f c o n s i s t a n t v a l u e s of \p which the system w i l l attempt t o assume. A few g e n e r a l s t a t e m e n t s can be made about the b e h a v i o u r of the system g i v e n a p l o t of /3i// v s . \p. An a r b i t r a r y r e l a t i o n s h i p i s s k e t c h e d i n f i g u r e 3.7. F i g u r e 3.7 )3iiV as an A r b i t r a r y F u n c t i o n of $ As l o n g as the c u r v e l i e s above the l i n e /3<jV = ^, > \p so t h a t \(/ w i l l be f o r c e d t o i n c r e a s e a t a r a t e which i n c r e a s e s or d e c r e a s e s as the c u r v e moves f u r t h e r from or c l o s e r t o the l i n e = ip. A l t e r n a t e l y , i f the c u r v e l i e s below the l i n e /3i/> = then < $ and $ w i l l be f o r c e d t o de c r e a s e a t 68 a r a t e depending upon the d i f f e r e n c e between j3\iV and \p. In a d d i t i o n we can i d e n t i f y two t y p e s of s o l u t i o n s . The f i r s t i s one i n which the p l o t of (l\p i n t e r s e c t s the l i n e R\li = \p w i t h a s l o p e of l e s s than one. That i s , i f the approach i s from the l e f t and hence above the l i n e P\p = \j/, then ^ w i l l keep i n c r e a s i n g towards the s o l u t i o n . In the same manner, i f the approach i s ' f r o m the r i g h t and hence below the l i n e = \}J w i l l keep d e c r e a s i n g u n t i l the s o l u t i o n i s r e a c h e d . Such a s o l u t i o n i s s t a b l e as any movement away from the r o o t r e s u l t s i n a d r i v i n g f o r c e which tends t o c a n c e l the e r r o r . The second type of r o o t i s c h a r a c t e r i z e d by a s l o p e i n the p l o t of R\jj v s . $ g r e a t e r than u n i t y . I f i// i n i t i a l l y r e s t s a t t h i s r o o t , no e r r o r s i g n a l i s g e n e r a t e d . Any movement away from t h i s p o i n t , however, g e n e r a t e s an e r r o r s i g n a l which tends t o d r i v e the system away from the r o o t . Such a r o o t i s m e t a s t a b l e and can i n a d v e r t a n t l y be o b t a i n e d i n p r a c t i c e w i t h the i n s e r t i o n of a 180° phase s h i f t i n one of the arms f e e d i n g the phase d e t e c t o r . I t i s v e r y i m p o r t a n t t o note a t t h i s p o i n t t h a t so f a r , no guarantee has been made t h a t a s t a b l e r o o t can be found a t the o r i g i n of the (S\p vs \p p l o t . That i s , t h e r e i s no guarantee t h a t $ w i l l a ttempt t o go t o z e r o a t the f r e q u e n c y co. W i t h r e f e r e n c e t o f i g u r e 3.7, \iV0 i s a s t a b l e s o l u t i o n w h i l e ^, i s o n l y a m e t a s t a b l e one. The system w i l l seek the s o l u t i o n a t \p0 i f \p i s i n i t i a l l y i n the r e g i o n \p < i/>,. I f , however, \jj i s i n i t i a l l y g r e a t e r than i/>1f then i// w i l l be 69 f o r c e d t o keep on i n c r e a s i n g u n t i l a t some p o i n t some component i n the feedback l o o p reaches the edge of i t s o p e r a t i n g r e g i o n and becomes p i n n e d t o a s t a t i c s t a t e . An i n v a l i d s o l u t i o n w i l l be found i n t h i s r e g i o n of h i g h n o n - l i n e a r i t y . In p r a c t i c e i f t h i s o c c u r s one must s t a r t over w i t h new i n i t i a l c o n d i t i o n s i n o r d e r t o u n l o c k the system. L e t us now examine a more r e a l i s t i c example u s i n g t y p i c a l v a l u e s f o r the p a r a m e t e r s . C o n s i d e r a system d e s i g n e d t o o p e r a t e about a c e n t r a l f r e q u e n c y f = 16.6 MHz. f i s used i n our n o t a t i o n r a t h e r than f 0 as our g o a l i s t o scan the r e s o n a n t f r e q u e n c y f 0 over a range c e n t e r e d a t f c« We r e q u i r e CJ = 1 (3.10) ° v/L(C + C j ) where C° = ^ ( V ^ ) and has been s e t t o c o r r e s p o n d r o u g h l y w i t h the c e n t r e of the o p e r a t i n g r e g i o n of t h e v a r a c t o r . In p r a c t i c e f i s s h i f t e d by s m a l l amounts by c h a n g i n g V^. I f f o r example we have a c o i l w i t h an i n d u c t a n c e of 1 5 M H , a c a p a c i t a n c e C + C° = 6 . 1 3 pF i s r e q u i r e d . A t y p i c a l v a r a c t o r might have a c a p a c i t a n c e v a r i a t i o n i n i t s o p e r a t i n g r e g i o n s i m i l a r t o 70 (3.11) ( i . e . K 2 = 3.0 pF-»/V and V, . = 0.6V) l e v e l i n g o f f t o a c o n s t a n t v a l u e o u t s i d e t h i s r e g i o n , as was shown i n f i g u r e 3.6. L e t us assume t h a t the l i m i t s f o r the v a r a c t o r c a p a c i t a n c e a r e about 1 pF and 5 pF. N e i t h e r the e x a c t shape of t h i s , c u r v e nor the v a l u e s a t which C v l e v e l s o f f a r e c r i t i c a l f o r t h i s example. A l s o note t h a t i t has been assumed t h a t the prop e r b i a s i n g i s V c > 0. I f we choose C v ( V f a ) = 3pF, V b = 1.6 V then C = 3.13pF and we have a t u n i n g range of about ±2pF or r o u g h l y ±2.5MHz. A t y p i c a l c i r c u i t might have a Q = wL/R 0 of about 100 r e s u l t i n g i n an e f f e c t i v e R 0 = 15.6 ohms. R c c i s perhaps two or t h r e e o r d e r s of magnitude l a r g e r , say 3 x 10 3 ohms. For an i n i t i a l c o n s i d e r a t i o n of the response of t h i s system l e t Us choose = 0.1V / r a d i a n . From e q u a t i o n 3.9, X i s g i v e n by where </  i s i n r a d i a n s . T h i s l e a d s t o the p l o t of (l\p = t a n ' 1 AX v s . \j/ shown i n f i g u r e 3.8. The f a c t t h a t t h i s p l o t p a s s e s t h r o u g h the o r i g i n i s a r e s u l t of the c h o i c e of c r i t e r i a s e t out f o r the g e n e r a l case we ob s e r v e one s t a b l e X = 7.213 - 14.13/0. 1ij/ +1.0 i/(0. ]xp + 1 .0) + .9590 (3.12) T+X2" f c such t h a t C° = C v ( V b ) (or v i c e v e r s a ) . R e f e r r i n g t o the 71 s e l f c o n s i s t e n t s o l u t i o n a t the o r i g i n . G i v e n any i n i t i a l i/>, the system w i l l i n time move towards t h i s s e l f c o n s i s t e n t s t a t e . Now l e t us c o n s i d e r the e f f e c t of c h a n g i n g co. O b v i o u s l y , f o r a c o n s t a n t t h e p l o t of (S\p v s . i/> w i l l no l o n g e r pass t h r o u g h the o r i g i n and the s e l f c o n s i s t e n t 72 s o l u t i o n ( i f i t s t i l l e x i s t s ) w i l l l i e e l s e w h e r e a l o n g t he l i n e /3\J/ = \p. For co > co^ w i t h no feedback s i g n a l b e i n g f e d t o the v a r a c t o r , the i n d u c t i v e impedance of the resonant c i r c u i t i n c r e a s e s and hence \jj becomes p o s i t i v e . T h i s i s r e f l e c t e d i n the p l o t of v s . \p as a s h i f t of the e n t i r e c u r v e towards more p o s i t i v e \p. S i m i l a r l y , l o w e r i n g co below OJQ r e s u l t s i n a s h i f t of t h i s p l o t toward more n e g a t i v e Such s h i f t s , however, a r e not s i m p l e t r a n s l a t i o n s a l o n g the \p a x i s as the shape of the cu r v e depends upon co. An example of the e f f e c t t h a t c h a n g i n g co has upon the /3</V v s . \p p l o t i s shown i n f i g u r e 3.9. T h i s may be extended i n t o a t h r e e d i m e n s i o n a l p l o t w i t h the d r i v i n g f o r c e b e i n g e x p r e s s e d as a f u n c t i o n of \p and CJ. F i g u r e 3.10 demonstrates t h i s f o r the parameters g i v e n e a r l i e r over the fr e q u e n c y i n t e r v a l 16.4 MHz t o 16.8 MHz. The a n a l y s i s of t h i s p l o t i s a s i m p l e e x t e n s i o n of the two d i m e n s i o n a l c a s e . G i v e n a frequ e n c y co and an i n i t i a l v a l u e f o r \JJ, the d r i v i n g f o r c e jl\p i s de t e r m i n e d and an attempt t o rea c h a s e l f c o n s i s t e n t s o l u t i o n i s made. Such a s o l u t i o n e x i s t s a t the i n t e r s e c t i o n of the /3<xV s u r f a c e w i t h the p l a n e P4/ = \p. T h i s i n t e r s e c t i o n d e s c r i b e s a l o c u s of p o i n t s p a s s i n g t h r o u g h the co a x i s a t co^ and i s g i v e n by a l l co and \jj which s a t i s f y e q u a t i o n 3.6 w i t h /? = 1. In the same manner as b e f o r e , i f 3 ( j 3 i / > ) / 3 \ / / < 1 i n the r e g i o n of the i n t e r s e c t i o n , then the s o l u t i o n i s s t a b l e . The s o l u t i o n i s o n l y m e t a s t a b l e i f a ( / 3 i / / ) / 3 i | / > 1 . 73 F i g u r e 3.9 v s . a s a F u n c t i o n o f F r e q u e n c y ( V b = C o n s t a n t ) G i v e n a f r e q u e n c y u> t h e s y s t e m w i l l i n t i m e move i n a p l a n e o f c o n s t a n t CJ a l o n g t h e /3t// v s . \j/ c u r v e t o w a r d t h i s l o c u s o f p o i n t s . I f we g i v e t h e s y s t e m s u f f i c i e n t t i m e t o r e a c h t h i s p o i n t and t h e n c h a n g e w, t h e s y s t e m w i l l r e s p o n d by s e e k i n g a new p o i n t on t h e l o c u s a l o n g a new v s . \p c u r v e . G i v e n s u f f i c i e n t t i m e t o r e s p o n d a t e a c h new u, t h e CO s F i g u r e 3.10 |3iiV as a F u n c t i o n of \p and Frequency 7 5 system can be made t o f o l l o w the l o c u s of p o i n t s , a s s i g n i n g a ^ t o e v e r y u> over a c e r t a i n range of a>. The fr e q u e n c y a t which the l o c u s of s o l u t i o n s passes t h r o u g h the p l a n e i|/ = 0 i s governed by the b i a s V ^ . Lowering has t h e e f f e c t of i n c r e a s i n g C° and hence r a i s i n g a>c. S i m i l a r l y , r a i s i n g has the e f f e c t of c a u s i n g the l o c u s of p o i n t s t o c r o s s the \p = 0 p l a n e at lower co. T h i s e f f e c t i s shown i n f i g u r e s 3 . 1 1 and 3 . 1 2 . So f a r ii/ has been f o r c e d t o behave d i f f e r e n t l y from the way i t would behave i f i t were b e i n g g e n e r a t e d by a s i m p l e s e r i e s r e s o n a n t c i r c u i t . At a l l f r e q u e n c i e s the measured phase s h i f t i s l e s s than t h a t which would be measured f o r the r e s o n a n t c i r c u i t a t t h a t f r e q u e n c y . O b v i o u s l y \p has not been c o n s t r a i n e d t o z e r o as we o r i g i n a l l y p l a n n e d (except a t u>c) . C o n s i d e r now the e f f e c t of i n c r e a s i n g the feedback g a i n G . O r i g i n a l l y K , = K , G was a r b i t r a r i l y s e t t o 0 . 2 V / r a d i a n t o show the s t r u c t u r e of the /3<i/ v s . \p r e l a t i o n s h i p . L e t us assume t h a t K , = 1 V / r a d i a n and v a r y the g a i n G . The e f f e c t i s t w o f o l d . I f we c o n s i d e r ($4/ a t a f i x e d CJ and i n c r e a s e G , the e n t i r e f5\jj v s . 4> p l o t i s compressed towards the (i\p a x i s . That i s , the f u n c t i o n becomes more s e n s i t i v e t o <//. In the l i m i t G - » we have the r e l a t i o n s h i p R\p = TT/2 - TTU(<//) ( 3 . 1 3 ) where the u n i t s t e p U ( x ) i s d e f i n e d as J si-ng CO by n9 V 1 2 low*,--V, 78 U(x) = 0 f o r x < 0 = 1 f o r x > 0 =1/2 f o r x = 0 B y , TL W '2 1 F i g u r e 3.13 /3\xV v s . $ as G Approaches » I t has been i m p l i c i t l y assumed t h a t t h e maximum change i n C v i s s m a l l enough t h a t X does not become v e r y l a r g e f o r any iiv. I n t h i s manner, t h e _ J b e h a v i o u r of the a r c t a n g e n t 1+X2 f u n c t i o n can e s s e n t i a l l y be n e g l e c t e d . I f we now c o n s i d e r a c o n s t a n t $ty and v a r y u> as G - °°, i t i s a s i m p l e m atter t o see t h a t , due t o t h e i n c r e a s e d s e n s i t i v i t y t o the whole c u r v e w i l l be compressed' towards the P4/-OJ p l a n e . I f we c o n s i d e r the i n t e r s e c t i o n of the IS\p s u r f a c e w i t h the p l a n e ^ = ^ as G - » i t i s c l e a r t h a t the l o c u s of the i n t e r s e c t i o n i s f o r c e d t o approach the w a x i s . 7 9 T h i s y i e l d s the d e s i r e d e f f e c t of a ' s t a b l e ' \JJ s o l u t i o n w i t h \p a r b i t r a r i l y c l o s e t o 0 over a range of co. T h i s i s shown i n f i g u r e 3.14. Note t h a t the range of convergence i s governed by the maximum p o s s i b l e d e v i a t i o n i n C^. For such a c i r c u i t , we need o n l y l e t G - » t o o b t a i n the r e s u l t \p = 0 f o r any i n p u t co over a c e r t a i n range of co. T h i s i s the same as s a y i n g t h a t , g i v e n an i n p u t f r e q u e n c y co near co^, co0 w i l l i n time a d j u s t i t s e l f so as t o c o i n c i d e w i t h co. Thus, the c i r c u i t w i l l r e s o n a t e a t the i n p u t f r e q u e n c y . T h i s i s the d e s i r e d system response as o u t l i n e d at the b e g i n i n g of t h i s c h a p t e r . R a i s i n g G t o v e r y h i g h v a l u e s tends t o make the system v e r y s u s c e p t i b l e t o n o i s e . Convergence t o a s t a b l e s o l u t i o n may be p o s s i b l e g i v e n s u f f i c i e n t t i m e . However, i n the presence of n o i s e which has the e f f e c t of s h i f t i n g \p s l i g h t l y , the system may encounter problems i n r e c o v e r i n g from such s h i f t s . I n t h i s way a s t a b l e o p e r a t i o n may be u n r e a l i z a b l e . The b i a s i n g c i r c u i t r y f o r the v a r a c t o r appears l i k e an RC i n t e g r a t o r t o the phase d e t e c t o r c i r c u i t r y . That i s , i n ord e r t o change the p o t e n t i a l V , a charge must be a p p l i e d t o , or removed from, the v a r a c t o r . T h i s p r o c e s s t a k e s a f i n i t e amount of time t o o c c u r . The r a t e a t which the measured phase \p can be changed i s r e l a t e d t o the r a t e a t which the b i a s p o t e n t i a l can be changed. F i g u r e 3.14 \\i F o r c e d t o 0 as G i s I n c r e a s e d 81 dj£ = d± dv, = _ J _ d v i (3.14) dt dV,dt R,G d t F i g u r e 3.15 Rate of Response t o an E r r o r S i g n a l Thus, as G - », d\p/dt - 0 which means t h a t i t t a k e s l e s s time t o r e c o v e r from a phase o f f s e t . These l a s t two p o i n t s w i l l be expanded upon i n the next s e c t i o n . F i n a l l y , i t has been s t a t e d t h a t \}/ may be f o r c e d t o become a r b i t r a r i l y c l o s e t o 0 over some f r e q u e n c y range. To some e x t e n t the bandwidth of the d e v i c e i s governed by G. T h i s range i s , of c o u r s e , l i m i t e d by the dynamic range of the v a r i o u s elements i n the c l o s e d l o o p . The l i m i t i n g f a c t o r tends t o be the maximum a v a i l a b l e change i n the v a r a c t o r c a p a c i t y . Other c o n s t r a i n t s imposed by the v a r a c t o r w i l l be d i s c u s s e d i n c h a p t e r 4. 82 C. TIME RESPONSE AND STABILITY To t h i s p o i n t o n l y the p r i n c i p l e of the o p e r a t i o n of the phase compensated NMR i n s t r u m e n t a t i o n scheme has been d i s c u s s e d . C e r t a i n c o n s t r a i n t s have been s e t which must be met b e f o r e an e q u i l i b r i u m c o n d i t i o n can e x i s t . By r a i s i n g G t o l a r g e v a l u e s the measured phase s h i f t c o u l d be m i n i m i z e d . T h i s had the e f f e c t of k e e p i n g the r e s o n a n t c i r c u i t a t resonance a t any f r e q u e n c y i n a c e r t a i n range. No mention has been made of the a c t u a l time e v o l u t i o n of the phase \jj toward i t s e q u i l i b r i u m v a l u e . C o n s i d e r a t i o n of t h i s time e v o l u t i o n l e a d s t o a d d i t i o n a l c o n s t r a i n t s which govern the s t a b i l i t y of the system and i t s a b i l i t y t o t r a c k the i n p u t f r e q u e n c y u>. In o r d e r t o model t h i s e v o l u t i o n , i t i s n e c e s s a r y t o i d e n t i f y which of the c i r c u i t elements r e q u i r e f i n i t e t i m e s i n o r d e r t o respond t o an i n p u t . In o r d e r t o be as g e n e r a l as p o s s i b l e , time r e s p o n s e s w i l l be a t t r i b u t e d t o each element. I f a t a l a t e r time some are found t o be u n n e c e s s a r y , i t i s a s i m p l e m a t t e r t o d e l e t e them. In the a n a l y s i s t h a t f o l l o w s , t h e a c t u a l form of the time response i s not as i m p o r t a n t as the t e c h n i q u e of a n a l y s i s . For t h i s r eason each response i s kept as s i m p l e as p o s s i b l e . A d d i t i o n 83 of more c o m p l i c a t e d responses i s s t r a i g h t f o r w a r d . C o n s i d e r the b i a s i n g c i r c u i t r y f o r the v a r a c t o r shown i n f i g u r e 3.16. E s s e n t i a l l y t h i s c o n f i g u r a t i o n c o n s i s t s of an RC i n t e g r a t o r or low pass f i l t e r w i t h a time c o n s t a n t R,C v = T, (3.15) F i g u r e 3.16 RC Time Constant Due t o V a r a c t o r C a p a c i t a n c e In a d d i t i o n , t h e r e may e x i s t a second RC time c o n s t a n t due to f i l t e r i n g c i r c u i t s and the l e n g t h of c o a x i a l c a b l e used t o s u p p l y the v a r a c t o r ^ w i t h the b i a s i n g p o t e n t i a l . i . e . R 2 C 2 = T 2 (3.16) The f r e q u e n c y response of the a m p l i f i e r G s h o u l d a l s o be c o n s i d e r e d . The g a i n of any m u l t i s t a g e a m p l i f i e r f a l l s o f f i n some manner above some l i m i t i n g f r e q u e n c y . T h i s i s u s u a l l y due t o the low pass f i l t e r s formed by s i g n a l s of f i n i t e s o u r c e impedance which d r i v e v a r i o u s c a p a c i t i v e l o a d s 84 V, R 2 v? R | 12 VW MA-IT l F i g u r e 3.17 RC Time Constant Due t o F i l t e r i n g and C a p a c i t i v e L o a d i n g w i t h i n the a m p l i f i e r s t a g e s . T h i s e f f e c t can be c h a r a c t e r i z e d by a t h i r d low pass f i l t e r w i t h a time c o n s t a n t T 3 . i . e . R 3 C 3 = T 3 (3.17) In f i g u r e 3.18, G r e p r e s e n t s an i d e a l a m p l i f i e r w i t h a g a i n which i s independent of f r e q u e n c y . F i g u r e 3.18 Frequency Dependent A m p l i f i c a t i o n F i n a l l y , the phase d e t e c t o r c i r c u i t r y i s most l i k e l y some form of double b a l a n c e d r i n g m o d u l a t o r , f o l l o w e d by an i n t e g r a t i n g s t a g e a f t e r the INT arm i n o r d e r t o p r o v i d e a dc 85 output p r o p o r t i o n a l t o the phase \[/ over some range of phase a n g l e s . We c h a r a c t e r i z e t h i s phase t o v o l t a g e c o n v e r s i o n as (3.18) v5 > R„C A - T n LO INT v 5 RF MODULATOR R 4 (3.19) V 4 (OUT) " C 4 m F i g u r e 3.19 I n t e g r a t i o n F o l l o w i n g Phase D e t e c t i o n I n t r o d u c t i o n of t h e s e time c o n s t a n t s i s s u f f i c i e n t t o demonstrate the time dependence of the response of the system. The a d d i t i o n of o t h e r time c o n s t a n t s or the m o d i f i c a t i o n of the form of the responses has l i t t l e e f f e c t upon the a n a l y s i s which f o l l o w s . L e t us make the s i m p l i f y i n g assumption t h a t the e f f e c t of l o a d i n g between v a r i o u s s t a g e s i n the feedback c i r c u i t can be n e g l e c t e d . T h i s a l l o w s us t o w r i t e 86 dV . = 1 i . d t o u t C.' 1 ( V i n " V o u t } • ( 3 ' 2 0 ) R / C / V- R i V * N/S/\A O U T F i g u r e 3.20 The i t h Low Pass F i l t e r where i ^ i s the c u r r e n t f l o w i n g i n P^ . . T h i s l e a d s t o the c o u p l e d s e t of e q u a t i o n s : dV = 1 ( V 2 - v ) (3.21) 3 T - c — c dt T 1 dV 2 = j _ ( V 3 - V 2 ) dt T 2 dV 3 = 1 (GV„ - V 3 ) dt T 3 dV, = 1 (K,^ - V,) dt T„ F i n a l l y , the l o o p can be c l o s e d by r e l a t i n g the measured phase $ t o the v a r a c t o r p o t e n t i a l V . U s i n g the 87 model d i s c u s s e d i n s e c t i o n 3-B we can r e w r i t e e q u a t i o n 3.6 w i t h /3 = 1 t o g i v e 4> = t a n " 1 AX 1 +z-(3.22) = t an -1 coL -CO C + K-( K ^ + V b - V M ) R cc R 0 ( R 0 + R ) cc 1 + coL - 1 CO C + K-+ v b - v M) R 0 ( R 0 + R C C ) O b v i o u s l y \p = \//(Vc,co) where co i s the i n p u t f r e q u e n c y which i s a f u n c t i o n of time f o r a f r e q u e n c y swept e x p e r i m e n t . i.e. 4> = tf[vcU),w(t)] . (3.23) The n o n l i n e a r i t y of t h i s response r e n d e r s an a n a l y t i c a l s o l u t i o n i m p o s s i b l e , and so a l i n e a r a p p r o x i m a t i o n of tf[V ( t f ) , u ( t ) ] w i l l be used. The T a y l o r e x p a n s i o n of \iV about V = V, and co = co i s 88 <//[V U) , u ( t ) ] <//(V, , u ) + 3V (v - v ) +M (CJ - co ) c V, (3.24) dV7 (V - V, ) 2 + C D b' c 32<// [CO R e t a i n i n g o n l y the l i n e a r terms, and r e a l i z i n g t h a t xpiV^.oo^) = 0 by d e f i n i t i o n , we o b t a i n <// * _3J/_ (V - V, ) + _3j> 3V c D dco C V, ,co b' c (co - u>c) V, ,co b c (3.25) cc  R 0 (Ro + R c c ) ' - 2 K 2 ( V b - V b.) [ v _ j oo[c(Vb - V b . ) 2 + K 2 ] 2 C b L + o r C + K-[ u ( t ) - ooc] M[V - V. ] + N[w(t) - w ] C D C where M = - 2 R c c K 2 ( V b - V b.) cc b b i coR 0(R 0 + R )[C(V, - V, . )z + K 2 J 2 N = R cc R 0 ( R 0 + R c c ) L + 1 co C + K 2  89 E q u a t i o n s 3.20 and 3.24 d e s c r i b e the e n t i r e system w i t h i n the s t a t e d a p p r o x i m a t i o n s . In terms of x, = V (3.26) c x 2 = V 2 x 3 = V 3 e q u a t i o n s 3.21 can be w r i t t e n as x , = ( x 2 - x , ) / r , (3.27) x 2 = ( x 3 - x 2 ) / r 2 x 3 = (Gx« - X 3 ) / T 3 x f l = (K,\p - X „ ) / T 4 = ( K j M ( x , - V. ) + N ( u ( t ) - u)] ~ x»)/r, . T h i s a l l o w s us t o use the s t a t e space r e p r e s e n t a t i o n 2 2 of t h i s l i n e a r system. The s t a t e e q u a t i o n i s x = Ax + B u ( t ) (3.28) 2 2 K a t s u h i k o Ogata, Modern C o n t r o l Systems, ( P r e n t i c e - H a l l , Englewood C l i f f s , N. J . , 1970), p. 666. 90 where x = x„ A = -1/T, 1/T, 0 0 0 "1/T 2 1/T 2 0 0 0 -1/T 3 G / r 3  K 1 M 0 0 -1/T„ (3.29) B = 0 0 0 1 u = u = - K i M V , + N[co(t) - co ] T B 7 c (3.30) ( i . e . the f o r c i n g f u n c t i o n ) and B and u have been w r i t t e n as v e c t o r s f o r the sake of g e n e r a l i t y . Note t h a t i n t h i s example u i s a v e c t o r of d i m e n s i o n a l i t y 1 and i s thus w r i t t e n as u. We a r e i n t e r e s t e d i n the s o l u t i o n of x , ( V c ) which g i v e s us it. Thus we can d e f i n e the o u t p u t e q u a t i o n t o be y = Cx where C = [ 1 0 0 0 ] (3.31) Note t h a t the i n c l u s i o n of a d d i t i o n a l elements i n t o t h i s a n a l y s i s s i m p l y r e q u i r e s an i n c r e a s e i n the d i m e n s i o n a l i t y of the v e c t o r space. In the example g i v e n above A and B a r e not f u n c t i o n s of t i m e . However i n the more g e n e r a l l i n e a r t i m e - v a r y i n g case t h e s e e q u a t i o n s become X = A ( t ) x + B ( t ) u (3.32) where x and u a r e n - d i m e n s i o n a l column v e c t o r s and A ( t ) and B ( t ) a r e n x m m a t r i c i e s . T h i s problem may be s o l v e d by r e p l a c i n g x ( t ) by x ( t ) = * ( t , t 0 ) * ( t ) (3.33) where $ ( t , t 0 ) s a t i s f i e s $ ( t , t 0 ) = A ( t ) * ( t , t 0 ) w i t h the i n i t i a l c o n d i t i o n $ ( t , t 0 ) = I the u n i t m a t r i x . T h i s y i e l d s * ( t ) = * " 1 ( t , t 0 ) B ( t ) u ( t ) (3.34) so t h a t t ¥ ( t ) = * ( t 0 ) + / * " 1 ( T , t 0 ) B ( r ) u ( r ) d T (3.35) to t or x ( t ) = < M t , t 0 ) x ( t 0 ) + ; * ( t , T ) B ( T ) u ( T ) d T (3.36) t o In t h i s way the s o l u t i o n f o r x ( t ) may be o b t a i n e d f o r v a r i o u s u ( t ) . 92 At t h i s p o i n t i t might be a p p r o p r i a t e t o say something about the stea d y s t a t e e r r o r of a system s i m i l a r t o the one we are d i s c u s s i n g . Our system c o n s i s t s of a c l o s e d l o o p c o n t a i n i n g f o u r i n t e g r a t i o n s (commonly r e f e r r e d t o as a type 4 c o n t r o l s y s t e m ) . G i v e n s u f f i c i e n t t i m e , t h i s system w i l l be a b l e t o t r a c k \p i f \p changes by as much as the t h i r d power of t i m e . 2 3 T h i s i s the b a s i s f o r the statement o f f e r e d i n the p r e v i o u s s e c t i o n , t h a t the system w i l l come t o e q u i l i b r i u m " g i v e n s u f f i c i e n t t i m e . " In o r d e r t o draw the r o o t l o c u s diagram f o r t h i s system we need t o o b t a i n the c h a r a c t e r i s t i c p o l y n o m i a l of the system. T h i s can e i t h e r be o b t a i n e d from the " t r a n s f e r f u n c t i o n " of the system ( d e f i n e d as the L a p l a c e t r a n s f o r m of the o u t p u t v a r i a b l e d i v i d e d by the L a p l a c e T r a n s f o r m of the i n p u t v a r i a b l e w i t h a l l i n i t i a l c o n d i t i o n s s e t t o z e r o ) or i t can be w r i t t e n down i m m e d i a t e l y i f A i s known. 2* The c h a r a c t e r i s t i c p o l y n o m i a l can be w r i t t e n |sI — A | = 0 where I i s the u n i t m a t r i x of d i m e n s i o n a l i t y n = 4 and s i s the L a p l a c e t r a n s f o r m v a r i a b l e . co - s t L { f ( t ) } = /e f ( t ) d t (3.37) 0 2 3 R i c h a r d C. D o r f , Modern C o n t r o l Systems, 3 r d ed. (Addison-Wesley, Don M i l l s , Ont., 1981), p.122. 2 a i b i d . p. 327. 93 Thus, s i - A I = d e t s+1 -]_ 0 0 0 T 1 s+1 0 0 0 s+1 -G -K,M 0 0 s+1 = 0 (3.38) I f we w r i t e the c h a r a c t e r i s t i c p o l y n o m i a l i n the form 1 + K F ( s ) = 0 where K i s the v a r i a b l e of i n t e r e s t ( i n our case the g a i n G ) , 1 + G (-K ,M) = 0 r,T2T3T4(s+1 A,)(s+1A 2) (S+1/T 3 )(s+1 Aa) i t can be seen t h a t F ( s ) has f o u r p o l e s s i t u a t e d a t s = - l A i r s = -1A 2, s = " 1 A 3 and s = -1A« on the n e g a t i v e r e a l a x i s i n the complex s - p l a n e . As G i n c r e a s e s from 0 t o =° the r o o t s of the c h a r a c t e r i s t i c p o l y n o m i a l d e s c r i b e the r o o t l o c u s which moves from the p o l e s of F ( s ) out t o t h e z e r o e s of F ( s ) a t i n f i n i t y . The l o c u s on the r e a l a x i s , of c o u r s e , always l i e s t o the l e f t of an odd number of p o l e s and z e r o e s . An example of t h i s r o o t l o c u s b e h a v i o u r i s shown i n f i g u r e 3.21 i n which the p o l e s have been p l a c e d i n an a r b i t r a r y f a s h i o n . A l s o note t h a t s = a + j o / where to' i s the f r e q u e n c y a t 94 i \C0 F i g u r e 3.21 Example Root Locus Diagram which t h e x. o s c i l l a t e ( i . e . i t i s not the i n p u t f r e q u e n c y co) . The p o i n t s a t which the r o o t l o c u s b r e a k s away from the r e a l a x i s can be found by w r i t i n g G = T 1 r 2 T 3 T n ( s + 1/>,)(s + 1 / T 2 ) ( S + 1 / r 3 ) ( s + 1/T„) 95 (3.39) and s e t t i n g dG = 0 t o f i n d a maximum i n G. The c e n t r o i d of ds the asymptotes i s g i v e n by c = I p o l e s of F ( s ) - Zzer o e s of F ( s ) (3.40) n - n P z where n^ i s the number of p o l e s of F ( s ) and n i s the number p z of z e r o e s of F ( s ) . For our example w i t h f o u r p o l e s , t h i s y i e l d s c = ~ 1/r, - 1/T 2 - 1 / T 3 - WTU (3.41) 4 The a n g l e s a t which the asymptotes c r o s s the a a x i s a r e g i v e n by <Pn = (2q + 1 ) TT (3.42) n - n p z where q = 0,1, ,(n„ - n„ - 1) For our example t h i s y i e l d s f o u r asymptote a n g l e s (7r/4, 3TT/4, 5TT/4 and 7TT/4) . 9 6 For s m a l l G the r o o t l o c u s l i e s on the n e g a t i v e r e a l a x i s and thus the system responds i n an e x p o n e n t i a l l y d e c a y i n g manner. At the breakaway p o i n t s , low fr e q u e n c y o s c i l l a t i o n s s t a r t t o d e v e l o p . The fr e q u e n c y of the s e o s c i l l a t i o n s i n c r e a s e s and the damping c o e f f i c i e n t d e c r e a s e s as G approaches °°. At some s p e c i f i c G d e t e r m i n e d by the Routh H u r w i t z c r i t e r i o n , the system breaks i n t o e x p o n e n t i a l l y growing s i n u s o i d s ( t h i s o c c u r s when the r o o t l o c u s c r o s s e s i n t o the r i g h t h a l f p l a n e ) . The v a l u e of G a t which t h i s o c c u r s i s the a b s o l u t e maximum v a l u e p e r m i t t e d f o r the system t o s u c c e s s f u l l y come i n t o e q u i l i b r i u m a t some \p g i v e n s u f f i c i e n t t i m e . T h i s v a l u e of G i s no more than a g u i d e l i n e as we r e q u i r e the system t o come i n t o e q u i l i b r i u m (or w i t h i n an a p p r o p r i a t e range of the e q u i l i b r i u m c o n d i t i o n ) b e f o r e co(t) can change s i g n i f i c a n t l y . In p r a c t i c e we r e q u i r e the damping f a c t o r a t o f o r c e the o s c i l l a t i o n s t o d i e out i n a time t h a t i s s h o r t when compared t o the f r e q u e n c y increment t i m e . The g o a l of t h i s d e s i g n then i s t o move each p o l e as f a r away from the o r i g i n as p o s s i b l e ( e s p e c i a l l y the dominant p o l e s which l i e the c l o s e s t t o the o r i g i n ) . T h i s has the e f f e c t of moving the e n t i r e l o c u s t o the l e f t and e f f e c t i v e l y i n c r e a s i n g the maximum a l l o w a b l e g a i n . T h i s i s of c o u r s e the u l t i m a t e g o a l as i t f o r c e s the r e s o n a n t f r e q u e n c y of the c i r c u i t t o t r a c k the i n p u t f r e q u e n c y more c l o s e l y . A nother p o s s i b l e t e c h n i q u e 2 5 2 6 i n v o l v e s the 2 5 i b i d . p. 327. 2 6 K a t s u h i k o Ogata, Modern C o n t o l Systems, ( P r e n t i c e - H a l l , CO I CO U J or s(t-to) (<r+ja/)(t-to) .—e = e 97 I— 3 Q. F i g u r e 3.22 O s c i l l a t o r y B e h a v i o u r of ^  i n t r o d u c t i o n of s e v e r a l z e r o e s so as t o compensate f o r some (or a l l ) of the p o l e s , d r i v i n g the r o o t l o c u s f u r t h e r t o the l e f t . T h i s i n t u r n a l l o w s the g a i n t o be i n c r e a s e d even f u r t h e r w h i l e s t i l l damping out any o s c i l l a t i o n s i n a time p e r i o d s h o r t enough t o be a c c e p t a b l e . Note t h a t i t has been assumed t h a t i n the attempt t o maximize G, damped o s c i l l a t o r y r e sponses have been deemed a c c e p t a b l e . Two comments w i t h r e g a r d t o n o i s e s h o u l d be made a t t h i s p o i n t . The system d e s c r i b e d i s s e n s i t i v e t o fre q u e n c y r e l a t e d n o i s e . T h i s i s p a r t i c u l a r l y n o t i c e a b l e when a d i g i t a l l y s t e pped f r e q u e n c y s o u r c e i s used r a t h e r than an 2 6 ( c o n t ' d ) Englewood C l i f f s , N. J . , 1970), pp.474-530. 98 a n a l o g s o u r c e , as some i n t e r v a l s may be wider than o t h e r s . A l a r g e f r e q u e n c y increment may s e t up o s c i l l a t i o n s which a r e not s u f f i c i e n t l y damped out b e f o r e the next increment o c c u r s . O b v i o u s l y t h i s may p r o p a g a t e the d i s t u r b a n c e or even s t a r t a much more wi d e s p r e a d o s c i l l a t i o n . These o s c i l l a t i o n s may be damped out by l o w e r i n g G w i t h a r e s u l t i n g l o s s i n the degree t o which co0 i s l o c k e d t o co. The second comment has t o do w i t h the c r o s s t a l k between the f r e q u e n c y co i n the r e s o n a n t c i r c u i t and the fr e q u e n c y to' of the c o n t r o l system. Because the v a r a c t o r i s a square law d e v i c e , the a p p l i c a t i o n of a s i n u s o i d a l l y v a r y i n g c u r r e n t a t fre q u e n c y co r e s u l t s i n a net dc p o t e n t i a l ( Z e r o t h o r d e r F o u r i e r component) b e i n g added t o the b i a s p o t e n t i a l V . F o r t u n a t e l y , due t o the n e g a t i v e feedback n a t u r e of the system, most d e l e t e r i o u s e f f e c t s such as t h i s can be. reduced t o a n e g l i g i b l e l e v e l . F i n a l l y , i t must be n o t e d t h a t i n our r o o t l o c u s c o n s i d e r a t i o n we have n e g l e c t e d the g a i n bandwidth p r o d u c t of the a m p l i f i e r . Assuming t h i s t o be a c o n s t a n t , then c l e a r l y i f G i s i n c r e a s e d , the p o l e a t s 3 = - 1 / T 3 moves c l o s e r t o the o r i g i n . For v e r y h i g h g a i n systems s 3 may become one of the dominant p o l e s f o r d e t e r m i n i n g the response of the system. I n c r e a s i n g G has the d u a l e f f e c t of moving the r o o t s f u r t h e r a l o n g the l o c u s and a l s o s h i f t i n g the e n t i r e l o c u s t o a more p o s i t i v e a. T h i s i s shown i n f i g u r e 3.23. O b v i o u s l y the g a i n bandwidth p r o d u c t of the a m p l i f i e r i s a 9 9 c r i t i c a l f a c t o r i n the d e s i g n of the system. A s i m i l a r e f f e c t a r i s e s from the f a c t t h a t the c a p a c i t a n c e of the v a r a c t o r changes c o n t i n u o u s l y . Thus the p o s i t i o n of the p o l e a t -1/T, i s a c t u a l l y time-dependent. T h i s p o l e i s more dominant a t lower f r e q u e n c i e s . F i g u r e 3.23 E f f e c t of I n c r e a s i n g A m p l i f i e r G a i n G D. THE SYSTEM AS A WHOLE In p r a c t i c e , once the feedback l o o p has been s u c c e s s f u l l y c l o s e d , the c i r c u i t r y d e s c r i b e d e a r l i e r can s i m p l y be r e g a r d e d as a b l a c k box h a v i n g one i n p u t and one o u t p u t . The i n p u t i s a c o n s t a n t p o t e n t i a l r f s i g n a l a t fr e q u e n c y c o ( t ) . The time b e h a v i o u r of co(t) i s r e s t r i c t e d t o 100 be a p i e c e w i s e c o n t i n u o u s f u n c t i o n w i t h dco'being r e s t r i c t e d dt t o some maximum r a t e i n any fr e q u e n c y r e g i o n of i n t e r e s t . T h i s maximum i s of c o u r s e s e t by the v a r i o u s parameters of the feedback l o o p i n o r d e r t o guarantee s t a b i l i t y . A f t e r any d i s c o n t i n u o u s change, s u f f i c i e n t time must be a l l o w e d f o r the system t o l o c k co0 t o co b e f o r e f u r t h e r o u t p u t d a t a i s c o n s i d e r e d . T h i s time p e r i o d depends upon the s i z e of the d i s c o n t i n u i t y . In p r a c t i c e , t y p i c a l s e t t l i n g t i m e s a re of the o r d e r of s e v e r a l m i c r o s e c o n d s . In p r a c t i c e , co i s ramped from a minimum v a l u e t o a maximum v a l u e and then e i t h e r dropped t o the o r i g i n a l v a l u e or ramped back down a g a i n . The a c t u a l f r e q u e n c y i n c r e m e n t s may be a n a l o g or d i s c r e t e i n n a t u r e as l o n g as the r e s t r i c t i o n s upon co a r e o b s e r v e d . The o u t p u t of the system c o n s i s t s of a v o l t a g e s i g n a l whose a m p l i t u d e i s p r o p o r t i o n a l t o the power l o s s i n the re s o n a n t c i r c u i t . As suc h , i t i s p r o p o r t i o n a l t o the sum of the l o s s e s i n the i n d u c t o r and the c a p a c i t o r and any a d d i t i o n a l l o s s e s e n c o u n t e r e d due t o power a b s o r p t i o n by n u c l e a r s p i n s i n t e r a c t i n g w i t h the f i e l d U- of the i n d u c t o r . That i s , the output i s a s i g n a l p r o p o r t i o n a l t o the t o t a l r e s i s t i v e impedance of the c i r c u i t . I t i s l i n e a r i n x" (co). The b e h a v i o u r d e s c r i b e d above i s i n d e e d the d e s i r e d b e h a v i o u r which we s e t out t o o b t a i n a t the b e g i n i n g of t h i s c h a p t e r . I t i s s t i l l n e c e s s a r y t o make an a d d i t i o n a l q u a l i f i c a t i o n b e f o r e a p p l y i n g t h i s concept t o a p r a c t i c a l system which u t i l i z e s a c o a x i a l c a b l e t o b r i n g the s i g n a l 101 out of the r e f r i g e r a t o r . In the p r a c t i c a l case the c o a x i a l c a b l e would connect the r e s o n a n t c i r c u i t t o the a m p l i f i e r A which i s shown i n f i g u r e 3.3. I f we were t o b l i n d l y s u b s t i t u t e the resonant c i r c u i t impedance R 0 + TJCJLX" (w) f o r Z i n e q u a t i o n 2.11, the imaged impedance Z' seen by the a m p l i f i e r A would o b v i o u s l y not be l i n e a r i n x" . Frequency dependent d i s t o r t i o n s s t i l l e x i s t though they a r e much s m a l l e r than t h o s e c h a r a c t e r i z i n g the uncompensated case ( e q u a t i o n s 2.16 and 2.17). These d i s t o r t i o n s a r e due t o the r e a c t a n c e of the c a b l e as d i s c u s s e d i n c h a p t e r two. I t i s n e c e s s a r y t o t r a n s f o r m the resonant c i r c u i t impedance R 0 + TJCJLX" t o the c a b l e impedance Z c. T h i s i s done by u s i n g an a m p l i f i e r w i t h a c o n s t a n t output impedance which i s matched t o the c a b l e impedance. E q u a t i o n 2.11 i s t h u s reduced t o Z' = Z c < T h i s e n s u r e s t h a t the o u t p u t p o t e n t i a l i s d i r e c t l y p r o p o r t i o n a l t o the t o t a l r e s i s t i v e impedance of t h e r e s o n a n t c i r c u i t and i s t h u s l i n e a r i n x"(o>). The next c h a p t e r c o n s i d e r s the o p e r a t i o n a l c h a r a c t e r i s t i c s of a p r a c t i c a l system i n more d e t a i l . V. CHAPTER 4 In c h a p t e r t h r e e , the t h e o r y of the o p e r a t i o n of a phase compensated NMR system was d e v e l o p e d . I t was d emonstrated t h a t such a system has the p o t e n t i a l t o p r o v i d e an u n d i s t o r t e d measurement of the a b s o r p t i v e NMR l i n e s h a p e of a c o l l e c t i o n of n u c l e a r s p i n s . T h i s type of i n s t r u m e n t has o b v i o u s a p p l i c a t i o n s i n the measurement of both "weak" and r e l a t i v e l y wide NMR l i n e s h a p e s such as those e n c o u n t e r e d i n p o l a r i z e d t a r g e t DMR work. W h i l e the d e u t e r o n t h e r m a l e q u i l i b r i u m p o l a r i z a t i o n i s s m a l l ( e q u a t i o n 1.15), the r e a l o b s t a c l e i n the measurement of the DMR l i n e s h a p e f o r a p o l a r i z e d t a r g e t i s i n the need t o r e t a i n a c o n s t a n t p o l a r i z a t i o n . That i s , the r f power s u p p l i e d t o , or removed from the d e u t e r o n s p i n system must be low enough so as t o e s s e n t i a l l y l e a v e t h e system u n d i s t u r b e d . The t r u e c h a l l e n g e i n the measurement of the t a r g e t p o l a r i z a t i o n l i e s i n the e x p e r i m e n t e r s a b i l i t y t o o b t a i n u s e f u l d a t a from the s m a l l e s t p o s s i b l e p e r t u b a t i o n of the s p i n system. T h i s i s where the phase compensated NMR system d e s c r i b e d e a r l i e r comes i n t o i t s own. By a v o i d i n g d i s t o r t i o n s t o the l i n e s h a p e and removing the r e a c t i v e components of the background s i g n a l , the measurement p r o c e s s i s o n l y l i m i t e d by the minimum a c c e p t a b l e s i g n a l t o n o i s e r a t i o . T h i s c h a p t e r i s concerned w i t h the development of a p r a c t i c a l NMR Q-meter d e s i g n e d a c c o r d i n g t o the p r i n c i p l e s o u t l i n e d i n the f i r s t c h a p t e r s of t h i s t h e s i s . When used i n 102 103 c o n j u n c t i o n w i t h the new TRIUMF P o l a r i z e d Deuterium T a r g e t 2 7 t h i s system has o p e r a t e d i m p r e s s i v e l y i n a demanding e x p e r i m e n t a l e n v i r o n m e n t . 2 8 Emphasis i s not p l a c e d so much upon the a c t u a l c o n s t r u c t i o n and performance of t h i s t a r g e t as i t i s upon the g e n e r a l i z e d b e h a v i o u r of the system. T h i s i s i n kee p i n g w i t h the purpose of t h i s t h e s i s , which i s t o i n t r o d u c e the phase compensation concept and t o p r o v i d e the n e c e s s a r y background i n o r d e r t o i n t e r p r e t i t s b e h a v i o u r i n an e x p e r i m e n t a l environment. W h i l e t h i s concept of phase compensation has found i t s f i r s t a p p l i c a t i o n i n p o l a r i z e d t a r g e t work, t h e r e i s no reason why i t s h o u l d be r e s t r i c t e d t o t h i s t y p e of work. The a c t u a l concept belongs t o the e n t i r e r e alm of NMR system d e s i g n and r f e l e c t r o n i c s . I t i s hoped t h a t the i d e a s p r e s e n t e d i n t h i s t h e s i s w i l l prove t o be u s e f u l i n f u r t h e r developments i n the s e a r e a s . A. THE TRIUMF DMR ELECTRONICS PACKAGE The TRIUMF system i s d e s i g n e d w i t h an f 16.6 MHz which c o r r e s p o n d s t o a s t a t i c magnetic f i e l d of B 0 - 2.5 T e s l a f o r d e u t e r o n s . The t a r g e t c o n s i s t s of s e v e r a l cm 3 of d e u t e r a t e d m a t e r i a l ( f o r example, 1.3 cm 3 of d e u t e r a t e d 2 7 G . D. W a i t , J . V. C r e s s w e l l , P. P. J . D e l h e i j , M. Hayden, D. C. H e a l e y , G. Waters, "The TRIUMF P o l a r i s e d Deuteron T a r g e t NMR System," t o be p u b l i s h e d i n the P r o c e e d i n g s of  the 5 t h I n t e r n a t i o n a l Workshop on P o l a r i z e d Sources and  T a r g e t s , Montana, S w i t z e r l a n d , 1986. 2 8 G . R. Smith e t a l . , "Measurement of Tensor O b s e r v a b l e s i n the ( r r + , 3 ) E l a s t i c S c a t t e r i n g R e a c t i o n , " TRIUMF Experiment 337 ( t o be p u b l i s h e d , 1986.). 104 b u t a n o l ( C a D 1 0 0 ) beads f o r the experiment of Smith e t a l . 2 9 ). The c o i l f o r t h i s f i r s t experiment c o n s i s t e d of 48 t u r n s of 0.01 cm d i a m e t e r copper w i r e wound on a r e c t a n g u l a r former whose dimensions were 15 mm x 16 mm x 5 mm. The c o i l was bonded t o a t h i n l a y e r of FEP t e f l o n used t o p r o v i d e s t r u c t u r a l r i g i d i t y . The b u t a n o l beads were h e l d i n p l a c e by the c o i l f ormer. The r e s o n a n t c i r c u i t c o n s i s t e d of t h i s c o i l , L, t o g e t h e r w i t h the p a r a l l e l c o m b i n a t i o n of C = 1 pF and two NE41137 FET's connected i n p a r a l l e l so as t o u t i l i z e the gat e t o d r a i n and gate t o so u r c e c a p a c i t a n c e s ( r e f e r t o f i g u r e 4.1). These c a p a c i t a n c e s are c o n t r o l l e d by the dc v o l t a g e a p p l i e d t o the gate as d i s c u s s e d i n appendix A. The b i a s i n g of t h e s e v a r a c t o r d e v i c e s i s done v i a a 1 MQ r e s i s t o r . A p p l i c a t i o n of v o l t a g e s from -1.0 V t o 0.5 V r e s u l t e d i n a change i n the r e s o n a n t f r e q u e n c y f 0 from 16.3 MHz t o 17.5 MHz. The r e s o n a n t c i r c u i t was l o c a t e d i n the m i x i n g chamber of the d i l u t i o n r e f r i g e r a t o r , o p e r a t i n g a t t e m p e r a t u r e s between 70 mK and 1.6 K. The c o n s t a n t c u r r e n t r f i n p u t was. p r o v i d e d v i a the 1 K$2 r e s i s t o r , R , l o c a t e d near the condenser of the ' c c ' r e f r i g e r a t o r . O p e r a t i n g t e m p e r a t u r e s i n t h i s r e g i o n a r e t y p i c a l l y 1.6 K. F o l l o w i n g R c c were two s e r i e s r e s i s t a n c e s of 27 Q and 10 r e s p e c t i v e l y . The 10 $2 r e s i s t o r (a Q 2 9 The f i r s t use of t h i s t a r g e t was an experiment t o i n v e s t i g a t e the s p i n dependence of the (7r +,3) s c a t t e r i n g and a b s o r p t i o n r e a c t i o n s . The 1 mm d i a m e t e r beads c o n t a i n e d 5% D 20 doped w i t h 6 x 1 0 1 9 m o l e c u l e s per ml of EHBA as a source of paramagnetic i m p u r i t i e s f o r the dynamic p o l a r i z a t i o n of the d e u t e r a t e d m a t e r i a l . 105 V R F INPUT DELAY Qf \ ' /AMPI PLIFIER & LIMITER 3" PHASE DETECTOR TO SIGNAL DETECTOR "DC"AMPLIFER "AC" AMPLIFIER Vb >" PHASE AMPLIFIER IK I -J | — E 7 2 0 8 4 -56&I I.6K K m E72084 IOOK SIGNAL AMPLIFIERHT1 OOK I CONDENSER ELECTRONICS PMIXING CHAMBER ELECTRONICS OIK .o^n I80JZ, 27& IM & - J - vW-l/i F I2PF t o i l 10/xH SAMPLE COIL 1 P V^NE4II37 IpF F i g u r e 4 . 1 The TRIUMF NMR E l e c t r o n i c s (The components i n s i d e t h e dashed l i n e s a r e l o c a t e d i n s i d e the d i l u t i o n r e f r i g e r a t o r ) 106 s p o i l i n g r e s i s t a n c e ) was used t o i n c r e a s e the c o i l r e s i s t a n c e R 0 as seen by the s i g n a l d e t e c t o r . That i s , t h i s t o t a l r e s i s t a n c e r e p r e s e n t s a c o n s t a n t term which i s added t o the background l e v e l . In the absence of phase compensation, t h i s Q s p o i l i n g r e s i s t a n c e reduced the Q of the r e s o n a n t c i r c u i t t o about 60. Under n e g a t i v e p o l a r i z a t i o n c o n d i t i o n s the a b s o r p t i v e (x") term i n e q u a t i o n 2.8 appears as a n e g a t i v e r e s i s t a n c e . Without the a d d i t i o n a l r e s i s t a n c e p r o v i d e d by the 10 0 r e s i s t o r i t would be p o s s i b l e f o r a s i g n a l from a n e g a t i v e l y p o l a r i z e d t a r g e t t o a c t u a l l y overmodulate the c a r r i e r s i g n a l . I f the phase s i g n a l was a c t u a l l y t a k e n from t h i s same p o i n t and o v e r m o d u l a t i o n o c c u r r e d , the e n t i r e l o c k i n g system would f a i l . O v e r m o d u l a t i o n r e s u l t s i n the i n t r o d u c t i o n of a 180° phase s h i f t i n t o the ou t p u t s i g n a l . W i th r e f e r e n c e t o comments made i n c h a p t e r t h r e e , t h i s has the e f f e c t of r e f l e c t i n g the v s . vxV p l o t t h r o u g h the o r i g i n and hence i n t e r c h a n g i n g a l l m e t a s t a b l e and s t a b l e s o l u t i o n s of e q u a t i o n 3.6. As a r e s u l t of the o v e r m o d u l a t i o n , f 0 i s no l o n g e r l o c k e d t o the i n p u t f r e q u e n c y . The output s i g n a l i s , of c o u r s e , no l o n g e r p r o p o r t i o n a l t o the magnetic resonance induced impedance m o d u l a t i o n and hence the i n t e g r a t e d l i n e s h a p e i s not p r o p o r t i o n a l t o the s p i n p o l a r i z a t i o n . Removal of the o v e r m o d u l a t i o n c o n d i t i o n i s u s u a l l y not s u f f i c i e n t t o r e s t o r e the system t o normal o p e r a t i n g c o n d i t i o n s as t h e r e i s the l i k e l y p o s s i b i l i t y t h a t the 1 07 system w i l l remain p i n n e d t o an anomalous s t a t e ( i . e . a h i g h l y n o n l i n e a r o p e r a t i n g r e g i o n where t h e r e i s an i n v a l i d s o l u t i o n t o e q u a t i o n 3.6). For t h e s e reasons the phase s i g n a l i s not taken from the same p o i n t as the o u t p u t . The second r e s i s t o r (27 fi) was used t o r a i s e the l e v e l of the c a r r i e r f e d t o the phase d e t e c t o r system t o an a p p r o p r i a t e l e v e l . T h i s had the e f f e c t of r a i s i n g the e f f e c t i v e R 0 as seen by the phase d e t e c t o r . W i t h r e f e r e n c e t o e q u a t i o n s 3.3 and 3.6 t h i s i s e q u i v a l e n t t o an e f f e c t i v e d e c r e a s e i n the l o o p g a i n . That i s , the measured phase iiV i s d e c r e a s e d w i t h r e s p e c t t o the a c t u a l phase <p. The r e s u l t i s a d e c r e a s e i n the l o c k i n g range of the d e v i c e ( i . e . a l a r g e r e r r o r between f 0 and the d r i v i n g f r e q u e n c y f was t o l e r a t e d ) . O b v i o u s l y a compromise must be made i n the c h o i c e of the v a l u e of t h i s r e s i s t o r between the l o o p g a i n and the minimum a c c e p t a b l e phase s i g n a l l e v e l . Care must a l s o be taken t o a v o i d o v e r m o d u l a t i o n of the c a r r i e r . Recent improvements i n the subsequent a m p l i f i c a t i o n s t a g e s suggest t h a t t h i s r e s i s t o r does not need t o be as l a r g e as 27 0. I d e a l l y one would want t o p l a c e the a m p l i f i e r s as c l o s e t o the c o i l as p o s s i b l e . Each of the a m p l i f i e r s however, d i s s i p a t e d on the o r d e r of 1 mW of e l e c t r i c a l power. T h i s d i s s i p a t i o n l e d t o an u n c e r t a n t y i n the t a r g e t t emperature d u r i n g the measurement of a TE s i g n a l due t o the p r o x i m i t y of t h e s e heat s o u r c e s t o the thermometers. In a d d i t i o n , when o p e r a t i n g i n the f r o z e n s p i n mode, l o c a l i z e d h e a t i n g can cause an inhomogeneous d e p o l a r i z a t i o n of the n u c l e a r s p i n s . 108 I n o r d e r t o a v o i d t h e s e p r o b l e m s , t h e a m p l i f i e r s w e r e p l a c e d n e a r t h e c o n d e n s e r o f t h e r e f r i g e r a t o r . T h i s n e c e s s i t a t e d t h e u s e o f t w o 40 cm l e n g t h s o f 50 ft B e C u s e m i r i g i d c o a x i a l c a b l e t h r o u g h w h i c h t h e p h a s e a n d i n p u t / o u t p u t s i g n a l s c o u l d b e c a r r i e d f r o m t h e m i x i n g c h a m b e r . A t 16 MHz t h i s l e n g t h r e p r e s e n t s o n l y a f e w p e r c e n t o f a w a v e l e n g t h . T h u s , t h e p h a s e s h i f t s w h i c h a r e i n t r o d u c e d b y t h e s e c a b l e s o v e r a 500 k H z s c a n w e r e o n l y a f r a c t i o n o f a d e g r e e . I n a d d i t i o n t o t h e s e c a b l e s , h o w e v e r , t h e r e a l s o e x i s t m i s m a t c h e d i m p e d a n c e s w h i c h w e r e n o t a c c o u n t e d f o r i n t h e d i s c u s s i o n l e a d i n g t o e q u a t i o n s 3 . 2 , 3 . 3 a n d 3 . 6 . T h e n e t r e a c t a n c e t h a t i s i n t r o d u c e d c a u s e s f r e q u e n c y d e p e n d e n t p h a s e s h i f t s w h i c h h a v e n o t b e e n a c c o u n t e d f o r . T h e r e s u l t i s a s m a l l a d d i t i o n a l s h i f t b e t w e e n t h e p h a s e s ^ a n d <j> w h i c h d o e s n o t n e c e s s a r i l y g o t o z e r o a s <j> g o e s t o z e r o . T h u s a s m a l l e r r o r w h i c h c a n n o t b e c o m p l e t e l y c a n c e l l e d i s i n t r o d u c e d i n t o t h e c o m p e n s a t i o n s y s t e m . T h e a v e r a g e e r r o r , h o w e v e r , c a n s t i l l b e c a n c e l l e d o u t b y a s h i f t i n t h e r e l a t i v e p h a s e b e t w e e n t h e t w o i n p u t a r m s o f t h e p h a s e d e t e c t o r . T h e r e m a i n i n g u n c a n c e l l e d e r r o r s t i l l i m p o s e s a n a d d i t i o n a l l i m i t u p o n t h e l o c k i n g r a n g e o f t h e s y s t e m . T h e r e a c t a n c e w h i c h i s i n t r o d u c e d i n t o t h e i m a g e d c o i l i m p e d a n c e c a u s e s a n a d d i t i o n a l m o d u l a t i o n t o b e s u p e r i m p o s e d u p o n t h e b a c k g r o u n d . E s t i m a t e s f o r t h e T R I U M F s y s t e m p l a c e a n u p p e r l i m i t o f a b o u t 3% u p o n t h e m a ximum c h a n g e i n t h e m a g n i t u d e o f t h e b a c k g r o u n d i m p e d a n c e ( 5 0 0 k H z d e v i a t i o n f r o m f ) d u e t o t h e sum o f t h e s e e f f e c t s a n d t h e c h a n g i n g 109 p o w e r d i s s i p a t i o n i n t h e v a r a c t o r ( s e e a p p e n d i x A ) . M e a s u r e m e n t s o f o u t p u t p h a s e a n g l e s ( t o b e d i s c u s s e d ) s u g g e s t t h a t t h e b u l k o f t h i s e f f e c t i s d u e t o t h e c h a n g i n g v a r a c t o r i m p e d a n c e . A l t h o u g h t h e e x i s t i n g c a b l e s h a v e a 50 fi c h a r a c t e r i s t i c i m p e d a n c e , i t h a s b e e n s u g g e s t e d t h a t c h a n g i n g t h e m t o 7 J2 c a b l e s w o u l d h a v e t h e e f f e c t o f r e d u c i n g t h e n e t i n d u c t a n c e i n p a r a l l e l w i t h t h e c o i l a n d h e n c e o f i n c r e a s i n g t h e e f f e c t i v e f i l l i n g f a c t o r r j . B o t h t h e p h a s e s i g n a l a n d t h e o u t p u t s i g n a l a r e a m p l i f i e d b e f o r e l e a v i n g t h e c r y o s t a t . T h i s i s d o n e i n o r d e r t o d r i v e t h e s i g n a l s t h r o u g h s e v e r a l m e t r e s o f c a b l e t o t h e e x t e r n a l e l e c t r o n i c s . T h i s l e n g t h o f c a b l e i s m ade n e c e s s a r y b y t h e e x p e r i m e n t a l e n v i r o n m e n t . B o t h a m p l i f i e r s c o n s i s t o f common s o u r c e a m p l i f i e r s b a s e d u p o n N E 7 2 0 8 4 M E S F E T ' s . T h e 1 mW p o w e r d i s s i p a t i o n c h a r a c t e r i z i n g e a c h a m p l i f i e r i s e a s i l y h a n d l e d a t t h e c o n d e n s e r s t a g e o f t h e r e f r i g e r a t o r . I n e a r l y v e r s i o n s o f t h i s p r o t o t y p e s y s t e m , p h a s e d e t e c t i o n w a s a c h i e v e d b y c o m m u t a t i n g t h e s i g n a l f r o m t h e p h a s e d e t e c t o r w i t h a r e f e r e n c e s i g n a l f e d d own i n t o t h e c r y o s t a t o n a n a d d i t i o n a l c o a x i a l c a b l e . T h e p h a s e o f t h e r e f e r e n c e s i g n a l was o f c o u r s e a r b i t r a r y . T h e c o m m u t a t e d s i g n a l w a s t h e n i n t e g r a t e d t o p r o v i d e a d c s i g n a l p r o p o r t i o n a l t o t h e d e t e c t e d p h a s e b e t w e e n t h e i n p u t s i g n a l s . T h e a c t u a l c o m m u t a t i o n was d o n e b y s w i t c h i n g a n N E 4 1 1 3 7 d u a l g a t e M E S F E T f e d b y a c u r r e n t s o u r c e . T h i s s y s t e m i s s h o w n i n f i g u r e 4 . 2 . 110 77) PHASE ANGLE REFERENCE SIGNAL _ NE72084 F i g u r e 4.2 E a r l y P h a s e D e t e c t i o n S y s t e m I t s h o u l d b e n o t e d t h a t b o t h t h e p h a s e a n d t h e r e f e r e n c e s i g n a l s m u s t b e a b l e t o t u r n t h e N E 4 1 1 3 7 c o m p l e t e l y o n a n d o f f . A n y a m p l i t u d e d e p e n d e n c e o f t h e i n p u t s i g n a l s i s e a s i l y m i s i n t e r p r e t e d a s a p h a s e s i g n a l . 111 W h i l e p e r f o r m a n c e o f t h i s s y s t e m was e x c e l l e n t i n t r i a l r u n s , i t was s e t a s i d e i n f a v o r o f a l e s s d i s s i p a t i v e s y s t e m . T h a t i s , p h a s e d e t e c t i o n i s now p e r f o r m e d e x t e r n a l t o t h e r e f r i g e r a t o r w i t h t h e r e q u i s i t e a d d i t i o n o f s e v e r a l m e t r e s o f c a b l e a n d h e n c e t h e a d d i t i o n o f p o s s i b l e p h a s e e r r o r s . T h e c u r r e n t T R I U M F s y s t e m u t i l i z e s a s t a n d a r d d o u b l e b a l a n c e d m i x e r t y p e p h a s e d e t e c t o r ( M i n i C i r c u i t s m o d e l S R A - 1 ) . T h e t w o s i g n a l s a r e f i r s t p a s s e d t h r o u g h s a t u r a t e d a m p l i f i e r s t o c r e a t e s q u a r e w a v e s r e m o v i n g a n y a m p l i t u d e d e p e n d e n c e f r o m t h e r e s u l t i n g p h a s e s i g n a l . T h e i n p u t s i g n a l s a r e f e d i n t o t h e L O a n d R F a r m s o f t h e m o d u l a t o r w i t h t h e r e s u l t i n g p h a s e s i g n a l b e i n g t a k e n f r o m t h e I N T a r m . T h i s s i g n a l i s t h e n i n t e g r a t e d t o d e t e c t t h e d c F o u r i e r c o m p o n e n t o f t h e s i g n a l w h i c h i s p r o p o r t i o n a l t o t h e p h a s e b e t w e e n t h e t w o i n p u t s . I n e a r l i e r v e r s i o n s o f t h e p r o t o t y p e t h e p h a s e s i g n a l w a s a m p l i f i e d b y a n a m p l i f i e r w i t h a t i m e c o n s t a n t o f a b o u t 10 MS. D i f f i c u l t i e s w e r e e n c o u n t e r e d i n t h a t f was o b s e r v e d t o d r i f t . T h i s h a d t h e e f f e c t o f s h i f t i n g t h e e n t i r e l o c k i n g r a n g e i n t h e same m a n n e r . T h i s i s e q u i v a l e n t t o a s h i f t i n g b a c k g r o u n d o r a l o w f r e q u e n c y s o u r c e o f n o i s e t r a n s l a t e d f r o m t h e f r e q u e n c y d o m a i n i n t o t h e a m p l i t u d e d o m a i n . T h e e f f e c t i s a t i t s w o r s t n e a r t h e l i m i t s o f t h e l o c k i n g r a n g e w h e r e t h e d i f f e r e n c e b e t w e e n f a n d t h e i n p u t f r e q u e n c y i s a m a x i m u m . 1 1 2 T h e p o s s i b i l i t y o f t h i s d r i f t b e i n g c o r r e l a t e d t o a n e n v i r o n m e n t a l c h a n g e w h i c h r e s u l t e d i n a f l u c t u a t i o n i n t h e c h a r a c t e r i s t i c s o f some c o m p o n e n t o f t h e c i r c u i t w as i n v e s t i g a t e d . W h i l e e n v i r o n m e n t a l f l u c t u a t i o n s w e r e b e l i e v e d t o c a u s e p a r t o f t h e p r o b l e m , t h e e f f e c t c o u l d n o t b e b l a m e d s o l e l y u p o n t h i s c a u s e . A s e c o n d a m p l i f i e r w i t h a m u c h l o n g e r t i m e c o n s t a n t ( 0 . 5 s ) was a d d e d i n p a r a l l e l t o t h e f i r s t i n o r d e r t o s t a b i l i z e t h e s y s t e m . T h i s s o l v e d t h e l o n g t e r m s t a b i l i t y p r o b l e m , a l l o w i n g t h e s y s t e m t o b e r u n f o r a t l e a s t t h r e e w e e k s w i t h o u t t h e n e e d f o r a d j u s t m e n t o f a n y p a r a m e t e r s . 3 0 T h e f a s t a m p l i f i e r i s r e f e r r e d t o a s t h e a c a m p l i f i e r w h i l e t h e s l o w a m p l i f i e r i s known a s t h e d c a m p l i f i e r . A s i n d i c a t e d i n f i g u r e 4 . 1 , i s a d d e d t o t h e v a r a c t o r b i a s d u r i n g t h e a m p l i f i c a t i o n o f t h e p h a s e e r r o r s i g n a l . A d j u s t m e n t s t o t h e v a l u e o f v" D r e s u l t i n a s h i f t i n f a n d h e n c e p r o v i d e t h e a b i l i t y t o t u n e t h e o p e r a t i n g f r e q u e n c y r a n g e . T h e common p r o c e d u r e f o r c l o s i n g t h e f e e d b a c k l o o p , o n c e a l l d e l a y s h a v e b e e n s e t t o t h e i r a p p r o p r i a t e v a l u e s , i s now p r e s e n t e d . T h e a c g a i n i s s e t t o z e r o a n d t h e d c g a i n i s d i s c o n n e c t e d f r o m t h e l o o p v i a a s w i t c h p r o v i d e d f o r t h a t p u r p o s e . i s a d j u s t e d w h i l e o b s e r v i n g t h e o u t p u t Q c u r v e o n a n o s c i l l o s c o p e o r s p e c t r u m a n a l y z e r u n t i l t h e c i r c u i t 3 0 G. D. W a i t , J . V . C r e s s w e l l , P. P. J . D e l h e i j , M. H a y d e n , D. C. H e a l e y , G. W a t e r s , " T h e T R I U M F P o l a r i s e d D e u t e r o n T a r g e t NMR S y s t e m , " t o b e p u b l i s h e d i n t h e P r o c e e d i n g s o f  t h e 5 t h I n t e r n a t i o n a l W o r k s h o p o n P o l a r i z e d S o u r c e s a n d  T a r g e t s , M o n t a n a , S w i t z e r l a n d , 1 9 8 6 . 1 13 r e s o n a t e s a t t h e d e s i r e d f . c T u r n i n g t h e a c g a i n u p h a s t h e e f f e c t o f l o w e r i n g t h e Q o f t h e r e s o n a n t c i r c u i t , f l a t t e n i n g t h e Q c u r v e b y b r i n g i n g t h e w i n g s o f t h e s i g n a l down t o t h e same l e v e l a s t h e r e s o n a n t l e v e l . T h e a c g a i n i s i n c r e a s e d u n t i l t h e d e s i r e d l o c k i n g r a n g e i s o b t a i n e d . T h i s r a n g e i s t y p i c a l l y 1 MHz o r m o r e i n t h e T R I U M F s y s t e m a n d i s s e t b y t h e p o i n t a t w h i c h t h e v a r a c t o r l e a v e s t h e a c c u m u l a t i o n r e g i o n . T o o l a r g e a v a l u e o f G c a r r i e s t h e d a n g e r o f o s c i l l a t o r y i n s t a b i l i y o r l o c a l i z e d o s c i l l a t i o n s i f , a t a n y f r e q u e n c y , t h e g a i n b e c o m e s t o o h i g h f o r t h e c o m p e n s a t i o n s y s t e m t o r e s p o n d q u i c k l y e n o u g h . V a r i a t i o n o f l o o p g a i n w i t h v a r a c t o r b i a s w i l l b e d i s c u s s e d i n g r e a t e r d e t a i l l a t e r i n t h i s c h a p t e r . I f t h e s e o s c i l l a t i o n s a r e n o t d a m p e d o u t , w i d e s p r e a d o s c i l l a t i o n s w i l l d e v e l o p . T y p i c a l l y , G i s i n c r e a s e d t o a l e v e l w h e r e t h e o n s e t o f o s c i l l a t o r y i n s t a b i l i t y i s i m m i n e n t s o a s t o l o c k f 0 t o t h e i n p u t f r e q u e n c y a s t i g h t l y a s p o s s i b l e , a n d t h e n b a c k e d o f f a s m a l l a m o u n t s o a s t o a l l o w t h e s y s t e m t o r e c o v e r f r o m s p u r i o u s n o i s e s i g n a l s . I n t h e T R I U M F s y s t e m , G i s o f t h e o r d e r o f 350 y i e l d i n g a n a c l o o p g a i n o f a b o u t 1 1 . T h e b a c k g r o u n d c a n b e s e e n t o d r i f t r a n d o m l y a t t h i s p o i n t i n t h e l o c k i n g p r o c e d u r e . A f t e r s w i t c h i n g o n t h e d c f e e d b a c k , h o w e v e r , t h e b a c k g r o u n d i s s t a b i l i z e d . T h e d c g a i n i s u s u a l l y s e t t o a s t a n d a r d v a l u e o f a b o u t 1 2 0 0 . T h i s y i e l d s a d c l o o p g a i n o f a b o u t 3 9 . C h a n g e s i n t h i s v a l u e a r e n o t u s u a l l y n e c e s s a r y o v e r a w i d e r a n g e o f o p e r a t i n g 1 1 4 f r e q u e n c i e s . T h e l o c k i n g p r o c e d u r e i s d e m o n s t r a t e d i n t h e p h o t o g r a p h s o f f i g u r e s 4.3 a n d 4 . 4 . F i g u r e 4.3 s h o w s a t y p i c a l 1 6 . 5 MHz s i g n a l b e i n g s c a n n e d o v e r a 5 1 2 k H z r a n g e w i t h o u t p h a s e c o m p e n s a t i o n . T h e o s c i l l o s c o p e i s t r i g g e r e d o n t h e r e f e r e n c e s i g n a l . T h e o b s e r v e d v a r i a t i o n i n p h a s e a n g l e i s r o u g h l y 3 5 % o f o n e p e r i o d o r 1 2 6 ° (0.77T r a d i a n s ) . I n a d d i t i o n t o t h e p h a s e d e p e n d e n c e , i t i s a l s o c l e a r t h a t t h e a m p l i t u d e o f t h e o u t p u t c h a n g e s d u e t o t h e Q o f t h e r e s o n a n t c i r c u i t . F i g u r e 4.3 O u t p u t W a v e f o r m S h o w i n g P h a s e V a r i a t i o n ( n o c o m p e n s a t i o n ) F i g u r e 4.4 s h o w s a s i m i l a r s c a n a f t e r l o c k i n g t h e r e s o n a n t f r e q u e n c y t o t h e i n p u t f r e q u e n c y . T h e r e i s n o o b s e r v a b l e p h a s e s h i f t o r a m p l i t u d e d e p e n d e n c e , i n d i c a t i n g t h a t t h e r e s o n a n t f r e q u e n c y h a s i n d e e d b e e n f o r c e d t o t r a c k 115 F i g u r e 4.4 Output Waveform Showing Phase V a r i a t i o n ( w i t h phase compensation) the i n p u t f r e q u e n c y . The same e f f e c t can be seen i n the f r e q u e n c y domain as a f l a t t e n i n g of t h e Q c u r v e . T h i s i s shown i n f i g u r e s 4.5 and 4.6 which a r e photographs of the same s i t u a t i o n s as o b s e r v e d on a spectrum a n a l y z e r . F i g u r e 4.5 shows t h e uncompensated s i g n a l w h i l e f i g u r e 4.6 shows the e f f e c t of c l o s i n g t h e feedback l o o p . The v e r t i c a l s c a l e i s 2 dB per d i v i s i o n . T h e o r e t i c a l c a l c u l a t i o n s based upon e q u a t i o n 3.6 p r e d i c t a maximum phase d e v i a t i o n of about 0.75° when f i s 500 kHz from f c f o r t h i s system. T h i s r e p r e s e n t s an improvement by a f a c t o r of about 84 i n the maximum phase d e v i a t i o n over the uncompensated system. Another d e v i a t i o n of the same o r d e r of magnitude must be added t o t h i s i n o r d e r t o account f o r t h e phase s h i f t i n t r o d u c e d by the 40 cm > -40C3T. 0016 HHZ 300PHZ ?£S F i g u r e 4.5 T h e U n c o m p e n s a t e d Q C u r v e j I > ~40a3ir 0016 MHZ 30OPHZ PES F i g u r e 4.6 T h e E f f e c t o f P h a s e C o m p e n s a t i o n i n t h e F r e q u e n c y D o m a i n 1 17 c a b l e s b e t w e e n t h e m i x i n g c h a m b e r a n d t h e c o n d e n s e r . T h i s b r i n g s t h e t o t a l e x p e c t e d p h a s e d e v i a t i o n t o a b o u t 1 . 5 ° . M e a s u r e m e n t s b a s e d u p o n o b s e r v a t i o n o f t h e o u t p u t w a v e f o r m s o f t h e s y s t e m p l a c e t h i s p h a s e d e v i a t i o n a s b e i n g l e s s t h a n a b o u t 2 . 5 ° . B. P O S T NMR E L E C T R O N I C S P R O C E S S I N G T h e a m p l i t u d e o f t h e o u t p u t v o l t a g e s i g n a l w h i c h i s a m p l i f i e d n e a r t h e c o n d e n s e r o f t h e 3 H e / " H e d i l u t i o n r e f r i g e r a t o r , i s p r o p o r t i o n a l t o t h e r e s i s t i v e i m p e d a n c e o f t h e r e s o n a n t c i r c u i t a t a n y f r e q u e n c y w i t h i n t h e l o c k i n g r a n g e o f t h e s y s t e m . I t i s t h i s s i g n a l w h i c h i s p r o c e s s e d a n d a n a l y z e d i n o r d e r t o m e a s u r e t h e NMR l i n e s h a p e a n d h e n c e t h e s a m p l e p o l a r i z a t i o n . A f t e r b e i n g b r o u g h t o u t o f t h e c r y o s t a t o n a l e n g t h o f B e C u c o a x i a l c a b l e t h e s i g n a l i s f u r t h e r a m p l i f i e d (91 d B ) a n d t h e n f e d i n t o a n I n t e r f a c e S t a n d a r d s m o d e l A D S - 1 2 0 a n a l o g u e - t o - d i g i t a l c o n v e r t e r . T h i s A t o D c o n v e r t e r , a s w e l l a s t h e m o d e l 5 6 0 0 R o c k l a n d f r e q u e n c y s y n t h e s i z e r w h i c h a c t s a s t h e f r e q u e n c y s o u r c e , a r e g a t e d b y t h e T R I U M F 8 0 8 5 m i c r o p r o c e s s o r ( T R I M A C 3 1 ) . T h e R o c k l a n d s y n t h e s i z e r i s p r o g r a m m e d t o p r o d u c e 83 c o m p l e t e f r e q u e n c y s c a n s p e r s e c o n d w i t h e a c h s c a n c o n s i s t i n g o f 2 5 6 f r e q u e n c y i n c r e m e n t s . I t s h o u l d b e n o t e d t h a t t h i s s c a n r a t e i s l i m i t e d b y t h e c y c l e t i m e o f t h e T R I M A C a n d n o t b y t h e r e s p o n s e t i m e o f t h e NMR e l e c t r o n i c s . 3 1 C . G l a z i e r , T R I M A C U s e r s M a n u a l , T R I U M F A - 1 3 7 4 1 1 18 E l e c t r o n i c s r f Frequency Synthesizer Output R c g l i t e r < 256 K ROC Gate R e s o n a n t C i r c u i t D i l u t i o n R e f r i g e r a t o r E l e c t r o n i c s associated with teaperature •easureaent > Caaac Oatauay Microwave Frequency Counter one nicrovsve Source ( flux Control Bus Starburst J-11 TRItlRC I TRIMflC Keyboard Crate C o n t r o l l e r POP-I1 F i g u r e 4.7 The E l e c t r o n i c s A s s o c i a t e d with the NMR System 1 19 A f t e r e a c h f r e q u e n c y i n c r e m e n t t h e NMR s y s t e m i s a l l o w e d e n o u g h t i m e t o c o m e t o e q u i l i b r i u m ( b r i n g t h e c i r c u i t i n t o r e s o n a n c e ) a f t e r w h i c h t i m e t h e A t o D c o n v e r s i o n i s c a r r i e d o u t . T h e T R I M A C t h e n i n c r e m e n t s t h e f r e q u e n c y a n d t h e p r o c e s s i s r e p e a t e d . O n c e t h e p h a s e c o m p e n s a t i o n s y s t e m i s i n o p e r a t i o n t h e S/N r a t i o f o r a T E s i g n a l i s a b o u t 1:5. T h i s n e c e s s i t a t e s c o n s i d e r a b l e s i g n a l a v e r a g i n g i n o r d e r t o o b t a i n s t a t i s t i c a l l y s i g n i f i c a n t i n f o r m a t i o n . O n e d a t a s e q u e n c e c o n s i s t s o f 1024 s c a n s o f 2 5 6 f r e q u e n c y i n c r e m e n t s r e c o r d e d o v e r t h e p e r i o d o f 1 2 . 5 s e c o n d s . T h e m e m o r y o f t h e d i g i t i z e r i s t h e n r e a d b y a S t a r b u r s t J - 1 1 m i c r o p r o c e s s o r . 3 2 A s s o o n a s t h i s s e q u e n c e i s r e c o r d e d , t h e T R I M A C r e s u m e s t h e d a t a t a k i n g p r o c e s s . W h i l e t h e n e x t s e t o f d a t a i s b e i n g m e a s u r e d , t h e S t a r b u r s t s p e n d s 5 s e c o n d s a v e r a g i n g t h e p r e v i o u s s e t o f d a t a i n t o 2 5 6 b i n s c o r r e s p o n d i n g t o t h e v a r i o u s f r e q u e n c y v a l u e s . A f t e r t h i s a v e r a g i n g t h e d a t a i s p a s s e d o n t o t h e h o s t L S I - 1 1 c o m p u t e r f o r s t o r a g e a n d f u r t h e r p r o c e s s i n g . I t i s a t t h i s l a s t s t a g e t h a t t h e e x p e r i m e n t e r i n t e r a c t s w i t h t h e s y s t e m , s e l e c t i n g s c a n w i d t h s , r a t e s , a n d t i m e l e n g t h s . V a r i o u s s p e c t r a c o r r e s p o n d i n g t o b a c k g r o u n d s o r s i g n a l s ( t h e r m a l e q u i l i b r i u m o r e n h a n c e d ) c a n b e r e c o r d e d . B a c k g r o u n d s c a n t h e n b e s u b t r a c t e d f r o m t h e s i g n a l d a t a , b i n b y b i n . T h e r e s u l t i n g s p e c t r u m h a s a n a r e a 3 2 G . W a t e r s , G. D. W a i t , D. A . H u t c h e o n , " E x p e r i e n c e w i t h t h e C E S S t a r b u r s t , " I E E E 1 9 8 5 N u c l e a r S c i e n c e S y m p o s i u m , S a n F r a n c i s c o C a l i f o r n i a . 1 20 p r o p o r t i o n a l t o t h e p o l a r i z a t i o n o f t h e n u c l e a r s p i n s . M e a s u r e m e n t o f s u c h a s p e c t r u m f o r a T E s p i n s y s t e m a n d f o r a d y n a m i c a l l y p o l a r i z e d s y s t e m p r o v i d e s a d i r e c t m e a s u r e m e n t o f t h e r e l a t i v e p o l a r i z a t i o n o f t h e t w o s y s t e m s ( s e e e q u a t i o n 1 . 4 5 ) . F u r t h e r o f f - l i n e a n a l y s i s i s c a r r i e d o u t u t i l i z i n g f i t t i n g p r o g r a m s . H o w e v e r , t h e e x p e r i m e n t e r h a s a t h i s o r h e r i m m e d i a t e d i s p o s a l s e v e r a l l e a s t s q u a r e s f i t t i n g r o u t i n e s f o r e s t i m a t i n g t h e b a s e l i n e o f t h e s i g n a l . B y f i t t i n g a l i n e a r o r q u a d r a t i c c u r v e t o t h e w i n g s o f t h e s i g n a l w h e r e t h e s i g n a l l e v e l i s a s s u m e d t o b e n e g l i g i b l e , a n e s t i m a t e o f t h e b a c k g r o u n d c a n b e o b t a i n e d . W i t h t h i s b a c k g r o u n d s u b t r a c t e d , t h e a r e a o f t h e s i g n a l c a n t h e n b e c a l c u l a t e d a n d h e n c e t h e r e l a t i v e p o l a r i z a t i o n o f v a r i o u s s i g n a l s c a n b e c a l c u l a t e d i m m e d i a t e l y . P r o v i s i o n i s made f o r a n y c o m b i n a t i o n o f s i g n a l , b a c k g r o u n d o r f i t t e d d a t a t o b e d i s p l a y e d a n d c o n t i n u o u s l y u p d a t e d o n a v i d e o s c r e e n . A b s o l u t e p o l a r i z a t i o n c a n a l s o b e p l o t t e d a s a f u n c t i o n o f t i m e . T a k i n g i n t o a c c o u n t a l l p r o c e s s e s , t h e a v e r a g e r a t e o f s i g n a l p r o c e s s i n g i s a b o u t 60 c y c l e s p e r s e c o n d . A t y p i c a l s e t o f 1 0 2 4 s c a n s t h u s r e q u i r e s a b o u t 17 s e c o n d s t o c a r r y o u t . 121 C. F A C T O R S WHICH C A U S E D E V I A T I O N S FROM I D E A L B E H A V I O U R . T h e r e a r e s e v e r a l f a c t o r s w h i c h c a u s e d e v i a t i o n s f r o m t h e i d e a l b e h a v i o u r o u t l i n e d i n c h a p t e r t h r e e . T h r e e o f t h e s e f a c t o r s p l a y e d a n i m p o r t a n t r o l e i n t h e d e v e l o p m e n t o f t h e T R I U M F p r o t o t y p e . T h e s e f a c t o r s a r e d i s c u s s e d b e l o w . 1. T H E I N F L U E N C E OF T H E N O N - I D E A L VARACTOR A s d i s c u s s e d i n a p p e n d i x A, t h e b i a s - d e p e n d e n t l e a k a g e c u r r e n t i n t h e v a r a c t o r d e v i c e l e a d s t o a b i a s - d e p e n d e n t r f p o w e r d i s s i p a t i o n w h i c h f o r a l l i n t e n t s a n d p u r p o s e s l o o k s a s i f i t h a s t h e s a m e o r i g i n a s t h e e f f e c t i v e c o i l r e s i s t a n c e R 0 . T h a t i s , R 0 b e c o m e s R 0 ( V c ) . O n c e a f r e q u e n c y r a n g e f o r o p e r a t i o n h a s b e e n c h o s e n , a m a p p i n g b e t w e e n t h e v a r a c t o r b i a s a n d t h e i n p u t f r e q u e n c y e x i s t s a l l o w i n g u s t o w r i t e R 0 = R 0 ( ^ ) « T h e i m p e d a n c e o b s e r v e d b y t h e c o n d e n s e r o u t p u t a m p l i f i e r w i t h p h a s e c o m p e n s a t i o n i s t h u s R 0 (OJ) + TJCJLX" (W) T h e g o a l i s t o s e p a r a t e t h e s e t w o t e r m s a n d t h u s i s o l a t e t h e m o d u l a t i o n d u e o n l y t o t h e n u c l e a r s p i n s . A t f i r s t g l a n c e t h i s p r o b l e m i s e s s e n t i a l l y t h e same a s t h e c a s e w h e r e R 0 i s a c o n s t a n t . U n f o r t u n a t e l y , t h e s e c o n d i m p e d a n c e m o d u l a t i o n c a n i n f l u e n c e t h e b e h a v i o u r o f t h e p h a s e c o m p e n s a t i o n s y s t e m . 1 22 The approximate frequency dependence of the power l o s s in the v a r a c t o r i s shown i n f i g u r e 4.8.' T h i s i s a s i m p l i f i c a t i o n of f i g u r e A.12 which allows us to i d e n t i f y f i v e d i f f e r e n t r e g i o n s of o p e r a t i o n (accumulation, d e p l e t i o n , i n v e r s i o n and two t r a n s i t i o n r e g i o n s ) . The r e l a t i v e h e i g h t s and widths of these regions are d i f f e r e n t f o r v a r i o u s d e v i c e s . ACCUMULATION o i — o < CO CO O -I G£ UJ o Q. DEPLETION INVERSION F i g u r e 4.8 Power D i s s i p a t i o n i n C v as a F u n c t i o n of B i a s In chapter three the e f f e c t of R 0 upon the loop gain of the feedback loop was analyzed (equation 3.25). An i n c r e a s e i n R 0 l e a d s to a decrease i n the loop gain which r e s u l t s i n a l a r g e r d e v i a t i o n of f 0 from the input frequency. With an R 0 which i s dependent upon frequency, s e t t i n g the loop gain i s no longer a t r i v i a l matter as t h i s gain i s dependent upon the r e g i o n of o p e r a t i o n . O b v i o u s l y o p e r a t i o n i n e i t h e r of the two t r a n s i t i o n r e g i o ns i s h i g h l y u n d e s i r a b l e as the loop 1 23 g a i n c h a n g e s c o n t i n u o u s l y a c r o s s t h e s e r e g i o n s . A s f a r a s m a i n t a i n i n g a c o n s t a n t l o o p g a i n , t h e r e m a i n i n g t h r e e r e g i o n s a r e r o u g h l y e q u i v a l e n t ( a c c u m u l a t i o n t e n d s t o g i v e t h e m o s t f r e q u e n c y i n d e p e n d e n t l o o p g a i n ) . E s t i m a t e s p l a c e t h e m a x i m u m c h a n g e i n g a i n a t a b o u t 5% o v e r t h e a c c u m u l a t i o n r a n g e f o r t h e T R I U M F s y s t e m . F r o m p o i n t o f v i e w o f m a x i m i z i n g t h e t o t a l l o o p g a i n a n d r e s t r i c t i n g t h e d o m i n a n t p o l e s o f t h e s y s t e m t o l i e a s f a r t o t h e l e f t o f t h e o r i g i n ( i . e . n e g a t i v e a a x i s ) a s p o s s i b l e , t h e a c c u m u l a t i o n r e g i o n i s t h e m o s t d e s i r a b l e r e g i o n t o o p e r a t e i n . T h a t i s , g i v e n a c o n s t a n t a m p l i f i e r g a i n , t h e l o o p g a i n i s t h e h i g h e s t u n d e r a c c u m u l a t i o n . C o m p e n s a t i n g f o r t h e l o s s i n l o o p g a i n d u e t o a n i n c r e a s e d R 0 b y i n c r e a s i n g t h e a m p l i f i c a t i o n G i s l e s s d e s i r a b l e a s t h i s a l s o t e n d s t o move t h e p o l e d u e t o t h e a m p l i f i e r c l o s e r t o t h e o r i g i n ( a s u b j e c t w h i c h was d i s c u s s e d i n T i m e R e s p o n s e a n d S t a b i l i t y , c h a p t e r 3 ) . I t h a s a l r e a d y b e e n m e n t i o n e d t h a t t h e a m p l i t u d e d e p e n d e n c e o f t h e s i g n a l s b e i n g f e d i n t o t h e p h a s e d e t e c t i o n c i r c u i t r y i s r e m o v e d b y s q u a r i n g - u p t h e w a v e f o r m s i n a s a t u r a t e d a m p l i f i e r . W e r e t h i s n o t d o n e , t h e c h a n g i n g i m p e d a n c e R 0 w o u l d l e a d t o a m i s i n t e r p r e t a t i o n o f t h e v a r i a t i o n i n t h e c a r r i e r h e i g h t . T h a t i s , i t w o u l d b e r e c o r d e d b y t h e p h a s e d e t e c t o r a s a p h a s e s h i f t w h i c h i n t u r n w o u l d l e a d t o a n i n c o r r e c t t u n i n g o f t h e r e s o n a n t c i r c u i t . O b v i o u s l y , r e m o v a l o f t h i s a m p l i t u d e d e p e n d e n c e i s a n e s s e n t i a l s t e p . 124 O n e o f t h e o r i g i n a l g o a l s b e h i n d t h e d e v e l o p m e n t o f t h e s y s t e m d e s c r i b e d i n t h i s t h e s i s w a s t o r e m o v e t h e f r e q u e n c y d e p e n d e n c e o f t h e a m p l i t u d e o f t h e c a r r i e r w a v e . T h e p u r p o s e was t o a v o i d p r o b l e m s i m p o s e d b y t h e d y n a m i c r a n g e o f t h e c a r r i e r . V a r i a t i o n s i n t h e r e s i s t i v e i m p e d a n c e d u e t o p o w e r l o s s i n t h e v a r a c t o r r e s u l t i n c h a n g e s i n t h e c a r r i e r w a v e w h i c h a r e a t l e a s t t w e n t y t i m e s s m a l l e r t h a n t h o s e e n c o u n t e r e d i n a r e s o n a n t c i c u i t w i t h a Q o f a b o u t 6 0 . E v e n s o , c a r e m u s t s t i l l b e t a k e n s o a s t o a v o i d s i t u a t i o n s i n w h i c h n o n l i n e a r a m p l i f i c a t i o n o c c u r s . A g a i n , t h i s s u g g e s t s a v o i d i n g t h e t w o t r a n s i t i o n r e g i o n s m e n t i o n e d e a r l i e r . F i n a l l y , i t s h o u l d b e n o t e d t h a t i n s e r t i o n o f a Q s p o i l i n g r e s i s t o r ( s u c h a s t h e 10 J2 r e s i s t o r i n t h e T R I U M F s y s t e m ) i n o r d e r t o a v o i d o v e r m o d u l a t i o n o f t h e c a r r i e r c a n r e d u c e t h e . e f f e c t d e s c r i b e d a b o v e . T h a t i s , r e l a t i v e c h a n g e s i n R 0 c a n b e r e d u c e d t o a f e w p e r c e n t w i t h l i t t l e r e s u l t i n g c h a n g e i n t h e l o o p g a i n a c r o s s t h e f r e q u e n c y s c a n . 2. T H E I N F L U E N C E OF T H E S T A T I C M A G N E T I C F I E L D O n e w o u l d e x p e c t t h a t a l a r g e s t a t i c m a g n e t i c f i e l d o f t h e o r d e r o f 2.5 T e s l a w o u l d p l a y some r o l e i n d e t e r m i n i n g t h e b e h a v i o u r o f t h e NMR e l e c t r o n i c s . I n d e e d , d u r i n g t h e d e v e l o p m e n t o f t h e p r o t o t y p e s y s t e m i t was o b s e r v e d t h a t t h e r f c a r r i e r a m p l i t u d e r e f l e c t e d c h a n g e s i n t h e f i e l d r e l a t e d t o b o t h B 0 a n d t o d B 0 . d t C o n s i d e r a s e m i c o n d u c t o r d e v i c e s u c h a s a t r a n s i s t o r w i t h a m a g n e t i c f i e l d a l i g n e d p e r p e n d i c u l a r t o i t s c h a n n e l 125 ( w h i c h i s a s s u m e d t o b e t h e d i r e c t i o n o f g r o s s c u r r e n t f l o w ) . W h i l e e x t e n s i v e m o d e l l i n g o f t h i s s y s t e m i s u n l i k e l y t o p r o v i d e m u c h i n t h e way o f p e r t i n e n t i n f o r m a t i o n f o r d e s i g n p u r p o s e s , i t i s i n s t r u c t i v e t o m a k e a f i r s t o r d e r a p p r o x i m a t i o n o f t h e r e l a t i v e m a g n i t u d e s o f t h e m a g n e t i c f o r c e Fg a n d t h e e l e c t r i c f o r c e F~g e x p e r i e n c e d b y a c h a r g e c a r r i e r . C o n s i d e r f o r t h e s a k e o f i l l u s t r a t i o n , a G a A s f i e l d e f f e c t t r a n s i s t o r a t 3 0 0 K w i t h a t w o v o l t p o t e n t i a l a c r o s s a 5 M l o n g c h a n n e l . N e x t a p p l y a 2.5 T e s l a m a g n e t i c f i e l d p e r p e n d i c u l a r t o t h e c h a n n e l o f t h e d e v i c e a n d h e n c e p e r p e n d i c u l a r t o t h e d r i f t v e l o c i t y v o f t h e c h a r g e c a r r i e r s . We c a n w r i t e A s s u m i n g E t o b e h o m o g e n e o u s t h r o u g h o u t t h e c h a n n e l , we h a v e Fg = q V x B a n d F*g = q E o r |F=| = v B ( 4 . 1 ) |3| = 2 V = 4 x 1 0 3 V / c m . 5 M F o r a n n t y p e s e m i c o n d u c t o r t h i s c o r r e s p o n d s 3 3 t o a n e l e c t r o n v e l o c i t y o f a b o u t 1.5 x 1 0 7 c m / s . T h u s , 3 3 S . M. S z e , P h y s i c s o f S e m i c o n d u c t o r D e v i c e s , ( J o h n W i l e y a n d S o n s , I n c . , T o r o n t o , O n t . , 1 9 8 1 ) , p . 4 6 . 126 |Fg| = {U5 x 1 0 7 (4 x 1 0 3 c m / s ) ( 2 . 5 T ) V / c m ) = 0.94 T h a t i s , t h e m a g n e t i c f o r c e s e x p e r i e n c e d b y t h e s e c h a r g e c a r r i e r s c a n b e o f t h e s a me r e l a t i v e s i z e a s t h e e l e c t r i c f o r c e s . T h i s e f f e c t was m i n i m i z e d b y a l i g n i n g t h e t r a n s i s t o r s w i t h t h e m a g n e t i c f i e l d s o a s t o m i n i m i z e t h e v x 1 f o r c e s . I n t u i t i v e l y , o n e w o u l d a l i g n t h e c h a n n e l o f a t r a n s i s t o r e i t h e r p a r a l l e l o r a n t i p a r a l l e l t o t h e m a g n e t i c f i e l d . H o w e v e r , s i n c e v i s a f u n c t i o n o f p o s i t i o n ( i . e . v = v ( r ) ) , i t s h o u l d n o t b e e x p e c t e d t h a t v a r i a t i o n s i n d e v i c e b e h a v i o u r c a n b e c a n c e l l e d c o m p l e t e l y . T h e s e e f f e c t s w e r e v e r i f i e d b y m e a s u r i n g t h e t r a n s c o n d u c t a n c e s a n d t h e l e a k a g e c u r r e n t s o f t h e t r a n s i s t o r s u s e d i n t h e p r o t o t y p e s y s t e m , b o t h a s a f u n c t i o n o f t h e m a g n i t u d e (0 t o 2.5 T e s l a ) a n d t h e r e l a t i v e o r i e n t a t i o n o f B 0 w i t h r e s p e c t t o t h e d e v i c e s . V a r i a t i o n s i n t h e s e q u a n t i t i e s f o r a p a r t i c u l a r d e v i c e w i t h B 0 p e r p e n d i c u l a r t o t h e c h a n n e l w e r e a b o u t 1 0 % o v e r t h e f u l l r a n g e o f t h e f i e l d . When t h e c h a n n e l s w e r e a l i g n e d p a r a l l e l t o t h e f i e l d , h o w e v e r , v a r i a t i o n s w e r e a b o u t 5% u p t o ± 2 . 0 T e s l a b u t l e s s t h a n 0 . 5 % o v e r t h e h i g h e r f i e l d r a n g e s . W i t h v a r i a t i o n s i n t h e s e c h a r a c t e r i s t i c s o f t h e t r a n s i s t o r s r e d u c e d t o b e l o w 0 . 5 % o v e r t h e r a n g e 2.0 T e s l a 1 27 t o 2.5 T e s l a , t h e i n f l u e n c e o f t h e m a g n e t i c f i e l d u p o n t h e r f c a r r i e r was r e d u c e d t o a n a l m o s t n e g l i g i b l e l e v e l . T E m e a s u r e m e n t s w i t h t h e t r a n s i t o r s i n t h i s o r i e n t a t i o n s h o w a m a r k e d i m p r o v e m e n t i n s t a b i l i t y , r e p r o d u c i b i l i t y a n d d i s t o r t i o n c o m p a r e d t o t h e o t h e r o r i e n t a t i o n s . E x t e n s i v e m e a s u r e m e n t s i n d i c a t e d t h a t t h e r e m a i n i n g s m a l l e f f e c t s c o u l d b e c a n c e l l e d b y a p p l y i n g a c o r r e c t i o n t o t h e b a c k g r o u n d s i g n a l . T h i s c o r r e c t i o n v a r i e s l i n e a r l y w i t h t h e f i e l d i n t h e r e g i o n a r o u n d 2.5 T e s l a . By t a k i n g b a c k g r o u n d m e a s u r e m e n t s w i t h t h e f i e l d a d j u s t e d s u c h t h a t t h e L a r m o r f r e q u e n c y i s j u s t a b o v e a n d j u s t b e l o w t h e f r e q u e n c y s c a n r a n g e a n d t h e n a v e r a g i n g t h e r e s u l t s , a b a c k g r o u n d e q u i v a l e n t t o t h a t w h i c h w o u l d b e o b t a i n e d i n t h e a b s e n c e o f t h e n u c l e a r s p i n s w i t h f 0 = f i s o b t a i n e d . T h i s p r o c e d u r e i s u n n e c e s s a r y f o r l a r g e d y n a m i c a l l y p o l a r i z e d s i g n a l s w h e r e t h e i n f l u e n c e o f t h e m a g n e t i c f i e l d v a r i a t i o n i s n e g l i g i b l e . 3. T H E R M A L , M E C H A N I C A L AND E L E C T R O M A G N E T I C N O I S E M a n y f o r m s o f n o i s e s i g n a l s c a n c a u s e a d e t e r i o r a t i o n o f t h e q u a l i t y o f t h e o u t p u t s i g n a l . T h i s t h i r d f a c t o r e n c o m p a s s e s t h e t h r e e m a j o r s o u r c e s o f n o i s e s i g n a l s c o m m o n l y e n c o u n t e r e d i n p o l a r i z e d t a r g e t NMR w o r k . A s m e n t i o n e d e a r l i e r , t h e p o l a r i z e d t a r g e t e n v i r o n m e n t i s i n d e e d a h a r s h t e s t i n g g r o u n d f o r a n e l e c t r o n i c s p a c k a g e . N o t o n l y a r e n o r m a l o p e r a t i n g t e m p e r a t u r e s v e r y l o w , b u t t h e y may c h a n g e b y a n o r d e r o f m a g n i t u d e o r s o d u r i n g t h e t i m e m e a s u r e m e n t s a r e m a d e . T h e u s e o f l a r g e v a c u u m p u m p i n g 1 28 f a c i l i t i e s i n c l o s e p r o x i m i t y t o t h e e x p e r i m e n t a l a r e a o n l y a d d s t o t h e a l r e a d y n o i s y a c c e l e r a t o r e n v i r o n m e n t . W h i l e a n e x h a u s t i v e d i s c u s s i o n o f t h e s e f a c t o r s i s n o t p o s s i b l e , some m e n t i o n w i l l b e made o f t h e m a j o r e f f o r t s w h i c h w e r e made i n o r d e r t o m i n i m i z e d e l e t e r i o u s e f f e c t s d u e t o t h e r m a l , m e c h a n i c a l a n d e l e c t r o m a g n e t i c n o i s e . T h e s e e f f o r t s r e s u l t e d i n a s y s t e m c a p a b l e o f m e a s u r i n g a d e u t e r o n T E s i g n a l a t 1 K w i t h a S/N r a t i o o f a b o u t 1:5. T h e r e a r e t w o f a c e t s t o t h e p r o b l e m o f t h e r m a l n o i s e i n t h e m i x i n g c h a m b e r o f t h e c r y o s t a t . T h a t i s , t h e r e a r e t w o c o m p e t i n g p r o c e s s e s a t w o r k . E i t h e r p r o c e s s c a n b e c o n s i d e r e d a s a n o i s y p e r t u r b a t i o n u p o n t h e p e r f o r m a n c e o f t h e o t h e r . F r o m t h e p o i n t o f v i e w o f o b t a i n i n g a h o m o g e n e o u s n u c l e a r p o l a r i z a t i o n i n t h e s a m p l e , a h o m o g e n e o u s t e m p e r a t u r e i s d e s i r e d . T h i s i s n o t c o m p a t i b l e w i t h a t t e m p t s made t o p l a c e d i s s i p a t i v e e l e c t r o n i c c o m p o n e n t s c l o s e t o t h e NMR c o i l t o r e d u c e e l e c t r o n i c n o i s e . F u r t h e r m o r e , t h e r e f r i g e r a t i o n s y s t e m m u s t b e a b l e t o s i n k t h e e x c e s s p o w e r c r e a t e d b y t h e e l e c t r o n i c s a n d s t i l l p r o v i d e t h e l o w t e m p e r a t u r e s c o m p a t i b l e w i t h t h e f r o z e n s p i n s t a t e . T h e r a n g e o f t e m p e r a t u r e s w h i c h a r e o f i n t e r e s t b e t w e e n T E m e a s u r e m e n t s a n d t h e f r o z e n s p i n s t a t e s p a n r o u g h l y a n o r d e r o f m a g n i t u d e . T h i s i s d e f i n i t e l y n o t c o m p a t i b l e w i t h m a i n t a i n i n g c o n s i s t e n t d e v i c e p e r f o r m a n c e c h a r a c t e r i s t i c s , e s p e c i a l l y w h en u s i n g s e m i c o n d u c t o r d e v i c e s w h o s e F e r m i l e v e l s a r e s t r o n g l y t e m p e r a t u r e d e p e n d e n t . 129 G a A s d e v i c e s w e r e u s e d i n o r d e r t o m i n i m i z e c h a n g e s i n d e v i c e p e r f o r m a n c e a n d t o a v o i d c a r r i e r f r e e z e o u t . 3 " T h e s e d e v i c e s t e n d t o h a v e r e l a t i v e l y h i g h c a r r i e r m o b i l i t i e s a t l o w t e m p e r a t u r e s . By u s i n g t h e s e d e v i c e s , d e v i a t i o n s i n d e v i c e c h a r a c t e r i s t i c s w e r e r e d u c e d t o a n e g l i g i b l e l e v e l . I n a s i m i l a r m a n n e r , r e s i s t o r s w i t h g o o d t e m p e r a u r e c o e f f i c i e n t s w e r e s e l e c t e d . T h e r e s i s t o r s u s e d w e r e P h i l l i p s t y p e S F R 1 6 T , 0.1 W r e s i s t o r s . D e v i a t i o n s f r o m t h e r o o m t e m p e r a t u r e r e s i s t a n c e v a l u e o f l e s s t h a n 1% w e r e n o t e d down t o 0.1 K w i t h e s s e n t i a l l y n o c h a n g e b e t w e e n 4 K a n d 0.1 K. F u r t h e r m o r e , i n o r d e r t o s t a b i l i z e t h e o p e r a t i n g t e m p e r a t u r e s o f s e v e r a l o f t h e c o m p o n e n t s a n d t o m i n i m i z e l o c a l i z e d h e a t i n g o f t h e t a r g e t m a t e r i a l , t h e s e t h e c o m p o n e n t s w e r e a t t a c h e d t o t h e c o n d e n s e r o f t h e r e f r i g e r a t o r ( o p e r a t i n g a t t h e w a r m e r a n d m o r e s t a b l e t e m p e r a t u r e o f 1.6 K ) . T h i s a l s o h a s t h e e f f e c t o f i n c r e a s i n g t h e maximum a l l o w a b l e p o w e r d i s s i p a t i o n o f t h e s e d e v i c e s . U n f o r t u n a t e l y i t r e q u i r e s t h e u s e o f t w o s h o r t l e n g t h s o f c o a x i a l c a b l e ( t o b r i n g t h e o u t p u t u p t o t h e c o n d e n s e r ) w h i c h i n t r o d u c e p h a s e e r r o r s i n t o t h e c o m p e n s a t i o n s y s t e m . A n o t h e r p r o b l e m l e a d i n g t o l o c a l i z e d h e a t i n g c o m e s f r o m t h e i n t r o d u c t i o n o f m i c r o w a v e p o w e r i n t o t h e m i x i n g c h a m b e r f o r t h e p u r p o s e o f o b t a i n i n g d y n a m i c n u c l e a r p o l a r i z a t i o n . S e l e c t i v e a b s o r p t i o n o f m i c r o w a v e p o w e r b y v a r i o u s c o m p o n e n t s s u c h a s t h e c a r b o n r e s i s t o r s u s e d f o r t e m p e r a t u r e 3* S. M. S z e , P h y s i c s o f S e m i c o n d u c t o r D e v i c e s , ( J o h n W i l e y a n d S o n s , I n c . , T o r o n t o , O n t . , 1 9 8 1 ) , p . 2 8 . 130 m e a s u r e m e n t s a l s o l e a d t o h e a t i n g p r o b l e m s . W h e n e v e r p o s s i b l e c o m p o n e n t s n e a r t h e t a r g e t w e r e m o v e d t o a r e a s o f l o w m i c r o w a v e p o w e r d e n s i t y . F i n a l l y , i t s h o u l d b e n o t e d t h a t t h e e f f e c t s o f t e m p e r a t u r e v a r i a t i o n s a r e e a s i l y o b s e r v e d e x t e r n a l l y . T h e y a r e s i m p l y r e f l e c t e d a s a c h a n g i n g b a c k g r o u n d s h a p e . M e c h a n i c a l v i b r a t i o n s h a v e o f t e n p l a g u e d l o w t e m p e r a t u r e e x p e r i m e n t e r s . W h i l e a t f i r s t s i g h t , v i b r a t i o n s may s e e m r e l a t i v e l y h a r m l e s s , t h e y a r e i n f a c t a n i m p o r t a n t c o n s i d e r a t i o n i n a n y m e c h a n i c a l l y n o i s y e n v i r o n m e n t . W h i l e a g o o d p a r t o f t h e c o u p l i n g o f t h e s e v i b r a t i o n s t o t h e o u t p u t o c c u r s v i a t h e c o i l , i t i s n o t a l w a y s p o s s i b l e t o i s o l a t e a l l o t h e r c a u s e s o f m i c r o p h o n i c p i c k u p . E a r l y v e r s i o n s o f t h e p r o t o t y p e NMR s y s t e m p r o v i d e d l i t t l e m e c h a n i c a l s u p p o r t f o r t h e c o i l . Human s p e e c h n e a r t h e c r y o s t a t r e s u l t e d i n a s i g n a l b e i n g r e c o r d e d o n t h e o u p u t c a r r i e r w a v e . T e c h n i q u e s s u c h a s s t a b i l i z i n g t h e c o i l a n d d a m p i n g o u t v i b r a t i o n s i n p u m p i n g l i n e s a n d t h r o u g h t h e c r y o s t a t f r a m e l e d t o m a r k e d i m p r o v e m e n t s i n s i g n a l q u a l i t y . V a r i o u s m e t h o d s f o r r e d u c i n g t h e c o u p l i n g o f m e c h a n i c a l v i b r a t i o n s t o a l o w t e m p e r a t u r e e x p e r i m e n t a r e g i v e n i n t h e r e f e r e n c e b y F r i e d m a n . 3 5 I t s h o u l d b e n o t e d t h a t n o i s e o f t h i s t y p e w h i c h i s c o u p l e d t o t h e c o i l c a n l e a d t o a s m a l l c h a n g e i n t h e i n d u c t a n c e o f t h e c o i l . W h i l e t h e p h a s e c o m p e n s a t i o n s y s t e m 3 5 L . F r i e d m a n , i n Low T e m p e r a t u r e T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n ( M a t e r i a l s S c i e n c e C e n t e r a t C o r n e l l U n i v e r s i t y , I t h i c a , New Y o r k , 1 9 8 2 ) , p p . 2 4 7 - 2 5 8 131 i s c a p a b l e o f r e s p o n d i n g t o t h i s c h a n g e a n d c a n c e l l i n g t h e e r r o r , a f i n i t e t i m e i s r e q u i r e d . T h e r e s u l t i s t h a t t h e r e s o n a n t f r e q u e n c y f 0 h a s a n o i s e s i g n a l i n t h e f r e q u e n c y d o m a i n a p p l i e d t o i t . T h i s f r e q u e n c y r e l a t e d n o i s e d u e t o t h e s h i f t i n g Q c u r v e i s r e f l e c t e d a s a n o i s e s i g n a l i n t h e a m p l i t u d e o f t h e c a r r i e r w a v e a t t h e o u t p u t . O b v i o u s l y , t h i s i s a n o t h e r r e a s o n f o r o p e r a t i n g i n a r e g i o n w h e r e t h e a m p l i t u d e o f t h e c a r r i e r w a v e h a s l i t t l e f r e q u e n c y d e p e n d e n c e . W e r e t h e r e n o f r e q u e n c y d e p e n d e n c e a t a l l , t h e o u t p u t w o u l d n o t r e f l e c t m i c r o p h o n i c p i c k u p v i a t h i s m e c h a n i s m . O u r o p e r a t i o n a l e x p e r i e n c e c o m e s c l o s e s t t o t h i s i d e a l s i t u a t i o n w h en t h e t u n n i n g F E T ' s a r e o p e r a t e d i n a c c u m u l a t i o n w h e r e R 0 ( V c ) , d e p e n d s t h e l e a s t u p o n b i a s . I t s h o u l d a l s o b e n o t e d , i n p a s s i n g , t h a t v i b r a t i o n s may a l s o c h a n g e t h e d i s s i p a t i v e l o a d t o t h e w a l l s ( r a d i a t e d p o w e r ) o f t h e m i x i n g c h a m b e r c a v i t y . T h e f i n a l t o p i c o f e l e c t r o m a g n e t i c i n t e r f e r e n c e i s f a r t o o b r o a d a t o p i c t o b e c o n s i d e r e d i n t h i s t h e s i s . S u f f i c e i t t o s a y t h a t c o n s i d e r a b l e e f f o r t m u s t b e p u t i n t o s h i e l d i n g v a r i o u s c o m p o n e n t s a n d e s t a b l i s h i n g g o o d g r o u n d c o n n e c t i o n s . O n e p r o b l e m p a r t i c u l a r l y r e l e v a n t t o t h e NMR a p p a r a t u s d e s c r i b e d i n t h i s t h e s i s c o n c e r n s t h e e m f ' s i n d u c e d i n t h e r e l a t i v e l y l a r g e g r o u n d l o o p s f o r m e d b y l e n g t h s o f c o a x i a l c a b l e . T h e u s e o f t e c h n i q u e s s u c h a s s e m i - f l o a t i n g i n p u t g r o u n d s 3 6 c a n b e o f g r e a t h e l p . 3 6 A n e x c e l l e n t r e f e r e n c e i s " G r o u n d i n g a n d S h e i l d i n g i n E l e c t r o c h e m i c a l I n s t r u m e n t a t i o n - S o m e B a s i c C o n s i d e r a t i o n s , " P r i n c e t o n A p p l i e d R e s e a r c h , A n a l y t i c a l I n s t r u m e n t D i v i s i o n . T e c h n i c a l n o t e 1 1 7 , 1 9 7 8 . 132 D. E X P E R I M E N T A L PERFORMANCE O F T H E T R I U M F P R O T O T Y P E T h i s f i n a l d i s c u s s i o n i s b a s e d u p o n t h e e x p e r i e n c e s o f t h e T R I U M F t a r g e t s g r o u p i n d e v e l o p i n g a n d o p e r a t i n g t h e p r o t o t y p e p h a s e c o m p e n s a t e d NMR i n s t r u m e n t a t i o n p a c k a g e a s d e s c r i b e d i n t h i s t h e s i s . T h i s d i s c u s s i o n r e f l e c t s a b o u t o n e a n d o n e h a l f y e a r s o f e x p e r i e n c e w i t h t h i s p a r t i c u l a r NMR s y s t e m . T h e w o r k r e c e n t l y c u l m i n a t e d i n t h e f i r s t u s e o f t h i s s y s t e m i n a n u c l e a r s c a t t e r i n g e x p e r i m e n t . U s i n g a t a r g e t o f d e u t e r a t e d b u t a n o l ( C « D 1 0 O ) a n d 5% D 2 0 d o p e d w i t h 6 x 1 0 1 9 m o l e c u l e s p e r m l o f E H B A , t h e s p i n d e p e n d e n c e o f (7r +,3) s c a t t e r i n g a n d a b s o r p t i o n r e a c t i o n s was s t u d i e d . 3 7 R e s u l t s o f t h i s e x p e r i m e n t y i e l d e d a n i n d e p e n d e n t m e a s u r e o f t h e t e n s o r p o l a r i z a t i o n o f t h e d e u t e r o n s i n t h e t a r g e t m a t e r i a l w h i c h a g r e e w i t h i n t h e e r r o r l i m i t s , w i t h t h e r e s u l t s c a l c u l a t e d f r o m t h e NMR m e a s u r e m e n t s . T h e n u c l e a r v e c t o r p o l a r i z a t i o n o f a c o l l e c t i o n o f d y n a m i c a l l y p o l a r i z e d n u c l e a r s p i n s i s m e a s u r e d b y i n t e g r a t i n g t h e NMR a b s o r p t i o n l i n e o f t h e s p i n s a n d c o m p a r i n g t h i s t o t h e a r e a o f t h e t h e r m a l e q u i l i b r i u m a b s o r p t i o n l i n e ( e q u a t i o n 1 . 4 6 ) . T h e a b s o l u t e p o l a r i z a t i o n o f t h e T E s p i n s c a n b e c a l c u l a t e d f r o m e q u a t i o n 1.7 a n d e i t h e r e q u a t i o n 1.12 ( s p i n 1/2 s y s t e m ) o r e q u a t i o n 1.13 ( s p i n 1 s y s t e m ) . T h e m e a s u r e m e n t s r e q u i r e d t o d e t e r m i n e t h e p o l a r i z a t i o n c o n s i s t o f a d e t e r m i n a t i o n o f t h e a r e a s o f t h e t w o s i g n a l s 3 7 G. R. S m i t h e t a l . , M e a s u r e m e n t o f T e n s o r O b s e r v a b l e s i n t h e (7r + , 3 ) E l a s t i c S c a t t e r i n g R e a c t i o n , T R I U M F E x p e r i m e n t 3 3 7 ( p r i v a t e c o m m u n i c a t i o n ) . 1 33 a t a k n o w n t e m p e r a t u r e , m a g n e t i c f i e l d a n d f r e q u e n c y . T h e a r e a a n d t e m p e r a t u r e m e a s u r e m e n t s a r e t h e l i m i t i n g f a c t o r s i n d e t e r m i n i n g t h e u n c e r t a n t y o f t h e m e a s u r e m e n t . E r r o r s i n t h e T E s i g n a l s a r e r o u g h l y 5% w h i l e e r r o r s i n t e m p e r a t u r e m e a s u r e m e n t s a r e a l s o a b o u t 5 % . T h e t o t a l e s t i m a t e d e r r o r i n a m e a s u r e m e n t o f t h e t a r g e t p o l a r i z a t i o n i s t y p i c a l l y 7%. O n c e t h e d i l u t i o n r e f r i g e r a t o r h a s b e e n s t a b i l i z e d a t a t e m p e r a t u r e o f a b o u t 0.8 K a n d t h e f r e q u e n c y s c a n r a n g e h a s b e e n s e l e c t e d , t h e L a r m o r f r e q u e n c y i s s e t t o a v a l u e c l o s e t o f b u t s u c h t h a t t h e a b s o r b t i o n l i n e o c c u r s b e l o w t h e c s c a n r a n g e . A b a c k g r o u n d m e a s u r e m e n t i s t a k e n a n d t h e n t h e L a r m o r f r e q u e n c y i s m o v e d a n e q u a l d i s t a n c e a b o v e f w h e r e a s e c o n d b a c k g r o u n d i s t a k e n . T h e t w o s i g n a l s a r e a v e r a g e d t o c o r r e c t f o r m a g n e t i c f i e l d e f f e c t s ( a s d i s c u s s e d i n t h e s e c t i o n o n t h e i n f l u e n c e o f t h e m a g n e t i c f i e l d , w h i c h w a s p r e s e n t e d e a r l i e r i n t h i s c h a p t e r ) . O n c e s u f f i c i e n t t i m e h a s p a s s e d f o r t h e n u c l e a r s p i n s t o r e a c h t h e r m a l e q u i l i b r i u m , t h e L a r m o r f r e q u e n c y i s m o v e d t o f w h e r e a m e a s u r e m e n t o f t h e s i g n a l i s m a d e . T h e L S I - 1 1 s u b s e q u e n t l y s u b t r a c t s t h e b a c k g r o u n d s i g n a l f r o m t h e NMR s i g n a l a n d f i t s a p o l y n o m i a l t o t h e b a s e l i n e o f t h e d i f f e r e n c e s i g n a l . T h e a r e a o f t h i s s i g n a l i s c a l c u l a t e d i n u n i t s o f f r e q u e n c y t i m e s v o l t s . T y p i c a l l y 80 s e t s o f 1024 s c a n s a r e made s o a s t o c h e c k f o r s y s t e m a t i c e r r o r s a n d s t a t i s t i c a l e r r o r s . A t y p i c a l t h e r m a l e q u i l i b r i u m s i g n a l i s s h o w n i n f i g u r e 4.9 ( 1 0 2 4 s c a n s w i t h b a c k g r o u n d s u b t r a c t e d ) . 134 J I L I I I J I ' I I I I I 1 J 1 1 1 1 1 1 1 1 1 —2 — j — i — i — i — i — | — i — i — i — i — | — i — i — i — i — j — i — i — i — i — | — i — i — i — i — 1 ~ ~ 1 52 103 155 206 257 TE SIGNAL AT 1.2 DEG KELVIN F i g u r e 4.9 T h e r m a l E q u i l i b r i u m S i g n a l i n D e u t e r a t e d B u t a n o l ( s e e t e x t f o r c o m p o s i t i o n ) 135 N e x t t h e n u c l e a r s p i n s a r e d y n a m i c a l l y p o l a r i z e d w i t h m i c r o w a v e p o w e r a t a b o u t 70 G H z . T h e m i c r o w a v e s o u r c e i s a 100 mW I m p a t t d i o d e w h o s e f r e q u e n c y i s m o n i t o r e d a n d m a i n t a i n e d w i t h i n 1 MHz o f t h e s e t f r e q u e n c y b y c o m p u t e r c o n t r o l . M a x i m u m m i c r o w a v e p o w e r s s u p p l i e d t o t h e m i x i n g c h a m b e r a r e o f t h e o r d e r o f 1 mW. L o c a l i z e d h e a t i n g o f c o m p o n e n t s b y m i c r o w a v e p o w e r i n t r o d u c e s some e r r o r i n t o t h e b a c k g r o u n d w h i c h i s n o t i c e d m o r e a t l o w e r p o l a r i z a t i o n s . T h e p o l a r i z a t i o n p r o c e s s c a n b e c o n t i n u o u s l y m o n i t o r e d w i t h t h e NMR s y s t e m . O n c e e n h a n c e d , t h e s a m p l e i s c o o l e d t o a b o u t 70 mK a t w h i c h p o i n t s p i n r e l a x a t i o n t i m e s a r e o f t h e o r d e r o f s e v e r a l h o u r s . I n p u t r f l e v e l s a r e t h e n d r o p p e d b y 20 d B s o a s t o a v o i d t h e m o d i f i c a t i o n o f t h e s i g n a l w h i c h c a n r e s u l t i f s i g n i f i c a n t a m o u n t s o f r f p o w e r a r e p u t i n t o t h e s p i n s y s t e m . I n s u c h a s i t u a t i o n , t r a n s i t i o n s t o w a r d s t h e t h e r m a l e q u i l i b r i u m s t a t e c a n o c c u r ( s e e c h a p t e r 1 ) . T h e r e l a t i v e r e s p o n s e o f t h e s y s t e m a t v a r i o u s i n p u t p o w e r l e v e l s h a s b e e n m e a s u r e d a n d i s l i n e a r l y r e l a t e d t o t h e p o w e r o v e r a t l e a s t a 30 dB c h a n g e i n t h e i n p u t r f l e v e l . A new b a c k g r o u n d m e a s u r e m e n t i s t h e n t a k e n a t t h e l o w e r t e m p e r a t u r e a s i t i s s i g n i f i c a n t l y d i f f e r e n t a t t h e s e t e m p e r a t u r e s . T h e s i g n a l i s t h e n m e a s u r e d a n d t h e a r e a c a l c u l a t e d a s b e f o r e . T h e a b s o l u t e p o l a r i z a t i o n i s t h e n c a l c u l a t e d u t i l i z i n g t h e r a t i o o f t h e s i g n a l a r e a s , t e m p e r a t u r e s , f r e q u e n c i e s a n d f i e l d s t r e n g t h s . A t y p i c a l e n h a n c e d DMR l i n e s h a p e i s s h o w n i n f i g u r e 4.10 ( 1 0 2 4 s c a n s 1 36 w i t h t h e b a c k g r o u n d s u b t r a c t e d ) . T h e l a r g e e n h a n c e d s i g n a l i s m e a s u r e d w i t h f a r m o r e e a s e t h a n t h e T E s i g n a l ( t y p i c a l T E p o l a r i z a t i o n s a r e o f t h e o r d e r o f 0 . 0 5 % a t a b o u t 1.0 K) p r o v i d i n g t h e o p p o r t u n i t y t o a s c e r t a i n t h e r e s p o n s e o f t h e s y s t e m a s a f u n c t i o n o f f r e q u e n c y . T h a t i s , t h e DMR s i g n a l h e i g h t a n d a r e a w e r e m e a s u r e d a s a f u n c t i o n o f t h e L a r m o r f r e q u e n c y f o r a f i x e d f . O b s e r v e d d e v i a t i o n s w e r e o f t h e o r d e r o f 2% a c r o s s t h e c w i d t h o f a DMR a b s o r p t i o n l i n e . A s e c o n d c h e c k o n t h i s l i n e a r i t y was made b y c a l c u l a t i n g t h e v e c t o r p o l a r i z a t i o n f r o m t h e p e a k a s y m m e t r y ( e q u a t i o n 1.56) a n d c o m p a r i n g t h i s w i t h t h e r e s u l t o b t a i n e d f r o m t h e s i g n a l a r e a r a t i o t e c h n i q u e ( e q u a t i o n 1 . 4 6 ) . S i m i l a r l y , m e a s u r e m e n t s w e r e made a t a c o n s t a n t f r e q u e n c y b y s c a n n i n g t h e m a g n e t i c f i e l d a n d c a l c u l a t i n g t h e p o l a r i z a t i o n . A l l t h r e e t e c h n i q u e s w e r e c o n s i s t e n t t o w i t h i n ± 3 % . T h e r e s u l t s o f t h e f i r s t e x p e r i m e n t a l r u n i n a n u c l e a r s c a t t e r i n g e x p e r i m e n t c o n f i r m t h a t t h e T R I U M F p h a s e c o m p e n s a t e d NMR s y s t e m i s r e l i a b l e . 3 8 T E s i g n a l s w e r e m e a s u r e d a t v a r i o u s t i m e s t h r o u g h o u t t h e t h r e e w e e k e x p e r i m e n t . T h e d a y t o d a y v a r i a t i o n i n t h e s t a n d a r d d e v i a t i o n s o f t h e s i g n a l a r e a s w e r e o f t h e o r d e r o f 4% t o 6%. E n h a n c e d v e c t o r p o l a r i z a t i o n s d u r i n g t h e e x p e r i m e n t w e r e r o u t i n e l y i n t h e r a n g e - 0 . 3 2 t o - . 3 4 . U n d e r l a b o r a t o r y 3 8 G . R. S m i t h e t a l . , " M e a s u r e m e n t o f T e n s o r O b s e r v a b l e s i n t h e (7r +,3) E l a s t i c S c a t t e r i n g R e a c t i o n , " T R I U M F E x p e r i m e n t 3 3 7 ( p r i v a t e c o m m u n i c a t i o n ) . 137 F i g u r e 4 . 1 0 D y n a m i c a l l y P o l a r i z e d S i g n a l ( n o t e s c a l e r e l a t i v e t o T E s i g n a l : s t a t i s t i c a l n o i s e c a n n o t b e s e e n • i n t h i s f i g u r e ) 138 c o n d i t i o n s , t h e b e s t p o l a r i z a t i o n s o b t a i n e d t o d a t e w e r e - 0 . 3 7 a n d + 0 . 3 1 . T h e f i n a l i n d e p e n d e n t t e s t o f t h e s y s t e m c o m e s f r o m a c a l c u l a t i o n o f t h e t e n s o r p o l a r i z a t i o n o f t h e s a m p l e s p i n s f r o m t h e n u c l e a r s c a t t e r i n g d a t a . K n o w i n g t h e T 2 o f o r t h e r e a c t i o n , t h e n u c l e a r a l i g n m e n t c a n b e c a l c u l a t e d f r o m t h e r a t i o o f t h e s c a t t e r i n g c r o s s s e c t i o n s w hen t h e t a r g e t i s i n t h e r m a l e q u i l i b r i u m c o m p a r e d t o w h e n t h e s p i n s a r e d y n a m i c a l l y p o l a r i z e d . T h e r e s u l t s c o m p a r e f a v o r a b l y w i t h v a l u e s c a l c u l a t e d f o r t h e a l i g n m e n t f r o m NMR d a t a . A s a n e x a m p l e o f t h i s c o r r e l a t i o n , t h e a v e r a g e d y n a m i c a l l y p o l a r i z e d n u c l e a r a l i g n m e n t c a l c u l a t e d f r o m t h e s c a t t e r i n g d a t a was 0 . 1 0 0 ± . 0 2 3 . 3 9 When e q u a t i o n s 1.46 a n d 1.57 w e r e u s e d t o c a l c u l a t e t h e a l i g n m e n t f r o m t h e r e l a t i v e a r e a s o f t h e e n h a n c e d a n d T E NMR l i n e s h a p e s , a r e s u l t o f A = 0 . 0 8 3 ± . 0 0 6 was o b t a i n e d . F i t t i n g t h e a b s o r p t i v e l i n e s h a p e o f t h e d y n a m i c a l l y p o l a r i z e d s p i n s t o d e t e r m i n e t h e a s y m m e t r y R a n d t h e n u s i n g e q u a t i o n s 1.56 a n d 1.57 t o c a l c u l a t e t h e n u c l e a r a l i g n m e n t y i e l d s a v a l u e o f 0 . 0 9 5 ± . 0 0 7 . E . C O N C L U D I N G REMARKS T h e T R I U M F p h a s e c o m p e n s a t e d NMR Q - m e t e r h a s d e m o n s t r a t e d t h a t a p r a c t i c a l d e v i c e b a s e d u p o n t h e p r i n c i p l e s o u t l i n e d i n t h i s t h e s i s c a n p e r f o r m r e l i a b l y i n t h e u s u a l e x p e r i m e n t a l e n v i r o n m e n t . T h e r e s o n a n t f r e q u e n c y 3 9 F . T e r v i s i d i s ( p r i v a t e c o m m u n i c a t i o n ) . 139 o f t h e t u n e d c i r c u i t i s f o r c e d t o t r a c k t h e i n p u t r f f r e q u e n c y a n d h e n c e t h e c i r c u i t i s f o r c e d t o s t a y a t r e s o n a n c e . T h i s e n a b l e s t h e s y s t e m t o d e t e c t t h e a b s o r p t i v e NMR l i n e s h a p e u n h a m p e r e d b y d i s t o r t i o n s a r i s i n g f r o m t h e f r e q u e n c y d e p e n d e n t n a t u r e o f t h e c i r c u i t e l e m e n t s . T y p i c a l l y t h e i n p u t r f f r e q u e n c y i s s w e p t o v e r a 512 k H z r a n g e c e n t e r e d a t f = 1 6 . 6 MHz a t a r a t e o f 83 s c a n s p e r s e c o n d . E a c h s c a n c o n s i s t s o f 2 5 6 f r e q u e n c y i n c r e m e n t s . T h e o u t p u t o f t h e Q - m e t e r i s s a m p l e d a f t e r e a c h o f t h e s e i n c r e m e n t s r e s u l t i n g i n a d a t a a c q u i s i t i o n r a t a o f 1024 s a m p l e s p e r b i n i n 1 2 . 5 s e c o n d s . When t i m e f o r f u r t h e r a n a l y s i s i s t a k e n i n t o a c c o u n t , i t i s p o s s i b l e t o r e p e a t t h i s p r o c e s s e v e r y 17 s e c o n d s . T h e m a ximum p h a s e d e v i a t i o n A\iV w h i c h i s m e a s u r e d a t t h e e x t r e m e s o f t h e f r e q u e n c y m o d u l a t i o n i s l e s s t h a n 2 . 5 ° . T h i s c o n f i r m s t h e a s s u m p t i o n t h a t o n c e o p e r a t i n g , t h e s y s t e m r e m a i n s a t r e s o n a n c e . I t i s a l s o i n a g r e e m e n t w i t h t h e t h e o r e t i c a l l y p r e d i c t e d m a ximum d e v i a t i o n o f a b o u t 1 . 5 ° . T e s t s i n d i c a t e t h a t i t i s p o s s i b l e t o o p e r a t e t h e s y s t e m f o r a t l e a s t t h r e e w e e k s w i t h o u t a d j u s t i n g a n y p a r a m e t e r s a n d y e t r e t a i n s t a b i l i t y a n d r e p r o d u c i b i l i t y . C h a n g e s i n t h e s t a t e o f t h e Q - m e t e r a r e i n s i g n i f i c a n t o v e r t h i s t i m e p e r i o d . R e p e a t e d i n t e g r a t e d t h e r m a l e q u i l i b r i u m l i n e s h a p e s y i e l d r e s u l t s w h i c h a r e c o n s i s t e n t t o w i t h i n 5%. T h i s r e s u l t h a s b e e n s h o w n t o b e r e p r o d u c i b l e o v e r t h e t h r e e w e e k p e r i o d . T h e s e r e s u l t s a r e s i g n i f i c a n t a s t h i s t y p e o f p e r f o r m a n c e i s o f t e n d e m a n d e d o f a p o l a r i z e d t a r g e t NMR 1 40 s y s t e m . M e a s u r e m e n t s o f t h e l i n e a r i t y o f t h e s y s t e m h a v e a l s o b e e n m a d e . T h e f r e q u e n c y d e p e n d e n t g a i n o f t h e s y s t e m w a s t e s t e d b y t h r e e d i f f e r e n t m e t h o d s . T h e a m p l i t u d e a n d i n t e g r a t e d a r e a s o f e n h a n c e d DMR l i n e s h a p e s w e r e f o u n d t o v a r y b y n o m o r e t h a n 2% o v e r t h e w i d t h o f t h e DMR a b s o r p t i o n l i n e . E q u a t i o n 1.56 was u s e d t o c a l c u l a t e t h e t a r g e t v e c t o r p o l a r i z a t i o n f r o m t h e a s y m m e t r y o f t h e l i n e s h a p e . T h i s w a s c o m p a r e d t o t h e r e s u l t c a l c u l a t e d f r o m t h e s i g n a l a r e a r a t i o t e c h n i q u e ( e q u a t i o n 1 . 4 6 ) . S i m i l a r p o l a r i z a t i o n m e a s u r e m e n t s w e r e a l s o made a t a c o n s t a n t f r e q u e n c y b y v a r y i n g t h e m a g n e t i c f i e l d . A l l t h r e e t e c h n i q u e s w e r e c o n s i s t e n t t o w i t h i n ± 3%. A t t e m p t s w e r e made t o m i n i m i z e t h e s u s c e p t i b i l i t y o f t h e s y s t e m t o m e c h a n i c a l , e l e c t r o m a g n e t i c a n d t h e r m a l n o i s e . I t w as n e c e s s a r y t o m a k e s m a l l c o r r e c t i o n s f o r t h e i n f l u e n c e o f t h e c h a n g i n g m a g n e t i c f i e l d ( r e f e r t o t h e d i s c u s s i o n i n s e c t i o n C-2 o f t h i s c h a p t e r ) . I t w a s p o s s i b l e t o o b s e r v e a T E l i n e s h a p e o n a n o s c i l l o s c o p e w i t h a s i g n a l t o n o i s e r a t i o o f 1:5. A f t e r a v e r a g i n g 1 024 s a m p l e s o f t h e l i n e s h a p e a t y p i c a l s p e c t r u m h a s a s i g n a l t o n o i s e r a t i o o f a b o u t 8 : 1 . R a n d o m n o i s e s i g n a l s w e r e n e g l i g i b l e i n c o m p a r i s o n t o t h e e n h a n c e d l i n e s h a p e ( a b o u t 3 0 % v e c t o r p o l a r i z a t i o n ) . I n t e g r a t e d t h e r m a l e q u i l i b r i u m l i n e s h a p e s w e r e m e a s u r e d t o a p r e c i s i o n o f a b o u t 5%. When t h e u n c e r t a i n t y i n t h e t e m p e r a t u r e m e a s u r e m e n t ( 5 % ) i s i n c l u d e d i n t h e c a l c u l a t i o n 141 o f t h e a b s o l u t e T E p o l a r i z a t i o n , a n e r r o r o f 7% w a s a t t r i b u t e d t o t h e c a l c u l a t e d e n h a n c e d v e c t o r p o l a r i z a t i o n s . T h e n u c l e a r s c a t t e r i n g d a t a d e s c r i b e d i n t h e p r e v i o u s s e c t i o n y i e l d e d a n i n d e p e n d e n t m e a s u r e m e n t o f t h e n u c l e a r a l i g n m e n t . T h e v a l u e o f 0.100 ± .023 i s i n a g r e e m e n t w i t h t h e v a l u e 0 . 0 8 3 ± .006 c a l c u l a t e d f r o m t h e s i g n a l a r e a r a t i o t e c h n i q u e a n d 0 . 0 9 5 ± .007 c a l c u l a t e d f r o m t h e l i n e s h a p e a s y m m e t r y t e c h n i q u e . T h e o v e r a l l p e r f o r m a n c e o f t h i s s y s t e m t o d a t e i s r e g a r d e d a s b e i n g v e r y s u c c e s s f u l . I t i s h o p e d t h a t t h e m a t e r i a l w h i c h h a s b e e n p r e s e n t e d i n t h i s t h e s i s w i l l b e o f u s e i n t h e d e v e l o p m e n t o f f u t u r e s y s t e m s b a s e d o n t h e s e c o n c e p t s . B I B L I O G R A P H Y A b b a s , D a n i e l C . " R e d u c t i o n o f S p u r i o u s B a s e l i n e E f f e c t s i n NMR," R e v . S c i . I n s t r u m . 50 ( 7 ) , 8 2 9 - 8 3 1 ( 1 9 7 9 ) . A b r a g a m , A . T h e P r i n c i p l e s o f N u c l e a r M a g n e t i s m , O x f o r d U n i v e r s i t y P r e s s , New Y o r k , 1 9 8 3 . A b r a h a m , M., M. A. H. M c C a u s e l a n d , a n d F . N . H. R o b i n s o n . " D y n a m i c N u c l e a r P o l a r i z a t i o n , " P h y s . R e v . L e t t . 2 ( 1 1 ) , 4 4 9 - 4 5 1 ( 1 9 5 9 ) . A l d e r m a n , D. W. " L i q u i d H e l i u m T e m p e r a t u r e cw NMR S/N I m p r o v e m e n t U s i n g a M O S F E T r f A m p l i f i e r , " R e v . S c i . I n s t r u m . 4J_ ( 2 ) , 1 9 2 - 1 9 7 ( 1 9 7 0 ) . A s l a m , J . a n d W. We y h m a n n . "A T u n n e l D i o d e NMR S p e c t r o m e t e r , " R e v . S c i . I n s t r u m . 4 4 ( 1 ) , 7 1 - 7 2 ( 1 9 7 3 ) . B l a n k e n b u r g , F r e d J . , R o y R. K n i s p e l a n d V . H u g o S c h m i d t . " S e n s i t i v e Low L e v e l T r a n s i s t o r i z e d NMR S p e c t r o m e t e r E m p l o y i n g F r e q u e n c y M o d u l a t i o n , " R e v . S c i . I n s t r u m . 3_7 ( 8 ) , 1 0 2 0 - 1 0 2 3 ( 1 9 6 6 ) . B l o c h , F . , W. W. H a n s e n a n d M. P a c k a r d . " T h e N u c l e a r I n d u c t i o n E x p e r i m e n t , " P h y s . R e v . 7 0 ( 7 a n d 8 ) , 4 7 4 - 4 8 5 ( 1 9 4 6 ) . B l o e m b e r g e n , N., E . M. P u r c e l l a n d R. V . P o u n d . " R e l a x a t i o n E f f e c t s i n N u c l e a r M a g n e t i c R e s o n a n c e A b s o r p t i o n , " P h y s . R e v . 7 3 ( 7 ) , 6 7 9 - 7 1 2 ( 1 9 4 8 ) . d e B o e r , W. a n d T . O. N i i n i k o s k i . " D y n a m i c P r o t o n P o l a r i z a t i o n i n i n P r o p a n e d i o l B e l o w 0 . 5 K," N u c l . I n s t r u m . a n d M e t h o d s . 1 1 4 , . 4 9 5 - 4 9 8 ( 1 9 7 4 ) . d e B o e r , W., M. B o r g h i n i , K. M o r i m o t o , T . 0. N i i n i k o s k i a n d F . U d o . " D y n a m i c P o l a r i z a t i o n o f P r o t o n s , D e u t e r o n s , a n d C a r b o n - 1 3 N u c l e i : T h e r m a l C o n t a c t B e t w e e n N u c l e a r S p i n s a n d a n E l e c t r o n S p i n - S p i n I n t e r a c t i o n R e s e v o i r , " J . Low Temp. P h y s . J_5, 2 4 9 - 2 6 7 ( 1 9 7 4 ) . d e B o e r , W., M. B o r g h i n i , K. M o r i m o t o , T . 0 . N i i n i k o s k i a n d F . U d o . " S i z a b l e P u r e T e n s o r P o l a r i z a t i o n o f D e u t e r o n s i n a S o l i d , " P h y s . L e t t . 4 6 A ( 2 ) , 1 4 3 - 1 4 4 ( 1 9 7 3 ) . B o r g h i n i , M. a n d K. S c h e f f l e r . "A B u t a n o l P o l a r i z e d D e u t e r o n T a r g e t , " N u c l . I n s t r u m . a n d M e t h o d s . 9 5 , 9 3 - 9 8 ( 1 9 7 1 ) . B r a n d w e i n , L e o n a r d a n d Max L i p s i c a s . " A p p l i c a t i o n o f F r e q u e n c y L o c k i n g a n d C o n t r o l t o a n A u t o d y n e O s c i l l a t i n g NMR D e t e c t o r , " R e v . S c i . I n s t r u m . 41_ ( 9 ) , 1 2 9 3 - 1 2 9 5 ( 1 9 7 0 ) . IM2L IH3 C a s h , A. R. " E l e c t r o n i c s f o r N u c l e a r M a g n e t i c R e s o n a n c e S p e c t r o m e t e r , " R u t h e r f o r d L a b o r a t o r i e s R e p o r t No. R L - 7 8 - 1 0 3 , 1 9 7 8 . C o u r t , G. " P o l a r i z a t i o n M e a s u r e m e n t w i t h Q - m e t e r , " i n P r o c e e d i n g s o f t h e S e c o n d W o r k s h o p o n P o l a r i s e d T a r g e t  M a t e r i a l s , e d i t e d b y G. R. C o u r t , S. F . J . C o x , D. A . C r a g g a n d T . 0. N i i n i k o s k i . R u t h e r f o r d a n d A p p l e t o n L a b o r a t o r i e s , C h i l t o n , D i d c o t , O x o n , E n g l a n d , 1 9 8 0 . C o l l i n , R. E . F o u n d a t i o n s f o r M i c r o w a v e E n g i n e e r i n g , M c G r a w - H i l l , T o r o n t o , 1 9 6 6 . D a v i e s , T . K. a n d B. C . M a n n i n g . "A n NMR P r e a m p l i f i e r w i t h I m p r o v e d S i g n a l t o N o i s e P e r f o r m a n c e , " R e v . S c i . I n s t r u m . 44 ( 1 0 ) , 1 5 1 3 - 1 5 1 4 ( 1 9 7 3 ) . D e n k e r , J . " E l e c t r o m a g n e t i c C o m p a t i b i l i t y , " i n Low T e m p e r a t u r e T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n . C o r n e l l U n i v e r s i t y M a t e r i a l s S c i e n c e C e n t e r , I t h i c a , New Y o r k , 1 9 8 1 , p p . 2 5 9 - 2 7 0 . D o r f , R i c h a r d C. M o d e r n C o n t r o l S y s t e m s , 3 r d e d . A d d i s o n - W e s l e y , D o n M i l l s , O n t a r i o , 1 9 8 1 . F r i e d m a n , L . "NMR T e c h n i q u e s , " i n Low T e m p e r a t u r e T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n . C o r n e l l U n i v e r s i t y M a t e r i a l s S c i e n c e C e n t e r , I t h i c a , New Y o r k , 1 9 8 1 , p p . 2 4 7 - 2 5 8 . G r o v e , A . S. a n d D. J . F i t z g e r a l d . " S u r f a c e E f f e c t s o n p - n J u n c t i o n s : C h a r a c t e r i s t i c s o f S u r f a c e S p a c e C h a r g e R e g i o n s U n d e r N o n - E q u i l i b r i u m C o n d i t i o n s , " S o l i d - S t a t e . E l e c t r o n . 9, 7 8 3 - 8 0 6 ( 1 9 6 6 ) . H i l l , D. A . , J . B. K e t t e r s o n , R. C . M i l l e r , A . M o r e t t i , R. C . N i e m a n n , L . R. W i n d m i l l e r , A . Y o k o s a w a , a n d C . F . H w a n g . " D y n a m i c P r o t o n P o l a r i z a t i o n i n B u t a n o l W a t e r B e l o w 1 K," P h y s . R e v . L e t t . 2_3 ( 9 ) , 4 6 0 - 4 6 2 ( 1 9 6 9 ) . H i l l , J o h n J . a n d D a n i e l A . H i l l . " S i m p l e C o r r e c t i o n s f o r Q - M e t e r M e a s u r e m e n t o f P r o t o n T a r g e t P o l a r i z a t i o n , " N u c l . I n s t r u m . a n d M e t h o d s . 1 1 6 , 2 6 9 - 2 7 4 ( 1 9 7 4 ) . H o r o w i t z , P a u l a n d W i n f i e l d H i l l , T h e A r t o f E l e c t r o n i c s , C a m b r i d g e U n i v e r s i t y P r e s s , New Y o r k , NY, 1 9 8 4 . J a c k s o n , J . D., C l a s s i c a l E l e c t r o d y n a m i c s , W i l e y , New Y o r k , NY, 1 9 7 5 . J o h n s o n , B. " H i g h F r e q u e n c y T e c h n i q u e s , " i n Low T e m p e r a t u r e  T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n . C o r n e l l U n i v e r s i t y M a t e r i a l s S c i e n c e C e n t e r , I t h i c a , New Y o r k , 1 9 8 1 , p p . 2 3 1 - 2 4 6 . • HH K l e v e r , H. a n d M. S c h l a a k . "A S i m p l e C i r c u i t f o r cw NMR M e a s u r e m e n t s , " R e v . S c i . I n s t r u m . 44 ( 1 ) , 2 5 - 2 7 ( 1 9 7 3 ) . M c Q u e e n e y , D. " E l e c t r o n i c N o i s e R e d u c t i o n T e c h n i q u e s , " i n Low T e m p e r a t u r e T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n . C o r n e l l U n i v e r s i t y M a t e r i a l s S c i e n c e C e n t e r , I t h i c a , New Y o r k , 1 9 8 1 , p p . 2 0 5 - 2 3 0 . M e s s i a h , A . Q u a n t u m M e c h a n i c s V o l u m e O n e , t r a n s l a t e d f r o m t h e F r e n c h b y G. M. Tem m e r . W i l e y , T o r o n t o , 1 9 7 6 . M e s s i a h , A . Q u a n t u m M e c h a n i c s V o l u m e Two, t r a n s l a t e d f r o m t h e F r e n c h b y J . P o t t e r . W i l e y , T o r o n t o , 1 9 7 6 . M i y o s h i , D. S. a n d R. M. C o t t s . " H e l i u m C o o l e d R a d i o F r e q u e n c y P r e a m p l i f i e r f o r U s e i n NMR," R e v S c i . I n s t r u m . 3 9 ( 1 2 ) , 1 8 8 1 - 1 8 8 4 ( 1 9 6 8 ) . N i i n i k o s k i , T . 0 . , a n d F . U d o . " ' F r o z e n S p i n ' P o l a r i z e d T a r g e t " N u c l . I n s t r u m . a n d M e t h o d s . J _34 , 2 1 9 - 2 3 3 ( 1 9 7 6 ) . N i i n i k o s k i , T . 0 . " M a t h e m a t i c a l t r e a t m e n t o f t h e S e r i e s Q - m e t e r , " i n P r o c e e d i n g s o f t h e S e c o n d W o r k s h o p o n P o l a r i s e d T a r g e t M a t e r i a l s , e d i t e d b y G. R. C o u r t , S. F . J . C o x , D. A . C r a g g a n d T . 0 . N i i n i k o s k i . R u t h e r f o r d a n d A p p l e t o n L a b o r a t o r i e s , C h i l t o n , D i d c o t , O x o n , E n g l a n d , 1 9 8 0 . N o g g l e , J o s e p h H. " S p i n D e c o u p l e r f o r NMR," R e v . S c i . I n s t r u m . 3_5 ( 9 ) , 1 1 6 6 - 1 1 6 9 ( 1 9 6 4 ) . N o i s e i n A m p l i f i e r s , P r i n c e t o n A p p l i e d R e a s e a r c h C o r p o r a t i o n T e c h n i c a l N o t e 2 4 3 , 1 9 7 6 . O g a t a , K a t s u h i k o . M o d e r n C o n t r o l E n g i n e e r i n g , P r e n t i c e - H a l l , T o r o n t o , 1 9 7 0 . P e t f i c e k , V . "A L i n e a r i z e d Q - M e t e r C i r c u i t f o r M e a s u r e m e n t o f H i g h P r o t o n P o l a r i z a t i o n i n a T a r g e t , " N u c l . I n s t r u m . a n d M e t h o d s . 5 8 , 1 1 1 - 1 1 6 ( 1 9 6 8 ) . P o l t u r a k , E . " V i b r a t i o n a l I s o l a t i o n , " i n Low T e m p e r a t u r e T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n . C o r n e l l U n i v e r s i t y M a t e r i a l s S c i e n c e C e n t e r , I t h i c a , New Y o r k , 1 9 8 1 , p p . 3 9 - 4 4 . R a g e s , K a t h y A . , R o b e r t E . S a w y e r a n d E d w a r d B. H a l e . " P r o p e r t i e s o f M i c r o w a v e C a v i t i e s C o n t a i n i n g M a g n e t i c R e s o n a n t S a m p l e s , " R e v . S c i . I n s t r u m . £ 4 ( 7 ) , 8 3 0 - 8 3 4 ( 1 9 7 3 ) . S c h e f f l e r , K. " D y n a m i c P o l a r i z a t i o n o f D e u t e r o n s i n n - B u t a n o l - ( D ! o ) a n d P r o t o n s i n A m m o n i a , " i n P r o c e e d i n g s o f t h e S e c o n d I n t e r n a t i o n a l C o n f e r e n c e o n P o l a r i z e d \H5 T a r g e t s , B e r k l e y , 1 9 7 2 , p p . 2 7 1 - 2 7 9 . S c h m u g g e , T . J . a n d C. D. J e f f r i e s . " H i g h D y n a m i c P o l a r i z a t i o n o f P r o t o n s , " P h y s . R e v . 138 ( 6 A ) , 1 7 8 5 - 1 8 0 1 ( 1 9 6 5 ) . S l i c h t e r , C. P. P r i n c i p l e s o f M a g n e t i c R e s o n a n c e , 2 n d e d . S p r i n g e r - V e r l a g , New Y o r k , 1 9 8 0 . S z e , S. M. P h y s i c s o f S e m i c o n d u c t o r D e v i c e s , 2 n d e d . W i l e y , T o r o n t o , 1 9 8 1 . W a i t , G. D., J . V . C r e s s w e l l , P. P. J . D e l h e i j , M. H a y d e n , D. C . H e a l e y , a n d G. W a t e r s . " T h e T R I U M F P o l a r i s e d D e u t e r o n T a r g e t NMR S y s t e m , " i n P r o c e e d i n g s o f t h e F i f t h  I n t e r n a t i o n a l W o r k s h o p o n P o l a r i z e d S o u r c e s a n d T a r g e t s , M o n t a n a , S w i t z e r l a n d , 1 9 8 6 ( t o b e p u b l i s h e d ) . Z i e r c h e r , E . " A c t i v e Low T e m p e r a t u r e S e m i c o n d u c t o r D e v i c e s , " i n Low T e m p e r a t u r e T e c h n i q u e s , e d i t e d b y B o b R i c h a r d s o n . C o r n e l l U n i v e r s i t y M a t e r i a l s S c i e n c e C e n t e r , I t h i c a , New Y o r k , 1 9 8 1 , p p . 2 7 1 - 2 7 9 . V I . A P P E N D I X A A. V A R A C T O R D E V I C E S I d e a l l y a v a r a c t o r i s s i m p l y a p u r e l y r e a c t i v e d e v i c e w i t h a v a r i a b l e i m p e d a n c e g i v e n b y X c = - j / w C ( V ) ( A . 1 ) w h e r e C ( V c ) i s t h e c a p a c i t a n c e o f t h e d e v i c e , w h i c h i s a f u n c t i o n o f t h e a p p l i e d b i a s v o l t a g e V . I n p r a c t i c e t h i s v o l t a g e v a r i a b l e c a p a c i t y i s o b t a i n e d v i a t h e m a n i p u l a t i o n o f t h e w i d t h o f a d e p l e t i o n r e g i o n a n d t h e a s s o c i a t e d c h a r g e , i n a s e m i c o n d u c t o r m a t e r i a l . T h i s i m m e d i a t e l y s u g g e s t s t h a t t h e r e s u l t i n g d e v i c e i s i n h e r e n t l y l e a k y . T h a t i s , u n d e r m o s t b i a s c o n d i t i o n s i t i s c o n c e i v a b l e t h a t a l e a k a g e c u r r e n t may f l o w a n d t h a t t h i s c u r r e n t i s a f u n c t i o n o f t h e b i a s . T h e p o w e r t h u s d i s s i p a t e d i s , f o r a l l i n t e n t s a n d p u r p o s e s , e q u i v a l e n t t o a r e s i s t a n c e w h o s e v a l u e i s b i a s d e p e n d e n t . T h i s i m p l i e s t h a t when a v a r a c t o r i s u s e d i n t h e r e s o n a n t c i r c u i t o f t h e p h a s e c o m p e n s a t e d NMR s y s t e m , t h e r e s i s t i v e b a c k g r o u n d i m p e d a n c e b e c o m e s a f u n c t i o n o f b i a s a n d h e n c e f r e q u e n c y . T h i s f r e q u e n c y d e p e n d e n t b a c k g r o u n d m u s t b e t a k e n i n t o a c c o u n t w hen t r y i n g t o u n d e r s t a n d t h e b e h a v i o r o f t h e p h a s e c o m p e n s a t e d NMR i n s t r u m e n t a t i o n s c h e m e d e s c r i b e d i n t h i s t h e s i s . T h i s a p p e n d i x t o t h e t h e s i s c o n t a i n s a b r i e f d i s c u s s i o n o f v a r a c t o r d e v i c e s a n d r e l e v e n t IM6 mi p h e n o m e n a . B. T H E S I M P L E V ARACTOR C o n s i d e r a s e m i c o n d u c t o r p n j u n c t i o n w i t h t h e p s i d e b e i n g h e a v i l y d o p e d . T h e o n e d i m e n s i o n a l P o i s o n e q u a t i o n d e s c r i b e s t h e p o t e n t i a l V a s a f u n c t i o n o f p o s i t i o n x i n t h e n r e g i o n . T h a t i s , 9 2 V = -gN a x 7 e ( A . 2 ) w h e r e e g i s t h e p e r m i t i v i t y o f t h e s e m i c o n d u c t o r m a t e r i a l a n d N i s t h e d o p i n g d i s t r i b u t i o n i n t h e n r e g i o n . N = B x m f o r x > 0 ( A . 3 ) I f m = 0, t h e j u n c t i o n i s r e f e r r e d t o a s a b r u p t w h i l e i f m = 1 i t i s r e f e r r e d t o a s a l i n e a r l y g r a d e d j u n c t i o n . H y p e r a b r u p t j u n c t i o n s a r e f o r m e d w h e n m < 0. W i t h t h e b o u n d a r y c o n d i t i o n s t h a t V ( x = 0 ) = 0 a n d V(x=W) = V + V ^ w h e r e W i s t h e d e p l e t i o n w i d t h , V ^ i s t h e b u i l t i n p o t e n t i a l a n d V i s t h e a p p l i e d v o l t a g e , we c a n s o l v e e q u a t i o n A.1 b y i n t e g r a t i o n . T h a t i s , t h e d e p l e t i o n w i d t h i s g i v e n b y 1MB ND-NA F i g u r e A . 1 p n J u n c t i o n D o p i n g w . [ e (m + 2 ) (V + V. . ) S ... D l 1/(m+2) ( A . 4 ) qB a n d t h e d i f f e r e n t i a l c a p a c i t a n c e p e r u n i t a r e a i s c - 1QC = 9V qB(e s) m+1 (m + 2 ) ( V + V b.) l/(m+2) ( A . 5 ) w h e r e Q_ = e _ E _ = v > i s t h e c h a r g e p e r u n i t a r e a a n d E „ „ i s C 5 U13X IUclX t h e m a x i m u m e l e c t r i c f i e l d w h i c h o c c u r s a t x = 0 . T h e e x p o n e n t o f e q u a t i o n A . 5 s = - d C V_ = - d ( l n C ) = _ 1 _ ( A . 6 ) C d V d ( l n V ) m+2 i s a m e a s u r e o f t h e s e n s i t i v i t y o f t h e c a p a c i t a n c e t o t h e a p p l i e d b i a s v o l t a g e V . R a t h e r t h a n a n a l y z e t h i s s i m p l e m o d e l o f v a r a c t o r b e h a v i o r f u r t h e r , i t i s m o r e i n s t r u c t i v e f o r t h e p u r p o s e s o f t h i s t h e s i s t o e x a m i n e t h e b e h a v i o r o f a M O S F E T s t r u c t u r e u s e d a s a v a r a c t o r d e v i c e . T h e s e d e v i c e s p r o b a b l y b e s t e x e m p l i f y t h e v a r i o u s t y p e s o f e f f e c t s w h i c h may b e o b s e r v e d a n d a r e c o m m o n l y u s e d i n l o w t e m p e r a t u r e e l e c t r o n i c w o r k . T h e p r o t o t y p e s y s t e m b u i l t a t T R I U M F u s e s t h e g a t e t o d r a i n a n d g a t e t o s o u r c e c a p a c i t i e s o f t h e N E 4 1 1 3 7 F E T . C . T H E M O S F E T AS A V A R A C T O R C o n s i d e r t h e n c h a n n e l M O S F E T s t r u c t u r e s h o w n i n f i g u r e A . 2 . T h e s o u r c e , d r a i n a n d b a c k s i d e o f t h e d e v i c e a r e g r o u n d e d w h i l e t h e g a t e i s h e l d a t some p o t e n t i a l V . T h e r e a r e t h r e e m a j o r i d e n t i f i a b l e b i a s i n g c o n d i t i o n s f o r t h i s d e v i c e . A p p l i c a t i o n o f a n e g a t i v e v o l t a g e ( V < 0) t o t h e g a t e c a u s e s t h e v a l e n c e b a n d i n t h e s u b s t r a t e t o b e n d u p w a r d s , c l o s e r t o t h e F e r m i l e v e l ( w h i c h r e m a i n s c o n s t a n t i n t h e s e m i c o n d u c t o r f o r z e r o c u r r e n t f l o w ) . T h e r e s u l t i s a n a c c u m u l a t i o n o f m a j o r i t y c a r r i e r s ( h o l e s i n t h i s c a s e ) u n d e r t h e o x i d e b e l o w t h e m e t a l g a t e . T h i s s i t u a t i o n i s r e f e r r e d t o a s a c c u m u l a t i o n a n d 1 5 0 Vc | — c . METAL p /77 vOX'DE GATE / ^7 /OHMIC CONTACT SOURCE n DRAIN F i g u r e A . 2 T h e M O S F E T i s s h o w n i n f i g u r e s A . 3 a n d A . 4 . GATE I V c >0 F i g u r e A . 4 M O S F E T i n A c c u m u l a t i o n A p p l i c a t i o n o f a s m a l l p o s i t i v e v o l t a g e d r i v e s t h e m a j o r i t y c a r r i e r s a w a y f r o m t h e g a t e r e g i o n . T h e e n e r g y 151 METAL p / / / / / / A / / /// V<0 p-SEMICONDUCTOR + + + + 4-+ - E, F i g u r e A . 3 M O S F E T i n A c c u m u l a t i o n b a n d s b e n d down a s s h o w n i n f i g u r e s A . 5 a n d A . 6 . T h i s s i t u a t i o n i s r e f e r r e d t o a s d e p l e t i o n . F i n a l l y w hen a l a r g e p o s i t i v e v o l t a g e i s a p p l i e d , i t i s p o s s i b l e t o d r i v e t h e e n e r g y b a n d s down f a r e n o u g h s o t h a t t h e i n t r i n s i c F e r m i l e v e l E . c r o s s e s t h e F e r m i l e v e l E , a t t h e s u r f a c e . T h e / f r e s u l t i s a c o l l e c t i o n o f m i n o r i t y c a r r i e r s ( e l e c t r o n s i n t h i s c a s e ) u n d e r t h e s u r f a c e . T h i s s i t u a t i o n i s k n o w n a s i n v e r s i o n . I t i s d i a g r a m m e d i n f i g u r e s A . 7 a n d A . 8 . C o n s i d e r n e x t t h e g e n e r a t i o n o f e l e c t r o n - h o l e p a i r s i n t h e d e v i c e w h i c h l e a d t o a l e a k a g e c u r r e n t . T h e r e a r e t h r e e m a i n r e g i o n s w h e r e t h e s e g e n e r a t i o n s i t e s o c c u r . T h e y i n c l u d e t h e d e p l e t i o n r e g i o n s d u e t o t h e m e t a l l u r g i c a l \5Z METAL p-SEMICONDUCTOR - E F i g u r e A . 5 M O S F E T i n D e p l e t i o n j u n c t i o n s n e a r t h e s o u r c e a n d t h e d r a i n o f t h e d e v i c e , t h e d e p l e t i o n r e g i o n o f t h e f i e l d i n d u c e d j u n c t i o n b e l o w t h e m e t a l g a t e , a n d g e n e r a t i o n c e n t r e s a t t h e o x i d e / s e m i c o n d u c t o r i n t e r f a c e . T o a f i r s t o r d e r a p p r o x i m a t i o n t h e c o n t r i b u t i o n t o t h e l e a k a g e c u r r e n t f r o m t h e d e p l e t i o n r e g i o n s d u e t o t h e m e t a l l u r g i c a l j u n c t i o n s i s i n d e p e n d e n t o f b i a s . T h a t i s , a s t h e v o l u m e o f t h e s e r e g i o n s i s r o u g h l y c o n s t a n t , t h e n u m b e r o f e l e c t r o n / h o l e g e n e r a t i o n c e n t r e s i s r o u g h l y c o n s t a n t ( r e f e r t o f i g u r e A . 9 ) . T h e c o n t r i b u t i o n d u e t o t h e f i e l d i n d u c e d j u n c t i o n i s n o n e x i s t e n t u n d e r a c c u m u l a t i o n a s t h e r e i s n o f i e l d i n d u c e d d e p l e t i o n r e g i o n . T h i s c o n t r i b u t i o n r i s e s t h r o u g h o u t d e p l e t i o n a s t h e f i e l d i n d u c e d d e p l e t i o n r e g i o n ( a n d h e n c e IS3 J F i g u r e A . 6 M O S F E T i n D e p l e t i o n t h e a s s o c i a t e d e l e c t r o n / h o l e p a i r g e n e r a t i o n c e n t r e s ) i s f o r m e d a n d i t may e i t h e r l e v e l o f f o r c o n t i n u e t o i n c r e a s e s l o w l y u n d e r i n v e r s i o n . T h i s e f f e c t i s i n d i c a t e d i n f i g u r e A . 1 0 . T h e t h i r d c o n t r i b u t i o n i s d u e t o t h e c e n t r e s c r e a t e d b y t h e i n t e r r u p t i o n o f t h e p e r i o d i c l a t t i c e s t r u c t u r e a t t h e s u r f a c e o f t h e s e m i c o n d u c t o r m a t e r i a l . O f c o u r s e , t h e s e c e n t r e s c a n n o t c o n t r i b u t e t o t h e l e a k a g e c u r r e n t w hen t h e s u r f a c e i s e i t h e r a c c u m u l a t e d o r i n v e r t e d . T h e r e s u l t i s a p e a k i n t h e l e a k a g e c u r r e n t d u r i n g d e p l e t i o n a s s h o w n i n f i g u r e A . 1 1 . T h e sum o f t h e t h r e e e f f e c t s i s s h o w n i n f i g u r e A . 1 2 . T h e t o t a l l e a k a g e c u r r e n t c a n b e r e l a t e d t o t h e p o w e r METAL V>0 E F TT77T71TTTTTT7 F i g u r e A . 7 M O S F E T i n I n v e r s i o n d i s s i p a t i o n i n t h e d e v i c e ( q u a l i t a t i v e l y ) a n d t h u s g i v e s a g o o d i n d i c a t i o n o f t h e e f f e c t i v e r e s i s t i v e i m p e d a n c e o f t h e d e v i c e a s a f u n c t i o n o f b i a s . T h e p h e n o m e n a w h i c h h a v e b e e n d e s c r i b e d a b o v e a r e e a s i l y o b s e r v e d w i t h t h e NMR Q m e t e r d e s c r i b e d i n t h i s t h e s i s . C h a p t e r 4 i n c l u d e s a d i s c u s s i o n o f how t h i s v o l t a g e v a r i a b l e r e s i s t a n c e l e a d s t o c e r t a i n r e s t r a i n t s u p o n t h e o p e r a t i o n o f t h e s y s t e m . ACCUMULATION z UJ a: tr. 3 O UJ CO cr UJ > UJ oc DEPLETION INVERSION F i g u r e A . 9 R e v e r s e C u r r e n t F l o w i n D e p l e t i o n R e g i o n o f M e t a l l u r g i c a l J u n c t i o n F i g u r e A . 1 0 R e v e r s e C u r r e n t F l o w i n D e p l e t i o n R e g i o n o f F i e l d I n d u c e d J u n c t i o n ACCUMULATION I DEPLETION , INVERSION F i g u r e A.1 1 R e v e r s e C u r r e n t F l o w D u e t o S u r f a c e G e n e r a t i o n 157 ACCUMULATION DEPLETION INVERSION F i g u r e A . 1 2 T o t a l L e a k a g e C u r r e n t ( Q - m e t e r h a s h i g h e s t Q i n a c c u m u l a t i o n 

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