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Spin-lattice relaxation in gaseous methane Beckmann, Peter Adrian 1971

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SPIN-LATTICE RELAXATION IN GASEOUS METHANE by  PETER ADRIAN BECKMANN B.Sc.  University of B r i t i s h Columbia, 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Physics We accept t h i s thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1971  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  I further  his  of  this  written  the  University  of  British  s h a l l make i t f r e e l y a v a i l a b l e  agree t h a t permission  for scholarly by  in p a r t i a l f u l f i l m e n t of  p u r p o s e s may  representatives. thesis for  be  for  requirements  Columbia, I agree r e f e r e n c e and  for extensive copying of  g r a n t e d by  the  Head o f my  It i s understood that  financial gain  s h a l l not  be  Physics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  April  16,  Columbia  1971.  that  thesis  Department  or  publication  allowed without  permission.  Department of  for  study.  this  copying or  Peter  Date  the  Beckmann  my  - ii ABSTRACT The i n gaseous CH^  s p i n - l a t t i c e r e l a x a t i o n time T-| has "been measured as a function of density at room temperature.  The  density region investigated i s from 0.006 to 7.0  amagats and T-j  passes through a minimum near 0.0k  spin-rotation  amagats.  The  i n t e r a c t i o n i s the dominant r e l a x a t i o n mechanism i n gaseous Since the spin-rotation constants are accurately known for  CH^. CH^,  the r e s u l t s provide a check on the e x i s t i n g theory of s p i n - l a t t i c e r e l a x a t i o n for spherical top molecules. was  An i n t e r e s t i n g feature  the f a i l u r e of commonly used t h e o r e t i c a l expressions for the  density dependence of T-j to f i t the experimental data. reasonable explanation  A  i s that the c e n t r i f u g a l d i s t o r t i o n of  CH^ molecule i s i n d i r e c t l y contributing to the s p i n - l a t t i c e relaxation.  the  - iii TABLE OF CONTENTS  Page Abstract  i i  L i s t of I l l u s t r a t i o n s  V .  Acknowledgements  vii  Chapter I  Introduction  1  Chapter II  Basic Concepts  3  2.1  Nuclear Magnetic Resonance ....  3  2.2  S p i n - l a t t i c e Relaxation  10  Chapter I I I  Chapter IV  The Experiment  Ik  3.1  Signal Detection  Ik  3.2  Models for Pulsed N.M.R. Experiments  15  3.2.1  Free Induction Decay  15  3.2.2-  Spin-echo  17  3.3  Experimental Procedure  18  3.4  T  Calculation  23  3-5  Error Analysis  25  1  The Apparatus  29  k.1  Pulsed N.M.R. Spectrometer ....  29  Transmission Stage  30  /f. 1.2  Tuned C i r c u i t Stage .,  3k  4.1.3  Receiving Stage  36  4.2  Gas Handling System  39  1.1  - i v-  Page Chapter V  S p i n - l a t t i c e R e l a x a t i o n i n CH^ and t h e I n t e r p r e t a t i o n o f the Experimental  W5  Results  C h a p t e r VI  Appendix Bibliography  5.1  Nuclear I n t e r a c t i o n s  Wb  5.1.1  Zeeman L e v e l s  kk  5.1.2  Perturbation Interactions  5.2  T 1 i n CH/,  hi  5.3  Data A n a l y s i s  60  Wi>  Summary and S u g g e s t i o n s f o r F u r t h e r Work  77  C i r c u i t Diagrams  79 95  - V -  LIST OF ILLUSTRATIONS  Page  Figure  3.1  Free induction decay  16  3.2  Spin-echo  19  4.1  Pulse spectrometer: schematic diagram  31  4.2  Pulse sequences  32  4.3  Tuned c i r c u i t stage  35  4.4  Free induction s i g n a l : phase s e n s i t i v e detection  37  4.5  Gas handling system  40  5.1  Relaxation e f f e c t s of c e n t r i f u g a l d i s t o r t i o n  55  5.2-  C o l l i s i o n induced f i e l d s  56  1 61  5.3  Experimental data:  5.4  Single term f i t around region of — T r  5.5  Q maximum  63 65  H  Single term f i t using l i n e a r region 'l  5.7  vs  1  P  5.6  —  vs  p  66 71  •1  1  5.8  vs IJ exp.  T1  P  73  - vi -  Page  Figure  5.9  vs 1J C^max  5.10  Pmax  vs  75  76  A. 1  D i f f e r e n t i a t o r D and Mixer E .  80  A.2  Pulse width generator F (or G)  81  A.3  Mixer H  82  A.k  Gating pulses a m p l i f i e r I ....  83  A.5  10 MHz C r y s t a l o s c i l l a t o r J ..  84  A. 6  Wideband a m p l i f i e r K  85  A.7-  Coherent gated o s c i l l a t o r L ..  86  A.8  Phase s h i f t e r M and T r i p l e r N  87  A.9  Gated power a m p l i f i e r 0  88  A. 10  Reference t r i p l e r P  89  A. 1 1  Reference a m p l i f i e r Q  90  A. 12  Preamplifier R  91  A. 13  Boxcar i n t e g r a t o r U  92  A. 1 ^  Power supply f o r Amplifier 0 .  93  A. 15  Power supply for J , K and P ..  94  - vii-  ACKNOWLEDGEMENTS  I s i n c e r e l y thank Professor Myer Bloom and Dr. E l l i o t t Burnell f o r t h e i r assistance over the past two years.  I find  Professor Bloom's conscientious a t t i t u d e toward h i s students most encouraging and I am g r a t e f u l for the many f r u i t f u l have had with him.  discussions I  Dr. Burnell aided me i n performing many of  the lengthy experiments. This work was supported, Research council o f Canada.  i n part, by the National  CHAPTER I INTRODUCTION The  aim o f t h i s work i s t o i n v e s t i g a t e t h e mechanism  o r mechanisms which  cause n u c l e a r s p i n r e l a x a t i o n i n gaseous  t e t r a h e d r a l m o l e c u l e s i n t h e v i c i n i t y o f t h e T^ minimum. t-echnique i s t h a t o f a p u l s e d n u c l e a r m a g n e t i c experiment.  The  resonance  The 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 T^ i s o b t a i n e d a s  a f u n c t i o n o f d e n s i t y a t c o n s t a n t room t e m p e r a t u r e .  I t turns  out t h a t T- goes t h r o u g h a minimum a t a p p r o x i m a t e l y 0.04 amagats and t h e experiment to 7-0 amagats.  i s performed  o v e r t h e range o f from 0.006 t o  The m o l e c u l e b e i n g i n v e s t i g a t e d i s methane (CH^) ip  which has a s p i n l e s s carbon n u c l e u s ( spin  C) a t i t s c e n t r e and f o u r  protons at the corners o f a tetrahedron. R e l a x a t i o n i n CHzj. has been i n v e s t i g a t e d u s i n g t h e s e  t e c h n i q u e s by Dorothy  ( 1 9 6 7 ) , Bloom and Dorothy  ( 1 9 6 7 ) , Bloom,  B r i d g e s and Hardy ( 1 9 6 7 ) , Dong (1969) and Dong and Bloom ( 1 9 7 0 ) . The  s p i n - r o t a t i o n i n t e r a c t i o n which can be shown t o be t h e  most i m p o r t a n t cause o f r e l a x a t i o n i n CH^ (Bloom, B r i d g e s and Hardy, 1967) has been i n v e s t i g a t e d u s i n g m o l e c u l a r beam t e c h n i q u e s by Anderson and Ramsey ( 1 9 6 6 ) , Y i ( 1 9 6 7 ) , Y i , O z i e r , K h o s i z and Ramsey ( 1 9 6 7 ) , O z i e r , Crapo and Lee (1968) and Y i , O z i e r and Ramsey ( t o be p u b l i s h e d ) .  The experiment  was o r i g i n a l l y  i n t e n d e d t o be a check on p r e v i o u s l y o b t a i n e d r e s u l t s b e f o r e  - 2 g o i n g on t o t e t r a h e d r a l m o l e c u l e s f o r which T i had n o t been p r e v i o u s l y measured.  However, w i t h c e r t a i n m o d i f i c a t i o n s to t h e  a l r e a d y e x i s t i n g a p p a r a t u s , g r e a t e r a c c u r a c y was a c h i e v e d as a r e s u l t o f g r e a t e r s i g n a l t o noise, and many more r u n s v/ere p e r f o r m e d t h a n i n p r e v i o u s works.  The r e s u l t s a r e more r e l i a b l e  and i n d i c a t e t h e p r e s e n c e o f some f i n e s t r u c t u r e i n t h e dependence of  T  1  on d e n s i t y . The r e m a i n i n g C h a p t e r s o f t h i s work a r e as f o l l o w s .  Chapter I I i n v o l v e s b a s i c concepts i n n u c l e a r magnetic and s p i n - l a t t i c e r e l a x a t i o n . case o f r a r e g a s e s .  resonance  The o n l y s p e c i a l i z a t i o n i s t o t h e  P e r s o n s r e l a t i v e l y new i n t h e f i e l d a r e  r e f e r r e d t o t h e c l a s s i c t e x t s o f Abragam ( 1 9 6 1 ) , Andrew (1955) and S l i c h t e r ( 1 9 6 3 ) . Chapter I I I i n v o l v e s o n l y the experiment; the procedures w h i c h a r e used t o measure and c a l c u l a t e T^ as a f u n c t i o n o f density.  T h i s does n o t i n v o l v e any t h e o r e t i c a l  considerations  o t h e r t h a n t h o s e which have to be met i n o r d e r t o s a t i s f a c t o r i l y perform the experiment.  I n Chapter IV, the apparatus i s  d i s c u s s e d i n d e t a i l and c i r c u i t diagrams f o r t h e p u l s e s p e c t r o m e t e r may be found i n t h e A p p e n d i x . The concept o f s p i n - l a t t i c e r e l a x a t i o n f o r CH^ through the s p i n - r o t a t i o n i n t e r a c t i o n i s d i s c u s s e d i n Chapter V and t h e e x p e r i m e n t a l d a t a i s i n t e r p r e t e d u s i n g t h e e x i s t i n g theory. The p o s s i b i l i t y o f f u t u r e f r u i t f u l work i n t h e f i e l d i s d i s c u s s e d , a l o n g w i t h a summary o f t h i s t h e s i s , i n Chapter VI.  CHAPTER I I BASIC IDEAS 2. 1 .  NUCLEAR MAGNETIC RESONANCE Some n u c l e i p o s s e s s an i n t r i n s i c m a g n e t i c p r o p e r t y o r c a l l e d t h e magnetic moment jX g i v e n by  observable  M = 7til  (2.D  where I i s t h e a n g u l a r momentum o f t h e n u c l e u s and ^  is its  gyromagnetic  ratio.  7 -  "  m c p  where e and nip a r e t h e charge and mass o f a p r o t o n and g i s t h e so-called g-factor. of l i g h t .  G a u s s i a n u n i t s a r e used and c i s t h e speed  When a m a g n e t i c f i e l d H  of I along H  0  Q  i sapplied,  the p r o j e c t i o n  i s q u a n t i z e d and can o n l y assume v a l u e s - I , -1+1,  , 1-1, I The i n t e r a c t i o n between t h e f i e l d H  Q  and t h e s p i n s  ( n u c l e i p o s s e s s i n g a magnetic moment) i s d e s c r i b e d by t h e Hamiltonian  -  W  h  -  (2  =-At-Ho  where fX i s u n d e r s t o o d t o be a quantum m e c h a n i c a l v e c t o r A laboratory coordinate that  H  0  d e f i n e d by assuming  Q  W  (2.1), e q u a t i o n  =  With,  (2.2) becomes  -7nH0Iz  p o s s i b l e s t a t e s o f t h e system a r e d e n o t e d by |m> and t h e  observables associated with the operator  L im> The  operator.  = H i i where k i s a u n i t v e c t o r i n t h e z d i r e c t i o n .  the a i d o f equation  The  system i s p a r t i a l l y  -  2)  allowed  I  a r e d e n o t e d by m.  = m im  energy v a l u e s a r e  W|m>  E  m  = E lm: m  = -7Ti H m 0  Note t h a t f o r ^ > 0 , m = - I has t h e h i g h e s t i n t e r a c t i o n energy . w i t h t h e f i e l d whereas m = +1 has t h e l o w e s t . Any  measurement made i n an e x p e r i m e n t w i l l  involve  many s p i n s and t h e p a r a m e t e r measured w i l l be an a v e r a g e o v e r an ensemble o f s p i n s .  F o r a system o f l i k e s p i n s i n e q u i l i b r i u m  w i t h a b a t h a t t e m p e r a t u r e T i t i s c o n v e n i e n t t o use t h e  - 5 Canonical Ensemble where one knows the t o t a l volume, the t o t a l number of spins and the temperature of the bath with which they are i n thermal contact.  This bath or l a t t i c e as i t i s c a l l e d  can be taken to be associated with the other degrees of freedom of the molecule i n which the n u c l e i e x i s t s .  In the Canonical  Ensemble, the p r o b a b i l i t y of a spin being i n the m^  h  state i s  given by  m  +1  I > [ - w ] m = -I In an equilibrium ensemble of N spins, there w i l l be a net magnetization along the z axis given by  M =£  ( c o n t r i b u t i o n of a spin i n mth state)  0  v A  ( f r a c t i o n of spins i n m . state) th  /_ v  a l l m values The entire i d e a of our experiment i s to measure how of spins returns to t h i s value of M  Q  the system  i f the system i s perturbed.  For completeness, we w i l l b r i e f l y evaluate equation (2.3) even though we do not use i t d i r e c t l y .  M  0  +1 = E m = -I  MN P Z  M  o  ^ ;  - 6 -  = i>7fimP  m  m =- l Because the sum i n the denominator of P t o t a l sum,  m  i s independent of the  we have  +1  L  M= 0  N  7  ^  m  ex  P[  kT  J  m =- l  . +1  E  r  7T\  HO m  m =- l I f one examines the magnitude of ^11H m/kT ^ 0  and laboratory conditions (room temperature), i s the order of 10"^ kilogauss.  for a spin  1  o r  t y p i c a l spins one f i n d s that i t  species i n a f i e l d of 7  With t h i s i n mind, i f one expands the  exponentials  and keeps terms of order m (high temperature approximation) the r e s u l t i s the Curie  Law:  N/VKI + D ,, M  °  3kT  =  Just as M  0  '  i s a sum over the jU. 's, the t o t a l z  magnetization M i s a vector sum over the i n d i v i d u a l yU's.  The  equation of motion for M i n a magnetic f i e l d r e s u l t s from the torque exerted on i t by i t s i n t e r a c t i o n with the main f i e l d i s given by  and  Note that H  i s i n the z d i r e c t i o n and  n  ^ dt  fvi  =  2  Equation ( 2 . 4 )  -  0  constant  M  =  0  implies that the vector c h a r a c t e r i z i n g the  magnetization i s precessing about the z axis.  To see t h i s ,  i t i s convenient to transform the equation of motion i n t o a reference frame r o t a t i n g with an a r b i t r a r y angular  u  about the d i r e c t i o n of the f i e l d H . Q  frequency  For our case the  equation of motion i n the r o t a t i n g reference frame i s given by  HM  -  —  A  = M x (7H -wk) 0  Equation (2.5)  dM  -j7 dt  (2.5)  i s conveniently rewritten as  -  = M x ( u0 -u  A  )k  (2.6)  where U0 =  7o H  defines the Larraor  equation (2.6) i t i s immediately frame r o t a t i n g at [J = LJo,  angular frequency.  From  obvious that i n the reference  the magnetization i s s t a t i c .  Working  backwards, i t i s evident that M i s precessing about the z axis i n the laboratory frame with an angular frequency of magnitude  UQ  .  Note that both frames of reference have the same z axis. The next step i n our development i s to introduce an  o s c i l l a t i n g f i e l d i n the x (or y) d i r e c t i o n of the laboratory frame.  This i s done by s e t t i n g up an o s c i l l a t i n g current i n a  solenoid, the sample under study being i n s i d e the solenoid.  One  can think of t h i s f i e l d as being the vector sum of 2 c i r c u l a r l y p o l a r i z e d f i e l d s with t h e i r p o l a r i z a t i o n s i n opposite senses. The r o t a t i n g frame we use i s defined by that one of these r o t a t i n g f i e l d s which i s constant i n the xy plane of the r o t a t i n g frame.  I f we define the x d i r e c t i o n i n the r o t a t i n g frame by the  d i r e c t i o n of the new f i e l d , we have  H, =  h/i  and the equation of motion i n the r o t a t i n g reference frame becomes  jY  '- M x [{u0 -u)k  where LJ^= ffR-\ defines LJ ].  + ufi]  V/e have ignored the other  (2 7)  -  component  of the o r i g i n a l o s c i l l a t i n g f i e l d which i s r o t a t i n g with an angular frequency -ZU  with respect to the r o t a t i n g frame we  - 9 have chosen.  In p r a c t i c e , H - | « H  0  and Abragam ( 1961)  shows that  t h i s f i e l d causes very small e f f e c t s which we can neglect. Equation (2.7)  can be misleading at f i r s t sight and one must  remember the following.  The magnitude of a f i e l d r o t a t i n g with  an angular frequency LJ with respect to the laboratory frame i s described by LJ\ .  The magnitude of the constant magnetic f i e l d i s  described by UQ  I f we write equation (2.7)  .  dM dt  = 7M  x  H  as  G f f  the magnetization i s now precessing about an e f f e c t i v e f i e l d i n the r o t a t i n g frame given by  Because U-\  ( ijJo - U  i s very small, for most values of LJ],  )  U]  «  and the magnetization i s almost unaffected by H] .  That i s , the s o l u t i o n to equation (2.7)  i s not very d i f f e r e n t  from the s o l u t i o n to equation ( 2 . 6 ) . The i n t e r e s t i n g case i s when we apply the r o t a t i n g f i e l d at U  = UQ  ^  .  Equation (2.7)  = M x  becomes  U/,f  (2-8)  The magnetization now precesses about the f i e l d H-|i i n the r o t a t i n g frame.  This i s a resonance  phenomenon and i s appropriately  -  10  -  named nuclear magnetic resonance.  I f we pulse the f i e l d H-j at  the resonance frequency for a predetermined time, M can be  left  at any o r i e n t a t i o n with respect  of  to the z axis.  A 7T pulse  the f i e l d  rotates M by 180° about the x axis and takes M  -M .  pulse e f f e c t i v e l y puts the magnetization i n the  z  A 77/2  plane and so The  into  z  xy  on. e f f e c t of the pulsed radio frequency i s to cause  the spin system to depart from i t s equilibrium p o s i t i o n providing the angle of r o t a t i o n i s not a multiple of 2.7T• Then i t w i l l tend to relax back to i t s equilibrium value.  Another way  of  s t a t i n g t h i s i s that we have perturbed the system from i t s lowest energy configuration and i t w i l l  s t r i v e for i t s equilibrium  s i t u a t i o n by g i v i n g up energy to the other molecular degrees of freedom which comprise the " l a t t i c e "  2.2.  or "bath".  SPIN-LATTICE RELAXATION Some kind of model i s required to explain the approach  to equilibrium of the spin system a f t e r i t has been perturbed. The  s p i n - l a t t i c e r e l a x a t i o n time T  1  i s l o o s e l y defined as a  c h a r a c t e r i s t i c time for the component of the magnetization along the f i e l d d i r e c t i o n to approach i t s equilibrium value.  A T-|  mechanism involves ah exchange of energy between the spins the l a t t i c e .  There may  and  be i n t e r a c t i o n s between spins which  r e d i s t r i b u t e energy among the spin system but do not change the t o t a l energy.  For simple systems t h i s r e l a x a t i o n i s described  by a s i n g l e constant T^,  the spin-spin r e l a x a t i o n time and  involves r e l a x a t i o n of the components of M perpendicular  to the  - 1 1 constant f i e l d .  Although T  2  i s not the subject of our discussion  we must l a t e r consider i t s e f f e c t s when v/e perform the pulsed nuclear magnetic resonance  experiments.  The microscopic mechanisms for s p i n - l a t t i c e r e l a x a t i o n for the s p e c i a l case of spherical top molecules such as CH^ are discussed i n Chapter V.  In our present discussion, the aim i s to  obtain a general formula for T-| i n terms of the t o t a l magnetization M which i s a macroscopic  and measureable observable.  The T^  expression we develop i n the microscopic theory i n Chapter V w i l l be used to i n t e r p r e t the experimental data which i s obtained by applying the expression for T^ we now macroscopic  develop using the  theory.  V/e can modify the equation of motion (2.7) r e l a x a t i o n a f t e r a perturbation.  to include  The modified equation of motion  which holds very v/ell for gases along with the concepts of Tj and T  2  i n nature and are due to Bloch ( 1 9 4 6 ) .  are phenomenological  dM  —  d t  =  -  r  M x [{u0 -u)k L  ^  i  A  + u,\]  T1 S t a r t i n g with the general equation (2.9)  1  T  2  i n the r o t a t i n g frame,  we can s p e c i a l i z e i t for our pulse techniques.  S t a r t i n g with  the equilibrium s i t u a t i o n where LJ-j has not been introduced, . we- have  -  dM dt  12 -  A  -u )k  M x ( u0  dM dt  M  z  7  . = 0  = M,  The system i s perturbed at the resonant frequency i n a time short enough that r e l a x a t i o n e f f e c t s are negligible.-  dM U  l  dt  — =  M  A X  (2.10)  I  As discussed previously, t h i s means that M precesses about the x d i r e c t i o n i n the r o t a t i n g frame.  Rather than solve equation ( 2 . 1 0 )  d i r e c t l y , we just leave LJ. on long enough to take M_ from i t s I z equilibrium value of M  0  to a non equilibrium value of M (0) .  In the experiment, the s p e c i a l case of a 7T pulse i s used which means M„(0) = -M_.  This i s as much as the system can be perturbed.  A f t e r the pulse, the equation of motion i n the frame r o t a t i n g at the Larmour angular frequency becomes  - 13 -  d t  Ti  T,  Note that i n the laboratory frame, the  (2.11)  part of equation  w i l l have factors i n v o l v i n g s i n ( L / t ) and cos(Uy t) but i n both 0  o  frames  d M d t  M T  n  -  M  A  ,  z  t  s  (2.12)  (2.12) i s e a s i l y solved and M (t) i s given by  Equation  M  z  ( t )=  2  M o - [Mo-M (0)][exp(-i)] z  and with the i n i t i a l condition, H (0) z  = -M  0  (2  .  imposed by a ff  13)  pulse  we have  M  z  ( t ) =  M o [ 1 - 2 e x p ( - y ) ]  Thus from a measurement of M  z  (2.H)  as a function of time a f t e r a  s u i t a b l e perturbation, v/e can uniquely determine T i .  CHAPTER I I I THE EXPERIMENT 3.1  SIGNAL DETECTION The  same c o i l i s used to d e l i v e r the radio frequency  pulse and detect the nuclear magnetic resonance s i g n a l . dependent net magnetization M(t)  A time  f o r the sample i n the c o i l  will  be proportional to a time dependent magnetic induction B ( t ) .  The  s i g n a l voltage induced i n the c o i l w i l l be  v  S  ( t )  = - j j f y  where the t o t a l magnetic f l u x (|)(t) i s given by  »(t)  = AB (t)  ( 3  x  -  2 )  The cross section area of the c o i l i s A and v/e have assumed the c o i l i s wound around the x axis i n the laboratory frame. The reference sine wave V (Cl/ ) from the reference r  q  a m p l i f i e r i s used for phase s e n s i t i v e detection i n the tuned amplifier.  Taking care to adjust V-^OJ^  such that the a m p l i f i e r  i s l i n e a r over a l l desired a m p l i f i c a t i o n regions, the output w i l l  15  -  -  be p r o p o r t i o n a l to the magnitude of Vs(U0 ) The s i g n a l Vg(tu/ )  f i g u r e 4-4.) of the UQ  0  component of the flux  and ( 3 . 2 )  at time t.  (See  i s p r o p o r t i o n a l to the magnitude (|)(t).  From equations  (3.1)  and the previous discussion i n t h i s section i t i s  c l e a r that Vg(L/ 0 ) i s p r o p o r t i o n a l to the magnitude of the component of M (t) x  with U  = L/Q.  To summarize, we can measure  a voltage p r o p o r t i o n a l to the Larmor  frequency component of the  magnetization i n the xy plane.  3.2  PULSED N. M. R.  EXPERIMENTS  The information we wish to obtain concerns Mz as a function of time a f t e r a s u i t a b l e perturbation.  However, v/e can  only measure a net magnetization i n the xy plane.  We  can prepare  a net magnetization i n the xy plane which i s p r o p o r t i o n a l to an i n i t i a l z component of the magnetization by using pulse techniques. V/e w i l l examine the'two types of pulse techniques used i n our experiments  3.2.1  by examining  some simple models.  FREE INDUCTION DECAY S t a r t i n g with the magnetization i n the equilibrium  p o s i t i o n with M take M  z  z  = M, Q  into M ( 0 ) . z  our experiments  the a p p l i c a t i o n of an r. f. pulse w i l l Figure 3.1  where a 7T pulse gives M ( 0 )  figure 3* 1 j we have A — > B . i n figure 3.1 time.)  shows the s p e c i a l case used i n z  = -M . 0  In terms of  (The primed and double primed  correspond to the same event for a second and  The spins w i l l now  relax according to equation  letters third  (2.13)  - 16 -  figure  3.1  free  induction  decay  t  —4  2  A  i  iC  A'  B  B'  i  A'  C D  (not  B'  to s c a l e )  D  At  A *  PS  -AP--y  i  i  A  B  which reduces to equation (2.14) for the s p e c i a l case of a pulse.  In a time t we  have B — > C i n figure 3.1-  pulse w i l l rotate M (t) z  by C — > D i n figure 3.1.  We  7T  At t h i s time  into the y d i r e c t i o n  indicated  are i n the r o t a t i n g frame and  perturbing f i e l d i s i n the x d i r e c t i o n . have the r e s u l t that a signal i s now  the  From Chapter 3-1»  detected.  v/e 77/2  After the  pulse, the s i g n a l w i l l be p o s i t i v e or negative depending on whether M„(t)  was  p o s i t i v e or negative.  This magnetization i n  Li  the xy plane w i l l decay according to the T (2.11).  2  part of equation  pulse, the  factor which describes the l o s s of signal due  to  e f f e c t s i n a rigorous expression for the amplitude of the becomes a constant and Chapter 3.1  the measured voltage Vs(t)  becomes proportional  to M_(t).  The  shows three measurements for increasing  t'' and i n d i c a t e s how  signal  discussed i n  s i g n a l i s sampled  by a measuring pulse indicated by E i n figure 3.1.  and  77/2  I f a measurement i s made a constant time a f t e r the  3.1  Figure  values of t, namely t, t'  the signal decays from -M  Q  to +M  0  as t  i s varied  from 0 to t , the r e p e t i t i o n rate of the pulse sequence  3.2.2.  SPIN-ECHO  R  The a f t e r the  77/2  spin-echo technique makes use of the fact that pulse discussed i n the previous section,  of signal i n the xy plane i s , i n part, r e v e r s i b l e . conditions,  the  loss  Under general  the decay of the x and y components of M through the  mechanism of mutual spin f l i p s r e s u l t i n g from the interaction i s irreversible. a r i s i n g from the  dipolar  However, that part of the decay  fact that the spins precess at s l i g h t l y d i f f e r e n  -  18  -  angular frequencies because of s l i g h t l y d i f f e r e n t constant l o c a l fields i s certainly reversible.  For our case, t h i s v a r i a t i o n i n  l o c a l f i e l d s i s c h i e f l y due to the inhomogeneity  of the magnet.  The spin-echo e f f e c t i s shown i n .figure 3.2 using three spins; , S  2  and S^,  for s i m p l i c i t y .  The p o s i t i o n s A, B, C and D are  the same as for the free induction decay s i t u a t i o n described i n the l a s t section.  At E, the spins are i n the p o s i t i o n shown  because the angular frequency of frequency of S^, than  i s greater than the angular  which i n turn has an angular frequency greater  . The a p p l i c a t i o n of another 7T pulse at t h i s time w i l l  f l i p the spins to p o s i t i o n F with the " f a s t e s t " spin now the others.  behind  As should be obvious with t h i s simple model, the  spins w i l l reunite at p o s i t i o n G and "pass through each other". The r e s u l t i s a spin-echo which i s i n e f f e c t two i n d u c t i o n decays back to back.  A measurement at the echo peak w i l l again give a  s i g n a l proportional to M ( t ) . z  3.3  EXPERIMENTAL PROCEDURE There are several experimental parameters to be set i n  performing a measurement of T-j.  In order that signal to noise be  maximized and signal d i s t o r t i o n be prevented, one must i n f l i c t l i m i t s on some variables and preserve r e l a t i o n s h i p s between others. The following i s a discussion of a t y p i c a l run which w i l l lead to the determination of T^ for a given pressure. Approximately the s i g n a l .  100 p. s. i . g. of CH^ i s used to tune  The phase of the r . f. pulses r e l a t i v e to the  reference voltage i s adjusted to give a negative free i n d u c t i o n  -  ure  3.2  19 -  spin-echo  - 20 decay and i f the three pulse sequence i s used, a p o s i t i v e spinecho i s obtained.  A negative i n d u c t i o n decay i s used because  i t i s easier to d i f f e r e n t i a t e between the i n d i s t i n c t end of the large  7T/2 pulse and the beginning of the s i g n a l .  The s i g n a l i s  tuned by maximizing the amplitude of the induction decay.  The  echo i s also maximized i n the same way, taking care not to disturb the induction decay.  The pressure i s then reduced to the desired  value. One must now decide on the values of f i v e parameters: tg i s the r e p e t i t i o n rate of the two or three pulse sequence, tg i s the width of the sampling pulse, and RC i s the value of the boxcar i n t e g r a t i n g constant.  These three parameters along with  the p o s i t i o n i n g of tg and the rate at which t i s varied must be determined.  This l a s t point decides hov; long a run w i l l take  and places an upper l i m i t on the e f f e c t i v e measuring time constant. This i s so, because the recorded signal must not l a g behind the detected s i g n a l .  However, the longer the e f f e c t i v e time constant  i s , the better the signal to noise w i l l be. Chapter k-3,  the e f f e c t i v e time constant X  As discussed i n i s given by  - RC R  r  =  l  s  For a t y p i c a l run of about 30 minutes, a value of 5 to 15 seconds seemed to be an optimum value for T .  T h e o r e t i c a l l y , the longer  the time taken f o r a run, the l a r g e r the T the better the r e s u l t i n g s i g n a l to noise.  that may be used and However, long term  - 21 drifts i nM  0  often make i t better i n p r a c t i c e to put up with  poorer s i g n a l to noise and use a shorter run time. Normally,  one f i x e s t  R  and t s independently and uses an  RC necessary to give an optimum X • enough that M (tjj)«M z  0  T l i e  time t% must be long-  or, i n other words, the spins must be very  close to equilibrium before the next pulse sequence begins. Within the s e n s i t i v i t y of the experiment} tp>10Ti i s a reasonable choice.  For higher values of t , signal to noise i s proportional R  1 /? to t g '  (among other things) so one does not wish to make i t  a r b i t r a r i l y long.  The width of t g depends on the width o f the  s i g n a l being sampled, (induction decay or echo). one centres t  s  For the echo,  symmetrically about the peak and adjusts the width  u n t i l s i g n a l to noise i s maximized; a l l other parameters being held constant.  For the induction decay, the sampling pulse i s  positioned as close as possible to the  pulse.  ( about 20 microseconds from the end of the  This l i m i t  pulse ) i s set  by the recovery time of the amplifier a f t e r the ^/2 pulse. Signal to noise i s then maximized by adjusting the width. that the width of either signal i s proportional to T i n turn i s approximately proportional to density . the lower the density i s , the smaller t s must be. i s proportional to density, but because t h i s T  2  2  Note  which  As a r e s u l t , Signal to noise  narrowing  effect  places an upper l i m i t on t g , there i s a further decrease i n signal to noise at lower d e n s i t i e s .  At d e n s i t i e s lower than about 0.03  amagats for CH^, s i g n a l to noise must be maximized on the chart recorder without seeing the s i g n a l , because i t i s completely buried i n noise.  I t should be noted that at several d e n s i t i e s  -  22 -  runs were performed sampling the echo and then the induction decay and i n each case,  was well within the scatter of  surrounding points. Attenuation occurs at three stages; the tuned a m p l i f i e r , the o s c i l l o s c o p e and the boxcar i n t e g r a t o r output. varied manually between 0 and almost t  R  The time t i s  and the three a m p l i f i e r s  are set to give f u l l scale d e f l e c t i o n on the chart M (~0) = -M  Note that M (00) = M z  z  0  i s the minimum s i g n a l and M (~tg)  gained by using  s i g n a l to noise. 2M «s  z  i s the maximum s i g n a l .  d e f l e c t i o n was not used was was  recorder.  The only time f u l l  «  scale  for very low d e n s i t i e s where nothing  f u l l scale d e f l e c t i o n because o f the very poor  The maximum s i g n a l to noise,  M (00) z  - M (0) z  N  0  varied depending on how much time was taken for various N  runs, but t y p i c a l s i g n a l to noise r a t i o s on the chart records were 3 at 0.006 amagats, 7  at 0.01 amagats, 13 at 0 . 0 2 amagats,  20 at 0 . 0 4 amagats and 30 for d e n s i t i e s above 0.1 amagats.  The  optimum bandwidth and d. c. a m p l i f i c a t i o n are determined by maximizing s i g n a l to noise and depend on the c h a r a c t e r i s t i c s of the i n d i v i d u a l a m p l i f i e r s .  This i s a matter o f t r i a l and error  and because the tuned a m p l i f i e r has a maximum a m p l i f i c a t i o n of about 10^, one must take care that the s i g n a l does not saturate its  f i n a l stage. The  and  sampling time t i s set as close to t  M„(00) « M z o  i s recorded.  chapter) which automatically  An ultra-slow  R  as possible  sweep (see next  varies t from 0 to about 2 or 3 T-|  (or u n t i l the s i g n a l becomes too close to M  0  to be useful) i s  used to measure M ( t ) . z  and M of  The ultra-slow sweep i s then turned o f f  i s again recorded.  Q  23 -  IfM  0  has varied more than about 5%  f u l l scale d e f l e c t i o n , the run i s r e j e c t e d .  This might happen  i f a temperature change during the run changes the constant magnetic f i e l d or temperature dependent components of the e l e c t r o n i c system.  The equilibrium magnetization Mo before and a f t e r the run  i s joined by a s t r a i g h t l i n e to enable one to compute M (t) 2  - M  0  as a function of t , which i s i n d i c a t e d by an event marker. I f systematic errors are present, one must t r y to account  for and eliminate as many as p o s s i b l e .  One e f f e c t  d e f i n i t e l y present i s an exponential s i g n a l superimposed on the i n d u c t i o n decay r e s u l t i n g from the a m p l i f i e r ' s recovery a f t e r the first  IX pulse.  The s i g n a l i s sampled a constant time a f t e r the  pulse and because only the d i f f e r e n c e s M ( t ) - M are z  involved i n the computation  0  of T-j, the constant voltage added to  the s i g n a l due to the a m p l i f i e r ' s recovery a f t e r the may be neglected. and the  pulse  However, as the time betv/een the f i r s t TTpulse  pulse i s varied, the boxcar sampling pulse e f f e c t i v e l y  sweeps the d i s t o r t i o n signal associated with the a m p l i f i e r ' s recovery a f t e r the f i r s t 7T pulse.  For a l l runs, the bomb was  evacuated  and a normal run performed to obtain t h i s c o r r e c t i o n .  3.4  T i CALCULATION  Having measured a voltage V"s(t) p r o p o r t i o n a l to M^(t) - M_, one has from equation (2.14)  - 2if -  V (t)  2M [exp(-J-)]  c<  s  (3.3)  e  A smooth curve i s drawn through the signal and equation (3-3)  is  f i t t e d on semi-log graph paper.  ln[v (t)]  =  s  - 4"  +  c  o  n  s  t  a  n  t  { 3  -  k )  n T  1  i s then calculated from the slope.  This was  performed  manually rather than using a l e a s t squares f i t ( f o r equation because of the author's b e l i e f that the experimenter  3.4)  can thus  study each p l o t c a r e f u l l y and take i n t o account more r e l i a b l y c e r t a i n problems which may  arise.  For example, when a curved  l i n e r e s u l t s from p l o t t i n g l n £ v ( t ) j s  against t over the entire  range of t, i t could always be traced to a poor choice of Vq(CO) , usually because M  Q  was not recorded long enough.  A scatter of  points s i g n i f i c a n t l y greater than that obtained from other runs i n s i m i l a r density regions could usually be a t t r i b u t e d to equipment i n s t a b i l i t y .  In both cases, the runs were r e j e c t e d .  I f there i s a s l i g h t error A^o  x  n  H 0  ^  e  curve might deviate  s l i g h t l y from a s t r a i g h t l i n e i n the high t regions because  A[M  - M (t)l  0  z  —=  =[M  0  may  be so small as to be unnoticeable at low  - M (t)J z  values of t  £large M (t)  - MJ  , but exceedingly large at high  values of t  £ small M (t)  - M]  .  z  z  0  0  In such a case, the high t  points were not used i f there was a considerable s t r a i g h t l i n e  -  25  region for lower t values.  Of about 300 runs, 294 s a t i s f i e d  the  condition that the change i n M before and a f t e r the run was  less  0  than 5% (discussed i n the previous section).  Of these 294  runs,  nine f a i l e d to meet the c r i t e r i a discussed i n t h i s section which means the experimental points i n figure 5.3  (Chapter V) number 285.  In the higher density regions, the r e l a x a t i o n v/as exponential within experimental error over 2 orders of magnitude, 2 decades being a l l that could be measured reasonably with the a v a i l a b l e equipment.  For one value of pressure near the Tj  minimum, a very lengthy run was performed Instrument  Computer and the r e l a x a t i o n was  using a Fabri-Tek found to be exponential  within experimental error over almost 3 orders of magnitude M (t) z  - M j Q  .  £in  I t should be noted that the Fabri-Tek, which i s a  d i g i t a l s i g n a l averager, was not used for very low pressure runs because; a) i t takes much longer to both perform the run and c a l c u l a t e T-| and b) T  2  becomes so short that the smallest time  per channel over which the Instrument  averages was  too long to  detect the s i g n a l .  3.5  ERROR ANALYSIS The r e s u l t s of the experiment  This graph of T-|^ -  vs  appear i n figure  5.3.  has the following s p e c i a l features.  F i r s t l y , a l l the experimental points appear as opposed to a few representative points and secondly, no error bars appear. l a t t e r point requires an explanation and upon g i v i n g t h i s ,  This the  reasoning behind the f i r s t point w i l l become obvious. In most error analysis, one computes a probable error  - 26 from a l e a s t squares f i t or some other well defined procedure. The error a n a l y s i s presented here d i f f e r s from the normal procedure i n that rather than c a l c u l a t i n g probable errors, the experimenter attempts to determine meaningful "possible errors". The meaning of the term "possible error" w i l l become clear i n the following paragraph. There are two quite d i f f e r e n t procedures one can use to obtain a reasonable estimate of the error i n Tj for a given run.  A. One can draw maximum and minimum slopes through the  points on the l n £ v s ( t ) j error i n T^".  vs t graph and c a l l these l i m i t s "the  B. One can go back to the trace from the chart  recorder and from s i g n a l to noise considerations put an error bar on each point of the l n j V s ( t ) j  vs t curve.  Both these  procedures v/ere c a r r i e d out for several runs i n a l l density regions.  For p > 0 . 2  was l e s s than 1%.  amagats, the error determined i n both ways  In the region  0.02 <  P  <  0.2,  the f i r s t  procedure y i e l d e d errors of about 5% and the second procedure y i e l d e d errors of about 3%  For the region p < 0 . 0 2 amagats,  a separate discussion i s given l a t e r i n t h i s section.  Because  the drawing of extreme l i n e s ( f i r s t method) always gave an error l a r g e r than the second method, t h i s procedure was adopted i n the e a r l i e r stages of the experiment. To make sure we were not "biased l i n e drawers", the following experiment was performed. persons i n the Physics department  Ten  (who knew nothing of the  experiment) were given 3 Xerox copies of each of 3 d i f f e r e n t runs (9 graphs).  They were asked to draw what they thought to be the  best l i n e , the l i n e of minimum reasonable slope and the l i n e of  maximum reasonable slope.  They were not given runs with a curved  departure from l i n e a r i t y at high t values because they would have no doubt used these high t points.  (See Chapter 3.4.)  The r e s u l t s  were the .following; a l l the best l i n e s ( i n c l u d i n g the author's) were within 1% of each other and the maximum and minimum slopes were " i n s i d e " the author's i n every case.  S a t i s f i e d that errors  were not being underestimated, t h i s method was used to determine errors.  That i s to say, i f anything, the errors were overestimated.  (This i s the author's philosophy.)  I t should be noted that t h i s  error i s c e r t a i n l y l a r g e r than the probable error r e s u l t i n g from a l e a s t squares f i t analysis. V/ith the exception of a very few cases, the errors computed i n t h i s manner were smaller than the spread i n points on the T}~^ vs ^0 graph.  This implies that for some unknown reason,  there i s a "systematic" error associated v/ith each run which i s "random" when considered over several runs.  For instance, the  error r e s u l t i n g from the spread i n points near the maximum i s about 20% whereas the error determined preceding . procedure i s about 3%.  for each T.,""^ by using the  Because the systematic  approaches  used i n determining a possible error for each T-j give errors smaller than the spread i n points on the T} ^ -  vs  graph, i t  seems reasonable to associate the spread i n points with the probable error. For the low density r e s u l t s ( p < 0 . 0 2 amagats) the preceding  arguments would probably .hold i f many more runs were  performed;  namely the spread would increase considerably.  Because  there are not many points i n t h i s region ( r e l a t i v e to the other  - 28  -  regions) the low density r e s u l t s look better than they are.  probably  I f one were to use the maximum-minimum slope method to  determine errors for each would probably  i n t h i s region, the r e s u l t i n g errors  exceed the spread i n points.  be the case for j Q < 0 . 0 1 amagats.  This would c e r t a i n l y  For instance, the lowest density  point (0.006 amagats) has 2 points, 4.24  and 5.26  msec.  two points r e s u l t from the same trace calculated by two Dr. Burnell and myself. f i t of the ln|Vs(t)J  One  might suspect  These people,  that a l e a s t squares  vs t graph r e s u l t i n g from a smooth curve  drawn through the trace might be s i g n i f i c a n t for t h i s point. However, the p o s s i b i l i t y of a large error has already been removed when the ln|Vcj(t)J  vs t plot has been made, namely other  smooth curves that could be drawn through the s i g n a l on the chart record.  Errors determined from the methods already  discussed  would c e r t a i n l y be l a r g e r and i n the author's opinion more meaningful than a l e a s t squares f i t approach. general remarks w i l l be made concerning  Because only very  t h i s low density region,  the errors are not included i n the experimental  p l o t of T i ~ ^ vs Q  ,  CHAPTER IV THE APPARATUS The apparatus used i n these experiments can be divided i n t o two parts, the nuclear magnetic resonance pulse and the gas handling system. John D. Noble ( 1 9 6 4 ) .  spectrometer  The o r i g i n a l apparatus was b u i l t by  Since that time, additions and improvements  have been made by Hardy ( 1 9 6 4 ) , Dorothy ( 1 9 6 7 ) , L a l i t a ( 1 9 6 7 ) , Dong ( 1 9 6 9 ) » Burnell and myself.  4. 1  N. M. R. PULSE SPECTROMETER The spectrometer  can be conveniently divided i n t o  three  stages; the transmission stage, the tuned c i r c u i t stage and the r e c e i v i n g stage.  The transmitter d e l i v e r s radio frequency  to the c o i l i n the tuned c i r c u i t . nuclear induction, s i g n a l .  pulses  The same c o i l receives the  This signal i s then detected, amplified,  displayed and recorded by the r e c e i v i n g stage. these three parts of the spectrometer.  Figure 4.1 shows  C i r c u i t d e t a i l s for a l l  the non commercial components can be found i n the Appendix. now consider the spectrometer  i n greater d e t a i l .  We  - 30 4.1.1  TRANSMISSION STAGE Components of the transmitter are designated by upper  case Arabic l e t t e r s whereas pulses and waveforms are i n d i c a t e d by underlined lower case Arabic l e t t e r s .  The reader may follow the  analysis more c l e a r l y by r e f e r r i n g to figures 4.1 and 4 . 2 . A Tektronix 162 waveform generator, A, d e l i v e r s a pulse, a, and a sawtooth, b, of period t same time.  R  to d i f f e r e n t channels at the  That i s , pulse a coincides with the leading edge of  sawtooth b, i n d i c a t e d by l i n e s 1 and 2 of figure 4 . 2 .  Another  Tektronix 162, B, i s modified to d e l i v e r an ultra-slow (or long period) sawtooth, c, i n d i c a t e d by l i n e 2 o f f i g u r e 4 . 2 . r e p e t i t i o n rate t  R  The  i s t y p i c a l l y i n the m i l l i s e c o n d range while  the period o f the ultra-slow i s usually about an hour.  These two  sawtopths, b and c_, are fed i n t o a Tektronix 163 wave generator, C.  C produces a square pulse, d, of width tj, when the voltages  of the two sawtooths crossover, i n d i c a t e d by l i n e s 2 and 3 of figure 4 . 2 . The r e s u l t i s a pulse, a, on one channel which defines the beginning of a sequence of pulses of period tg.  On another  channel there i s a square, pulse of width t-g which s t a r t s a time t a f t e r the i n i t i a l pulse, a.  As dicussed i n Chapter 3 . 3  S  t and R  tg are fixed and t increases automatically as the ultra-slow sawtooth decreases i n amplitude. The square pulse, d, i s d i f f e r e n t i a t e d , D, producing a p o s i t i v e pulse, e, from the leading edge of d and a negative pulse, f, from the t r a i l i n g edge of d, i n d i c a t e d by l i n e 4 of figure 4 . 2 .  The two pulses are channeled separately and the sign  - j>l -  figure pulse  spectrometer  schematic  transmission stage A  sawtooth + pulse  B  4.1  diagram  ultraslow  C pulse generator  X H.P. time i n t e r v a l unit  Y digital recorder  V 3: E  mixer  D  differentiater  V  sawtooth  Z  chart recorder  ffi  Va, f F  pulse generator  G  pulse generator  W  boxcar pulses  U boxcar integrator  •a',f H  S  mixer  detectoram p l i fi er  I gating pulses amp.  J 10 MHZ. oscillator  R  L gated oscillator  K amplifier  receiving stage  preamp.  3  phase shifter  P frequency triplfir  N frequency tripler  V Q reference amplifier  FD111's  7? 0  power ampli f i er  scope  7K  V ^, e; f  M  T  —  r  J_  magnet  tuned circuit stage  - 32 -  figure  4.2  pulse sequences  33  -  of the negative pulse i s reversed, i n d i c a t e d by l i n e s 5 and 6 of figure 4 . 2 .  Pulse a from A and pulse f from D are mixed, E, and  channeled into a pulse width generator, F, where they t r i g g e r 2 square waves, d/and f ' r e s p e c t i v e l y . 7 and 8 of figure 4 . 2 .  This i s i n d i c a t e d by  This unit i s c a l l e d a  7TP  because a'and f ' w i l l eventually be 77" pulses.  lines  ulse  generator  Pulse e from D  t r i g g e r s another square pulse, e'in the 7T/  2  pulse width G, i n d i c a t e d by l i n e 9 of figure 4 . 2 .  generator  The three pulses are mixed,  H, and amplified, I, i n d i c a t e d by l i n e 10 of figure  4.2.  The three pulse sequence, a' e' and f ' i s repeated v/ith period t . R  The widths of the f i r s t and t h i r d may  independently  from the width of the second.  be varied  The time betv/een  a and e ' i s t and the time between e and f ' i s t .  Depending on  which pulse sequence i s desired (see Chapter 3-2)  pulse f can be  7  E  turned on or o f f . A 10 MHz  c r y s t a l o s c i l l a t o r , J , provides the radio  frequency sine wave which i s amplified, K, and superimposed on the three gating pulses, a' e' and f ' i n a gated o s c i l l a t o r , L. The three radio frequency pulses pass through a phase s h i f t e r ,  M,  where the phase of the radio frequency voltage can be adjusted relative  to the o r i g i n a l o s c i l l a t o r , J .  Finally,  the  frequency  i s t r i p l e d , N, and the pulses are amplified to about 1000 peak to peak, 0.  volts  A reference signal from the o s c i l l a t o r , J , i s  amplified, K, t r i p l e d , P, and amplified again, Q.  This s i g n a l  i s used for phase s e n s i t i v e detection i n the r e c e i v i n g stage. Having superimposed the radio frequency voltage on the three gating pulses, we rename them to be consistant v/ith the  - 34 -  former discussions i n t h i s work.  That i s , the pulses a^ e'and f'  plus the radio frequency sine wave become K  7T/2 and 7^2 . The  transmitter's output, then, i s these three r . f. pulses on one channel and a reference r . f. on another channel.  4.1.2  TUNED CIRCUIT STAGE The physics takes place i n the tuned c i r c u i t stage.  This i s perhaps the most important part of the apparatus and i t i s here where advances i n s i g n a l to noise have been made which yielded the lower density r e s u l t s . The lead from the transmitter to the tuned c i r c u i t i s one-half wavelength to reduce the l o s s of r . f. power.  A 4.7  picofarad capacitor decouples the transmission and tuned c i r c u i t stages.  The tuned c i r c u i t i t s e l f e n t a i l s a c o i l of fixed inductance  and a v a r i a b l e capacitor to meet the resonance condition, (LC)~ . 2  tJ0 =  A factor o-f at l e a s t 2 i n s i g n a l to noise has been  gained over previous experiments by having the tuning capacitor as close to the c o i l as p h y s i c a l l y possible.  UQ  ( = *^H )  i s fixed  0  by the permanent magnetic f i e l d , H , of about 7000 gauss which may be varied amout t 2 5 gauss with a set of d. c. c o i l s .  A  diagram of the tuned c i r c u i t stage i s shown i n figure 4.3. A general discussion of the tuned c i r c u i t stage f o r an N. M. R. spectrometer i s given by Clark ( 1 9 6 4 ) .  Clark's excellent  discussion on " c o i l strategy" explains how one goes about optimizing signal to noise which i s dependent on several factors. The c o i l i n t h i s p a r t i c u l a r experiment was made from 10 turns of #14 guage copper wire with a diameter of 3/4" and a  - 35  figure  4.3  tuned  ^.7 pf  r. f. p u l s e s  in <>  I  -  circuit  '  stage  U.7pt  -° s i g n a l -aluminum  out shield  d. c. c o i I s kovar o  ring  sealseal  epoxi -— glass --brass  seal cylinder bomb  permanent magnet copper  coil  length of l-g-".  The wire was well cleaned, i n c l u d i n g removal of  the enamal and surrounded  by a 26 mm. outside diameter glass  cylinder to prevent breakdown between the c o i l and the bomb. The c o i l and glass were placed i n the pressure t i g h t brass bomb with the r . f. lead coming through a kovar s e a l .  The glass was  fixed to the bomb with epoxi, which extended to the kovar again to prevent breakdown.  seal,  The gas enters the system at the  other end o f the bomb. The inductance of the c o i l was 2 microhenries 30 MHz the c i r c u i t  tuned at 15 picofarad.  The output  and at signal  passes through a 4 . 7 picofarad decoupling capacitor en route to the r e c e i v i n g stage.  4.1.3  THE RECEIVING STAGE The r e c e i v i n g stage detects, amplifies, displays and  records the s i g n a l . "As discussed i n Chapter 3 * 2 , the signal i s an induction decay immediately  following the  pulse and, i f  desired, an echo following the second 7T pulse (7T2) • The signal from the c o i l would look something l i k e the picture i n figure 4.4A. A set of crossed diodes to ground cuts the high voltage pulses down to the back voltage of the diodes. The signal i s very small ( 1 microvolt) and i s unaffected. Quarter wavelength leads j o i n the diodes to the tuned c i r c u i t and to the preamplifier, R, and cut down the l o s s o f the radio frequency (30 MHz) s i g n a l .  The reader i s again r e f e r r e d to figure 4 » 1 •  The 2 stage preamplifier, R, has crossed diodes between i t s stages which cut the voltages of the pulses to the point where the main  - 37  figure  4.4  --very  free  -  induction  signal  large  * rvery  small  rT. f.  N  signal  — —signal envelope  A  before phase s e n s i t i v e d e t e c t i o n and a m p l i f i c a t i o n - y pulse ( not to s c a l e )  B  after p h a s e s e n s i t i v e d e t e c t i o n and a m p l i f i c a t i o n  - 38 amplifier, S, can recover i n about 20 microseconds. i n Chapter 3.3 important how  As discussed  the recovery time of the tuned a m p l i f i e r , S, i s  at low d e n s i t i e s where T2 i s short because t h i s l i m i t s  close to the  pulse the signal can be measured.  The main amplifier, S, i s a commercial, low noise L. E. L. amplifier, model 21B.S.  This i s a multi-staged tuned  a m p l i f i e r with a bandwidth of 2 MHz at 3 db.  centred around 30  MHz  The r . f. signal i s amplified i n the f i r s t 3 stages  where the reference voltage from the reference a m p l i f i e r , Q, i n the transmitter i s introduced for phase s e n s i t i v e detection.  The  reference voltage was kept at about 2 v o l t s (d. c. l e v e l ) and  the  output  from the 30 MHz  a m p l i f i e r was kept below 0.2  volts i n  order that the a m p l i f i e r operate i n i t s l i n e a r region.  After  phase s e n s i t i v e detection, the signal becomes the r . f. envelope shown i n figure  4.4B.  The signal from the tuned a m p l i f i e r i s again amplified by, and displayed on a Tektronix 531A o s c i l l o s c o p e , T, with a type Z p l u g - i n .  I t should be noted that t h i s combination  of  o s c i l l o s c o p e and plug-in i s a very low noise wideband a m p l i f i e r . The s i g n a l from the o s c i l l o s c o p e i s fed i n t o a boxcar i n t e g r a t o r , U.  A pulse from the  pulse generator, G, t r i g g e r s a Tektronix  162 waveform generator, V, which produces a sawtooth. time a f t e r i t s beginning,  At a set  t h i s sawtooth t r i g g e r s p o s i t i v e and  negative square pulses of duration tg i n a modified Tektronix pulse generator, W.  161  These square pulses, or sample pulses, gate  the signal from the o s c i l l o s c o p e i n the boxcar i n t e g r a t o r . boxca'r accepts the signal over the time t  q  The  and then averages using  - 39 an RC i n t e g r a t i n g c i r c u i t . boxcar i s f = ( t / tg) RC. R  The e f f e c t i v e time constant of the An excellent account of the  t h e o r e t i c a l aspects and experimental techniques involved i n the boxcar i n t e g r a t o r may be found i n Hardy ( 1 9 6 4 ) .  The output of  the boxcar i s displayed on a Varian s t r i p chart recorder, Z. The time t between the f i r s t  pulse and the  pulse i s measured by a Hewlett Packard E l e c t r o n i c Counter, X, with a .time i n t e r v a l unit plug-in.  The count s t a r t s on a t r i g g e r  pulse from the TC pulse generator, F, and stops on a t r i g g e r puis from the ^7/2 pulse width generator, G.  At convenient times, a  Hewlett Packard D i g i t a l Recorder, Y, i n d i c a t e s the time t on a printed output and at the same time t r i g g e r s an event marker on the s t r i p chart recorder, Z. The input of the r e c e i v i n g stage i s an r . f. s i g n a l detected by the c o i l and the output i s the measured signal on a chart recorder.  4.2  THE GAS HANDLING SYSTEM Methane i s a r e l a t i v e l y easy gas to work v/ith because  i t may be expelled i n t o the a i r as long as the v e n t i l a t i o n i n the room i s reasonable and there i s no flame near the pump o u t l e t . The schematic diagram shown i n figure 4-5 i s s e l f explanatory. For pressures greater than 2 amagats, a c a l i b r a t e d 0 to 100 pound per square i n c h guage was used.  In the region from 1 to 2 amagat  the mercury manometer was used with stopcock 1 open and stopcock 2 closed.  Atmospheric pressure was added to the pressure i n the  system to get the absolute pressure.  For pressures below 1  - 40 -  figure gas  4.5  handling  system  bombmagnet  1_  r  methane  vacuum pump 0=  0-100 p s i guage  mercury manometer  6  valve <[]> stopcock  1  stopcock  2  - 41 amagat, s t o p c o c k 1 was c l o s e d and s t o p c o c k 2 opened i n o r d e r t o pump o u t t h e r i g h t hand s i d e o f t h e manometer.  The p r e s s u r e o f  t h e system i s t h e n d e t e r m i n e d ( w i t h s t o p c o c k 2 c l o s e d ) w i t h no a d d i t i o n o r s u b t r a c t i o n due t o a t m o s p h e r i c  absolutely  pressure.  Note t h a t t h i s l a t t e r method r e d u c e s t h e p o s s i b l e e r r o r i n pressure  by a f a c t o r o f V I .  completely  The p r o b a b l e e r r o r i n p r e s s u r e i s  n e g l i g i b l e down t o 0.01 amagats and f o r t h e p o i n t s  below 0.01 amagats t h e p r o b a b l e e r r o r i s s m a l l compared w i t h t h e spread. ( i n T-j~1) i n p o i n t s . A l l pressures  i n mm Hg were c o n v e r t e d t o d e n s i t y i n  amagats w h i c h i s t h e r a t i o o f t h e d e n s i t y a t a p a r t i c u l a r t e m p e r a t u r e and p r e s s u r e and  to the density a t standard  temperature  The maximum d e n s i t y , 7 amagats, i s w e l l below t h e  pressure.  d e n s i t y v/here 3 body c o l l i s i o n s become s i g n i f i c a n t and t h e P e r f e c t Gas Law i s an e x c e l l e n t a p p r o x i m a t i o n .  PV-  d  nkT  =  = A' V  PkT  =  p (amagats)  =  where t h e s u b s c r i p t zero r e f e r s t o s t a n d a r d pressure.  Amagats a r e o b v i o u s l y  t e m p e r a t u r e and  dimensionless.  Although temperature d i f f e r e n c e s f o r d i f f e r e n t runs w i l l n o t a f f e c t t h e c o m p u t a t i o n o f d e n s i t y i n amagats, t h e y  will  - hz  -  a f f e c t T-j by changing the constant, of p r o p o r t i o n a l i t y between and  as discussed i n the next chapter.  For t h i s reason, no  runs were performed i f the temperature i n the magnet gap outside the l i m i t s  20°C<T<23°C  (!%• change i n °K) .  was  T  c  CHAPTER V SPIN-LATTICE RELAXATION IN CH^ AND THE INTERPRETATION OF THE EXPERIMENTAL DATA In the following discussion, the possible mechanisms for nuclear spin r e l a x a t i o n are considered with emphasis placed on the s p i n - r o t a t i o n i n t e r a c t i o n which i s the dominant r e l a x a t i o n mechanism i n CH^.  The r e l a x a t i o n rate 1/T  1  i s then formulated i n  terras of the combined e f f e c t s of the s p i n - r o t a t i o n i n t e r a c t i o n and  the molecular motion.  F i n a l l y , having reviewed the microscopic  theory f o r r e l a x a t i o n , we use the theory to i n t e r p r e t the experimental data.  5.1  NUCLEAR INTERACTIONS The nuclear hyperfine  i n t e r a c t i o n s f o r spherical top  molecules such as CH^ are discussed Ozier and Anderson ( 1 9 6 7 ) .  i n considerable  d e t a i l by Y i ,  In the following discussion, we pick  out those i n t e r a c t i o n s which play a r o l e i n the theory of spinl a t t i c e relaxation.  5.1.1  -  ZEEMAN LEVELS A discussion has been given i n Chapter 2.1 o f the Zeeman  energy l e v e l s a r i s i n g from the i n t e r a c t i o n of a free spin v/ith a constant magnetic  field.  Of the many possible modifications to  these l e v e l s r e s u l t i n g from the molecular environment  of the  n u c l e i , only the effect o f the r o t a t i o n a l moment i s important * This i s an i n t e r a c t i o n between the molecular magnetic moment associated v/ith the r o t a t i o n a l angular momentum "hJ and the constant field H  0  and must be added to the Zeeman Hamiltonian. As  discussed by Gordon ( 1 9 6 6 ) , the molecular r o t a t i o n i s very fast compared v/ith the Larmor precession and although several J states are occupied i n CH^ at room temperature, B.yi  average fcoiDst-snt^  of foft;.  on the N.M.R. experiment) from E L / to "h( U0 0  - LJj).  the r o t a t i o n has only  The ^ o ^ i fi^stiO!"  t  i*-b.<? effect-  amounts to changing the Zeeman s p l i t t i n g s The parameter Uj  by Anderson and Ramsey (1966) to be 0 . 0 5 6 L / . 0  has been measured Note that t h i s  does not concern the i n t e r a c t i o n between the nuclear spins and a f i associated with the molecular r o t a t i o n , but rather the e f f e c t on the N.M.R. experiment  of the i n t e r a c t i o n betv/een the r o t a t i o n a l  moment and the f i e l d . Although there are other i n t e r a c t i o n s which have to be taken into account f o r c e r t a i n v/ork with molecular beams ( Y i , Ozier and Anderson, 1967) they can be neglected i n our case i n which the Zeeman l e v e l s already discussed completely dominate i n determining  -  k5 -  the unperturbed energy l e v e l s of the spin system.  5.1.2  PERTURBATION INTERACTIONS The  following i n t e r a c t i o n s are denoted as  because i n a general discussion, the l a t t i c e and, therefore,  perturbations  they a l l may l i n k the spins with  represent possible mechanisms f o r  nuclear spin r e l a x a t i o n . As a r e s u l t of i t s importance, we consider r o t a t i o n i n t e r a c t i o n i n somewhat more d e t a i l .  the spin-  The r o t a t i o n o f a  free CH^ molecule w i l l r e s u l t i n a magnetic f i e l d at the s i t e of a nuclear spin because of the periodic motion of the other three spins i n the tetrahedron.  The general Hamiltonian describing  t h i s i n t e r a c t i o n may be written  w  S R  = -IM,-HP i=1  where we must sum over the 4 spins i n the molecule and include the p o s s i b i l i t y that each may see d i f f e r e n t f i e l d s .  The r o t a t i o n a l  f i e l d s are r e l a t e d to the r o t a t i o n a l state by  H-r = 4 ? C , - J 9  which defines the s p i n - r o t a t i o n tensor C. (5.2)  i n t o equation (5.1) y i e l d s  S u b s t i t u t i n g equation  - 46  W  SR  -  = -27rnE  I,C,J  (5.3)  I t i s convenient to s i m p l i f y t h i s Hamilton!an  by considering the  r e s t r i c t i o n s required by the tetrahedral symmetry. i n d e t a i l by Anderson and Ramsey ( 1966)-  This i s done  There are two  physically  given by equation ( 5 . 3 ) .  d i f f e r e n t contributions to Wg  R  The  first  i s the average i n t e r a c t i o n between the spins and the r o t a t i o n and can be written see a constant f i e l d  -27T^C I'J a  277c J/'Y . a  which i m p l i e s that a l l 4 spins The second term, denoted  i s an a n i s o t r o p i c tensor i n t e r a c t i o n and may departure from the average.  be described as the  The exact form i s somewhat complicated  and does not concern us d i r e c t l y ; i t may Ramsey ( 1 9 6 6 ) .  by  be found i n Anderson and  For reasons discussed l a t e r , i f the C  a  and  terms can not be separated i n the r e l a x a t i o n experiments,  one  speaks of an e f f e c t i v e s p i n - r o t a t i o n coupling constant,  ^ ff e  For purposes of data analysis, the numerical values for C , a  and C  e f  £ are taken from the molecular beam experiments  Ozier, Khosla and Ramsey(1967).  These values have been v e r i f i e d  by V/ofsy, Muenter and Klemperer ( 1970) (to  of Y i ,  and Y i , Ozier and Ramsey  be published). There w i l l , i n general, be a magnetic d i p o l a r i n t e r a c t i o n .  The intermolecular dipolar i n t e r a c t i o n can be completely neglected because of the e f f e c t of the r ~ ^ factor i n the Hamiltonian. Bloom, Bridges and Hardy (1967) i n v e s t i g a t e d the intramolecular  - 47  -  dipolar i n t e r a c t i o n ' s contribution to the r e l a x a t i o n and to be about 5% of the spin-rotation i n t e r a c t i o n ' s With t h i s i n mind, we neglect  found i t  contribution.  t h i s contribution to the r e l a x a t i o n  and assume the spin-rotation i n t e r a c t i o n i s dominant.  I t should  be noted that t h i s assumption i s made purely on the t h e o r e t i c a l evidence of Bloom, Bridges and Hardy (1967) because there i s no information  i n the experimental r e s u l t s concerning the  of the intramolecular  5.2  T  1  IN  contribution  dipolar i n t e r a c t i o n .  CH,,  Nuclear spin r e l a x a t i o n i n spherical top molecules to the s p i n - r o t a t i o n i n t e r a c t i o n has been i n v e s t i g a t e d thoroughly and we  due  quite  give here only a b r i e f review v/ith an emphasis  on the physical processes involved.  The i n t e r e s t e d reader i s  r e f e r r e d to the following s i x publications.  Hubbard (1963)  and  B l i c h a r s k i (1963) a r r i v e d at expressions for T-j i n l i q u i d s for symmetric top molecules.  Bloom, Bridges and Hardy (1967) extended  t h i s to the case of gases for symmetric top and molecules.  spherical  top  These papers use the treatment of r e l a x a t i o n i n the  c l a s s i c text of Abragam (1961) as the s t a r t i n g point.  Dong (1969)  and Dong and Bloom (1970) s i m p l i f i e d the expression for T-| for CH^  using the experimental evidence that the same c o r r e l a t i o n  time could be associated  with both the C  a  and  C  d  terms.  The r e l a x a t i o n rate w i l l involve the p r o b a b i l i t y per unit time that a t r a n s i t i o n betv/een spin states w i l l occur i n the  spin  system.  the  In order that energy be conserved, we must consider  l a t t i c e as well as the spins because a change of state of  the  - 48 l a t t e r implies a change of state of the former.  I f the l a t t i c e  states are denoted by |1> and the spin states by |s> , the unperturbed energies are given quite generally by  w n>  =  E: 11 >  W is>  =  E  t  s  In a rigorous treatment,  L  q  ls>  w i l l involve the r o t a t i o n a l and  t r a n s l a t i o n a l energies and E^, the nuclear Zeeman l e v e l s .  I f we  consider the case of a free molecule and assume the s p i n - r o t a t i o n i n t e r a c t i o n i s a small perturbation on the unperturbed l e v e l s , we can use f i r s t order perturbation theory to determine the p r o b a b i l i t y per unit time that the system goes from a state to a state |l's'>. form  -h where Wg  R  II,s>  This p r o b a b i l i t y w i l l contain terms o f the  — 2 r < l,s|W U',s'>| 6[(E .-E ) SR  s  s  +(E -Ei>]  i s the perturbing s p i n - r o t a t i o n Hamiltonian  usual the unperturbed energies are used.  K  and as  The t r a n s i t i o n  p r o b a b i l i t y per unit time f o r the spin system w i l l i n v o l v e an ensemble average over the l a t t i c e the form  states and w i l l give terms of  cs.  - 49 where W, ,/ / contains terms l i k e those given i n expression l,s->l,s (5.4)  and the P-j_ are the normal exponential factors i n the  Canonical Ensemble.  (5-function  Using the i n t e g r a l form of the  for expression ( 5 . 4 ) ,  an expression for the r e l a x a t i o n r a t e . w i l l  involve terms l i k e  2  E E P , | < l , s | \ A y i ; s ' > | i 'i A where  HL/Q/  dropped.  Noting that  unperturbed  HUss /=  = E-j/ - E-j_,  LJSs /  E/ - E S  S  must be (*/  0  / e x p [ i (  and constants have been - U j  because the  spin states are equally spaced, expression (5«5)  may  be expressed as a sum over l a t t i c e t r a n s i t i o n s rather than i n i t i a l and f i n a l states.  V/ith t h i s i n mind, we write  the p h y s i c a l causes of the  T 11  as a sum  LJ^,  Z G^W/exprifd^'-L/j) + L/J tjdt u  J  over  L  J  (5.6)  A great deal has been omitted i n the t r a n s i t i o n from expression (5.5)  to equation ( 5 . 6 ) ,  but what equation ( 5 . 6 )  i s that one can associate an amplitude G^(0) frequency component LJ^.  ^(0)  e f f e c t i v e l y says  with each l a t t i c e  i s c a l l e d the time  independent  c o r r e l a t i o n function and i t i s the sum of squares of matrix elements of the s p i n - r o t a t i o n Hamiltonian between l a t t i c e states separated by energy  Each matrix element squared i s  weighted by an appropriate Boltzmann factor.  - 50 Equation ( 5 . 6 ) of c o l l i s i o n s .  must be modified to include the e f f e c t  The " e f f e c t i v e l a t t i c e states" take into account  a l l the l a t t i c e degrees of freedom and may be thought of as the discrete  rotational  states,  band by the t r a n s l a t i o n a l  each of which i s broadened i n t o a  energies associated with the Boltzmann  d i s t r i b u t i o n of v e l o c i t i e s . experience anisotropic  During a c o l l i s i o n , a molecule w i l l  forces which w i l l change the f i e l d s  associated with molecular r o t a t i o n at a nuclear spin s i t e .  We  can include the e f f e c t of t h i s c o l l i s i o n a l modulation by associating  a "reduced c o r r e l a t i o n  function"  g^Ct) with each term  i n equation ( 5 . 6 ) .  This w i l l , perhaps, become clearer i f one defines the density  J ((J)  spectral  as  k  exp[\[u + u )t] g ( t ) d t K  and i n t e r p r e t s  (5.7)  k  t h i s as the frequency d i s t r i b u t i o n of l o c a l f i e l d s  provided by the broadening due to c o l l i s i o n s .  The  relaxation  rate then becomes proportional to the components of t h i s d i s t r i b u t i o n with  U  = UQ  -  Uj-  field  - 51  T  I  The time independent  -  G (0) j ( ^ - ^ j ) k  k  (5.8)  0  c o r r e l a t i o n function Gi_(0) and the constants  of p r o p o r t i o n a l i t y i n equation ( 5 . 8 )  are known and i t remains to (5.7).  obtain an expression for g ( t ) i n order to solve equation k  7*k,  I f one associates a c o r r e l a t i o n time  which describes a  c h a r a c t e r i s t i c time for the e f f e c t of a c o l l i s i o n , with each i n t e r a c t i o n and assumes that the c o l l i s i o n s are random, g^(t) may  be written  g.(t)  =  ill  exp  (5.9)  T h e o r e t i c a l l y , one can only say that g^(t) must be a  monotomically  decreasing function of time at long times and the form given i n equation ( 5 . 9 )  must be i n t e r p r e t e d as a reasonable attempt  explain the data.  to  This i s discussed by Dong and Bloom ( 1 9 7 0 ) .  I f v/e perform the i n t e g r a t i o n i n equation (5*7)> using equation ( 5 . 9 ) ,  1  equation ( 5 . 8 )  becomes  Y~'  r  G (0) u  ~  r,  k ;  w^?  (5 10)  -  I t i s i n t e r e s t i n g to look at some s p e c i a l cases of equation ( 5 . 1 0 ) .  I f c e n t r i f u g a l d i s t o r t i o n i s n e g l i g i b l e but we  - 52 must associate d i f f e r e n t c o r r e l a t i o n times with the C and a  terms, we have  2  1 T,  4jr r2  "  OC  +  U  a  T  i  1 +  (Uc-Uj) ^ 2  4  ATT  45  Of  2  2  r  2  n  1 + {Uo-ujHT^)  d  2  C5  -'°  The constants i n equation ( 5 . 1 1 ) are taken from Dong and Bloom (1970) and the choice of notation for the T's i s i n keeping v/ith the l i t e r a t u r e .  The parameter QC i s given by  . where I  0  2I kT 0  i s the moment o f i n e r t i a f o r the s p h e r i c a l l y  symmetric  molecule. I f the same c o r r e l a t i o n time may be associated with both terms, equation ( 5 . 1 1 ) reduces to  Ti  47TV  r.  2  1 +  2^2 {Uo-UjfTi  (5.12)  where  2  rs  C ff =  C  e  2  Q  _4_^ 2 + — 45 C,  I f c e n t r i f u g a l d i s t o r t i o n i s not n e g l i g i b l e , the anisotropic  c o n t r i b u t i o n t o t h e r e l a x a t i o n r a t e raay  have non  zero u/^'s and we can w r i t e  -  1  47T  T]  2 2  2^2 C5.13)  _4_ 47r .. 9 V " 4 5 oc 2  +  where F  k  k [(u -uj) T  1  +  0  +  4j r 2  k  2  0  i"  i s a n o r m a l i z e d f u n c t i o n . and t h e sum o v e r k i n c l u d e s k  k A s u p e r f i c i a l examination o f the experimental r e s u l t s g i v e n i n f i g u r e 5.3 i n d i c a t e s t h a t 1/T-j i s o f t h e form g i v e n by equation (5.12).  T h i s i m p l i e s t h a t t h e r e l a x a t i o n r a t e 1/Tj i s  d e s c r i b e d by; A) e q u a t i o n ( 5 . 1 2 ) , o r B) e q u a t i o n (5.11) n o t  v e r y d i f f e r e n t from  term dominant.  7*i >  o  r  with  C) e q u a t i o n ( 5 . 1 3 ) v a t h one  I f t h e r e l a x a t i o n r a t e were g i v e n by, s a y , two  e q u a l l y dominant, b u t v e r y d i f f e r e n t terms,  we would expect t o  -  see a n o t h e r  54 -  maximum o r a t l e a s t a bump where t h e second term has  i t s maximum.  I f the s p e c t r a l d e n s i t i e s  j k (L/) a s s o c i a t e d v/ith  t h e c e n t r i f u g a l d i s t o r t i o n a r e c e n t r e d around LJ ^ s w h i c h a r e ••far away" from LJQ  - LJ j  v/e c a n c e r t a i n l y i m a g i n e  their  c o n t r i b u t i o n to the r e l a x a t i o n r a t e being very s m a l l .  This i s  shown s c h e m a t i c a l l y i n f i g u r e 5 . 1 where o n l y a s i n g l e frequency  i s shown f o r s i m p l i c i t y .  d e s c r i b e d by  The dominant C  a  distortion term i s  j^(Cv/) and t h e d i s t o r t i o n term i s d e s c r i b e d by  j^iLJ)  As i s e v i d e n t i n e q u a t i o n (5.12) o r a dominant term i n (5.11) and (5.13)> t h e f a s t e s t r e l a x a t i o n r a t e  equations when  1  LJQ  T^~  - LJ j  This i s the c h a r a c t e r i s t i c  1/T-j maximum  o r T j minimum and m a n i f e s t s i t s e l f i n t h e e x p e r i m e n t a l i n f i g u r e 5.3. a s ; A)  The r e l a x a t i o n r a t e t h e n d e c r e a s e s  decreases  T]"^ > LJQ  i n the r e g i o n  7*i i n c r e a s e s i n t h e r e g i o n  Tf  ]  < U0  - Uj.  - LJj  occurs  results  monotomically and, B)  Case A i s  c o n s i d e r e d t h e h i g h d e n s i t y r e g i o n and c a s e B t h e l o w d e n s i t y region.  The maximum r e l a x a t i o n a l o n g v/ith t h e two extreme  cases  a r e shown s c h e m a t i c a l l y i n f i g u r e 5.2 f o r t h e c a s e o f t h e j ^ L / ) term c o m p l e t e l y  dominant.  E x p e r i m e n t a l l y , one measures t h e d e n s i t y T  r e l a t e d to the c o r r e l a t i o n times  1  -=r  where 0"  k  -  k  which i s  by  D<CTkV>  <5.i«  i s an e f f e c t i v e c o l l i s i o n c r o s s s e c t i o n f o r t h e  p a r t i c u l a r i n t e r a c t i o n and v i s t h e speed o f a m o l e c u l e .  The  C a n o n i c a l Ensemble a v e r a g e < ( 5 ' j > depends o n l y on t h e mean v  c  figure  5.1  r e l a x a t i o n e f f e c t s of centrifugal distortion  - 56 -  figure  5.2  collision  induced  r  c  u0  fields  »1  low  p  u  v.- uT  u  r  c  high  uQ «1 p  u  - 57 v e l o c i t y which, i n turn, i s a function of the temperature o f the lattice.  I f temperature i s held constant,  r,  <k  =  we can write  Ai  —  (5.i5)  k  p  where we allow d i f f e r e n t constants,  A , f o r the d i f f e r e n t lc  cross  sections associated v/ith each i n t e r a c t i o n . For purposes o f analysing the experimental data i t i s convenient to express the r e l a x a t i o n rate as a function of density through equation (5.15) and furthermore noting that the density at which the expression  1 •  [ ( u - 4 ) * Mj r 2  0  2 k  i s a maximum occurs when  [(u0 -uj)  + uk ]rk  = 1  Using equation (5«15)> we have  Ai<  v/hich can be taken as the d e f i n i t i o n of jO^.  V/e have then  - 58 -  r  k  -  _  2  [(u -Uj)  _  _  +  0  (5.16)  P  ' and by s u b s t i t u t i n g e q u a t i o n (5.16) i n t o the t h r e e p o s s i b l e f o r 1/Tj g i v e n by e q u a t i o n s ( 5 . 1 1 ) , (5.12) and the f o l l o w i n g Case 1.  (5-13) we  cases  obtain  formulae.  I f c e n t r i f u g a l d i s t o r t i o n i s n e g l i g i b l e and t h e same  c o r r e l a t i o n time may  be a s s o c i a t e d w i t h t h e C a  r e l a x a t i o n rate i s given  and  C^ t e r m s , the  by  2  n Cf T, " oau-uj)  j_  4  P  f  p 1  where p,  P  +  =  ( I ^ u l  = 0,  p  c o r r e l a t i o n t i m e s a r e r e q u i r e d f o r the Ca by  , 7 >  2  ( 5  I f c e n t r i f u g a l d i s t o r t i o n i s n e g l i g i b l e , but  r e l a x a t i o n rate i s given  -  C 5  i s g i v e n by e q u a t i o n (5.16) w i t h U.  T l  Case 2  2  and  -'  8 )  different  C^ t e r m s , t h e  - 59 -  ]_  _  T,  =  47T C 2  B .P  2  Q  Ot(U -Uj)  1  0  I  p2 —I  +  p 2  p  4  +  45  4 7T C  x  1J2  2  P  2 D  "  a(u -uj) 0  +  , , RQ  (5.19)  { p { ? )  P where P  1  i s given by equation (5«18) and P  equation ( 5 . 1 6 ) with L/12  2  Case 3  2  i s given by  = 0'  P; 1 2  12  2  (Uo-Uj)  1 p  I f r e l a x a t i o n due to c e n t r i f u g a l d i s t o r t i o n i s small  but not n e g l i g i b l e ( i . e . measurable), the r e l a x a t i o n rate i s given by  - 60 -  P1 1  4 7T C 2  _^  2 Q  P  P  2  (5.21)  167T C 2  +  2 d  V^  F  P  k  1 with yO^. given by equation 5.3  *  P  2  (5-16).  DATA ANALYSIS  The experimental data shown i n figure 5 . 3 i s the relaxation rate l/T as a function of density jO. 1  The simplest  f i t of the data i s to assume that the relaxation rate i s given by equation  (5.17).  Equation  (5.17)  can be subjected to a least  squares f i t using  -r  P  -  0  —o  +  D  (5.2a)  o if)  - 62 -  where  Ot(u -Uj)  py  0  a  4 7T C f 2  2  ef  CX(L/ -Uj ) 0  47T C §P 2  8  1  or  P  2  ^eff  A determination The  =  a(u0-L/,)  "  47T  -vab  2  o f a and b w i l l  most f r u i t f u l  1  give values  f o r C fj» and jO^. e  approach i s to f i t over a c e r t a i n r e g i o n and  note t h e f i t f o r t h e r e s t o f the c u r v e .  S e v e r a l p l o t s were  performed i n t h i s manner and t h e agreement i n t h e r e g i o n s the  figure fit  fit  The e r r o r s f o r t h e t h e o r e t i c a l  5«4 a r e v e r y  curves  such as i n  s m a l l because o f t h e f a c t t h a t t h e r e g i o n  c o n t a i n s many p o i n t s .  unknown, in  F i g u r e 5*4 i s an example o f  f i t was v e r y poor i n each case.  such a f i t .  outside  If C  2 e f f  and p  i  are both  considered  one can n o t f i t t h e l i n e a r h i g h d e n s i t y r e g i o n because  terms o f e q u a t i o n  (5.22),  b »  a/p  2  and the l e a s t  does n o t c o n t a i n s u f f i c i e n t i n f o r m a t i o n .  squares  However, because  - 6k C  = 137.60 k H z  2  -  i s accurately known, we can use the fact  Gil  portrayed i n figure 5.3 that for D > 1  amagats, —  i s constant.  ft  P,  In terms of equation (5.17), t h i s ' i m p l i e s that — ^ ft  1 r e s u l t that — i s given by b i n equation ( 5 . 2 2 ) .  2  <<  1 with the  T  The  experimental  P r e s u l t that  II p  =  21.9  uniquely determines f)^ now  be p l o t t e d .  1  0.4  m  s  e  c  (5.23)  amagats t  with the r e s u l t that equation (5.17) can  This i s done i n figure 5 - 6 .  The  theoretical  value of jOj agrees very well with the experimental value, but the t h e o r e t i c a l curve gives f a r too strong r e l a x a t i o n i n the region of the maximum.  This high density f i t of the data using  equation ( 5 . 1 7 ) provides the l o g i c a l conclusion that whatever i n t e r a c t i o n s contribute to the r e l a x a t i o n at higher d e n s i t i e s there are some e f f e c t s which do not contribute as much i n the region of the maximum. Having eliminated equation ( 5 . 1 7 ) as a reasonable f i t of the data (but none the l e s s having gained considerable i n s i g h t i n t o the problem) the next step i s to allow two c o r r e l a t i o n times. Using the accepted values of C equation ( 5 . 1 9 )  a  = 10.4 kHz and  contains two unknowns; fc)-\ and  = 18.2 ft{2.'  kHz,  This i s the  same as saying that the two c o r r e l a t i o n times are unknown (given by equations ( 5 . 1 8 ) and ( 5 . 2 0 )  ).  Rather than randomly picking  figure  P  34r o  a  en in o E E o  1  5.5  vs  30 26  22 1 p  I 1  1  (amagats)  i _ J _ l  10  1  J  I I •  .01  0  .1 (amaaats)  10  - 67 v a l u e s f o r t h e s e two p a r a m e t e r s i t i s r e a s o n a b l e t o f i t t h e h i g h d e n s i t y r e g i o n under t h e a s s u m p t i o n t h a t  P»  P]>P\2.  i n v e s t i g a t e t h e consequences i n t h e a r e a o f t h e maximum.  a  n  d  For the  r e g i o n p > 1 amagats, e q u a t i o n (5-19) r e d u c e s t o  1  47T  T,  a{u -Uj) (5.23)  and  w i t h o n l y one v a r i a b l e . of  p  0  Using equations  (5.2if)  (5.24),  e q u a t i o n (5.19) c a n be w r i t t e n  That i s t o s a y Pjg becomes a f u n c t i o n  E q u a t i o n (5.19) v/as t h e n p l o t t e d f o r a range o f v a l u e s  Py  for  P .  for  i n a l l cases there i s too strong r e l a x a t i o n i n the r e g i o n o f  1  I t v/as a b s o l u t e l y i m p o s s i b l e t o f i t t h e d a t a  t h e maximum.  We do n o t g i v e an example o f such a f i t because a  t y p i c a l f i t l o o k e d something l i k e to  adequately,  f i g u r e 5.6.  I f we had d e c i d e d  f o r c e a f i t i n t h e r e g i o n o f the maximum, we would c e r t a i n l y  be l e f t t o o weak r e l a x a t i o n i n t h e r e g i o n o f P > 1  amagats.  V/e a r e l e f t w i t h t h e t a s k o f i n t e r p r e t i n g t h e d a t a i n terms o f a m e a s u r a b l e c o n t r i b u t i o n t o t h e r e l a x a t i o n r a t e by centrifugal distortion effects.  V/e now want t o a n a l y s e t h e d a t a  i n terms o f e q u a t i o n (5.13) whose i n d e p e n d e n t v a r i a b l e s a r e t h e T^,  o r t h e i d e n t i c a l e q u a t i o n (5.21) whose i n d e p e n d e n t v a r i a b l e  is p .  Bloom and O z i e r ( p r i v a t e communication) m a i n t a i n t h a t one  can c a l c u l a t e t h e  and F  k  and t i m e consuming p r o c e d u r e .  but that i t i s , perhaps, a lengthy C e r t a i n l y , the experimental  data  s h o u l d be r e - a n a l y s e d when t h i s i s done, b u t we c a n a r r i v e a t r e a s o n a b l e q u a l i t a t i v e r e s u l t s by a p p r o x i m a t i n g e q u a t i o n (5«13)  - 68 (or 5.21).  F i r s t we assume that the s p e c t r a l d e n s i t i e s  j^(L/)  associated with each peak frequency U/^ add i n such a way at LA - U that we can approximate them by one s p e c t r a l density o <j T  which we denote by  j( (L/)  peaked around U^.  2  i n terms o f equation ( 5 . 1 3 )  (or 5.21), Fk = 1 f o r  and F^ = 0 f o r a l l other frequency  i  + L/^] . 2  =  L/^  Also, we assume that the angular  s u f f i c i e n t l y high that ( U ^ )  F i n a l l y , we assume that both the C  single time  s  That i s to say,  2  a  ^  [(^o-  ^j)  term and the now  term may be analysed i n terms of a s i n g l e c o r r e l a t i o n  7"i •  With these approximations and assumptions, equation  (5.13) becomes  1  4 7T C 2  m  +  I6 7 f 2 c d 2 45  r,  2 Q  0C  r}  •• 1  +  v/ith the v a r i a b l e parameters T-j and becomes  7 —2 0 2 (U{2 ) T}  C5.25)  2 and equation ( 5 . 2 1 )  - 69 -  4  P  n Cg 2  T, P  2  P'  16 7T C 2  H2  2  D  45a u, 2  1  [Phi2 P  +  with, the v a r i a b l e parameters P^,1 ' P{^f\2 given by  =  T j and equation  (5.26)  a  n  ^12  d  a n (  ^ ^  e  (5.16)  (5.27)  U  A 1-2  restriction  12  V/e now u t i l i z e another r e s t r i c t i o n ; namely the slope of the high P^ »  For p  s a t i s f i e d for p  > 1 amagatj equation ( 5 - 2 6 )  J ^ _  »  P^,  density region.  w  n  2  c  16 7T C 2  D  i-  2  D  h  i  s  assumed to be  becomes  ,  (5.28)  p and using equation  (5.27)  -  _1_  70 -  47T Co  2  45  +  p  1  47T Ceff (U -Uj ) a 2  P  0  Using the numerical value given i n equation ( 5 . 2 3 ) , we have the following r e s u l t  P, P,'  amagats  =  0.039  =  (2.2 X 1 0 "  sec amagats) L/^  1 0  2  Having determined a l l the parameters for the C  a  contribution to  equation (5.26) we v/rite equation (5«26) as  _1_ T  T  +  "1 "  (5.29)  T C  i s plotted i n figure 5 . 7 .  and  d  As expected, the C  a  contribution  L ta. T  to the r e l a x a t i o n rate i s dominant and as proposed by figure 5 . 7 i t i s the only mechanism that i s important at the maximum. V/e now have a reasonable explanation  of why the t h e o r e t i c a l curve f o r  a s i n g l e term f i t shown i n f i g u r e 5 . 6 gives too strong i n the region o f the maximum, namely the C  d  relaxation  contribution to CQ ff  p  1 •  '  J  figure  j  5.7  vs  p  i o in  .01  .01 p  .1 (amagats)  10  - 72 -  i s not present.  Having accounted for the C  c o n t r i b u t i o n to 1/T^  a  we now p l o t  T Jexp.  TJ  vs  (5.30)  and attempt to i n t e r p r e t i t i n terms o f  16 7T C 2  A1 :2 2  D  (5.31)  45Q!Uf2 P  1 2  =  (2.2 X 10"  1 0  (Pi 2> sec a m a g a t s ) U ^  (5.32)  The p l o t given by equation (5.30) i s given i n f i g u r e 5 . 8 . The fact that the C„ a term i s dominant accounts f o r the large apparent spread i n points i n figure 5 . 8 .  There i s a maximum somewhere i n  the region between 0 . 1 and 0 . 4 amagats.  I f the low density  r e s u l t s are r e l i a b l e there seems to be another  term which  peaks at a density lower than we are presently able to observe. I f v/e neglect the very low density r e s u l t s momentarily, v/e can analyse figure 5 . 8 i n terras of equation ( 5 « 3 D and ( 5 . 3 2 ) . Instead of randomly varying r e s t r i c t i o n given by equation ( 5 . 3 2 ) ,  a  n  d  ^\2.  ^ ne  v/e note that the best f i t  w i l l occur when the maximum of equation (5.31) corresponds to a possible maximum of figure 5 . 8 .  To f i n d v/here t h i s might be, the  density  figure  region  5.8  vs  .08  .04  0  -.04  L  .01  .1 (amagats)  10  - 74 -  l o c u s o f maxima g i v e n by  1_  J6HfCdL  1  LT J c m a x  ,.33)  4 5 a u{2  2  d l  (  and e q u a t i o n (5.32) i s p l o t t e d i n f i g u r e 5.9 a s a f u n c t i o n o f / .  Vi n l T A  /  1 2 *  1 8  °  n e t v/ U  ,-\w-.-i  4" 4" .-> ,~!  Ouia. v  v^. KJ.  the r e g i o n o f i n t e r e s t .  4" It  *1 A  m  v / i i ^ ^_ w vii  4 - -t -  ,-1 AT,  v-i. b i i o j . v j  v i n rm'  , i-»  x o £,0.  n  ,-%->->  J- n  . - v 4 " O  u o i  o-U  -.v. r-. ~ n n  -Ptr  iu&4.£j.i.i.x. x j  S u r p r i s i n g l y enough, t h i s l o c i o f maxima  do pass t h r o u g h p o s s i b l e c a n d i d a t e p o i n t s f o r t h e e x p e r i m e n t a l maxima. H i g h l y q u a n t i t a t i v e c o n c l u s i o n s a r e i m p o s s i b l e because o f the s p r e a d ' i n p o i n t s , but t h e allov/ed range o f LJ-j^  c  a  n  be  narrowed by p l o t t i n g e q u a t i o n (5.31) f o r a few r e a s o n a b l e o f LJ^'  T h i s i s done i n f i g u r e 5.10.  values  N o t i n g t h a t t h e t o p and  bottom c u r v e s i n f i g u r e 5.10 can p r o b a b l y be r u l e d o u t , i t seems r e a s o n a b l y s a f e t o say thafc  5(LJ  0  - LJj) •<  LJ^p  <  &(U  Q  - LJj)  I f t h e l o w d e n s i t y e f f e c t i s r e a l , we c a n n o t s a y a n y t h i n g about i t .  The tremendous s p r e a d i n p o i n t s a l o n g w i t h  the f a c t t h a t a maximum has n o t been r e a c h e d way o f p u t t i n g l i m i t s on a n o t h e r L/k  does n o t p e r m i t any  for this effect.  I t should  be n o t e d , however, t h a t i f t h e e f f e c t i s r e a l , t h e l i m i t s s e t on (*/|2  w i l l be s l i g h t l y m o d i f i e d because  f o r LJ^p w i l l be  s l i g h t l y l e s s t h a n 1. I t would n o t be s u r p r i s i n g , t h e n , i f t h e t h e o r e t i c a l p l o t s i n f i g u r e 5.10 s h o u l d be s l i g h t l y  lowered.  .08  r  figure  o  5.9 vs  (j1 )' e x p v  CD CO  E  p  .04  >  .04  amagat  VJ1  / — 25 Z.J B u Q_  X CD  0 B = K>- u  ;  c max d i  -04  J  l_I  J  .1  P  (amagats)  i  I  I  vs  T  (p)  max .0  .08  figure  o  (1)  CD  5.10  -(1)  vs  p  if)  E  p  .04  >  .04  amagat  o  3,2 fl  X 0)  9.2 B  0  -.04  I  I  L  l .1 p  (amagats)  i l l  J  L_l  10  CHAPTER VI SUMMARY AND SUGGESTIONS FOR FURTHER WORK Using n u c l e a r magnetic resonance pulse  techniques, the  s p i n - l a t t i c e r e l a x a t i o n time T^ i n gaseous methane has been measured as a f u n c t i o n o f d e n s i t y a t c o n s t a n t t e m p e r a t u r e .  The  e x p e r i m e n t a l r e s u l t s c o u l d n o t be i n t e r p r e t e d i n terms o f t h e simplest  t h e o r e t i c a l framework which s u g g e s t s t h a t t h e r e l a x a t i o n  r a t e 1/T^ i s a L o r e n t z i a n density.  f u n c t i o n o f ]/p  where p  i s the  The " f i n e s t r u c t u r e " w h i c h u p s e t s t h e s i m p l e t h e o r e t i c a l  model has been a t t r i b u t e d t o a c o n t r i b u t i o n t o t h e r e l a x a t i o n r a t e by c e n t r i f u g a l d i s t o r t i o n .  I n a r r i v i n g at this  s e v e r a l a s s u m p t i o n s have been made.  conclusion  The i m p o r t a n t r e s u l t i s t h a t  a more r i g o r o u s a p p r o a c h would p r o b a b l y a f f e c t t h e r e s u l t s i n a q u a n t i t a t i v e and n o t q u a l i t a t i v e manner. The  q u a l i t a t i v e conclusions  o f t h i s t h e s i s s h o u l d be  checked w i t h o t h e r t e t r a h e d r a l m o l e c u l e s and t h i s i s p r e s e n t l y b e i n g done w i t h S i H ^ ( s i l a n e ) .  A s t u d y o f t h e r e l a x a t i o n as a  f u n c t i o n o f t e m p e r a t u r e would a l s o be a u s e f u l A rigorous  experiment.  t h e o r e t i c a l treatment should i n v o l v e a study  o f r e l a x a t i o n e f f e c t s a r i s i n g from t h e i n t r a m o l e c u l a r  dipolar  i n t e r a c t i o n and s p i n d i f f u s i o n i n and o u t o f t h e s o l e n o i d . symmetry e f f e c t s , w h i c h have been c o m p l e t e l y n e g l e c t e d  Spin  i n the  - 78 conventional theory have been introduced i n t h i s thesis as the high frequency LJ^  (k £ 0) terms.  Whereas a proper  treatment  of the intramolecular dipolar i n t e r a c t i o n and spin d i f f u s i o n would probably not a f f e c t the conclusion concerning r e l a x a t i o n due to c e n t r i f u g a l d i s t o r t i o n , spin symmetry e f f e c t s are c e n t r a l to t h i s conclusion and an expression for 1/T  1  should be  developed  which includes the higher order d i s t o r t i o n frequencies i n the r o t a t i o n a l states.  In p r i n c i p l e , one could calculate the  s p e c t r a l d e n s i t i e s for a l l the d i s t o r t i o n frequencies and as a r e s u l t t h e i r contribution to the relaxation could be ( i . e . t h e i r contribution at the Larmor frequency).  determined The rigorous  r e s u l t s would be s i m i l a r to equation (5.13) i n form with the F^'s and  U^s  calculated exactly.  In the a n a l y s i s presented here  we have assumed only one d i s t o r t i o n frequency simply because of l a c k of information.  A re-analysis of the experimental data on  completion of these t h e o r e t i c a l suggestions would c e r t a i n l y reveal f r u i t f u l information. The most f r u i t f u l approach i s , probably, to search for s i m i l a r e f f e c t s i n other s p h e r i c a l l y symmetric gaseous molecules and to engage i n a t h e o r e t i c a l programme of some r i g o u r i n order to pinpoint the o r i g i n of the experimentally observed e f f e c t which we have a t t r i b u t e d to c e n t r i f u g a l d i s t o r t i o n of the molecule.  APPENDIX CIRCUIT DIAGRAMS Some of the o r i g i n a l components have been changed i n a trivial  fashion since they were f i r s t assembled i n the early  1960's and some components have been recently added.  Although  some of the c i r c u i t diagrams may be found i n t h e i r o r i g i n a l form i n previously published theses, the following diagrams represent the  spectrometer i n i t s e n t i r e t y .  has been omitted.  Only the commercial  equipment  The upper case Arabic l e t t e r s r e f e r to the  d e s c r i p t i o n of the apparatus i n Chapter IV.  y12AU7  5670  150 V  12K I—'WV-  100 pf  from -If— square pulse generator C  from pulse generator A  .01  470K  1M  A CO  1N307  o  to p u l s e w i d t h generator F  K .01  1N195  to p u l s e w i d t h g e n e r a t o r G  12 K  x figure  A1  differentiator  D  and  mixer  E  5725  12AU7  12AU7  - 170 V  figure pulse  A 2  width generator  F  (or  G)  + 150 V 5670  from pulse width generator F  f r o m pulse generator  .047  width G  047  i n to g a t i n g amplifier  680K  figure mixer  A3 H  Co  pulses I  ro  5687  5670  33K .AAAAr—•» 225 V  18K + 150V  •  -AAA/  1  4K  1 47K  -AAAA/  A<W 7 ^  ZD/  +  i—'\AAA-  1  200 o  680K  A A A A r IT .05 1N307  FROM MIXER  figure  -170 V  H  A4  gating  pulses  TO GATED P O W E R AMPLIFIER 0  amplifier  I  TO GATED OSCILLATOR  L  CO  OUTPUT T O ° WIDEBAND K  AMPLIFIER  CO  l CRYSTAL  figure 10 m h z c r y s t a l  oscillator  A5 J  (10  mhz)  -L  .005  o—1|  w/v  .01  1.5K  from crystal oscillator  -o-6V  10^h  Co  to g a t e d oscillator L and tripler P  OUTPUT  J  o+6V  figure  A6  wide  band a m p l i f i e r  K  225 V  to p h a s e shifter M  from wideband amplifier K  Co  from gating  pulses  amplifier  figure coherent gated  I  A 7 oscillator  L  170V Q  +150V+225V 9  O  .01 .01  o  120  __) A 5687  WV-  FROM GATED OSCILLATOR  L 2 5763 180  >180  TRIPLES  DELAY LINE: BEL FUSE vs-250 PHASE  SHIFTER  M  figure  A8  N  from gating  pulses  amplifier  90V GATE I N  •*+750V  .01  9  from tripler N  I r.f.  829  6SN7  22 K  I  /TTTTN I  "° OUTPUT  10 :56K  r i  ^ W v V  10  10K  •150  22K  figure  3.15 V  -300V  A9  gated  power  amplifier  -<*1200V  /RiO  0  Co Co  to 0.5 uH  0  ± 1 0 PF  A-30  4-30  .01  f rom wide band amplifier  •22K  K  H H .01  .01  h .01  .01  5.1 K  5.1K<  t figure  0 5uH  2N3323  2N3323  H l -  reference amplifier  A10  +6V -6V  reference tripler P  2  6  D  J  8  from t r i p l e r  figure  A11  P  reference amplifier  Q  8-10  HI-  10K  100uH I  | 10UH A  V  150 V  FD111  .01  -Ih  .01  ± -  ^•6DJ8  .22 7722  from crossed diodes  .01  3uH  .01  figure  A12  100K  1K  —T—  pregmplifier  R  -o to t u n e d amplifier  from  oscilloscope  T  to c h a r t r e c o r d e r  4  Z  220 K  /from pulse • / generator  7266  AAA—O 1M 330K "lOOK° —AAA—o  50K 150K 2W  .  1 ± 1 ± T T T .1 .01 .001 b o x c a r "= integrator  figure A13  50K 150K 2W  12AU7  3 3 K ~ ^ j J  1  0-10K  I  U  1  8  8  HAMMOND 1166T  QD3 's •O+1200V  ~2K(20Yv7  HAMMOND 196C  6  © ©  1.5KV  HAMMOND 776  high  voltage  figure power supply  A14  for a m p l i f i e r  0  •*»*750V  4  + 20V  1N540's  + 6V  •A/WVr  1.8 K  1.8 K  M Z 1000-29  1N968  M Z 1000-9  hammond transformer 166L6  -OKr 1N753  T 100uF -6V  f i g u r e A15  power supply  J , K and  P  - 95 BIBLIOGRAPHY  Abragam, A. 1961 The p r i n c i p l e s of nuclear magnetism (Oxford Univ. Press, London). Anderson, C.H. and Ramsey, N.F. 1966 Phys. Rev. 149, 14Andrew, E.R. 1955 Nuclear magnetic resonance (Cambridge Univ. Press, London). B l i c h a r s k i , J.S. 1963 Acta Phys. Polon. 24, 817. Bloch, F. 1946 Phys. Rev. 7 0 , 460. Bloom, M., Bridges, F. and Hardy, W.N. 1967 Can. J . Phys. 45_, 3533. Bloom, M and Dorothy, R.G. 1967 Can. J . Phys. 45_, 3411. Clark, W.G. 1964 Rev. S c i . Instru. 3J?, 316. Dong, R.Y. 1969 Ph.D. Thesis, U.B.C. (unpublished). Dong, R.Y. and Bloom, M. 1970 Can. J . Phys. 48, 793. Dorothy, R.G. 1967 Ph.D. Thesis, U.B.C. (unpublished). Gordon, R.G. 1966 J . Chem. Phys. 44, 1184. Hardy, W.N. 1964 Ph.D. Thesis, U.B.C. (unpublished). Hubbard, P.S. 1963 Phys. Rev. Jj51_, 1155L a l i t a , K. 1967 Ph.D. Thesis, U.B.C. (unpublished). Noble, J.D. 1964 Ph.D. Thesis, U.B.C. (unpublished). Ozier, I . , Crapo, L.M. and Lee, S.S. 1968 Phys. Rev. 172, 63 S l i c h t e r , C P . 1963 P r i n c i p l e s o f magnetic resonance (Harper and Row, New York). Wofsy, S.C., Kuenter, J.S. and Klemperer, W. 1970 J . Chem Phys. 53_, 4005. Y i , P. 1967 Ph.D. Thesis, Harvard Univ. (unpublished). Y i , P., Ozier, I . and Anderson, C.H. 1968 Phys. Rev. 165, 9 2 .  - 96 -  Y i , P., O z i e r , I . , K h o s i z ,  A. and  Ramsey, N.F. 196?  Phys. Soc. J_2, 509. Y i , P., O z i e r , I . and Ramsey, N.F. ( t o be  published)  Bull.  Am.  

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