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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 British 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r poses may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . P e t e r Beckmann Department o f P h y s i c s The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 1 6 , 1 9 7 1 . - i i -ABSTRACT The spin-lattice relaxation time T-| has "been measured i n gaseous CH^ 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 amagats. The spin-rotation interaction i s the dominant relaxation mechanism i n gaseous CH^. Since the spin-rotation constants are accurately known for CH^, the results provide a check on the existing theory of spin-lattice relaxation for spherical top molecules. An interesting feature was the failure of commonly used theoretical expressions for the density dependence of T-j to f i t the experimental data. A reasonable explanation i s that the centrifugal distortion of the CH^ molecule i s indirectly contributing to the spin-lattice relaxation. - i i i -TABLE OF CONTENTS Page Abstract i i Li s t of Illustrations V . Acknowledgements v i i Chapter I Introduction 1 Chapter II Basic Concepts 3 2.1 Nuclear Magnetic Resonance .... 3 2.2 Spin-lattice Relaxation 10 Chapter III 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 1 Calculation 23 3-5 Error Analysis 25 Chapter IV The Apparatus 29 k.1 Pulsed N.M.R. Spectrometer .... 29 1 . 1 Transmission Stage 30 /f. 1.2 Tuned Circuit Stage ., 3k 4.1.3 Receiving Stage 36 4.2 Gas Handling System 39 - 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 the I n t e r p r e t a t i o n o f the Experimental R e s u l t s W5 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 P e r t u r b a t i o n I n t e r a c t i o n s Wi> 5.2 T1 i n CH/, hi 5.3 Data A n a l y s i s 60 Chapter VI Summary and Suggestions f o r F u r t h e r Work 77 Appendix C i r c u i t Diagrams 79 B i b l i o g r a p h y 95 - V -LIST OF ILLUSTRATIONS Figure Page 3.1 3 . 2 4.1 4 . 2 4 . 3 4 . 4 4 . 5 5.1 5.2-5.3 5.4 5 .5 5.6 5.7 5.8 Free induction decay Spin-echo Pulse spectrometer: schematic diagram Pulse sequences Tuned c i r c u i t stage Free induction signal: phase sensitive detection Gas handling system Relaxation effects of centrifugal distortion Collision induced fields 1 Experimental data: — vs Q Single term f i t around region of — maximum T r 1 P H Single term f i t using linear region 'l vs p •1 IJ exp. 1 T1 16 19 31 32 35 37 40 55 56 61 63 65 66 71 vs P 73 - vi -Figure Page 5.9 vs 5.10 A. 1 A.2 A.3 A.k A.5 A. 6 A.7-A.8 A.9 A. 10 A. 1 1 A. 12 A. 13 A. 1 ^  A. 15 Pmax 75 1J C^max vs Differentiator D and Mixer E . Pulse width generator F (or G) Mixer H Gating pulses amplifier I .... 10 MHz Crystal o s c i l l a t o r J .. Wideband amplifier K Coherent gated o s c i l l a t o r L .. Phase shifter M and Tripler N Gated power amplifier 0 Reference t r i p l e r P Reference amplifier Q Preamplifier R Boxcar integrator U Power supply for Amplifier 0 . Power supply for J, K and P .. 76 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 - v i i -ACKNOWLEDGEMENTS I sincerely thank Professor Myer Bloom and Dr. E l l i o t t Burnell for their assistance over the past two years. I find Professor Bloom's conscientious attitude toward his students most encouraging and I am grateful for the many f r u i t f u l discussions I have had with him. Dr. Burnell aided me i n performing many of the lengthy experiments. This work was supported, i n part, by the National Research council of Canada. CHAPTER I INTRODUCTION The aim of t h i s work i s to i n v e s t i g a t e the mechanism or 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 molecules i n the v i c i n i t y of the T^  minimum. The t-echnique i s t h a t of a pulsed n u c l e a r magnetic resonance experiment. 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 s obtained as a f u n c t i o n o f d e n s i t y at constant room temperature. I t t u r n s out t h a t T- goes through a minimum at approximately 0.04 amagats and the experiment i s performed over the range of from 0.006 to to 7-0 amagats. The molecule being i n v e s t i g a t e d i s methane (CH^) i p which has a s p i n l e s s carbon nucleus ( C) at i t s centre and four s p i n 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 these techniques by Dorothy (1967), Bloom and Dorothy (1967), Bloom, Br i d g e s and Hardy (1967), Dong (1969) and Dong and Bloom (1970). 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 to be the most important cause of r e l a x a t i o n i n CH^ (Bloom, Bridge s and Hardy, 1967) has been i n v e s t i g a t e d u s ing molecular beam techniques by Anderson and Ramsey (1966), Y i (1967), Y i , O z i e r , Khosiz and Ramsey (1967), O z i e r , Crapo and Lee (1968) and Y i , O z i e r and Ramsey (to 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 to be a check on p r e v i o u s l y obtained r e s u l t s before - 2 -going on to t e t r a h e d r a l molecules f o r which Ti had not 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 the a l r e a d y e x i s t i n g apparatus, g r e a t e r accuracy was achieved as a r e s u l t of g r e a t e r s i g n a l to noise, and many more runs v/ere performed than i n previous works. The r e s u l t s are more r e l i a b l e and i n d i c a t e the presence of some f i n e s t r u c t u r e i n the dependence of T 1 on d e n s i t y . The remaining Chapters of t h i s work are 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 resonance and s p i n - l a t t i c e r e l a x a t i o n . The only s p e c i a l i z a t i o n i s to the case of r a r e gases. Persons r e l a t i v e l y new i n the f i e l d are r e f e r r e d to the c l a s s i c t e x t s o f Abragam (1961), Andrew (1955) and S l i c h t e r (1963). Chapter I I I i n v o l v e s o n l y the experiment; the procedures which are used to measure and c a l c u l a t e T^  as a f u n c t i o n of d e n s i t y . T h i s does not i n v o l v e any t h e o r e t i c a l c o n s i d e r a t i o n s other than those which have to be met i n order to s a t i s f a c t o r i l y perform the experiment. In Chapter IV, the apparatus i s di 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 the p u l s e spectrometer may be found i n the Appendix. The concept of 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 the experimental data i s i n t e r p r e t e d using the e x i s t i n g theory. The p o s s i b i l i t y of f u t u r e f r u i t f u l work i n the f i e l d i s d i s c u s s e d , along w i t h a summary of 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 possess an i n t r i n s i c magnetic property or observable c a l l e d the magnetic moment jX given by M = 7 t i l ( 2 . D where I i s the angular momentum o f the nucleus and ^ i s i t s gyromagnetic r a t i o . 7 - " m p c where e and nip are the charge and mass of a proton and g i s the s o - c a l l e d g - f a c t o r . Gaussian u n i t s are used and c i s the speed of l i g h t . When a magnetic f i e l d H Q i s a p p l i e d , the p r o j e c t i o n o f I along H 0 i s quantized and can on l y assume values - I , -1+1, , 1-1, I-The i n t e r a c t i o n between the f i e l d H Q and the sp i n s ( n u c l e i p o ssessing a magnetic moment) i s de s c r i b e d by the Hamiltonian - h -W = - A t - H o (2-2) where fX i s understood to be a quantum mechanical v e c t o r operator. A l a b o r a t o r y coordinate system i s p a r t i a l l y d e f i n e d by assuming t h a t H 0 = H Q i i where k i s a u n i t v e c t o r i n the z d i r e c t i o n . With, the a i d o f equation (2.1), equation (2.2) becomes W = - 7 n H0Iz The p o s s i b l e s t a t e s of the system are denoted by |m> and the observables a s s o c i a t e d w i t h the operator I are denoted by m. L i m > = m im The allowed energy val u e s are W | m > = E m l m : E m = - 7T i H 0 m Note t h a t f o r ^ > 0 , m = -I has the high e s t i n t e r a c t i o n energy . w i t h the f i e l d whereas m = +1 has the lowest. Any measurement made i n an experiment w i l l i n v o l v e many s p i n s and the parameter measured w i l l be an average over an ensemble o f s p i n s . For a system of 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 bath at temperature T i t i s convenient to use the - 5 -Canonical Ensemble where one knows the total volume, the total 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 called can be taken to be associated with the other degrees of freedom of the molecule i n which the nuclei exists. In the Canonical Ensemble, the probability 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 M0 = £ (contribution of a v (fraction of spins /_ ^ spin i n mth state) A i n mth. state) v o ; a l l m values The entire idea of our experiment i s to measure how the system of spins returns to this value of MQ 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 directly. +1 M0 = E MZN PM m = - I - 6 -= i > 7 f i m P m m = - l Because the sum i n the denominator of P m i s independent of the total sum, we have + 1 L N 7 ^ m e x P [ kT J m = - l M0 = . +1 E r 7T\ HO m m = - l I f one examines the magnitude of ^11H0m/kT ^ o r typical spins and laboratory conditions (room temperature), one finds that i t i s the order of 10"^ for a spin 1 species i n a f i e l d of 7 kilogauss. With this i n mind, i f one expands the exponentials and keeps terms of order m (high temperature approximation) the result i s the Curie Law: N/VKI + D ,, M ° = 3 k T ' Just as M0 i s a sum over the jU.z's, the total magnetization M i s a vector sum over the individual yU's. The equation of motion for M i n a magnetic f i e l d results from the torque exerted on i t by i t s interaction with the main f i e l d and i s given by Note that H n i s i n the z direction and ^ = 0 d t fvi2 - constant = M 0 Equation (2.4) implies that the vector characterizing the magnetization i s precessing about the z axis. To see this, i t i s convenient to transform the equation of motion into a reference frame rotating with an arbitrary angular frequency u about the direction of the f i e l d HQ. For our case the equation of motion i n the rotating reference frame i s given by H M — - A = M x ( 7 H 0 - w k ) (2.5) Equation (2.5) i s conveniently rewritten as dM - A -j7 = M x ( u0-u ) k (2.6) dt where U0= 7 Ho defines the Larraor angular frequency. From equation (2.6) i t i s immediately obvious that i n the reference frame rotating at [J = LJo, the magnetization i s static. 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) direction of the laboratory frame. This i s done by setting up an o s c i l l a t i n g current i n a solenoid, the sample under study being inside the solenoid. One can think of this f i e l d as being the vector sum of 2 circu l a r l y polarized f i e l d s with their polarizations i n opposite senses. The rotating frame we use i s defined by that one of these rotating f i e l d s which i s constant i n the xy plane of the rotating frame. I f we define the x direction i n the rotating frame by the direction of the new f i e l d , we have H, = h/i and the equation of motion i n the rotating reference frame becomes jY '- M x [{u0-u)k + ufi] (2-7) where LJ^= ffR-\ defines LJ ]. V/e have ignored the other component of the original o s c i l l a t i n g f i e l d which i s rotating with an angular frequency -ZU with respect to the rotating frame we - 9 -have chosen. In practice, H-|«H 0 and Abragam ( 1961) shows that this f i e l d causes very small effects 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 rotating 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) as dM dt = 7M x H G f f the magnetization i s now precessing about an effective f i e l d i n the rotating frame given by Because U-\ i s very small, for most values of LJ], U] « ( ijJo - U ) and the magnetization i s almost unaffected by H] . That i s , the solution to equation (2.7) i s not very different from the solution to equation ( 2 . 6 ) . The interesting case i s when we apply the rotating f i e l d at U = UQ . Equation (2.7) becomes ^ = M x U/,f ( 2 - 8 ) The magnetization now precesses about the f i e l d H-|i i n the rotating frame. This i s a resonance phenomenon and i s appropriately - 10 -named nuclear magnetic resonance. If we pulse the f i e l d H-j at the resonance frequency for a predetermined time, M can be l e f t at any orientation with respect to the z axis. A 7T pulse of the f i e l d rotates M by 180° about the x axis and takes Mz into -Mz. A 77/2 pulse effectively puts the magnetization i n the xy plane and so on. The effect of the pulsed radio frequency i s to cause the spin system to depart from i t s equilibrium position providing the angle of rotation 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 stating this i s that we have perturbed the system from i t s lowest energy configuration and i t w i l l strive for i t s equilibrium situation by giving up energy to the other molecular degrees of freedom which comprise the " l a t t i c e " or "bath". 2.2. SPIN-LATTICE RELAXATION Some kind of model i s required to explain the approach to equilibrium of the spin system after i t has been perturbed. The spin-lattice relaxation time T 1 i s loosely defined as a characteristic time for the component of the magnetization along the f i e l d direction to approach i t s equilibrium value. A T-| mechanism involves ah exchange of energy between the spins and the l a t t i c e . There may be interactions between spins which redistribute energy among the spin system but do not change the total energy. For simple systems this relaxation i s described by a single constant T^, the spin-spin relaxation time and involves relaxation 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 later consider i t s effects when v/e perform the pulsed nuclear magnetic resonance experiments. The microscopic mechanisms for spin-lattice relaxation for the special 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 total 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 interpret the experimental data which i s obtained by applying the expression for T^  we now develop using the macroscopic theory. V/e can modify the equation of motion (2.7) to include relaxation after a perturbation. The modified equation of motion which holds very v/ell for gases along with the concepts of Tj and T 2 are phenomenological i n nature and are due to Bloch (1946). dM - r ^ A i — = M x [{u0-u)k + u,\] d t L 1 T1 T2 Starting with the general equation (2.9) i n the rotating frame, we can specialize i t for our pulse techniques. Starting with the equilibrium situation where LJ-j has not been introduced, . we- have - 12 -dM d t A M x ( u0 - u ) k d M z . dt = 0 M 7 = M, The system i s perturbed at the resonant frequency i n a time short enough that relaxation effects are negligible.-dM — A U l = M X I (2.10) dt As discussed previously, this means that M precesses about the x direction i n the rotating frame. Rather than solve equation (2.10) directly, we just leave LJ. on long enough to take M_ from i t s I z equilibrium value of M0 to a non equilibrium value of M (0) . In the experiment, the special case of a 7T pulse i s used which means M„(0) = -M_. This i s as much as the system can be perturbed. After the pulse, the equation of motion i n the frame rotating at the Larmour angular frequency becomes - 13 -d t Ti T, Note that i n the laboratory frame, the part of equation (2.11) w i l l have factors involving sin(L/ 0t) and cos(Uy ot) but i n both frames d Mz Mn- Mz t s A (2.12) d t T , Equation (2.12) i s easily solved and M (t) i s given by M2( t ) = M o - [ Mo- M z ( 0 ) ] [ e x p ( - i ) ] ( 2. 13) and with the i n i t i a l condition, H z(0) = -M0 imposed by a ff pulse we have Mz( t ) = M o [ 1 - 2 e x p ( - y ) ] ( 2 . H ) Thus from a measurement of Mz as a function of time after a suitable perturbation, v/e can uniquely determine T i . CHAPTER III THE EXPERIMENT 3.1 SIGNAL DETECTION The same c o i l i s used to deliver the radio frequency pulse and detect the nuclear magnetic resonance signal. A time dependent net magnetization M(t) for the sample i n the c o i l w i l l be proportional to a time dependent magnetic induction B(t). The signal voltage induced i n the c o i l w i l l be vS( t ) = - j j f y where the total magnetic flux (|)(t) i s given by »(t) = A B x ( t ) ( 3 - 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 Vr(Cl/q) from the reference amplifier i s used for phase sensitive detection i n the tuned amplifier. Taking care to adjust V-^OJ^ such that the amplifier i s linear over a l l desired amplification regions, the output w i l l - 15 -be proportional to the magnitude of Vs(U0) at time t. (See figure 4-4.) The signal Vg(tu/ 0) i s proportional to the magnitude of the UQ component of the flux (|)(t). From equations ( 3 . 1 ) and ( 3 . 2 ) and the previous discussion i n this section i t i s clear that Vg(L/0) i s proportional to the magnitude of the component of M x(t) with U = L/Q. To summarize, we can measure a voltage proportional 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 after a suitable 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 proportional 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 by examining some simple models. 3.2.1 FREE INDUCTION DECAY Starting with the magnetization i n the equilibrium position with Mz = MQ, the application of an r. f. pulse w i l l take Mz into M z ( 0 ) . Figure 3.1 shows the special case used i n our experiments where a 7T pulse gives M z (0) = -M0. In terms of figure 3* 1 j we have A—>B. (The primed and double primed letters i n figure 3.1 correspond to the same event for a second and third time.) The spins w i l l now relax according to equation ( 2 . 1 3 ) - 16 -f i gu re 3.1 f ree i nduct ion decay —4 t 2 A iC B D i i A' B' C D A' B ' (not to sca le) A t - A P - - y i i A * A B PS which reduces to equation (2.14) for the special case of a 7T pulse. In a time t we have B—>C i n figure 3.1- At this time pulse w i l l rotate M z(t) into the y direction indicated by C—>D i n figure 3.1. We are i n the rotating frame and the perturbing f i e l d i s i n the x direction. From Chapter 3-1» v/e have the result that a signal i s now detected. After the 77/2 pulse, the signal w i l l be positive or negative depending on whether M„(t) was positive or negative. This magnetization i n Li the xy plane w i l l decay according to the T 2 part of equation ( 2.11). I f a measurement i s made a constant time after the 77/2 pulse, the factor which describes the loss of signal due to effects i n a rigorous expression for the amplitude of the signal becomes a constant and the measured voltage Vs(t) discussed i n Chapter 3.1 becomes proportional to M_(t). The signal i s sampled by a measuring pulse indicated by E i n figure 3.1. Figure 3.1 shows three measurements for increasing values of t, namely t, t' and t'' and indicates how the signal decays from -MQ to +M0 as t i s varied from 0 to t R , the repetition rate of the pulse sequence 3.2.2. SPIN-ECHO The spin-echo technique makes use of the fact that after the 77/2 pulse discussed i n the previous section, the loss of signal i n the xy plane i s , i n part, reversible. Under general conditions, the decay of the x and y components of M through the mechanism of mutual spin f l i p s resulting from the dipolar interaction i s irreversible. However, that part of the decay arising from the fact that the spins precess at s l i g h t l y differen - 18 -angular frequencies because of sli g h t l y different constant local fields i s certainly reversible. For our case, this variation i n local fields i s chiefly due to the inhomogeneity of the magnet. The spin-echo effect i s shown i n .figure 3.2 using three spins; , S 2 and S^, for simplicity. The positions A, B, C and D are the same as for the free induction decay situation described i n the last section. At E, the spins are i n the position shown because the angular frequency of i s greater than the angular frequency of S^, which i n turn has an angular frequency greater than . The application of another 7T pulse at this time w i l l f l i p the spins to position F with the "fastest" spin now behind the others. As should be obvious with this simple model, the spins w i l l reunite at position G and "pass through each other". The result i s a spin-echo which i s i n effect two induction decays back to back. A measurement at the echo peak w i l l again give a signal proportional to M z(t). 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 distortion be prevented, one must i n f l i c t l i m i t s on some variables and preserve relationships between others. The following i s a discussion of a typical run which w i l l lead to the determination of T^  for a given pressure. Approximately 100 p. s. i . g. of CH^ i s used to tune the signal. The phase of the r. f. pulses relative to the reference voltage i s adjusted to give a negative free induction - 19 -ure 3.2 s p i n - e c h o - 20 -decay and i f the three pulse sequence i s used, a positive spin-echo i s obtained. A negative induction decay i s used because i t i s easier to differentiate between the in d i s t i n c t end of the large 7T/2 pulse and the beginning of the signal. The signal 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 five parameters: tg i s the repetition 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 integrating constant. These three parameters along with the positioning of tg and the rate at which t i s varied must be determined. This last point decides hov; long a run w i l l take and places an upper l i m i t on the effective measuring time constant. This i s so, because the recorded signal must not lag behind the detected signal. However, the longer the effective time constant i s , the better the signal to noise w i l l be. As discussed i n Chapter k-3, the effective time constant X i s given by r = -RRC l s For a typical run of about 30 minutes, a value of 5 to 15 seconds seemed to be an optimum value for T . Theoretically, the longer the time taken for a run, the larger the T that may be used and the better the resulting signal to noise. However, long term - 21 -d r i f t s i n M0 often make i t better i n practice to put up with poorer signal to noise and use a shorter run time. Normally, one fixes t R and t s independently and uses an RC necessary to give an optimum X • T l i e time t% must be long-enough that M z(tjj)«M 0 or, i n other words, the spins must be very close to equilibrium before the next pulse sequence begins. Within the sensitivity of the experiment} tp>10Ti i s a reasonable choice. For higher values of t R , signal to noise i s proportional 1 /? to tg ' (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 tg depends on the width of the signal being sampled, (induction decay or echo). For the echo, one centres t s symmetrically about the peak and adjusts the width u n t i l signal 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. This l i m i t ( about 20 microseconds from the end of the pulse ) i s set by the recovery time of the amplifier after the ^ /2 pulse. Signal to noise i s then maximized by adjusting the width. Note that the width of either signal i s proportional to T 2 which i n turn i s approximately proportional to density . As a result, the lower the density i s , the smaller t s must be. Signal to noise i s proportional to density, but because this T 2 narrowing effect places an upper l i m i t on tg, there i s a further decrease i n signal to noise at lower densities. At densities lower than about 0.03 amagats for CH^, signal to noise must be maximized on the chart recorder without seeing the signal, because i t i s completely buried i n noise. I t should be noted that at several densities - 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 amplifier, the oscilloscope and the boxcar integrator output. The time t i s varied manually between 0 and almost t R and the three amplifiers are set to give f u l l scale deflection on the chart recorder. Note that M z(~0) = -M i s the minimum signal and M z(~tg) « Mz(00) = M0 i s the maximum signal. The only time f u l l scale deflection was not used was for very low densities where nothing was gained by using f u l l scale deflection because of the very poor Mz(00) - M z(0) signal to noise. The maximum signal to noise, N 2M 0 «s varied depending on how much time was taken for various N runs, but typical signal to noise ratios on the chart records were 3 at 0.006 amagats, 7 at 0.01 amagats, 13 at 0.02 amagats, 20 at 0.04 amagats and 30 for densities above 0.1 amagats. The optimum bandwidth and d. c. amplification are determined by maximizing signal to noise and depend on the characteristics of the individual amplifiers. This i s a matter of t r i a l and error and because the tuned amplifier has a maximum amplification of about 10^, one must take care that the signal does not saturate i t s f i n a l stage. The sampling time t i s set as close to t R as possible and M„(00) « M i s recorded. An ultra-slow sweep (see next z o chapter) which automatically varies t from 0 to about 2 or 3 T-| (or u n t i l the signal becomes too close to M0 to be useful) i s - 23 -used to measure M z(t). The ultra-slow sweep i s then turned off and MQ i s again recorded. I f M0 has varied more than about 5% of f u l l scale deflection, the run i s rejected. 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 electronic system. The equilibrium magnetization Mo before and after the run i s joined by a straight l i n e to enable one to compute M 2(t) - M0 as a function of t, which i s indicated by an event marker. If systematic errors are present, one must try to account for and eliminate as many as possible. One effect definitely present i s an exponential signal superimposed on the induction decay resulting from the amplifier's recovery after the f i r s t IX pulse. The signal i s sampled a constant time after the pulse and because only the differences M z(t) - M0 are involved i n the computation of T-j, the constant voltage added to the signal due to the amplifier's recovery after the pulse may be neglected. However, as the time betv/een the f i r s t TTpulse and the pulse i s varied, the boxcar sampling pulse effectively sweeps the distortion signal associated with the amplifier's recovery after the f i r s t 7T pulse. For a l l runs, the bomb was evacuated and a normal run performed to obtain this correction. 3.4 Ti CALCULATION Having measured a voltage V"s(t) proportional to M^(t) - M_, one has from equation (2.14) - 2if -V s ( t ) c< 2 M e [ e x p ( - J - ) ] (3.3) A smooth curve i s drawn through the signal and equation (3-3) i s f i t t e d on semi-log graph paper. l n [ v s ( t ) ] = - 4" + c o n s t a n t { 3 - k ) n T1 i s then calculated from the slope. This was performed manually rather than using a least squares f i t (for equation 3.4) because of the author's belief that the experimenter can thus study each plot carefully and take into account more reliably certain problems which may arise. For example, when a curved lin e results from plotting l n £ v s ( t ) j against t over the entire range of t, i t could always be traced to a poor choice of Vq(CO) , usually because MQ was not recorded long enough. A scatter of points significantly greater than that obtained from other runs i n similar density regions could usually be attributed to equipment i n s t a b i l i t y . In both cases, the runs were rejected. I f there i s a slight error A^o x n H 0 ^ e curve might deviate slightly from a straight line i n the high t regions because A[M 0 - M z(t)l —= =- may be so small as to be unnoticeable at low [M 0 - M z(t)J values of t £large M z(t) - M0J , but exceedingly large at high values of t £ small M z(t) - M0] . In such a case, the high t points were not used i f there was a considerable straight line - 25 -region for lower t values. Of about 300 runs, 294 satisfied the condition that the change i n M 0before and after the run was less than 5% (discussed i n the previous section). Of these 294 runs, nine failed to meet the c r i t e r i a discussed i n this section which means the experimental points i n figure 5.3 (Chapter V) number 285. In the higher density regions, the relaxation v/as exponential within experimental error over 2 orders of magnitude, 2 decades being a l l that could be measured reasonably with the available equipment. For one value of pressure near the Tj minimum, a very lengthy run was performed using a Fabri-Tek Instrument Computer and the relaxation was found to be exponential within experimental error over almost 3 orders of magnitude £in M z(t) - M Qj . It should be noted that the Fabri-Tek, which i s a d i g i t a l signal averager, was not used for very low pressure runs because; a) i t takes much longer to both perform the run and calculate 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 signal. 3.5 ERROR ANALYSIS The results of the experiment appear i n figure 5.3. This graph of T-|-^ vs has the following special 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. This latte r point requires an explanation and upon giving 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 least squares f i t or some other well defined procedure. The error analysis presented here diff e r s from the normal procedure i n that rather than calculating 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 different 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 ln£vs(t)j vs t graph and c a l l these limits "the error i n T^". B. One can go back to the trace from the chart recorder and from signal to noise considerations put an error bar on each point of the lnjVs(t)j vs t curve. Both these procedures v/ere carried out for several runs i n a l l density regions. For p > 0 . 2 amagats, the error determined i n both ways was less than 1%. In the region 0.02 < P < 0 .2 , the f i r s t procedure yielded errors of about 5% and the second procedure yielded errors of about 3% For the region p < 0 . 0 2 amagats, a separate discussion i s given later i n this section. Because the drawing of extreme lines ( f i r s t method) always gave an error larger than the second method, this procedure was adopted i n the earlier stages of the experiment. To make sure we were not "biased line drawers", the following experiment was performed. Ten persons i n the Physics department (who knew nothing of the experiment) were given 3 Xerox copies of each of 3 different runs (9 graphs). They were asked to draw what they thought to be the best line, the line of minimum reasonable slope and the line of maximum reasonable slope. They were not given runs with a curved departure from li 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 results were the .following; a l l the best lines (including the author's) were within 1% of each other and the maximum and minimum slopes were "inside" the author's i n every case. Satisfied that errors were not being underestimated, this 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 this error i s certainly larger than the probable error resulting from a least squares f i t analysis. V/ith the exception of a very few cases, the errors computed i n this 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 resulting from the spread i n points near the maximum i s about 20% whereas the error determined for each T.,""^  by using the preceding . procedure i s about 3%. 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 results ( 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 this region (relative to the other - 28 -regions) the low density results look better than they probably are. I f one were to use the maximum-minimum slope method to determine errors for each i n this region, the resulting errors would probably exceed the spread i n points. This would certainly be the case for jQ<0.01 amagats. For instance, the lowest density point (0.006 amagats) has 2 points, 4.24 and 5.26 msec. These two points result from the same trace calculated by two people, Dr. Burnell and myself. One might suspect that a least squares f i t of the ln|Vs(t)J vs t graph resulting from a smooth curve drawn through the trace might be significant for this 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 signal on the chart record. Errors determined from the methods already discussed would certainly be larger and i n the author's opinion more meaningful than a least squares f i t approach. Because only very general remarks w i l l be made concerning this low density region, the errors are not included i n the experimental plot of T i ~ ^ vs Q , CHAPTER IV THE APPARATUS The apparatus used i n these experiments can be divided into two parts, the nuclear magnetic resonance pulse spectrometer and the gas handling system. The original apparatus was bui l t by John D. Noble (1964). Since that time, additions and improvements have been made by Hardy (1964), Dorothy (1967), L a l i t a (1967), Dong (1969)» Burnell and myself. 4. 1 N. M. R. PULSE SPECTROMETER The spectrometer can be conveniently divided into three stages; the transmission stage, the tuned c i r c u i t stage and the receiving stage. The transmitter delivers radio frequency pulses to the c o i l i n the tuned c i r c u i t . The same c o i l receives the nuclear induction, signal. This signal i s then detected, amplified, displayed and recorded by the receiving stage. Figure 4.1 shows these three parts of the spectrometer. Circuit details for a l l the non commercial components can be found i n the Appendix. We now consider the spectrometer i n greater detail. - 30 -4.1.1 TRANSMISSION STAGE Components of the transmitter are designated by upper case Arabic letters whereas pulses and waveforms are indicated by underlined lower case Arabic let t e r s . The reader may follow the analysis more clearly by referring to figures 4.1 and 4 .2. A Tektronix 162 waveform generator, A, delivers a pulse, a, and a sawtooth, b, of period t R to different channels at the same time. That i s , pulse a coincides with the leading edge of sawtooth b, indicated by lines 1 and 2 of figure 4 .2 . Another Tektronix 162, B, i s modified to deliver an ultra-slow (or long period) sawtooth, c, indicated by line 2 of figure 4.2. The repetition rate t R i s typically i n the millisecond range while the period of the ultra-slow i s usually about an hour. These two sawtopths, b and c_, are fed into a Tektronix 163 wave generator, C. C produces a square pulse, d, of width tj, when the voltages of the two sawtooths crossover, indicated by lines 2 and 3 of figure 4 .2. The result 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 starts a time t after the i n i t i a l pulse, a. As dicussed i n Chapter 3 . 3 S t R and tg are fixed and t increases automatically as the ultra-slow sawtooth decreases i n amplitude. The square pulse, d, i s differentiated, D, producing a positive 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, indicated by line 4 of figure 4 .2 . The two pulses are channeled separately and the sign - j>l -f i g u r e 4.1 pulse spect rometer s chemat i c d i ag ram transmission stage B ul t r a -slow A sawtooth + pulse C pulse generator V 3: E mixer Va, f F pulse generator •a',f H mixer V ^, e; f I gating pulses amp. L gated osc i l l a t o r M phase shifter N frequency t r i p l e r 0 power ampli f i er D d i f f e r -entiater G pulse generator J 10 MHZ. osc i l l a t o r K amplifier P frequency triplfir V Q reference amplifier X H.P. time interval unit Y d i g i t a l recorder V sawtooth Z chart recorder ffi W boxcar pulses U boxcar integrator S detector-am p l i f i er 7K T scope R preamp. receiving stage — r 3 FD111's J_ 7? magnet tuned circuit stage f igure 4.2 - 32 -pulse sequences 33 -of the negative pulse i s reversed, indicated by lines 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 trigger 2 square waves, d/and f' respectively. This i s indicated by lines 7 and 8 of figure 4 .2. This unit i s called a 7TPulse generator because a'and f ' w i l l eventually be 77" pulses. Pulse e from D triggers another square pulse, e'in the 7T/2 pulse width generator G, indicated by line 9 of figure 4 .2. The three pulses are mixed, H, and amplified, I, indicated by li 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 third may be varied independently from the width of the second. The time betv/een a 7 and e'is t and the time between e and f ' i s t E . Depending on which pulse sequence i s desired (see Chapter 3-2) pulse f can be turned on or off. A 10 MHz crystal 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 shifter, M, where the phase of the radio frequency voltage can be adjusted relative to the original o s c i l l a t o r , J. Finally, the frequency i s tripled, N, and the pulses are amplified to about 1000 volts peak to peak, 0. A reference signal from the oscil l a t o r , J, i s amplified, K, tripled, P, and amplified again, Q. This signal i s used for phase sensitive detection i n the receiving 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 this 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 signal to noise have been made which yielded the lower density results. The lead from the transmitter to the tuned c i r c u i t i s one-half wavelength to reduce the loss 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 entails a c o i l of fixed inductance and a variable capacitor to meet the resonance condition, tJ0= (LC)~ 2. A factor o-f at least 2 i n signal to noise has been gained over previous experiments by having the tuning capacitor as close to the c o i l as physically possible. UQ ( = *^H0) i s fixed 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. coi 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 for an N. M. R. spectrometer i s given by Clark (1964). 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 this particular experiment was made from 10 turns of #14 guage copper wire with a diameter of 3/4" and a - 35 -f i gu re 4.3 tuned c i rcu i t stage ^.7 pf ' U.7pt r. f. p u l s e s in <> I d. c. c o i I s -kovar s e a l -o r ing s e a l pe rmanent magnet -° s i g n a l out - a l u m i n u m s h i e l d epox i s e a l - — g l a s s c y l i n d e r - - b r a s s bomb copper co i l length of l-g-". The wire was well cleaned, including 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 tight brass bomb with the r. f. lead coming through a kovar seal. The glass was fixed to the bomb with epoxi, which extended to the kovar seal, again to prevent breakdown. The gas enters the system at the other end of the bomb. The inductance of the c o i l was 2 microhenries and at 30 MHz the circ u i t tuned at 15 picofarad. The output signal passes through a 4.7 picofarad decoupling capacitor en route to the receiving stage. 4.1.3 THE RECEIVING STAGE The receiving stage detects, amplifies, displays and records the signal. "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 join the diodes to the tuned c i r c u i t and to the preamplifier, R, and cut down the loss of the radio frequency (30 MHz) signal. The reader i s again referred 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 -f igure 4.4 free induction s i gna l - -very large * r very s m a l l rT. f. s i gna l - y pulse — A N — s i g n a l envelope before phase sens i t i ve detect ion and a m p l i f i c a t i o n ( not to scale ) B after phase s e n s i t i v e detect ion and a m p l i f i c a t i o n - 38 -amplifier, S, can recover i n about 20 microseconds. As discussed i n Chapter 3.3 the recovery time of the tuned amplifier, S, i s important at low densities where T2 i s short because this limits how 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 amplifier with a bandwidth of 2 MHz centred around 30 MHz at 3 db. The r. f. signal i s amplified i n the f i r s t 3 stages where the reference voltage from the reference amplifier, Q, i n the transmitter i s introduced for phase sensitive detection. The reference voltage was kept at about 2 volts (d. c. level) and the output from the 30 MHz amplifier was kept below 0.2 volts i n order that the amplifier operate i n i t s linear region. After phase sensitive detection, the signal becomes the r. f. envelope shown i n figure 4.4B. The signal from the tuned amplifier i s again amplified by, and displayed on a Tektronix 531A oscilloscope, T, with a type Z plug-in. I t should be noted that this combination of oscilloscope and plug-in i s a very low noise wideband amplifier. The signal from the oscilloscope i s fed into a boxcar integrator, U. A pulse from the pulse generator, G, triggers a Tektronix 162 waveform generator, V, which produces a sawtooth. At a set time after i t s beginning, this sawtooth triggers positive and negative square pulses of duration tg i n a modified Tektronix 161 pulse generator, W. These square pulses, or sample pulses, gate the signal from the oscilloscope i n the boxcar integrator. The boxca'r accepts the signal over the time t q and then averages using - 39 -an RC integrating c i r c u i t . The effective time constant of the boxcar i s f = ( t R / tg) RC. An excellent account of the theoretical aspects and experimental techniques involved i n the boxcar integrator may be found i n Hardy (1964). The output of the boxcar i s displayed on a Varian strip chart recorder, Z. The time t between the f i r s t pulse and the pulse i s measured by a Hewlett Packard Electronic Counter, X, with a .time interval unit plug-in. The count starts on a trigger pulse from the TC pulse generator, F, and stops on a trigger puis from the ^7/2 pulse width generator, G. At convenient times, a Hewlett Packard Digital Recorder, Y, indicates the time t on a printed output and at the same time triggers an event marker on the strip chart recorder, Z. The input of the receiving stage i s an r. f. signal 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 relatively easy gas to work v/ith because i t may be expelled into the a i r as long as the ventilation i n the room i s reasonable and there i s no flame near the pump outlet. The schematic diagram shown i n figure 4-5 i s self explanatory. For pressures greater than 2 amagats, a calibrated 0 to 100 pound per square inch 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 4.5 gas hand l ing s y s tem bomb-magnet methane 0 = 0-100 p s i guage 1 _ r vacuum pump 6 valve mercury manometer <[]> stopcock 1 stopcock 2 - 41 -amagat, stopcock 1 was c l o s e d and stopcock 2 opened i n order to pump out the r i g h t hand s i d e o f the manometer. The pressure of the system i s then determined ( w i t h stopcock 2 closed) a b s o l u t e l y w i t h no a d d i t i o n or s u b t r a c t i o n due to atmospheric pr e s s u r e . Note t h a t t h i s l a t t e r method reduces the 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 . The probable e r r o r i n pressure i s completely n e g l i g i b l e down to 0.01 amagats and f o r the p o i n t s below 0.01 amagats the probable e r r o r i s sm a l l compared w i t h the spread. ( i n T-j~1) i n p o i n t s . A l l p r essures i n mm Hg were converted to d e n s i t y i n amagats which i s the r a t i o of the d e n s i t y at a p a r t i c u l a r temperature and pressure to the d e n s i t y at standard temperature and pressure. The maximum d e n s i t y , 7 amagats, i s w e l l below the 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 the P e r f e c t Gas Law i s an e x c e l l e n t approximation. PV- = nkT d = A ' = P-V kT p ( a m a g a t s ) = where the s u b s c r i p t zero r e f e r s to standard temperature and pressure. Amagats are o b v i o u s l y 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 not a f f e c t the computation o f d e n s i t y i n amagats, they w i l l - hz -affect T-j by changing the constant, of proportionality between T c and as discussed i n the next chapter. For this reason, no runs were performed i f the temperature i n the magnet gap was outside the limits 2 0°C<T< 2 3°C (!%• change i n °K) . CHAPTER V SPIN-LATTICE RELAXATION IN CH^ AND THE INTERPRETATION OF THE EXPERIMENTAL DATA In the following discussion, the possible mechanisms for nuclear spin relaxation are considered with emphasis placed on the spin-rotation interaction which i s the dominant relaxation mechanism i n CH^. The relaxation rate 1/T1 i s then formulated i n terras of the combined effects of the spin-rotation interaction and the molecular motion. Finally, having reviewed the microscopic theory for relaxation, we use the theory to interpret the experimental data. 5.1 NUCLEAR INTERACTIONS The nuclear hyperfine interactions for spherical top molecules such as CH^ are discussed i n considerable detail by Yi, Ozier and Anderson (1967). In the following discussion, we pick out those interactions which play a role i n the theory of spin-l a t t i c e relaxation. - -5.1.1 ZEEMAN LEVELS A discussion has been given i n Chapter 2.1 of the Zeeman energy levels arising from the interaction of a free spin v/ith a constant magnetic f i e l d . Of the many possible modifications to these levels resulting from the molecular environment of the nuclei, only the effect of the rotational moment i s important * This i s an interaction between the molecular magnetic moment associated v/ith the rotational angular momentum "hJ and the constant f i e l d H 0 and must be added to the Zeeman Hamiltonian. As discussed by Gordon ( 1966) , the molecular rotation i s very fast compared v/ith the Larmor precession and although several J states are occupied i n CH^ at room temperature, the rotation has only B.yi average fcoiDst-snt^ of foft;. The ^o^i fi^stiO!" t i*-b.<? effect-on the N.M.R. experiment) amounts to changing the Zeeman splittings from EL/ 0 to "h( U0 - LJj). The parameter Uj has been measured by Anderson and Ramsey (1966) to be 0 . 0 5 6 L / 0 . Note that this does not concern the interaction between the nuclear spins and a f i associated with the molecular rotation, but rather the effect on the N.M.R. experiment of the interaction betv/een the rotational moment and the f i e l d . Although there are other interactions which have to be taken into account for certain v/ork with molecular beams (Yi, Ozier and Anderson, 1967) they can be neglected i n our case i n which the Zeeman levels already discussed completely dominate i n determining - k5 -the unperturbed energy levels of the spin system. 5.1.2 PERTURBATION INTERACTIONS The following interactions are denoted as perturbations because i n a general discussion, they a l l may link the spins with the l a t t i c e and, therefore, represent possible mechanisms for nuclear spin relaxation. As a result of i t s importance, we consider the spin-rotation interaction i n somewhat more detail. The rotation of a free CH^ molecule w i l l result i n a magnetic f i e l d at the site of a nuclear spin because of the periodic motion of the other three spins i n the tetrahedron. The general Hamiltonian describing this interaction may be written wS 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 different fi e l d s . The rotational fields are related to the rotational state by H-r = 4 ? C , - J 9 which defines the spin-rotation tensor C. Substituting equation ( 5 . 2 ) into equation (5.1) yields - 46 -WSR = -27rnE I,C,J (5.3) I t i s convenient to simplify this Hamilton!an by considering the restrictions required by the tetrahedral symmetry. This i s done i n detail by Anderson and Ramsey ( 1966)- There are two physically different contributions to WgR given by equation ( 5 . 3 ) . The f i r s t i s the average interaction between the spins and the rotation and can be written -27T^C aI'J which implies that a l l 4 spins see a constant f i e l d 277c aJ/'Y . The second term, denoted by i s an anisotropic tensor interaction and may be described as the departure from the average. The exact form i s somewhat complicated and does not concern us directly; i t may be found i n Anderson and Ramsey (1966). For reasons discussed later, i f the C a and terms can not be separated i n the relaxation experiments, one speaks of an effective spin-rotation coupling constant, ^eff For purposes of data analysis, the numerical values for Ca, and C e f£ are taken from the molecular beam experiments of Yi, Ozier, Khosla and Ramsey(1967). These values have been verified by V/ofsy, Muenter and Klemperer ( 1970) and Yi, Ozier and Ramsey (to be published). There w i l l , i n general, be a magnetic dipolar interaction. The intermolecular dipolar interaction can be completely neglected because of the effect of the r ~ ^ factor i n the Hamiltonian. Bloom, Bridges and Hardy (1967) investigated the intramolecular - 47 -dipolar interaction's contribution to the relaxation and found i t to be about 5% of the spin-rotation interaction's contribution. With this i n mind, we neglect this contribution to the relaxation and assume the spin-rotation interaction i s dominant. I t should be noted that this assumption i s made purely on the theoretical evidence of Bloom, Bridges and Hardy (1967) because there i s no information i n the experimental results concerning the contribution of the intramolecular dipolar interaction. 5.2 T 1 IN CH,, Nuclear spin relaxation i n spherical top molecules due to the spin-rotation interaction has been investigated quite thoroughly and we give here only a brief review v/ith an emphasis on the physical processes involved. The interested reader i s referred to the following six publications. Hubbard (1963) and Blicharski (1963) arrived at expressions for T-j i n liquids for symmetric top molecules. Bloom, Bridges and Hardy (1967) extended this to the case of gases for symmetric top and spherical top molecules. These papers use the treatment of relaxation i n the classic text of Abragam (1961) as the starting point. Dong (1969) and Dong and Bloom (1970) simplified the expression for T-| for CH^ using the experimental evidence that the same correlation time could be associated with both the C a and C d terms. The relaxation rate w i l l involve the probability per unit time that a transition betv/een spin states w i l l occur i n the spin system. In order that energy be conserved, we must consider the 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 t n > = E: L11 > W s i s > = E q l s > In a rigorous treatment, w i l l involve the rotational and translational energies and E^ , the nuclear Zeeman levels. I f we consider the case of a free molecule and assume the spin-rotation interaction i s a small perturbation on the unperturbed levels, we can use f i r s t order perturbation theory to determine the probability per unit time that the system goes from a state II,s> to a state |l's'>. This probability w i l l contain terms of the form -h < — 2 r l,s|W S RU',s '>| 6[(Es.-Es) +(E K -E i>] cs. where WgR i s the perturbing spin-rotation Hamiltonian and as usual the unperturbed energies are used. The transition probability per unit time for the spin system w i l l involve an ensemble average over the l a t t i c e states and w i l l give terms of the form - 49 -where W, ,/ / contains terms li 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. Using the integral form of the (5-function for expression ( 5 . 4 ) , an expression for the relaxation rate.will involve terms l i k e E E P , | < l , s | \ A y i ; s ' > |2/ e x p [ i ( us s^ u /u0 t ] d t ( 5 . 5 ) i ' i A where HL/Q/ = E-j/ - E-j_, HUss/= ES/ - E S and constants have been dropped. Noting that LJSs/ must be (*/0 - U j because the unperturbed spin states are equally spaced, expression (5«5) may be expressed as a sum over l a t t i c e transitions rather than i n i t i a l and f i n a l states. V/ith this i n mind, we write as a sum over the physical causes of the LJ^, T Z G^W/exprifd^'-L/j) + L/J tjdt ( 5 . 6 ) 11 u J L J A great deal has been omitted i n the transition from expression ( 5 . 5 ) to equation ( 5 . 6 ) , but what equation ( 5 . 6 ) effectively says i s that one can associate an amplitude G^(0) with each l a t t i c e frequency component LJ^. ^ ( 0 ) i s called the time independent correlation function and i t i s the sum of squares of matrix elements of the spin-rotation 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 ) must be modified to include the effect of collisions. The "effective 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, each of which i s broadened into a band by the translational energies associated with the Boltzmann distribution of velocities. During a c o l l i s i o n , a molecule w i l l experience anisotropic forces which w i l l change the fields associated with molecular rotation at a nuclear spin site. We can include the effect of this c o l l i s i o n a l modulation by associating a "reduced correlation function" g^Ct) with each term i n equation ( 5 . 6 ) . This w i l l , perhaps, become clearer i f one defines the spectral density J k((J) as exp[\[u + u K )t] g k (t)dt ( 5 . 7 ) and interprets this as the frequency distribution of lo c a l fields 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 this f i e l d distribution with U = UQ - Uj-- 51 -T I G k(0) j k ( ^ 0 - ^ j ) ( 5 . 8 ) The time independent correlation function Gi_(0) and the constants of proportionality i n equation ( 5 . 8 ) are known and i t remains to obtain an expression for g k(t) i n order to solve equation ( 5 . 7 ) . I f one associates a correlation time 7*k, which describes a characteristic time for the effect of a c o l l i s i o n , with each interaction and assumes that the c o l l i s i o n s are random, g^(t) may be written g.(t) = exp ill ( 5 . 9 ) Theoretically, 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 interpreted as a reasonable attempt to explain the data. This i s discussed by Dong and Bloom ( 1 9 7 0 ) . I f v/e perform the integration i n equation (5*7)> using equation ( 5 . 9 ) , equation ( 5 . 8 ) becomes 1 Y~' r k Gu(0) ~ r, ; w^? (5-10) I t i s interesting to look at some special cases of equation ( 5 . 1 0 ) . I f centrifugal distortion i s negligible but we - 52 -must associate different correlation times with the C a and terms, we have 1 4jr2r2 Ti  T, " OC U a 1 + (Uc-Uj)2^2 4 ATT2 2 rn  + 45 Of d 1 + {Uo-ujHT^)2 C5-'° The constants i n equation (5 .11) are taken from Dong and Bloom (1970) and the choice of notation for the T's i s i n keeping v/ith the literature. The parameter QC i s given by . 2 I 0 kT where I 0 i s the moment of i n e r t i a for the spherically symmetric molecule. I f the same correlation time may be associated with both terms, equation (5 .11) reduces to Ti 4 7 T V 2 r . 2^ 2 1 + {Uo-UjfTi (5 .12) where 2 rs 2 _4_^ 2 45 Ceff = CQ + — 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 a n i s o t r o p i c 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 raay have non zero u/^'s and we can w r i t e 1 - 47T2 2 T] 2^2 C5.13) _4_ 47r2.. 9 V " T k  + 45 oc 1 + [ ( u 0 - u j ) + 4 j 2 r k 2 0 i" where F k i s a normalized f u n c t i o n . and the sum over k i n c l u d e s k k A s u p e r f i c i a l examination of the experimental r e s u l t s given i n f i g u r e 5.3 i n d i c a t e s that 1/T-j i s of the form given by equation ( 5 . 12 ) . This i m p l i e s that the r e l a x a t i o n r a t e 1/Tj i s desc r i b e d by; A) equation ( 5 .12 ) , or B) equation (5.11) w i t h n o t very d i f f e r e n t from 7*i > o r C) equation ( 5 . 1 3)vath one term dominant. I f the r e l a x a t i o n r a t e were given by, say, two e q u a l l y dominant, but very d i f f e r e n t terms, we would expect to - 54 -see another maximum or at l e a s t a bump where the 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 jk(L/) a s s o c i a t e d v/ith the c e n t r i f u g a l d i s t o r t i o n are centred around LJ ^ s which are ••far away" from LJQ - LJ j v/e can c e r t a i n l y imagine t h e i r 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 . T h i s 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 d i s t o r t i o n frequency i s shown f o r s i m p l i c i t y . The dominant C a term i s de s c r i b e d by j^ (Cv/) and the 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 evident i n equation (5.12) or a dominant term i n equations (5.11) and (5.13)> the f a s t e s t r e l a x a t i o n r a t e occurs when T^ ~1 LJQ - LJ j T h i s i s the c h a r a c t e r i s t i c 1/T-j maximum or Tj minimum and mani f e s t s i t s e l f i n the experimental r e s u l t s i n f i g u r e 5.3. The r e l a x a t i o n r a t e then decreases monotomically as; A) decreases i n the r e g i o n T]"^ > LJQ - LJj and, B) 7*i i n c r e a s e s i n the r e g i o n Tf] < U0 - Uj. Case A i s considered the h i g h d e n s i t y r e g i o n and case B the low d e n s i t y r e g i o n . The maximum r e l a x a t i o n along v/ith the two extreme cases are 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 the case o f the j^L/) term completely dominant. E x p e r i m e n t a l l y , one measures the d e n s i t y which i s r e l a t e d to the c o r r e l a t i o n times T k by 1 -=r - D<CTkV> <5.i« where 0" k 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 the 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 the speed o f a molecule. The Canonical Ensemble average < ( 5 ' j c v > depends o n l y on the mean f igure 5.1 r e l a xa t i on e f f ec t s of cent r i f uga l d i s tor t ion - 56 -f i gu re 5.2 co l l i s i on induced f i e l d s r c u0  » 1 low p u v.- uT u r c uQ«1 high p u - 57 -velocity which, i n turn, i s a function of the temperature of the l a t t i c e . I f temperature i s held constant, we can write Ai r, = — k (5.i5) <k p where we allow different constants, Alc, for the different cross sections associated v/ith each interaction. For purposes of analysing the experimental data i t i s convenient to express the relaxation rate as a function of density through equation (5.15) and furthermore noting that the density at which the expression 1 • [ ( u 0 - 4 ) * Mj 2r k 2 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 definition of jO^. V/e have then - 58 -r - _ 2 _ _ (5.16) k [(u0-Uj) + P ' and by s u b s t i t u t i n g equation (5.16) i n t o the three p o s s i b l e cases f o r 1/Tj given by equations ( 5 . 1 1 ) , (5.12) and (5-13) we o b t a i n the f o l l o w i n g formulae. Case 1. 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 the 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 the Ca and C^ terms, the r e l a x a t i o n r a t e i s given by j _ 4 n2 C f f P T, " oau-uj) p 2 C 5 - , 7 > 1 + P 2 where p, i s given by equation (5.16) w i t h U. = 0, T l = ( I ^ u l p ( 5 - ' 8 ) Case 2 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 d i f f e r e n t c o r r e l a t i o n times are r e q u i r e d f o r the Ca and C^ terms, the r e l a x a t i o n r a t e i s given by - 59 -B ]_ _ 47T2CQ2 . P T, = Ot(U0-Uj) p 2 I 1 + — I p 2 p x 1J2 4 4 7T2CD2 P ,R Q, + " (5 . 19) 45 a(u0-uj) + { p { ? ) 2 P2 where P1 i s given by equation (5«18) and P12 i s given by equation (5 . 16) with L/12 = 0 ' P;2 1 1 2 (Uo-Uj) p Case 3 I f relaxation due to centrifugal distortion i s small but not negligible ( i . e . measurable), the relaxation rate i s given by - 60 -P 1 4 7T2CQ2 _^  P 1 P2 ( 5 . 2 1 ) + 167T2 C d 2 V ^ Fk P 1 * P 2 with yO^ . given by equation (5-16). 5 . 3 DATA ANALYSIS The experimental data shown in figure 5 . 3 i s the relaxation rate l/T 1 as a function of density jO. The simplest f i t of the data i s to assume that the relaxation rate i s given by equation ( 5 . 1 7 ) . Equation ( 5 . 17 ) can be subjected to a least squares f i t using - r - 0 —o + D ( 5 . 2 a ) P o if) - 62 -where a Ot(u0-Uj) py 4 7T 2 C e f f 2 CX(L/0-Uj ) 47T 2C 8§P 1 or P 2 a ( u 0-L/,) 1 ^ef f = " 47T 2 -vab A determination of a and b w i l l give values for Cefj» and jO^. The most f r u i t f u l approach i s to f i t over a c e r t a i n region and note the f i t for the r e s t of the curve. Several p l o t s were performed i n t h i s manner and the agreement i n the regions outside the f i t was very poor i n each case. Figure 5*4 i s an example of such a f i t . The errors for the t h e o r e t i c a l curves such as i n figure 5«4 are very small because of the fact that the region f i t contains many points. I f C e f f 2 and p i are both considered unknown, one can not f i t the l i n e a r high density region because i n terms of equation ( 5 . 2 2 ) , b » a/p 2 and the l e a s t squares f i t does not contain s u f f i c i e n t information. However, because - 6k -C = 137.60 kHz 2 i s accurately known, we can use the fact G i l portrayed i n figure 5.3 that for D > 1 amagats, — i s constant. ft P, 2 In terms of equation (5.17), this'implies that — ^ << 1 with the ft T1 result that — i s given by b i n equation ( 5 . 2 2 ) . The experimental P result that II = 21.9 1 0.4 m s e c t ( 5 . 2 3 ) p amagats uniquely determines f)^ with the result that equation (5.17) can now be plotted. This i s done i n figure 5 - 6 . The theoretical value of jOj agrees very well with the experimental value, but the theoretical curve gives far too strong relaxation i n the region of the maximum. This high density f i t of the data using equation (5 .17) provides the logical conclusion that whatever interactions contribute to the relaxation at higher densities there are some effects which do not contribute as much i n the region of the maximum. Having eliminated equation (5 .17) as a reasonable f i t of the data (but none the less having gained considerable insight into the problem) the next step i s to allow two correlation times. Using the accepted values of C a = 10.4 kHz and = 18.2 kHz, equation (5 . 19) contains two unknowns; fc)-\ and ft{2.' This i s the same as saying that the two correlation times are unknown (given by equations (5 . 18) and ( 5 . 2 0 ) ). Rather than randomly picking f i g u r e 5.5 o a en in o E E o 3 4 r 30 26 22 1 I 1 P - 1 vs 1 p ( a m a g a t s ) i _ J _ l 10 1 J I I • .01 .1 0 ( amaaat s ) 10 - 67 -v a l u e s f o r these two parameters i t i s reasonable to f i t the high d e n s i t y r e g i o n under the assumption t h a t P » P]>P\2.  a n d i n v e s t i g a t e the consequences i n the area o f the maximum. For the r e g i o n p > 1 amagats, equation (5-19) reduces to 1 47T T, a{u0-Uj) p (5.2if) Using equations (5.23) and (5.24), equation (5.19) can be w r i t t e n w i t h o n l y one v a r i a b l e . That i s to say Pjg becomes a f u n c t i o n o f Py Equation (5.19) v/as then p l o t t e d f o r a range o f values f o r P 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 to f i t the data adequately, f o r i n a l l cases there i s too st r o n g r e l a x a t i o n i n the r e g i o n o f the maximum. We do not give an example of such a f i t because a t y p i c a l f i t looked something l i k e f i g u r e 5.6. I f we had decided to f o r c e a f i t i n the 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 too weak r e l a x a t i o n i n the r e g i o n o f P > 1 amagats. V/e are l e f t w i t h the task o f i n t e r p r e t i n g the data i n terms o f a measurable 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 by c e n t r i f u g a l d i s t o r t i o n e f f e c t s . V/e now want to analyse the data i n terms o f equation (5.13) whose independent v a r i a b l e s are the T^, or the i d e n t i c a l equation (5.21) whose independent v a r i a b l e i s 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 the and F k but th a t i t i s , perhaps, a lengthy and time consuming procedure. C e r t a i n l y , the experimental data should be re-analysed when t h i s i s done, but we can a r r i v e at reasonable q u a l i t a t i v e r e s u l t s by approximating equation (5«13) - 68 -(or 5.21). F i r s t we assume that the spectral densities j^(L/) associated with each peak frequency U/^ add i n such a way at LA - UT that we can approximate them by one spectral density o <j which we denote by j( 2(L/) peaked around U^. That i s to say, i n terms of equation (5 .13) (or 5.21), Fk = 1 for = L/^ and F^ = 0 for a l l other Also, we assume that the angular frequency i s s u f f i c i e n t l y high that ( U ^ ) 2 ^ [ ( ^ o - ^ j ) + L/^] 2 . Finally, we assume that both the C a term and the now single term may be analysed i n terms of a single correlation time 7"i • With these approximations and assumptions, equation (5.13) becomes 1 m 4 7T2CQ2 r, I6 7 f 2 c d 2 r} + •• 7 — 0 C5.25) 45 0C 1 + (U{2)2T}2 v/ith the variable parameters T-j and 2 and equation (5 .21) becomes - 69 -T, 4 n2Cg P P 2 16 7T2CD2 45a u, 2 P ' H2 1 + [Phi P2 ( 5 . 2 6 ) with, the variable parameters P^, P{^ a n d ^ 1 2 a n (^ ^ e r e s t r i c t i o n 1 ' f\2 given by = Tj and equation (5 .16) A -12 U12 ( 5 . 2 7 ) V/e now u t i l i z e another restriction; namely the slope of the high density region. For p » P^, P^2» w n i - c h i s assumed to be satisfied for p > 1 amagatj equation (5-26) becomes J ^ _D 16 7T2CD2 , p ( 5 . 2 8 ) and using equation ( 5 . 2 7 ) - 70 -_1_ 47T Co 2 + 45 p 47T2 Ceff 1 a (U0-Uj ) P Using the numerical value given i n equation ( 5 . 2 3 ) , we have the following result P, = 0.039 amagats P,'2 = (2.2 X 1 0 "1 0 sec amagats) L/^ 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 " T ( 5 . 2 9 ) Cd and L T t a . i s plotted i n figure 5 . 7 . As expected, the C a contribution to the relaxation 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 theoretical curve for a single term f i t shown i n figure 5 .6 gives too strong relaxation p i n the region of the maximum, namely the C d contribution to CQff 1 • ' J o in .01 .01 f i g u r e 5.7 .1 p (amagat s ) j v s p i 10 - 72 -i s not present. Having accounted for the C a contribution to 1/T^ we now plot T J exp. T J vs ( 5 . 3 0 ) and attempt to interpret i t i n terms of 16 7T2CD2 45Q!Uf2 A : 12 (Pi 2> (5.31) P 1 2 = ( 2 . 2 X 1 0 " 1 0 sec amagats)U^ ( 5 . 3 2 ) The plot given by equation (5.30) i s given i n figure 5 . 8 . The fact that the C„ term i s dominant accounts for the large apparent a 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 results are reliable 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 results momentarily, v/e can analyse figure 5 .8 i n terras of equation ( 5«3D and ( 5 . 3 2 ) . Instead of randomly varying a n d ^\2. ^ ne r e s t r i c t i o n given by equation ( 5 . 3 2 ) , 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 find v/here this might be, the .08 dens i ty region f i gu re 5.8 vs .04 0 - .04 L .01 .1 ( amagat s ) 10 - 74 -l o c u s of maxima given by 1_ LT J c d l max 1 J6HfCdL (,.33) 2 45a u{2 and equation (5.32) i s p l o t t e d i n f i g u r e 5.9 as a f u n c t i o n of / . / Vi n l T A ,-\w-.-i 4" 4" .-> ,~! 4" It *1 A m ,-1 AT, 4- -t - v i n r m ' , i-» ,-%->-> . - v 4 " O -.v. r-. ~ n n -Ptr 1 2 * 1 8 ° n e t v/ U O u i a . v v^. KJ. v / i i ^ ^ _ w vii v-i. b i i o j . v j x o £,0. n J- n u o i o-U iu&4.£j.i.i.x. x j the r e g i o n of i n t e r e s t . S u r p r i s i n g l y enough, t h i s l o c i of maxima do pass through p o s s i b l e candidate p o i n t s f o r the experimental 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 are i m p o s s i b l e because of the s p r e a d ' i n p o i n t s , but the allov/ed range of LJ-j^ c a n be narrowed by p l o t t i n g equation (5.31) f o r a few reasonable values o f LJ^' This i s done i n f i g u r e 5.10. Noting t h a t the top and bottom curves i n f i g u r e 5.10 can probably be r u l e d out, i t seems reasonably safe to say thafc 5(LJ0 - LJj) •< LJ^p < &(UQ - LJj) I f the low d e n s i t y e f f e c t i s r e a l , we can not say anything about i t . The tremendous spread i n p o i n t s along w i t h the f a c t t h a t a maximum has not been reached does not permit any way of p u t t i n g l i m i t s on another L/k f o r t h i s e f f e c t . I t should be noted, however, t h a t i f the e f f e c t i s r e a l , the l i m i t s set 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 than 1. I t would not be s u r p r i s i n g , then, i f the 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 should be s l i g h t l y lowered. .08 r o CD CO E .04 Q_ X CD 0 - 0 4 (1) v j ' e x p f i g u r e 5.9 vs p > .04 a m a g a t 25 B / — Z.J uB = K>- u T J l_I ; c d i m a x J i I I vs ( p ) m a x .1 P ( a m a g a t s ) VJ1 .0 .08 o CD if) E .04 o X 0) 0 - . 04 3,2 fl f i g u r e 5.10 (1) - ( 1 ) p > .04 a m a g a t vs p 9.2 B I I L l i l l .1 J L _ l 10 p ( a m a g a t s ) 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 of d e n s i t y at constant temperature. The experimental r e s u l t s could not be i n t e r p r e t e d i n terms of the si m p l e s t t h e o r e t i c a l framework which suggests t h a t the 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 f u n c t i o n of ]/p where p i s the d e n s i t y . The " f i n e s t r u c t u r e " which upsets the simple t h e o r e t i c a l model has been a t t r i b u t e d to a 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 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 t h i s c o n c l u s i o n s e v e r a l assumptions have been made. The important r e s u l t i s that a more r i g o r o u s approach would probably a f f e c t the r e s u l t s i n a q u a n t i t a t i v e and not q u a l i t a t i v e manner. The q u a l i t a t i v e c o n c l u s i o n s of t h i s t h e s i s should be checked w i t h other t e t r a h e d r a l molecules and t h i s i s p r e s e n t l y being done w i t h S i H ^ ( s i l a n e ) . A study of the r e l a x a t i o n as a f u n c t i o n of temperature would a l s o be a u s e f u l experiment. A r i g o r o u s t h e o r e t i c a l treatment should i n v o l v e a study of 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 the i n t r a m o l e c u l a r d i p o l a r 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 out of the s o l e n o i d . Spin symmetry e f f e c t s , which have been completely neglected i n the - 78 -conventional theory have been introduced i n this thesis as the high frequency LJ^ (k £ 0) terms. Whereas a proper treatment of the intramolecular dipolar interaction and spin diffusion would probably not affect the conclusion concerning relaxation due to centrifugal distortion, spin symmetry effects are central to this conclusion and an expression for 1/T1 should be developed which includes the higher order distortion frequencies i n the rotational states. In principle, one could calculate the spectral densities for a l l the distortion frequencies and as a result their contribution to the relaxation could be determined ( i . e . their contribution at the Larmor frequency). The rigorous results would be similar to equation (5 .13) i n form with the F^'s and U^s calculated exactly. In the analysis presented here we have assumed only one distortion frequency simply because of lack of information. A re-analysis of the experimental data on completion of these theoretical suggestions would certainly 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 similar effects i n other spherically symmetric gaseous molecules and to engage i n a theoretical programme of some rigour i n order to pinpoint the origin of the experimentally observed effect which we have attributed to centrifugal distortion of the molecule. APPENDIX CIRCUIT DIAGRAMS Some of the original components have been changed i n a t r i v i a l 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 ci r c u i t diagrams may be found i n their original form i n previously published theses, the following diagrams represent the spectrometer i n i t s entirety. Only the commercial equipment has been omitted. The upper case Arabic let t e r s refer to the description of the apparatus i n Chapter IV. y 1 2 A U 7 5670 150 V f r o m squa re pu l se gene ra to r C 100 pf - I f — 12K .01 I — ' W V -1M 4 7 0 K A K 1N307 .01 1N195 f r om p u l s e gene ra to r A CO o to p u l s e w i d th generator F 12 K to pu l se w i d t h generator G x f i g u r e A1 d i f f e r e n t i a t o r D and m i x e r E 12AU7 12AU7 5725 - 170 V f i gu re A 2 pulse w i d th g e n e r a t o r F (o r G ) 5670 f r om pu l s e .047 w i d t h g e n e r a t o r F 680K + 150 V f r o m pu l se w i d t h g e n e r a t o r G 047 i n -to ga t i ng pu l s e s a m p l i f i e r I Co ro f igure A 3 m i x e r H 5670 + 150V • -AAA/ 1 1 4K 5687 18K i—'\AAA-A<W ^7 ZD/ 200 680K -AAr IT .05 1N307 FROM M I X E R H 47K -AAAA/ 1 33K .AAAAr—•»+225 V o TO GATED OSCILLATOR L -170 V TO GATED POWER AMPLIFIER 0 CO f igure A 4 ga t i n g pu l se s a m p l i f i e r I OUTPUT TO ° W I D E B A N D A M P L I F I E R K CO l C R Y S T A L (10 m h z ) f igure A 5 10 mhz c r y s t a l o s c i l l a t o r J o—1| w / v .01 1.5K f rom c r y s t a l o s c i l l a t o r J -L 10^h .005 -o-6V Co OUTPUT to gated o s c i l l a t o r L and t r i p l e r P o+6V f igure A6 w ide band a m p l i f i e r K f r om w ideba nd a m p l i f i e r K 225 V to phase s h i f t e r M from gat ing p u l s e s a m p l i f i e r I Co f igure A 7 coherent gated o s c i l l a t o r L 170V Q 120 o WV-FROM GATED OSCILLATOR L +150V+225V 9 O .01 .01 __) A 5687 180 >180 2 5763 TRIPLES N DELAY LINE: BEL FUSE vs-250 PHASE SHIFTER M f i g u re A 8 f r o m ga t i ng p u l s e s a m p l i f i e r I 9 0 V G A T E I N 9 f rom t r i p l e r N 22 K 22K 6 S N 7 10K •150 .01 I • * + 7 5 0 V 8 2 9 /TTTTN 1 0 I ^ W v V 10 :56K / R i O r.f. "° OUTPUT -<*1200V r i Co Co 3.15 V - 3 0 0 V f i gu re A 9 gated power a m p l i f i e r 0 .01 f rom wide band a m p l i f i e r K 0.5 uH 4-30 2N3323 •22K H l -.01 5.1K< H H .01 t ± 1 0 PF A-30 2N3323 h .01 5.1 K .01 to r e f e r e n c e 0 a m p l i f i e r 0 5uH + 6 V - 6 V reference f i gu re A10 t r i p l e r P 2 6 D J 8 from tr ipler P f i g u r e A11 re fe rence amp l i f i e r Q 8-10 H I -100uH I | 10UH A V FD111 -± .01 f rom crossed d i odes - Ih .22 7722 3uH .01 — T — 10K f i gu re A12 p r e g m p l i f i e r R 10K 150 V .01 ^•6DJ8 .01 1 K -o to t uned a m p l i f i e r f rom osc i l loscope T 4 to chart recorder Z 220 K 7266 / f rom pul se • / generator AAA—O 1M 330K " lOOK° —AAA—o . 3 3 K ~ ^ j J 1 ± 1 ± 1 I T T T "= .1 .01 .001 50K 0-10K 50K 150K 2W 150K 2W 12AU7 f i gure A13 boxcar integrator U 8 1 H A M M O N D 1 1 6 6 T H A M M O N D 1 9 6 C ~2K(20Yv7 1.5KV H A M M O N D 7 7 6 f igure A14 h igh vo l tage power supply for a m p l i f i e r 0 8 Q D 3 ' s 6 © © •O+1200V •*»*750V 4 1N540 ' s hammond t rans former 166L6 f i gure A15 power supply •A/WVr 1.8 K 1.8 K + 20V + 6V MZ 1000-29 - O K r 1N753 1N968 T MZ 1000-9 100uF - 6V J , K and P - 95 -BIBLIOGRAPHY Abragam, A. 1961 The principles of nuclear magnetism (Oxford Univ. Press, London). Anderson, C.H. and Ramsey, N.F. 1966 Phys. Rev. 149, 14-Andrew, E.R. 1955 Nuclear magnetic resonance (Cambridge Univ. Press, London). Blicharski, J.S. 1963 Acta Phys. Polon. 24, 817. Bloch, F. 1946 Phys. Rev. 70, 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. Sci. 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_, 1155-L 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 Slichter, CP. 1963 Principles of magnetic resonance (Harper and Row, New York). Wofsy, S.C., Kuenter, J.S. and Klemperer, W. 1970 J. Chem Phys. 53_, 4005. Yi, P. 1967 Ph.D. Thesis, Harvard Univ. (unpublished). Yi, P., Ozier, I. and Anderson, C.H. 1968 Phys. Rev. 165, 9 2 . - 96 -Y i , P., Ozier, I . , Khosiz, A. Phys. Soc. J_2, 509. Y i , P., Ozier, I . and Ramsey, and Ramsey, N.F. 196? B u l l . Am. N.F. (to be published) 

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