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Proton magnetic resonance in methane and its deuterated modifications Sandhu, Harbhajan Singh 1964

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PROTON MAGNETIC RESONANCE IN METHANE AND ITS DEUTERATED MODIFICATIONS by HARBHAJAN SINGH SANDHU B . A . , Panjab U n i v e r s i t y , India , 1951 B.Sc . (Hons.) , Panjab U n i v e r s i t y , India , 1952 M . S c . , Panjab U n i v e r s i t y , India , 1953 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept th i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1964 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives,. I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of PhySJCS  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date J a n u a r y 2h? 196^-. The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY HARBHAJAN SINGH SANDHU B.A., Panjab University, India, 1951 ,Sc.(Hons.). Panjab U n i v e r s i t y , India, 195 M.Sc, Panjab University, India, 1953 FRIDAY9 SEPTEMBER 27, 1963.AT 9s00 A.M. IN ROOM 303, PHYSICS BUILDING of COMMITTEE IN CHARGE Chairman%• F.H. Soward M. Bloom J.W. Bichard L.G. de Sobrino L.W. Reeves R.F. Snider D.LI. Williams External Examiner: B.H. Muller U n i v e r s i t y of .'Wyoming Laramie PROTON MAGNETIC RESONANCE IN METHANE AND ITS DEUTERATED MODIFICATIONS ABSTRACT Proton magnetic resonance has been studied i n methane and i t s deuterated modifications. Measure-ments of relaxation time were c a r r i e d out at a f r e -quency of 30 Mc/sec. using 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 has been measured i n l i q u i d and s o l i d samples of CH4, CH3D, CH2D2 and CHD3 between 110°K to 56 CK. The simplest possible i n t e r p r e t a t i o n of our results i n both the l i q u i d s and s o l i d s i s that only one mechanism, that asso-ciated with the inter-molecular i n t e r a c t i o n s , i s probably predominant in causing re l a x a t i o n . The spin-rotational, and intra-molecular dipolar i n t e r -actions do not seem to contribute appreciably to relaxation. The e f f e c t of dissolved paramagnetic ions has also been studied i n samples of CH4 between 110°K - 78°K and we have developed a very simple and r e l i a b l e technique f or obtaining oxygen-free samples i n order to study T]_ i n pure samples because T^ has been found by us to be very s e n s i t i v e to small, amounts of oxygen. The re s u l t s v e r i f y the invers p r o p o r t i o n a l i t y of T i to the oxygen concentration. The spin-spin relaxation time T2 has been measured i n l i q u i d and s o l i d samples between 110°K and 56°K. The r e s u l t s show that the l i n e shape i s predominantly due to inter-molecular i n t e r a c t i o n s . Furthermore the resonance l i n e has a gaussian shape at temperatures below 65°K and changes to a Lorentzlan shape at higher temperatures. GRADUATE STUDIES F i e l d of Study; Physics Electromagnetic Theory Nuclear Physics Quantum Mechanics D i e l e c t r i c s and Magnetism Related Studiess Theory and a p p l i c a t i o n of D i f f e r e n t i a l Equations Applied E l e c t r o n i c s E l e c t r o n i c Instrumentation R, Barrie J.B. Warren F.A. Kaempffer G. Bate C.A. Swanson M.P. Beddoes F.K. Bowers PUBLICATIONS H.S, Sandhu and M. Bloom. Nuclear spin relaxation i n l i q u i d Methane. Bull.Am.Phys.Soc. 3, 324, 1958. H.S. Sandhu, J . Lees and M. Bloom. Removal of oxygen from methane and the use of nuclear magnetic resonance to measure oxygen concentration. Can.Jour.Chem. 38, 493, 1960. M. Bloom and H.S. Sandhu. Proton s p i n - l a t t i c e relaxation i n pure methane and i t s deuterated modifications. Can.Jour.Phys. 40, 289, 1962. M. Bloom and H.S. Sandhu. N.M.R. l i n e shape studies in methane using pulse techniques. Can.Jour.Phys. 40, 292, 1962. ABSTRACT The work reported here i s a study of proton magnetic resonance i n methane and i t s deuterated modi f icat ions . There i s a strong isotope e f fec t on the proton sp in re laxa t ion which makes i t poss ible to d i s t i n g u i s h , at l east p a r t i a l l y , between the various proposed sp in in terac t ions leading to nuclear sp in r e l a x a t i o n . The measurements of nuclear sp in re laxa t ion times were c a r r i e d out at a frequency of 3G Mc/sec. using pulse techniques i n samples of CH , CH D, CH D and CHD . ~r O £ £ *J The s p i n - l a t t i c e re laxa t ion time 1\ has been measured i n o o l i q u i d and s o l i d samples between 110 K to 56 K. Our re su l t s show that T^ i s temperature dependent and t h i s temperature dependence i s independent of the number of deuterons i n the samples. We can make only p laus ib l e statements about the mechanisms that contr ibute to re laxa t ion because our r e s u l t s have not been able to provide conclusive evidence about the r e l a t i v e contr ibut ions of d i f f eren t mechanisms. The simplest poss ib le i n t e r p r e t a t i o n of our r e s u l t s i n both the l i q u i d s and s o l i d s i s that only one mechanism, that associated with the inter-molecular d ipo lar i n t e r a c t i o n s , i s probably predomi-nant i n causing r e l a x a t i o n . The s p i n - r o t a t i o n a l in terac t ions and intra-molecular in terac t ions do not seem to contribute appreciably to the r e l a x a t i o n . C e r t a i n l y they are not as important as predic ted by conventional theory which pred ic t s - i i i -that intra-molecular in terac t ions should be more important than the inter-molecular in terac t ions i n the l i q u i d s (90°K - 1 1 0 ° K ) and i n the so l ids below about 80°K. The e f fec t of d i sso lved paramagnetic ions has also been studied and we have developed a very simple and r e l i a b l e technique for obtaining oxygen-free samples i n order to study T^ i n pure samples because T has been found by us to be very sens i t ive to small amounts of oxygen i n the sample. T has also been measured i n l i q u i d and s o l i d samples of CH^ (110°K -o 78 K) conta ining 1.08% and 2.54% oxygen. The r e s u l t s v e r i f y the inverse p r o p o r t i o n a l i t y of T.^  to the concentrat ion of oxygen. The s p i n - s p i n re laxa t ion time has been measured i n o o l i q u i d and s o l i d samples between 110 K and 56 K. The re su l t s show that the l i n e shape i s predominantly due to inter-molecular i n t e r a c t i o n s . Furthermore, the resonance l i n e has a Gaussian o shape at temperatures ^ 65 K and changes to a Lorentz ian shape at higher temperatures. The a c t i v a t i o n energies from T 2 measurements are found to be 3.2 kcal/mole independent of the number of deuterons i n the samples. These r e s u l t s agree with the values of a c t i v a t i o n energies obtained from measurements i n the s o l i d s jus t o o below the melt ing point (80 K - 90 K) . Since the l ine -width i s due to inter-molecular i n t e r a c t i o n s , i t confirms that o o between 80 K and 90 K i s predominantly due to inter-molecular i n t e r a c t i o n s . M. Bloom - v i i i -ACKNOWLEDGEMENT I wish to express my s incerest grat i tude to Dr. M. Bloom for h i s continuous encouragement, patience and guidance throughout the progress of t h i s work. I am extremely indebted to Mr. J . Lees for his continuous assistance and advice i n the construct ion and operation of the experimental equipment. My thanks are also due to the members of the Physics Workshop for t h e i r help i n the construct ion of the experimental equipment. I wish to express my apprec iat ion to Dr. D. L . Wil l iams for c r i t i c a l l y reading the manuscript of th i s thes i s . The author i s grate fu l to the National Research Counci l for f i n a n c i a l assistance i n the form of grants to Dr. M. Bloom and the award of a National Research Counci l Studentship. - i v -TABLE OF CONTENTS Page ABSTRACT i i LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENT v i i i CHAPTER I INTRODUCTION 1 CHAPTER II EXPERIMENTAL APPARATUS AND MEASUREMENT OF RELAXATION 9 TIMES 2-1 Apparatus 9 2- 2 Measurement of 25 Relaxation Times CHAPTER III THEORY OF SPIN-LATTICE 32 RELAXATION 3- 1 Methane (CHj) 32 3- 2 Theory 33 CHAPTER IV SPIN-LATTICE RELAXATION 45 MEASUREMENTS 4- 1 Est imation of Accuracy 46 4-2 Results 46 4-3 In L iqu ids (90°K - 110°K) 46 4-4 In Sol ids (90°K - 56°K) 54 4-5 Ef fec t of Paramagnetic 62 Impurities on T-^  4-6 Est imation of O2 Concentration i n a Pure 67 C H 4 Sample CHAPTER V N.M.R. LINE-SHAPE STUDIES 5-1 Measurement of Tg 5-2 Results and Discuss ion CHAPTER VI SUMMARY AND CONCLUSIONS BIBLIOGRAPHY LIST OF ILLUSTRATIONS Figure 1. Proton energy l e v e l s i n a magnetic f i e l d 2. Block diagram of n .m.r . spectrometer 3. 3 Pulse time base generator 4. 30 Mc/sec. gated pulsed o s c i l l a t o r 5. The tuned c i r c u i t 6. Low temperature system 7. Bomb 8. Schematic diagram of the sample container 9. Osc i l loscope d i sp lay - two pulse n .m.r . experiment 10. A C ^ 0 ) - A(t) versus time showing uncerta inty i n a T-^  measurement 11. (1/T-L) versus 1 0 0 0 / T ° K i n l i q u i d and s o l i d samples of C H 4 , CR^D, C H 2 D 2 and CHT>3 between 56°K and 110°K 12. (1/T., ) / (1/T..) versus no. of deuterons l n l o i n l i q u i d samples 13. (1/T-. ) / (1/T..) versus no. of deuterons I n 1 o i n s o l i d samples 14. (1 /T , ) versus 1 0 0 0 / T ° K for C H . , CH_D, J. x 4 o CH„D„ and CHD_ below 80°K - v i i -Figure 1 5 ' ^ T l > x ( n ) / ( 1 / T l > * ( o ) V e r s u s no. of deuterons 16. versus temperature for CH^ samples containing known amounts of oxygen 17. 1/T^ versus oxygen concentrat ion i n CH\ at 108°K 4 18. Corrected amplitude of the induct ion 2 s igna l versus time and (time) for C H 4 at 7 8 . 1 5 ° K 19. Corrected amplitude of the induct ion 2 s igna l versus time and (time) for 20, 22 C H 4 at 6 1 . 4 ° K T 2 versus 1000/T K for s o l i d CE^, CH^D C H 2 D 2 and CHT>3 between 56°K and 90°K 21. Diagram i l l u s t r a t i n g the motion of the vector r between two nuclear spins ( uyp)n / ( u r p ) 0 versus no. of deuterons i n r i g i d l a t t i c e CHAPTER I INTRODUCTION The nuclear magnetic resonance technique makes poss ible the study of the in terac t ions between a nuclear spin system and i t s magnetic environment. The mechanism of the exchange of energy between a system of nuclear spins immersed i n a strong magnetic f i e l d , and the heat re servo i r cons i s t ing of the other degrees of freedom (the l a t t i c e ) of the substance containing the magnetic n u c l e i , i s of considerable in teres t s ince i t provides valuable information about phys ica l processes which are going on i n the system. The system of nuclear spins re fers to an assembly of a l l the nuc le i of the same type contained i n a bulk sample. The sample may be i n any of the three states of matter; gas, l i q u i d or s o l i d . Our work i s the study of proton magnetic resonance i n samples of C H 4 , CHgD, CHgDg and CHD 3 i n the l i q u i d and s o l i d o o phases from 110 K to 56 K. There i s a strong isotope ef fect on the proton sp in r e l a x a t i o n . This e f fect enables us to d i s t i n g u i s h , at l eas t p a r t i a l l y , between the various proposed spin in terac t ions leading to nuclear sp in r e l a x a t i o n . We w i l l , f i r s t of a l l , discuss b r i e f l y the n .m.r . technique e s sent ia l for our d i scuss ion . Consider an assembly of i d e n t i c a l atomic nuc le i i n thermal equ i l ibr ium i n the presence of a steady magnetic f i e l d H Q i n the z d i r e c t i o n ( H q = H z ) . Since our work i s l i m i t e d to proton resonance, the nuclear spin I = \ . Each -2 -nucleus has two poss ib le energy l eve l s a and b, corresponding to the proton spin po int ing i n the upward and downward d i rec t ions respec t ive ly as shown i n F i g . 1. H o I-T-E = W Q = 2 A : H 0 F i g . 1 The d i f ference i n energy between the two l eve l s i s equal to 2yWj H Q w h e r e i s the magnetic moment of the protons. The populat ion of the lower l e v e l exceeds that of the upper l e v e l by the Boltzmann factor exp(2vUj.HQ/kT) where k i s Boltzmann's constant and T i s the e q u i l i b r i u m temperature of the sample. This excess of populat ion i n the lower l e v e l corresponds to a net macroscopic magnetization M oriented p a r a l l e l to H Q . The system i s then subjected to an r f magnetic f i e l d at the resonance cond i t ion uy- urQ = 7 H o > where 7 i s the gyromagnetic r a t i o of the protons and uj^ i s the Larmor frequency. Trans i t ions are induced between the Zeeman l eve l s which corresponds to the transfer of some of the excess populat ion from the l e v e l a to b. This causes a change i n the z component of macroscopic nuclear magnetization which, i n general , acquires a non-e q u i l i b r i u m o r i e n t a t i o n . After the removal of the r f magnetic f i e l d , M precesses f r e e l y with components M , M and M . The x y z nuclear magnetization vector M then recovers towards i t s e q u i l i -brium value.* M - 0, M = 0 and M„ = M . The recovery of the x y z o nuclear magnetization vector M towards equ i l ibr ium i s af fected by (1) in terac t ions of the sp in system with the l a t t i c e , which -3-involve an exchange of energy from the spin system to the l a t t i c e ; (2) interactions of the spins among themselves which do not involve any change i n the t o t a l energy of the system. Since the dominant part of the t o t a l spin energy i s caused by the strong f i e l d H Q i n the z d i r e c t i o n , major changes of the t o t a l energy are due to a change of the z component of nuclear magnetization vector M. If Mz be the instantaneous value of the z component of M, then i n simple systems M w i l l tend to the z equilibrium value M at a rate determined by the degree of o thermal motion according to the equation dM M„ - IW _ £ = 5 ° ( i . i ) dt T x The s o l u t i o n of equation (1.1) i s M z = MQ x [l - exp(- t/T 1)J (1.2) taking the i n i t i a l value of M_ . 0. T i s c a l l e d the z 1 " s p i n - l a t t i c e " or " l o n g i t u d i n a l " relaxation time. The interactions which do not change the t o t a l energy of the spin system a f f e c t only the transverse components and My, and the approach to equilibrium of these components may be represented by the following equations for a spin system which obeys the Bloch equations + — = o dt T2 (1.3) dM M _ y + -JL = o dt T 0 -4-where Tg i s termed the "spin-spin" or "transverse" relaxation time. The solutions of equation (1.3) can be written as 1L = ML(O) exp ( - t/T 9) X 2 (1 .4) My = M y(0) exp ( - t/T 2) Thus ^ and My w i l l tend to a t t a i n zero values at a rate deter-mined by Tg. Therefore, the study of recovery of Mx, M^  and Mz by n.m.r. technique provides a measure of T and T and hence provides a means of studying the mechanisms that enable the spin system to relax back into thermal equilibrium with the l a t t i c e . We now consider the possible mechanisms which cause relaxation. We have seen that the spin system i s disturbed from thermal equilibrium by the application of an external r f magnetic f i e l d . In order to relax there must be some in t e r a c t i o n mechan-ism between the spins and the l a t t i c e by virtue of which energy can be continually transferred to the l a t t i c e from the spin system, leading to equilibrium conditions i n a time of the order of T-^  a f t e r the a p p lication of the r f magnetic f i e l d . In most diamagnetic substances, the dipole-dipole i n t e r a c t i o n among the nuclear magnetic moments seems to be dominant. We neglect e l e c t r i c quadrupole interactions because they are important for spins I > 1 and our work i s confined to the proton resonance (I s | ) . A simple picture i s to consider each spin as "seeing" a f l u c t u a t i n g l o c a l magnetic f i e l d produced by a neighbour which induces t r a n s i t i o n s among i t s l e v e l s tending to restore the system to i t s equilibrium value. These fluctuations i n the l o c a l magnetic f i e l d s are brought about by molecular motions. These motions may be r o t a t i o n a l , t r a n s l a t i o n a l or - 5 -of some other form. The three basic contributions to the l o c a l f i e l d due to molecular motions are (i) intra-molecular dipolar interactions which involve interactions between nuclear spins on the same moleculej ( i i ) inter-molecular dipolar interactions which involve interactions between nuclei on d i f f e r e n t molecules or between nuclear spins and paramagnetic ions; ( i i i ) s p in-rotational interactions between the nuclear spin and r o t a t i o n a l angular momentum of the molecule. There has been some evidence that spin-rotational i n t e r -(2) actions are predominant i n systems such as l i q u i d CHFg and (3) Johnson and Waugh have suggested that these interactions may also be important for l i q u i d CH^. Our study of T-^  has not been able to provide conclusive evidence about the r e l a t i v e c o n t r i -butions of ( i ) , ( i i ) and ( i i i ) to the relaxation rate 1/T-^ . We have only been able to give the upper and lower l i m i t s for each of them and therfore, we can make only plausible state-ments about the mechanisms that contribute to relaxation. The simplest possible i n t e r p r e t a t i o n of our re s u l t s i n both the l i q u i d s and s o l i d s i s that only one mechanism, that assoc-iated with the inter-molecular dipolar interactions, i s probably predominant i n causing relaxation. The spin-r o t a t i o n a l interactions and intra-molecular interactions do not seem to contribute appreciably to the relaxation. Certainly, they are not as important as predicted by conventional theory which predicts that intra-molecular interactions should be - 6 -more important than inter-molecular interactions i n the l i q u i d s (90°K - 110°K), and i n the s o l i d s below about 80°K. We w i l l now consider b r i e f l y the e f f e c t of dissolved paramagnetic ions on T^. The magnetic moment of the paramag-3 netic ion i s about 10 times larger than the nuclear magnetic moment. This r e s u l t s i n larger f l u c t u a t i n g l o c a l f i e l d s and considerably shorter T^, which however depends upon the concen-t r a t i o n of paramagnetic ions i n the sample. Previous measure-(4) ments of i n CH^ gave values much shorter than these obtained here and were due to dissolved oxygen which i s para-magnetic. A c r u c i a l step i n our experiments was the estab-lishment of a simple and r e l i a b l e technique for obtaining oxygen-free samples. As a by-product of measurements, we have been able to v e r i f y the dependence of T-^  on the concentration of oxygen and also to estimate an upper l i m i t for the oxygen concentration i n our pure sample of CH^. The measurement of T 2 can provide valuable information i n the study of the n.m.r. l i n e shape since T 2 i s a measure of the spread i n Larmor frequencies due to the l o c a l magnetic f i e l d s produced by the neighbouring nuclear magnets. A magnetic resonance l i n e of a spin system i n an inhomogeneous magnetic f i e l d has a c e r t a i n width owing to the spread of Larmor frequencies which i s due to the differences among the resonance frequencies of the ind i v i d u a l spins, rather than the int e r a c t i o n among them. We w i l l , however, consider the case when the line-width i s due to the existence of coupling between neighbouring spins. The i n t e r a c t i o n between two nuclear spins -7-depends upon the magnitude and orientation of th e i r magnetic moments and also on the length and orient a t i o n of the vector describing t h e i r r e l a t i v e positions. The e f f e c t of t h i s i n t e r a c t i o n depends strongly on whether the vector i s fix e d i n space or changes because of motion of the nuc l e i . At very low temperatures where very l i t t l e t r a n s l a t i o n a l motion of the molecules i s taking place, the nuclei may be considered as fix e d ( r i g i d l a t t i c e ) . At each instant the microscopic d i s t r i b u t i o n of l o c a l f i e l d s throughout the sample i s of a stationary character. This r i g i d l a t t i c e l i n e shape i s often c l o s e l y approximated by a Gaussian shape and the nuclear magnetic resonance absorption as a function of frequency i s usually independent of temperature. When rapid molecular motion sets i n , the l o c a l f i e l d produced at any resonant nucleus by i t s neighbouring dipoles i s now time-varying. If the v a r i a t i o n i s s u f f i c i e n t l y rapid, the time average of the l o c a l f i e l d over a l l permitted orientations of the dipole p a i r during a time Tg can, i n general, be expected to be les s than for a r i g i d system. This r e s u l t s i n a narrowing of the resonance l i n e and also the observable l i n e shape changes to a Lorentzian form. The second moment of the resonance l i n e i s unaffected by the narrowing "motion". The c a l c u l a t i o n of second moment from r i g i d l a t t i c e l i n e shape enables us to study the temperature dependence of 7* (the c o r r e l a t i o n time for changes i n l o c a l c f i e l d s due to t r a n s l a t i o n a l motions) i n the region where the l i n e shape i s Lorentzian, using the theory of motional -8 -"narrowing". The contribution to second moments comes from intra-molecular dipolar interactions and inter-molecular dipolar interactions. One can, therefore, study the mechanism that may be contributing to the observable l i n e shape. We f i n d that intra-molecular interactions do not contribute to the observa-ble second moments because our experimental values agree with the values of second moments predicted from inter-molecular interactions. We can, therefore, say that the l i n e shape i s predominantly due to inter-molecular interactions. We also observe that the resonance l i n e has Gaussian shape at tempera-o tures <65 K and changes to a Lorentzian shape at higher temperatures. Thus, we f i n d that the n.m.r. technique enables us to study the l i n e shape and to gain some insight into the d i f f e r e n t mechanisms that contribute to i t . In Chap. II, we describe the experimental equipment i n v o l -ved i n t h i s work and the techniques of the measurements of the relaxation times T^ and . In Chap. I l l , the theory of spin-l a t t i c e r elaxation i s presented i n greater d e t a i l . In Chap. IV, the experimental r e s u l t s of T-^  are presented and interpreted on the basis of the theory outlined i n Chap. III. We also discuss the e f f e c t on T-^  of the dissolved paramagnetic ions and present our measurements of T-^  i n the presence of known amounts of oxygen i n CH^. In Chap. V, we discuss the theory of n.m.r. l i n e shape and present our experimental r e s u l t s of T 2 along with t h e i r interpretation. In Chap. VI, we review the main re s u l t s and make some suggestions for further work, although work along the suggested l i n e s i s already i n progress. -9-CHAPTER II EXPERIMENTAL APPARATUS AND MEASUREMENT OF RELAXATION TIMES 2-1 APPARATUS A block diagram of the equipment used i n thi s work i s shown i n Fi g . 2. In addition to t h i s , the equipment not shown i n the diagram, consisted of: (a) A low temperature system capable of maintaining the samples at any temperature between 56°K and room temperature. (b) A vacuum system required for the p u r i f i c a t i o n of various samples used i n t h i s work. The equipment was s p e c i a l l y constructed for t h i s work and a general description of each item w i l l now be given. 1. MAGNET The magnet made available for the preliminary work was the 7000 gauss permanent magnet already described by (5) Watermann . The frequency of the spectrometer had to be matched to the magnetic f i e l d , leading to a choice of a frequency of approximately 30 Mc/sec. Most of the l a t e r work was done with a Varian Model V-4007-1 six inch Rotatable Laboratory Electromagnet with 2-inch a i r gap. 2. TIME BASE GENERATOR The c i r c u i t diagram of t h i s timer i s shown i n Fig. 3, and i s of standard design. This unit gives out pulses to TRIGGER » OSCILLOSCOPE TIME - BASE. GENERATOR. SO HC/S. OSCtLLATER. SAMPLE COIL HALF-WAVE LINE. I COPPER BOMB LEL . AMPLIFIER. \ 7 MAGNET. FIG. 2 BLOCK DIAGRAM OF N.M.R. SPECTROMETER FIG. 3 3 PULSE TIME-BASE GENERATOR -12-trigger the gate which drives the r f power unit. It also triggers the oscilloscope. The sequence of pulses could be repeated automatically at any i n t e r v a l up to 4 seconds or at longer i n t e r v a l s with the help of a manual switch. 3. RADIO FREQUENCY CIRCUITRY The c i r c u i t diagram of the 30 Mc/sec. pulsed o s c i l l a t o r i s shown i n Fig. 4. This o s c i l l a t o r has already been described by Bloom, Hahn and Herzog v . The o s c i l l a t o r gives r i s e to a peak to peak 30 Mc/sec. r f voltage of about 200 volts across the sample c o i l while gating time was variable between 10 and 150,asec. During most of the work pulse lengths used were about 40Asec. However, some work was done with pulse lengths of about 15^csec. since T becomes of the order of 20 xtsec. at the lowest temperatures studied. 4. AMPLIFIER The r f amplifier i s b u i l t commercially by Linear Equipment Q Laboratory. It has a maximum gain of approximately 10 and a band width of approximately 2 Mc/sec. The f i n a l stage of the amplifier has a IN34 diode detector. The signal to noise r a t i o was of the order of 20:1 for methane and was further improved by a factor of 2 by using a simple R-C f i l t e r at the input of the oscilloscope. This improvement was p a r t i c u l a r l y useful i n the s o l i d samples when the relaxation time became longer than 50 sec. The signals observed on the Tektronix 545 oscilloscope +290 V +190 V 6 ° X -I90V IOV. 629-B r II• > 6L6, 6J6. F I FIG. 4 30 MC/SEC. GATED PULSED OSCILLATOR. Co I -14-were of the order of § v o l t amplitude. 5. THE TUNED CIRCUIT The tuned c i r c u i t consists of a sample c o i l L of 11 turns and inner diameter of 6 mm. The sample c o i l shown i n Fig. 5 was connected to the tuning capacitor C by a \ wave l i n e RG58U, approximately 3 meters i n length. The tuned c i r c u i t was i s o l a t e d from the o s c i l l a t o r and amplifier by means of capacitors ^ ( 1 0 pF) and C 2 (22 pF) . 6. LOW TEMPERATURE SYSTEM The low temperature system described here was used for making measurements between 56°K and 110°K. The magnet had a two inch a i r gap and t h i s imposed some r e s t r i c t i o n s i n designing such a system. A "blow cryostat" system was f i n a l l y decided upon which w i l l be described now. A schematic diagram of the low temperature system i s shown i n Fig. 6. The copper bomb i s surrounded by two dewars A and B. Dewar A i s evacuated by a mercury d i f f u s i o n pump while dewar B i s f i l l e d with l i q u i d nitrogen. Dewar C i s the storage dewar which i s f i l l e d with l i q u i d nitrogen. Dewars A and B are made vacuum tight with brass caps C-, and C ; the vacuum tight seal i s provided with "0" rings. Liquid nitrogen i s pushed into the copper bomb via F, the jacket around F being evacuated by a mer-cury d i f f u s i o n pump. The flow of l i q u i d nitrogen i s con t r o l l e d by valves (11) and (12), which provide coarse and fine flow controls respectively. . TO L E L AMPLIFIER c 2 OSCILLATOR 'i LINE S A M P L E COIL FIG. 5 T U N E D CIRCUIT. FIG. 6 LOW TEMPERATURE SYSTEM. -17-The flow rate i s monitored by an o i l manometer, , attached to an o r i f i c e place flowmeter G inserted i n the return l i n e , by means of which the gas after cooling the copper bomb either escapes into the a i r v i a valve (10) or to the recovery tank v i a valves (13) and (14). The mercury manometer M provides pressure measurements either on the flow system or i n any other part of the system. The copper bomb shown i n Fig. 7 i s jacketed by a concen-t r i c brass c y l i n d e r . The four brass s t r i p s shown i n section A - B of F i g . 7 force the l i q u i d nitrogen entering from the top to t r a v e l a l l around the outside of the copper bomb, before i t can enter the inside of the bomb from the bottom. The lower i portion of the bomb i s f i l l e d with copper wool to provide a good thermal contact. The sample c o i l i s held inside the bomb by t e f l o n spacer so that a sample tube can be inserted into i t from the top, each sample tube having a ground glass seal attached to i t . One end of the sample c o i l i s soldered to the bomb and grounded. The other end of the c o i l i s f i r s t brought out of bomb through a German s i l v e r tubing (3 mm. diameter) through a t e f l o n spacer and then through a kovar glass seal soft soldered to the end of Germen s i l v e r tubing. It was found i n practice that a constant flow of nitrogen gas could not be maintained through the bomb because of pressure i n s t a b i l i t y generated by l i q u i d nitrogen b o i l i n g i n the dewar C. Numerous unsuccessful attempts were made but ultimately t h i s technique was dropped. However, i t was found that i f the bomb was cooled to l i q u i d nitrogen temperature, the warm up rate of FRONT VIEW. SECTION A-B. SECTION C 106 mm. 96 mm. BRASS, f BRASS 86 m m. X - D 10 m r FIG. 7 BOMB -19-the bomb could be cont r o l l e d by c o n t r o l l i n g the pressure inside the dewar A and the space between i t s walls. o o For temperatures i n the range of 80 K - 110 K, dewar B was f i l l e d with l i q u i d oxygen because l i q u i d oxygen i n B was found to give better temperature s t a b i l i t y than l i q u i d nitrogen i n th i s temperature range. Dewar A, the space between the walls of dewar A and the jacket around F were evacuated by mercury d i f f u s i o n pump. The bomb was cooled to l i q u i d nitrogen tempera-ture i n about 10-15 minutes by pushing l i q u i d nitrogen over from dewar C as described e a r l i e r and then valve (13) was closed while valves (10) and (11) were opened to allow the rest of l i q u i d nitrogen i n the dewar C to b o i l o f f into the a i r . The pressure inside the dewar A and i t s walls was maintained at -4 -5 10 - 10 mm. The bomb was then allowed to warm up and meas-urements of relaxation times were made during the warm up. The temperature s t a b i l i t y during the warm up was as follows: 80°K - 95°K r i s e of 0.1°K to 0.15°K i n about 10 minutes. 95°K - 110°K r i s e of 0.15°K to 0.25°K i n about 5 minutes. However, the relaxation measurements could be completed i n about 5 minutes. o o For temperatures i n the range 56 K - 80 K, dewar A was f i l l e d with l i q u i d nitrogen. The space between i t s walls had -6 been pumped hard to a pressure of approximately 10 mm., baked and then sealed o f f permanently. Dewar C was taken o f f and the opening H was closed by a rubber stopper. Tube F leading to the bomb was connected to valve (7) by a rubber tubing and helium gas at one atmosphere pressure was introduced into -20-the bomb via valves (8) and (7), which were then closed. Liquid nitrogen i n the dewar A was then pumped o f f by a rotary pump via valves (1) and (5) and i t s pressure was observed v i a valves (6) and (9). The rate of pumping was con t r o l l e d by valve (1) and by introducing a variable leak v i a valve (2) i n order to obtain a desired temperature. The bomb temperatures were always constant for a considerable length of time (20 - 30 minutes) to within + .05°K. 7. PURIFICATION TECHNIQUE The basic problem i n our work was to f i n d the most e f f e c -t i v e way of removing oxygen from the samples of CH^ and i t s deuterated compounds. It was found by us that T^ i n CH^ was very sen s i t i v e to small amounts of oxygen i n the sample, because the oxygen molecule i s paramagnetic. The e f f e c t of paramagnetic (7) impurities on T^ i n diamagnetic l i q u i d s has long been known Our preliminary measurements with samples of CH supplied 4 commercially by Fisher S c i e n t i f i c Instruments, gave us T values of the order of 10 msec. These values agreed with the values reported e a r l i e r ( 4 ) The vapour pressure readings for (8) such samples, however, disagreed with those published for pure CH^. It was suspected that t h i s disagreement could be due to the presence of impurities, oxygen and nitrogen being the chief impurities. It was considered necessary to eliminate such impurities i n order to study T^ i n the pure liquid^®^. An attempt was f i r s t made to use the fact that the normal b o i l i n g point of oxygen (90°K) i s lower than that of CH 4(112°K) The l i q u i d was maintained at a temperature (106 K) such that most of the oxygen might b o i l o f f . The vapours were then pumped o f f . This process was repeated a number of times. Each time we pumped o f f the vapours we obtained a longer relaxation time. Although T^ values as long as 10 seconds were obtained, t h i s technique for p u r i f i c a t i o n was f i n a l l y dropped for three reasons. 1. This method was very wasteful of methane. 2. T^ values so obtained were not reproducible. It was suspected that t h i s change i n T^ values with time could be due to the small amounts of oxygen adsorbed i n the walls of the glass system. 3. It was also found that methane attacked the vacuum grease on the stop cocks i f i t was l e f t i n the glass system for a couple of days. It was, therefore, considered necessary to bake the system thoroughly and then to store the p u r i f i e d sample i n a sealed glass bulb. The second technique employed was to c i r c u l a t e methane over heated copper. This method was not s a t i s f a c t o r y because the gas decomposed during the process. The technique found to be most suitable for our purposes involved the use of a getter, misch metal, which consists of rare earths derived from monazite and alloyed with iron. This metal was selected because i t was found by Ehrke and Slack i n t h e i r study of the gettering power of various metals l i k e aluminum, magnesium, barium and misch metal that misch metal i s the most e f f e c t i v e for removal of oxygen. As described i n -22-the following paragraph, our p r o c e d u r e i s a s l i g h t l y s i m p l i f i e d version of that described by Ehrke and Slack A ^ l i t r e f l a s k drawn into a long tube on one side i s used as shown i n Fig . 8. The end of the tube f i t s into an NMR probe to make T^ measurements on the condensed l i q u i d . A tungsten c o i l of f i v e or six turns of 0.01 inch diameter wire with two or three small pieces of misch metal i n the c o i l i s sealed into the f l a s k . The f l a s k i s then connected to a d i f f u s i o n pump through a trap and evacuated to a pressure of -6 the order of 10 mm. The f l a s k i s baked by flaming several times during pumping to get r i d of oxygen adsorbed on the walls. The tungsten c o i l i s also degassed during pumping. When the fl a s k has been thoroughly baked and pumped, i t i s f i l l e d with argon to a pressure of 1-2 mm. The getter i s then flashed by passing current through the tungsten c o i l using a variac to control the current, u n t i l a c h a r a c t e r i s t i c diffused layer of the deposit i s produced. The presence of argon i s necessary to produce a diffused deposit, the gettering power of which i s much higher than the bright shiny t y p e ^ ^ . The argon i s then pumped o f f , leaving an uncontamiriated gettering layer of high a c t i v i t y . Methane at one atmosphere pressure i s introduced into the f l a s k which i s then sealed o f f from the system. When t h i s procedure was followed, was always found to increase to 16 seconds at 108°K, t h i s value being reproduced whether the measurements were done on d i f f e r e n t samples or the same sample at d i f f e r e n t times during a period of several days. Since the o r i g i n a l measurements, T^ has been found to be FIG. 8 SCHEMATIC DIAGRAM OF THE SAMPLE CONTAINER -24-reproducible over a period of several years '. It would seem that the technique described here would be well suited to remove oxygen from other hydrocarbons and deuter-ated samples of CH^ have been also prepared using t h i s technique. 8. TEMPERATURE MEASUREMENTS A l l temperature measurements were made using a platinum resistance thermometer (Hartman-Braun). The thermometer was mounted i n a copper tube soldered deep into the main body of the copper bomb. The thermometer was im-mersed into copper tube which contained some glycerine to ensure good thermal contact at low temperatures. A Rubicon Mueller bridge was used for accurate measurement of the temperature. The thermometer was ca l i b r a t e d against the platinum r e s i s -tance thermometer (No. 60-597) which was very accurately c a l i b r a t e d by National Bureau of Standards. The c a l i b r a t i o n was done at three temperatures by immersing the two thermometers together i n l i q u i d nitrogen, l i q u i d oxygen and an ice-water mixture. The resistances of the two thermometers were read on the Mueller bridge. The temperatures were found from the thermometer c a l i b r a t e d by NBS. These temperature values were then plotted against the measured value of resistance of the thermometer used i n t h i s work. A l i n e drawn through these points turned out to be p a r a l l e l to the l i n e plotted for the cal i b r a t e d thermometer. The l i n e was then extended to lower temperatures; the lowest temperature used i n these experiments o being 56 K. 2-2 MEASUREMENT OF RELAXATION TIMES A l l measurements reported i n t h i s work were c a r r i e d out (13) usxng the pulse method developed by Hahn at a frequency of 30 Mc/sec. Since t h i s method has already been discussed i n d e t a i l by various authors, only a b r i e f account w i l l be given i n t h i s chapter. We consider an ensemble of nuclear spins placed i n a strong steady dc magnetic f i e l d H Q oriented i n the z d i r e c t i o n . When thermal equilibrium i s attained, the nuclear ensemble gets d i s t r i b u t e d among the d i f f e r e n t spin states according to the Boltzmann law of d i s t r i b u t i o n . As a r e s u l t there w i l l be a net magnetic moment vector M = MD oriented p a r a l l e l to H q. The spin ensemble i s then subjected to a l i n e a r l y polarized radio frequency f i e l d of amplitude 2H, normal to H , where |H,|«JH | x o i o and has the frequency Ut~o = 7H Q, 7 being the gyromagnetic r a t i o for protons. The t o t a l external f i e l d vector has the components H x = 2 ^ c o s i x ^ t , H y = 0, H z = H Q (2.1) In practice, t h i s radio frequency f i e l d i s applied i n the form of intense, short pulses which excite a c o i l surrounding the sample containing nuclei and having i t s axis i n the x d i r e c t i o n . This o s c i l l a t i n g f i e l d i n the x d i r e c t i o n i s equivalent to two c i r c u l a r l y polarized radio frequency f i e l d s H x = H x cos U>Qt, H y = + E1 s i n UTQt (a) (2.2) H x = E1 cos (*rot, H y = - H x s i n (XrQt (b) -26-rot a t i n g around the z d i r e c t i o n . It i s convenient to transform to a coordinate system rotating at the Larmor frequency around z axis. The gyromag-netic r a t i o of the protons being p o s i t i v e , f i e l d (a) w i l l be synchronous with the gyromagnetic precession of the protons about the fix e d f i e l d H Q while f i e l d (b) being far from synch-ronous with the gyromagnetic precession of the protons, w i l l (14) have very l i t t l e e f f e c t and may be neglected . As a r e s u l t of the f i e l d (a) applied for a time t , the macroscopic magneti-zation MQ w i l l be deflected by an angle 9 = from i t s equi-librium d i r e c t i o n along H q. M w i l l then precess about H Q at the Larmor frequency which i n the rota t i n g coordinate system may be represented by the vector M being tipped from the z axis. Due to the precessional motion of the vector M af t e r the pulse, an alt e r n a t i n g voltage i s set up by magnetic induction i n the same c o i l as i s used to produce the r f magnetic f i e l d , and the magnitude of t h i s voltage w i l l be maximum when 6 = i . e . when the nuclear magnetization i s completely i n the transverse (x-y) plane. This induced voltage i s then ampli-f i e d , detected and applied to the y axis of the oscilloscope, the sweep of which has been triggered j u s t before application of the pulse. This induced voltage decays because of (1) di f f e r e n t rates of precession throughout the sample caused by the s t a t i c " l o c a l " f i e l d s or by an inhomogeneous magnetic f i e l d (2) s p i n - l a t t i c e relaxation (3) processes which cause random f l u c t u a t i o n of precessional frequency and phase a r i s i n g from spin-spin coupling and, p a r t i c u l a r l y i n the case of l i q u i d s , -27-molecular s e l f - d i f f u s i o n i n an external inhomogeneous f i e l d . Since the amplitude of the induction voltage immediately following the pulse i s proportional to the magnetization of the nuclear spins at the time the pulse i s applied, the measure-ment of the maximum amplitudes of the induction signals following each of a pair of i d e n t i c a l pulses as a separation of the two pulses i s used to measure T . 1 If, i n the absence of an r f f i e l d , M ( t ) , the component of magnetization of the nuclear spin system i n the d i r e c t i o n of H Q, returns from M (T^) to i t s equilibrium value III according to z ° the equation and i f M (0) = M i . e . the spin system i s i n thermal equilibrium before the f i r s t pulse i s applied, one gets the following r e l a t i o n s h i p between the values of the detected r f signals A ( 0 ) immediately following the f i r s t pulse, and afte r the pulse A(T) at a time T l a t e r M z(t) = MQ - |_Mo - M, [z( f±)] exp [- ( (2.3) o A ( 0 ) - A ( T ) = constant exp ( - T/T, ) (2.4) AW ; T E I_L. A(2T) O F i g . 9 F i g . 9 i s a drawing of oscilloscope display showing amplitude -28-of the detected r f signal as a function of time for the two-pulse nuclear magnetic resonance experiment. Following the pulses at 0 and T indicated by shaded regions are induction signals of maximum amplitude A(0) and A ( T ) respectively. At 2 Y a spontaneous pulse appears which has been c a l l e d "spin echo" (13) by Hahn . It has maximum amplitude at 2T . TECHNIQUE A Thus, a simple method of measuring T-^ i s to apply two im-pulses separated by a known time T . The pulse sequence of two >V2 pulses i s repeated at a r e p e t i t i o n rate which allows the system to recover to equilibrium before each sequence. The f i r s t pulse disturbs the system from equilibrium and the second pulse i s a measure of the regrowth of the z component of magnetization. A ( T ) i s obtained by varying the separation between pulses. The slope of the plot of log (A(0) - A(T)J versus T i s a measure of T-j^ . This method was used for samples of CH^ containing known amounts of oxygen, where T-^  happens to be f a i r l y short (of the order of msec.). TECHNIQUE B Since the relaxation times were f a i r l y long i n the pure samples of methane used i n t h i s work, the method described above was s l i g h t l y modified. A t r a i n of pulses each of which rotated the magnetization vector through an angle of 7/2, approximately was applied i n a time short compared with T . The condition M = 0 was established by applying a t r a i n of many pulses -29-within a time short compared with T^ . At a time t l a t e r a sim i l a r t r a i n was applied and the height A(t) of the induction signal immediately after the f i r s t pulse was measured as a function of time t. was deduced from the r e l a t i o n A ( o o ) - A(t) = constant exp (- t/T±) (2.5) which one gets from equation (2.4) by substituting oo for 0. A ( o o ) i s the value of A(t) for t >> . Usually we waited a time t > 5 T to 10 to obtain A ( o o ) . The virtue of t h i s method i s that i t i s not necessary to wait for a time long compared with T-^  between successive measurements, since each t r a i n gives M = 0 regardless of the previous history. TECHNIQUE C Spin-spin relaxation time i n s o l i d samples for temper-atures near the melting point was measured from the amplitude A ( 2 T ) of the "echo" at t = 2 T as a function of 2T(see Fig. 9). It has been shown that i f the Bloch equations h o l d ^ ^ and i f (13) one s a t i s f i e s the condition t < T < T n , T Q , the height of the "echo" obeys the r e l a t i o n A ( 2 T ) = constant exp (- 2T/T 2) (2.6) where t w i s the width of the pulse and T i s the i n t e r v a l between two pulses. Equation (2.6) has to be modified i f d i f f u s i o n e f f e c t s are to be taken into account. However, d i f f u s i o n i n s o l i d s i s too slow to a f f e c t the "echo" d i r e c t l y and equation (2.6) can be used as i t stands. From the plot -30-of the In A ( 2 T ) as a function of 2T , one gets a straight l i n e graph with slope 1/T2. TECHNIQUE D At lower temperatures, i t becomes extremely d i f f i c u l t to observe the "echo" and, therefore, T 2 cannot be measured by the method j u s t described above. This i s because the l o c a l f i e l d due to l a t t i c e neighbours (AH^1/7T 2) i s superimposed on the externally applied f i e l d (^H^1/7T 2) at the p o s i t i o n of the precessing nuclei and t h i s l o c a l f i e l d i s spread over a width much greater than the width due to the magnet ( i . e . T 2 < T 2 ) . Extremely intense r f power i s required to excite a l l of the spins over a broad spectrum of Larmor frequencies i n a pulse time t and the condition T 0 << T < T Q has to be s a t i s f i e d i n W £1 & order to prevent damping down of the echo. Because of our l i m i t a t i o n i n s a t i s f y i n g these conditions,' T 2 was measured from the decay of the induction signal following the pulse (the t a i l ) . This was done by taking photographs of the induct-ion signal i n the s o l i d and l i q u i d phases of the sample. The two signals were f i r s t normalized af t e r applying appropriate corrections due to non-linearity of the amplifier. Then the amplitude A(t) of the induction signal i n s o l i d for values of t was divided by the corresponding amplitudes of the induct-ion signal i n l i q u i d phase. This correction was necessary because the inhomogeneity i n the externally applied f i e l d brings about an attenuation of observed induction signals i n s o l i d s i n addition to the decay due to T whereas i n l i q u i d -31-state, the decay of the induction signal i s governed by the spread caused by the external f i e l d inhoraogeneity over the sample; any contributions to the l o c a l f i e l d at the nucleus by neighbours i n the l a t t i c e averages out completely. In making t h i s correction i n t h i s way, i t i s assumed that the influence of the l o c a l f i e l d s i n the s o l i d s i s independent of the inhomogeneity of the applied f i e l d . It can be shown that the induction signal has the form A(t) = constant exp (- t/T^) i f the l i n e shape i s Lorentzian. If the l i n e shape i s Gaussian, the form of the induction signal i s 2 * A(t) = constant exp (- t /2T2 ) (to be discussed i n Chap. V). Plot of the logarithm of A(t) o for d i f f e r e n t values of t or t gave a straight l i n e depending upon the l i n e shape being Lorentzian or Gaussian. In either case T 2 was derived from the slope of the l i n e . -32-CHAPTER III THEORY OF SPIN-LATTICE RELAXATION In t h i s chapter the theory of nuclear s p i n - l a t t i c e relaxation w i l l be discussed i n order to explain our re s u l t s i n methane and i t s deuterated modifications. It i s necessary to discuss the properties of methane before proceeding to the theory i t s e l f . 3-1 METHANE (CH 4) Normal B o i l i n g Point 111.67°K Normal Melting Point 90.66°K The methane molecule i s a highly symmetrical molecule. It i s composed of one carbon atom and four atoms of hydrogen arranged i n a tetrahedral array about the carbon atom. There are three types of modifications of the molecule: meta (I =2), ortho (I = 1) and para (I = G) with s t a t i s t i c a l weights 5:9:2. Since the ro t a t i o n a l states associated with (15) each of these modifications are not i d e n t i c a l , we might expect, i n general, to observe a relaxation process described by two time constants a r i s i n g from the ortho and meta (16 ) molecules respectively. Hubbard has also shown i n his analysis of the s p i n - l a t t i c e relaxation process i n methane, that one would expect two time constants for a molecule with four i d e n t i c a l spin \ nuclei even i f the quantum mechanical considerations (Fermi-Dirac s t a t i s t i c s of the spin \ nuclei) -33-leading to the existence of ortho, para and meta modifica-tions were neglected. In the work reported here the relaxation of Mz towards equilibrium was always describable i n terms of a single exponential, and no evidence for more than one time constant was found within experimental error. In t h i s connection i t should be said that i n Hubbard's analysis, the two exponential decays have r e l a t i v e amplitudes of 50:1 and time constants which d i f f e r by only about 2 5%. The differences of t h i s order are too d i f f i c u l t to resolve experimentally because of signal to noise considerations. The dominant relaxation term i n Hubbard's analysis gives a r e s u l t which d i f f e r s from the r e s u l t s quoted below by only a few percent for the special case of CH^ to which Hubbard's analysis applies. 3-2 THEORY T-^  may be expressed i n terms of Fourier transforms of the c o r r e l a t i o n functions of those interactions which enable energy to be exchanged between the nuclear spin system and the other degrees of freedom ( l a t t i c e ) of the material i n which ' the spins are located. For a system of non-identical nuclei a and b of spins I and S and gyromagnetic r a t i o s 7j (17) and 7g respectively, Abragam (page 295, eqs. 87 and 88) has given the following equations to describe the approach to equilibrium of the z components of the average angular momenta per spin <TI_> and <S_> respectively: (Note that the Z z z component of magnetization i s proportional to < I z> and -34-<S > respectively for the a and b type spins.) Z d<I,> dt d<5z> 3.1(b) with 0) <2-> -i I -!, = § r H « 8(8 + i)[J„"- > + v7« a H + T 4 + T ; - W , 2 t 2 s(s + D ^ - ^ f J ^ H J ^ H T, I S to) , ^ ~\ y \ i ? i ( . + ! ) [ - ! J „ ' H , - « . . ^ J „ ( - x * ^ J and s i m i l a r expressions for — s s and — 5 I by interchanging the indices I and S. I and S are the equilibrium values o o of the z components of the angular momenta per spin. We have added two terms — T and — t o include contributions T, T, to relaxation due to interactions between spins I and S on the same molecule respectively. Also, one must include contributions to relaxation which do not involve interactions between a and b type n u c l e i . Therefore, we have added two terms —~~SR » ~^SR a n t i ~~« which include contributions T, T, T, to relaxation due to spin-rotational and quadrupole interactions respectively. A l l these terms have been added because Abragam's o r i g i n a l expressions assume only one -35-nucleus of type a i n t e r a c t i n g with one nucleus of type b, whereas i n the deuterated methanes we may have more than one spin of each type per molecule. The spectral densities are given by: J (U) = J b (T) & *7 m . 0, 1 and 2 where ,3 G (T) = < P (o) P ( r ) > The external magnetic f i e l d H Q i s oriented along z axis and the vector r ^ j o i n i n g the nuclei a and b has polar and azimuthal angles and 0 a b respectively. Equations (3.1 a) and (3.1 b) can be greatly s i m p l i f i e d (18") for our purposes, i f an experimental r e s u l t of G. de Wit v ' i s used. de Wit has measured the deuteron spin relaxation i n CD^ and CHDg. He finds that the relaxation of <C S z> i s exponential with a time constant T-^  = 10 seconds for each molecule roughly independent of temperature between 55°K and 105°K. This means that the deuteron spin relaxation i s independent of the number of protons. If the relaxation were -36-due to d ipo l e -d ipo l e i n t e r a c t i o n s , the deuteron T-^  for CHDg would be shorter than for CD^ since 7?- 1(1 + 1) S»" 20 7^ S(S + 1) . The deuteron s p i n - l a t t i c e r e l a x a t i o n i s due to the quadrupolar i n t e r a c t i o n between the deuteron quadrupole moment and the e l e c t r i c f i e l d gradient of the molecule. This i n t e r a c t i o n i s present for the deuteron but not for the proton since S = 1 and I = | . This r e s u l t means that i n equation 3 .1(b) , the term —„ i s dominant and equation 3.1(b) i s wel l approximated by d < S Z > (—s)< < S z > " So> <3.2) dt » T In the experiments reported here < I z >- i s dis turbed from i t s equ i l ibr ium value I Q by a pulse . I n i t i a l l y < S Z > = S Q . Therefore, according to equation (3.2) i t remains at th i s value for a l l times and the second term i n equation 3.1(a) can be dropped. Equation 3.1(a) can now be rewri t ten for our purposes as fo l lows: o C < I z > I I-which has an exponential s o l u t i o n [e .g . equation (2.3) , Chap. I I j with -37-T' T' to w 1 1 . { Y ^ U I + ^ J ^ + J J ^ ] T, + —SR " <3-3> For non-metallic l i q u i d s the important mechanism i s that a r i s i n g from the thermal motion of the atoms or molecules which constitute the l a t t i c e . The atoms or molecules are regarded as vehicles conveying the nuclei from point to point. Thus each nuclear magnetic moment takes part i n the random tr a n s l a t i o n a l and r o t a t i o n a l Brownian motion of the molecules. In consequence, the l o c a l magnetic f i e l d at any point contributed by the neighbouring nuclear magnetic moments and by any e l e c t r o n i c magnetic moments which may also be present, i s a rapidly f l u c t u a t i n g function of time. The component at the resonant frequency uyQ of the Fourier spectrum of t h i s f l u c t u a t i n g f i e l d , then, i s capable of inducing t r a n s i t i o n s between the nuclear magnetic energy l e v e l s and causing relaxation. The interactions which contribute to T-^  are divided into three groups: (a) dipole-dipole interactions between nuclei on the same molecules whose contribution to 1/T-^  i s usually denoted by ^ / ^ " l ^ r o t a n a t h : i- s contribution i s included i n the f i r s t two terms of equation (3.3). -38-(b) dipole-dipole interactions between nuclei on d i f f e r e n t molecules or between nuclear spins and paramagnetic and t h i s contribution i s included i n the f i r s t two terms of equation (3.3). (c) spin-rotational i n t e r a c t i o n between the nuclear spin and r o t a t i o n a l angular momentum of the molecule, whose contribution to 1/T^ i s denoted by ( 1 / T ^ ) s p i n _ r o t a n d t h i s contribution i s given by the t h i r d term i n equation (3.3). We have then The reasons for using the above notation i s c l e a r l y that fluctuations i n (a) are brought about by e f f e c t s which cause the molecule to reorient or rotate, while fluctuations i n (b) are mainly brought about by t r a n s l a t i o n a l motion of the molecules r e l a t i v e to each other. Fluctuations i n (c) are also brought about by molecular reorientations. (a) ROTATIONAL CONTRIBUTION ( M X T l ' r o t In the Debye model (BPP^^ ) the molecule i s regarded as a r i g i d sphere undergoing random reorientations. The o r i e n t a t i o n of the vector j o i n i n g the two nuclear spins then varies randomly. If the d i f f u s i o n equation describes the motion and the c o r r e l a t i o n functions are taken to be of the (7) following form as i s commonly done ions, whose contribution to 1/T i s denoted by (1/T]_) t r a n s l . (3.4) -39-(3.5) where i s the c o r r e l a t i o n time of the order of the average time for appreciable reor i en ta t ions , then using the fact that the in ter -nuc lear separat ion r i s f ixed i n a r i g i d sphere, equation (3 .5) gives -G (o) = 8 X fir < j ' " = - 1 , _ '6 %. /5 -6 x /t, -6 and s i m i l a r expression for J j S - In the spec ia l case of the ro ta t ion being described by s o l u t i o n s to the r o t a t i o n a l d i f f u s i o n equation , 7/ i s given by r e l a t i o n r = 4 A ^ a 3 'r 3kT where ^ i s the c o e f f i c i e n t of v i s c o s i t y and a i s the radius of sphere. Therefore, i f one has N a nuc le i of type a per molecule, nuc l e i of type b per molecule and i f one s a t i s f i e s the short c o r r e l a t i o n time approximation 2 2 (UQ "Tp < < 1) the expression for t > using equation ( 3 . 3 ) , becomes GlLt " I ^ T r [s("" - 1 ) 1 ( 1 + 1 ) + l 7 ^ S ( S + a -40-We define a quantity F a = | 7j h. T r ~ which has the dimensions sec In our work, since nuclei a are protons for which I = \ and nuclei b are deuterons for which S = 1, we get (!) = K - 1 > + 1 4 R x N bl x F £ A l rot L J 7o where R = £ b | ~*a 2 S 2x F a (3.6) 7 I 3 _ i _! 7 C for deuteron = 4.1 x 10 gauss sec 4 -1 -1 7j for proton = 2.7 x 10 gauss sec This gives (I) - f ( N a " x> + 4'1 x 1 0 " 2 N b J x l rot L (b) TRANSLATIONAL CONTRIBUTION ( i ) ^Tl'^transl. We assume that the motion i s described by the solutions of the d i f f u s i o n equation. If we consider an int e r a c t i n g nucleus at a distance r from the nucleus under study, then we may regard the c o r r e l a t i o n time for t h i s nucleus as the time i n which t h i s nucleus moves a distance r i n any d i r e c t i o n r e l a t i v e to the other nucleus. The r e l a t i v e motion i s by d i f f u s i o n and may be described by means of the d i f f u s i o n c o e f f i c i e n t D of the l i q u i d , which has the approximate value kT/6;*v^a for spherical molecules of radius a and -41-c o e f f i c i e n t of v i s c o s i t y ^ . If short c o r r e l a t i o n time l i m i t s are again s a t i s f i e d and the r a d i a l d i s t r i b u t i o n funct ion i s approximated by a f l a t d i s t r i b u t i o n from a distance of c loses t approach d = 2a to (17) exo , then the expressions for spec tra l dens i t i es are ' (0) JJ JI T - 3 2 7 r a 0 l 1 = ~ 5 X d D T ( l > = 16* r N a N 225 dD x W 64K N a N x 22 5 dD and s i m i l a r expressions for J^g by subs t i tu t ing for N a . N i s the number of molecules per cm , so that N a N i s the number of n u c l e i per cm . The expression for 1/T^ , using equation (3.3) , becomes = ! i r H 2 1(1 + 1) N a N l ' t r a n s l 1 5 1 d D + 16.7,r2 2 ^ 2 g ( s + 1 ) »b_N 45 I b dD We define again a quantity 2 * 7? -n 2 N _ x F = z which has the dimensions sec b 5 dD Subst i tu t ing the values for I, S, 7 j , and 7g, we get ( U = (N a + 4.1 x 10 2 N b ) F b (3.7) 1 1 / t r a n s l -42-(c) SPIN-ROTATIONAL CONTRIBUTION 1 spin-rot This i n t e r a c t i o n , i n general, i s of the form A l . J between the nuclear spin I and the rot a t i o n a l angular momentum of the molecule J, A being a constant. S p i n - l a t t i c e relaxation occurs because molecular c o l l i s i o n s cause nij and J to change thereby producing fluctuations i n the ro t a t i o n a l magnetic f i e l d at the nu c l e i . There has been some evidence that t h i s mechanism i s probably predominant for flu o r i n e nuclei i n systems such as l i q u i d C HF 3^ 2\ Recently Johnson and Waugh have suggested that i t i s also important for l i q u i d CH^. In making th i s suggestion, they have surmised that the relaxation rate due to t h i s mechanism i s proportional to < J ( J + 1)> . If P i s J the equilibrium t r a n s i t i o n p r o b a b i l i t y , then < J ( J + 1)> 51 J ( J + 1) P where (2J +1) exp [- J ( J + l)-h 2/2I okT] SI(2J +1) exp [- J ( J + 1) 1i 2/2I 0kT] i s the moment of i n e r t i a of the molecules. For high J, we can write oo < J ( J + 1)> = dJ dJ 0 OC T I (3.8) o where T i s the absolute temperature. -43-If we accept t h i s assumption, then the complete expression for 1/T^ from equations (3.6), (3.7) and (3.8), i s ^ = [(N a-I)+ VU/0 2 N b j F t t +[N a +^-*xto 2 N b ]F t + ( U N b ) F c where F c i s the contribution to 1/T^ due to spin-rotational i n t e r a c t i o n i n CH.(N, = 0). 4 b If we put N W = n b N - 4 - n a where n i s the number of deuterons i n our case, then 1/T^ in the molecule CH^ D q has the following dependence on n at constant temperature. (2)^ : | ( 3 - T i ) + ^ x i 5 V ] k + ^ (3.9) Equation (3.9) can be written i n the form ( ^ r f o - ^ ^ ^ - ^ ^ V j V C ' * ^ —- <3.10) where Ra, R b and Rc are the contributions to l / T ^ for n = 0 due to the mechanisms a, b and c respectively. In writing equation (3.10), we neglect the changes i n the c o r r e l a t i o n functions appearing i n Ra, R^ , R c due to changes (17) in n. In the conventional theories of T, , R and R, l a 0 would depend on the d i f f u s i o n c o e f f i c i e n t D, and both are proportional to 1/D. If D were proportional to -44-(molecular mass) 2 , then for our system R and R^ would be 1 m u l t i p l i e d by (1 + TT.)2, having at most an influence of 16 9% for CHDg. The expression for 1 /T X i s then +{(1-1)+^"}^] + ( i + n ) R . c ( 3 . i i ) -45-CHAPTER IV  SPIN-LATTICE RELAXATION MEASUREMENTS A series of measurements of the s p i n - l a t t i c e relaxation time T^ i n CH^, CHgD, CHgDg and CHDg were performed i n l i q u i d o and s o l i d samples covering a range of temperature from 110 K to about 56°K. In addition to t h i s , some measurements of T-^  were made by adding known amounts of oxygen i n samples of CH^ i n the temperature range 78°K - 110°K. Measurements of T^ i n pure samples were c a r r i e d out at 30 Mc/sec. using the technique (B) for long relaxation described i n Chap. II. The samples were prepared very c a r e f u l l y using the p u r i f i c a t i o n technique described i n Chap. II. These measurements w i l l now be presented and the re s u l t s w i l l be interpreted i n terms of the theory outlined i n Chap. III. If ( 1 / T 1 ) n i s f i t t e d to a l i n e a r function of n at each temperature as implied by equation (3.10), one obtains only two parameters from the equations (- ) = R + R, + R (4.1) V T T / O a b C and ^ (- ) - - 0.32 R - 0.24 R. + 0.25 R„ (4.2) -BAIT / a b c 1 n Since i t i s not possible to obtain a unique solution of these equations, one cannot evaluate the exact order of magnitude of R , R, and R . However, the fact that RQ, Ry, and R_ must a l l a' b c ' a' b c -46-be p o s i t i v e , enables us to es t a b l i s h the upper and lower bounds for R , R, and R„. a' b c Before presenting our experimental r e s u l t s , we w i l l mention the estimation of accuracy of our r e s u l t s . 4-1 ESTIMATION OF ACCURACY If we take into account an uncertainty of + .2 of a d i v i s i o n on the oscilloscope screen i n our measurement of T^, a t y p i c a l plot of T-^  at high temperatures (Fig. 10) shows a maximum v a r i a t i o n of about ± 10%, although at low temperatures t h i s figure was improved to ^ ± 5%. 4-2 RESULTS The r e s u l t s are shown i n Fi g . 11 where l / T ^ i s plotted versus 1000/T°K for l i q u i d and s o l i d CH., CHQD, CH D and 4 o 2 2 CHD between 56°K and 110°K. 3 These r e s u l t s show that T-^  i s strongly dependent on temper-ature both i n l i q u i d and s o l i d samples and undergoes an abrupt change at the melting point, which i s due to the change i n phase. Furthermore, t h i s temperature dependence i s independent of the number of deuterons both i n the l i q u i d and s o l i d samples of CE^. 4-3 IN LIQUIDS (90°K - 110°K) Since the plot of In d/T^) versus reciprocal of temperature has been found experimentally to be a straight l i n e whose slope i s independent of the number of deuterons within experimental error, we can therefore, express (1/T^) n i n -U7-9-0-8-0-7.0-6.0-5.0-4.0-3.0-9. < 2.0-1.5- - T -5 10 —r—• 20 —r 25 15 TIME (t) IN SEC FIG. 10 PLOT OF rA(OO)-A(tf i VERSUS TIME SHOWING  UNCERTAINTY IN A T ( MEASUREMENT -48-HG. II Plot of y T versus I U U U / T in liquid and solid C H 4 , C H 3 D , CH 2 D 2  and CHD 3 between 56° K and I IO°K -49-terms of the following r e l a t i o n : ( 1 / T 1 ) n = (a + bn) exp(- E a/RT) (4.3) where E a i s the a c t i v a t i o n energy independent of n and a and b are constant. Equation (4.3) leads to ( 1 / T l ) n = 1 + | n = 1 + M k _ „ ... (4.4) <1/T 1) 0 ( 4 1 In order to e s t a b l i s h l i m i t s on Ra, R^ and Rc, we plot (1/Ti)n// ( 1 / T 1 ) Q versus n (Fig. 12). Since the r a t i o i s independent of temperature, the slope of the plot gives a reasonable f i t of the experimental data over the entire temperature range for ^ • ( 1 / T 1 ) n = - ( 0.21 ±0.02 ) ( 1 / T 1 ) 0 (4.5) This l i m i t of ± 10% has been established on the basis of the uncertainty i n our measurements of T^. a' b R , R, ~ 0 give the following approximate l i m i t s for R (0.06 ± .04) (1 ) ^ Rc ^ (0.19 ± .04) ( i ) T r o N T r o Now l i m i t s on RQ and R. are obtained as follows: a b For R a = 0, R c = (0.06 i . 0 4 ) ( i ) , Rfe = (0.94 ± - O ^ l ? ) For R b = 0, Rc = (0.19 ± .04)(lJ^, R a = (0.81 ± .04)(|) If we take into account the change i n c o r r e l a t i o n functions -50-0 1 2 3 4 5 NO. OF DEUTERONS (n) FIG. 12 PLOT O F (± ) *J_, V E R S U S NO OF D E U T E R O N S IN LIQUID S A M P L E S . -51-appearing i n R , R H and R due to change i n n, then the x correction factor [ l + 2 g J 2 = 1 + 0.03 n + explained i n equation (3.11) for R_ and R. i s applied, and a D equation (4.2) i s changed to A ( l \ = _ 0.29 R a - 0.21 R b + 0.25 R c — (4.6) 11 n Then the l i m i t s for Rc are 0 < R c < (0.15 + .04) ( 1 / ^ ) 0 thereby causing a change of 4 - 6%. On the basis of the above analysis i f we take the upper (3) l i m i t on Rc we may say that Johnson and Waughv ' may be correct i n predicting that R c may be contributing to the relaxation mechanism but the upper experimental l i m i t on R c i s at least a factor of 2 lower than t h e i r predicted value which represents a th e o r e t i c a l upper l i m i t assuming no quenching of r o t a t i o n a l angular momentum. This means that the maximum possible contribution to relaxation from the spin-rotational i n t e r -actions i s (19 ± 4)% i n which case the contribution from the intra-molecular interactions i s (81 ± 4)% and the i n t e r -molecular interactions do not contribute at a l l (R D = 0). H. S. Gutowsky, I. J. Lawrenson and K. Shimomura^2^ have shown that spin-rotational interactions are strongly temperature dependent, giving an increasing contribution to the relaxation rate with increasing temperature. (18) G. de Wit has measured deuteron spin relaxation i n -52-CD^ and CHDg. He found that the deuteron i n these samples i s independent of temperature within 10% between 55°K - 105°K. The c o r r e l a t i o n functions associated with the quadrupolar interactions, which produce the deuteron spin relaxation, are i d e n t i c a l to the c o r r e l a t i o n functions associated with the intra-molecular i n t e r a c t i o n s v ' . Since the observed relaxation p r o b a b i l i t y i n our methane proton spin relaxation studies decreases with increasing temperature, i t seems highly improbable that the l i m i t s obtained with R^ = 0 are close to the correct values. It i s more l i k e l y that the l i m i t R a = 0 i s closer to being correct, i n which case the p r i n c i p a l contribution (94 ± 4)% to relaxation would come from i n t e r -molecular interactions and the contribution from spin-rotational interactions i s (6 ± 4)%. Taking the lowest l i m i t of 2% we can say that spin-rotational interactions do not seem to contribute appreciably to T-^  i n our case and may well be zero. In addition, i t should be noted that our T^ measurements give the same a c t i v a t i o n energy for d i f f e r e n t n i . e . we do not observe d i f f e r e n t temperature dependence for d i f f e r e n t n within experimental error. Since the r e l a t i v e contributions to the relaxation rate from d i f f e r e n t mechanisms change with n, t h i s also indicates that only one mechanism i s probably predominant. Thus we may say that inter-molecular interactions are predominant i n causing the relaxation. However, further experi-ments w i l l have to be done to s e t t l e t h i s point conlusively. (17") If we use the conventional theory of r e l a x a t i o n v ' as applied to l i q u i d CH4, we can calculate d / T i ) r o t u s i n S -53-equation (3.6). Taking 4 - 1 1 7j = 2.7 x 10 gauss sec - J-t = i o - 2 7 ergs-sec k = 1.38 x 10~ 1 6 ergs/deg. a = 2.0 x 10" 8 cm r = 1.78 x 10~ 8 cm at 91°K = .2005 x 10~ 2 poise Y\, at 110°K • .1214 x 10" 2 poise we f i n d that ^ / ^ i ^ r o t v a r :*- e s from .37 to .18 s e c - 1 between 91°K to 110°K respectively. ( l / V t r a n s l c a n ^ e c a^- c u ; L a"te<i from equation (3.7). Taking d = 4.0 x 10" 8 cm D at 91°K = 1.66 x 10" 5 cm 2/sec D at 110°K = 3.316 x 10" 5 cm 2/sec N at 91°K = 1.7 x 1 0 2 2 N at 110°K = 1.59 x 1 0 2 2 we f i n d that (1/T-^)+- r a n sj varies from 0.1 to 0.05 s e c - 1 between 91°K to 110°K respectively. The values of D are calculated from measured v i s c o s i t y values using the Stokes-E i n s t e i n r e l a t i o n s h i p (D = kT/6 7*v ,^ a). This leads to the r a t i o of ( l / T 1 ) r o t to ( V T ^ t r a n s l as 4:1 approximately, which indicates that i n t r a - molecular interactions should be dominant. This i s contrary to what we expect from our experimental r e s u l t s . This discrepancy -54-(19) has also been found by B. H. Muller and J. D. Noble i n th e i r measurements of l i q u i d ethane. They f i n d that (17) although the conventional theory of relaxation ' applied to l i q u i d ethane predicts that the r a t i o of d / T i ) r o t t o (1/T,). . i s 4:1, the experimental r e s u l t s show that 1 t r a n s l ' v spin-rotational interactions may well be zero and the p r i n c i p a l contribution comes from inter-molecular interactions. They further suggest that t h i s depression of the intra-molecular contribution r e l a t i v e to the inter-molecular contribution i s due i n part to the averaging-out e f f e c t of either internal motion or rot a t i o n of the molecule i t s e l f . This may well be the true i n our case. 4-4 IN SOLIDS (90°K - 56°K) The experimental r e s u l t s show that from 90°K to approxi-mately 80°K, the plot of In (1/T X) versus 1000/T i s a straight l i n e and the slope i s independent of n. In t h i s region i n order to es t a b l i s h l i m i t s on R , R, and R , we can a b c use again equation (4.4). We plot again ( 1 / T i ) n / ( 1 / T i ) 0 versus no. of deuterons i n Fig. 13. Since t h i s r a t i o i s again independent of n, the slope of t h i s plot gives a reasonable f i t of the experimental data given i n Fig. 11 for i L ( i ) = "(0.24 + .02) (I) (4.7) ° n T n 1 o Using equations (4.1) and (4.2), the following approximate l i m i t s are obtained for R„ -55 0 1 2 3 4 NO. OF DEUTERONS (n) FIG. 13 PLOT O F (7;) A±) V ERSUS NO O F D E U T E R O N S IN SOLID S A M P L E S . -56-0 < R ^ c (0.14 + .04) There i s much evidence that Rc i s considerably quenched i n the l i q u i d s due to the quenching of ro t a t i o n a l angular momentum. It should, therefore, be absent i n the s o l i d s as indicated by these r e s u l t s . In t h i s temperature region, there i s additional evidence available which enables us to choose R c = 0. The a c t i v a t i o n energies are found to be 3.2 k cal/mole independent of n, from the plot of In (1/Tj) versus 1000/T (Fig. 11). This confirms that mechanism (b) i s predominant i n t h i s region since the a c t i v a t i o n energies are the same as obtained form T 2 measur-ements (to be discussed i n Chapter V) and the contributions to T 2 can be said with considerable certainty to be so l e l y i n t e r -molecular dipolar interactions. At temperatures below 80°K, the plot of In (1/T-^) versus 1000/T departs from a straight l i n e r e l a t i o n s h i p . It appears that a new relaxation mechanism i s dominant at these lower temperatures (81°K - 56°K). If we assume that the contribution of (1/T-j^) from t r a n s l a t i o n a l d i f f u s i o n ( 1 / T i ) d i f f a n d t h e contribution to (1/T-^) from t h i s new mechanism d / T i ) x a d d to give (1/T.. ) , then we can write xpt or (4.8) -57-If we extend the straight l i n e portion just below the melting point (Fig. 11), we obtain ^ d i f f * Subtracting t h i s contribution from (1/T.) , we obtain (1/T 1) . In F i g . 14, we then plot In (1/T,) versus 1000/T . This plot shows X X that within experimental errors, the slopes are independent of n, although there seems to be some deviation at the higher temperature end. But t h i s deviation can be attributed to subtracting two larger quantities from each other i n equation (4.8). We now plot the r a t i o ( 1 / T i ) x ( n ) / ^1//Tl^x(o) v e r s u s no. of deuterons i n F i g . 15. Within experimental error, the slope of t h i s plot i s the same as obtained from second moments using line-width data (to be discussed i n Chapter V). If t h i s new mechanism i s predominant at lower temperatures ( ^ 80°K), i t seems that t h i s mechanism i s associated with dipolar inter-molecular interactions modulated by the reorien-t a t i o n of the molecules. Using the r i g i d - l a t t i c e line-width data (Chap. V), we can calculate the c o r r e l a t i o n times for molecular reorientation required to give these r e s u l t s . In order to calculate these c o r r e l a t i o n times, we must calculate terms of the form F ( t ) F ( t +T) for interactions between pairs of nuclei on d i f f e r e n t molecules. If T i s the c o r r e l a t i o n c time for reorientation of a single molecule and i f we assume that the c o r r e l a t i o n functions are well approximated by assuming uncorrelated motions of the members of a pair, then F ( t ) F ( t + T ) s iFOOTj2 exp(- p ) (4.9) •e -58-.06-.05-. 0 4 -. 0 3 -AND CHD 3 BELOW 80 ° K. -59-NO. OF DEUTERONS (n) FIG. 15 P L O T OF . , '/( ! ) VERSUS NO. O F D E U T E R O N S • i x tn>x M | 'x(o) — -60-We must, therefore, calculate an average of the form y — The average must be calculated for a l l possible rotations of the in d i v i d u a l molecules and i n the case of a c r y s t a l l i n e powder or a l i q u i d , over a l l possible orientations of the carbon-carbon vector. The averaging may be done i n any order. If the l a s t average i s done f i r s t , keeping r fixed, we see that _ L . < I > (4.10) Abragam (17) (page 289 - 291) gives the following expression for T-^  for a system of i d e n t i c a l spins Using equations (4.9) and (4.10) and assuming ( T ) 1, J±k ( U V A N D J±V ( ^ o 5 a r e given by J i k ^o> ='c 4 A < - 6 / x T 5 = 2AT ^ ""6 ' J ( 2 ) (2UT o) = % L < l f i > x 32* ik o c 4 ^ ^ r e ' 15 ik S < I > 15 ^ r6 ^ ik This gives (4.11) For the r i g i d - l a t t i c e , Abragam (17) (page 112, equation 39) -61-gives the following expression for second moment (M2) (4.12) Combining equations (4.11) and (4.12) (4.13) We w i l l show i n Chap. V that intra-molecular interactions do not contribute to the observable second moments. Second moments calculated from intra-molecular interactions have much larger values than those determined from experimental r e s u l t s , while second moments calculated from inter-molecular i n t e r -actions agree with the experimental values. Using our 9 -2 experimental value of the second moment = 4.8 x 10 sec (see Chap. V) for a sample of CH^ and taking = 50 s e c , we get T ,^-12 ' c = 10 sec. Thus, i f the new mechanism i s causing relaxation at these low temperatures ( < 80°K) and the motion can be described by a single reorientation time , one would expect l/T^ values to r i s e as the temperature i s lowered and *X to r i s e u n t i l a maximum i s reached for oJ 2 T 2 ^ 1 • Then the values of o ' c 1/T-j^  should f a l l as the temperature i s s t i l l lowered where < < » From our experimental r e s u l t s of 1/T^ (Fig. 11), we do observe that 1/T-, values s t a r t r i s i n g at our lowest -62-temperature (56°K). The a c t i v a t i o n energies obtained from a plot of In ( 1 / T 1 ) X versus 1000/T (Fig. 14) are 0.2 kcal/mole independent of n. As mentioned e a r l i e r we also observe that the plot of the r a t i o ^ / ^ ^ ( n ) ^ ( 1 / / T l ^ x ( o ) versus n (Fig. 15) gives the same slope as obtained from the second moments using line-width data (Chap. V) within experimental error and the second moments are due to i n t e r -molecular interactions. Since the c o r r e l a t i o n times for -12 molecular reorientation (10 sec.) are quite reasonable, we may probably be j u s t i f i e d i n suggesting that the new mechanism which contributes to relaxation i n t h i s region i s due to dipolar inter-molecular interactions modulated by the reorientation of the molecules. However, our re s u l t s seem to be i n c o n f l i c t with the conventional theory which predicts that for such reorientationa l motions R_ should be several times larger than R^. Therefore, further work w i l l have to be done to c l a r i f y t h i s . It should be emphasized that i n drawing the conclusion that R a = 0, the assumption has been made that the spin-rotational i n t e r a c t i o n i s not e f f e c t i v e i n the so l i d s o (R c =0). It i s well established that Rc = 0 between 80 K o o and 90 K, but below 80 K i t i s only a guess based on the fact that (• L/ Ti^ x(n) ^ ^ 1 / / T l ^ x ( o ) i s i n d e P e n d e n t o f temperature. 4-5 EFFECT OF PARAMAGNETIC IMPURITIES ON Tn The addition of paramagnetic impurities to substances can influence markedly the proton relaxation time. The magnetic moment of a paramagnetic ion i s of the order of one Bohr -63-3 magneton and t h i s i s some 10 times larger than a nuclear magnetic moment. Therefore, larger 7-values of the e l e c t r o n i c moments w i l l r e s u l t i n a larger i n t e r a c t i o n of the nuclear moments with the el e c t r o n i c moments. The f l u c t u a t i n g l o c a l magnetic f i e l d w i l l , therefore, be correspondingly larger and the relaxation time T-^  shorter. The relaxation produced by the ions i s inter-molecular. Assuming that the d i f f u s i o n equation describes the r e l a t i v e motions of the nuclei and paramagnetic ions c o r r e c t l y and the c o r r e l a t i o n time T* i s small compared with the Larmor (17) period 1/w of the paramagnetic ion, Abragam (page 304, equation 118) has given the following formula for T i where \ i s the c o e f f i c i e n t of v i s c o s i t y and 7p i s the gyromagnetic r a t i o of protons. S i s a dimensionless number defined by j \ "n2 S(S + 1) = </jt?> o <xxf > being the mean square of the magnetic moment of the 2 3 ion and i s denoted by yiA. . N i s the number of ions per cm e f f This leads to i = x 7 2 x M „ x ^ N i o n — (4.15) T-L 15 P ' e f f kT l o n Of course, we should add to t h i s expression, the c o n t r i --64-bution of the protons i n the s o l u t i o n , which i n pure CH^ are 2 s o l e l y responsible for the re laxa t ion time. But as 7. i s J ion 6 2 about 10 times larger than 7 p , the influence of paramagnetic -3 ions i s predominating even i n a concentrat ion of 10 N. According to the above expression, the re laxa t ion time should be inverse ly proport iona l to the concentrat ion of the paramagnetic ions . We have studied the e f fec t of d i sso lved paramagnetic ions i n CH^. Roughly known amounts of oxygen were added to a pure sample of CH^. These samples were then analysed by a mass spectrometer at the Department of Chemistry, Univers i ty of Washington, Seat t l e , Washington, U. S. A. The oxygen concentrations were 1.08% and 2.54% respec t ive ly . T^ was measured i n these samples from 110°K - 78°K with the technique (A) described i n Chapter I I . The r e s u l t s of these measurements are shown i n F i g . 16. We f i n d that above the melting point (90°K - 1 1 0 ° K ) , T^ has the same temperature dependence as observed i n our purest sample of CH^. F i g . 17 shows a p lo t of 1/T^ as a funct ion of oxygen concentrat ion at 1 0 8 ° K . The dotted l i n e shows the values ca l cu la ted from equation (4.15), taking 7 p = 2.68 x 10 4 gauss" 1 sec" 1 -20 _ i "^•eff = 1-4 x 10 erg-gauss -3 7^ = 1.262 x 10 poise 20 3 N i o n (1.08% 0 ) = 1.5 x 10 / cm N i o n < 2 - 5 4 % 0 2 ) = 4.0 x 1 0 2 0 / cm 3 -16 o -1 k = 1.38 x 10 ergs ( K) -65--67-Our experimental values agree reasonably well with the calculated values and our r e s u l t s also v e r i f y the inverse proportionality of and oxygen concentration. This shows that T 1 i n the l i q u i d can be well represented by equation (4.15). In the s o l i d samples (90°K - 78°K), T x i s f a i r l y short and independent of temperature. Bloembergen^^ has shown that i f the concentration of paramagnetic ion i s greater than 1%, the spin d i f f u s i o n process i s unnecessary; within a few l a t t i c e spacings of every nuclear dipole there i s an ion with which energy can be exchanged d i r e c t l y . Moreover, the magnetic i n t e r a c t i o n of the ions, now r e l a t i v e l y close to each other, may endow t h e i r energy l e v e l s with a breadth greater than "hur^ , the magnitude of the quanta which the nuclear dipoles wish to exchange. The system of ionic spins can now absorb these small quanta d i r e c t l y merely by rearrangement of the spins, a process which i s independent of l a t t i c e vibrations; the s p i n - l a t t i c e relaxation i s then short and independent of temperature. This i s what we have found experimentally. Nitrogen, which i s not paramagnetic, was also added i n a CH^ sample, the concentration of nitrogen being about 3%. As anticipated i t was found to have no measureable e f f e c t on T-^ . 4-6 ESTIMATION OF 0 o CONCENTRATION IN A PURE CH\ SAMPLE 2 4 Because of oxygen background i n the mass spectrometer i t i s impossible to analyse the oxygen concentration with i t i n a region where T-^  measurements indicate that the oxygen concentration i s le s s than about 0.1%. Therefore, our T-^  -68-measurements represent a more sensitive means of detecting oxygen than a mass spectrometer unless special steps are taken to cut down the oxygen background i n the mass spectrometer. We can calculate the percentage of oxygen concentration i n our pure sample of CH4 using the r e l a t i o n : where 1/T n represents the natural relaxation time of pure CH4 and 1/T° i s the contribution due to oxygen. o o The measured - 16.5 sec. at 108 K whereas T-^  = 40 msec. and 16.7 msec, with oxygen concentration of 1.08% and 2.54% o o respectively. If were about 16.5 sec. at 108 K, t h i s -3 would correspond to oxygen concentration f = 2.6 x 10 %. Since 1/^, > J 1/T°, we can say that f < 2.6 x 10~ 3 % i n our pure sample of CH^. Our measurements of T^ were reproducible from day to day and i n d i f f e r e n t samples. Since we would have been able to detect a change of less than 10% i n T-^ , we are probably j u s t i f i e d i n saying that our p u r i f i c a t i o n procedure described i n Chap. II resulted i n f < 2.6 x 10" 4 %. -69-CHAPTER V  N.M.R. LINE-SHAPE STUDIES The purpose of t h i s chapter i s to discuss measurements of the nuclear magnetic resonance l i n e shape i n methane where the measurements have been made using pulse techniques. Line-width may be characterized i n several ways: by a width i n magnetic f i e l d or frequency, or, a l t e r n a t i v e l y , by a time conventionally c a l l e d T 2, which i s known as the spin-spin relaxation time and i s the inverse of the width i n frequency (T 2 = • C l a s s i c a l l y the time Tg represents the time for a group of spins i n i t i a l l y precessing about H Q i n phase to dephase as a r e s u l t of t h e i r s l i g h t l y d i f f e r e n t precession frequencies. The reason for the differences i n the precession frequency of i n d i v i d u a l spins i s due to the fact that each spin i s acted upon by the l o c a l magnetic f i e l d of i t s environment. The l o c a l f i e l d may be considered to give r i s e to a spread of Larmor frequencies given by A. UT where .A uJ may have value ranging from zero to a c e r t a i n value determining the width of the Larmor frequency d i s t r i b u t i o n . The dephasing time T 2 depends on the spread of precessional frequencies and on the rate at which these change due to molecular motions. The various types of motions such as molecular rotation or t r a n s l a t i o n etc., which give r i s e to relaxation processes, influence also the width of the resonance l i n e s . With a few exceptions these motions r e s u l t i n a narrowing of the resonance -70-l i n e s and t h i s i s given the name of "motional narrowing". The physical explanation of the motional narrowing i s that i f the spins are i n rapid r e l a t i v e motion, the l o c a l f i e l d "seen" by a given spin w i l l fluctuate rapidly i n time. Only i t s average value taken over a time of the order of Larmor period i n the l o c a l magnetic f i e l d i s observable. If t h i s time i s long compared with the duration of a flu c t u a t i o n , t h i s average value of the l o c a l f i e l d w i l l be much smaller than the instantaneous value of the l o c a l f i e l d . This gives r i s e to line-width much smaller than the width obtained from the steady l o c a l f i e l d for a r i g i d system. The narrower l i n e corresponds to a longer dephasing time T . In contrast, when there i s no motion, a given spin experiences a constant l o c a l f i e l d . It either precesses at a fixed rate faster than the average, or slower. The dephasing of a group of spins arises from the accumulation of positive or negative phase. Since the l o c a l f i e l d s "seen" by a given spin are much greater than the mean l o c a l f i e l d "seen" by that spin when there i s motion, the l i n e i s broadened. When diamagnetic s o l i d s are cooled to s u f f i c i e n t l y low temperatures so that very l i t t l e t r a n s l a t i o n a l motion of the molecules i s taking place, the nuclear magnetic resonance abso-rpt i o n as a function of frequency I(W) i s usually independent of temperature. This r i g i d l a t t i c e l i n e shape, that i s , i n a sample where the lengths and orientations of the vectors describing the r e l a t i v e positions of the spins do not change i n time, i s often c l o s e l y approximated by a Gaussian function i . e . -71-where LW - 7H i s the Larmor frequency of the nuclear spins i n o - o 2 the external f i e l d H , and Lcr i s the second moment of the o p r (17) 1 l i n e for r i g i d l a t t i c e [ Abragam , page 107 equation 24] The amplitude of the free induction signal after a 90° r f (13) pulse observed i n a pulse experiment i s proportional to the "relaxation function" G(t) ^Abragam^ 1 7^ page 114^ which i s the Fourier transform of I(UT) i . e . CO G(t) ^ J I ( U ^ c o s / O S t d u r which gives G(t) •—s exp 2 2 for a Gaussian l i n e shape 2 With the onset of rapid molecular motion the observable l i n e shape or relaxation function changes to a Lorentzian form, G(t) exp [- U>-p t j [Abragam ( 1 7 ) page 433, eq. 22] where *T i s the c o r r e l a t i o n time for changes i n l o c a l f i e l d s 2 2 due to t r a n s l a t i o n a l motion with U p f c << 1. If one assumes that at each instant the microscopic d i s t r i -bution of l o c a l f i e l d s through the sample i s the same as for the r i g i d l a t t i c e , but that the l o c a l f i e l d at each point fluctuates at a rate described by a c o r r e l a t i o n function G^CT) = Up g^ ( T ) , we can predict the relaxation function for a l l times to be [Abragam^ l 7^ page 432 eq. 19J G(t) — exp -ojp Jb(t - T ) g w . ( T ) d r ] where g w (T ) i s the reduced c o r r e l a t i o n function of U/(t) such that Sux (0) = 1. -72-Because of s e l f - d i f f u s i o n which e x i s t s i n many s o l i d s except at very low temperatures, the e f f e c t on the resonance l i n e can be described by assuming the reduced c o r r e l a t i o n function g ^ ( T ) for the frequency uKt) of the random l o c a l f i e l d that fluctuates on account of the d i f f u s i o n , to be of the form g w ( T O = exp JT] 1 This leads to the following expression for the relaxation function G(t): G ( t ) ^ e x p [-u/ {exp ( =^.) - 1' + * } •c c J - (5.1) The general form of G(t) i n equation (5.1) has not been extensively used i n evaluating experimental r e s u l t s . Normally, i n order to obtain 1^ as a function of temperature i n a system where d i f f u s i o n takes place, one works i n a region where f i s so short that G(t) i s only studied for t » T so c c that the Lorentzian form i s obtained. When the motion has slowed down to the point that the " r i g i d l a t t i c e " l i n e shape i s obtained, i t i s usually assumed that no further information on T can be obtained, c (17) Abragam (page 455) has pointed out that from the general form for G(t) i n equation (5.1), i t i s possible to obtain, the detailed shape of the resonance curve, to correlate the observed line-width parameter with the c o r r e l a -t i o n time T and obtain the v a r i a t i o n of T with temperature, c c Thus i t enables us to interpret our experimental r e s u l t s . In -73-fact, we have found i n studying the proton resonance i n CH^ CHgD, CH 2D 2 and CHDg that the use of the pulse technique enables the d i r e c t study of equation (5.1) over a range of "T not usually accessible to absorption methods, c _5 When T i n such systems becomes of the order of 10 c J seconds or longer, the absorption signal gives the temperature-independent, r i g i d l a t t i c e l i n e shape. For t/ T 1, equation (5.1) reduces to 2 G(t)—>exp | - us t /2 I — exp t 2 (T ) 2 2 r i g i d l a t t i c e where (T ) 2 = _ i - (5.2) 2 r i g i d l a t t i c e ^ P G(t) i s Gaussian and the measurement of T 2 enables to calculate the (temperature-independent) second moment. However, the proton induction signals i n such systems are so large that we can also study G(t) for t/'T' 1. Equation (5.1) reduces to G(t) exp w 2 T c t J '—- exp where T 2 = — i — (5.3) P c Actual measurements of T 2 enables us to obtain 'Y as a function of temperature. Here, the l i n e shape i s indeed found to be Lorentzian. The corresponding observation i n absorption experiments -74-i s that I(W) i s Lorentzian for ( k-T _ us ) •< i which o c agrees with our r e s u l t s . However, the observation of a Lorentzian pip near the center of an absorption l i n e i s d i f f i c u l t since one must subtract the large Gaussian c o n t r i -bution to I(ur), In the pulse experiment, the Gaussian portion i s allowed to die away, leaving the Lorentzian t a i l to be studied separately. 5.1 MEASUREMENT OF T g T i n the various samples of methane was measured by the technique C or D described i n Chap. II. In the region where the "echo" was observable, T 2 was measured from the "echo" amplitude while i n the region where echo was unobservable, T 2 was measured from the induction signal ( t a i l ) . In F i g . 18, we have compared plots of In A(t) versus t and t 2 for a sample of CH^ at 78.15°K. We f i n d that (a) gives a straight l i n e only, which confirms that the l i n e shape i s indeed Lorentzian. o In F i g . 19, we again plot In A(t) versus t and t , for a sample of CH^ at 61.4°K. A straight l i n e graph i s obtained only i n (b). Here the l i n e shape i s Gaussian. 5.2 RESULTS AND DISCUSSION The experimental r e s u l t s are shown i n Fig. 20 where T 2 i s plotted versus 1000/T°K. The crosses show T 2 evaluated from free precession signals observed at t / T_ « 1 f i t t i n g the curve G(t)*~N exp The c i r c l e s are obtained -75-.1 1 1 1 1 1 — — r — 5,000 10,000 I5P00 20,000 25j000 30,000 TIME t* ( / i s ec ) 2 — » • FIG. 18 PLOTS OF THE CORRECTED AMPLITUDE OF THE INDUCTION  SIGNAL A(t) VERSUS (TIME) AND (TIME)2 FOR CH 4 AT 78.I5°K 1— 1 1 1 ! ! 1) 100 200 300 4 0 0 500 600 700 t 2 (/Asec)2 — » • FIG. 19 PLOTS OF THE CORRECTED AMPLITUDE OF THE INDUCTION SIGNAL A(t) VERSUS TIME AND (TIME)2 FOR C H 4 AT 61.4°K. -77-FIG. 20 PLOT OF T 2 VERSUS FOR SOLID CH 4 . C i ^ D , CJi 2D 2 AND CHD 3 BETWEEN 56° K AND 9 0 ° K. -78-from spin echo experiments at higher temperatures where T c i s short . The squares are obtained from free precession s ignals for t / f » 1, f i t t i n g G ( t ) — exp ( - - ) . The c T 2 r e s u l t s show that for the molecule CH. D„ , the c o r r e l a t i o n 4-n n ' times accurate ly fol low the a c t i v a t i o n energy curve over the en t i re temperature range < % > n = < T ° > n « p [ j j | ] « • « > where E a i s the a c t i v a t i o n energy corresponding approximately to the height of the po tent ia l b a r r i e r between two equivalent molecular p o s i t i o n s . As the temperature increases , T decreases and the resonance l ikewise narrows. From the p lo t of In T 2 versus 1000/T (F ig . 20), the a c t i v a t i o n energies E a are found to be 3.2 kcal/mole indepen-dent of n, i n disagreement with previous measurements of (21) 1.5 kcal/mole for CH^ . However, our value i s i n agreement with the value obtained from T^ measurements. Professor Waugh's estimates are based on the l ine -width data of Thomas, Alper t (4) and T o r r e y v . The lack of agreement between our a c t i v a t i o n energies and those obtained from the l ine -width data i s probably due to the change i n l i n e shape as the temperature i s changed. A knowledge of second moments provides useful information on the l i n e shape and the in terac t ions that contr ibute to the l i n e shape. The second moments can be ca l cu la ted from equation (5.2) using T 2 measurements for the r i g i d l a t t i c e l ine -width data. The experimental values of T 2 for the -79-samples of CR"4, CH 3D, CH 2D 2 and CHD 3 from Fig. 20, are 14.5, 16.0, 18.0 and 27.0 ^ise c . respectively. The values of the 9 -2 second moments are 4.8, 3.95, 3.1 and 1.4 i n units of 10 sec for n = 0, 1, 2 and 3 respectively corresponding to the r a t i o s 1 : 0.82 : 0.64 : 0.29. The contributions to second moments come from i n t r a -molecular dipolar interactions and inter-molecular dipolar interactions. For a system of two unlike spins I and S, (17) Abragam (page 122) gives the following expression for the second moments for the case of dipolar broadening. where and 2s , 2 N ( 6U\J = ( UX_) 2 (U> )-•. p II. ( U " p > I S P II + ( uf) p IS (5.5) = - h 2 i d + 1 ) 5 " 4 I fc-Cl - 3 cos 0)' 1 ..2 _.2 = - r- r- i 2 s(s + i ) y~ ( 1 - 3 c o s 2 9 ) 2 3 1 8 nT r 6 6 i s the angle of vector r with the applied magnetic f i e l d H q as shown i n the Fig . 21, and the sum ^ i s to be taken over a l l the s i t e s of the spins S surrounding a spin I. Fig. 21 -80-The contributions to second moments from inter-molecular interactions and intra-molecular interactions can be evaluated separately by using equation (5.5). Let us f i r s t evaluate the contribution to the second moments from inter-molecular interactions. In our case spins I r e f e r to protons and spins S to deuterons. For a polycrys-t a l l i n e material, there i s an i s o t r o p i c d i s t r i b u t i o n of the axes of reorientation and i t i s permissible to average 2 2 ( 1 - 3 cos 0) over a l l d i r e c t i o n s . Replacing i t by i t s mean value of 4/5 and taking 4 -1 -1 7 T = 2.67 x 10 gauss sec i 3 - 1 1 7 g = 4.1 x 10 gauss sec" Equation (5.5) can be written as ( Or ) = 228.7X 10 V(4-n)i- + 6.39 x 10 5* n — ___ (5.6) P n r 6 r 6 In order to calculate ]T 1/r , l e t us assume a uniform d i s t r i b u t i o n of protons or deuterons over a sphere of radius (a - 1.09 A) given by C-H distance i n CH 4. This model assumes that the centres of mass of the molecules are fixed (no d i f f u s i o n ) , and that the molecules reorient very r a p i d l y . If r represents the distance between a proton and a deuteron situated at A and B, 6^ and 6 2 are the polar angles on the two spheres as shown and jz^ and 0 2 are the corresponding azimuthal angles, then -2 2 , 2 , 2 r = x + y + z -81-where x = a (s in 0^ cos jz^  - s i n cos jzfg) y = a (s in s i n 0^ - s i n 0 2 s i n 0^) z = R - a (cos 0-^  + cos 6 2 ) (22) If we use the known c r y s t a l s tructure of CH^ and take the carbon-carbon separation R = 4.09 A, we get 6 48 £~ 1/r = .0057 x 10 taking into account the nearest neighbours ( = 12). Subst i tut ing th i s value i n equation (5.6) , we get ( W p ) n = 1.3 x 10 9 (4 - n) + .036 x 10 9 n 9 = 5.2 x 10 (1 - 0.24 n) (5.7) In w r i t i n g equation (5.7) we neglect i so topic changes i n inter-molecular separat ions . Equation (5.7) predic t s second moments for n = 0, 1, 2 and 3 to be 5.2, 3.95, 2.70 and 9 —2 1.45 i n un i t s of 10 sec , corresponding to the r a t i o s 1 : 0.76 : 0.52 : 0.28. We can now evaluate the contr ibut ion to the second moments from intra-molecular i n t e r a c t i o n s . These would contr ibute to the observable second moments only i f the rate of r e o r i e n t a t i o n of the molecules was slow compared with the Larmor periods of the nuc le i i n the l o c a l magnetic f i e l d s . Since r i s f i xed , we have to average only ( 1 - 3 c o s 2 © ) 2 over a l l s p a t i a l d i r e c t i o n s . Subst i tu t ing the values of various quant i t i e s i n equation (5.5) , we get -82--83-( CO-p)n = 21.6 x 10 9 (1 - 0.32 n) (5.8) Equation (5.8) predicts second moments for n = 0, 1, 2 and 3 to be i n the r a t i o s of 1 : 0.68 : 0.36 : 0.04. In F i g . 22, we show our re s u l t s of the observable second moments calculated from equations (5.2), (5.7) and (5.8). We f i n d that equation (5.8) gives much larger values of the observable second moments than our experimental values. It seems u n l i k e l y that intra-molecular interactions are contributing to the observable second moments. Since the ra t i o s predicted by equation (5.7) agree with our experimental values to within a few percent, the contribution to the second moments comes from inter-molecular interactions. It seems that i s o t r o p i c reorientation reduces the intra-molecular c o n t r i -bution to zero, intra-molecular l o c a l f i e l d s being averaged to zero. Local f i e l d s which are inter-molecular i n o r i g i n do not average to zero so long as the centres of mass of the molecules remain fi x e d . It then follows that the observed change i n with increase i n temperature above 69°K i s associated with the inter-molecular interactions which become time-dependent due to s e l f - d i f f u s i o n . Therefore, the act i v a t i o n energy E a associated with the v a r i a t i o n of T 2 with .temperature i s a s e l f - d i f f u s i o n a c t i v a t i o n energy. In the region where the l i n e shape i s Lorentzian, T l can be evaluated from equation (5.3), which can be written as ( f c > . " (urp% ( T J „ (5.9) -84-The r a t i o s of ( LW ) / ( UJ ) obtained from r i g i d p n p o l a t t i c e line-width data are 1 : 0.82 : 0.64 : 0.29. The r a t i o s of ( T Q V /(T 0) above 69°K (Fig. 20) are & O & Ii 1 : 0.75 : 0.54 : 0.31 for n « 0 , 1, 2 and 3 respectively. Substituting these values i n equation (5.9), the r a t i o s of ( T c ) n / ( T •) for n = 0, 1, 2 and 3 are 1 : 0.92 : 0.86 : 1.0 respectively. Since our values of a c t i v a t i o n energies from T£ measure-ments agree with the values obtained from T^ measurements and we have shown that inter-molecular interactions contribute to the line-width, we can say that Tj between 80°K and 90°K i s predominantly due to inter-molecular interactions. This r e s u l t has been used i n Chap. IV. -85-CHAPTER VI  SUMMARY AND CONCLUSIONS The work described i n the preceding chapters may be regarded as a part of a program for the study of proton and deuteron nuclear magnetic resonances i n methane. Our chief aim i s to explore the various possible mechanisms that cause s p i n - l a t t i c e relaxation. In addition, one can obtain informa-t i o n on the l i q u i d and s o l i d - s t a t e properties of the system being investigated. Our work has been concerned with the measurements of T-^  from 110°K to 56°K and T 2 from 90°K to 56°K i n samples of CH 4, CHgD, CH 2D 2 and CHD^. We have been able to show that: (1) T^ i n a pure sample of CH^ i s more than 1000 times (4) longer than the previous reported values v . Our measurements have shown that T-^  i s very sensitive to small amounts of oxygen and the previously reported shorter values of T^ may be attributed to dissolved oxygen. (2) We have been able to develop a very simple technique for the removal of oxygen from our samples. This technique i s well suited for the removal of oxygen from other gases and has (23) already been extended to l i q u i d s . On the basis of our T^ measurements i n CH 4, we are probably j u s t i f i e d i n saying that _4 our p u r i f i c a t i o n procedure resulted i n oxygen £ 2.6 x 10 %. (3) Our measurements of T^ i n samples of CH 4 with d i f f e r e n t concentrations of oxygen have confirmed that T-, i s -86-inversely proportional to oxygen concentration. (4) Contrary to the observations of Thomas, Alpert and Torrey^ 4^, we have been able to observe a marked discontinuity i n T-j^  across the melting point i n a l l our samples. (3} (5) Johnson and Waughv ' have predicted that i n l i q u i d s , s pin-rotational interactions may be contributing to the relaxation process, but i n our case we f i n d that inter-molecular dipolar interactions probably play a predominant role i n l i q u i d s and s o l i d s , and the contribution to relaxation from spin-rotational interactions appears to be small. There i s additional evidence that the inter-molecular dipolar interac-tions are predominant i n the region j u s t below the melting point, since we get the same value of a c t i v a t i o n energies from our T-^  and T 2 measurements. (6) At lower temperatures ( — 80°K), i t seems that relaxation i s caused by inter-molecular dipolar interactions modulated by the reorientation of the molecules. Our r e s u l t s are i n c o n f l i c t with the conventional theory which predicts that intra-molecular interactions should be more important i n l i q u i d s and i n s o l i d s below 65°K. (7) The l i n e shape i s Lorentzian i n the region just below the melting point which changes to Gaussian at lower temperatures. The l i n e shape i s predominantly due to i n t e r -molecular interactions. It would, however, be most valuable to extend the present work i n several d i r e c t i o n s : (1) One would l i k e to have data on T^ and T 2 over -87-the entire range extending down to l i q u i d helium temperatures so as to get a complete picture of the various mechanisms that cause relaxation. (2) Since we have only been able to predict the most predominant mechanism causing relaxation and we cannot exclude completely the p o s s i b i l i t y of others, as predicted by conventional theory, i t would be useful to perform some more experiments by mixing CH 4 and CD 4 i n d i f f e r e n t concentrat-ions to obtain more precise values of inter-molecular and intra-molecular interactions. (3) It may be of inte r e s t to study proton resonance at some other frequency (say 4-5 Mc/sec.) . The r e s u l t s should show the same dependence i n temperature as do our r e s u l t s . This w i l l provide a check on our r e s u l t s . (4) Coir a c t i v a t i o n energies are i n disagreement with (21) previous measurements for CH 4 . This lack of agreement between our values and those obtained by Professor Waugh from (4) the line-width data of Thomas, Alpert and Torrey i s probably due to the change i n l i n e shape with temperature, which i s not taken into account i n t h e i r data. However, steady state experiments on methane should be repeated to check whether or not t h i s conjecture i s true. If i t i s true, many of the published a c t i v a t i o n energies derived from line-width measurements may have to be re-examined. We have shown that, with the equipment constructed for n.m.r. work, we have been able to gain insight into some aspects of relaxation mechanisms i n methane and i t s -88-deuterated modifications. Some work along the suggested l i n e s i s already being done by students under the supervision of Dr. M. Bloom but further work w i l l have to be done to get a cle a r picture of the problem. -89-BIBLIOGRAPHY 1. F. Bloch, Phys. Rev., 70, 460, 1946. 2. H. S. Gutowsky, I. J. Lawrenson, and K. Shimomura, Phys. Rev. Letters, 6, 349, 1961. 3. C. S. Johnson and J. S. Waugh, J. Chem. Phys., 35, 2020, 1961. 4. J. T. Thomas, N. L. Alpert, H. C. Torrey, J. Chem. Phys. 1_8, 1511, 1950. 5. H. Waterman, Thesis, U.B.C., 1954. 6. M. Bloom, E. L. Hahn and B. Herzog, Phys. Rev., 97, 1699, 1955. 7. N. Bloembergen, E. Purc e l l and R. Pound, Phys. Rev., 73, 679, 1948. 8. G. T. Armstrong, F. G. Brickwedde and R. B. Scott, J. Research Natl. Bur. Standards, 55, 39, 1955. 9. J. H. Simpson and N. Y. Carr, Phys. Rev., I l l , 1201, 1958. 10. L. F. Ehrke and CM. Slack, J. Appl. Phys., 11, 129, 1940. 11. H. S. Sandhu, J. Lees and M. Bloom, Can. J. Chem., 38, 493, 1960. 12. M. Bloom - Private Communication 13. E. L. Hahn, Phys. Rev., 80, 580, 1950. -90-14. F. Bloch and A. Siegert, Phys. Rev., 57, 522, 1940 15. A. W. Maue, Ann. de Physik, 30, 555, 1937. 16. P. S. Hubbard, Phys. Rev., 109, 1153, 1958. 17. A. Abragam, The P r i n c i p l e s of Nuclear Magnetism. (Oxford University Press, London, 1961) 18. G. de Wit, Thesis, U.B.C., 1963. 19. B. H. Muller and J. D. Noble, J. Chem. Phys., 38, 777, 1963. 20. N. Bloembergen, Physica, 15, 386, 1949. 21. J. S. Waugh, J. Chem. Phys., 26, 966, 1957. 22 . H. M. James and T. A. Keenan, J. Chem. Phys., 31, 12, 1959. 23. J. Lees, B. H. Muller and J. D. Noble, J. Chem. Phys., 34, 341, 1961. 

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