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Nuclear spin relaxation and Overhauser effects in polyatomic gases Dong, Yi-Yam Ronald 1969

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NUCLEAR SPIN RELAXATION AND OVERKAUSER EFFECTS IN POLYATOMIC GASES BY YI-YAM RONALD DONG •B.A.Sc, U n i v e r s i t y o£ Toronto, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1'969 In present ing th i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the 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 for reference and Study. I further agree that permission for extensive copying of this thesis for s c h o l a r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or p u b l i c a t i o n of this thes,is for f i n a n c i a l gain sha l l not be allowed without my wri t ten permiss ion. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 3 ^ ABSTRACT Using modern s i g n a l averaging techniques, the proton and f l u o r i n e s p i n - l a t t i c e r e l a x a t i o n time T^ has been measured, i n CH^, CF^, CHF^, ^H^F, and CF^Cl gases at low d e n s i t i e s and 297°K. By measuring, the dependence of T^ on de n s i t y near the c h a r a c t e r i s t i c T^ minimum, we have been able to o b t a i n new in f o r m a t i o n on the s p i n - r o t a t i o n i n t e r -a c t i o n c o u p l i n g constants i n CF^,' CHF^, and CH^F. The CH^ system was used to t e s t the v a l i d i t y of t h i s method s i n c e the s p i n - r o t a t i o n c o u p l i n g constants are a c c u r a t e l y known f o r CH^. The c o r r e l a t i o n f u n c t i o n f o r the s p i n - r o t a t i o n i n t e r a c t i o n was found to be exponential w i t h i n e x p e r i -mental e r r o r f o r a l l of the molecules s t u d i e d . The temperature dependence of the f l u o r i n e T^ i n CHF^, C-^F, and CF^ was i n v e s t i g a t e d at higher d e n s i t i e s . The proton T^ i n CHF^ and CH^F have a l s o been s t u d i e d i n the same d e n s i t y r e g i o n and at s e v e r a l temperatures. A very s t r i k i n g d e n s i t y dependence of the proton T^ i n these two symmetric-top molecules was discovered. A p l o t o f T^/p versus p shows "steps". Steady-state Overhauser e f f e c t s have been s t u d i e d i n experiments performed at 297°K i n both CHF^ and CH^F gases to demon-s t r a t e the importance of the i n t r a - m o l e c u l a r magnetic d i p o l a r i n t e r -a c t i o n at moderate d e n s i t i e s . This i n t e r a c t i o n i n CHF^ and CFI^F i s found to be r e s p o n s i b l e f o r the p e c u l i a r d e n s i t y dependence of the proton T^. A phenomenological i n t e r p r e t a t i o n of the above proton r e s u l t s was given using a high temperature approximation r e l a x a t i o n theory i n which the c o r r e l a t i o n s between the spin-dependent i n t e r a c t i o n s of th d i f f e r e n t n u c l e i and the existence of three d i s t i n c t molecular symmetry species i n CX^Y molecules were p r o p e r l y accounted f o r . A d e t a i l e d molecular theory f o r polyatomic molecules i s s t i l l needed to e x t r a c t i n f o r m a t i o n on the a n i s o t r o p i c p a r t of the i n t e r - m o l e c u l a r p o t e n t i a l . - i v -TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES - .. . v i i LIST OF ILLUSTRATIONS v i i i ACKNOWLEDGEMENTS x i CHAPTER I INTRODUCTION 1 CHAPTER I I EXPERIMENTAL METHODS AND APPARATUS 10 11.1 Measurement of S p i n - l a t t i c e R e l a x a t i o n Time T 10 1 11.2 Measurement of Steady-state Overhauser E f f e c t s 13 I I . 3 Apparatus ; 16 i ) General 16 i i ) Pulsed NMR Spectrometers 17 a) Timing C i r c u i t 17 b) Transmitters , 19 '. c) Receivers 20 d) Boxcar I n t e g r a t o r and Fabri-Tek Instrument Computer 21 e) Sample C o i l and Sample Holder 21 f ) Heater and Temperature Measurements .. 22 I I . 4 Sample P u r i t y and Density Measurements' 22 CHAPTER I I I THEORY OF SPIN-LATTICE RELAXATION 25 I I I . 1 Intra-molecular I n t e r a c t i o n s ' 25 i ) Magnetic D i p o l a r I n t e r a c t i o n 26 i i ) S p i n - r o t a t i o n I n t e r a c t i o n 33 - v -Page I I I . 2 Inter-molecular I n t e r a c t i o n 44 CHAPTER IV LOW DENSITY EXPERIMENTAL RESULTS AND DISCUSSION 4 7 •I.V..1 Methane 5 0 i ) The T Minimum i n CH 4 5 0 IV. 2" Carbon T e t r a f l u o r i d e 5 4 i ) The-T^ Minimum, i n CF^ ...... 54 IV. 3 Fluoroform 60 i ) Proton T Measurements i n CHF^ 60 i i ) The F l u o r i n e T^ Minimum i n CHF^ 62 IV. 4 Methyl F l u o r i d e 6 6 i ) F l u o r i n e T^ Measurements i n CH^F 66 i i ) The proton T^ Minimum i n Clt^F 66 IV. 5 Chl o r o t r i f l u o r o m e t h a n e 7* i ) F l u o r i n e T^ Measurements i n CF^Cl .- 71 CHAPTER V PHENOMENOLOGICAL THEORY OF SPIN RELAXATION FOR CX 3Y MOLECULES 74 V. l Nuclear Spin Symmetry and T r a n s i t i o n P r o b a b i l -i t i e s 74 V.2 R e l a x a t i o n M a t r i x f o r CX^Y Molecules 79 V.3 D i s c u s s i o n f o r CHF 3 . . .' 8 7 V.4 D i s c u s s i o n f o r CH^F 91 CHAPTER VI EXPERIMENTAL RESULTS AT HIGHER DENSITIES AND DISCUSSION 9 5 - v i -Page VI. 1 General Remarks 95 VI.2 Carbon T e t r a f l u o r i d e 100 i ) F l u o r i n e Measurements 100 VI.3 Fluoroform 103 i ) F l u o r i n e Measurements 103 i i ) Proton Measurements 103 i i i ) Overhauser Measurements I l l i v ) I n t e r p r e t a t i o n 114 VI.4 Methyl F l u o r i d e 119 i ) F l u o r i n e Measurements 119 i i ) 'Proton and Overhauser Measurements ... 122 i i i ) I n t e r p r e t a t i o n 127 CHAPTER VII SUMMARY AND SUGGESTIONS FOR FURTHER WORK ... 130 Appendix A C i r c u i t D e t a i l s of the Pulsed Spectrometer .. 134 Appendix B E v a l u a t i o n of <f(J,K)> -.• 144 Appendix C D i p o l a r Hamiltonian f o r CX^Y Molecules i n Convenient Form 146 Appendix D E q u i l i b r i u m Magnetizations f o r CX^Y Molecules ^49 BIBLIOGRAPHY • ' 151 - v i i -LIST OF TABLES Table Page 1. Molecular Constants o f CHF 3 and CH^F... 33 2. Summary of Molecular Parameters and Experimental 2 Values o f C , , 70 e f f . 3. Summary of Various Cross Sections a", ,o, . , a and a 73 re s . 4. Mole c u l a r Wavefunctions of CX^Y Molecules 7 7 5. Submatrix R... . . 8 4 Am;Am' OA 6. Submatrix , E,m; E,m' 7. .Submatrix R. „ , Am;Em' 8. Submatrix R„ „ . E m; E ^m' 9. R e l a x a t i o n M a t r i x R" 10. C o n t r i b u t i o n s to p/T^ of some l o w - l y i n g K t r a n s i t o n s i n CHF 3 at 297°K 85 85 86 120 - v i i i -LIST OF ILLUSTRATIONS Figure Page 1 S i n g l e c o i l arrangement f o r the Overhauser e f f e c t measurements 15 2 Block diagram of 30 MHz pulsed spectrometer 18 3 Proton T^ versus d e n s i t y f o r CH^ at room tempera-t u r e and low d e n s i t i e s . ... 51 4 P l o t of ( l - 4 . 6 8 d T 1 / T 1 ) " 1 versus ( l + p 2 / p 2 . ) _ 1 f o r CH^ at room temperature 55 5 F l u o r i n e T^ versus d e n s i t y f o r CF^ at room tempera-t u r e and low d e n s i t i e s . . 56 6 Proton T^ versus d e n s i t y f o r CHF^ at room tempera-t u r e and low d e n s i t i e s 61 7 F l u o r i n e T^ versus d e n s i t y f o r CHF^ at room tempera-t u r e and low d e n s i t i e s 63 8 P l o t o f Cjj versus f o r the f l u o r i n e spins i n CHF 3 65 9 F l u o r i n e T^ versus d e n s i t y f o r CH^F at room tempera-t u r e and low d e n s i t i e s 67 10 Proton T^ versus d e n s i t y f o r CH^F at room tempera-t u r e and low d e n s i t i e s 68 11 F l u o r i n e T^ versus d e n s i t y f o r CGIF^ at room tempera-tur e 72 12 F l u o r i n e as a f u n c t i o n of d e n s i t y f o r CF^ at room temperature - i x -Figure Page 13 Temperature dependence of T^/p f o r CF^ 102 14 F l u o r i n e T as a f u n c t i o n of de n s i t y f o r CHF at room temperature 104 15 Temperature dependence of f l u o r i n e T^/p f o r CHF . . 105 16 Proton T^ as a f u n c t i o n of d e n s i t y for. CHF^ at room temperature 106 17 P l o t of T /p versus p f o r CHF3- The value of T^/p f o r the f l u o r i n e spins i s 3.5 msec/Amagat 109 18 Temperature dependence of proton T^/p f o r CHF^ at low d e n s i t i e s H O 19 Proton T^ versus p f o r CHF^-He and CHF^-Ar mixtures at room temperature H 2 20 Overhauser e f f e c t measurements f o r CHF at room temperature H 3 21 F l u o r i n e T^ as a f u n c t i o n of d e n s i t y f o r CH^F at room temperature 121 22 Temperature dependence of f l u o r i n e T^/p f o r CH F 1 2 3 23 Proton T^ as a f u n c t i o n of d e n s i t y f o r CH^F at room temperature -^4 24 P l o t , of T /p versus p f o r CH 3F ' 1 2 5 25 Temperature dependence of proton T^/p f o r CH 3F at low d e n s i t i e s 26 Overhauser e f f e c t measurements f o r CH 3F at room 1 28 temperature Figure Page A l . A2 136 A3 137 A4 138 A5 ... 139 A6 140 A7 141 AS 142 A9 143 CI 148 - XI -ACKNOWLEDGEMENTS I wish to express my si n c e r e g r a t i t u d e to Professo r Myer Bloom f o r h i s guidance and encouragement throughout the work. I would l i k e to thank Professor S. Alexander f o r h i s s t i m u l a t i n g d i s c u s s i o n s and h i s i n t e r e s t i n t h i s work i n the past two years. Thanks are due to Mr. W i l l i a m Morrison f o r c o n s t r u c t i n g the sample holder and h i s ready help and to Mr. John Lees, our glassblower, f o r h i s a s s i s t a n c e i n t h i s work. The f i n a n c i a l support provided by the N a t i o n a l Research C o u n c i l of Canada i n the form of s c h o l a r s h i p s over the past, three years i s g r a t e f u l l y acknowledged. The research was supported by the N a t i o n a l Research C o u n c i l of Canada. CHAPTER I INTRODUCTION The techniques of Nuclear Magnetic Resonance (NMR) have been a powerful t o o l i n the study of molecular c o l l i s i o n s i n gases (e.g., Bloom 1957, Hardy 1966, Bloom and Oppenheim 1967, Johnson J r . et a l . 1961, Armstrong et a l . 1968). I t i s the cou p l i n g between the nuclear spins and the molecular degrees of freedom which f a c i l i t a t e s the i n v e s t i g a t i o n of p r o p e r t i e s i n molecules using the nuclear moment as a probe. The aim of t h i s work i s to e x t r a c t q u a n t i t a t i v e molecular i n f o r m a t i o n from measurements of the r a t e at which the d i s t u r b e d nuclear s p i n system approaches the thermodynamic e q u i l i b r i u m w i t h i t s surround-i n g s . S i m i l a r i n f o r m a t i o n can be, i n p r i n c i p l e , obtained from the study of microwave pressure broadening in. nonresonant or r e l a x a t i o n s p e c t r a (Birnbaum 1967). Indeed the c o l l i s i o n diameter obtained from nonresonant absorption i n gases a r i s i n g from e l e c t r i c d i p o l a r t r a n s i -t i o n s should be s i m i l a r to th a t determined by NMR methods (Gordon 1966). The s p i n - l a t t i c e r e l a x a t i o n time T^  i n gases with r o t a t i o n a l degrees of freedom depends on the mean c o r r e l a t i o n time x c between c o l l i s i o n s , the Larmor frequency oi and the stren g t h of the spin-dependent molecular i n t e r a c t i o n s . At very low d e n s i t i e s where T £ i s comparable to the Larmor p r e c e s s i o n p e r i o d ( i . e . w T ^ 1) , a c h a r a c t e r i s t i c minimum i n T, e x i s t s (Hardy 1966, Dorothy 1967) which i s very s e n s i t i v e to the i n t r a - m o l e c u l a r i n t e r a c t i o n s . Thus the d e n s i t y dependence of i n t h i s r e gion provides i n f o r m a t i o n on the coupling constants of the i n t e r a c t i o n s . In most molecular f l u i d s , the T^ may be w r i t t e n (1.1) = RA' + R B + R c where R i s the r e l a x a t i o n r a t e due to i n t r a - m o l e c u l a r d i p o l a r and A. quadrupolar ( f o r I > 1/2 only) i n t e r a c t i o n s i . e . , those i n t e r a c t i o n s which transform as Y„ (Q) under r o t a t i o n s of the molecule where 0. denotes 2mv J the o r i e n t a t i o n o f . a vector f i x e d to the molecule, R n i s the r e l a x a t i o n D r a t e due to i n t e r - m o l e c u l a r d i p o l a r i n t e r a c t i o n s and R^ , i s the r e l a x a -t i o n r a t e due to s p i n - r o t a t i o n i n t e r a c t i o n s , whose dominant term transforms as J , the r o t a t i o n a l angular momentum. In molecular gases which are not too dense, R n i s n e g l i g i b l e . The i n t r a - m o l e c u l a r i n t e r -is a c t i o n s (denoted by ) i n R and R„ are modulated i n time through f l u c t u a t i o n s i n ^2m^ a n c^ ^ ^ ^ u e t 0 m 0 i e c u i a r c o l l i s i o n s which cause t r a n s i t i o n s between molecular s t a t e s having d i f f e r e n t quantum numbers, denoted by J , K and M f o r symmetric-top and s p h e r i c a l - t o p molecules. Using a p e r t u r b a t i o n expansion i n the small c o u p l i n g term ', the nuclear s p i n r e l a x a t i o n time T^ can be c a l c u l a t e d i n terms of the F o u r i e r transforms of the c o r r e l a t i o n f u n c t i o n s of J ( s J + + x i J ) and Y„ (Q). These c o r r e l a t i o n f u n c t i o n s c o n t a i n terms that are y J 2mv ' independent of time and, i n a d d i t i o n , terms that o s c i l l a t e at high frequencies a r i s i n g from molecular t r a n s i t i o n s between non-degenerate l e v e l s . For very d i l u t e gas, only the time-independent terms are - 3 -important i n r e l a x i n g nuclear s p i n system and the high-frequency terms must be dropped i n s o f a r as nuclear s p i n r e l a x a t i o n i s concerned. This reduces the c o n t r i b u t i o n of the d i p o l a r i n t e r a c t i o n s to the s p i n r e l a x a t i o n r a t e i n s p h e r i c a l - t o p molecules by a f a c t o r of e x a c t l y 5 . This r e d u c t i o n f a c t o r i n symmetric-top molecules can be much grea t e r . As the d e n s i t y of a gas i s i n c r e a s e d , the r o t a t i o n a l l e v e l s of molecules broaden more and more owing to i n c r e a s i n g l y frequent molecular c o l l i s i o n s . When groups of r o t a t i o n a l l e v e l s are c o l l i s i o n broadened by an amount of order of the c h a r a c t e r i s t i c frequencies of the o s c i l l a t i n g terms, the s p i n r e l a x a t i o n r a t e should gain a d d i t i o n a l c o n t r i b u t i o n from these high-frequency terms. A study of i n two symmetric-top molecules CHF^ and CTijF at moderate d e n s i t i e s was made to l o o k . f o r t h i s e f f e c t . I t i s b e l i e v e d t h a t , as i n the case of diatomic gases (Bloom and Oppenheim 1 9 6 7 ) , T^ measurements i n polyatomic molecules can be u s e f u l i n determining a n i s o t r o p i c i n t e r m o l e c u l a r p o t e n t i a l s . A complete d i s c u s s i o n of the p r i n c i p l e s of nuclear magnetism has been given by Abragam (1961) and- only an o u t l i n e i s given below f o r readers not f a m i l i a r w i t h the NMR. An e x t e r n a l magnetic f i e l d H along the Z a x i s a p p l i e d t o a nucleus with s p i n angular momentum I and magnetic moment "p =y#I produces an i n t e r a c t i o n Hamiltonian given by j{ = .-lM? = -ynH I ^ o ' o z where y i s the gyromagnetic r a t i o . The allowed energies are ( 1 . 2 ) - 4 -(1.3) E = -YfiI-1 m v J m 1 o where m = - I , - I + 1, . .., I . The f r a c t i o n a l populations of these energy l e v e l s at e q u i l i b r i u m are given by (1.4) -P. exp (-E A T ) m m Z exp (-En)/kT) m where k i s the Boltzmann constant and T i s the absolute temperature of the gas. The e q u i l i b r i u m magnetization f o r a system of N weakly i n t e r a c t e d spins i s given by • (1.5) M = NEP Yfm v J o m' m = N Y 2 r i 2 I ( I + l)/3kT where the high temperature approximation y ^  ^c/^T. K < ^ u s e c*' The approach to e q u i l i b r i u m of the magnetization a f t e r being d i s t u r b e d i s o f t e n d e s c r i b a b l e by the phenomenological Bloch equations -» M i + M j M - M (1.6) ^ ^ yM x ^—i— * — k where T^ and 1 ^ are l o n g i t u d i n a l and tra n s v e r s e r e l a x a t i o n times. I f r e l a x a t i o n e f f e c t s are negl e c t e d , the equation o f motion f o r M i n a refer e n c e frame r o t a t i n g w i t h angular frequency to may be w r i t t e n (1.7) — = YM x (H + - ) M i s s t a t i o n a r y i n the r o t a t i n g frame f o r H = H k i f u = -yH k the • o . ' o. angular frequency w = ~yH i s c a l l e d the "Larmor frequency". I f H now c o n s i s t s of a s t a t i c magnetic f i e l d H k and a time v a r y i n g f i e l d (t) where (1.8) H^(t) = H^[ i cos wt + j s i n cot] represents the a p p l i e d radio-frequency magnetic, f i e l d , the equation of motion f o r M i n the r o t a t i n g frame with the X-axis along H^(t) i s (1.9) = Y M x [ ( H q + ^ ) k + H ^ i ) = Y M x H j . ~ \ e f f where (1.10) H = (H +— ) k + H.i \ 1 e f f o y 1 I t can be seen from equation (1.9) that M precesses about H ^ with an angular frequency "YUgf-f When w = coo, = H^i and the magnetization t h e r e f o r e precesses about H^. This phenomenon i s nuclear magnetic resonance. I f the time v a r y i n g f i e l d i s a p p l i e d f o r a time t , the M w i l l process at resonance about through an angle 0 = yH^t w. and the pulse width t can be chosen so that 6 = 180° or 90°. The former 1 w i s termed as a u pulse and the l a t t e r , a IT/2 p u l s e . A f t e r the a p p l i c a -t i o n of a IT/2 p u l s e , M precesses i n the X-Y plane about H and - 6 -induces an e.m.f. i n the c o i l which i s r e f e r r e d to as a f r e e i n d u c t i o n decay. I f a IT p u l s e i s a p p l i e d at a time t = T a f t e r the I T/2 p u l s e , the spins tend to repha.se i n the X-Y plane at a time t = 2x r e s u l t i n g i n a f r e e i n d u c t i o n ^signal known as a s p i n echo. When r e l a x a t i o n e f f e c t s are considered, the s o l u t i o n s to eq. (1.6) f o r (1.11) H = kll + i 2I-Lcos tot , I-L « H v J o 1 1 o are found to be (1.12) M x ( t ) = M (0) [cos (cot + <)>)] exp ( - t / T 2 ) (1.13) H ft) = M (0) [sin(wt + <j>)] exp (-t/T ) y xy i. ( 1 . 1 4 ) M z ( t ) = M q + [M z(0) - M q ] exp (-t/T^ where (1.15) M = / M 2 + M 2  K xy . x y Hence and can be determined experimentally by observing M^(t) and M x ( t ) (or M ( t ) ) , r e s p e c t i v e l y , as a f u n c t i o n of time. In t h i s t h e s i s , only the measurement of and i t s i m p l i c a t i o n s w i l l be discussed. When two d i f f e r e n t kinds of spin-1/2 n u c l e i , and I and S, are c o n s t i t u e n t s of molecules of a molecular f l u i d and i f magnetic d i p o l a r - 7 -i n t e r a c t i o n s are important between I and S, the l o n g i t u d i n a l r e l a x a t i o n of each type of s p i n i s . i n ge n e r a l , not governed by a s i n g l e time constant and, furthermore, an Overhauser e f f e c t occurs. To w i t h i n our experimental err o r , . b o t h proton and f l u o r i n e T^ i n CHF^ and CH F obey an exponential r e l a x a t i o n i n the d e n s i t y r e g i o n s t u d i e d . The f l u o r i n e spins i n these gases are predominantly r e l a x e d by s p i n - r o t a t i o n i n t e r -a c t i o n and hence an exponential r e l a x a t i o n i s expected. A q u a n t i t a t i v e measure of an Overhauser e f f e c t i s to examine the f r a c t i o n a l change of the s t e a d y - s t a t e magnetization of one kind o f nucleus from i t s e q u i l i -brium value when the other k i n d i s saturated by radio-frequency ( r . f . ) power at i t s Larmor frequency. The magnitude of the steady-state Over-hauser e f f e c t depends on a l l r e l a x a t i o n mechanisms between the two s p i n systems and among themselves, but f o r a. non-zero e f f e c t p r i m a r i l y depends on those i n t e r a c t i o n s which couple the d i f f e r e n t types of s p i n and produce s p i n t r a n s i t i o n s among them, such as magnetic d i p o l a r i n t e r a c t i o n s . The f l u o r i n e s t e a d y - s t a t e magnetization i n both CHF^ and CH^F i s unaffe c t e d by whether or not the proton spins are i r r a d i a t e d because the f l u o r i n e spins are r e l a x e d s t r o n g l y by s p i n - r o t a t i o n i n t e r a c t i o n . We have observed the Overhauser e f f e c t s of the proton i n these'gases at room temperature. This enabled us to separate the c o n t r i b u t i o n s from s p i n - r o t a t i o n and magnetic d i p o l a r i n t e r a c t i o n s to the proton s p i n r e l a x a t i o n . S i m i l a r double resonance experiments (both s t e a d y - s t a t e and t r a n s i e n t Overhauser e f f e c t s ) on l i q u i d CHF^, have been done p r e v i o u s l y by C h a f f i n , I I I and Hubbard (1967). In Chapter I I , we present the experimental methods and apparatus. The term R and R i n eq. (1.1) are discussed i n d e t a i l f o r - . 8 -polyatomic molecules i n Chapter I I I . Equivalent spins i n a molecule are t r e a t e d as independent of each other i n spin-dependent i n t e r a c t i o n . Spin r e l a x a t i o n theory aiming to i n c l u d e c o r r e l a t i o n between equivalent spins i s formulated l a t e r i n Chapter V. In Chapter IV, the low de n s i t y r e s u l t s i n Cti^, CF'4, CHF , CH F, and CF 3C1 at 297°K and t h e i r i n t e r -p r e t a t i o n s using the theory o f Chapter I I I are presented. The determin-a t i o n of s p i n - r o t a t i o n constants i n methane and i t s f l u o r i n a t e d m o d i f i c a -t i o n s i s given. Symmetric-top molecules of the form CX Y have :C^ molecular symmetry. When nuclear s p i n symmetry e f f e c t s are considered, t h e i r r o t a t i o n a l s t a t e s can be c l a s s i f i e d according to the A, E^ and rep r e s e n t a t i o n s of the C^ r o t a t i o n p o i n t group. A l l s t a t e s w i t h K = 3n belong to the A r e p r e s e n t a t i o n and have the X-type s p i n i n s t a t e s w i t h t o t a l n uclear s p i n I = 3/2. States and E^ have K = 3n + 1 and -(3n + 1 ) , r e s p e c t i v e l y , and both have I = 1/2. The e f f e c t s of nucl e a r s p i n symmetry and molecular symmetry i n nuclear s p i n r e l a x a t i o n are discussed i n Chapter V from a phenomenological approach. The theory should be a p p l i c a b l e i n general to any CX^Y molecule c o n t a i n i n g spin-1/2 n u c l e i . The f l u o r i n e T^ data and the s t r i k i n g proton T^ r e s u l t s at se v e r a l temperatures i n both CHF^ and CH^F and t h e i r Overhauser e f f e c t measurements are presented i n Chapter VI. The phenomenological theory of nuclear s p i n r e l a x a t i o n developed i n Chapter V i s used to i n t e r p r e t * _e. K i s the p r o j e c t i o n of the angular momentum J on the symmetry a x i s of a molecule. - 9 -the proton data and the Overhauser e f f e c t s i n these two gases. F i n a l l y , a summary of the main r e s u l t s of t h i s t h e s i s and suggestions f o r f u r t h e r s t u d i e s are given i n Chapter V I I . Appendix A contains c i r c u i t d e t a i l s of the pulsed spectrometer. Appendix B contains e v a l u a t i o n of <f(J,K)> which i s defined as CO j (1.16) <f(J,K)> = I E (2J + 1)P f ( J , K ) J=0 K=-J J K M Appendix C contains a d e r i v a t i o n of d i p o l a r Hamiltonian f o r CX^Y molecules i n convenient form. Appendix D contains e v a l u a t i o n of e q u i l i b r i u m magnetizations f o r CX Y molecules. CHAPTER I I EXPERIMENTAL METHODS AND APPARATUS II . 1 . Measurement of S p i n - l a t t i c e R e l a x a t i o n Time T^ The method of measuring s p i n - l a t t i c e r e l a x a t i o n time T^ by the pulse technique i s w e l l - e s t a b l i s h e d i n the l i t e r a t u r e (Abragam 1961, Halm 1950) and w i l l be described here b r i e f l y . When a bulk sample c o n t a i n i n g n u c l e a r moments i s placed i n a constant magnetic f i e l d H along the z a x i s say, a thermodynamic e q u i l i b r i u m s t a t e i s a t t a i n e d with a macroscopic magnetization M along H . To measure T^, the system must be d i s t u r b e d from e q u i l i b r i u m . As shown i n the previous chapter, the magnetization M q can be r o t a t e d from the z a x i s through any d e s i r e d angle 6 by app l y i n g a radio-frequency f i e l d H^ perpendicular to H at the Larmor frequency of the s p i n s . The nuclear s p i n system, a f t e r being perturbed, w i l l r e l a x to e q u i l i b r i u m w i t h a time constant T^ by exchanging energy with i t s surroundings l o o s e l y c a l l e d the " l a t t i c e " . The r . f . pulse used to t i p M must be intense enough to r o t a t e i t i n a time much s h o r t e r than T^ or T so that the r e l a x a t i o n e f f e c t s w h i l e 1-1^  i s on can be neglected. The inhomogeneity of the r . f . f i e l d H^ over the volume of the sample r e s u l t s i n a r o t a t i o n of M Q which i s not " c l e a n " , i . e . , one obtains a d i s t r i b u t i o n of angles 0. Care has been taken to e l i m i n a t e any sample outside the sample c o i l to ob t a i n a " c l e a n " r o t a t i o n . However, the "unclean" r o t a t i o n by no means e f f e c t s the measurement but does e f f e c t the s i g n a l amplitude. Values of s h o r t e r than or equal to one second were measured by f i r s t i n v e r t i n g M (a TT pulse) and then monitoring the recovery of the from -M to the e q u i l i b r i u m value +Mo by subsequent a p p l i c a t i o n of a TT/2 and a ?r pulses separated by a short time x. The s p i n echo, whose TT magnitude i s p r o p o r t i o n a l to M j u s t before the - j p u l s e , forms at a time 2x a f t e r the — pulse. The three-pulse sequence was repeated a f t e r a time' long compared with T^ (> 10T^) so that the nuclear s p i n system was always i n e q u i l i b r i u m at the s t a r t of a sequence. A "boxcar i n t e g r a t o r " (Blume 1961) was used to improve the s i g n a l to noise 77 r a t i o . During a T^ measurement, the time of a p p l i c a t i o n of the — - TT sequence with f i x e d x was swept very s l o w l y from the f i r s t T: p ulse to s e v e r a l T^. The y pulse was used to t r i g g e r the boxcar i n t e g r a t o r and the sample gate which was adjusted to sample the maximum echo amplitude. The echo amplitude and thus the recovery of was p l o t t e d on a V a r i a n s t r i p - c h a r t recorder. The time t between the f i r s t TT pulse and the ^ pulse was measured with a Hewlett-Packard 524-C e l e c t r o n i c counter and the time was p r i n t e d out a u t o m a t i c a l l y by a Hewlett-Packard 526-B d i g i t a l recorder which.simultaneously t r i g g e r e d the event-marker of the s t r i p - c h a r t recorder. The M at various times could then be i d e n t i f i e d . For an exponential approach to e q u i l i b r i u m , the magnitude of the echo as a f u n c t i o n of time i s described by (2.1) M z ( t ) =.MQ + (M z(0) - M Q) exp (-t/T^ where M i s obtained by s e t t i n g t to i n f i n i t y i . e . , > 10T . A p l o t of - 12 -l°g (M 0 - M ( t ) ) against t then gives a s t r a i g h t l i n e with a negative slope equal to 'T^'. When was greater than one second, the procedure d e s c r i b e d above became tedious and time consuming, and a d i f f e r e n t p u l s e sequence was used. In order to avoid the need of w a i t i n g ten a f t e r each measurement, M (0) was decreased to zero u s i n g a sequence IT of c l o s e l y spaced (> 10 in sec) — pulses (about 40) before each measure-ment. The recovery of M at a l a t e r time t was again monitored by a — - IT sequence. The boxcar i n t e g r a t o r was incapable of measuring T^ gr e a t e r than one second owing to the l i m i t a t i o n of i t s short h o l d i n g time (^  10 minutes). Hence the echo was recorded on p o l a r o i d f i l m u s i n g a DuMont type 2620 camera. The time t between the end of the burst of y pulses and the monitoring pulse was again measured by a Hewlett-Packard e l e c t r o n i c counter. The photographic r e c o r d i n g does not provide as much s i g n a l averaging as the boxcar so t h a t the T^ measurements are not as accurate. This was remedied i n the l a t t e r part of t h i s work by using a Fabri-Tek instrument computer which i s j u s t a s i g n a l averager l i k e a boxcar i n t e g r a t o r , but w i t h an i n f i n i t e h o l d i n time. With a Fabri-Tek instrument computer,' the echo can be repeated s e v e r a l times and averaged to improve s i g n a l to noise d e s p i t e long T^ values. The number of times the echo must be repeated depends on the d e s i r e d accuracy and on the length of time one i s w i l l i n g to spend on a T^ measurement. Long T^ measurements were made using both methods and were found to be c o n s i s t e n t w i t h each other. At very low d e n s i t i e s i t i s again unable to make T^ measurements using the boxcar because of i t s l i m i t e d h o l d i n g time and the long time r e q u i r e d to e x t r a c t the very weak s i g n a l from the n o i s e . However, the Fabri-Tek instrument computer was i d e a l l y s u i t e d to making T, measurements at these low - 13 -d e n s i t i e s . Owing to the small at low d e n s i t i e s , the echo can be 4 repeated many many times (up to 10 ) i n a reasonable time to achieve the d e s i r e d accuracy of 5% i n each measurement. In t h i s r e s p e c t , the Fabri-Tek instrument computer has proven to be a very powerful t o o l f o r pulsed NMR experiments. II . 2 Measurement of Steady-state Overhauser E f f e c t s Mien two d i f f e r e n t types of nuclear spins are i n a molecule, an Overhauser e f f e c t may occur. The st e a d y - s t a t e Overhausereffect measurements discussed here i n v o l v e s a t u r a t i n g f l u o r i n e spins u n t i l the proton spins reach a stea d y - s t a t e i n the new environment i n both gaseous CHF^ and CH^F. The proton magnetization thus obtained i s compared with i t s e q u i l i b r i u m value to o b t a i n the f r a c t i o n a l change. A gated power t r a n s m i t t e r operating at the f l u o r i n e Larmor frequency of 28.2 MHz i s used to pulse the f l u o r i n e s p i n to i n f i n i t e temperature. I t can be shown that the most e f f e c t i v e pulses f o r satux-ating the s p i n system correspond to IT p u l s e s . The temporal s e p a r a t i o n between TT pulses must be short compared to the f l u o r i n e T^ so t h a t the f l u o r i n e spins dp not a p p r e c i a b l y r e l a x towards ther/modynamic e q u i l i b r i u m . This temporal s e p a r a t i o n i n our experiments was set to 10 m sec, which i s much sm a l l e r than the f l u o r i n e T^ at a l l the d e n s i t i e s we encountered i n the Overhauser e f f e c t measurements. The TT p u l s e width was chosen by minimizing the f l u o r i n e f r e e i n d u c t i o n s i g n a l i n a s i n g l e pulse experiment and was found to vax~y between 15-30 y sec depending on the sample d e n s i t y . The reason f o r t h i s v a r i a t i o n was th a t the impedance of the sample c o i l changed with d e n s i t y - 14 -because both CHF^ and CH^F are p o l a r molecules. When the r e p e t i t i o n -r a t e o f the TT pulses was set to 10 rn sec, the f r e e i n d u c t i o n s i g n a l of the f l u o r i n e spins was too small to be detected so th a t w i t h i n experimental e r r o r , „its s p i n temperature was i n f i n i t e . Instead o f usin g separate c o i l s f o r feeding r . f . power i n t o the sample at the proton and f l u o r i n e resonance f r e q u e n c i e s , which would have r e q u i r e d a c r o s s e d - c o i l set up, we used a s i n g l e c o i l i n which the r . f . f i e l d H^ was generated f o r the f l u o r i n e and the proton resonances at d i f f e r e n t times. In order to improve the s i g n a l to noise of the proton echoes, which had to be measured i n order to measure the magnitude o f the Overhauser e f f e c t , the c o i l was always tuned to the proton frequency with a v a r i a b l e c a p a c i t o r i n p a r a l l e l . Since the Larmor frequencies of the f l u o r i n e and proton spins i n a magnetic f i e l d of about 7000 gauss only d i f f e r by 1.8 MHz, the resonant tank c i r c u i t i s tuned c l o s e enough to the f l u o r i n e r . f . to enable a r e l a t i v e l y narrow TT pulse width to be used. A block diagram showing the s i n g l e c o i l arrangement i n these measurements i s given i n Figure 1. Se r i e s tuned c i r c u i t s were used to give a low impedance at one frequency and very high impedance at the other. H a l f wavelength t r a n s m i s s i o n l i n e s were used between the outputs o f the gated t r a n s m i t t e r s to the s e r i e s tuned c i r c u i t s to minimize r . f . power l o s s i n the l i n e . When s a t u r a t i o n of the f l u o r i n e spins was not d e s i r e d , the ga t i n g pulses to the 28.2 MHz t r a n s m i t t e r were cut o f f manually. The e q u i l i b r i u m proton echoes f o r both the f l u o r i n e being saturated and not saturated were repeated a few times ( u s u a l l y 8 or 16 samples f o r each) and averaged s e p a r a t e l y u s i n g the Fabri-Tek instrument computer. The percentage increase of the proton s i g n a l s was thus determined. The experiment was repeated gating pulses 30 MHZ transmitter gating pulses 28.2 MHZ transmitter tuned to 30 MHZ 3.3 pf t u ned to; 28.2MHZ' 7.5pf 3.3pf 12pf mag net sample c o i l arnpti f ier i Figure 1. Single c o i l arrangement for the Overhauser effect measurements. - 16 -s e v e r a l times to o b t a i n higher accuracy. Due to the long proton T^  , t h i s procedure was tedious and time consuming and each measurement of the Overhauser e f f e c t reported i n Chapter VI took about 2 days. The Over-hauser enhancement was measured as a f u n c t i o n of d e n s i t y f o r CHF^ and CH 3F at 297°K. In a d d i t i o n to having an i n f l u e n c e v i a the Overhauser e f f e c t , the proton spins can be a f f e c t e d d i r e c t l y by the f l u o r i n e TT pulses t r a i n f o r the reason that t h e i r Larmor frequencies only d i f f e r by 1.8 MHz. This e f f e c t on the proton magnetization has been checked e x p e r i m e n t a l l y to be unimportant. The experiment i n v o l v e d t u r n i n g on a small b i a s i n g f i e l d along H during the TT p u l s e s . The b i a s i n g f i e l d can be adjusted by monitoring the f l u o r i n e f r e e i n d u c t i o n s i g n a l so that the e f f e c t i v e f i e l d i n the f l u o r i n e r o t a t i n g frame becomes 2H^, 4H^, or 6H^ etc. For these cases the pulses produce no net e f f e c t on the f l u o r i n e spins f o r homogeneous H q and s i n c e they r o t a t e the f l u o r i n e magnetization by an i n t e g r a l m u l t i p l e of 2TT . Measurements at 4H^ f o r the e f f e c t i v e f i e l d are p r e f e r a b l e to measurements at 2H^ s i n c e they reduce the e f f e c t s of the inhomogeneity of H^. The proton magnetization was monitored i n the presence of such a t r a i n o f f l u o r i n e 4-rr pulses and was found to be unchanged from that without any f l u o r i n e pulses to an accuracy of about i 1%. II . 3 Apparatus i ) General The 30 MHz coherent pulsed spectrometer used i n these experiments was o r i g i n a l l y built by J . Noble (1964) and modified by K. L a l i t a (1967) to - 17 -detect weak NMR s i g n a l s using phase-coherent d e t e c t i o n . A c r y s t a l c o n t r o l l e d o s c i l l a t o r and a t r i p l e r stage formed part of the t r a n s m i t t e r . I t could be converted i n t o a 28.2 MHz spectrometer by r e p l a c i n g a new c r y s t a l of 9.4 MHz and r e t i m i n g to the appropriate frequency to measure f l u o r i n e T^. This was necessary, because a permanent magnet was used i n t h i s work. In making T^ measurements, the absence of any long term f i e l d d r i f t o f the permanent magnet was c e r t a i n l y an advantage f o r long T^ measurements though a more homogeneous f i e l d would have been h e l p f u l . In order to measure the Overhauser e f f e c t s , a 28.2 MHz coherent pulsed t r a n s m i t t e r was constructed s i m i l a r to the 30 MHz t r a n s m i t t e r . Minor m o d i f i c a t i o n s were made and the c i r c u i t d e t a i l s are presented i n Appendix A. A block diagram of the spectrometer i s shown i n Figure 2. A l l the major items w i l l be described b r i e f l y . i i ) Pulsed NMR Spectrometers a) Timing C i r c u i t The timing c i r c u i t f o r measuring T^ < 1 sec was i d e n t i c a l to t h a t described i n L a l i t a ' s t h e s i s (1967). For long T^ measurement a pulse sequencer was constructed using FET as shown i n Figure A l . A Te k t r o n i c 162 waveform generator was used i n a r e c u r r e n t mode at 10 m sec r e p e t i t i o n r a t e as input to the pulse sequencer. Two gatin g pulses of v a r i a b l e width were generated manually or r e c u r r e n t l y i n the pulse sequencer t o gate the i n p u t . The gated output o f the f i r s t gating pulse was fed to a IT/2 pulse width generator to produce a burst of I T/2 pulses f o r d e s t r o y i n g M at t = 0. The t r a i l i n g edge of the f i r s t g a t i n g pulse was used to t r i g g e r the s t a r t of the e l e c t r o n i c counter and TEK i> pulse TEK modified modified 162 sequencer 162 TEK163 TEK162 !_ s w i tch pul se sequencer HP. time interval unit A mixer ± repulse w i d t h ge nerator 3 ex. c T T pul se w i d t h ge nerator m i x e r V2 tripler ^ gat i ngjpulses gated power amplifier oscillator! amplifier phase shifter t r ip ler Z 3 a. c CX o in HP. digital recorder eve nt marker boxcar integrator strip-chart recorder tr igger - A 7T FABRI-TEK inst. cornput. CO I amplifier detector A magnet r e f e r e n c e vo l t age Figure 2.. Block diagram of 30 MHz pulsed spectrometer. 19 -i n i t i a t e the second gating pulse whose width could be v a r i e d from zero to t e n minutes. The t r a i l i n g edge of the second gat i n g pulse was then used to t r i g g e r a sawtooth i n a Tek 162, which i n t u r n t r i g g e r e d a p a i r o f - j - TT puls.es as described by L a l i t a (1967) . When making Overhauser measurements, the gated output of the second g a t i n g pulse was fed to a Tek 161 pulse generator whose output was used to gate the 28.2 MHz t r a n s m i t t e r . b) T r a n s m i t t e r s The t r a n s m i t t e r was b u i l t (see Appendix A) to detect weak NMR s i g n a l s using phase-coherent d e t e c t i o n i . e . , to detect s i g n a l s i n the l i n e a r r e g i o n of a diode. A c r y s t a l c o n t r o l l e d o s c i l l a t o r , o s c i l l a t i n g c o n t i n u o u s l y at e i t h e r 9.4 or 10 MHz,was housed i n a r . f . l e a k - t i g h t copper can (Blume 1961) w i t h a s t a b l e wideband a m p l i f i e r and a t r i p l e r . The t r i p l e r was used to generate a r e f e r e n c e marker f o r b i a s i n g the diode at 1-2 v o l t s D.C. p o s i t i v e f o r phase-coherent d e t e c t i o n . A power supply, w e l l - r e g u l a t e d by Zener diodes, was a l s o put i n s i d e the can. A l l u n i t s i n the can were completely t r a n s i t o r i z e d to achieve compactness. The r . f . leakage was kept to a minimum by using a s i n g l e lead to feed a 6.3 v o l t s A.C. supply to the power transformer i n the can. The r . f . power from the c r y s t a l c o n t r o l l e d o s c i l l a t o r , a f t e r being a m p l i f i e d by the wideband a m p l i f i e r , leaked out of the can to a conventional phase s h i f t e r (Clark 1964) and a t r i p l e r whenever a 90 v o l t s gate was a p p l i e d . The output from the t r i p l e r was fed i n t o a gated power a m p l i f i e r stage tuned at 28.2 or 30 MHz. Reasonably - 20 -re c t a n g u l a r r . f . pulses of 1200 v o l t s peak to peak were produced by the power a m p l i f i e r . With t h i s power output, the pulse widths r e q u i r e d to give TT/2 and TT pulses were about 6 y sec and 13 y sec, r e s p e c t i v e l y . The r . f . pulses were taken to the sample c o i l through a small decoupling c a p a c i t o r of 3.3 pf. c) Receivers A commercial low-noise L.E.L. a m p l i f i e r model I.F. 21B.S. was used to detect proton s i g n a l . I t had a vo l t a g e gain of lOOdb with a bandwidth at 3db of 2 MHz, and a center frequency of 30 MHz. A 1N295 diode was used to detect r . f . s i g n a l s at t h i s frequency. Tine 30 MHz reference v o l t a g e to b i a s the diode at two v o l t s p o s i t i v e was introduced to the r . f . s i g n a l at a stage i n the r e c e i v e r where the stagger tuned t r i p l e t stages were coupled together. A low-noise, narrow band 28 MHz a m p l i f i e r was designed and b u i l t by S. Koskennon of T e l e - S i g n a l E l e c t r o n i c s , Vancouver, B.C. This a m p l i f i e r had a bandwidth of 2 MHz at the 3db p o i n t s and a voltage gain of 130 db. I t had a noise f i g u r e of 1.5db and a source impedance of 50 ohms. A 1N541 diode was used i n t h i s r e c e i v e r to detect f l u o r i n e s i g n a l s i n a l l the measurements. The 28.2 MHz reference s i g n a l was i n s e r t e d i n the l a s t a m p l i f i e r stage. This gave a reference voltage which biased the diode at one v o l t p o s i t i v e to assure l i n e a r d e t e c t i o n . The s i g n a l was always kept at a tenth of the reference v o l t a g e to avoid any s i g n a l d i s t o r t i o n . The l i n e a r i t y of these ampli-f i e r s was checked by means of the NMR s i g n a l s i n a gas and a c a l i b r a t e d s i g n a l generator and was found to be l i m i t e d only by the vo l t a g e c h a r a c t e r i s t i c s of the diodes. - 21 -d) Boxcar I n t e g r a t o r and Fabri-Tek Instrument Computer The boxcar i n t e g r a t o r was the same one used by W.N. Hardy based on the design of R.J.Blume (1961) and was thoroughly discussed i n h i s t h e s i s (Hardy 1964). A Fabri-Tek Model 1062 instrument computer was used to make measurements whenever the boxcar was i n a p p l i c -able. This instrument'computer, having a t o t a l memory of 1024 channels, could be d i v i d e d i n t o four quadrants. I t could be t r i g g e r e d to sweep both i n t e r n a l l y and e x t e r n a l l y and made to stop a u t o m a t i c a l l y by s e t t i n g the auto-stop to a d e s i r e d number of sweeps.. The channel width could be v a r i e d from 50 y sec to 200 m sec. For a l l our measurements, the 50 y sec channel width was always used and only a quarter of the memory was used at a time. A r i t h m e t i c a l operations could be c a r r i e d out i n t h i s instrument computer. The IT/2 pulse was used to t r i g g e r the instrument computer and the TT/2 - TT pulses together with an echo were recorded i n the memory a f t e r each sweep. In making T^ measurements, the computer stored, both M and M(t) i n two quadrants and s u b t r a c t i o n between them was then executed. The M -M(t) were p r i n t e d out i n d i g i t a l form by means of a high speed p r i n t e r Fabri-Tek Model 201. The echo spanned about 20-25 channels and the s i g n a l s f o r M -M(t) were taken by summing the same eight channels centered around the echo peak. Any systematic e f f e c t due to the instrument computer was checked against the boxcar by making measurements of a gas at s e v e r a l d e n s i t i e s . The agreement was good to one percent or b e t t e r . e) Sample C o i l and Sample Holder The sample c o i l was made of approximately twelve turns of 22 S.W.G. wire w i t h a diameter of l e s s than 3/8" and a length of about - 22 -an i n c h . Since the sample c o i l was immersed i n the sample, the enamel on the wire had to be s t r i p p e d away and the sample c o i l was p r o p e r l y cleaned by organic s o l v e n t . A t h i n walled g l a s s tube was used to f i t the sample c o i l t i g h t l y i n order to i n s u l a t e i t from the sample holder. Glass tubing was used f o r i n s u l a t i o n elsewhere whenever necessary. Care was taken to block a l l the volume outside the sample c o i l to o b t a i n a " c l e a n " TT p u l s e . The sample holder was designed f o r high pressure work and was adequately described i n the t h e s i s of L a l i t a . The only m o d i f i c a -t i o n was to e l i m i n a t e the volume outside the sample c o i l . The standard techniques used i n the gas c o n t r o l system have been discussed by K. L a l i t a (1967). f) Heater and Temperature Measurements The heater wound on the high-pressure v e s s e l was the one used by K. L a l i t a . The temperature was measured with a chromel-alumel thermocouple. The " h o t - j u n c t i o n " of the thermocouple was placed i n a hole made f o r i t at the bottom of the high-pressure v e s s e l . The " c o l d -j u n c t i o n " was immersed i n an i c e bath. The thermo E.M.F. was measured with a "Honey-Well" potentiometer and a n u l l i n d i c a t o r . The temperature was accurate to w i t h i n a degree once the current through the heater c o i l reached an e q u i l i b r i u m . A p e r i o d of ten hours was allowed to l e t the sample reach e q u i l i b r i u m before measurements were made. A l l temperature measurements were accurate to b e t t e r than h a l f a percent. II .4 Sample P u r i t y and Density Measurements A l l gases were purchased from the Matheson of Canada Ltd. and were used i n t h i s work without any f u r t h e r p u r i f i c a t i o n . The 23 -s p e c i f i e d minimum p u r i t i e s ' i n CM,, CILF,' CHF„, C!;., and CP. CI were 99.99, J 4' 3 • .6 4 6 99.0, 98.0, 99,7, and 99.0%, r e s p e c t i v e l y . Hy means of a' mass-spectrometer a n a l y s i s , the oxygon c o n c e n t r a t i o n s i n CI!J"•'^  and C!i 7F were found to be l e s s than 0.01 and (1.1%, r e s p e c t i v e l y . Pressures were simply measured with a mechanical gauge and .with a mercury .manometer above and below two atmospheres, r e s p e c t i v e l y . A l l pressure measurements were accurate to w i t h i n h a l f a percent. For pressures below two atmospheres, the i d e a l gas.law was assumed to f i n d d e n s i t i e s i n i d e a l Amagats, one such u n i t 19 3 corresponding to 2.69 x 10 molecules/cm . The dependence of d e n s i t y on temperature and pressure (up to 500 p s i ) f o r CHF^ was s u p p l i e d to us by the General Chemical D i v i s i o n , A l l i e d Chemical Co., New J e r s e y . These values were confirmed to w i t h i n few percent e r r o r by measurements of .the proton or f l u o r i n e ' m a g n e t i c resonance s i g n a l s t r e n g t h . This method was used to o b t a i n the d e n s i t y f o r pressures above 500 p s i . The c r i t i c a l temperature and c r i t i c a l pressure f o r CHF^ i s 33°C and 691 p s i , r e s p e c t i v e l y . For CH^F, the Van der IVaals gas equation (a = 2 * * 4.631 atm./(mol) , b = 0.05264 1/mol) was used to get the d e n s i t y at. v a r i o u s pressures and temperatures. This was again checked to w i t h i n a few percent e r r o r by means of the proton magnetic resonance s i g n a l s t r e n g t h . The c r i t i c a l temperature and pressure f o r CH F i s 44,6°C and 852 p s i r e s p e c t i v e l y . The d e n s i t y as a f u n c t i o n of temperature and pressure f o r CF were obtained from the c o m p r e s s i b i l i t y f a c t o r Z given by D o u s l i n Z et. a l . (1961) and p 18.6 P/T where Z i s Z at N.T.P. and P i s pressure i n p s i . For CF^Cl, the d e n s i t y was determined by comparing the * * Handbook of Physics and Chemistry - 24 -f l u o r i n e magnetic resonance s i g n a l s t r e n g t h of CF^Cl and CF^H whose d e n s i t y was known. CHAPTER I I I THEORY OF SPIN-LATTICE RELAXATION I I I . l .Intra-molecular I n t e r a c t i o n s As mentioned i n Chapter I , the s p i n - l a t t i c e r e l a x a t i o n r a t e T^ ^ f o r most molecular f l u i d s r e c e i v e s c o n t r i b u t i o n s from the i n t e r - m o l e c u l a r and the i n t r a - m o l e c u l a r i n t e r a c t i o n s . For d i l u t e gases, the i n t e r - m o l e c u l a r , c o n t r i b u t i o n R i s n e g l i g i b l y small because of the la r g e average s e p a r a t i o n between molecules. However, i f paramagnetic i m p u r i t i e s are present i n the sample, T^ can be dominated, at d e n s i t i e s which are not too low, by the i n t e r - m o l e c u l a r i n t e r a c t i o n w i t h the im p u r i t y s p i n . I t should be noted t h a t i n very d i l u t e gases, the most important spin-dependent i n t e r a c t i o n s are always i n t r a -molecular so t h a t the i m p u r i t i e s then play no r o l e i n T^ measurements. The c o n t r i b u t i o n s to the s p i n r e l a x a t i o n r a t e from the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n R A and the s p i n - r o t a t i o n i n t e r a c t i o n R^ w i l l be discussed s e p a r a t e l y i n t h i s s e c t i o n f o r symmetric-top molecules. The r o t a t i o n a l Hamiltonian f o r a symmetric-top molecule i s denoted by The eigenfunctions |JKM> of ji^ are c h a r a c t e r i z e d by the quantum numbers J,K, and M, which are the values of the t o t a l angular momentum J , the p r o j e c t i o n of J on the symmetry a x i s of the molecule and the — A p r o j e c t i o n of J on a spaced-fixed z a x i s , r e s p e c t i v e l y . - 26 -(3.1) %R |JKM> =-R(oJK|JKM> where 2 2 (3.2) -h U j K = L J(J+1) + \ ( j - - ^ ) K 2 o A o hBJ(J+l) + h(A-B)K 2 and I , I are the two p r i n c i p a l moments of i n e r t i a of the symmetric-top molecule. Of course, we have assumed that the molecule i s i n the ground e l e c t r o n i c and v i b r a t i o n a l s t a t e and has no angular momentum as s o c i a t e d w i t h these degrees o f freedom. i ) Magnetic D i p o l a r I n t e r a c t i o n The Hamiltonian f o r the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n i n a CX^Y molecule i s given by (C i s s p i n l e s s ) 2 (3.3) ft.. ^ i ^ ) 1 / 2 t I l i l i l . I T ( 2 ) ( T . , T . ) Y 9 d l P 5 i=2 j = l r . . 3 y=-2 P 1 J 2>~]i . . IT 1 > J (8. .,•..) where r. ., 6. . and d>. . are the s p h e r i c a l p o l a r coordinates o f the vec t o r j o i n i n g spins i and j which have angular momentum 1^ and I_. and gyromagnetic r a t i o y^ and y_., r e s p e c t i v e l y . Let the i n d i c e s i = 1, 2,3 l a b e l the X spins and i = 4 l a b e l the Y s p i n so tha t y-,=y~ =Y? =Y, (2) -*• and y^-Yy• The s p h e r i c a l tensor T^ J ( I ^ , I _ . ) i s given by - 27 (3.4) . T ^ J ( I I ) = z C(112|p y-y ) T J J ( I )T 1 J ( I ) y 1 .1 p 1 J- 1 y-y^ j where C(112|y ,y-y ) i s a Clebsch-Gordan c o e f f i c i e n t (Rose 1957) and (3.5) = I , T[V= - — (I i i i ) ^ • ' o z ±1 pj x y' From the general theory of nuclear s p i n r e l a x a t i o n (Abragam 1961), the d i p o l a r c o n t r i b u t i o n to the r e l a x a t i o n of nuclear spins is_ governed by the " s p e c t r a l d e n s i t i e s " ^ (to) of the c o r r e l a t i o n f u n c t i o n s ' G ^ f r ) of the s p h e r i c a l harmonics ^m^^1-'^ which are de f i n e d as f o l l o w s (3.6) J 2 m ^ = j G 2 m ^ e x p ( : " ^ (3.7) G 2 m ( x ) = Re < («(())) (Q(x))> where the bracket < > denotes an ensemble average and ft(x) denotes the o r i e n t a t i o n of a m o l e c u l e - f i x e d v e c t o r V (e.g. r..) as a f u n c t i o n of time. The r e l e v a n t angular frequencies to i n equation (3.6) are those a s s o c i a t e d w i t h the Larmor frequencies to and to of the X and Y n u c l e i n x y i . e . , f o r those terms i n i n v o l v i n g p a i r s of X n u c l e i , the character-i s t i c frequencies are zero, to r and 2to , wh i l e f o r the p a i r s of X-Y n u c l e i , they are to^, co^ and - to . The time dependence of G ^ C T ) i s a f f e c t e d both by molecular r o t a t i o n and by c o l l i s i o n s w i t h other molecules. The e f f e c t of molecular c o l l i s i o n s on (-^(x) ^ s a s s u m e d by Bloom, Bridges and Hardy (1967) to be adequately accounted f o r by a - 28 -"reduced c o r r e l a t i o n f u n c t i o n " g 2 ( T ) s o that (3.8)- G 2 M ( T ) = [ G 2 M ( T ) ] F R E E g 2 ( T ) where [G 2 ( T)]£ r e e denotes the c o r r e l a t i o n f u n c t i o n f o r a f r e e l y r o t a t i n g molecule. The !'G„ ( T ) I ^ has been evaluated by Bloom et a l . L 2mv / J r r e e J (1967) as f o l l o w s , 0 0 J J . .Ah _. At (3.9) [ G 2 m ( x ) ] f r e e = Re E E E P J K M < J K M | e 1 ™ 1 R T Y ( f i ) e ' ^ R * J=0 K=-J M=-J Y^(f i)|jKM> = T£ • P J m l < J K M l Y 2 m ^ ! J ' K ' M ' > | 2 c o s [ ( W j K - a > J 1 K 1 ) T ] K, K' M,M' where (3.10) p B. e x p [ - a J ( J + l ) -v ' JKM oo J • E E (2J+1) exp[-aJ(J+l)-3K ] j=0 K=-J and a a M , 0 = h(A-B) kT p kT and the sums over K and M range from - J to J , w h i l e the sum over K' and M' range from - J ' to J ' . Using the tra n s f o r m a t i o n property of Y 2 m ( f t ) from a space-fixed coordinate system to a mo l e c u l e - f i x e d system i n which 6 of f! = (6 ,d> ) i s the angle between the vect o r V and the o o ^ o cr • • molecular symmetry a.xis, i . e . - 29 -(3.H) Y 2 m(Q) = E ( a . B . Y ) Y ^ ) 1< where Dj^(a,g,Y) denote the r o t a t i o n matrices and a, 3, Y are the Euler angles f o r the t r a n s f o r m a t i o n , we o b t a i n (3-12) [ G 2 m ( x ) ] f r e e = P J K J v ! | Y 2 j _ k ( % ) | 2 [ C ( J 2 J ' | K k ) ] 2 [ C ( J ' 2 j | M - m ) ] H,W X C ° S [ C U J K " U J « K ' ) T ] . Z 2 ^ i p j K N . ] [ C ( J 2 J - | K k ) ] 2 | Y 2 j _ k ( V | 2 c o s K y K ! f , . [ ( U 3 J K - C 0 J I K , ) where the sum over M drops out f o r a f i x e d M' and m, and the sum over M' has been performed using eq. (3.7) i n Rose (1957). I t appears that the [ G 2 m ( T ) ] f r e e i R e c l - (3.12) i s independent o f m. Hardy (1966) has shown under q u i t e g e n e r a l . c o n d i t i o n s , which should apply to gaseous methane, that (3.13) G l m ( T ) = G i ( ) ( T ) e x p W J T ^ where to, i s the Larmor p r e c e s s i o n frequency of the molecule. The f r e e molecule " s p e c t r a l d e n s i t y " o f [ G 2 m ( T ) ] f r e e i s a s e r i e s of 6-functions at the frequencies (toT -toT,„.) a s s o c i a t e d w i t h p a i r s of r o t a t i o n a l J K J K. s t a t e s between which " A , , has non-zero matr i x elements. The e f f e c t 0 d i p of c o l l i s i o n s among molecules i s to l i m i t the l i f e t i m e o f the molecules - 30 -i n t h e i r r o t a t i o n a l s t a t e s and hence to spread the " s p e c t r a l d e n s i t y " a s s o c i a t e d w i t h each 6-function over a range of frequencies of the order of the c o l l i s i o n frequency. Since the lowest r o t a t i o n a l frequency i s o f t e n much l a r g e r than w and ai , t h i s means that at low x y d e n s i t i e s where the c o l l i s i o n frequency i s much.smaller than the lowest r o t a t i o n a l frequency, only those terms i n ^ ^^p which couple r o t a t i o n a l s t a t e s of the same energy give appreciable c o n t r i b u t i o n s to the nuclear s p i n r e l a x a t i o n r a t e . Hence the [ G 2 m ( T ) ] f r e e ^ n e c l ' (3.12) becomes time-independent. The " s p e c t r a l d e n s i t y " of (T) c a n he w r i t t e n as (3.14) J 2 m ( u 0 G 2 Q ( T ) exp ( imtOj i ) exp (-icox)dx where nr f 2 J 2 C u j - m u j ) (3.15) j 2(a)) i : g 2 ( x ) exp (-iwx)dx 2 T , . 2 2 1 + 0) X 2 i f the e f f e c t of molecular c o l l i s i o n i s assumed to give g 2("0 = exp(-x/x 2) This i s indeed to be expected f o r strong c o l l i s i o n s i n which the molecular r o t a t i o n a l s t a t e i s completely u n c e r t a i n a f t e r each c o l l i s i o n . The q u a n t i t y f 2 d e f i n e d i n eq. (3.14) must take on the value between zero - 31 -and one. Mien the o s c i l l a t i n g terms i n eq. (3.12) are dropped, i t has been shown by Bloom et a l . (1967) that f takes the value of 0.2 f o r s p h e r i c a l - t o p molecules. For symmetric-top molecules, using eqs. (3.12)-(3.14), we ob t a i n °° J (3-16) ^ f 2 = j | Y 2 j 0 0 y | 2 E E ( 2 J + 1 ) P J K M [ C ( J 2 J | K 0 ) ] 2 J=0 K=-J 2 ? + 5 l Y 2 , 2 ( ^ o ) l j f 0 ( 2 J + l ) P J 1 M [ C ( J 2 J | l , - 2 ) ] 2 In t h i s chapter, we s h a l l neglect any c o r r e l a t i o n between d i f f e r e n t s p i n - s p i n vectors i n a CX Y molecule. In other words, the motions of the three X spins and the Y s p i n are completely u n c o r r e l a t e d , i.e. < Y 2 m ( f i i j ( 0 ) ) Y 2 _ m ^ k i C T ) ) > = 0 unless i=k, j = l, or i = l , j=k. With t h i s assumption, the i n t r a - m o l e c u l a r d i p o l a r c o n t r i b u t i o n to the nuclear s p i n r e l a x a t i o n r a t e of a X nucleus can be w r i t t e n as <3-17> fVx = 4>xx + ^xy and (3-18) [ R A ] y = 3 ( i - ) x y f o r the Y nucleus i n the molecule. From the general theory of nuclear s p i n r e l a x a t i o n (Abragam 1961), 4 2 Y Ii 1(1+1) -(3-19) ( - ) = f - 1 [ J 2 l K ^ + 4 J 2 2 ^ 2 V ] 1 r x xx 32 2 2 2 Y y "ft S(S + 1) 1 (3.20) (4-) = ~ - X — ^ — ^ -a) ) + J 0 1 ( O J ) + 2J„„(oo +to )] v *T, xy 5 6 L3 20 x y J 21K x J 22y x y J 1 r J • xy where 1=1=1=1 and S=I^. In the short c o r r e l a t i o n time l i m i t , we can r e w r i t e eqs. (3.19) and (3.20) using eq. (3.14) as 4 ? Y V l ( I + l) (3-21) (-) = _ £ 2 ( 6 x x ) j 2 ( 0 ) 1 xx r xx 2 2 2 Y Y f i S(S+1) (3-22) ( l - ) x y = § f 2 ( e x y ) 3 2 ( 0 ) r x y For the proton i n CHF ( i . e . , Y H H , X E F ) molecule, u s i n g eqs. (3.18), (3.22), Y,, = 2.67xl0 4 e.m.u., Y,- = 2.52xl0 4 e.m.u. and I=S=l/2, we n r o b t a i n 10 -2 (3.23) R A = 1.193x10 f (8 ) J 2 ( 0 ) sec where r u r , 0,.„ are given i n Table 1 . S i m i l a r l y we o b t a i n f o r the nr Hr proton i n CH^F molecule i n 9 -2 (3.24) R A = [ 2 . 4 9 3 x l 0 i U f 2 ( 6 H H ) + 3.613x10 f 2 ( 6 ^ ) ] j 2 (0) sec by us i n g eqs. (3.17),(3.21), (3.22) and Table 1. From eq. (3.15), J 2 ( 0 ) = 2 T 2 > E v a l u a t i n g the sums i n f 2 of eq. (3.16) by the technique of Birnbaum (1957) given i n Appendix B and Table 1, we ob t a i n f o r CHF^ (3.25) £ 2 C 0 H F 3 = 4.335x10 and f o r CH 3F (3.26) F 2 C 8 H H ^ = 6 - 8 4 5 x 1 0 " 2 f 2 ( 6 H p ) = 5.38xl0~ 2 TABLE 1: Molecular Constants of CHF and CH F o o 1000a 10003 R H F C A ) r H H ( A ) °HF °HH < J > < K > CHF 1.67 -0.77 1.977 -- 141°13' -- ^33 ^19 CH F 4.13 20.1 2.03 1.8 149°9' 90° M2 ^ 4 T = 297°K i i ) S p i n - r o t a t i o n I n t e r a c t i o n The i n t e r a c t i o n of a nuclear magnetic moment with the magnetic f i e l d generated at the p o s i t i o n of the nucleus by the r o t a t i o n of the molecule c o n t a i n i n g the nucleus i s c a l l e d a s p i n - r o t a t i o n i n t e r a c t i o n . F l u c t u a t i o n s i n the i n t e r a c t i o n r e s u l t from c o l l i s i o n a l molecular r e o r i e n t a t i o n and the corresponding c o n t r i b u t i o n to the nuclear s p i n r e l a x a t i o n contains i n f o r m a t i o n about r o t a t i o n a l r e l a x a t i o n of the molecule p r o v i d i n g that the s p i n - r o t a t i o n i n t e r a c t i o n i s known. In polyatomic molecules, the s p i n - r o t a t i o n i n t e r a c t i o n i s i n general c h a r a c t e r i z e d by s i x independent parameters, the three p r i n c i p a l values of the s p i n - r o t a t i o n tensor and the three E u l e r angles r e q u i r e d to s p e c i f y the o r i e n t a t i o n of the p r i n c i p a l a x i s coordinate system. In - 34 -s p h e r i c a l - t o p molecules such as C H ^ and C F ^ , the number of independent parameters i s reduced to only two, which are denoted by C J J and ( Y i , O z i e r and Anderson, 1 9 6 S ) . I t i s customary to represent the spin-r o t a t i o n tensor by q u a n t i t i e s C & and defined as f o l l o w s , (3.27) C A = - CC:,, + 2 C j_), C D = C j_ - C„ Accurate values of C_. may be obtained with r e l a t i v e ease f o r s p h e r i c a l -top molecules from an a n a l y s i s of molecular beam spec t r a but the q u a n t i t y i s more d i f f i c u l t to measure. In the case of symmetric-top molecules of the form CX^Y , the s p i n - r o t a t i o n i n t e r a c t i o n of the Y s p i n i s a l s o d e s c r i b a b l e under c e r t a i n c o n d i t i o n s by two independent parameters as defined i n eq. (3.27). For the o f f - a x i s X s p i n , the s p i n -r o t a t i o n i n t e r a c t i o n i s i n general c h a r a c t e r i z e d by four independent parameters, the three p r i n c i p a l values of the s p i n - r o t a t i o n tensor C r ) XX Cy and C^z and an Euler a n g l e d . One of the p r i n c i p a l axes, say y', i s p e r p e n d i c u l a r to the YCX plane. Another, say z', i s at an angle ^ w i t h the molecular symmetry a x i s . A f t e r r o t a t i n g about the y' a x i s through an a n g l e d such that the z' a x i s c o i n c i d e s w i t h the molecular symmetry a x i s , the s p i n - r o t a t i o n tensor of a s i n g l e X s p i n becomes (3.28) C 0 xx 0 C yy c o xz to 0 0 0 iC o \ xz xz 0 0 / 'l/2(C -C ) 0 0 V xx yy J 0 -1/2(C -C ) 0 xx yy \ 0 0 0/ /l/2(C +C ) 0 0 ^ xx yy' 0 1/2(C +C ) 0 xx yy 0 0 C zz - 35 -The f i r s t two components of C i n eq. (3.28) have matr i x elements connecting AK=tl t r a n s i t i o n s and the l a s t term of C only involvesAK=0. By n e g l e c t i n g matrix elements of the s p i n - r o t a t i o n i n t e r a c t i o n between d i f f e r e n t K s t a t e s , , t h e s p i n - r o t a t i o n tensor of a X s p i n only r e t a i n s the l a s t term of C i n eq. (3.28) and i s t h e r e f o r e c h a r a c t e r i z e d by two independent parameters a f t e r d e f i n i n g Cjj = C^^ and C^= 1 / 2 ( 0 ^ + C ) . The theory of nuclear spi n r e l a x a t i o n due to the s p i n - r o t a t i o n i n t e r a c t i o n was f i r s t discussed by Hubbard (1963) and B l i c h a r s k i (1963) and was extended to the case of d i l u t e gases by Bloom, Bridges and Hardy (1967) . For the Y spin i n a symmetric-top molecule of the form CX^Y, i t s p r i n c i p a l a x i s coordinate system of the s p i n - r o t a t i o n tensor c o i n c i d e s w i t h the p r i n c i p a l a x i s coordinate system of the molecule. The s p i n - r o t a t i o n Hamiltonian $1 ^ of such an on-axis nucleus i s then given by Hubbard (1963), (3.29) f i j i =C E ( - l ) k I , ' J ' 1 - C, ( I ' J ' - I ( - l ) k I ' J ' ) v J s . r . a , , J k -k d v o o 3 , , J k -kJ k=-l k=-i where (3.30) 1 = 1 , v J o z and ,/2 (I, t i l ) y (3.31) J = J , v o z J . = + — (J + i J ) - 36 -The primes i n eq. (3.29) i n d i c a t e that the components of J and I are r e f e r r e d to a mo l e c u l e - f i x e d coordinate system with i t s z a x i s along the molecular symmetry a x i s . C & and defined i n eq. (3.28) are i n terms of Cj^ and C J J , the p r i n c i p a l values of the s p i n - r o t a t i o n tensor. A f t e r transforming J ' and I' i n t o a space-fixed coordinate system, Hubbard (1963) has obtained (3.32) fiH = E I K v s. r . m lm m where (3.33) K = ( - l ) m C J - (|-) 1 / 2C, E (-1) P C(112|my)D 2 , , , J J lm v J a -m  VV d ^ J v 1 1 o,-(m+y) y y 2 The operators D and J and hence are time-dependent because 1 m i , m 2 ]1 of changes of molecular s t a t e as a r e s u l t of c o l l i s i o n s among molecules, From the general theory o f nuclear s p i n r e l a x a t i o n (Abragam 1961) , (3.34) R c = 4TT2 J n O o ) = 4TT2 f 00 „ , . -ico T G n ( x ) e o d x J -co where G ^ ( T ) i s the c o r r e l a t i o n f u n c t i o n of K ^ ( x ) , defined as (3.35) _ G l m ( x ) = Re < K ^ o ) h ^ ) > The bracket < > again denotes an ensemble average of molecules at the temperature T. We need only to evaluate G ^ Q ( T ) (see eq. (3.13), which i s given by - 37 -2 2 1/2 (3.36) G i n ( x ) = R e[C <J (o)J (x)> -2(-f-) 1 C C, x v 10 • a o v o 3 a d x E (-1) m C(112|O I J ) < D 2 (o)J (o)J (x)> + H + | C 2 E (-l) y + y ,C(112!Ou)C(112|Op') x x < D 2 (o) J (o) J 1', (x) D 2 + , (x)> ] o,-y v 1 y v J y 1 ^ J o,-y ' v J J where (3.37) J T ( T ) = e j e ^ V In e v a l u a t i n g G ^ ( x ) , Hubbard (1963) has assumed that J and 2 D q _^ are s t a t i s t i c a l l y independent which gives a s i g n i f i c a n t s i m p l i f i c a -t i o n . The assumption i s not tru e f o r a gas, though i t may be a reasonable one f o r a l i q u i d where the a n i s o t r o p i c i n t e r - m o l e c u l a r i n t e r a c t i o n s are l a r g e . A l e s s r e s t r i c t i v e approximation has been used by Bloom et a l . (1967). They assume that the i n f l u e n c e of molecular c o l l i s i o n s on the time e v o l u t i o n of each of the c o r r e l a t i o n f u n c t i o n s i n eq. (3.36) can be accounted f o r by m u l t i p l y i n g the f r e e molecular c o r r e l a t i o n f u n c t i o n s by a monotonically decreasing f u n c t i o n of time i . e . (3.38) < J o ( 0 ) J Q ( x ) > = [ < J q ( 0 ) J o ( X ) > ] £ r e e g l ( x ) (3.39) < D 2 j _ m ( 0 ) J m ( 0 ) J o ( X ) > = [ < D 2 j _ ) j ( 0 ) J i j ( 0 ) J o ( x ) > ] f r e g g ^ x ) - 3 8 -( 3 . 4 0 ) <D2 ( 0 ) J ( 0 ) J ^ ( x ) D 2 t , ( x ) > = [<D2 ( 0 ) J ( 0 ) J + , ( T ) 0,-y y y' d,-y 0,-y y y where g-^(x), g^ 2(' r) a n ^ ^12^T^ a r e ^'ie " r e < ^ u c e ^ c o r r e l a t i o n f u n c t i o n s " w i t h g-^  ( 0)=g^ 2 ( 0 ) =gj 2 ( 0 ) = 1 • We see that the angular momentum operator J has a reduced c o r r e l a t i o n g ^ ( x ) , the tensor of rank one formed by the product of J and the s p h e r i c a l harmonic Y2m(°-) has a reduced c o r r e l a t i o n f u n c t i o n g ^ ( x ) and the cross term between J and (Q) has a reduced c o r r e l a t i o n f u n c t i o n g ^ C x ) . The f r e e molecule c o r r e l a t i o n f u n c t i o n s i n eqs. ( 3 . 3 8 ) - ( 3 . 4 0 ) can be evaluated as f o l l o w s ( 3 , 4 1 ) [ < V ° > V T > W JKM P J W 1 1 J 2 ( 0 ) | J ' K - M' >< J ' K 'M M J z (x ) | JKM> J'K'M' E n T , T l 1 , r n f „ T,„nM2, JKM PJKM J C J + 1 ) [ C ( J 1 J l M 0 ) ] S i " c e [ J z ' \ ] = ° I J K P J K M J ^ J + 1 ^ 2 J + 1 ) ( 3 . 4 2 ) [<D2 ( 0 ) J ( 0 ) J (x)>], =• E PT[/..<JKM|D2 • jJ'K'H'xJ'K'M' JJ ( 0 ) J (0)|JKM> L o,-]i~ v, J J f r e e J K M JKM 1 . 0,-y 1 1 y o 1 J'K'M' % L p ^ ( i ^ ) 1 / 2 c ( J , 2 j | K O ) c ( J ' 2 j | M ' - ^ x JKM J'M' x <J'K'M'|J (0) J Q ( 0 )|JKM> = E P T„.J(J+l)C ( J 2 J|KO)C ( J 2 J|M+y,-y)C(JlJ|M )y)C(JlJ|M , 0 ) JKM J K M - 39 -[3.43) [<D2 (0)J ( 0 ) / , (x)D 2 + (x)>] , L o,-y ^  ' y v ' y 0,-y*- y J£ree = E PTI_,<JKM|D2 ( 0 ) J (0) I J'K'M'xJ'K'M' I (D 2 , (T ) J ,(x))' I JKM> J K M o.-M y 1 1 o>-y' y 1 J'K'M' • =E P..„.,<JKM|D2 [ 0 ) J (0) I J'K'M'xJKMlD 2 , (x)J , (x) IJ 1 K 'M.' >* J K M JKM ' 1 0 , - V y' " ' 0 3 - y ' v y " 1 J'K'M' = Z P J K M 2 ( | j ^ ) J ' ( J ' + l ) [ C ( J ' 2 J | K O ) ] 2 C ( J ' 2 J | M + y 3 - y ) C ( J ' U ' |M,y)x JKM J ' x C(J'2J|M+y'^y^CCJ'U' | M J - y ' ) e x p [ - i ( o j J K - c o J ( K )x] Let us d e f i n e F(J,M) as (3.44) F(J,M) = E(-l) yC.(112|Oy)C(J2j|M+y,-y)C(JlJ|M,y) y ( - . l ) J 1 _ M ( | ) 1 / 2 ( 2 J + 1 ) C ( J J 1 |M,-M)W(1J2J;J1) where W(1J2J;J1) i s a Racah c o e f f i c i e n t and the sum over y i n eq. (3.44) has been performed using eqs. (6.23a) and (6.24) i n Rose (1957). S u b s t i t u t i n g eqs. (3.38)-(3.44) i n t o eq. (3.36) and summing over M, we ob t a i n G ^ Q ( X ) a f t e r some c a l c u l a t i o n s CO J • (3.45) G 1 Q ( x ) = | c 2 E E ( 2 J + l ) J ( J + l ) P J 1 ( M { g l ( x ) -J = 0 K =-J _ 2 Cjl r S ^ - J C J ^ l j , ( T ) + l ( g d 2 3K 2-J(J+1) 2 + 3 C 1 J ( J + 1) J 8 1 2 L T J 9 lC' J 1 J(J+1) J 8 1 2 L T J a ^ ' a - 4 0 -+ ^ 2 ' . ^ ^ [ C ^ J - | K O ) ] 2 c o s [ ( . J K - . J 1 K ) x ] x a J?J ' ( J + J ' H - 3 ) ( - J + J ' + 2 ) ( J - J ' H - 2 ) ( J + J ' - - 1 ) 4 J ' ( J'+l) • *12K J s As i n the c a l c u l a t i o n of [G„ ( T ) I ^ , the f r e e molecular 2m rree c o r r e l a t i o n f u n c t i o n s appearing above i n v o l v e terms t h a t are time-independent and terms that o s c i l l a t e at frequencies corresponding to r o t a t i o n a l t r a n s i t i o n A J = i l . The A J = t2 t r a n s i t i o n s do not co n t r i b u t e here because the s p i n - r o t a t i o n Hamiltonian i s formed by two v e c t o r s (tensors of rank one) as opposed to the magnetic d i p o l a r i n t e r a c t i o n which i s contracted from two tensors of rank two. The reason that there are no A K = t l o s c i l l a t i n g terms i n eq. ( 3 . 4 5 ) i s that the p r i n c i p a l ' a x i s coordinate system f o r the s p i n - r o t a t i o n tensor i s the same as the p r i n c i p a l a x i s coordinate system of the molecule i n the above d e r i v a t i o n . For the X s p i n s , the G ^ Q ( X) would be modified to c o n t a i n o s c i l l a t i n g c o n t r i b u t i o n s at the molecular frequencies which correspond to A J = 0 , tl and A K = 0 , t l . With the assumption that the m a t r i x elements of the s p i n - r o t a t i o n i n t e r a c t i o n between d i f f e r e n t K s t a t e s can be negle c t e d , the above c a l c u l a t i o n of G ^ ( x ) which i s v a l i d f o r the Y s p i n i s t h e r e f o r e a p p l i c a b l e to the o f f - a x i s n u c l e i such as the X s p i n s . For d i l u t e gases, the o s c i l l a t i n g terms do not c o n t r i -bute a p p r e c i a b l y to the s p e c t r a l d e n s i t y of G ^ m ( x ) at'the nuclear Larmor frequency and can be neglected. N e g l e c t i n g the high frequency terms i n eq. ( 3 . 4 5 ) , we simply o b t a i n ** The i n f l u e n c e of those terms f o r dense f l u i d s w i l l be discussed i n Chapter VI - 41 -(3.46) 1 .2 J Gi 0CT) = y C a Z Z ( 2 J + l ) J ( J + l ) P J K M { g l ( T ) J ~ U k — J ' C 2 — J± i-3K - J ( J + 1) 1 3 C L J(J+1) •I812(-T-} + I ( ^ ) 2 l l ^ J f J ^ , 2 } 9 lC J L J(J+1) J G 1 2 L J J a ' The s p e c t r a l d e n s i t y of G J Q ( X ) i s then given by (3.47) J i 0 ( ^ ) G | 0 ( T ) e " l a 3 T dx where p(oj)y C 2 < J ( J + 1 ) >J 1(a J) .(3.48) J ^ O J ) g x ( x ) e dx (3.49) <f(J,K)> = E E (2J+1JP f ( J , K ) j=0 K=-J J K M and (3.50) . 2 ,C d , <3K2-J(J+1)> 3 1 2Cw) = :-3 < j ( j + D > — j ^ r + 4 9K 2 ' 1 (Cd,2 <J1J>IT ~ 6K-+J(J+i) > 312(u») + 9 V <J(J+1)> where j ^ O d ) a n d J { 2 ( c o ) a r e t n e s p e c t r a l d e n s i t i e s of g^C 1') and g^^CT) r e s p e c t i v e l y . Using eq. (3.34) and eq. (3.47), we ob t a i n - 42 -2 (3.51) R c = p C ^ - U j ) C 2 < J(J+1) > j ^ ^ - U j ) i n which the high frequency terms are neglected. When the assumption that there i s no net molecular p o l a r i z a t i o n ( i . e . , fiu) ^  << kT, where bij i s the Larmor frequency of the molecular angular momentum) i s used, one sees that Gn (x) = G.,_(x) and hence lm J 10 • (3.52) R c = ^f- p(a) o) C2& < J ( J + 1 ) > j ^ ) For the s p e c i a l case i n which C^ , i s zero, or small compared with C g, p(co) = 1. Two main assumptions have to be made i n usin g eq. (3.47) or (3.52). I t i s o f t e n assumed that the "reduced c o r r e l a t i o n f u n c t i o n s " g c ( x ) decay e x p o n e n t i a l l y , i . e . , (3.53) g c ( x ) = exp(-x/x c) In our a n a l y s i s of the low de n s i t y experimental data i n Chapter VI we s h a l l examine the consequences o f the assumption that a l l s p e c t r a l d e n s i t i e s i n p(co) are equal, i . e . 2x (3.54) j ^ u O = j 1 2 ( u > ) = j J 2 ( o O = — T " 2 l + 0 ) X This simply i m p l i e s that x^ = x^ 2 = T J 2 ~ T c - As w e s h a l l see, the experimental r e s u l t s i n d i c a t e t h a t t h i s i s probably a good approximation. The e v a l u a t i o n of <f(J,K)> f o r f ( J , K ) = K " m [ J ( J + l ) ] n i s given i n appendix B. A3 -Using eqs. (3.46)-(3.51) and eqs. (B.1)-(B.3), R can be w r i t t e n i n terms of the e f f e c t i v e s p i n - r o t a t i o n constant, as where A 2 _ x 4TT Jl c ( 3.55) K c = — ~C e f f — -y-2 ].+ ( a ) o - W j ) x c (3.56) C 2 f £ = (1 - | i ) C 2 + i y 2 C aC d + [ 9 ( A ^ ) 2 ( - 1 - f . + ^_ l n 0) y and -1 + 3 y ] C d (3.57) y= ( J / / 2 = ( 1 - ^ ) 1 / 2 o For s p h e r i c a l - t o p molecules, CX^, the two p r i n c i p a l moments of . 2 i n e r t i a are equal and so y = 0. Since <3K - J(J+1)> vanishes f o r 2 t h i s case, CQ££ f o r s p h e r i c a l - t o p molecules s i m p l i f i e s to 2 2 4 2 (3.58) C = C + C, ^ ' e f f a 45 d by u s i n g eq. (B.4) - 44 -I I I . 2 I nter-molecular I n t e r a c t i o n In d i l u t e gases, the i n t e r a c t i o n s between nuclear spins i n d i f f e r e n t molecules can be neglected. At l i q u i d l i k e d e n s i t i e s , however, t h e i r c o n t r i b u t i o n s to may be comparable to those of i n t r a - m o l e c u l a r i n t e r a c t i o n s . The t r a n s i e n t d i p o l a r i n t e r a c t i o n between magnetic moments on d i f f e r e n t molecules, f i r s t proposed by Bloembergen (1948), occurs when they c o l l i d e . Since the magnetic moments of para-magnetic atoms or molecules are about 1000 times as l a r g e as those of n u c l e i , i t can be the dominant r e l a x a t i o n mechanism i n gases when la r g e amounts of paramagnetic i m p u r i t i e s are present. At low d e n s i t i e s , i t i s obvious that the presence or absence of paramagnetic i m p u r i t i e s such as i s of no importance. At s u f f i c i e n t l y high d e n s i t i e s , T^ w i l l no longer increase l i n e a r l y w i t h d e n s i t y p even i n pure gases, f o r the i n t e r - m o l e c u l a r d i p o l a r couplings during molecular c o l l i s i o n s should become important as the c o l l i s i o n frequency i n c r e a s e s . To f a c i l i t a t e the o bservation of t h i s , one can d e l i b e r a t e l y add 0 i n t o pure sample. The e f f e c t of 0^ on T^ i n gases i s to be o u t l i n e d below. For d i l u t e gases, the l i n e a r i t y between T^ and p can be expressed as (3.59) T x = p/a where a i s dependent on temperature only. In CX Y-G^ mixtures, the f i r s t d e v i a t i o n s from constancy of T/p are expected to r e s u l t from t r a n s i e n t magnetic d i p o l e - d i p o l e i n t e r a c t i o n . The i n t e r - m o l e c u l a r - 45 -d i p o l a r i n t e r a c t i o n s are s u f f i c i e n t l y weak and of s u f f i c i e n t l y short d u r a t i o n that the t r a n s i t i o n p r o b a b i l i t i e s per u n i t time f o r a nuclear s p i n t r a n s i t i o n from a s t a t e M to a s t a t e M' can be roughly described by the quantum mechanical t r a n s i e n t approximation. That i s , where t ^ d/v i s the d u r a t i o n of a c o l l i s i o n , d i s the d i s t a n c e of o c l o s e s t approach,v i s the r e l a t i v e v e l o c i t y of the c o l l i d i n g molecules and Z i s the number of c o l l i s i o n s per second made by a molecule. i s the i n t e r - m o l e c u l a r d i p o l a r i n t e r a c t i o n l e a d i n g to a M t r a n s i t i o n . The matrix .element, of the magnetic i n t e r a c t i o n during a c o l l i s i o n i s estimated t o be of the order of (Bloembergen 1948, Bloembergen, P u r c e l l and Pound 1948) . (3.60) W ^ (3.61) 2 ,-3 where the gyromagnetic r a t i o s r e f e r to the nucleus and to the c o l l i d i n g paramagnetic molecule. The r e l a x a t i o n r a t e a r i s i n g i n t h i s manner i s estimated to be (3.62) _ 2 2,2 2Y Y TI n p 2 A v a Z by using eqs. (3.60) and (3.61). I t i s seen i n eq. (3.62) that only Z depends on the d e n s i t y . The p r o b a b i l i t y of b i n a r y c o l l i s i o n s between - 46 -oxygen molecules and CX Y molecules i s p r o p o r t i o n a l to p n and p v Thus Z i s given (3.63) P 0 2 P C X 3 Y Z cc = p « p PCX„Y 2 Hence we see th a t T^ i s i n v e r s e l y p r o p o r t i o n a l to d e n s i t y p due to t h i s mechanism. Using eq. (1.1), we o b t a i n (3-64) i - = i + ( b X o ) P where X i s the mole f r a c t i o n of 0„ and a and b are f u n c t i o n s of o 2 temperature only. CHAPTER IV LOW DENSITY EXPERIMENTAL RESULTS AND DISCUSSION We now present the low den s i t y experimental r e s u l t s f o r T^ as a f u n c t i o n of d e n s i t y at 297°K f o r CH , CF 4, CHF 7, CH F (Dong and Bloom 1969) and CF^Cl, r e s p e c t i v e l y , and di s c u s s t h e i r i m p l i c a t i o n s . T^ measurements i n polyatomic molecules can provide i n f o r m a t i o n on C and C, because c o l l i s i o n a l molecular r e o r i e n t a t i o n modulates the a d s p i n - r o t a t i o n i n t e r a c t i o n and thereby gives r i s e to nuclear s p i n t r a n s i t i o n s . I t i s known that t h i s i n t e r a c t i o n i s the dominant proton s p i n r e l a x a t i o n mechanism i n CH^ (Bloom, Bridges and Hardy 1967). For the f l u o r i n e s p i n s , t h e i r s p i n - r o t a t i o n i n t e r a c t i o n i s always much great e r than t h e i r d i p o l a r i n t e r a c t i o n . Hence i t i s an e x c e l l e n t approximation f o r f l u o r i n e s p i n r e l a x a t i o n to ignore the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s completely. That t h i s i s a good approximation f o r proton spins i n the above symmetric-top molecules at low d e n s i t i e s i s not obvious and w i l l be . j u s t i f i e d by the Overhauser measurements (Hubbard 1965) presented l a t e r and the c a l c u l a t i o n s of the c o n t r i b u t i o n from R^ at low d e n s i t i e s . In order to o b t a i n q u a n t i t a t i v e i n f o r m a t i o n on C & and C^ from T^ measurements, i t i s necessary to know the c o r r e l a t i o n f u n c t i o n of the r o t a t i o n a l angular momentum J as w e l l as that of the product o f J - 48 -and Y 2 m ( f i ) • Although these c o r r e l a t i o n f u n c t i o n s are not known p r e c i s e l y from f i r s t p r i n c i p l e s , Bloom and Dorothy (1967) were able to use the T. measurements i n CH. gas at low d e n s i t i e s to d i s t i n g u i s h 1 4 ° 6 unambiguously between the two a l t e r n a t i v e assignments f o r the s p i n -r o t a t i o n c oupling constants l i s t e d by Anderson and Ramsey (1966) and give = 21 KHz. This was the assignment favoured by Anderson and Ramsey and has been confirmed by subsequent molecular beam measure-ments which were analysed i n a more d e f i n i t i v e manner than was p o s s i b l e before ( O z i e r , Crapo and Lee 1968). These authors have a l s o measured C and 3. C^ i n SiH^, GeH^, CF^, SiF^ and GeF^ and these measurements represent a l l the molecules f o r which both C and C, have been determined. I t i s a d. noteworthy that the accuracy obtained by Ozier et a l . f o r C i s of a the order of a few percent i n a l l the cases s t u d i e d , but only about 30% at best f o r C^, In some cases, i t was only p o s s i b l e to p l a c e an upper l i m i t on C^. For example, i n the case of CF^, i t was found that C a = 6.85 t 0.35 KHz, while f o r i t was only p o s s i b l e to set the l i m i t s -13 KHz ^ C^ st 17 KHz. The most p r e c i s e value of = 18.2 t 0.5 KHz has r e c e n t l y been obtained f o r CH 4 ( Y i et a l . 1967) by studying the magnetic resonance spectrum of i n d i v i d u a l r o t a t i o n a l s t a t e s at low magnetic f i e l d s . This method, however, i s only a p p l i c a b l e to a few molecules i n which only a few r o t a t i o n a l l e v e l s are a p p r e c i a b l y populated. In view of the d i f f i c u l t y of o b t a i n i n g accurate values of C^ using molecular beam techniques, i t i s our purpose to f u r t h e r i n v e s t i -gate the accuracy w i t h which can be determined from T^ measurements. This i n v o l v e s a study of the c h a r a c t e r i s t i c minimum i n the d e n s i t y - 49 dependence of T, at. low d e n s i t i e s (Hardy 1966, Dorothy 1967). The data f o r the d i f f e r e n t molecules s t u d i e d were used i n v a r i o u s ways depending on the amount of i n f o r m a t i o n on the s p i n - r o t a t i o n c o u p l i n g constants now a v a i l a b l e from molecular beam experiments. Since the C and C, r a d f o r CH^ have been determined i n molecular beam measurements with greater p r e c i s i o n than p o s s i b l e by other methods, a comparison of theory and experiment i n CH^ can provide a strong t e s t on the assumptions made i n eqs. (3.53) and (3.54) i n the previous chapter on the form of the c o r r e l a t i o n f u n c t i o n . Thus we use the CH^ system to t e s t the v a l i d i t y of our method of o b t a i n i n g i n f o r m a t i o n on the s p i n - r o t a t i o n tensor. In CF., C i s known very w e l l , w h i l e C. i s only known to w i t h i n an order 4 a J ' d J of magnitude. Our measurements provide the more accurate determination of f o r t h i s molecule. The cases of CHF , F and CF^Cl are q u i t e d i f f e r e n t , f o r very l i t t l e independent i n f o r m a t i o n on the s p i n - r o t a t i o n •tensor i s a v a i l a b l e f o r these'molecules. As has been pointed out i n Chapter I I , the apparatus was o r i g i n a l l y designed f o r high pressure work so that optimum matching c o n d i t i o n s f o r s i g n a l to noise r a t i o were not achieved i n the experiments d e s c r i b e d i n t h i s chapter. Furthermore, the magnetic f i e l d was not very homogeneous, as can be seen from the f a c t that the width of our s p i n echo was of the order of 800 usee. By improving these two experimental c o n d i t i o n s and using l a r g e samples, i t should be p o s s i b l e to achieve a c o n s i d e r a b l e improvement i n s i g n a l to noise r a t i o , thus making i t p o s s i b l e to extend our measurements to lower d e n s i t i e s and to achieve gr e a t e r accuracy i n the d e n s i t y range covered by our experiments. - 50 -IV.1 Methane i ) The 7 Minimum i n CH 4 The experimental data f o r T^ as a f u n c t i o n of d e n s i t y i n CH are p l o t t e d i n Figur-e 3 where e r r o r bars denote i one standard d e v i a t i o n . Since the s p i n - r o t a t i o n c o u p l i n g constants are known to be C = 10.4 i 0.1 KHz and C. = 18.2 t 0.5 KHz from molecular beam a d measurements ( Y i et a l . 1967), we f i r s t c a l c u l a t e d T^ as a f u n c t i o n of p under the assumption that the molecular r e o r i e n t a t i o n i s governed by a s i n g l e c o r r e l a t i o n time i . e . , = x^ = f j ^ * which leads to eqs. (3.55) and (3.58) f o r s p h e r i c a l - t o p molecules. In using these equations at low d e n s i t i e s , we may put (4.1) x x = A/p where p i s the d e n s i t y i n Amagat u n i t s and A i s independent of d e n s i t y x^ can al s o be w r i t t e n as (4.2) x, 1 = P <ov> = P a,v v 1 1 o 1 o 1 3 19 where P i s the number of molecules per cm ( i . e . , p = P x 2.69x10 ), o o \7 i s the r e l a t i v e v e l o c i t y of a c o l l i d i n g p a i r of molecules and i s the average cross s e c t i o n f o r s p i n - r o t a t i o n r e l a x a t i o n . The r e l a t i v e v e l o c i t y V can be taken to have i t s k i n e t i c theory value "v = / 8 k T = 4 A l (4.3) /Try /irm - 52 -where u i s the reduced mass of the c o l l i d i n g p a i r , T i s the absolute temperature and m the molecular mass. Thus ™ can be evaluated once A i s determined. This can be compared with the k i n e t i c cross s e c t i o n a k i n b e t w e e n c o l l i s i o n s d e fined by ( H i r s c h f e l d e r et a l . 1954) (4.4) o k i = l t d 2 = 45.5 A 2 where d i s the "hard-sphere" diameter of CH^, which we have taken to o be 3.82 A. The asymptotic value of T^/p = 0.020 sec/Araagat above the T^ minimum obtained from our data and those of Bloom et a l (1967) f i x e s the value of A to be A(CH 4) '•= 2.36 x 1 0 - 1 0 sec-Amagat so t h a t no a d j u s t a b l e parameters are l e f t i n the t h e o r e t i c a l determina-t i o n o f T^ as a f u n c t i o n of P using eqs. (3.55) and (3.58), p r o v i d i n g o 2 ° 2 that a) i s known.. oT i s found to be 17.3 A as compared to 45.5 A f o r a k i n ' S ° ^ i a / t t' i e c o l i : ^ s i o n s are n e i t h e r very strong nor very weak. I f the c o l l i s i o n s were very weak, then the angle through which J. was r o t a t e d during a s i n g l e c o l l i s i o n would be small and we would o b t a i n , 0 ^ << a i , - ^ n - T n e s o l i d curve shown i n Figure 3 i s p l o t t e d using the value of g = 0.3133 nuclear magneton (n.m.) obtained by Anderson and Ramsey (1966) to o b t a i n = 0.056 COq. The agreement with experiment i s good. In order to i n d i c a t e the s e n s i t i v i t y of T^ to the value of C^, we have a l s o p l o t t e d T^ as a f u n c t i o n of P using = 21.0 KHz, and we see that the agreement with experiment -is poorer. This r e s u l t does - 53 -not imply, however, that the p r e s e n t l y a v a i l a b l e data could r e a l l y d i s t i n g u i s h between these two values of C ^ . By r e l a x i n g the assumption that T-^ = t^2' ^"e" P u t t i n 2 (4.5) T{2 = T X (1 + 5) the data could be adequately f i t f o r = 21.0 KHz using a s u i t a b l e value of 6. Indeed, f o r = 18.2 KHz, the s o l i d curve can be made to f i t our experimental data b e t t e r by. p i c k i n g a s u i t a b l e 6 as f o l l o w s . For x^ i- ^Y2' w e O D t a i n , a . 1 r2 1 , * /-I i • ( 7 T } Tn = °eff — 72 2 + A (f7 } 4Tf 1 1+ (lO -CO T J T, 1 O J 1 2 where i s given by eq. (3.58) and 2 2 A f K T l 4 p2 / ^ V V T i T i 2 (4.6) A ^ ) = ~ - ~ R R - ^ C D [ — -j-^ x — - 1] Using eq. (4.5) as w e l l as C q = 10.4 KHz and CA = 18.2 KHz, i t can be e a s i l y shown t h a t ( 4 . 7 ) L_ - i . H2 + 6) 1 1 - 4.68 d T l 1 + 6 1 + 6 1 + (p/p . ) 2 = — • mm 1 where p . •= 0.0419, at which T, minimum occurs, i s given by mm 1 to J (4.8) P . = A(co -wT) mm o J - 54 -i f eq. (3.55) i s assumed to h o l d . Taking T the values given by the s o l i d curve i n Figure 3 and dT^ as the d e v i a t i o n s of the experimental -1 d T l -1 data from the s o l i d curve below 10 Amagat, we p l o t (1 - 4.68 TT.— ) J 2 7 - 1 versus (1 + p /p". ) as shown i n Figure 4. For small 6 , s t r a i g h t l i n e s with slope equal to 26 and i n t e r c e p t equal to."(1-6) are drawn and mean square d e v i a t i o n s a are c a l c u l a t e d f o r v a r i o u s f i t s . By p l o t t i n g a versus 6 thus obtained from Figure 4, we o b t a i n a parabola w i t h a minimum a at 6 = -0.14. Thus a "best f i t " of the experimental data i s obtained w i t h t h i s va.lue of 6. However, with the l i m i t e d data of Figure 3, i t i s not c l e a r whether the d i f f e r e n c e between x j ^ " T ] _ and T_2 = 0.86 x^ i s s t a t i s t i c a l l y s i g n i f i c a n t and hence f u r t h e r measure-ments are needed to a c c u r a t e l y determine the value of 6. IV.2 Carbon T e t r a f l u o r i d e 297°K and d e n s i t i e s below 5 Amagats i n Figure 5 along with those of Armstrong and Tward (1968). The r e g i o n of d e n s i t y i n which T^ i s found t o be p r o p o r t i o n a l to p i s i n d i c a t e d by the s o l i d l i n e . In f a c t we have found t h a t t h i s l i n e a r r e l a t i o n s h i p holds up to at l e a s t 50 Amagats, but the data at higher d e n s i t i e s are not shown here. Assuming t h a t eq. (3.55) holds i n t h i s r e g i o n , we o b t a i n i ) The T^ minimum i n CF^ Our experimental data of T are p l o t t e d as a f u n c t i o n o f p at (4.9) a = (2.1 I 0.05) x 10~ 3 sec/Amagat e f f A - 5 5 -Fluorine T, vs p 297 °K o Armstrong and Tward i t i j J_ J . . I L 1 0 -I 1 0 p (AMAGATS) gure 5. F l u o r i n e T versus d e n s i t y f o r CF 4 at room temperature and low d e n s i t i e s 2 The procedure used to evaluate C rr. i n CF„, and l a t e r i n CHF_ 1 e f f 4 3 and CF1,F, i s to f i r s t determine p . from the data of Figure 5. This . 3 mm then gives the value o f A using eq. (4-8) and the a n t i c i p a t e d f a c t that uij <<u>Q. Indeed, th.e | g j | value f o r CF^ has been found to be small and equal to 0.031 t 0.005 n.m. (Cederberg, Anderson and Ramsey 1964). Some in f o r m a t i o n on the g tensor f o r the symmetric-top molecules have a l s o 2 been obtained p r e v i o u s l y by Cox and Gordy (1956). The value of C can t h e r e f o r e be obtained from eq. (4.9). Using eqs. (3.55) and (4.1) i t i s e a s i l y shown that 2 2 2 _ P 1 pT, - p p'T ' (4.10) pmin . = — p ' T i " P T i f o r any T^ and T^ at the d e n s i t i e s p and p', r e s p e c t i v e l y . F i x i n g T|, p' i n the l i n e a r r e g i o n at high d e n s i t i e s , we then evaluate P m^ n from each of measured values of T^ at d e n s i t i e s below 10 * Amagat. As s i g n i n g equal s t a t i s t i c a l weight to each, of these measurements, we de f i n e the experimental value of P m ^ n as the average of the values so obtained. This gives p . = 0.048 I 0.002 Amagat mm which y i e l d s the value A(CF 4) = (2.71 1 0.10) x 10 sec-Amagat - 58 -Using the value a i n Table 2, eq. (4.9) gives C 2 £ £ = 41.0 t 1.7 (KHz) 2 On the b a s i s of the molecular beam measurement of C = 6.85 t a 0.35 KHz and the good f i t of the dependence of T^ on p using eq. (3.55) 4 2 as i n d i c a t e d by the dotted curve of Figure 5, we conclude that 2 << C and that a C = 6.41+0.12 KHz a The value of t h e r e f o r e agrees w i t h that of the molecular beam experiment w i t h i n experimental e r r o r . I t i s d i f f i c u l t to pla c e a p r e c i s e upper l i m i t on at t h i s time, s i n c e our a n a l y s i s i s based on the assumption that x^ = x j ^ • I f t h i s assumption i s a good r e p r e s e n t a t i o n of the system (that i t i s not so bad, i s i n d i c a t e d by the f i t of the experimental data of Figure 5 ) , then we can be more p r e c i s e as f o l l o w s . The e r r o r s quoted by O z i e r et a l . (1968) f o r C correspond to the 95% confidence l e v e l . a 2 Thus there i s a 95% p r o b a b i l i t y that C & > 6.50 KHz, i . e . that C e £ £ > 2 2 2 42.2 (KHz) . Our standard d e v i a t i o n of about . 1.7 (KHz) f o r C g £ £ 2 gives upper l i m i t to C g £ £ (95% confidence l e v e l ) o f approximately 2 44.4 (KHz) . Therefore, we can say, using eq. (3.58), that | | < 5 KHz i f the 95% confidence l e v e l of each experiment i s used. This can be considered as an extreme upper l i m i t to the value of |C^|. - 59 -An important i m p l i c a t i o n of t h i s low upper l i m i t to | | i s that the r o t a t i o n a l d i f f u s i o n model used to de s c r i b e molecular r e o r i e n t a t i o n i n l i q u i d CF^ by Rugheimer and Hubbard (1963) i s d e f i n i t e l y not a good one. The,reason i s simply because t h e i r T data i n l i q u i d CF^ could only be f i t t e d with c e r t a i n s p e c i f i c values of Cjj and under the c o n s t r a i n t of a known C value which i n turn gave a l a r g e value. . An a l t e r n a t i v e procedure f o r d e s c r i b i n g molecular r e o r i e n t a -t i o n i n l i q u i d s composed of "nearly s p h e r i c a l molecules" such as CF^ has been given by Bloom (1966). The average cross s e c t i o n "a" f o r s p i n - r o t a t i o n r e l a x a t i o n °2 i n CF 4 i s found, u s i n g eqs. (4.1)-(4.3) and the A ( C F 4 ) , to be 37.2 A . °2 This i s to be compared with the k i n e t i c cross s e c t i o n = 69.5 A o obtained using the value of CF 4 "hard-sphere" diameter d = 4.7 A ( H i r s h f e l d e r et a l . 1954). I t should be noted that the average cross s e c t i o n given by us was deduced d i r e c t l y from our T^ minimum data, w h i l e the value of obtained by Armstrong and Tward (1968) depends on the a v a i l a b l e i n f o r m a t i o n on the s p i n - r o t a t i o n coupling constants. Thus our value of i s considered to be more r e l i a b l e . To make sure that the magnetic d i p o l a r c o n t r i b u t i o n R^ i n CF 4 i s indeed n e g l i g i b l e , we simply use eq. (3.21), f = 0.2, I = 1/2 and o r„„ = 2.16 A to o b t a i n I-r Y F V l ( I + l ) T p/T, = 6 -± ^ - f 2 A -1 r F F 1 T2 -1 = 1.075 — Amagat-sec T l - 60 -This i n d i c a t e s that c o n t r i b u t e s only 0.23% of T ^ f o r - T ^ . IV.3 Fluoroform i ) Proton. T^  Measurements i n CHF^ Due to s i g n a l to noise c o n s i d e r a t i o n s , the proton T^ minimum i n CHFg has not been, observed. A p l o t of the proton T versus p f o r d e n s i t i e s below 10 Amagats i s shown i n Figure 6. On the b a s i s of T l these measurements we o b t a i n the value of — = 0.42 sec/Amagat, which i s the same as that obtained e a r l i e r at d e n s i t i e s below 20 Amagats (Johnson et a l . 1961, Dong and Bloom 1968). We are unable to 2 determine C rj~ f o r . p r o t o n d i r e c t l y because the value of A ^ (CHF„) e f f r proton 3 i s u n a v a i l a b l e due to the lack of a proton T^ minimum. However, i f we. argue that the values of the c o r r e l a t i o n time x^ f o r the proton and 2 f l u o r i n e spins are equal, an assignment to C e £ £ f o r proton i s p o s s i b l e wit h the a i d of the f l u o r i n e T^ minimum data. But the argument seems to c o n t r a d i c t the experimental f a c t presented i n Chapter VI that the temperature dependence f o r the f l u o r i n e and proton spins at low 2 d e n s i t i e s i s d i f f e r e n t . Thus the C „ r f o r proton assigned below i s e f f r • d o u b t f u l and only a proton T^ minimum i n t h i s molecule can provide a 2 d e f i n i t i v e determination of i t s C r j :. e f f The importance of the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n f o r proton i n t h i s low d e n s i t y regime can be determined using eqs. (3.23) and (3.25) as f o l l o w s , (4.11) (T—),- = 1.03 x 10 9 A _ ( C H F - ) ( — ) .. Amagat-sec" 1  v J ^ T ^ d i p proton^ V 7 proton 6 - 61 -p (AMAGATS) Figure 6. Proton versus d e n s i t y f o r CHF at room temperature and low d e n s i t i e s . - 62 -Using the value of A_n . determined below f o r the A , i n eq. f l u o r i n e proton (4.11), we estimate that the c o n t r i b u t i o n of R to the proton * i n T2 t h i s d e n s i t y r e g i o n i s (—) 11%. Therefore we may say that the dominant T l i n t e r a c t i o n f o r proton r e l a x a t i o n i n CHF i s the s p i n - r o t a t i o n i n t e r a c t i o n , i i ) The F l u o r i n e T^ Minimum i n CHF The f l u o r i n e T versus p plot,- shown i n Figure 7, does r e v e a l T 1 -3 a T^ minimum. The asymptotic value of — = 3.5 x 10 -sec/Amagat. i s i n agreement with p r e v i o u s l y reported values f o r the f l u o r i n e spins i n CHF (Johnson et a l . 1961, Dong and Bloom 1968). The F l u o r i n e T data at low d e n s i t i e s are t r e a t e d i n the same manner as f o r CF^ and the f o l l o w i n g r e s u l t s are obtained, p . = 0.045 t 0.002 Amagat Kmin & A £ l u o r i ! i e ( C H F 3 ) = (2.56 i 0.11) x 10 1 0 sec-Amagat C 2 f f (CHF ) = 48.1 + 2.1 (KHz) 2 f l u o r i n e °2 0 ^ ( f l u o r i n e ) = 33.3 A o 2 o The k i n e t i c cross s e c t i o n a, . i s equal to 58.9 A s i n c e d = 4.33 A k i n 1 (Birnbaum 1967). In the low d e n s i t y regime i n which the matrix elements o f the s p i n - r o t a t i o n i n t e r a c t i o n between d i f f e r e n t K s t a t e s p l a y no r o l e in nuclear s p i n r e l a x a t i o n , the value of f o r the proton s p i n i s expected to be i d e n t i c a l to that of t h e ' f l u o r i n e s p i n s , i . e . 1 0 Fluorine T i *s p CHF 3 2 9 7 °K Figure 7. F l u o r i n e T versus d e n s i t y f o r CHF at room temperature and low d e n s i t i e s . - 64 -A . (CHF„) = k r i . (CHF_: proton 3 f l u o r i n e 3 I f t h i s i s t r u e , we see i n the extreme narrowing l i m i t from eqs. (3.55) and (4.1) th a t the asymptotic values of T^/p f o r the proton 2 and f l u o r i n e spins enable us to determine C £ £ (proton) = 0.40 - 0.05 2 2 (KHz) i n terms o f the deduced C £ £ ( f l u o r i n e ) value i n t h i s s e c t i o n . In the absence o f any other i n f o r m a t i o n on the s p i n - r o t a t i o n i n t e r a c t i o n constants, we are only able to give the f o l l o w i n g r e l a t i o n -ship between C j | and f o r the proton and f l u o r i n e spins i n CHF^ using eq. (3.56) and the value of a and y i n Table 2. (4.12) C 2 £ f ( C H F 3 ) = 0.433 C|(2 + 0.366 C^C^ + 0.484 c£ The locus of allowed values of C J J and f o r the f l u o r i n e 2 spins based on eq. (4.12) and C £ £ £ f o r f l u o r i n e i s given m Figure 8. The shaded.areas i n t h i s p l o t i n d i c a t e some r e p r e s e n t a t i v e u n c e r t a i n t i e s i n the r e l a t i o n s h i p between and i n our experiments A s i m i l a r p l o t f o r the proton s p i n can be made too, but i t seems 2 p o i n t l e s s to present i t here as the C G £ £ f o r proton i s not yet d e f i n i t i v e l y determined. I t should be emphasized t h a t our i n a b i l i t y to determine C|j and s e p a r a t e l y i s due to the f a c t that the dependence of T^ on p i s compatible with the c o r r e l a t i o n f u n c t i o n s of the s p i n - r o t a t i o n i n t e r a c t i o n being governed by a s i n g l e time constant. I f we d i d not have = = ^thxT) experimental e r r o r , i t would be p o s s i b l e , - 66 -i n p r i n c i p l e , to determine both and C^. I t may be that more accurate measurements of T-^  as a f u n c t i o n of de:isity w i l l enable such a program to be c a r r i e d out. Even those measurements we are now able to make should be u s e f u l sin.ce they may help to r e s t r i c t the range of values of C|j and C^ over which molecular beam experimenters must search i n order to f i t t h e i r experimental data. Any great d i s c r e p a n c i e s which may occur between our r e s t r i c t i o n on Cj| and and molecular beam r e s u l t s would be of great i n t e r e s t , f o r they would r e q u i r e a r e - e v a l u a t i o n of the theory of nuclear spin r e l a x a t i o n due to molecular r e o r i e n t a t i o n i n gases. IV.4 Methyl F l u o r i d e i ) F l u o r i n e Measurements i n CH^F Again due to s i g n a l to noise c o n s i d e r a t i o n s , the f l u o r i n e minimum i n CH F has not been observed. The dependent of T^ on p f o r the f l u o r i n e s p i n i s given i n Figure 9. The value of T^/p f o r the f l u o r i n e s p i n i s found to be 0.044 sec/Amagat f o r d e n s i t i e s below 10 Amagats which i s the same as reported e a r l i e r (Dong and Bloom 1968). 2 Again the C r r f o r f l u o r i n e has not been determined d e f i n i t i v e l y and i t s b e f f d o u b t f u l assignment given i n Table 2 is.because of the d i f f e r e n t temperature dependence f o r the proton and f l u o r i n e spins at low d e n s i t i e s presented i n Chapter VI. i i ) The Proton T^ Minimum i n CH^F A p l o t of proton T^ versus p i s shown i n Figure 10 i n which a T^ minimum i s observed. The a n a l y s i s proceeds i n e x a c t l y the same way as f o r CHF . The asymptotic value of T /p f o r the proton spins - 68 -- 69 -i s 0.292 sec/Amagat which i s true up to about 24 Amagats, not shown i n the f i g u r e . Our data enable us to get p . = 0.018 1 0.002 Amagat and • ° mm • 11 2 A p r o t o n ( C H 3 F ) = (9.64 1 0.10) x 10 ' sec-Amagat. These lead to C g f f ( p r o t o n ) given i n Table 2 i f R f o r proton spins i s neglected. We may a l s o deduce 2 a value f o r C £^ ( f l u o r i n e ) given i n Table 2 as i n the case of CHF^ i f we assume that. A = Aj.. . . The n o n - o s c i l l a t i n g i n t r a - m o l e c u l a r proton t l u o r m e • d i p o l a r c o n t r i b u t i o n f o r the proton spins can be estimated using eqs. (3.24), (3.26), (3.27) and A ^ (CILF) as f o l l o w s , v J proton 3 T2 -1 (4.13) (p / T J , . - = 0.366 (—) _ Amagat-sec v j i ^ d i p proton b T 2 ' This c o n t r i b u t e s (—-) 10.7% of the proton s p i n r e l a x a t i o n r a t e at low d e n s i t i e s and should be accounted f o r i n determining C ^(proton) . 2 2 I f we assume that = x^, the (proton) becomes 3.46 (KHz) i n s t e a d of 3.86 (KHz) 2. The average cross s e c t i o n f o r s p i n - r o t a t i o n r e l a x a t i o n "o^ °2 ( i g n o r i n g the .10% e f f e c t of R ) i s found to be 61.7 A as compared to A . o *y o °kin = 5 0 " 1 computed with d = 4.0 A (Birnbaum 1967). Thus the c o l l i s i o n s i n CH^F are q u i t e strong which i m p l i e s t h a t - x^ may be a good approximation. The r e s t r i c t i o n s on Cj| and f o r the proton and f l u o r i n e spins due to our measurements are governed by (4.14) C 2 f f ( C H 3 F ) = 0.017 C2, + 0.079 C (j C ± + 0.627 - 70 -2 TABLE 2: Summary of molecular parameters and experimental values of C ^ The r e s u l t s f o r CHF^ .and CH F are p r o v i s i o n a l , as discussed i n the text 2 2 Molecule lOOOct y C e £ £ ( p r o t o n ) C £ £ ( f l u o r i n e ) (KHz) 2 (KHz) 2 CH 25.86 0 137.60* CF 0.92 0 — . 1.7 CHF 1.67 0.92 /T 0.40 + 0.05 48.1 + 2.1 3 '3 CH F 4.13 0.911- 3.86 t .39* 25.621 3.85 * From molecular beam experiments ( Y i et a l , 1967). # 2 2 R^ i s neglected. When R^ i s accounted f o r , C^^ (proton) = 3.46 (KHz) hB f l ZA ,1/2 « = w , y = ( l • o Again u s i n g the C e £ £ (proton) and C £ £ ( f l u o r i n e ) v a l u e s , we can p l o t the l o c i of allowed values of C|| and Cj^ based on eq. (4.14) f o r the proton and f l u o r i n e s p i n s , r e s p e c t i v e l y . However i t seems p o i n t l e s s to present these curves, here i n view of the absence of any p u b l i s h e d data on C J J and by other methods. A summary of v a r i o u s cross s e c t i o n s ' o f the above molecules i s given i n Table 3. IV.5 C h l o r o t r i f l u o r o m e t h a n e i ) F l u o r i n e T^ Measurements i n CF^Cl The f l u o r i n e spins i n CF^Cl are governed predominantly by the s p i n - r o t a t i o n i n t e r a c t i o n s . One would t h e r e f o r e expect the f l u o r i n e T^ i n CF^Cl to behave s i m i l a r l y to that i n CF^H. The dependence of T^ on p at 297°K i s shown i n Figure 11 f o r d e n s i t i e s between 0.3 and 50 Amagats. The value of T^/p i s found to be (3.45 1" 0.1) x 10 sec/ Amagat which i s the same as t^/p. f o r f l u o r i n e spins i n CHF^. The f l u o r i n e 2 T, minimum i n CF,C1 has not been observed and thus i t s C ~~ value cannot 1 . 3 e f f be determined. - 72 -TABLE 3: Summary of va r i o u s cross s e c t i o n s a , a... c and c Molecule °l °kin Vm-res. a r e s , (A ) (A Z) (A Z) (A") CH 17.3 45.5 CF, 37.2 69.5 4 CHF ' 33.3 58.9 88.3 1778 (J=0-> 1) 824 (J=l->2) CH F 61.7 50.1 132.7 633 (J=0-»-l) 546 (J=2^3) •k From microwave non-resonant a b s o r p t i o n and resonant measurements (Birnbaum 1967). CHAPTER V PHENOMENOLOGICAL THEORY OF SPIN RELAXATION FOR CX Y MOLECULES V.1 Nuclear Spin Symmetry and T r a n s i t i o n P r o b a b i l i t i e s In Chapter I I I , we have discussed the theory of nuclear s p i n r e l a x a t i o n f o r u n c o r r e l a t e d spins i n CX Y molecules. Mien nuclear s p i n symmetry e f f e c t s are i n c l u d e d f o r these molecules, which have Cg molecular symmetry, t h e i r r o t a t i o n a l s t a t e s ca.n be c l a s s i f i e d according to the three r e p r e s e i i t a t i o n s of the C^ r o t a t i o n a l p o i n t group. A l l s t a t e s with K = 3n belong to the A r e p r e s e n t a t i o n and have the X-type spins i n s t a t e s w i t h t o t a l nuclear s p i n I = 3/2. States E^ and E 2 have K = 3n + 1 and -(3n + 1 ) , r e s p e c t i v e l y and both have the X-type spins i n s t a t e s w i t h t o t a l s p i n I = 1/2. In f a c t , s t a t e s E^ and E 2 are equivalent and degenerate i n p a i r s . As a r e s u l t , t r a n s i t i o n s between A and E s t a t e s and between E^ and E^ s t a t e s except between K = 1 and K = -1 are between non-degenerate molecular s t a t e s so that the r e l e v a n t c o r r e l a t i o n f u n c t i o n s have only high frequency terms at molecular f r e q u e n c i e s . In t h i s chapter, we develop a theory f o r s p i n r e l a x a t i o n i n such a manner as to d i s t i n g u i s h terms i n ""Hi- a n d ri b e d i p 0 s . r . which couple molecular s t a t e s belonging to the same or d i f f e r e n t n u c l e a r s p i n symmetry sp e c i e s . The Overhauser e f f e c t s i n CX^Y molecules ( i . e . CHF^ and CH^F) a r e a l s o discussed w i t h i n the context of the above theory. - 75 -Now the appropriate molecular wavefunctions jJ,K,M;a, m_> m > X \ i n v o l v e the usual r o t a t i o n a l quantum numbers J,K,M, the nuclear s p i n symmetry species a.=A,E , or E and the nuclear magnetic quantum numbers 3 m (m = I m.) and m . For our purposes, i t i s a good approximation to x x i = i 1 -y w r i t e these wavefunctions as a product of a r o t a t i o n a l part and a s p i n p a r t , i . e . (5.1) |jKM;a,m> = |JKM>|a,m> where the r o t a t i o n a l eigenfunctions |JKM> are expressed i n terms of the usual r o t a t i o n matrices D?" as km (5.2) |JKM> = ( ^ i / V 2 and the ma t r i x elements o f are given by (Rose 1957, eq. (4.62)) (5.3) -<JKM|D^ J J ' K ' M ^ = ( ^ j ^ 1 ) 1 / 2 C ( . J 1 2J | K' k) C (J ' 2J | M'm) The s p i n wavefunctions |a, m^ , m > may be obtained as f o l l o w s . We f i r s t note t h a t (5.4) |A, 3/2, l/2> = |' + > 1 | + > 2 | + > 3 | + > 4 where | i >. i s the IIU = 1 1 / 2 s t a t e f o r the s p i n . Now we define a s p i n operator I f o r the X s p i n s , which i s appropriate to the C 7 symmetry of the molecule (see Figure C3 f o r molecular geometry), i . e . , - 76 -(5.5) t j ; k ) = 7 1 + e x p ( i |^ k) "t, + e x p ( - i |^ k)"?3 The wavefunctions |a,l/2,l/2> , f o r i n s t a n c e , are obtained by operating —J>f k) on |A,3/2,l/2> with-the lowering operator l ^ v , the s t a t e s corresponding to ct=A, and E^ being generated by k =0, 1 or -2 and -1 or 2, r e s p e c t i v e l y . A l l these s p i n wavefunctions |a,m> are given by Townes and Schawlow (1955) and the molecular wavefunctions are l i s t e d i n Table 4. Using the tr a n s f o r m a t i o n i n eq. (3.11), the i n t r a - m o l e c u l a r d i p o l a r K a m i l t o n i a n of eq. (3.3) can be r e w r i t t e n as (s.6, Kdip - j_2£^lp]k - j _ 2 C ^ ) 1 / 2 J_2 ( - D ^ V U where C5.7) i f \ - * [L<2>1? and [ L ^ ] * ^ and [ L ^ ' ' ] ^ X are expressed i n terms of the Y-spin operator —*» — " ~ * r k l 1^ = I y and the e f f e c t i v e X-spin operators I x defined i n eq. (5.5) as f o l l o w s , •r2 (5.8) uPhly = c-Dk ^ f - | Y , . c e ) i T ( 2 ^r,^ k ) ) > } L p Jk v J .3 1 2k v xy 1 p y x 1 x y Y V T ( 2 ) t ( 0 ) -(k) (5-9) [L M C 2>]« = (-D k - i y |Y 2 k C^| - V ^ - ^ 5 F — -• M r 17- k i - r - k xx e 6 + e o The d e r i v a t i o n s of eqs. (5.5)-(5.9) are presented i n Appendix C. - 77 -TABLE 4: Molecular Wavefunctions o f CX^Y Molecules m m Molecular Wavefunctions a y x • i rid t 1/2 3 / 2 {|j'KM>}{^}a 1a 2a 3 1/2 - { |JKM>}{.^} ( B 1 a 2 a 3 + o ^ B ^ + o ^ a ^ ) •1 /2 J • ( I J K M ^ ^ C a ^ ^ + B ^ B , + ) - 3 / 2 ' {|JKM>'}{gJ}B 1 3 2 f i 1/2 1/2 - {g 4 :}[|JKM>(3 1a 2a 3 + e 1 3 c ^ B ^ + e - 1 3 a ^ B , ^ ^ J U — u ^ ^ • - - . " 1 ^ 3 ° " " 1 " 2 M 3 ) ] - 1 / 2 i { ^ } [ | j K M > ( a 1 3 2 B 3 + e 1 3 B ^ ^ + e" 1 3 B ^ o ^ ) ] /3~ * 1/2 1/2 Same as E but w i t h s i g n i n exponentials reversed E 2 1 - 1 / 2 a = + > M | - > - 78 -To c a l c u l a t e t r a n s i t i o n p r o b a b i l i t i e s f o r the d i p o l a r i n t e r -a c t i o n between p a i r o f X-Y s p i n s , we note from the p e r t u r b a t i o n theory and the part of d i p o l a r Hamiltonian f o r X-Y spins given i n eqs. (5.6) and (5.8) that 2 2,2 Y Y "n R R UA'A 24TT x y „ i . , i „ (2 ) , - r * T^f-k), i ^ | 2 T , (5.10) W , = — £ <m'm' 1 J (I ,1' J) rn m > J„ (w , ) v ' m'm 5 6 1 y x 1 y ^ y x J ' y x ' zy m'm r y ^ H J J xy ' ct where W , denotes t r a n s i t i o n p r o b a b i l i t i e s f o r t r a n s i t i o n s between m'm nuclear s t a t e s la 'rn' > and |a,m>, and (5.11) J 2 y ( . ) = Re T [ C ( J 2 J ' | K k ) ] 2 | Y 2 r k ( e x y ) l 2 P J K M J K J'K' e - i [ u J - y a , J - C ( o J K - o ) j l K I ) ] T d x The k i n J (w) i s determined by the nuclear t r a n s i t i o n s a'<—KX i n eq. (5.10). Now PJJ^J' have to take i n t o account the nuclear s p i n syimnetry and they are given by Townes and Schawlow (1955) i n the high temperature l i m i t as f o l l o w s , 1/2 (5.12) Pjkm = I °C« ± e ) exp(-aJ 2-gK 2) f o r K^3n 1/2 ?JKM:zl~~~l3~ exp(-aJ 2-(3K 2) forK=0,3n S i m i l a r l y , t r a n s i t i o n p r o b a b i l i t i e s f o r d i p o l a r i n t e r a c t i o n between p a i r of X-X spins are obtained, using eqs. (5.6) and (5.9), as f o l l o w s , - 79 -4 f 2 , c 1 7 , ...a'a 12TT Y X „• , , i . „ (2 ) ,-p-(0) - f(-k), , , 2 , , ( 5 . 1 3 ) V , = —f— I < i ' m ' i MI- , 1 v J ) m m > J „ (OJ v 7 m'm 5 . 1 y x 1 u x ' x J 1 y x. 1 2u m r (ln-cos—rr- ) xx ^ 3 I t i s noted that W . i s non-zero only i f n=Aui. In w r i t i n g down the m'm J • above t r a n s i t i o n p r o b a b i l i t i e s , we ignore any i n t e r f e r e n c e e f f e c t between [ L ^ l f ^ and [ L ^ l f ^ i n the d i p o l a r Hamiltonian. This i s u k U k u* H indeed a good approximation to f i r s t order i n ~ — , because, f o r in s t a n c e , the matrix element<a'm'm | l z I + + I z I + i a m m > vanishes when x y 1 y x x x ! x y the sum over m i s performed. V . 2 R e l a x a t i o n M a t r i x f o r CX^Y Molecules To r e l a t e our experimental r e s u l t s at moderate d e n s i t i e s to the molecular c o r r e l a t i o n f u n c t i o n s , we have to di s c u s s e x p l i c i t l y the s i x t e e n s p i n l e v e l s system of a molecule of the type CX^Y. Suppose the system i s p r o p e r l y d e s c r i b a b l e by r a t e equations w r i t t e n as (5.14) ^ n = R(n - rf Q) where n i s a vect o r whose components n^ are the p o p u l a t i o n of the d i f f e r e n t l e v e l s and n Q i s the thermal e q u i l i b r i u m value of n and R i s a r e l a x a t i o n matrix. A formal d i s c u s s i o n of R has been given by Zussmann and Alexander (1968) . We w i l l f r e q u e n t l y use a n o t a t i o n (5.15) n. = n l a ,m - 8 0 -where m represents a l l n u c l e a r magnetic quantum numbers ( i . e . m, and m ) and a represents the molecular symmetry s p e c i e s , as before. In the h i g h temperature approximation i m p l i e d by (5.14), R i s a. symmetric m a t r i x , i . e . (5.16) R. .. = R. . and i t s o f f - d i a g o n a l elements are the r e s p e c t i v e t r a n s i t i o n r a t e s , F u r t h e r , i t i s obvious that (5.17) R.. = -I R. . K J n . I J 3 In a magnetic resonance experiment, we a l s o have time r e v e r s a l symmetry which i s (5.18) R , , = R , , am;a'm' a-m;a'-m' where the signs of a l l m i n the R.H.S. are reversed. Because of eqs. (5.16) and (5.18), the r a t e equations f o r the antisymmetric combinations , r , n. a n t i (5.19) n = n -n J a,m a,m a,-m separate from those of the symmetric ones. Furthermore, the and E^ s t a t e s are equivalent so that - 81 -(5.20) L l m ; A m b 2 m ; A m PE^m;E2m' , ^E^jE^m' As a r e s u l t , a l l the i n f o r m a t i o n we need i s contained i n the s i x dimensional v e c t o r n whose components are symmetric under E^, E 2 and antisymmetric under time r e v e r s a l , i . e . ^ • 2 1 ) nA.,m= nA , m - n A , - m 2 E,m = . \ (nE.,m-nE.,-m^ i = l l I To be s p e c i f i c , we w r i t e (5.22) A n A,3/2,1/2 A,1/2,1/2 ft A,-1/2,1/2 A,-3/2,1/2 ^E,1/2,1/2 ' E , - l / 2 , l / 2 y V n S n 2 - n ? V n 6 V n 5 V n 1 2 + n l 3 - n 1 6 n 1 0 - n l l + n . 1 4 - n l 5 where s t a t e s 1-12 are d e f i n e d i n t a b l e s f o r R and st a t e s 13-16 are equiv a l e n t to sta t e s 9-12, r e s p e c t i v e l y . Then, (5.23) %- if = R' (ff-lf) dt , v o where the r e l a x a t i o n m a t r i x R' can be shown to be given simply by - 82 -(5.24) R» , , = [6 (a, A) +{1/2)5 (a ,E) 6 (a ' , A) | (R , ,-R , ,) + <S (a, E) 6 (a' , E) Z ) K=i i K i K and 6(a,A) denotes a value of 1 i f a=A and zero otherwise. - The r a t e equations can be transformed f u r t h e r to (5.25) ^ M = R» ( M - M Q ) where (5.26) M x M M and M i s the magnetization along the e x t e r n a l magnetic f i e l d of the r n u c l e a r s p i n species (r = X , Y ) f o r molecules i n s t a t e s of symmetry a(a=A,E). n and p are de f i n e d i n (5.27) Tf = - j t If where (5.28) t 1 3 0 0 1 1 1 1.0 0 - 1 - 3 1-1 0 0-1 3 1-3 0 0 1-1 o o i - i o o \0 0 1-1 0 0 - S 3 -The p e r t i n e n t r e l a x a t i o n , matrix R" can be shown to be given by (5.29) R" = t R ' t X 2 where X i s a diagonal n o r m a l i z a t i o n matrix defined so as t X i s u n i t a r y . In Tables 5-8, we give the r e l e v a n t submatrices of R. A l l other elements i n R f o l l o w from eqs. (5.16)- (5 .18) and (5.20). We have included the more obvious nuclear matrix elements e x p l i c i t l y where t h i s was p o s s i b l e . We a l s o have t r i e d to s t r e s s s i m i l a r matrix elements by using s i m i l a r l e t t e r s . Thus, f o r example, R^  and d i f f e r only i n the e x p l i c i t dependence of the X - Y d i p o l a r i n t e r a c t i o n c o n t r i b u t i o n on m and would be equal when the t r a n s i t i o n s are dominated by s p i n - r o t a t i o n i n t e r a c t i o n . S i m i l a r l y corresponding terms i n Tables 5 and 6 can d i f f e r only because of the e x p l i c i t K dependence of the matrix elements. This i s probably unimportant i n the experimental s i t u a t i o n s because of the large number of d i f f e r e n t l e v e l s occupied f o r each species. There i s however no reason to assume that the o f f diagonal elements i n R (Tables 7 and 8) are s i m i l a r . R" i s given i n Table 9. I t should be emphasized that eq. (5.25) i s the simplest general form f o r the r e l a x a t i o n equations. Experimentally we cannot observe n and p and we" a l s o only observe the sum M^ +M^  and M^ +M^ . In g e n e r a l , one should expect s i x time constants x x y )' i n t h i s s o r t of problem. Since the expressions f o r R" are f a i r l y i n v o l v e d , i t seems worthwhile to make some general observations on the d i f f e r e n t terms i n R. As i s mentioned p r e v i o u s l y , the t r a n s i t i o n s between A and E s t a t e s i n v o l v e molecular c o r r e l a t i o n f u n c t i o n s having - . 8 4 -TABLE 5: Submatrix Am;Am' m X 1 3/2 2 1/2 3 -1/2 4 -3/2 5 3/2 6 1/2 7 -1/2 8 -3/2 m y 1/2 -1/2 • A 0 R l 3B 0 0 • 4W? A 3S R2 4B 0 • '3W 0 4S R 2 3B • 0 0 3S R l • 3W1 A 0 • 4W2 A 3W, • TABLE 6: Submatrix R E^m;E^m' 9 10 11 12 m x 1/2 -1/2 1/2 -1/2 y 1/2 •1/2 • • R 2 S' • R2 W2 • - 85 -TABLE 7: Submatrix R ^ , Am;Em' 1/.2 1/2 -1/2 1/2 1/2 -1/2 -1/2 -1/2 3/2 1/2 3D G. 3F 0 1/2 2E D H F •1/2 D 2E K H •3/2 G 3D 0 3K TABLE 8: Submatrix R„ _ , E m;E2m' 1/2 1/2 , -1/2 1/2 1/2 -1/2 -1/2 -1/2 1/2 1/2 2E" D" H" F" -1/2 D" 2E" K" H" 1/2 -1/2 H" K" 2E" D" -1/2 F" H" D" 2E" - 86 -TABLE 9: R e l a x a t i o n M a t r i x R' 1 U A , < A < u b , , b M M M M n p y x y x l f/ - ( A A + Y ) -Ce+a) Y A X 0 M A ~5(0+a) -(^ A +-|) "5a £ • -4a+23 * X A. J M b Y . ' .'-a - Og+Y) <J> 6 0 E 1 1 a ((; - (eE+£) 2a A •n x - 4a+ -1(3 e 2a -T - i(3a+6g) 5 b n 5 P 0 ^ 0 A -(3a+6g) . -T Where A^R-j+R +5S+5B+2H+4F+4K A =2R^+S'+B'+2(H+H")+4F+F"+4K+K" 3W +17W eA= — — - +B+S+| A+4D+— G + 2(F+K) eE=2W2-2(F+K)+4G-4D+B*+S'+2D"+F"+Kn T E6W,+R,+R +G+2F+H+2A+2B+2S+2K+2E+4D n 1 1 2 T H ~ W,+ +6B+| D-4 + ~ E + | H + | (K+F) p 5 1 5 5 5 5 5 5 5 a=F-K, 3=B-S3 Y=4D+G-H+2(E-F-K) 8 = 2 ( D - E ) + G + H - F - K , £ = 8 D - 3 G + 4 ( F + K ) + 2 E + H X = - R 1 + R 2 - G - F - K + H + 2 S + 2 B + 2 E - 2 D A = 6 ( D - E ) - G + 3 ( F + K - H ) ~ V •p" W2+ | A+ | (E-D)- | (G+K+F-H) P E-K+F+S' -B' -F" + K,,=a-S' -a' - 87 -high frequency terms at molecular frequencies. A l l elements i n R" E E ~* coupling M and M to the other components of M are of t h i s type. I t x . y i s a l s o obvious that a l l terms coupling the to M o r i g i n a t e only from the d i p o l a r i n t e r a c t i o n between the two s p i n species. I t i s f a i r l y s t r a i g h t f o r w a r d to use eq. (5.25) to o b t a i n e x p l i c expressions f o r the r e s u l t s of steady-state s a t u r a t i o n and Overhauser experiment. The general s o l u t i o n s f o r t r a n s i e n t experiments i n v o l v e s i x r e l a x a t i o n times, but can be obtained f o r m a l l y . These general expressions are however of l i t t l e p r a c t i c a l value. The i n t e r p r e t a t i o n of the a c t u a l experiments on C F H and C H F i s g r e a t l y s i m p l i f i e d by the f a c t that the measured f l u o r i n e T^ i s always s h o r t e r than the proton T^. Since the d i p o l a r i n t e r a c t i o n s f o r the proton and f l u o r i n e spins are comparable, t h i s i m p l i e s that the f l u o r i n e r e l a x a t i o n i s dominated by the f l u o r i n e s p i n - r o t a t i o n i n t e r a c t i o n s . I t •has been shown (Johnson et a l . 1961, Dong and Bloom 1968) that t h i s i n t e r a c t i o n i s indeed much l a r g e r f o r f l u o r i n e spins than f o r proton spins i n both molecules. V . 3 D i s c u s s i o n f o r C F ^ H ( i . e . Y E H , X = F ) Since the f l u o r i n e s p i n r e l a x a t i o n i s short and dominated by s p i n - r o t a t i o n i n t e r a c t i o n , i t f o l l o w s that (5 .30) W 2% .w ^ W2 = W and t h a t W i s much l a r g e r than a l l other elements i n R. The consequence - 88 -i s then t h a t n and p are not e f f e c t i v e l y coupled to the M so that n ^ n^ and p ^ p Q to lowest order i n 1/W. To t h i s order, we have from eq. (5.25) (5.31) ^ Mp = -2W(M™ - ' M ™ ) o si n c e a l l other terms i n the equation f o r M are s m a l l . For the proton s p i n s , we have to d i s t i n g u i s h two types of experiments. In a T^ measurement, we only i r r a d i a t e the proton spins a CL so t h a t M ^ M . In the same approximation, we o b t a i n f o r the proton r r O spins (5.32) ^ - - ( A A + y ) ( M £ - M * )+y(M^-M^ )-(3+a) ( M ^ - M ^ ) + a Q i * - M * ) O O J O (5.33) ^-M^ - Y ( M „ - M ^ ) - ( X E + Y ) ( M ^ - M J 2 ] )-a(Mp-M^ )+cb(Mp-Mp ) o o o o where a, 3, y and cb are defined i n Table 9, Mp i s determined by eq. (5.31) and (5.34) X = R + R + SS + 5B + 2H + 4F + 4K (5.35) X = 2R 2 + S' + B' + 2(H + H") + 4F + F" + 4K + K" For the T^ measurement, the l a s t two terms i n the R.H.S. of eqs. (5.32) and (5.33) can be neglected. In general one then expects two time constants - 89 i n a pulsed measurement. We observe only one. This i s c o n s i s t e n t with eqs. (5.32) and (5.33) i n two cases, a) I f | x ; - x E | , Y « x A , x E the two time constants would be very c l o s e and could be d i f f i c u l t to d i s t i n g u i s h e x p e r i m e n t a l l y , b) when I V X E I , << 1 Y only one of the time constants shows up i n the t o t a l magnetization A E M„ = M. + M,.. In t h i s l i m i t , H i i Ii ' dM X A + A r (5-36) H _ _A E _ _ F , _ , F . _ _ C M H ^ , H ) - R „ ( M H M H ) o o and At low d e n s i t i e s one probably has case ( a ) , f o r the i n t e r - s p e c i e s t r a n s i t i o n s r a t e y should be s m a l l . Moreover X^ ^ X^ i f they are dominated by the s p i n - r o t a t i o n i n t e r a c t i o n s and the d i p o l a r c o n t r i b u t i o n are s m a l l . We w i l l see that t h i s i s confirmed by the Overhauser experiments. At high d e n s i t i e s t h i s e x p l a n a t i o n seems i n a p p l i c a b l e . - 90 -However, one notes that D and E i n y a r e probably dominated by the AK = i 1 pa r t of the f l u o r i n e s p i n - r o t a t i o n i n t e r a c t i o n and could be large compared to the proton t r a n s i t i o n r a t e s i n |X^  - A^|. One thus probably has the case (b) at "high d e n s i t i e s . The s t e a d y - s t a t e Overhauser experiment i s adequately described as a measurement of M i n a steady-state experiment w i t h the f l u o r i n e spins being s a t u r a t e d at i t s Larmor frequency. Therefore, K ( 5 . 3 8 ) M£ = 0, ^ = 0 As a. consequence, we o b t a i n ( 5 . 3 9 ) - ( V Y ) ( M H ~ M H ) + Y ( M H _ M H ) + ^ + A ) M F _ A M F = ° o 'o 0 0 ( 5 . 4 0 ) Y ( M H - M H ) - X A E + Y ) C M I B r M H ) + A M F _ < ( ) M F = ° O 0 0 0 These give M„-M., 2 53+8a+3l.+a"-[-=-^r ] (53-B-'-aM). ( 5 . 4 1 ) o _ F M _ 2 2 H o 2 Y H X- —6— 2y+X where (5.42) X = f ( A A + A E ) , 6 = f ( A A - X E ) and we have used the f a c t that " u Yu 1 U E H 1 ri M o when a la r g e number of r o t a t i o n a l l e v e l s are populated (Appendix D). At very low d e n s i t i e s , only AK=0 t r a n s i t o n s e x i s t which give ( i . e . a l l t r a n s i t i o n p r o b a b i l i t i e s are zero except R, R, R', B, S, B',S') (5.44) AM|- I B-S Y F 1 B'-S' Y F M.. 2 1 . . „ . . . 2 2 2 R 1 + B'+S» 2 V s O y i y + B + s yH 2 Y H AA" AE At high d e n s i t i e s , we probably have — << 1 and thus can r e w r i t e eq. (5.41) as (-5 4 5 ^ A M H YF. SB+ 8a + 3'+a" M „ 0 2 X H o H which i s not a p p l i c a b l e to AK=0 t r a n s i t i o n s only because of y =0. From A M H eq. (5.45), TI—- a r i s i n g from t r a n s i t i o n s between A and E species i s H o f r 4 f i 1 A M H _ 8(F-K>5(B-S) + (B'-S') Y F 1 J M„ 2R1+R +R +5(B+S)+B!+S'+2(2H+4F+4K) 2 H 2 1 2 Y,, o H A s i m i l a r expression f o r E^ <—>- E^. t r a n s i t i o n s can be obtained from eq. (5.45) too. V.4 D i s c u s s i o n f o r CH F ( i . e . , X=H, Y=F) Because of the short f l u o r i n e , R^, R^ and R^ are much l a r g e r than the r e s t of t r a n s i t i o n p r o b a b i l i t i e s i n the r e l a x a t i o n - 92 matrix R. R i s equal to R because the mechanism f o r the f l u o r i n e i s the s p i n - r o t a t i o n i n t e r a c t i o n which i s of course unaffected by the proton m values. By examining Table 9, we note that Mp and MpJ are not e f f e c t i v e l y coupled^to n sin c e x a n d 6 a r e s m a l l . Also n r e l a x e s to n very f a s t . Thus n and p do not deviate from t h e i r thermodynamical e q u i l i b r i u m values a p p r e c i a b l y . To f i n d the proton T^ i n CK^F, we o' cy need to solv e the f o l l o w i n g two equations where the f a c t Mp ^ Mp has o been used. (5.47) —M = -fe + — n ('M -M )+E(M -M ) L } dt H A 5 s J U l i H J H H 1 o o dM E (5.48) ^ J L ) - C V O C ^ ) o o where 3W +17W (5.49) e A = —~0—- + B + S . + | A ' + 4 D + - | 2 - G + 2(F + K ) (5.50) e„ = 2WI - 2(F + K) + 4G ~ 4D + B' + S' + 2D" + F" + K" h 2 From eqs. (5.47) and 5.48), we o b t a i n dM (5.51) = - e C M H _ M H ^- AC MH- MH 3 o o dM' (5.52) ^ = - (A- fc)(M H-M H ) - ( e + ^ ) (M^-M^ ) o o where - 93 -A ,,E eA + EE EA - EE (5.53) MJ|J = M H - M H , E 2 ' " 2 I t i s , however, not obvious from eqs. (5.51) and (5.52) that the proton s p i n r e l a x a t i o n i n CH^F should be governed by a s i n g l e exponential decay s i n c e there are no AK = t 1 t r a n s i t i o n s i n the f l u o r i n e s p i n -r o t a t i o n i n t e r a c t i o n . The ste a d y - s t a t e Overhauser e f f e c t s i n t h i s molecule can be obtained i n the manner described i n l a s t s e c t i o n , i . e . p u t t i n g M = 0 H and =0. Noting that (5.54) M A /Mp = 1 o o we ob t a i n the f o l l o w i n g expression f o r AM /M , o M M (5B+8a+B'+a") A^-~ (50+12a-g'-a") (5.55) H" H Q e + - £ Yp' M u • * 4 r , 2 ° A / - 5 g , S i E - A ( - ) A + - K For t r a n s i t i o n s between E^ and E^ only , i t can be e a s i l y shown th a t E E M -M ^ ' 5 6 ) _^ ^o = B' - S 1 +F"-1(" ^ 2 M,E, 2Wi+B'+S'+2D"+F"+K" YH n 2 O At low d e n s i t i e s , o n ly the AK=0 t r a n s i t i o n s (both A and E species) c o n t r i b u t e to the Overhauser e f f e c t s and t h i s i s given by eq. (5.55) as - 94 -(5.57) For A <-(5.58) AM MTT 5(B-S) "3W +17W _____ + ~ A+B+S 2WI B ' - S' : B' + S' 6Y, E t r a n s i t i o n s , eq. (5.55) gives AM H 5e E(B-S)+e A(B ,-S l)+i0e F(F-K)-2_ A(F-K ) + | c[5(B-S)+8(F-K)+(B t-S')] = . _____ E E £ A + S E A + J 5 e I Y F , 2 6'H CHAPTER VI EXPERIMENTAL RESULTS AT HIGHER DENSITIES AND DISCUSSION VI.1 General Remarks Before p r e s e n t i n g the experimental r e s u l t s i n CF^, ^F I-I and CFH as a f u n c t i o n of temperature and pressure, i t i s worthwhile to comment on the high frequency terms mentioned i n our previous t h e o r e t i c a l , d i s c u s s i o n s of the s p i n - r o t a t i o n and i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s . The q u a n t i t y f , d e f i n e d i n eq. (3.14), i s f o r the case that [G ( T ) ] £ R E E i s time-independent. While the o s c i l l a t i n g components i n [G2 m("0 ] f r e e are not neglected, we can s i m i l a r l y d e f i n e ^ a s (-6'1-' J 2 m ^ = G 2 0(T)exp(imw JT)exp(-iwT)dx •£ where j^(w) i s given by eq. (3.15). The q u a n t i t y ^ must take on a value between f and 1, i . e . - ^> ~ ^' ^'ie u P P e r l i m i t 1 S readied i n the l i m i t of i n f i n i t e l y l a r ge " c r y s t a l f i e l d " (Bloom 1967) i n which case the molecules are " o r i e n t e d " i n the c l a s s i c a l sense. At l i q u i d -l i k e d e n s i t i e s , one would expect ^ to approach t h i s upper l i m i t and so the c o n t r i b u t i o n to nuclear spi n r e l a x a t i o n i n c l u d e s both the zero - 96 -frequency ( n o n - o s c i l l a t i n g ) component and the high frequency terms at molecular frequencies i n the i n t e r a c t i o n s . In d i l u t e gases, = f ? s i n c e only the broadened zero frequency component i n the i n t e r a c t i o n s i s e f f e c t i v e i n r e l a x i n g nuclear s p i n system. For s p h e r i c a l - t o p molecules, f ^ has been shown to take the value of 0 .2 (Bloom et a l 1967). For symmetric-top molecules, i s given by eq. (3.16) and i s found i n Chapter I I I , f o r CHF^ and CH^F, to take an even smaller value. That i s why we have seen that the proton spins i n CHF^ and CH^F are r e l a x e d predominantly by the s p i n - r o t a t i o n i n t e r a c t i o n at low d e n s i t i e s . I f the s p i n - r o t a t i o n i n t e r a c t i o n f o r the proton i n these molecules i s not too strong and i f we examine them to higher d e n s i t y , i t i s expected that increases i t s value from f ^ by gathering some high frequency terms from the d i p o l a r i n t e r a c t i o n . The change i n the value of ^ should be manifested by the proton T^ of these symmetric-top molecules. The high frequency terms i n the s p i n - r o t a t i o n i n t e r a c t i o n can i n p r i n c i p l e c o n t r i b u t e to the nuclear s p i n r e l a x a t i o n at high enough d e n s i t y . To examine t h e i r importance, we have to c a l c u l a t e the c o n t r i -b u t i o n to from the l a s t term i n G ^ q ( T ) of eq. (3.45) and compare to the zero frequency c o n t r i b u t i o n R^(0) of eq. (3.55). In the high temperature approximation, the AJ = i ' l terms i n eq. (3.45) are the same and can be e a s i l y shown to give (6 .2) R r(2JOj ) = 4T T 2 C ? [ * - -r-J- ] j . (u + 2JQ ) /• J • CK oJ d L6(a+3) 10a J Jl^ o o where Q, = -7—7— • Therefore, we o b t a i n - 97 -,r „, R (2Jfi ) C2, , • i , (03 + 2JQ ) (6.3) f. o _ __d ,- g 1 -, • 1 o o' R "(0) = c2 L6(a+3) " KV j ^ ) e f f This r a t i o i s of course small i f C, << C r r . In any case, i,(to + 2Jfi ) << d e f t 3 J l o o J^(DJ O ) i n d i l u t e gases s i n c e i s i n general much gr e a t e r than a> . Thus i t i s u s u a l l y an e x c e l l e n t approximation to neglect the h i g h frequency c o n t r i b u t i o n R^,(2Jfi Q) to nuclear s p i n r e l a x a t i o n . For n u c l e i which l i e o f f of the.symmetry a x i s i n a symmetric-top molecule, high frequency c o n t r i b u t i o n to n uclear s p i n r e l a x a t i o n can a l s o come from AK = 1 1 t r a n s i t i o n s which have not yet been worked out t h e o r e t i c a l l y . I t i s u s e f u l to make some general observations on the h i g h frequency components i n J„ (to) of eq. (5.11). These a r i s e from molecular t r a n s i t i o n s at j = AJ .= il, i 2 and k = AK = t l , 12 allowed by the Clebsch-Gordan c o e f f i c i e n t s i n eq. (5.11). The t r a n s i t i o n frequencies (6.4) Aw = j [ 2 J + j + 1)] fiQ + k[2K + k]0 where Q. = -—^— , vary f o r v a r i o u s i n i t i a l r o t a t i o n s t a t e s designated A 4TT1. A by t h e i r quantum numbers J and K. However, the K t r a n s i t i o n s w i t h j = 0 and a f i x e d k and K have the same frequency f o r d i f f e r e n t J . Hence t h e i r c o n t r i b u t i o n s , each at a p a r t i c u l a r frequency, to nuclear s p i n r e l a x a t i o n are enhanced by i n t e g r a t i n g over a l l J . By examining the Clebsch-Gordan c o e f f i c i e n t s C(J2J|Kk) and computing the i n t e g r a l s over J shown below, we note that f o r l o w - l y i n g K s t a t e s , the k = _2 t r a n s i t i o n s c o n t r i b u t e more to ^ 0 than the k = -1 t r a n s i t i o n s . This - 98 -i s because of the lead i n g term f o r k = t2 i n i n t e g r a t i o n over J i s p r o p o r t i o n a l to J . while f o r k = tl i t i s i n v e r s e l y p r o p o r t i o n a l to J The f o l l o w i n g i n t e g r a l s over J enter i n t o (6.5) ^ CO 2 J e dJ = — exp (-ay ) y (6.6) I " = j e " a J dJ = - \ E i ( - y 2 a ) Jy sCD 2 (6.7) I'" = e " a J dJ = | [~- exp (-y2a) + a E i ( - y 2 a ) ] Jy J • y where E i i s an exponential i n t e g r a l . Using eq. (6.5)-(6.7), we e a s i l y o b t a i n 2 2 2 , = i . - a E i ( - y a) . a a E:i.(-y a) 2 ' 2 2 exp(-y a) y exp(-y a ) and that I " / I ' and are << 1. For high K s t a t e s , the i n t e g r a l s 2 4 I" and I'" a f t e r being m u l t i p l i e d by the f a c t o r K and K , r e s p e c t i v e l y , from the Clebsch-Gordan c o e f f i c i e n t may become comparable to I' though t h e i r molecular frequencies are l a r g e r . We w i l l examine the various l o w - l y i n g K t r a n s i t i o n s l a t e r i n connection wi t h CHF^ to see i f they are capable of c o n t r i b u t i n g enough i n t e n s i t y to e x p l a i n i t s proton T^ at moderate d e n s i t i e s . F i n a l l y , we would l i k e to comment on the r o l e of nuclear s p i n symmetry i n e v a l u a t i n g f ^ of eq. (3.16). This i s necessary when we want to evaluate t r a n s i t i o n p r o b a b i l i t i e s ^ m i m « Mien nuclear spi n symmetry i s taken i n t o account, the c o r r e c t expression f o r PJJ,.-^ i n eq. (3.16) i s given by eq. (5.12) i n s t e a d of eq. (3.10). I t can be shown that the value of f so obtained i s the same using eqs. (5.12) or AA (3.10). When e v a l u a t i n g W , , the sum over K m f„ has to be r e s t r i c t e d ^ • a m'm 2 to K = t 3n only. The appropriate should be P = exp(-aJ 2-gK 2) JKM I E _ • . T2 7, J K = - 3 n l 2 J + 1 ) e x P ( " a J -3R ) 30(0+6) 1 / 2 2 2 exp(-aJ -0K ) EF where the spin-dependent part j u s t cancels. S i m i l a r l y , f o r W , the sum over K i n f„ has to be r e s t r i c t e d to K / 3n and i t s P T l„. i s 2 J K M P J K M • .xpC - 2 - B K 2 > 2 V TT I t i s found that there i s no e x p l i c i t K dependence i n f f o r both A and E species and that they are equal to the f of eq. (3.16). As a consequence, the primed and unprimed t r a n s i t i o n p r o b a b i l i t i e s denoted by the same l e t t e r i n Tables 6 and 5 are equal when only the i n t e r a c t i o n s between I and 1 ^ are considered, i . e . W' = W_, R' = R„, S' = S and y x 2 2 2 2 B' = B. Hence at low d e n s i t i e s , i f one can measure the average of the r e l a x a t i o n r a t e s of A and E s p e c i e s , the value of T^ due to d i p o l a r i n t e r a c t i o n thus c a l c u l a t e d w i t h nuclear s p i n symmetry i s the same as that without nuclear s p i n symmetry. This can be shown to be the case •in CHF- using eq. (5.37). - 100 -VI. 2 Carbon T e t r a f l u o r i d e i ) F l u o r i n e Measurements A p l o t o f f l u o r i n e data as a f u n c t i o n of d e n s i t y up to 50 Amagats at 297°K is"shown i n Figure 12. The value o f T^/p = (2.1 t 0.05) x _ 3 10 sec/Amagat. (Dong and Bloom 1968, Armstrong and Tward 1968) i s obtained from t h i s p l o t . This i s the same as obtained at low d e n s i t i e s presented e a r l i e r . For d i l u t e gases, i s p r o p o r t i o n a l to d e n s i t y as des c r i b e d i n eq. (3.59) where a.seT11 and T i s the absolute temperature i n °K, i . e . The temperature dependence of the f l u o r i n e T^ i n CF^ has been s t u d i e d above room temperature at s e v e r a l temperatures. The f l u o r i n e T^ at these temperatures shows a l i n e a r dependence on p as that shown i n Figure 12. By p l o t t i n g log T^/p versus l o g T shown i n Figure 13, we ob t a i n n=1.65 t 0,1 (Dong and Bloom 1968). In the same f i g u r e , we have i n c l u d e d the data of Armstrong and Tward (1968) who found that n=1.5. To w i t h i n experimental e r r o r s , our experimental r e s u l t s agree with those of Armstrong et a l . r o t a t i o n i n t e r a c t i o n . The t h e o r e t i c a l temperature dependence of T^ due to s p i n - r o t a t i o n i n t e r a c t i o n can be obtained from eq. (3.55). The 2 *"eff S-"-ven ky eq. (3.56) only depends on the moments of i n e r t i a o f the molecule considered. An e x p l i c i t T dependence comes from a i n R^,. I t 1/2 i s e a s i l y shown that x , given by eq. (4.2), i s p r o p o r t i o n a l to T (6.8) -n The f l u o r i n e spins are predominantly r e l a x e d by the s p i n -- K)2 -3 5 u E Log Tj //o vs Log T x C R o Armst rong and Tward 2-5 2-6 L o g T ( ° K ) Figure 13. Temperature dependence of T^/p f o r CF^, 2-7 - 103 -i f "o^is i n v e r s e l y p r o p o r t i o n a l to temperature (e.g. Armstrong and 3/2 Hanrahan 1968). Hence R i s found to p r o p o r t i o n to T . This i s i n agreement w i t h our f l u o r i n e T^ data'. VI.3 Fluoroform i ) F l u o r i n e T^ Measurements The f l u o r i n e T^ versus p p l o t at 297°K, shown i n Figure 14, obeys a simple l i n e a r r e l a t i o n p r e d i c t e d by eq. (3.59) up to p = 90 Amagats. The value of T^/p i s determined as 3.5x10 ' sec/Amagat as reported e a r l i e r . S i m i l a r p l o t s f o r f l u o r i n e T^  at higher temperatures were obtained i n order to f i n d the temperature dependence of the f l u o r i n e T^. A p l o t of l o g T^/p versus log T i s shown i n Figure 15 from which the value of n i s found to be 2.5 i 0.1 (Dong and Bloom 1968). Though the f l u o r i n e spins i n CHF are no doubt r e l a x e d by the s p i n -r o t a t i o n i n t e r a c t i o n , i t s temperature dependence deviates c o n s i d e r a b l y from the conventional 3/2 law described above and no p l a u s i b l e explana-t i o n f o r t h i s i s a v a i l a b l e a t . t h i s moment. i i ) Proton T^ Measurements While the f l u o r i n e T^ i s found p r o p o r t i o n a l to p over the range of d e n s i t y s t u d i e d , the proton 7 behaves i n a s t r i k i n g manner over the same d e n s i t y region (Dong and Bloom 1968). The proton T^ data as a f u n c t i o n of p at 297°K i s shown i n Figure 16. Some of the T^ measurements shown here were made w i t h a Fabri-Tek instrument computer to achieve b e t t e r accuracy and to confirm that the proton s p i n i s indeed r e l a x e d with a s i n g l e time constant. The novel f e a t u r e of the Figure 14. F l u o r i n e T as a f u n c t i o n of d e n s i t y f o r CHF at room temperature. .0 20 4 0 6 0 80 p (A MAG ATS) Figure 16. Proton T as a f u n c t i o n of d e n s i t y f o r CHF at room temperature. - 107 -proton i s checked against any paramagnetic impurity e f f e c t by performing proton measurements i n a gas sample c o n t a i n i n g 0.35% of oxygen. These measurements are presented i n the same f i g u r e and found to obey eq. (3.63).- Therefore i t i s c e r t a i n that paramagnetic i m p u r i t i e s cannot be a cause to the observed d e n s i t y dependence of the proton T^. P o s s i b l e r e l a x a t i o n of the proton spins due to c o l l i s i o n s w i t h the surface of the sample c e l l has a l s o been considered. Assuming th a t we have a one-dimensional d i f f u s i o n problem, we can c a l c u l a t e the time r e q u i r e d f o r molecules to d i f f u s e the rad i u s of the c e l l and compare w i t h the value of T^ at the den s i t y where T^/p s t a r t s to deviate from constancy. U n f o r t u n a t e l y , the d i f f u s i o n time t ^ i s not much l a r g e r than the T^ value as shown i n the f o l l o w i n g c a l c u l a t i o n . The s e l f d i f f u s i o n c o e f f i c i e n t D can be taken to have i t s k i n e t i c theory value (Sears 1954) (6.9) D = < V > 3p a, . o k m where <v> i s the average v e l o c i t y of the molecule and i s equal to — . Using o, . = 58.9 A of the CHF,, the value of D at p = 25 Amagats can 6 k i n 3 be c a l c u l a t e d from eq. (6.9) and i s used to f i n d (6.10) t < A 'd 2D 2 2 where <x > = 0.16 cm . The d i f f u s i o n time t ^ at p = 25 Amagats i s equal to about 31 sec and i s about four times longer than the T^ at t h i s d e n s i t y . The f a c t o r of four i s probably an over e s t i m a t i o n s i n c e - ICS -the molecules can d i f f u s e i n any d i r e c t i o n . Thus i t i s hard to conclude that there i s no surface e f f e c t on the proton T^ based on t h i s c a l c u l a -t i o n . To t e s t t h i s e x p e r i m e n t a l l y , we increased the surface area and decreased the average d i s t a n c e of the molecules from the surface by i n s e r t i n g s e v e r a l glass tubes (O.D. = 2 mm, I.D. = 1.5 mm) at the center of the gas c o n t a i n e r . The proton T^ at s e v e r a l d e n s i t i e s were found to be a b s o l u t e l y u n a f f e c t e d by t h i s change i n geometry. This leads us to conclude that the novel f e a t u r e of the proton T^ as a f u n c t i o n of p i s not due to any surface e f f e c t of the c o n t a i n e r ' s w a l l . Measure-ments of the proton T^ as a f u n c t i o n of p were made at s e v e r a l tempera-tures above the room temperature. They are presented i n Figure 17 as p l o t s of T^/p versus p i n which the f l u o r i n e T^ r e s u l t s at 297°K are i n c l u d e d f o r comparison. These s o l i d l i n e s f o r the proton have no t h e o r e t i c a l s i g n i f i c a n c e , however, and we have i n d i c a t e d , by the dashed l i n e drawn through the room-temperature p o i n t s i n Figure 17, an a l t e r n a -t i v e curve which f i t s the data as w e l l as the s o l i d l i n e . The tempera-tu r e dependence of the proton T^ at low d e n s i t i e s i s shown i n Figure 18 as a p l o t of log T^/p versus log T. The value of n i s found to be q u i t e d i f f e r e n t from that of the f l u o r i n e n u c l e i i n the same molecule and i s equal to 1.23. Part of t h i s discrepancy may be accounted f o r by the p a r t i a l c o n t r i b u t i o n to the proton s p i n r e l a x a t i o n due to the d i p o l a r i n t e r a c t i o n . For d i p o l a r i n t e r a c t i o n , the value of n can be shown to be 1/2 i n the weak c o l l i s i o n l i m i t . In any case, the d i f f e r e n t temperature dependence of the proton and f l u o r i n e spins i n the same molecule i s not to be expected i f they are predominantly r e l a x e d by the p (AMAGATS) Figure 17. P l o t of T^/p versus p f o r CHF„. The value of T^/p f o r the f l u o r i n e spins i s 3.5 msec/Amagat. 110. -G 6 < 0 < 5 0 - 4 u <D L o g T, /p v s L o g T P r o t o n C H F . 0 - 2 ± 2-4 2 - 5 L o g T ( ° K ) Figure 18. Temperature dependence of proton T^/p f o r CHF 3 d e n s i t i e s . - I l l same kind of i n t e r a c t i o n , at l e a s t at low d e n s i t i e s . We have s t u d i e d the proton as a f u n c t i o n of the t o t a l d e n s i t y p i n a CHF--He and a CHF -Ar mixture at 297°K. To prepare a 61.6% CHF_ mixture w i t h He or Ar, we f i r s t put 200 p s i of He or Ar i n t o a c o ntainer and then added i n CHF- to give a t o t a l f i n a l pressure of 600 p s i . The d e n s i t y of CHF_ i n the mixture as a f u n c t i o n of pressure was determined by the proton magnetic resonance s i g n a l s t r e n g t h . The proton T^ data i n these mixtures are presented i n Figure 19. They behave i n a s i m i l a r f a s h i o n as the CHF- sample. The f l u o r i n e T\ i n the CHF„-He mixture was again found to obey a l i n e a r 1 5 • ° J dependence on p i n the same de n s i t y r e g i o n s t u d i e d and the value of T./p i s equal to 2.73 msec/Amagat. i i i ) Overhauser Measurements In order to see whether the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s t a r t s to play a stronger r o l e at high d e n s i t i e s i n the region of d e n s i t y s t u d i e d , the steady s t a t e Overhauser experiment was performed at room temperature and v a r i o u s . d e n s i t i e s by monitoring the proton M s i g n a l w h i l e the f l u o r i n e spins were e i t h e r saturated or not saturated. A p o s i t i v e Overhauser enhancement was observed i n t h i s gas f o r d e n s i t i e s higher than 20 Amagats which i n d i c a t e s that the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s do c o n t r i b u t e to the proton s p i n r e l a x a t i o n i n that d e n s i t y r e g i o n . In Figure 20, we show the Overhauser e f f e c t experimental r e s u l t s . The e r r o r bars shown denote one standard d e v i a t i o n i n our measurements. For the sake of d i r e c t comparison, a p l o t of p/T^ versus p f o r the proton at 297°K i s a l s o shown i n the same f i g u r e . The observed s i z e p ( A M A 6 A T S ) Figure 19 Proton T , . versus 0 f o r CHF_-He and CHF_-Ar mixtures at room temperature. 1 y 5 3 - 114 -of the Overhauser enhancement w i l l be seen to be smaller than p r e d i c t e d from the data. i v ) I n t e r p r e t a t i o n We have demonstrated that the proton s p i n i n CHF., i s re l a x e d predominantly by the s p i n - r o t a t i o n i n t e r a c t i o n at the low den s i t y r e g i o n . Inference i s drawn from the zero Overhauser enhance-ment and the c a l c u l a t i o n of the zero frequency c o n t r i b u t i o n from the d i p o l a r i n t e r a c t i o n . Furthermore, at high d e n s i t i e s , both the s p i n -r o t a t i o n and the d i p o l a r i n t e r a c t i o n s are r e s p o n s i b l e f o r i t s r e l a x a t i o n . As discussed i n s e c t i o n VI.1, the low de n s i t y d i p o l a r c o n t r i b u t i o n to the proton s p i n r e l a x a t i o n r a t e can be e a s i l y c a l c u l a t e d w i t h or without n u c l e a r s p i n symmetry c o n s i d e r a t i o n s and the r e s u l t i s the same f o r both cases i n CHF^. I t i s estimated i n eq. (4.11) that c o n t r i b u t e s only 11% of the proton T ^~ below 20 Amagats i n CHF^ at 297°K i f x„ = t n and A A„, . . The percentage would be 2 1 proton f l u o r i n e 1 lower i f the c o l l i s i o n s among CHF^ molecules were weak sin c e x 2 would then be sma l l e r than x (Bloom and Oppenheim 1963). Using eqs. (5.10) and (5.13), we can c a l c u l a t e a l l the t r a n s i t i o n p r o b a b i l i t i e s i n R. In the short c o r r e l a t i o n time l i m i t , i . e . ( U K ^ - W J , ^ , ) T << 1, we ob t a i n (6.11) F=fC, K=kC, H=hC (6.12) F"=4fC", K"=4kC", HM=hC" (6.13) B 1 =B=f L, S'=S=kL, 4R2=|-(R1 + R 2)=hL where - 115 -2 2 2 2A TT YH Y F * (6.14) £=h=A»/4, k=A'/24, A'= ——g rHF and (6.15) L = J 2 ( 0 ) / C"=J 2'(0), C=JI(0) AMH Using eqs. (6 .11) - (6.15) , we o b t a i n - — = 44.2% f o r AK=0 and f o r ''H A <—> E t r a n s i t i o n s from eqs. (5.44) and (5.46), r e s p e c t i v e l y , and s i m i l a r l y f o r t r a n s i t o n s between E and E„ sp e c i e s . In f a c t , the AM H same value f o r - — i s obtained by n e g l e c t i n g s p i n symmetry e f f e c t s ii (Abragam 1961, pg?•295). The maximum t h e o r e t i c a l Overhauser enhancement one expected to observe below p= 20 Amagats should be 0.44 x 11% = 4.85%. That we d i d not detect t h i s may i n d i c a t e a smaller R value at these d e n s i t i e s than 11% of the proton T^1. Our Overhauser experiments tend to support the idea of high-frequency d i p o l a r c o n t r i b u t i o n s i n R^ being brought i n to r e l a x the proton s p i n at p> 20 Amagats. These high frequency d i p o l a r terms A^ are accomplished e i t h e r w i t h nuclear s p i n conversions between A and E or E^ and species when k/0, or without nuclear s p i n conversions when k=0 and j / 0 . .The t h e o r e t i c a l Overhauser enchancement due to high frequency d i p o l a r terms i s given A (6.16) - ^ - r — 44.2% A + A . s . r . d where A s_ r_ = R P u t t i n g A d = ^  - A = X^^ + A d = ^ from - 116 -the proton data, we obt a i n a p r e d i c t e d value of 12% f o r the f i r s t " s t e p " i n the Overhauser measurements. A lower bound value of 8.4% i s p r e d i c t e d i f T^/p takes an e r r o r of t 5%. The experimental value of Overhauser enhancement given i n Figure 19 seems to overlap with t h i s extreme lower bound value. The a d d i t i o n a l increase i n the Overhauser enhancement at even higher d e n s i t y i s A , + A ' X . (6.17) ( I + A f - ) 44.2% ' • d A L A = r T T T ( 1 - r-) 4 4 - 2 % a Using A^ = — j - j j j j ' ^ + ~ "22~0 ^ r o m t n e P r o t o n ^ data, we get a t h e o r e t i c a l change i n Ovethauser enhancement of 11%. Thus we should see a maximum t o t a l of 23% Overhauser enchancement at d e n s i t i e s above the f i r s t "step". That the Overhauser enhancement p r e d i c t e d t h e o r e t i -c a l l y i s l a r g e r than the experimental Overhauser enhancement i s not yet understood. I t may be becuase the proton T^ i n CHF^ a l s o r e c e i v e s c o n t r i b u t i o n from the j/0 terms i n the s p i n - r o t a t i o n i n t e r a c t i o n at high d e n s i t y . We w i l l see l a t e r that a s i n g l e l o w - l y i n g K t r a n s i t i o n (say K=0, k=*2) i s incapable of g i v i n g the f i r s t "step" i n Figure 17. We b e l i e v e that the breaks i n the dependence of T^/p on p a r i s e when groups of r o t a t i o n a l l e v e l s undergo c o l l i s i o n s roughly at the same r a t e as the c h a r a c t e r i s t i c frequencies of the o s c i l l a t i n g terms i n the d i p o l a r i n t e r a c t i o n s . Perhaps there i s an averaging e f f e c t between groups of p o s i t i v e and negative frequency components i n the i n t e r a c t i o n s so as to give a lower frequency component which must then be broadened - 117 -i n order to be e f f e c t i v e i n r e l a x i n g the s p i n system. The c o l l a p s i n g of the p o s i t i v e and negative i n v e r s i o n l i n e s to a non-resonant abs o r p t i o n at the zero frequency i n NH_ as the pressure i s increased has been observed i n microwave measurements and discussed t h e o r e t i c a l l y (Ben-Reuven 1965, Birnbaum 1967). We know that molecular c o l l i s i o n s cannot induce nuclear s p i n t r a n s i t i o n s because t h e i r processes are purely e l e c t r i c . Thus i t seems impossible to couple a p o s i t i v e frequency component with i t s negative counterpart i n the K t r a n s i t i o n s as t h i s r e q u i r e s c o l l i s i o n s to change nuclear spin species. I t remains to be examined how those high frequency terms i n the d i p o l a r i n t e r a c t i o n s s t a r t to c o n t r i b u t e to nuclear spin r e l a x a t i o n as the d e n s i t y i s increased. To c a l c u l a t e the t o t a l d i p o l a r c o n t r i b u t i o n R without s p i n symmetry c o n s i d e r a t i o n , we merely r e p l a c e f ^ by (jf i n eq. (3.23) and put ^  = 1 to o b t a i n (6.18) P R = 2.39 x 10 1 0A (CHF_) — v ' A proton 3 Assuming that A A~, . = 2.56 x 10 1 0 sec-Amagat and b proton f l u o r i n e T2 = T l ' w e t n e r e r " o r e o b t a i n p/T^ = 6.1 Amagat-sec 1 which i s l a r g e r than R ^ of the proton. To account f o r the proton s p i n r e l a x a t i o n r a t e at low d e n s i t i e s by R ^ o n l y , we need ^ _ 0.4. However we have found that excluding a l l the o s c i l l a t i n g terms which are unimportant at low d e n s i t i e s f = 0.043 f o r CHF-. To get the f i r s t break at 297°K i n Figure 17, has to increase i t s value by only 0.15. Hence i t appears that R ^ has the s t r e n g t h to r e l a x the proton s p i n i n CHF as the density i n c r e a s e s . 1.18 F i n a l l y , we wish to show how one can, i n p r i n c i p l e , c a l c u l a t e the c o n t r i b u t i o n to as s o c i a t e d with the i n d i v i d u a l l o w - l y i n g K t r a n s i t i o n s . For i n s t a n c e , l e t us consider the t r a n s i t i o n from a = A, K = 0 to a = E 2 , K = 2. From eqs. (5.34)-(5.37), we see that (6.19) -p- = 2H + 2F + 4K 1 due to A •<—*- E conversions. To c a l c u l a t e the q u a n t i t i e s H, F and K, we cx' a merely use the expression f o r W , of eqs. (5.10)- (5.12). Now 3F = WAE 1/2 3/2;-1/2 1/2 2 2f[2 = ~- YX lY 1 l Y 2 j - 2 ( e x y ) l 2 | < : 1 / 2 V 2 | T ( 2 ) ( f , lJ" 2 ) )f- l/2 l/2> r x y 0 0 1 / 2 2 x I ^ [ C ( J 2 J | 0 2 ) ] 2 3/2 e " a J j.(4fi.) J=2 b A" where the Larmor frequencies t'oT, w , to <<|Q.|, and are neglected i n 1 J x y 1 A 1 o 2 ^ 1/2 2F - — Y x Y y I Y re il 2 ( a + 3 ) i f4« ) 2 F " 100 6 1 ^2,-2 l xy J 1 r J 2 l 4 V r V'TT xy S i m i l a r l y , we o b t a i n H = 6K = F 2 .2,2 xy - 119 -Using.the molecular constants f o r CHF given i n Table 1 and assuming th a t p_„ = A.-, . (CHF„1 = 2,56 x 10 sec-Amagat, we o b t a i n i n the K 2 f l u o r i n e v V  b short c o r r e l a t i o n time l i m i t ( i . e . , j 2 - 2T 2 ) p/T = 3.13 x 10 3 Amagat-sec 1 For K=0 to K=-2,a =E^, p/T^ takes the same value as above. We note that the t r a n s i t i o n s from K- = _2 to K=0 have a frequency of - 4 ^ and can be shown to c o n t r i b u t e about 1/2 of the K=0 to K=2 c o n t r i b u t i o n to p/T^ because of P j ^ and eqs. (6.5)-(6.7). Suppose we can add them -3 -1 together, then p/T^ = 9.4 x 10 Amagat-sec due to K=0, k=i2 t r a n s i -t i o n s . In Figure 19, the f i r s t " s tep" i n p/T correspondings to b,{~)--1 1 0.89 Amagat-sec . Therefore the i n t e n s i t y from K=0, k=_2 only accounts f o r 1.04- of. the f i r s t "step". I t can be shown that p/T^ due to A <—> E t r a n s i t i o n s with k=_2 decreases monotonically w i t h K. S i m i l a r c a l c u l a t i o n s can be done f o r A •<—> E t r a n s i t i o n s with k=_l. A few of them are given i n Table 10. I t can be shown that the maximum c o n t r i b u t i o n to p/T^ due to t h i s type of t r a n s i t i o n occurs at K - 15. The l o w - l y i n g K t r a n s i t i o n s f o r E^ <—> E^ s p i n conversion are expected to be s i m i l a r to those of A '<—> E t r a n s i t i o n s . VI.4 Methyl F l u o r i d e i ) F l u o r i n e T Measurements The f l u o r i n e T^ data at 297°K as a. f u n c t i o n of d e n s i t y below p = 45 Amagats are presented i n Figure 21. A l i n e a r r e l a t i o n between -3 T^ and p is.obeyed and the value of T^/p i s determined as 44 x 10 - 120 -TABLE 10: C o n t r i b u t i o n s to p/T^ of some l o w - l y i n g K t r a n s i t i o n s i n CHF, at 297°K T r a n s i t i o n s Spin Molecular • Conversions Frequency-C o n t r i b u t i o n Percent of to p/Tq Assuming the j 2 = 2 r 2 (Amagat-sec" ) F i r s t "Step 1 K=±&>K=±2 K=±2->K=0 A -> E E -> A 2 x 3.13 x 10" 2 x 1.56 x 10" 1. 04% K=±0->K=+1 K=±_»-K=0 A -> E E + A ±fl. 2 x 1.92 x 10 2 x 0.82 x 10 -4 0.06% K=±2+K=±3 K=±3-*K=±2 E -+ A A E 2 x 1.3 x 10 2 x 2 . 1 x 10" 0.76% 1(=±3->K=±4 K=±4->K=±3 A -> E E + A ±7fi; 2 x 4.0 x 10 2 x 1.64 x 10 -3 1.34% K=±9+K=±10 K=±10->K=±9 A ->- E E -> A ±195. A 2 x 1.01 x 10 2 x 4.4 x 10" 3.26% K=±15->-K=±16 K=±16->K=±15 A -y E E -y A ±31Q A 2 x 1.06 x 10 2 x 4.52 x 10" 3.40% K=±18+K=±19 K=+19->K=±18 A -y E E -»- A ±37Q A 2 x 9.8 x 10 2 x 3.84 x 10" -3 3.06% - 122 sec/Amagat as reported e a r l i e r . S i m i l a r p l o t s f o r the f l u o r i n e at higher temperatures were obtained i n order to study i t s temperature dependence. A p l o t of l o g T^/p .versus log T i s shown i n Figure 22. The value of n i s fo_und to be 1.63 ± 0.05 (Dong and Bloom 1968) and i s t h e r e f o r e c l o s e to the conventional 3/2 law f o r the s p i n - r o t a t i o n i n t e r a c t i o n . Thus we can say that the f l u o r i n e s p i n i n CFi^F i s pre-dominantly r e l a x e d by the s p i n - r o t a t i o n i n t e r a c t i o n . i i ) Proton T^ and Overhauser Measurements The proton T^ i n CH^F behaves i n a s i m i l a r manner as the proton T^ i n CHF 3 d e s c r i b e d e a r l i e r . A p l o t of the proton T^ versus p at 297°K i s shown i n Figure 23. The e f f e c t of paramagnetic i m p u r i t i e s was s t u d i e d by adding 0.6% of oxygen to a gas sample and i t s proton T^ was shown a l s o i n Figure 22 as a f u n c t i o n of d e n s i t y . The novel feature of the proton T^ i s again not a t t r i b u t a b l e to paramagnetic i m p u r i t i e s . A p l o t of T^/p versus p at three d i f f e r e n t temperatures f o r the proton spins i s shown i n Figure 24. In the same f i g u r e , T^/p remains constant f o r a l l the d e n s i t i e s s t u d i e d at 297°K f o r the f l u o r i n e spin which i s compared to the novel " s t e p s " presented i n the proton T^ data over the same d e n s i t y r e g i o n . The temperature dependence of the proton T^ at low d e n s i t i e s i s shown i n Figure 25 as a p l o t of log T^/p versus log T. The value of n i s found to be 1.13 which i s smaller than the n value of the f l u o r i n e s p i n . .It has been shown i n Chapter IV that c o n t r i b u t e s 10.5% of the proton s p i n r e l a x a t i o n r a t e which may decrease the value of n f o r the proton s p i n s . The time to d i f f u s e to the w a l l s can be c a l c u l a t e d as i n the case of CHF ' I t i s found to be about 22 sec at 60 Figure 23. Proton T as a f u n c t i o n of d e n s i t y f o r CF13F at room temperature. 3 0 0 ~ © _ 2 0 0 1 0 0 T , / p v s p C H 3 F P r o t o n 2 9 7 ° K o o P r o t o n 3 5 0 ° K A - A ^ P r o t o n 4 3 3 ° K F l u o r i n e 2 9 7 ° K 0 1 0 2 0 3 0 ( A M A G A T S ) 4 0 Figure 24. P l o t of T^/p versus p f o r CH_F . - 126 -0 - 4 r 0-3 k < u 0 - 2 0 - 1 2 5 L o g T j / p v s L o g T P r o t o n C H 3 F 2 - 6 L o g T ( ° K ) 2 - 7 Figure 25. Temperature dependence of proton T /p f o r CH F at low d e n s i t i e s . - 127 -p = 30 Amagats which i s about three times of the at the same d e n s i t y The steady s t a t e Overhauser experiment, i s performed at the room tempera-t u r e and various d e n s i t i e s by again monitoring the proton M s i g n a l w h i l e the f l u o r i n e .spin i s e i t h e r saturated or not saturated. A p o s i t i v e Overhauser enhancement i s observed f o r d e n s i t i e s above 25 Amagats which i n d i c a t e s that the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s do c o n t r i b u t e to the proton s p i n r e l a x a t i o n i n that r e g i o n of d e n s i t y . The Overhauser measurements are presented i n Figure 26. In the same f i g u r e , a p l o t o f p/T^ versus p f o r the proton at 297°K i s a l s o presented f o r comparison. i i i ) I n t e r p r e t a t i o n Again we have demonstrated that the proton spins i n CH-F are predominantly r e l a x e d by the s p i n - r o t a t i o n i n t e r a c t i o n at low d e n s i t i e s . At high d e n s i t i e s , both the s p i n - r o t a t i o n and the d i p o l a r i n t e r a c t i o n s are r e s p o n s i b l e f o r the proton spi n r e l a x a t i o n . To evaluate,the t o t a l d i p o l a r c o n t r i b u t i o n R^, we merely rep l a c e both i n eq. (3.24) by ^ 2 = 1 to get p/T^ = 5.5 Amagat-sec ^ i f x. = x i s assumed. At low d e n s i t i e s , i f we de f i n e f such that 1 1 Av f 5.5 = pR (0) Av then f = 0.067. To account f o r the step i n the proton T. at 297°K. Av / A M A G A T V sec Overhauser (o/o i n c r e a s e ) enhancement - 821 -we need to acquire from those high frequency terms an a d d i t i o n a l value of 0.26. One would expect to f i n d a smaller Overhauser e f f e c t i n t h i s molecule than i n CHR^, s i n c e the d i p o l a r i n t e r a c t i o n between the proton and f l u o r i n e spins i n v o l v e s only one f l u o r i n e s p i n . As shown e a r l i e r , the c o n t r i b u t i o n to the proton s p i n r e l a x a t i o n r a t e from R - A M H at.low d e n s i t i e s i s 10.7% f o r = T ^ . The ^ — i n t h i s d e n s i t y r e g i o n i s found to be 7.28% by using eqs. (5.57), (6.13), and = 0.206A'L, W = (1/16)A'L, A = 3A'L and W = 0.206A'L. The p r e d i c t e d experimental Overhauser enhancement i s t h e r e f o r e equal to 0.073 x 10% = 0.73% which A M has not been detected i n our double resonance experiments. The ^r-^-H f o r A *- E t r a n s i t i o n s found from eq. (5.58) using L = C ( i . e . neglect any e x p l i c i t K dependence), H = 1/4 A'C, G = 3/8 A'Cj E = D = 0.134A'C' and eqs. (6.11), (6.13) i s equai to 5.7% only. This gives an Over-hauser enhancement of 0.294 x 5.7% = 1.68% i f the "step" i n p/T^ versus p a r i s e s from A <—>• E t r a n s i t i o n s . However we have observed a much l a r g e r value f o r the Overhauser enhancement at that d e n s i t y r e g i o n . Thus i t i s b e l i e v e d that the "step" i s p o s s i b l y not due to A-< E^ AM H t r a n s i t i o n s . The n— f o r t r a n s i t i o n s between E-, and E„ species i s H . 1 2 * found to be 36.6% from eq. (5.56) with D" = 0.394A'C", eqs. (6.12), (6.13) and L = C". This can give a "step" i n the Overhauser e f f e c t measurements of 10.8% which i s about double our experimental value.. This d i s c r e p -ancy may perhaps be accounted f o r by the h i g h frequency terms i n the proton s p i n - r o t a t i o n i n t e r a c t i o n s a r i s i n g from AK = tl or/and AJ = 11. CHAPTER VII • SUMMARY AND SUGGESTIONS FOR FURTHER WORK Using modern s i g n a l averaging techniques, we have been able to measure the s p i n - l a t t i c e r e l a x a t i o n , time T^ i n low d e n s i t y gases even at r e l a t i v e l y low magnetic f i e l d s . In t h i s work, we have i n v e s t i g a t e d the accuracy w i t h which s p i n - r o t a t i o n c o u p l i n g constants i n polyatomic molecules can be determined from T^ measurements near i t s c h a r a c t e r i s t i c minimum. A d i r e c t measure of the average cross s e c t i o n f o r s p i n - r o t a t i o n r e l a x a t i o n was obtained from the T^ minimum i f the s p i n system i s predominantly r e l a x e d by the s p i n -r o t a t i o n i n t e r a c t i o n . Tt i s of i n t e r e s t to compare our value of "o^ with the non-resonant absorption cross s e c t i o n a i n microwave 1 non-res. pressure broadening measurements.. In p r i n c i p l e , values f o r and a i n polyatomic molecules can y i e l d v a l u a b l e i n f o r m a t i o n con-non-res. cerning a n i s o t r o p i c i n t e r - m o l e c u l a r f o r c e s , though the necessary advances i n molecular theory f o r t h i s to be p o s s i b l e have not yet been achieved. Since the s p i n - r o t a t i o n tensor f o r CH^ has been determined w i t h g r e a t e r accuracy by molecular beam measurements, we t h e r e f o r e used i t as a t e s t system to check our methods, of o b t a i n i n g information on the s p i n - r o t a t i o n c oupling constants. Our experimental data, a f t e r being f i t t e d with the known C and C,, support the assumptions that the - 131 -c o r r e l a t i o n f u n c t i o n s can. be w r i t t e n as a product of a " f r e e " molecule c o r r e l a t i o n function, and an exponential "reduced" c o r r e l a t i o n f u n c t i o n , and t h a t , w i t h i n the experimental accuracy, x^ =t]l2' ^ e observation of a f l u o r i n e minimum i n CF^ enabled us to place an upper bound to i t s |C^| value which i s lower than that obtained by molecular beam methods. I t i s t h e r e f o r e f r u i t f u l to i n v e s t i g a t e at low d e n s i t i e s i n those s p h e r i c a l - t o p molecules whose C but not C, have been measured J- 1 a d f a i r l y a c c u r a t e l y by the molecular beam techniques. I t seems to us that our methods of o b t a i n i n g i n f o r m a t i o n concerning s p i n - r o t a t i o n coupling constants are r e l a t i v e l y e a s i e r than the molecular beam techniques. With improvement i n the s i g n a l to noise r a t i o as indicated, i n Chapter IV, a program can be c a r r i e d out to see i f measurements can. se p a r a t e l y determine C and C , . So f a r , we can determine a r e l a t i o n between C a d a and C^ or between Cj| and Cj_ si n c e only a s i n g l e c o r r e l a t i o n time x i s r e q u i r e d to f i t our data at mimimum w i t h i n our experimental accuracy. F l u o r i n e and proton minima were observed i n CHF^ and CH^F, r e s p e c t i v e l y , which enable us. to o b t a i n some p r o v i s i o n a l i n f o r m a t i o n on t h e i r r e s p e c t i v e s p i n - r o t a t i o n tensors. Due to s i g n a l to noise consid-e r a t i o n s , we were unable to observe minima of the proton i n CHF^ and the f l u o r i n e i n CK^F. I t i s necessary to perform f u r t h e r measure-ments near the proton and f l u o r i n e minima i n CHF^ and CH^F at various temperatures i n order to examine the form of c o r r e l a t i o n f u n c t i o n s and t h e i r temperature dependences. Only when such i n f o r m a t i o n are accurately' found w i l l i t be p o s s i b l e to o b t a i n d e f i n i t i v e i n f o r m a t i o n on t h e i r r e s p e c t i v e s p i n - r o t a t i o n tensors. The cross s e c t i o n a of CHF„ '- 1 non-res. 3 and CHgF were found to be about 50% l a r g e r than t h e i r determined i n - 132 -Chapter IV. This seems to i n d i c a t e that the c o r r e l a t i o n time increases with J s i n c e the s p i n - r o t a t i o n r e l a x a t i o n r a t e i s weighted by <J(J+1)^ but i t i s not so i n the non-resonant absorption measurements. In Chapter, V, we have • developed a high temperature nuclear-s p i n r e l a x a t i o n theory f o r CX_Y molecules, c o n t a i n i n g spin-1/2 n u c l e i X and Y, from a phenomenological approach. The molecular symmetry of the CX,Y molecules and hence the nuclear spi n symmetry e f f e c t s are in c l u d e d i n our t h e o r e t i c a l treatment of the problem. The general r e s u l t s were then a p p l i e d to discus s the T^ and Overhauser e f f e c t measure-ments i n both CHF- and CH F molecules. While the f l u o r i n e spins i n CHF and 'CH_F give a l i n e a r r e l a t i o n between T^ and d e n s i t y p over the d e n s i t y r e g i o n s t u d i e d , t h e i r proton spins e x h i b i t q u i t e s t r i k i n g l y d i f f e r e n t features over the same d e n s i t y r e g i o n . A p l o t of T^/p versus p f o r the proton i n these two symmetric-top molecules d i s p l a y s "steps". We have demonstrated i n Chapter VI by means of the Overhauser e f f e c t measurements that the proton spins are r e l a x e d predominantly by the s p i n - r o t a t i o n i n t e r a c t i o n at low d e n s i t i e s . At higher d e n s i t i e s , the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n s p l a y a r o l e i n r e l a x i n g the proton s p i n system and are shown to account at l e a s t p a r t l y of the observed "steps". The mechanisms f o r c o l l e c t i n g those high frequency terms i n the i n t e r a c t i o n s to c o n t r i b u t e at a common c o l l i s i o n frequency have not yet been understood. The tempera-tur e dependence of the f l u o r i n e T^ i n CHF_, CH_F and CF^ have been s t u d i e d and found to s a t i s f y the r e l a t i o n T^ /paT n . The value of n f o r the s p i n - r o t a t i o n i n t e r a c t i o n i s u s u a l l y equal to 1.5. The values of n f o r CH,F and CF were found to be c l o s e to 1.5, but the value of n f o r CHF„ has an anomalous value of 2.5. The proton T n i n CHF- and CH F were a l s o s t u d i e d at s e v e r a l temperatures. In the d i s c u s s i o n of the nuclear s p i n r e l a x a t i o n by the s p i n -r o t a t i o n and d i p o l a r i n t e r a c t i o n s , we have to evaluate c o r r e l a t i o n f u n c t i o n s of ^ 2 (^ 2) and J whose power s p e c t r a are r e s p o n s i b l e f o r the s p i n r e l a x a t i o n . These c o r r e l a t i o n f u n c t i o n s have been assumed to be a product of the c o r r e l a t i o n f u n c t i o n f o r f r e e - r o t a t i n g molecule (which i s e a s i l y evaluated) and a, "reduced" c o r r e l a t i o n f u n c t i o n . This assumption may be v a l i d f o r very d i l u t e gases but i t s v a l i d i t y at moderate d e n s i t i e s may be questionable. Perhaps the c o r r e l a t i o n times f o r these c o r r e l a t i o n f u n c t i o n s have a f u n c t i o n a l dependence on the r o t a t i o n a l l e v e l s which has not been incorp o r a t e d i n our simple p i c t u r e f o r molecular r e o r i e n t a t i o n s i n polyatomic molecules. The nature of over l a p p i n g r o t a t i o n a l l e v e l s as a r e s u l t of c o l l i s i o n s i n l i q u i d l i k e gases remains to be examined both t h e o r e t i c a l l y and experimentally. The deuterated m o d i f i c a t i o n s of CH^ may c o n s t i t u t e a good system f o r the f u r t h e r search f o r these high frequency o s c i l l a t i n g terms i n the i n t e r a c t i o n s . - 134 -Appendix A C i r c u i t D e t a i l s of the Pulsed Spectrometer Figure A l . Pulse sequencer HAMMOND 775 Figure A 2 . High v o l t a g e DC power supply. .01 9-.-.MC OSCILLATOR WIDEBAND AMPLIFIER POWER S U P P L Y T R I P L E R + 90V GATE Figure A 3 . Coherent gates o s c i l l a t o r . 100 -o * 225V | 6 ^ h <2.7K 120K 668ft .01 TO PHASE ' SHIFTER , <150K 120 >120 + 1 50V T 3 3 P f < OUTPUT TO ° W I D E B A N D A M P L I F I E R CO CRYSTAL +20V Figure A4. 9.4 MHz c r y s t a l o s c i l l a t o r . 22 ph - W V -.02 330 12pf 1.5/*h .005 •^ww— 10,uh 2N502 IK £l0 4 v -02 OU" PUT -o -6 V Figure A5. Stable wide band a m p l i f i e r . \ INPUT 4-30? /Ah :10pf # 5 0.5uh z4-30 * 28.2MC OUTPUT o - 6 V Figure A6. 28.2 MHz t r i p l e r . +1 50V 5670 W \ A - - -) 47 r~ZY 2 0 0 p f ^ <V 4^.05 1 N307 GATE I N 22 K 9K , A / V \ A ~ i > -A A A A A 47K T ={=0.1-33 170V TO 6SN7 Figure A 7 . Gating pulses a m p l i f i e r . H D * 2 2 5 V -r90V GATE - 1 7 0 V + 225V 9 FROM GATED Q S C I L U A J O B +150V + 225V 9 9 .01 22 K 7 I ioopf T <«70pf DELAY L I N E : BEL F U S E vs-2 50 P H A S E S H I F T E R Figure A8. o i i ? POWER ? f ~ f jg AMPLIEIER TO Phase s h i f t e r and t r i p l e r . o-90V GATE I N 22 K U L 6 S N 7 22K •150 3.15 V 6 -300 V .01 Hi-l t : -a +7S0V 2^f 829 if 10 -AVW 10 56K 40 Figure A9. Gated power a m p l i f i e r . r.T. •° OUTPUT « 9+1200V -.01 - , - 144 -Appendix 13 E v a l u a t i o n of <f(J,K)> Let us consider f ( J , K ) = K 2 r a [ J ( J + l ) ] n co J I n s <f(J,K)> = i E E K 2 r a [ J ( J + l ) ] n [ 2 J + l ) e x p [ - a J ( J + l ) - p J K 2 ] ^ J=0 K=-J where a = Bh (A-B)h kT ' p kT ~ J 2 Q • E E ( 2 J + l ) e -"JCJ+D-BK J=0 K=-J A and B are def i n e d i n eq. (3.2), and Q i s the r o t a t i o n a l p a r t i t i o n f u n c t i o n . At high temperatures, i . e . , a, 3 <<1, <f(J,K)> may be c a l c u l a t e d by r e p l a c i n g the sum over K by an i n t e g r a l from - J to J , 2 the sum over J by an i n t e g r a l from 0 to °° and J ( J + 1) by J . A f t e r changing v a r i a b l e s (Birnbaum 1967) K =' ^ y " 6 , J = , we o b t a i n 3  1 a' „ _ -1,3.1/2 A r 0 0 1 n2 f tan (—) T 4 n 2 ( m + n + 1 ) - R J D I = r-7^r r- R e dR mn ^ m+1/2 n+1 Q3 a J o a i 0 • 2 m o (2n+l), d9 s i n 6cos ( /"IT S i m i l a r l y , Q can be e a s i l y found to be r y y . Now by simple a(a+3) i n t e g r a t i o n , we get the f o l l o w i n g r e s u l t s : - 145 -2 <K > = I 10 2(a+3) (B.2) <3K2 - J(J+1)> - 1 3 a a+6 (;B--3) < J ( J + 1 ) > = I2,-l B 1/2 = 2a" ( 3 3 t ' 1 - 3 5+1 + 2 ( T ~ } l n ( , f _ J L _ , l/2 ) ] Mien 3 i s negati v e , (B3) i s s t i l l a p p l i c a b l e s i n c e 1 i i + i y + - i ~ In -—^- = tan y 2 i 1-iy When 3 i s zero, i . e . A = B as i n s p h e r i c a l - t o p molecules 10a When n = -m, we w r i t e R = I and a general r e c u r s i o n r e l a t i o n f o r ' m m,-m. & R can be obtained as m CB-5) 2m+T " R m = _ £ _ $ R m + l - 146 -Appendix C Di p o l a r Hamiltonian f o r CX Y Molecules i n Convenient Form The Hamiltonian f o r the i n t r a - m o l e c u l a r d i p o l a r i n t e r a c t i o n i n a CXgY molecule has been given i n eq. (3.3). Using the t r a n s f o r m a t i o n property of Y ? (fi) given i n eq. (3.11), we o b t a i n 94 i / 9 4 3 Y-Y - "R2 2 ^ 2 _ ( C I ) Hd. = {^-)L/~ E E E (-1)^ T | ( I . ,~T.) E D^  Y (P; ) d l P J i = 2 j = l r . . 3 ,a = -2 y 1 3 k=-2 k ," 1 J 2 k 0 X 3 I J i>l 2 where (C.2) [ f t . ]. = ( 2 p V / 2 Z C - 1 ) U [ L ( 2 ; J ] I D? L d i p J k 5 u = -2 u k 2 4 3 Y • Y **R (C.3) [L ^  ]. = ( - l ) k E E ^ V T ( 2 ] ( " , t . ) L ( f l - . ) e l k < i > o i j J 1 u J k v . „ . . 3 u v I i • 2k v o i i J J i=2 i = l r . . J • . • i i i > 1 J (2) For CX^Y molecules, i t i s convenient to separate [L^ i n t o two terms i . e . (C.4) [ L C 2 ) k = [ L ( 2 ) l ^ + [ L ( 2 ^ l f v. J L p J k L y J k L u J k Using the geometry of CX^Y i n F i g . C 1, we deduce from eqs. (C.3) and (C.4) CCS) ( V ^ V ! v 2 kte x y)| f > (T y ,T™) r x y where - 147 -(C.6) l—-k - f e I. e 3 I. and 2 2 ( C 7 ) [ L C 2 ) ] J X = C-D k ^4 l Y 2 k ^ l [ ^ i H 0 ) T C 2 ) ( f , t ) + • r 1? xx . + exp(ikA ) T ^ 2 ) ( t ? )+exp(ikcp ) T ( 2 ) ( t " f ) ] °13 1 ° 2 3 y = C-l) k"^Hr | Y 2 ] f C | ) i [e^J [e -^V 2 3 ( t ^ V r xx Noting that (c.8) 1 ^ ( 1 . , I . ) = T ^ j ( I . , I . ) i / j = 0 i=j (spin 1/2 n u c l e i ) We o b t a i n , (C.9) T ^ l l ^ ^ ' ^ l e 1 ! k + e ' 1 ^ k ) [ e ' 1 ^ k T ^ l t l . T , ) + ^ J ]s ^ x x J J L y v 1 2 + e 1 3 T ^ C I ^ D + T y ( 2 ^ ( I 2 , I 3 ) ] Hence we can r e w r i t e eq. (C.7) as Y2 2 T(2) T ( 0 ) 7 ( k ) ) ( C . X 0 ) . ^ . ( - ! ) " - ^ - | . Y 2 k C f ) l " * r i-r k 1-— k xx e 6 + e 6 - 148 -Let the. three X atoms be i n the (X-Y) plane and the Y atom on the Z-axis (molecular symmetry a x i s ) Figure CI. Geometry of CX Y molecule. - 1 4 9 -Appendix D E q u i l i b r i u m Magnetizations f o r CX^Y Molecules Denoting the molecular l e v e l s of a CX^Y molecule by quantum numbers £p, m^  and m where <f> = J,K,M, we note that i t s f r a c t i o n a l p opulations at thermodynamic e q u i l i b r i u m are given by e x p ( - 6 E J e x p [ - e ( y m + Y m )fi H] rr, i •> TT = 1 v d> . r L 'x x 'y y (D.l) d),m ,m * J — 1  x y • E exp (-0E ) e x p [ - 3 ( Y m + y m )* H] 6,m ,m ™ - / .> x y where 3= The number n^ of molecules belonging to species A and m^  and m i s given by (D.2) n. = E TT , A.m ,m ,, • » >m 5 x' y a l l (j>jA v' x' .y P u t t i n g (D.4) Z F = E exp (-3E ) 1 <f)9E1 . ^  We o b t a i n from eqs. (D.l)-(D.2) t h a t ( D - 5 ) nA,m ,m = 8(Z A+Z r) e X ? ^ ^ A + V V * H ] x y v A E 2Z E (D.6) n r = — — v exp [-3(Y M + Y m )"R H] ^ J E,m ,m 8(Z +Z„) 1 L V 1 x x y y x y A E y using Z, = Z„ = Z... In the high temperature approximation, we note b l b2 h that the molecular l e v e l s are populated up to high J value and thus - 150 -Z. = Z . Of course, Z >> Z at 4°K say because only J = 0 i s populated, A n A c The e q u i l i b r i u m magnetization of X-type spins i n A species i s c l e a r l y given as f o l l o w s , (D.7) M A = E y m n. ^ J x x x A,m ,m o mx-,m x y • . ' ' = -5q • Y x 2 f i H where q = —^f— • S i m i l a r l y , we o b t a i n C D . 8 ) = -q o (D.9) = -q' o (D.IO) = -q 1 y o . , Y y 2 f i H where q< = — ^ . Thus we o b t a i n eq. (5.43) and eq. (5.54) using eqs. (D.7)-(D.8) and eqs. (D.9)-(D.10), r e s p e c t i v e l y . BIBLIOGRAPHY Abragam, A. 1961 T h e ' p r i n c i p l e s of nuclear magnetism (Oxford Univ. Press, London). 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(Unpublished); 1966 Can. J . Phys. 44, 265. H i r s c h f e l d e r , J.O., C u r t i s s , C.F. and. B i r d , R.B. 1954 Molecular theory of gases and l i q u i d s (Wiley, New York). Hubbard, P.S. 1963 Phys. Rev. 131, 1155; 1965 J . Chem. Phys. 42_, 3546. Johnson J r . , C.S., Waugh, J.S. and P i n k e r t o n , J.N. 1961 J . Chem. Phys. 35, 1128. L a l i t a , K. 1967 Ph.D. T h e s i s , U.B.C. (Unpublished). Noble, J . 1964 Ph.D. Th e s i s , U.B.C. (Unpublished). O z i e r , I . , Crapo, L.M. and Lee, S..S. 1968 Phys. Rev. 172, 63. Rugheimer, J.H. and Hubbard, P.S. 1963 J . Chem. Phys.' 3£, 552. Sears, F.W. 1954 An i n t r o d u c t i o n to thermodynamics, the k i n e t i c theory of gases and s t a t . mech. (Addision Wesley). Townes, C H . and Schawlow, A. L. 1955 Microwave Spectroscopy (McGraw-H i l l ) . - 153 -Y i , P., O z i e r , I . , Khosiz, A. and Ramsey, N.F. 1967 B u l l . Am. Phys. Soc. _12, 509. Y i , P., O z i e r , I. and Anderson, C.H. 1968 Phys. Rev. 165, 92. Zussman, A. and Alexander, S. 1968 J . Chem. Phys. 49, 5179. 

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