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Nuclear magnetic resonance study of ethane near the critical point Noble, John Dale 1964

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The U n i v e r s i t y of B r i t i s h C o l u m b i a FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY o f JOHN DALE NOBLE B . S . , U n i v e r s i t y o f Wyoming, 1956 M . S . , U n i v e r s i t y o f Wyoming, 1959 TUESDAY, FEBRUARY 2, 1965 AT 1 ;30 P . M . IN ROOM 302, HENNINGS BUILDING (PHYSICS) COMMITTEE IN CHARGE C h a i r m a n : I . M c T . Cowan P . R„ C r i t c h l o w , J . L i e l m e z s B . A . D u n e l l G . M . V o l k o f f K a W. G r a y D . L l . W i l l i a m s E x t e r n a l E x a m i n e r : J . G„ Powles U n i v e r s i t y of Kent C a n t e r b u r y , K e n t E n g l a n d A NUCLEAR. MAGNETIC RESONANCE STUDY OF ETHANE NEAR. THE CRITICAL POINT ABSTRACT A nuclear magnetic resonance study of the c r i t i c a l region has been made i n ethane which was chosen as the working substance for i t s convenient c r i t i c a l tempera-ture and pressure. Standard radio frequency pulse techniques were used to measure the spin™lattice re-laxation time Ti and the s e l f d i f f u s i o n constant D by the method of spin echoes. A spectrometer having good s t a b i l i t y and very f l e x i b l e timing c i r c u i t s was designed and constructed. An auto-matic temperature control system capable of holding the sample temperature constant to better than 0,01°C f or long periods of time was also designed and constructed. The s p i n - l a t t i c e r e l a x a t i o n time i n ethane has been measured along the vapor pressure curve over the enti r e l i q u i d temperature range, as well as i n the equilibrium vapor from 0°C to the c r i t i c a l temperature (T c = 32.32°C) and i n the dense gas from T c to 60°C. In the l i q u i d Tj r i s e s rapidly with increasing temperature and goes through a maximum at about 0°C a f t e r which i t begins to f a l l . In the vapor Ti i s always less than i n the l i q u i d and increases with increasing temperature. In the dense gas above T c the rel a x a t i o n time decreases slowly with increasing temperature. These r e s u l t s are compared with the conventional theory f o r relaxation i n l i q u i d s and dense gases. The theory gives the rel a x a t i o n rate 1/T] i n terms of three relaxation mechanisms: The dipole-dipole intermolecular i n t e r a c t i o n , the dipole-dipole intramolecular i n t e r a c t i o n and the sp i n - r o t a t i o n a l i n t e r -action. In view of the gross approximations made i n the theory a very reasonable f i t to the experimental data i s obtained. For the low temperature l i q u i d the dipole-dipole interactions are s u f f i c i e n t to account f o r the rel a x a t i o n . At high temperatures the sp i n - r o t a t i o n a l i n t e r a c t i o n seems to contribute s i g n i f i c a n t l y to the relaxation and near the c r i t i c a l point i t i s the dominant re l a x a t i o n mechanism. No anomalous behaviour was observed i n the relaxation near the c r i t i c a l point: and to within the error of measurement i t i s adequately described i n terms of changes i n density and s e l f d i f f u s i o n constant. T] was also measured .i.n d i l u t e ethane gas over a temperature range of 180°R to 300°R. It was observed that Tj I S proportional to density /o and the tempera-ture dependence of T/p i s about T"l*3-7„ Measurements of the d i f f u s i o n constant: reveal that for low temperatures the product D/° for l i q u i d ethane varies approximately as T^.^As the temperature approaches the c r i t i c a l temperature there appears to be anomalous behavior i n D. For both the l i q u i d and vapor the product Dp begins to decrease and goes through a mini=-mum. and then, increases r a p i d l y as the c r i t i c a l point i s reached- Oxygen has been added to these samples to decrease t h e i r r e l a x a t i o n time and t h i s may well be an impurity e f f e c t . P a r t i c u l a r attention was devoted to the question of the equilibrium state i n the c r i t i c a l region and measurements were made on the time taken to achieve equilibrium.. The approach of the r a t i o of l i q u i d to vapor density to i t s equilibrium value, was found to vary i n a roughly exponential manner with a time constant of the order of several hours. S u f f i c i e n t time was allowed a f t e r changing the sample temperature f o r equilibrium to be established and a l l measurements of d i f f u s i o n constant and s p i n - l a t t i c e r e laxation time reported here are thought to be equilibrium values. GRADUATE STUDIES F i e l d of Study: Nuclear Magnetic Resonance Elementary Quantum Mechanics F„ A. Kaempffer Electromagnetic Theory G» M„ Volkoff Nuclear Physics J„ B. Warren So l i d State Physics R„ Barrie Magnetism M» Bloom Related Studies: Theory and Applications of D i f f e r e n t i a l Equations C. W. Clark PUBLICATIONS 1 . D e g a s s i n g o f L i q u i d s f o r N u c l e a r S p i n - L a t t i c e R e l a x a t i o n S t u d i e s . J . L e e s , B . H . M u l l e r and J . D„ N o b l e . J . C h e m . P h y s . 34, 341, ( 1 9 6 1 ) . 2. P r o t o n S p i n - L a t t i c e R e l a x a t i o n I n P u r e L i q u i d E t h a n e and Some o f i t s D e u t e r a t e d M o d i f i c a t i o n s . B„ H . M u l l e r and J . D„ N o b l e . J . C h e m . P h y s . _38, 777 ( 1 9 6 3 ) . 3 . S p i n - L a t t i c e R e l a x a t i o n i n E t h a n e N e a r t h e C r i t i c a l P o i n t . J . D„ N o b l e . B u l l . A m e r . P h y i s . S o c . 9, 568 (1.964). A NUCLEAR MAGNETIC RESONANCE STUDY OF ETHANE NEAR THE CRITICAL POINT hy JOHN DALE NOBLE B.S., U n i v e r s i t y of Wyoming, 1956 M.S., Un i v e r s i t y of Wyoming, 1959 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 required standard THE UNIVERSITY OF BRITISH COLUMBIA December, I96U In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t , c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* • Department of / fj 1/^ S < The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 S Canada Date i i ABSTRACT A n u c l e a r m a g n e t i c r e s o n a n c e s t u d y o f t h e c r i t i c a l r e g i o n has "been made i n e t h a n e w h i c h was c h o s e n as t h e . w o r k i n g .• s u b s t a n c e f o r i t s c o n v e n i e n t c r i t i c a l t e m p e r a t u r e . a n d p r e s s u r e . S t a n d a r d r a d i o f r e q u e n c y p u l s e t e c h n i q u e s were u s e d t o measure t h e s p i n - l a t t i c e r e l a x a t i o n ' t i m e T, and t h e s e l f d i f f u s i o n c o n s t a n t D b y t h e method o f s p i n e c h o e s . A s p e c t r o m e t e r h a v i n g good s t a b i l i t y and v e r y • f l e x i b l e t i m i n g c i r c u i t s was d e s i g n e d ; a n d c o n s t r u c t e d . . A n . a u t o m a t i c t e m p e r a t u r e c o n t r o l s y s t e m c a p a b l e o f h o l d i n g t h e sample t e m p e r a t u r e c o n s t a n t t o b e t t e r t h a n Q,,01° C f o r l o n g p e r i o d s o f t i m e was a l s o d e s i g n e d and c o n s t r u c t e d . The s p i n - l a t t i c e r e l a x a t i o n t i m e i n e t h a n e has b e e n m e a s u r e d , a l o n g t h e v a p o r p r e s s u r e c u r v e o v e r t h e e n t i r e , l i q u i d t e m p e r a t u r e r a n g e , , as w e l l as i n t h e e q u i l i b r i u m v a p o r f r o m 0° C t o t h e c r i t i c a l t e m p e r a t u r e ( T c =32-32° C) and i n t h e dense gas f r o m T c t o 60°C In t h e : l i q u i d T ( r i s e s r a p i d l y w i t h i n c r e a s i n g t e m p e r a t u r e . a n d goes t h r o u g h a maximum a t a b o u t 0°C a f t e r w h i c h i t b e g i n s 0 t o f a l l . I n t h e v a p o r T , i s a l w a y s l e s s t h a n i n t h e l i q u i d and i n c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e . I n t h e . d e n s e gas above T c t h e r e l a x a t i o n t i m e d e c r e a s e s s l o w l y w i t h i n c r e a s i n g t e m p e r a t u r e . T h e s e r e s u l t s a r e - c o m p a r e d w i t h t h e c o n v e n t i o n a l t h e o r y f o r r e l a x a t i o n i n l i q u i d s - a n d dense g a s e s . . T h e . t h e o r y g i v e s t h e r e l a x a t i o n r a t e — i n t e r m s o f t h r e e r e l a x a t i o n m e c h a n i s m s : t h e T, d i p o l e - d i p o l e 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 . , .the d i p o l e - d i p o l e . 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 and the . s p i n - r o t a t i o n a l I n t e r a c t i o n . I n v i e w o f t h e g r o s s , a p p r o x i m a t i o n s made i n t h e t h e o r y a v e r y r e a s o n a b l e f i t t o t h e e x p e r i m e n t a l d a t a i s o b t a i n e d . F o r t h e low t e m p e r a t u r e • l i q u i d t h e d i p o l e - d i p o l e i n t e r a c t i o n s a r e s u f f i c i e n t t o : a c c o u n t f o r t h e r e l a x a t i o n . A t h i g h t e m p e r a t u r e s the. s p i n -r o t a t i o n a l I n t e r a c t i o n seems t o c o n t r i b u t e s i g n i f i c a n t l y ; t o t h e r e l a x a t i o n a n d n e a r t h e c r i t i c a l p o i n t i t i s t h e d o m i n a n t r e l a x a t i o n mechanism. No i i i anomalous, behaviour was observed i n the rel a x a t i o n near the c r i t i c a l point and to-within the error of measurement i t i s adequately described i n terms of changes i n density and s e l f d i f f u s i o n constant. T, was also measured i n d i l u t e ethane gas over a temperature range of l 8 0°K to 300°K. I t was ob served that T( i s proportional to density p and the temperature dependence of T i s about T"-'-*'37. Measurements of the d i f f u s i o n constant reveal that f o r low temperatures the product Dp f o r l i q u i d ethane v a r i e s approximately.as T 3 . -As the temperature approaches the c r i t i c a l temperature.there appears.-. to be ; anomalous behaviour i n D. For both the l i q u i d and vapor the product Dp begins to decrease and goes through a minimum and then increases ..rapidly-as the c r i t i c a l point I s reached. Oxygen has been added to these samples to decrease t h e i r r e l a x a t i o n time and t h i s may we l l be an impurity e f f e c t . P a r t i c u l a r attention was devoted to the question of the equilibrium state i n t h e . c r i t i c a l region and measurements.were made on the time taken to achieve equilibrium. The approach of the r a t i o of l i q u i d to vapor density .to i t s equilibrium value was found to vary i n a. roughly exponential manner with.a time constant of the order of several hours. S u f f i c i e n t time was allowed.after changing the sample temperature f o r equilibrium to be established and a l l measurements of d i f f u s i o n constant and s p i n - l a t t i c e r e laxation time reported here are thought to be equilibrium values. i v TABLE OF CONTENTS Page Abstract i i L i s t of Tables v i L i s t of I l l u s t r a t i o n s v i i Acknowledgements i x Chapter I INTRODUCTION 1 I I THE THEORY OF SPIN LATTICE RELAXATION 11 2.1 General Theory 11 2.2 The Intramolecular Contribution R^ 1'3 2- 3 The Intermolecular Contribution Rg 17 2.4 The Spin-Rotational Contribution BQ l 8 I I I THE EXPERIMENTAL METHOD 22 3.1 The N.M.R. Spectrometer 22 3.1.1 • The magnet 22 g.l.2 Timing c i r c u i t s 22 3-I-3 Radio-frequency ..transmitter 25 3-1.4 Sample c o i l s 26 3.I.5 Receiver 26 3-1.6 Oscilloscope 27 3- 2 Temperature, I t s Measurement and Control 27 3.2.I Thermometer 27 3-2.2 Potentiometer system 29 •3-2.3 Temperature c o n t r o l system 32 3-3 Sample Holder 35 .3.4 Sample Preparation 37 C h a p t e r .Page 3-5 ' E x p e r i m e n t a l T e c h n i q u e s 39 3-5-1 I n t r o d u c t i o n 39 3-5-2 Measurement o f . s p i n - l a t t i c e r e l a x a t i o n t i m e 2+2 3- 5 • 3 Measurement o f s e l f - d i f f u s i o n c o n s t a n t - I4.3 IV THE APPROACH TO EQUILIBRIUM 50 V THE EXPERIMENTAL- RESULTS 57 .5.1 S e l f D i f f u s i o n R e s u l t s 57 5,1.1 The e f f e c t o f d e n s i t y 6 l 5--1-2 T e m p e r a t u r e dependence o f D 63 5-2 S p i n r L a t t i c e R e l a x a t i o n Measurements 70 5-2.1 A c c u r a c y - . o f Tj measurements 72 5-2.2 C o m p a r i s o n w i t h t h e o r y 75 5-2.3 Measurements n e a r t h e c r i t i c a l p o i n t 80 5-2.k Some p o s s i b l e d i s t u r b i n g e f f e c t s 87 '5-2-5 I m p u r i t y ; r e l a x a t i o n 88 ,5.2.6 D i l u t e e t h a n e gas measurements .91 V I CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 96 A p p e n d i x A C i r c u i t D e t a i l s o f t h e S p e c t r o m e t e r 99 B i b l i o g r a p h y .106 VI -LIST GF TABLES Table Page I Interpr.oton distances for the ethane molecule 15 II Best f i t values for A and .B 8U III Summary of dilute gas results 95 v i i LIST OF ILLUSTRATIONS Figure . Page 1 Energy, levels for a proton-in a magnetic field . •• .. •. • • . '5', •2 The spin-lattice interaction . . . . . . . . .. . .. .... 6 •3 Block diagram of spectrometer .... . . . .. .. .. ..... . 23. . k Thermometer control and measurement circuit . . . . . . .... 31 5 Temperature control system .... .. . . . . . . . . . . . 3^ 6 Sample holder .. . . . . . . . . ..... .. .. .. .. ...... 36 7 A typical calibration plot for the vapor coil . . .. kQ 8 Temperature dependence of .density ratio . •. . .. . .. . •• . 5.2 9 Time dependence of density ratio . . . . . . . . . . . . . . •10 Self diffusion in ethane . . .. •. .. •. .. . •. .. •• •. ... 58 11 Self dif fusion near the critical point in ethane . . 60 12 The effect of density on diffusion constant for "non-critically" loaded samples of ethane . . . . . . . .. 62 •13 Temperature dependence of"D/° for liquid ethane ....... . 6.5 •lk Temperature dependence of D^5 near the critical point . . 67 15 Spin-lattice relaxation - in pure liquid ethane . . . . . . 71 16 Spin-lattice relaxation in pure ethane . •• .. . .'• .. . • 73 17 .Temperature dependence of R^, R-g and RQ . . . . . . . . . .. 77 18 Theoretical temperature dependence of relaxation rate . . 79 19 -Spin-lattice relaxation near the critical point in '"}. ethane ....... . ... . .. .. .. . .. .. .. . .. •. .. .. .. . .. •• . 8 l 20 The coexistance curve for ethane in the critical temperature region . . . . . . . 82 •21 T| as a function of density for "non-critical" samples . . 85 22 Impurity.controlled spin-lattice relaxation in ethane . . 89 23 Temperature dependence of T, Jp for ethane at low densities . . . .. .. .. .. • • . .. • .. • • .. . . .. •• •. •. .. 92 v i i i Figure Page IA Pulse sequence sub-unit ........ .. .. . .. .. .. .. .. . .. .. .. \ 100 i 2A Gate and clamp circuit .... .. .. . .. -.. . .. .. .. . .. . .. . J. 101 3A Switch details .. •'. 102 • UA -Preset counter for repetition rate ..................... . 103 5A • Pulse width generator . -. . .. . .. -. .. .. .. . .. .. . . .. •• . . 10k 6 A Radio frequency transmitter ....................... ....... 105 i x . ACKNOWLEDGEMENTS I wish to express my deep, appreciation;to Dr. Myer Bloom f o r h i s i n t e r e s t , guidance-and counsel during the course of t h i s work. I t i s a pleasure to thank Dr. K.W. Gray f o r many h e l p f u l conversations. His continued i n t e r e s t i n t h i s work i s greatly appreciated. I would l i k e to thank Mr. Walter Hardy-who gave me many :ho.urs of enjoyable discussion e s p e c i a l l y - w i t h respect to some of the t h e o r e t i c a l aspects of nuclear magnetic resonance. For assistance i n obtaining some of the low temperature measurements I wish to thank Mr. Frank Bridges. Of .the many members of the Physics Department who contributed -a great deal to t h i s work I must ;mention Mr. John Lees. For h i s indispensable assistance i n sample preparation and f o r h i s constant encouragement I am indeed g r a t e f u l . I also wish to thank Mr. William Morrison who constructed the sample holder and was always most h e l p f u l . I am indebted to Miss Ruth Hogan f o r programming the T| data and ca r r y i n g out the machine computations, I wish to thank my wife, Peggy;, without her manifold s a c r i f i c e s t h i s work would have been impossible. The research was supported by a grant from the National Research Council of Canada to Dr. M. Bloom. I wish also to thank Dr.' Bloom'for the - award of a Research Assistantship during the course of the work from h i s A l f r e d P. Sloan Research Fellowship, 1 Chapter I INTRODUCTION For a l l gases there exists ..a critical temperature 1C such that at temperatures;above Tc the transition from dense gas to liquid takes place, as far as has been determined•experimentally, .without any.singularity : in the density or other thermodynamic variables. At temperatures below T c a gas can be condensed into the liquid state by isothermal compression. As the temperature is increased to Tc the difference in density between the co-existing liquid -and vapor tends continuously to.zero as Tc is approached. The limiting density and corresponding pressure may be used to define the critical point.. •The discovery of the critical point was made in 1&22 when"Cagniard de ;la Tour was experimenting with heated .alcohol, ether and other liquids in sealed glass tubes. Since that time many-attempts have been made to-obtain.accurate experimental data concerning the properties of a pure substance in its critical region. Interpretation of the experimental results has rested chiefly upon the concepts of molecular clustering .in gases .and local order in liquids, and upon knowledge of the effect of gravity upon a compressible fluid. The exact nature of the critical point between liquid,and gas does not yet seem to be established. Certainly the classical ideas based on the theory of van der Waals have never been demonstrated to represent the true state of affairs beyond question. In a recent review ar t i c l e ^ ) Fisher gives an excellent discussion, of the theories about the critical point and of their experimental and theoretical validity. Fisher points out that there are three principal predictions which follow from the classical van der Waals theory. The first prediction is that the coexistance curve follows a square-root law of the form ft. 'ft, ** A (Tc -T)^ ( l - l ) where pu and pv are the d e n s i t i e s of the liquid.and vapor along the vapor pressure curve at temperature T and A i s a constant. Experimentally i t i s found that the shape of the coexistance curve -is much f l a t t e r than the van der Waals curve and a b e t t e r f i t i s found by replacing the index •§ by 1/3. The second p r e d i c t i o n i s that the isothermal c o m p r e s s i b i l i t y K T - ± {$?\ ( 1 . 2 ) along the c r i t i c a l isochore (constant density) diverges as K T « ( P « A , T - * T c + ) . ( 1 . . 3 ) The c o m p r e s s i b i l i t y of the vapor and of the' l i q u i d . a l s o diverge as a simple pole as T—*"TC"" but the constant B i s smaller. This p r e d i c t i o n does not seem to have been adequately tested. However the. experimental evidence suggests that the c o m p r e s s i b i l i t y might diverge more sharply.than (T - T^)""1" . The t h i r d p r e d i c t i o n i s that the s p e c i f i c heat at constant volume Cy(T) along the c r i t i c a l isochore increases to a maximum and then decreases discontinuously as T passes through T c . That i s , C$(T) C c + - Dt I T - T c| , T > T c , C V(T) ~ C~ D~ I T - T c/ , T < T c , (l.k) with the constants Cc Ccf = A c >0 • "It has long been'known that there e x i s t s a very pronounced maximum.-in the s p e c i f i c heat near the c r i t i c a l point. Fisher discusses recent measurement of C V(T) for-argon along the c r i t i c a l isochore which strongly:suggest that C v ( . T ) - » o o as T - ^ T / * ( 1 . 5 ) Such a r e s u l t i s inconsistant with c l a s s i c a l theory. The o p t i c a l p r o p e r t i e s o f f l u i d s i n t h e c r i t i c a l r e g i o n have a l s o b e e n s t u d i e d e x t e n s i v e l y . As a f l u i d a p p r o a c h e s i t s c r i t i c a l t e m p e r a t u r e t h e s c a t t e r i n g o f l i g h t becomes v e r y p r o f u s e and a t h i c k c o l o u r e d f o g a p p e a r s . T h i s b e a u t i f u l phenomenon i s c a l l e d c r i t i c a l o p a l e s c e n c e . I t i s c a u s e d b y m a c r o s c o p i c v a r i a t i o n s i n d e n s i t y o v e r d i s t a n c e s o f t h e o r d e r o f • t h e wave l e n g t h o f l i g h t . The c l a s s i c a l t h e o r y o f c r i t i c a l s c a t t e r i n g i s t h a t d e v e l o p e d b y O r n s t e i n and Z e r n i k e . T h i s t h e o r y i s d i s c u s s e d b y b o t h F i s h e r ^ " ^ (2) and Hemmer, Kac and U h l e n b e c k ^ O f c e n t r a l i m p o r t a n c e i n t h e t h e o r i e s o f s c a t t e r i n g i s t h e t i m e d e p e n d e n t p a i r c o r r e l a t i o n f u n c t i o n G ( r , t ) . T h i s c o r r e l a t i o n f u n c t i o n d e s c r i b e s t h e c o r r e l a t i o n b e t w e e n p o s i t i o n s o f two d i s t i n c t m o l e c u l e s a t d i f f e r e n t t i m e s . F o r t u n a t e l y , t h e p a i r c o r r e l a t i o n f u n c t i o n c a n be s t u d i e d d i r e c t l y b y s c a t t e r i n g e x p e r i m e n t s w h i c h a r e u s u a l l y (2) p e r f o r m e d w i t h l i g h t o r w i t h X r a y s . V a n - H o v e has p o i n t e d o u t t h a t t h e r m a l n e u t r o n s o f wave l e n g t h i l A m a y - a l s o b e u s e d . I t i s f o u n d t h a t t h e c o r r e l a t i o n f u n c t i o n e x h i b i t s l o n g r a n g e c o r r e l a t i o n s r e s u l t i n g f r o m s p o n t a n e o u s d e n s i t y f l u c t u a t i o n s o f m a c r o s c o p i c s i z e . T h i s a c c o u n t s f o r t h e c r i t i c a l s c a t t e r i n g w h i c h becomes v e r y . l a r g e as t h e c r i t i c a l p o i n t i s a p p r o a c h e d . I t i s e v i d e n t t h a t e x p e r i m e n t a l work has n o t a l l b e e n r e c o n c i l e d w i t h t h e o r e t i c a l s t u d i e s o f t h e c r i t i c a l r e g i o n and so i t was f e l t t h a t t h e a p p l i c a t i o n o f a n o t h e r e x p e r i m e n t a l t e c h n i q u e m i g h t p r o v e d e s i r a b l e . The work r e p o r t e d h e r e i s a p r o t o n m a g n e t i c r e s o n a n c e s t u d y o f t h e c r i t i c a l p o i n t i n e t h a n e . T h i s s u b s t a n c e was c h o s e n b e c a u s e i t has a c o n v e n i e n t c r i t i c a l t e m p e r a t u r e (32.32°C) a n d c r i t i c a l p r e s s u r e ( 4 8 . 8 A t m ) . F o r t h e s e same r e a s o n s o t h e r w o r k e r s have a l s o s t u d i e d t h e c r i t i c a l p o i n t i n e t h a n e and t h e i r d a t a i s most u s e f u l . P a l m e r ^ ) made, e x t e n s i v e v i s u a l o b s e r v a t i o n s a n d o b t a i n e d t h e d e n s i t y - d i s t r i b u t i o n . a s .a f u n c t i o n o f h e i g h t i n a c e l l c o n t a i n i n g e t h a n e , u s i n g a S c h l l e r e n o p t i c a l s y s t e m . M a s o n , N a l d r e t t and M a a s s ^ ^ have a l s o made, a d e t a i l e d s t u d y o f e t h a n e i n . t h e c r i t i c a l r e g i o n a n d t h e y o b t a i n e d t h e c o e x i s t a n c e c u r v e a t t e m p e r a t u r e s b e l o w t h e c r i t i c a l p o i n t . The p r e s e n t work i s c o n c e r n e d w i t h a s t u d y o f t h e s p i n - l a t t i c e r e l a x a t i o n t i m e T j a n d t h e s e l f d i f f u s i o n c o n s t a n t m e a s u r e d as a f u n c t i o n o f t e m p e r a t u r e a l o n g t h e v a p o r p r e s s u r e c u r v e . Measurements were made i n b o t h t h e v a p o r a n d l i q u i d a t t e m p e r a t u r e s l o w e r t h a n T c a n d i n t h e d e n s e gas a t t e m p e r a t u r e s g r e a t e r t h a n T c . P a r t i c u l a r a t t e n t i o n was d e v o t e d t o t h e p r o b l e m o f t h e e q u i l i b r i u m s t a t e i n t h e c r i t i c a l r e g i o n as w e l l as t o t h e p r o c e s s o f e q u i l i b r i u m a t t a i n m e n t . P o w l e s and h i s co-workersC^ have s t u d i e d s p i n - l a t t i c e r e l a x a t i o n i n s e v e r a l s u b s t a n c e s o v e r n e a r l y . t h e e n t i r e l i q u i d t e m p e r a t u r e r a n g e . I n some e x p e r i m e n t s t h e y s t u d i e d s a m p l e s l o a d e d t o t h e c r i t i c a l d e n s i t y f r o m low t e m p e r a t u r e s r i g h t t h r o u g h t h e c r i t i c a l p o i n t t o t e m p e r a t u r e s g r e a t e r t h a n T c . T h e y were c o n c e r n e d w i t h more g e n e r a l f e a t u r e s o f t h e r e l a x a t i o n p r o c e s s a n d t h e r e f o r e do n o t p r e s e n t d e t a i l e d r e s u l t s i n t h e c r i t i c a l r e g i o n . The r e s u l t s o b t a i n e d f o r e t h a n e e x h i b i t t h e same g e n e r a l f e a t u r e s a s f o u n d b y P o w l e s and his c o l l e a g u e s f o r s e v e r a l o t h e r s u b s t a n c e s . (6) S c h w a r t z ^ h a s s t u d i e d t h e s p i n - l a t t i c e r e l a x a t i o n t i m e i n s u l f e r h e x a f l o r i d e i n t h e c r i t i c a l r e g i o n . H i s r e s u l t s a r e v e r y s i m i l a r t o t h e r e s u l t s r e p o r t e d h e r e , . e x c e p t t h a t he o b s e r v e d h y s t e r e s i s e f f e c t s . T h e s e e f f e c t s may be due t o . a l a c k o f s u f f i c i e n t t i m e b e i n g s p e n t f o r t h e s y s t e m t o a c h i e v e t h e e q u i l i b r i u m s t a t e . H i s t e m p e r a t u r e c o n t r o l s y s t e m was r a t h e r p o o r and he o n l y - a c h i e v e d a l o n g t e r m s t a b i l i t y o f a b o u t 0 . 1 e C . U n f o r t u n a t e l y d e n s i t y and d i f f u s i o n d a t a were u n a v a i l a b l e f o r s u l f e r . h e x a f l o r i d e , a n d so S c h w a r t z c o u l d n o t make a d e t a i l e d c o m p a r i s o n b e t w e e n h i s r e s u l t s a n d t h e o r y . The f o l l o w i n g c h a p t e r s d e s c r i b e t h e e x p e r i m e n t a l method a n d t h e r e s u l t s o b t a i n e d f o r e t h a n e . C h a p t e r I I p r e s e n t s t h e c o n v e n t i o n a l t h e o r y f o r s p i n r l a t t i c e r e l a x a t i o n i n l i q u i d s a n d dense g a s e s i n some d e t a i l . The e x p e r i m e n t a l a p p a r a t u s and methods o f measurement a r e d e s c r i b e d i n C h a p t e r I I I . Measurements c o n c e r n i n g t h e a p p r o a c h t o . - e q u i l i b r i u m a r e d i s c u s s e d i n C h a p t e r I V . D i f f u s i o n a n d s p i n - l a t t i c e r e l a x a t i o n r e s u l t s - a r e p r e s e n t e d a n d compared w i t h 5 theory i n Chapter V. Some suggestions are made f o r f u r t h e r work i n Chapter VI. Appendix A co n t a i n s c i r c u i t diagrams f o r v a r i o u s p a r t s of the apparatus. The remainder of t h i s chapter w i l l now be devoted to a b r i e f i n t r o d u c t i o n t o some of the elementary aspects o f nuclear magnetic resonance. Nuclear magnetic resonance i s concerned w i t h the resonant exchange of energy between a r a d i o frequency c i r c u i t and a system of nuclear spins immersed i n a magnetic f i e l d , H Q ., as a r e s u l t of t r a n s i t i o n s among energy l e v e l s of the spin system, These ene r g y ; l e v e l s are set up because of the magnetic p r o p e r t i e s of the s p i n system. In any given s t a t e of t o t a l angular momentum J , a nucleus possesses a t o t a l magnetic moment ~P -rt (1.6) where y i s a s c a l a r c a l l e d the "gyromagnetic r a t i o " . N u c l e i which possess a magnetic moment can be expected t o i n t e r a c t w i t h an e x t e r n a l magnetic f i e l d . The energy of t h i s i n t e r a c t i o n w i l l be -/< • H Q . This immediately leads to a Zeeman s p l i t t i n g of the energy l e v e l s . For protons there are two l e v e l s which correspond c l a s s i c a l l y t o the proton moment being e i t h e r , a l i g n e d p a r a l l e l or a n t i p a r a l l . e l to the f i e l d . The spacing between these two l e v e l s i s JT+>H 0. N-A E = y H H Q = TLW N+ F i g . 1. Energy l e v e l s f o r a proton i n -a magnetic f i e l d . There are many n u c l e i i n a macroscopic sample and the number .aligned w i t h the f i e l d i s s p e c i f i e d by N + w h i l e the number a l i g n e d i n the opposite d i r e c t i o n i s s p e c i f i e d by N_. The lower s t a t e i s more h i g h l y populated'than the upper. The r e l a t i v e p o p u l a t i o n s are given by the Boltzmann f a c t o r N + _ exp N-H Q ' k T . s . (1 -7) where k i s B o l t z m a n n ! s c o n s t a n t and T g i s t h e t e m p e r a t u r e o f t h e s p i n s . -S i n c e t h e d i f f e r e n t o r i e n t a t i o n s o f t h e s p i n s w i t h r e s p e c t t o the. f i e l d c o r r e s p o n d t o d i f f e r e n t m a g n e t i c e n e r g i e s t h e r e a p p e a r s i n a sample c o n t a i n i n g a l a r g e number o f e l e m e n t a r y m a g n e t i c moments a n e t m a c r o s c o p i c m a g n e t i z a t i o n •MQ w h e n . t h e sample i s •placed i n a m a g n e t i c f i e l d . I t I s t h i s m a c r o s c o p i c m a g n e t i z a t i o n w h i c h i s d e t e c t e d a n d m e a s u r e d • i n a n u c l e a r r e s o n a n c e e x p e r i m e n t . The p r o c e s s o f m a g n e t i z a t i o n o f an u n m a g n e t i z e d sample must t h e r e f o r e r e q u i r e a n e t number o f t r a n s i t i o n s f r o m t h e u p p e r t o t h e l o w e r e n e r g y s t a t e . I n o t h e r w o r d s , i f a sample i n e q u i l i b r i u m i n z e r o f i e l d i s s u d d e n l y s u b j e c t e d t o a l a r g e f i e l d some o f . the e l e m e n t a r y moments must a l i g n t h e m s e l v e s w i t h t h e f i e l d . I n t h e p r o c e s s t h e s e moments g i v e up e n e r g y a n d t h e r e i s a " h e a t t r a n s f e r " . T h e r e must t h e r e f o r e be a n o t h e r s y s t e m c a p a b l e o f a c c e p t i n g t h i s e n e r g y . T h i s o t h e r s y s t e m i s commonly c a l l e d t h e " l a t t i c e " a n d i t c o n s i s t s o f a l l t h e o t h e r r o t a t i o n a l and t r a n s l a t i o n a l e n e r g i e s o f t h e s y s t e m . The. e l e m e n t a r y magnets m a y b e c o n s i d e r e d as a s y s t e m i n t h e thermodynamic sense o f t h e w o r d , c o n n e c t e d b y - a h e a t c o n d u c t i n g l i n k t o t h e w o r l d a r o u n d them. S p i n S y s t e m I n t e r a c t i o n L a t t i c e T — — — — — _ — . — ^ F i g . 2 . The s p i n - l a t t i c e i n t e r a c t i o n I t i s sometimes p o s s i b l e t o d e s c r i b e t h e s p i n . s y s t e m i n t e r m s o f a t e m p e r a t u r e . F o r t h i s t o be t r u e some c o u p l i n g b e t w e e n t h e n u c l e i must be rassumed so t h a t t h e . a s s e m b l y m a y . b e c o n s i d e r e d i n t h e r m a l e q u i l i b r i u m a t . a t e m p e r a t u r e T s . F r e q u e n t l y t h e i n t e r a c t i o n s w i t h i n t h e s p i n . s y s t e m and w i t h i n t h e l a t t i c e a r e c o n s i d e r a b l y g r e a t e r t h a n any i n t e r a c t i o n between t h e two s y s t e m s . I n t h i s c a s e i t i s p o s s i b l e t o d e f i n e a s e p a r a t e s p i n t e m p e r a t u r e a n d - a l a t t i c e t e m p e r a t u r e . The i n t e r a c t i o n b e t w e e n t h e two systems tends to create thermal equilibrium.at a common temperature. Except at extremely:low temperatures the. heat capacity .of the spin system is much less than that of the lattice. At temperatures greater ..than about 1°K the heat capacity of the lattice can be .assumed infinite comparedto that of the spin:system, 'The spin-lattice interaction.acts to bring the spin system to the same temperature;as the lattice-with negligible effect on the lattice. The coupling of the spins with .the lattice allows for the possibility of transitions between the N+ and N_ levels. In this case the rate equation may be written •"tit .= . + N- "Wj, - W+ Wf (1.8) d t " where is the probability per second of a spin transition downward in energy (from - to +) • and '.is the probability for the reverse process-. This equation may be restated in terms of the total number of spins N .= N+:.+ N_ and the difference in populations n = N + - N_ , d n d t Equation (l-9) can be further rewritten as d n N ( W+ - Wf ) - n ( + Wf ) , (1-9) d t n n - n T l (1,10) where n Q =:N ( W» - W» f = (V* + ) , (1.11) The solution of Equation (l.lO) is n •= n D .+ A e ~t:/.<j> (1..12) and i t is .apparent that. nQ.- represents the thermal equilibrium difference in populations, while Ti is a characteristic time associated with the approach to thermal equilibrium, T-^  is usually ..called the •'•"spin-lattice.-relaxation •time*' and i t characterizes the time necessary to magnetize an unmagnetized sample. 8 To observe the magnetization of a sample a means must be provided to couple to the..spin system. The most commonly used coupling is an alternating magnetic field applied perpendicularly to the static field. The frequency of this field must of course be matched to the Zeeman splitting i f a resonance is to be observed. A resonance occurs when the angular frequency, *jj , of the alternating field is adjusted to satisfy or ^ = *H 0 (1.13) This same result may be derived by classical argument. If a magnetic dipole possessing angular momentum is placed in. a magnetic field, the dipole precesses about the direction of the applied field. The rate of precession is given by the well-known Larmor angular frequency " = *H Q where y* is the gyromagnetic ratio of the dipole. The applied alternating field will only.have an effect i f i t remains in synchronization with the precessing dipole. The application of an alternating field-at the resonance frequency may be expected to cause transitions from N + to N_ and thereby change the macro-scopic magnetization vector of the sample.. For the proton the -value of if is 2 .68 x l c A sec"-'- gauss"-*- so that the frequency of the alternating field falls in the radio .frequency range for normal laboratory magnetic fields. •After the removal of the r.f. field the magnetization vector has in general acquired a non-equilibrium orientation and i t precesses freely in the static field. This precessing magnetization induces a signal in the same coil used to produce the alternating field. It is this signal which is amplified and finally measured. A study of how the magnetization returns to its equilibrium value can clearly give information about the spin-lattice, interaction. There are two types of interactions which must be examined-when considering the approach to equilibrium after disturbing the sample. First there are the interactions between the spins themselves. This interaction conserves energy in the spin system and allows the spins to come to a common temperature. It is monitored by the.transverse components, Mx and My , Of the magnetization. Secondly .there are the interactions between the spin system and the lattice. This interaction allows the spin system to lose energy.and come into thermal equilibrium with the lattice. It is monitored by the longitudinal component Mz of the magnetization.. The approach to equilibrium can very often be described by the well known Bloch equation; l l = r M x% - t ^ - " My- k (Mz - MG ) (LiU) T 2 T 2 T l The first term describes the motionof free spins, while the last terms describe the motion due to the interactions. Solutions of this, equation may be written.as follows: M2(t) = M 0 ( I - e't/r' ) M/(0 = Mxy(o) t S<n Cut ttf)] e " ^ M*(t) = fr*y (o) I CosCwt + <p)] e '^ a (1,15) with M*y - ( Mx t f The time constant T 2 is called the transverse relaxation time and T-^  is the spin-lattice relaxation time mentioned above. A description in terms of Bloch's equation is usually-adequate for dense fluids, although a single time constant does not always describe properly either the transverse or longitudinal approach to equilibrium. The n.m.r. technique provides a means of monitoring M^, My and Mz and thus a measure of the relaxation times T-j_ and T 2 • This then yields a method of studying the mechanisms which allows :the spin system to•relax back into thermal equilibrium with the lattice. 10 Any theory of relaxation must therefore discover the important mechanisms which produce transitions. As shown above the spin system was disturbed from thermal equilibrium by the application of an .external Larmor frequency magnetic field. This provides a clue as to.the origin of possible relaxation mechanisms. Any.fluctuating .internal Larmor frequency magentic field would also be capable of producing transitions. Since the energy ;in this case must come from the lattice i t is important to consider the thermal motions of the molecules which make up the lattice. The molecules may.be considered..as vehicles carrying the nuclei from point to point. Each nucleus thus takes part in the random translational and rotational Brownian motion of the molecules. As a result, the local magnetic field at the position of. any spin contributed by the neighbouring nuclei is a rapidly fluctuating function of .time. The component at the Larmor frequency of the Fourier spectrum of this fluctuating field induces transitions in just the same way as does .an applied r-f field. The. interactions usually - assumed to be responsible for thermal relaxation of spin •§• nuclei in polyatomic molecules such as ethane :are: (a) The intramolecular dipolar interactions between neighbouring nuclei on the same molecule. •(b) The intermolecular dipolar interactions between nuclei on two different molecules. (c) The spin-rotational interactions between the nuclear spin and the rotational angular momentum of the molecules. In the conventional theories for T^_ the-assumption is usually made that a l l three.of the above mechanisms contribute independently to .the relaxation process, so that i t is possible to write l = RA + RB + Rc ( 1 . 1 6 ) l l where R^ , Rg, and-RQ are the contributions -to the relaxation rate, T^ ""'",due to -mechanisms a, b and c respectively. These contributions will be examined in detail in Chapter II. Chapter II THE THEORY OF SPIN-LATTICE RELAXATION As discussed i n Chapter I s p i n - l a t t i c e r e laxation i s a process i n which energy i s exchanged between the nuclear spin.system-and i t s surroundings. This tends to b r i n g the two systems i n t o thermal equilibrium at a common temperature. The mechanism of i n t e r a c t i o n i s b a s i c a l l y magnetic. However f o r n u c l e i with spin greater than a f l u c t u a t i n g e l e c t r i c f i e l d gradient acting on the quadrupole moment of a nucleus i s another powerful r e l a x a t i o n mechanism. This w i l l not be considered here since protons were'the only n u c l e i studied. Every nucleus then experiences the f i e l d produced by the magnetic moments of i t s neighbours. I t i s t h i s magnetic dipole-dipole (7) i n t e r a c t i o n which was considered i n d e t a i l by Bloembergen, P u r c e l l and Pound .in t h e i r q uantitative theory of relaxation. A good deal of e f f o r t has gone into the theory of r e l a x a t i o n since the pioneering work of Bloembergen, P u r c e l l and Pound. Perhaps the most complete treatment of these developments may be found i n the book by;A. Abragam^^. This theory w i l l now be sketched b r i e f l y following Abragam. 2-1 General Theory To derive a value f o r T^_ i s i s important to f i n d how t h e ' t r a n s i t i o n p r o b a b i l i t y can be r e l a t e d to the random perturbations brought on by molecular motion. Abragam shows (page 273) that wot,8 = J*<* (<*Vp) (2-1) where ^  i s the average (taken over a s t a t i s t i c a l ensemble) p r o b a b i l i t y of t r a n s i t i o n per u n i t time between energy l e v e l s of the system l a b e l l e d o< and (8 and J ( < * J ) i s the Fourier transform of the c o r r e l a t i o n functions of the i n t e r a c t i o n s which enable energy to be exchanged between the spin system and the l a t t i c e . Using t h i s t r a n s i t i o n p r o b a b i l i t y i t i s then possible to write 12 down an equation f o r the rate of change of the populations of the energy l e v e l s of the spin system. Once the populations are known i t i s possible to c a l c u l a t e the magnetization of the system of nuclear spins as a function of time. Abragam pr e f e r s to use a more general formalism which avoids the necessity of making c e r t a i n assumptions which must be made using the procedure out l i n e d above. He introduces the s t a t i s t i c a l operator a- represented by a so c a l l e d density matrix which contains a l l the information necessary .for the d e s c r i p t i o n of a s t a t i s t i c a l ensemble of i d e n t i c a l systems. The time dependence, i s then given by the time dependence of the s t a t i s t i c a l operator. Using t h i s formalism-Abragam•derives the macroscopic equation f o r s p i n - l a t t i c e r e l a x a t i o n due to the d i p o l a r coupling between two l i k e spins I and l'. • It has the following form (Abragam, page 291, Eq. 75)? (2.2) and where ^ I z + I z ' ^ i s proportional to the macroscopic magnetization. The s p e c t r a l d e n s i t i e s are given by ) 1 = 1, 2. (2.*0 with (2-5) The random functions F are r e l a t e d to the normalized s p h e r i c a l harmonics YzyC&jfi) by (2.6) l > (2.7) 13 In w r i t i n g these equations the external magnetic f i e l d i s assumed along the z axis. The vector r a b j o i n i n g the two n u c l e i a and b has polar and.azimuthal angles 6ab and&ib respectively. In.order to a r r i v e at a f i n a l expression f o r TQ_ there remains the task of . /a ) c a l c u l a t i n g the c o r r e l a t i o n • f u n c t i o n s of the random functions -F " and f i n a l l y . t h e Fourier transforms of these c o r r e l a t i o n functions. Conventional theory u s u a l l y distinguishes between the intramolecular, i n t e r a c t i o n s inside a molecule, and the intermolecular i n t e r a c t i o n s among •. spins of d i f f e r e n t molecules. This convention w i l l be followed here and the various contributions to T-^.will now be discussed i n turn. 2.2 The Intramolecular Contribution In. order to evaluate the intramolecular contribution' to T]_ f o r ethane i t i s necessary to c a l c u l a t e the s p e c t r a l d e n s i t i e s given by Eq. (2.k) and then substitute into Eq. (2.3')- The r o t a t i o n of the molecule i n the . f l u i d w i l l change the angle between the magnetic f i e l d H Q and 'the vector connecting any .two protons i n - a random fashion. To c a l c u l a t e the c o r r e l a t i o n functions of the random.functions F ^ i t i s necessary to choose some model to describe the motion. The usual convention i s to choose.-a r o t a t i o n a l random •walk model that can be described by a r o t a t i o n a l d i f f u s i o n equation. I f t h i s convention i s followed:then the c o r r e l a t i o n functions i n Eq. (2.5) maybe approximated by F ( q ) ( t ) F(<i)#(t + - T ) = F(<i) (t) F^^Ct) e ~ V r * (2.8): where fz , the -correlation:time, I s usually very much shorter than the Larmor period. The -form e~*/Tt of .the c o r r e l a t i o n function i s a consequence of the r o t a t i o n a l d i f f u s i o n ^equation. In.the case of a r i g i d molecule the interproton distances ;are f i x e d -and time independent. Using the diffusion-equation Abragam c a l c u l a t e s the rel a x a t i o n rate f o r two protons at a f i x e d distance b^apart. He writes (Eq. 106, page 300') .For a molecule containing N protons t h i s .result may be generalized to where yzij i s the separation between the i x n proton and -any neighbouring protons j on the same molecule. S t r i c t l y speaking, the r e l a x a t i o n caused by dip o l a r coupling i n a r i g i d molecule i s .not describable by a single r e l a x a t i o n time. ' This complication a r i s e s because of the c o r r e l a t i o n between :the r e l a t i v e motion of p a i r s of spins when the molecule moves .as a r i g i d body. 'Hubbard^) has ca l c u l a t e d the rel a x a t i o n due to three or four protons equidistant from one another i n a r i g i d molecule. He fi n d s that .the r e l a x a t i o n • i s governed by two exponentials i n both.cases. However the departure predicted by Hubbard from a single exponential would be very d i f f i c u l t to observe experimentally. This may be taken as j u s t i f i c a t i o n , along with the f a c t that only.one time constant was observed experimentally, f o r relating'T-^ to 75 f o r ethane. From the known structure of the C2Hg m o l e c u l e t h e interproton distances may be calculated. The C-H bond "length i s 1.107 A/and the C-C bond-length i s I.536 A. The angle :H-C-H "is IO9.3 0 . Using .this information i t i s pos s i b l e to construct a table g i v i n g a l l 'the interproton-distances f o r both configurations of the ethane molecule. ( 2 . 9 ) 15 Table I. Interproton distances for the ethane molecule staggered configuration eclipsed •configuration same methyl r i l .1-810 A 1.810 A group ri2 1-810 1.810 other methyl r13 .2-909 2.5.O5 group Tik •2-909 2.505 r15 •2-27:7 3.091 Substituting the interproton distances listed in Table I into Equation • (2. 10) yields = I.77 x 10 x x sec^ staggered configuration (2.11) • T-]_ Tr = I.74 x 10"11 sec 2 eclipsed configuration It is unlikely that the difference between the two configurations could be detected experimentally. Tn order to make contact vith the macroscopic properties of the fluid i t is necessary to relate T g to some macroscopic parameter. • Abragam does this by noting that the correlation:time is related to the, diffusion constant for rotation Ds. Taking.a to be the diameter of the rigid sphere he shows that L2 -Tz •- (fj (2.12) 2 If the rigid sphere is in.a medium of viscosity ^  then the diffusion constant •for rotation is given by Stokes's formula Ds (2.13-) k rr. a q This gives f o r "7"^  , 6kT .The v i s c o s i t y may be e l i m i n a t e d by use of Stokes formula f o r t r a n s l a t i o n a l d i f f u s i o n D = kT v a l i d f o r a r i g i d sphere, .this gives T V = ^— (2.15) 18 D Thus, w i t h i n the scope of the above theory, "T^  may be w r i t t e n as a 2 r o t ] 8 (I.76 x 10--L-L) D (2.16) •It i s by no means obvious t h a t the t r a n s i t i o n from the microscopic equation (2.10) to the macroscopic (2-l6) .is v a l i d . The use of the macro-scopic v i s c o s i t y and d i f f u s i o n on a molecular scale must c e r t a i n l y be questioned. Ivanov^^-) has p o i n t e d out th a t the experimental values of D g g e n e r a l l y exceed those evaluated t h e o r e t i c a l l y by means of Equation (2.-13) by. a f a c t o r of 3 to 10. This o f course means th a t Tj_ evaluated by Equation (2.l6) may be out by t h i s same f a c t o r . The theory of r o t a t i o n a l Brownian motion given by. Ivanov can perhaps be a p p l i e d t o the i n t r a m o l e c u l a r r e l a x a t i o n i n dense f l u i d s . Bloembergen, P u r c e l l and Pound j u s t i f y .their use of the r o t a t i o n a l d i f f u s i o n , equation by. . i t s success. Ethane i s not a r i g i d molecule and i t i s , a l s o necessary to take : i n t o account the i n t e r n a l motions of the molecule. There e x i s t s the p o s s i b i l i t y of each of the two methyl groups undergoing independent r o t a t i o n s about the C-C a x i s of the molecule. The a c t i v a t i o n energy f o r t h i s process i s about (12) 3 Kcal/mole^ .'so t h a t , at high temperatures t h i s r o t a t i o n c o u l d take place t o an.appreciable degree. A complete theory would take i n t o account the i n t e r n a l degrees of freedom of the molecules. In the absence of such a theory.-it i s s t i l l p o s s i b l e to make a statement about the l i m i t i n g case where the i n t e r n a l motion i s much f a s t e r than the motion of the molecule ,as a whole. This motion 17 always tends -to make longer and Muller^-^). has-pointed out that for rapid rotation of a methyl group about its :symmetry axis-T^ is multiplied by.a factor k. ;2..3 The Intermolecular Contribution Eg In relating Tf to correlation functions of the form given by'Equation ( 2 . 5 ) for spins on different molecules, the assumption is usually made that only pair spin correlations are important. Hubbard has shownx ' that this assumption is reasonable. This means that the ethane molecule can be considered to be :a structureless, sphere with a l l six spins at the center. Another approximation usually made is the so called low frequency.limit. Since the Larmor frequencies of the nuclei are much smaller than the characteristic frequencies of the molecular motions i t is possible to write 00} (_SL)2 l (2.-17.) where a is taken to be the diameter of the molecules and D is the self diffusion constant. The time ( a ) is a measure of the time taken for a \ 2D / molecule to diffuse out of the range of the dipolar interactions. It is further assumed that the diffusion equation is a valid description of the motion of the molecules. Using these assumptions Abragam (page 302 , Equation 115) finally writes for protons, = R B = 2 n N ( 2 . 1 8 ) trans 5 aD 3 where N is the density of spins per cm . (15) The more general theory of Oppenheim and Bloomv y' gives :a result quite close to Equation ( 2 . 1 8 ) for Rg. 18 2-k The Spin-Rotational Contribution EQ The appearance of something l i k e free r o t a t i o n ;of simple, nearly spherical molecules i n a dense f l u i d may permit important contributions to the rel a x a t i o n from the collision-modulated i n t e r a c t i o n between spins and the l o c a l magnetic f i e l d s produced by molecular r o t a t i o n . This mechanism i s known to be important f o r many d i l u t e diatomic and polyatomic gases. Recently.interesting evidence has been accumulating that I t may-also be important f o r l i q u i d s at high (16) temperatures. -Powles and Smith^ be l i e v e that the spi n - r o t a t i o n i n t e r a c t i o n dominates ; i n pure water above.260°C. • In l i q u i d hydrogen ch l o r i d e and l i q u i d hydrogen.bromi.de decreases with increasing temperature above, about - 50°C ( IT) and K r y n i c k i and Powles suggest that t h i s I s due to the. increasing effectiveness of the spin- r o t a t i o n mechanism. For nonlinear molecules the s p i n - r o t a t i o n a l i n t e r a c t i o n between nuclear spin • I and molecular angular momentum "J can be expressed by a-Hamiltonian °H' of the f o r i r / 1 ^ -Vi' - -&/ h l - j - 1 ( c+K - Cj tj.,) i-T„ (2 .19) where C, = 1 £ ( , * - l} 2, 3 3 *, / t l ) . The C j are the s p i n - r o t a t i o n a l constants.related to the l o c a l magnetic f i e l d H by the equation c T .= U L (2.20) 1 2TT A (19) This Hamiltonian i s .considered by B l i c h a r s k i v y / and he ca l c u l a t e s the spinr r o t a t i o n c o n t r i b u t i o n to T-^  i n a manner s i m i l a r to that given.by Abragam, page 3^9- For the case of l i q u i d s B l i c h a r s k i presents.the formula (- ) = 8 TTZ cZx q < J(J.+ 1 ) > T C [ l + 2 (±cjfl (2.21) ^ T l ' s.r 3 9 \ C j / • 19 The q u a n t i t y q w h i c h i s c a l l e d t h e ' " q u e n c h i n g f a c t o r " i s p o s i t i v e a n d l e s s t h a n u n i t y . I t t a k e s i n t o a c c o u n t t h e f a c t t h a t t h e s p i n - r o t a t i o n i n t e r a c t i o n i n . a l i q u i d may be c o n s i d e r a b l y , i n f l u e n c e d b y s t r o n g 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 . The q u a n t i t y N A i s d e f i n e d b y 2 (Aci) 2 = ( c z z - C x x ) + ( c Z 2 - -QyyXcxx - C y y ) , (2.22) The e t h a n e m o l e c u l e . i s a p r o l a t e symmetr ic t o p m o l e c u l e , so t h e l a s t t e r m i n E q u a t i o n (2.22.) v a n i s h e s . A n d e r s o n ^ 2 ^ . has m e a s u r e d t h e s p i n - r o t a t i o n a l c o n s t a n t s f o r e t h a n e . H i s r e s u l t s . a r e c x x = C y y = 0.70+ 0.20 kc c z z •= '9-1 :+ -Ivk kc . (2.23) T h i s t h e n g i v e s Ac-j. = 8.4,+ 1.-6 kc and So t h a t c T ••= 1 (c + c .+ c : ) = 3-5 + 0.6 kc I - ^ x x . y y .-zz' J ' — . c j 2 £ l . + 2 ^ A c T j 2 j= 2.8 x T O 7 s e c " 2 . '(2.-2k) The r a t h e r T a r g e e r r o r s g i v e n f o r t h e s p i n - r o t a t i o n a l c o n s t a n t s mean t h a t t h i s l a s t q u a n t i t y i s o n l y known t o w i t h i n a b o u t a f a c t o r o f '2. An a p p r o x i m a t e f o r m u l a f o r ^ J ( . J .+ l ) ^ i s g i v e n b y B l i c h a r s k i ^ 1 - ^ , < j ( j + l ) ? ^ k T ( I ,+ 2 l J (2.25) A B w h i c h i s c o r r e c t f o r A h c « 1, B h c ^ < 1, where A , B . a n d I . , ' T g a r e k T k T t h e r o t a t i o n a l c o n s t a n t s and t h e p r i n c i p a l moments o f i n e r t i a r e s p e c t i v e l y . The r o t a t i o n a l c o n s t a n t s . a n d moments o f i n e r t i a a r e g i v e n f o r e t h a n e b y H e r z b e r g ^ 2 1 ^ •: A = 2.538 c m " 1 ' B = 0.6621 c m " 1 . T A = II.03 xl0"^° g m - c m 2 I B = U2.-28 x T O - 1 * 0 g m - c m 2 Equation (2.25) i s then v a l i d i n the temperature range studied. Using the values given f o r the moments of i n e r t i a Equation (2.25) "becomes <J("J .+ 1) > = 1.19 T (2.26) Using these numerical r e s u l t s Equation (2.2T) gives = (26.3)(2.8 x 107)(1.19 T) q re o (2-27) = 8.8 x 10 Tq 7t (k) -If Ti i s known i t i s then possible using this:equation•to f i n d q values from measured values of T-^  assuming that T-^  depends mainly on. the .spin-r o t a t i o n a l i n t e r a c t i o n . T t i s perhaps be t t e r to make a p r i o r i estimates f o r q and using these estimates to deduce how e f f e c t i v e the. s p i n - r o t a t i o n a l i n t e r a c t i o n i s i n producing relaxation, (17) The quenching f a c t o r i s given by Kr y n i c k i and Powles as q = ^r-^ (2.28) Tj -t Tc where TJ i s the average l i f e t i m e of the r o t a t i o n a l J states .and TQ i s again the e f f e c t i v e molecular c o r r e l a t i o n time. A rough estimate can'be made of Tj following a s i m i l a r argument to that of Kr y n i c k i and Powles. I f 9 i s the angle of some vector i n the molecule, then AOl = AO^ i = (u,r^ ± (2..29) a f t e r .time t. When t = * T c i t i s expected that the average change i n 0 w i l l be about 1 radian. This gives (ujjf g « 1 (2.30) .The .average angular v e l o c i t y may.be crudely estimated by -*2 4. 2 y W. T . n ^2 JJtJi = J fl = <J(J.+ 1)> = 9 k -T ( 2 , 3 1 ) I * ^ A + 2 IB->] 2 JA + 2 I B This then gives Tj = T A * 2 I B = 3-9 x 10-2h ( 2 > 3 2 ) 9 :k • T Tc T Te It i s hard to judge how v a l i d an estimate i s obtained f o r Tj by these arguments. A r e a l l y s a t i s f a c t o r y theory would derive the r e l a t i o n s h i p between TJ and the molecular properties from f i r s t p r i n c i p l e s . A s t a r t i n t h i s d i r e c t i o n has been made by Oppenheim and Bloom^-^. In the case of l i q u i d s K r y n i c k i and Powles state that T c ^ y. and t h i s gives, using Equation (2.28) and Equation (2-32) q = 3-9 x 10-^ (2..33) •T T* and t h i s y i e l d s = 3-k x l O " 1 5 (2.'34) ay sr Tc The molecular c o r r e l a t i o n time may be r e l a t e d to the d i f f u s i o n constant using Equation (2.15) and t h i s gives -lk . sr a1 = 6.1 x-10-^ D (2.35) where .a i s the molecular•diameter. 22 Chapter III THE;EXPERIMENTAL METHOD 3-1 The N.M.R. Spectrometer The basic apparatus and techniques of measurement using spin echoes have (22 2^ 2h 25) been described by many authors v . ' .•'>>••:" ' . Only a short d e s c r i p t i o n of the apparatus w i l l be given here and t e c h n i c a l d e t a i l s may be found .in Appendix A. -A block diagram of the apparatus i s given i n Figure 3- Each of the major items w i l l now be discussed b r i e f l y . 3-1.1 The magnet The magnet made ava i l a b l e f o r t h i s work was the .7.OOO gauss permanent . a g n e t a l r e a d y d e s c r i e d m d e t a i l » y W a t e r m a r k 2 6 >. A permanent H a g„et tos a r e a l advantage over an electromagnet f o r work of t h i s nature. I t allows almost complete freedom from long term d r i f t i n the magnetic f i e l d . This s i m p l i f i e s considerably the tuning of the spectrometer, which i s always d i f f i c u l t when long re l a x a t i o n times are involved. Also many of the measurements took at l e a s t an hour to complete and a slow d r i f t would have been very disturbing. The magnet does have a s l i g h t temperature c o e f f i c i e n t , but with a i r conditioning i n the room t h i s was never noticeable. Unfortunately the magnetic f i e l d i s not very homogeneous, but t h i s i s compensated by the fa c t that the magnetic f i e l d gradient i s constant i n time. 3-1-2 Timing c i r c u i t s I t was known from the beginning that the rel a x a t i o n times to be measured would be rather long (about 30 sec). .Consequently a timing system capable of the accurate measurement of long r e l a x a t i o n times was e s p e c i a l l y designed - and constructed. The system consists of several sub-units: a Tektronic time mark generator (model RM - l8 l ) f o r generating fundamental time units-; a.pulse sequence sub-unit to produce two channels with c o n t r o l l e d pulse spacing; and TIME MARK GENERATOR TIMING CIRCUITS CH.I PULSE GENERATOR TRIGGER INPUT C H , n PULSE GENERATOR MIXER PULSED R.F, T R A N S M I T T E R TU_NING HEAD _ _ _ _ _ _ _ HI-i i v-It i INLINE SAMPLE MAGNET Figure 3- Block diagram of spectrometer two independent pulse generators used to gate the r - f o s c i l l a t o r . A timing system based on e l e c t r o n i c counters was^chosen since t h i s i s capable of producing a l a r g e r range of time i n t e r v a l s with given accuracy than other types of c i r c u i t r y and, moreover, several independent time channels can be added with l i t t l e d i f f i c u l t y . 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 i n the time mark generator controls the o v e r a l l accuracy of the timing system. A l l time i n t e r v a l s are f i x e d by t h i s o s c i l l a t o r and they are known to an accuracy of a few parts i n 10 . Following the o s c i l l a t o r i n the time mark generator are f i v e scale-of-ten m u l t i v i b r a t o r s . The output of each m u l t i v i b r a t o r i s a v a i l a b l e i n the.form of sharp pulses on'the front panel, Fundamental time u n i t s of duration.1, 10, 100, 1000 and 10,000 microseconds are provided. These time units are then counted by a cascade bank of 6 magnetron beam switching tube scales of ten. Each of these counters has an output corresponding to each d i g i t . Outputs from the f i r s t three counters.are connected to two independent t r i p l e coincidence c i r c u i t s using selector switches. 'One row of three•selector switches i s associated with channel I and the other with channel II. With the channel I selector switches set at 123, f o r example, the coincidence c i r c u i t produces .an output pulse whose leading edge occurs only when i t s three input c i r c u i t s .are simultaneously-.activated, which i s whenever the counter reads 123- Depending on which fundamental time u n i t i s selected t h i s may be a time of 123 /(sec, 1.23 msec, 12-3 msec, ect. Channel II works on the same p r i n c i p l e as channel I, only at d i f f e r e n t times selected by i t s own set of switches. These two pulses are then used to activa t e the transmitter gating c i r c u i t s . The l a s t three counters are also connected to a t r i p l e coincidence c i r c u i t by-a set of three switches. This set of counters i s used to c o n t r o l the r e p e t i t i o n rate, .that i s , how often the two pulse sequence described above i s generated. When the counters and selector switches are i n coincidence ••25 an output pulse i s generated which t r i g g e r s the os c i l l o s c o p e , resets the counters to zero and opens gates f o r the channel I and II pulses. These gates determine how many pulses are generated by each channel. This system i s extremely f l e x i b l e . More/technical .details .can ;be found in-Appendix .A. The pulses from channels I and I I are fed to two separate pulse width generators -used to co n t r o l the r - f transmitter. The two i d e n t i c a l pulse width generators were e s p e c i a l l y designed f o r t h i s a p p l i c a t i o n . Pulse lengths are determined by l i n e a r sweep c i r c u i t r y (a phantastron.), rather .than by mult i v i b r a t o r s , to make possible a 1$ s t a b i l i t y . The-pulse widths are continuously v a r i a b l e from.less than 1 microsecond,to more than.100 micro-seconds. These pulses are then mixed and .used to gate the transmitter. 3-1-3 Radio-frequency transmitter The rectangular outputs of the pulse width generators :were used to turn a radio frequency o s c i l l a t o r on-and o f f , rather than an a m p l i f i e r following i t , because.leakage past the gated-amplifier would .ordinarily cause d i f f i c u l t i e s with the receiver, which i s tuned to the o s c i l l a t o r frequency ;.and i s extremely .sensitive -•A simple g a t e d • o s c i l l a t o r takes several microseconds to reach f u l l amplitude-, so a combination of-a r i n g i n g c i r c u i t and an . o s c i l l a t o r has been used which reaches f u l l amplitude i n about 0.1 microsecond. The pulsed Hartley o s c i l l a t o r used i n t h i s equipment was operated-at low. l e v e l to improve i t s frequency s t a b i l i t y . The o s c i l l a t o r i s electron coupled to furth e r i s o l a t e i t from i t s load-and improve i t s s t a b i l i t y . Following the o s c i l l a t o r are a voltage amplifying . stage and -a power ampli f i e r . The. power-.amplifier produces reasonably rectangular r - f pulses of about 1000 v o l t s peak to peak.. The o s c i l l a t o r frequency must of course match the magnetic f i e l d available-and was fo r t h i s reason set at 30 megacycles. The output power from the transmitter i s fed through a small capacitor (about 3 P f ) to the tuned circuit.. 26 Sample c o i l s The sample c o i l s were t u n e d r e m o t e l y b y m a k i n g u s e o f a h a l f wave l e n g t h t r a n s m i s s i o n l i n e . T h i s l i n e c o u l d be p l u g g e d i n t o e i t h e r t h e ' l i q u i d o r v a p o r c o i l , t h u s f a c i l i t a t i n g t h e change f r o m a l i q u i d t o a v a p o r measurement . T h i s meant t h a t a change c o u l d be made r a p i d l y . a n d w i t h o u t d i s t u r b i n g t h e sample i n . a n y way. B o t h , sample c o i l s were i d e n t i c a l a n d t h e y were a b o u t 1 cm l o n g . 'They were made o f 10 t u r n s No.18 w i r e wound a r o u n d -a 6 mm g l a s s t u b e w i t h a w i r e s p a c i n g a b o u t e q u a l t o t h e d i a m e t e r o f t h e w i r e . The c o i l n o t i n . use was l e f t open c i r c u i t e d so t h a t no c u r r e n t w o u l d be i n d u c e d i n t o i t . T h i s was done b e c a u s e t h e c o i l s , were o n l y - s e p a r a t e d b y - a d i s t a n c e o f a b o u t t h r e e c e n t i m e t e r s and t h i s c l o s e p r o x i m i t y w o u l d c a u s e some d i f f i c u l t y b e c a u s e o f c o u p l i n g b e t w e e n ' t h e two c o i l s . The s i g n a l s i n d u c e d b y t h e p r e c e s s i n g n u c l e i i n t h e sample were c o u p l e d t o t h e r e c e i v e r t h r o u g h a 12 , p f . c a p a c i t o r . 3-1-5 R e c e i v e r I n t h i s a p p a r a t u s t h e r e c e i v e r u s e d was a c o m m e r c i a l u n i t . I t was o b t a i n e d f r o m L E L , I n c . and was t h e i r m o d e l I F '21 B S. T h i s i s a w i d e - b a n d S y n c h r o n o u s l y t u n e d a m p l i f i e r , w i t h a maximum g a i n o f a b o u t ' 1 0 0 db a n d a b a n d w i d t h o f 2 m e g a c y c l e s c e n t e r e d a r o u n d 30 m e g a c y c l e s . A s p e c i a l e f f o r t was made t o m a t c h t h e i n p u t impedance o f the : r e c e i v e r f o r b e s t s i g n a l t o n o i s e - r a t i o . The i n p u t was m o d i f i e d t o a h i g h i m p e d a n c e , h i g h Q t u n e d c i r c u i t t o c o u p l e t o t h e " h i g h impedance sample c i r c u i t . The a m p l i f i e r was c a r e f u l l y n e u t r a l i z e d t o o b t a i n t h e b e s t s i g n a l t o n o i s e r a t i o . A 1N295 d i o d e d e t e c t o r i s n o r m a l l y ; s u p p l i e d w i t h t h i s u n i t a n d i t was u s e d f o r a l l measurements . A "diode d e t e c t o r s u c h as t h i s d o e s n o t o p e r a t e l i n e a r l y o v e r i t s e n t i r e o u t p u t r a n g e . T h e r e i s a v e r y p r o n o u n c e d n o n - l i n e a r i t y f o r s m a l l s i g n a l s .and t h i s becomes e x t r e m e l y . important - .when, d e a l i n g w i t h p o o r s i g n a l t o n o i s e r a t i o s . I t i s t h i s n o n - l i n e a r i t y w h i c h o f t e n d r i v e s one t o c o n s i d e r t h e use o f p h a s e s e n s i t i v e d e t e c t i o n , w h i c h - a l l o w s o p e r a t i o n i n t h e l i n e a r r e g i o n o f t h e diode conductance curve. There are many problems associated with phase sensitive detection at 3 0 megacycles and so possible•alternatives should be considered. One possibility is to make corrections for the non-linearity using a calibration curve of output voltage vs. input voltage. This works fine i f noise is not present to any significant extent in the output but loses its value in the presence of noise. Another possibility is .to increase the amplifier gain to keep the signal plus noise up out of the non-linear region and to be careful not to use signals of small amplitude. This is the method used here and .while i t is not completely satisfactory i t was'.found to be adequate. The detector linearity was carefully checked in the presence of noise and a working criterion ,was established emperically. -It is this -discard a l l signals whose .amplitude-is less than twice the.d.c. component of noise or whose amplitude is less than 0 . 2 volts, -whichever is greater. As long as this rule is followed ruthlessly, non-linearity of the diode will not be a problem, however i t does-limit somewhat the range of useful output voltage. 3 - 1 - 6 Oscilloscope A Tektronix Model 5 3 1 oscilloscope with P7 phosphor screen was used for a l l measurements reported here. The. long persistence of the P7 phosphor was very useful since a l l measurements of echo amplitude were made visually, directly from the face of the CRT- A type Z plug in unit was used with the oscilloscope. The type Z is a calibrated differential comparator and i t allows a considerable expansion to be made in the vertical scale. Maximum echo amplitudes of 2 or 3 volts were usually used and this corresponds to a vertical scale of 2 0 or 3 0 cm which could be measured to about 0 . 1 cm. • 3 - 2 Temperature, its Measurement and Control 3 - - 2-,l Thermometer All measurements of temperature were made using a platinum.resistance 2.8 thermometer. This type of thermometer', the ."work horse" of the International Temperature Scale, was chosen because i t makes a particularly versatile thermometer. When properly made i t is extremely reproducible and is many times more sensitive than the gas thermometer. The thermometer used was a Hartman-Braun model W 4 8 7 1 suitably modified to N.R.C. specification. Since most of the measurements were to be made in the room temperature range the thermometer was calibrated at•two fixed points, the freezing and the boiling points of water. First the thermometer was immersed in a well stirred bath prepared by freezing distilled water. The thermometer was left in the bath for some time and many measurements were made of its resistance, with a current of about 1 ma flowing through the thermometer first one direction and then the other. This was done to cancel out the effects due to thermal emfs. A resistance at 0°C of 99-725 ohms was obtained with an estimated uncertainty of about 0 . 0 0 1 ohm. The.ice point is very reproducible and the uncertainty in the temperature is about 0 . 0 0 1°C. The thermometer was next placed in a hypsometer to establish its resistance at the boiling point of water. Again distilled water was used and the heating power input to the hypsometer was set at several different values to determine that the resistance of the thermometer was independent of heating power. Since the temperature of boiling water is strongly dependent on the atmospheric pressure this was measured simultaneously. To obtain the atmospheric pressure a large bore manometer was read using a precision cathetometer. The current in the thermometer was again kept at 1 ma and readings were taken for currents in both directions. A resistance of 138.127 ohms was obtained with an uncertainty of about 0 . 0 0 1 ohm. The atmospheric pressure was 76U.78 mm Hg with an uncertainty of 0 . 0 5 mm Hg. This then gives a temperature of 100.175°C with an uncertainty of about 0 . 0 0 2 ° C 29 For temperatures between these two f i x e d points "the standard i n t e r p o l a t i o n formula of Callendar was used. The i n t e r p o l a t i o n formula as given i n the d e f i n i t i o n of the International Temperature Scale, = R D(1 + At + B t 2 ) (3-1) may be written i n the Callendar form where o< = J L / Rioo - i I . (3.3) 100 I j - - — 1 v The r e l a t i o n s between the c o e f f i c i e n t s are A = oi ( l + 6 \ B = - S K Tool 100 2 • -(3-4) Normally i t then requires three f i x e d points to completely determine the scale (Rj_ce> -^steam* ^ s u l f u r ) * However since no measurements were to be taken above 100°C only two were used and the value, of >$ was assumed be I.50 as i s the case f o r pure platinum. In t h i s temperature region the c a l i b r a t i o n i s not very sensiti^sre to the value of S . Using the measured values of resistance and the assumed value of & gives f o r the i n t e r p o l a t i o n formula R t •= 99.725 ( l + 3.9OI8 X--10-3 t - 5-7663 x 10-7 t f t) (3.5) Values f o r R^ were then c a l c u l a t e d at 0.1° i n t e r v a l s and tabulated, then l i n e a r i n t e r p o l a t i o n was used between these i n t e r v a l s . This c a l i b r a t i o n i s thought to be within 0.01°C on an absolute scale. 3.2.2 Potentiometer system A l l measurements of the platinum thermometer resistance were made using a high p r e c i s i o n Rubicon model 2781 potentiometer- This instrument i s e s p e c i a l l y designed f o r the accurate measurement of p o t e n t i a l difference and 30 i n c o r p o r a t e s c o m p l e t e e l e c t r o s t a t i c s h i e l d i n g and s p e c i a l g u a r d i n g c i r c u i t s . P e r h a p s t h e most u s e f u l f e a t u r e o f t h i s p o t e n t i o m e t e r i s t h e p r o v i s i o n o f a f i x e d r e f e r e n c e emf. The two b i n d i n g p o s t s marked "1.0" c o n n e c t t o f i x e d p o i n t s i n t h e m e a s u r i n g c i r c u i t t o p r o v i d e a r e f e r e n c e v o l t a g e o f 0 . T v o l t w i t h t h e r a n g e s e l e c t o r s w i t c h s e t a t " M E D " . O p e r a t i o n o f t h e "1.0-E M F " Key t o t h e "1.0" p o s i t i o n c o n n e c t s t o t h e 0.1 v o l t p o i n t s t h r o u g h t h e r e g u l a r g a l v a n o m e t e r c i r c u i t a v o i d i n g t h e n e e d f o r e x t e r n a l s w i t c h i n g . T h i s r e f e r e n c e emf i s compared w i t h t h e emf d e v e l o p e d b y t h e c u r r e n t f l o w i n g t h r o u g h a p r e c i s i o n s t a n d a r d r e s i s t o r ( M i n n e a p o l i s - H o n e y w e l l , Type N . B . S . , M o d e l No, 1102-99'999 ohms) i n s e r i e s w i t h t h e t h e r m o m e t e r . See F i g u r e k. The p o t e n t i o m e t e r c u r r e n t i s s t a n d a r d i z e d a g a i n s t t h e c u r r e n t f l o w i n g i n t h e s t a n d a r d r e s i s t o r i n s t e a d o f u s i n g ' a ' s t a n d a r d c e l l . " T h i s s p e e d s up t e m p e r a t u r e measurements c o n s i d e r a b l y and . s i n c e t h e s t a n d a r d r e s i s t o r i s e q u a l t o 100 ohms t h e p o t e n t i o m e t e r i s d i r e c t r e a d i n g i n r e s i s t a n c e . F o r p r e c i s i o n measurements a p o t e n t i o m e t r i c method o f measurement o f r e s i s t a n c e i s much p r e f e r r e d t o s a y , a W h e a t s t o n e b r i d g e , s i n c e l e n g t h a n d r e s i s t a n c e o f l e a d w i r e does n o t m a t t e r . The s t a n d a r d r e s i s t o r i s m o u n t e d . i n a 2" t h i c k S t y r o - f o a m e n c l o s u r e t o p r o t e c t i t f r o m t e m p e r a t u r e c h a n g e s . I t s v a l u e was t a k e n t o be 100.000 ohms and t h e t e m p e r a t u r e c o r r e c t i o n was f o u n d t o b e i n s i g n i f i c a n t . The w o r k i n g c u r r e n t f o r t h e thermometer was s u p p l i e d b y a 1.5 v o l t d r y c e l l mounted i n t h e c o n t r o l b o x . A l s o i n "the c o n t r o l b o x were t h e r e v e r s i n g s w i t c h e s t o r e v e r s e b o t h t h e c u r r e n t i n t h e thermometer a n d t h e p o t e n t i o m e t e r •and c o n t r o l s f o r a d j u s t i n g and m o n i t o r i n g t h e c u r r e n t i n t h e t h e r m o m e t e r . T h i s c u r r e n t was a l w a y s k e p t c o n s t a n t a t a b o u t 1 ma. The i n t e r n a l g u a r d c i r c u i t s o f t h e p o t e n t i o m e t e r were e x t e n d e d o v e r a l l t h e e x t e r n a l c i r c u i t s t o p r o v i d e f r e e d o m f r o m t h e a d v e r s e e f f e c t s o f l e a k a g e c u r r e n t s b e t w e e n c r i t i c a l c i r c u i t p o i n t s . 1.5V | o-i CONTROL BOX I K < ^ I5T HELIPOT < 402 fl [% - M A . 0 — o NBS TYPE 100 f l STANDARD - O - A A / V - Q - 0 1-0-1 MA 6V L X — +6V BA SHIELD o I o PLATINUM RESISTANCE THERMOMETER + — 1.0 + — EMF RU8IC0N MODEL 2781 POTENTIOMETER GA L o-T NULL INDICATOR Figure k. Thermometer control and-measurement circuit. A measure of the s e n s i t i v i t y of the potentiometer system i s the l e a s t count f o r the instrument, i . e . the reading which may he taken without i n t e r p o l a t i o n by eye. For t h i s instrument the l e a s t count i s 5 microvolts which corresponds to a resistance of 0.005 ohms and a temperature of about 0.01°C I t was found however that r e l i a b l e readings could be taken to 1 microvolt or 0.002°C so that temperature changes of t h i s order were detectable. In spite of t h i s , the o v e r a l l accuracy i s not f e l t to be b e t t e r than •about 0.01°C 3.2.3 Temperature c o n t r o l system From the outset i t was f e l t that the temperature c o n t r o l should be as good as pos s i b l e . This meant of course that an automatic c o n t r o l system must be used. The major requirement f o r the system i s that i t be stable, yet s e n s i t i v e to small changes of temperature. It i s because of these two requirements that the f i r s t compromise must be made. A platinum resistance thermometer could again be used to sense the temperature and t h i s would c e r t a i n l y be stable, however the s e n s i t i v i t y of such a thermometer i s not very high. Indeed, none of the pure m e t a l l i c conductors have a very high temperature c o e f f i c i e n t of r e s i s t i v i t y . A possible a l t e r n a t i v e i s the use of a semi-conductor thermometer. These thermistors are, i n p r i n c i p l e , extremely u s e f u l because t h e i r resistance varies approximately exponentially with temperature, so that a properly chosen thermistor may be many times as se n s i t i v e as a m e t a l l i c thermometer. Unfortunately i t i s found that t h e i r s t a b i l i t y i s not as good as a m e t a l l i c thermometer. I t was f i n a l l y decided to use.a w e l l aged thermistor as the sensing.element i n the c o n t r o l system. A Western E l e c t r i c 17A thermistor was selected since i t s resistance (about 1000 ohms at room temperature) i s high enough to eliminate the problem of lead resistance and low enough so that noise does not become a problem. The resistance was measured by a Wheatstone bridge so that a slow drift in the battery potential would not cause a drift in temperature. The output of the Wheatstone bridge is the error signal which is used by the electronic system for making corrections to the heating power applied. This output is fed directly into an electronic Null Indicator which is used as a high gain direct current amplifier. An output was added to the Null Indicator as shown in Figure 5- This output is fed into the temperature control circuit, which consists of two difference amplifiers, a cathode follower driver and an output stage. High quality stable components were used throughout in the construction of these circuits. Power is fed to the heaters directly from the cathode of the output 6L6 power amplifier. A recording milliammeter provides a continuous record of the power supplied to the sample holder, so that the stability of the system may be checked over long periods of time. Heat was supplied to the sample holder by two 100G ohm bifilar wound resistors. One of these resistors is wound around the outside of the sample holder and is the main source of heat. The other resistor is located inside the sample holder, very close to the thermistor. The second heater is needed to speed up the overall response of the system. With no power supplied to this heater, the time lag between the application of heat and the thermistor's response is so long that the system oscillates at a very low frequency. However with the inside heater supplying just the proper ratio (this may be adjusted) of the total heat, the thermistor will follow the sample holder in temperature almost exactly. This adjustment is made by observing the transient response of the system to a change in setting on the Wheatstone bridge. The sensitivity of the system proved to be more than adequate; however the long term stability achieved was not quite so gratifying. A steady drift in temperature of about 0.01°C per day had to be tolerated; this drift came almost entirely from the thermistor. Actually two thermistors were used during the course of this work. Drift + 225 V -Figure 5- Temperature control system. 35 due to the f i r s t thermistor became s t e a d i l y worse with time u n t i l i t was replaced. The second thermistor was much better and the d r i f t remained constant at about 0.01°C per day. Short term d r i f t could not be detected by the thermometer and was therefore l e s s than 0.002°C per hour. With some manual intervention the temperature could be held constant at better than O.OI°C i n d e f i n i t e l y . 3-3 The Sample Holder The sample holder (see Figure 6) was made from.a s o l i d rod of copper l p " i n diameter and 6" long with a 5/8" hole bored 5" deep. This means that the sample was completely surrounded by a c y l i n d e r of copper 5/l6" thick. At one end of t h i s c y l i n d e r was a closed chamber f i l l e d with o i l as a heat t r a n s f e r medium into which the thermistor and one of the heaters were inserted. At the other end, the c y l i n d e r was capped with a brass plug with three holes i n i t . Two of the holes were f o r c o a x i a l cables to the l i q u i d and vapor c o i l s and the other hole allowed the samples to be inserted into the holder. The brass plug was then attached to a s t a i n l e s s s t e e l tube which supported the sample holder i n a vacuum jacket. A vacuum of about 1 micron was found to be s u f f i c i e n t f o r purposes of thermal i s o l a t i o n . S t a i n l e s s s t e e l i s a r e l a t i v e l y good i n s u l a t o r so that the major heat leak from the sample holder was by r a d i a t i o n . To reduce t h i s heat l o s s , the copper sample holder and the inside of the vacuum jacket were highly polished. A l l of these measures were taken to reduce the thermal gradient along the length of the sample. It required about 100 mw to hold the sample at the c r i t i c a l temperature. The worst case would be i f t h i s t o t a l heat flowed ; a x i a l l y down the. c y l i n d e r . In t h i s case the thermal gradient i n the copper sample holder would be about 0.00U°C per cm. An e f f o r t was made to measure t h i s gradient using thermocouples placed 3 6 Coaxial Line Kovar Seal Thermomete Heater Thermistor NOT TO SCALE Stainless Steel Vacuum Jacket Glass Tube Copper Vapor Coil Liquid Coil Sample Tube Oil Filled Chamber Vacuum Line Figure 6 . Sample holder. i n s i d e a g l a s s t u b e a t t h e p o s i t i o n s o f t h e two c o i l s . The t h e r m o c o u p l e s e n s i t i v i t y was n o t s u f f i c i e n t f o r a g r a d i e n t t o be d e t e c t e d . An u p p e r l i m i t o f 0.05°C was p l a c e d on t h e t e m p e r a t u r e d i f f e r e n c e b e t w e e n t h e s e two p o s i t i o n s . The p l a t i n u m thermometer was imbedded deep i n t o t h e c o p p e r c y l i n d e r (as shown i n F ig .6' ) and t h i s t e m p e r a t u r e was t a k e n t o be t h e t e m p e r a t u r e o f t h e s a m p l e . 3.U Sample P r e p a r a t i o n A l l o f t h e e t h a n e samples were p r e p a r e d b y J o h n L e e s , t h e p h y s i c s d e p a r t m e n t g l a s s b l o w e r , f r o m P h i l l i p s r e s e a r c h - g r a d e 99-99-mole $ p u r e m a t e r i a l . P h i l l i p s s t a t e t h a t t h e most p r o b a b l e i m p u r i t y - i s p r o p a n e . T n t h i s c o n c e n t r a t i o n t h e p r o p a n e i m p u r i t y w o u l d be i n s i g n i f i c a n t . O f t h e o t h e r u b i q u i t o u s i m p u r i t i e s s u c h as n i t r o g e n , o x y g e n and w a t e r , o n l y p a r a m a g n e t i c o x y g e n w o u l d be i m p o r t a n t . D i s s o l v e d - o x y g e n c o n t r i b u t e s a v e r y p o t e n t r e l a x a t i o n mechanism a n d o n l y s m a l l , q u a n t i t i e s o f o x y g e n a r e n e e d e d f o r t h i s mechanism t o d o m i n a t e t h e s p i n - l a t t i c e r e l a x a t i o n . The s a m p l e s were- t h e r e f o r e f u r t h e r p u r i f i e d t o remove t h e o x y g e n i m p u r i t y . The. p u r i f i c a t i o n t e c h n i q u e •• (21 was t h e same g e t t e r i n g - t e c h n i q u e - a s t h a t d e s c r i b e d b y L e e s , M u l l e r and N o b l e v e x c e p t t h a t a f t e r t h e gas h a d b e e n i n c o n t a c t w i t h t h e g e t t e r f o r s e v e r a l d a y s t h e e t h a n e was c o n d e n s e d i n t o s m a l l t u b e s u s i n g l i q u i d n i t r o g e n and t h e n s e a l e d o f f : f r o m t h e g e t t e r i n g b u l b . T h i s t e c h n i q u e y i e l d e d s a m p l e s f o r w h i c h T]_ was r e p r o d u c i b l e f r o m sample t o sample o v e r l o n g p e r i o d s o f t i m e . M o s t o f t h e s a m p l e s s t u d i e d were c o n t a i n e d i n g l a s s t u b i n g w i t h 1 m m . w a l l t h i c k n e s s a n d 6 mm o u t s i d e d i a m e t e r a n d were a b o u t 8 o r 9 cm l o n g . E t h a n e i s a s o l i d a t l i q u i d n i t r o g e n t e m p e r a t u r e a n d t h i s makes f i l l i n g t h e t u b e s somewhat d i f f i c u l t . The s o l i d i s v e r y p o r o u s and i t s d e n s i t y i s d i f f i c u l t t o d e t e r m i n e . To overcome t h i s d i f f i c u l t y t h e t u b e s were a l l o w e d t o warm up u n t i l t h e s o l i d m e l t e d a n d a l i q u i d l e v e l c o u l d be s e e n . The p r o b l e m i s t o d e t e r m i n e t h e a v e r a g e o v e r a l l ( v a p o r + l i q u i d ) d e n s i t y O f l o a d i n g p , so t h a t t h e sample c o u l d be c r i t i c a l l y l o a d e d . U s i n g t h e known d e n s i t y o f t h e l i q u i d a t t h e m e l t i n g p o i n t and a s s u m i n g t h a t t h e v a p o r d e n s i t y i s z e r o t h e o v e r a l l d e n s i t y may be c a l c u l a t e d . F o r c r i t i c a l l o a d i n g , t h i s g i v e s t h a t t h e sample t u b e s h o u l d be a b o u t one t h i r d f u l l o f l i q u i d a t t h e m e l t i n g p o i n t . Samples were f i l l e d b y t r i a l and e r r o r e a c h t i m e w i t h a b o u t one t h i r d o f t h e t o t a l volume o c c u p i e d b y t h e l i q u i d . The samples were t h e n a l l o w e d t o warm up a n d t h e b e h a v i o u r o f t h e m e n i s c u s o b s e r v e d . T h e r e a r e t h r e e t y p e s o f b e h a v i o u r , d e p e n d i n g on t h e o v e r a l l d e n s i t y o f l o a d i n g : W h e n : p < pc The m e n i s c u s f a l l s , f i n a l l y d i s a p p e a r i n g a t the b o t t o m o f t h e v e s s e l a n d t h e e n t i r e t u b e i s f i l l e d w i t h g a s . p > f>c The m e n i s c u s r i s e s , f i n a l l y d i s a p p e a r i n g a t t h e t o p o f t h e v e s s e l and t h e e n t i r e t u b e i s f i l l e d w i t h l i q u i d . ^ S pz The m e n i s c u s d i s a p p e a r s a t a p o i n t a b o u t midway up t h e c o n t a i n e r . The l a s t c o n d i t i o n may be t a k e n as a c r i t e r i o n f o r a t l e a s t p a r t o f t h e sample p a s s i n g t h r o u g h t h e c r i t i c a l p o i n t . The e f f e c t o f t h e g r a v i t a t i o n a l f i e l d on t h e f l u i d i s much g r e a t e r t h a n one w o u l d s u s p e c t . I n d e e d , t h e b e h a v i o u r o f a r e a l gas i n i t s c r i t i c a l r e g i o n i s c r u c i a l l y d e p e n d e n t on t h e e f f e c t s o f g r a v i t y . The i s o t h e r m a l c o m p r e s s i b i l i t y K j j i — -d i v e r g e s t o i n f i n i t y a t t h e c r i t i c a l p o i n t ^ ) . T h i s means t h a t t h e f l u i d becomes v e r y c o m p r e s s i b l e . Now t h e p r e s s u r e i n a f l u i d v a r i e s w i t h h e i g h t due t o t h e w e i g h t o f t h e f l u i d above a c c o r d i n g t o t h e r e l a t i o n f6L = - /O (z) g (3-7) dz T h e n , s i n c e t h e t e m p e r a t u r e i s c o n s t a n t t h r o u g h o u t t h e f l u i d , t h e d e n s i t y g r a d i e n t i s .2/ _£ - -.fU) g. = -/o 2 ( - . )Kp g (3-8) d z - (^hf> )T This leads to a macroscopic density gradient due to hydrostatic pressure in the fluid. Only one part of the sample can therefore be at the critical density and this is usually the part of the sample about midway up the container. The sample inside the signal coils does not, strictly speaking, pass through the critical point, but only near to i t . At the critical temperature the sample in the upper coil is slightly less dense than the critical density and that in the lower coil is slightly more dense. No real difficulty was experienced using this method of f i l l i n g the sample tubes and about one in four was found to be correctly fi l l e d . A reasonable amount of caution was exercised in handling the sample tubes because of the high pressures involved. Of the many samples prepared only three exploded spontaneously and these were a l l at temperatures above room temperature. However since these samples did explode i t is not felt that the margin of safety is sufficient and great care should be used in handling and storing samples of this nature. After determining that a sample was properly f i l l e d by visual observation of the meniscus at the critical point the samples were cemented to a long glass tube using epoxy resin cement. The glass tube was used to insert and position accurately the samples in the sample holder. 3-5 Experimental Techniques All of the measurements reported here were made using the spin echo ( 2 2 ) technique discovered by E.L. Hann in 19^9 • The use of spin echoes in the measurement of spin lattice relaxation times and self diffusion constants ( 2 3 ) is discussed in detail by Hahnand by Carr and;Purcellv :SO that only.a brief description of these methods will be given here. 3.5.I Introduction A description of the magnetic properties of. bulk samples of nuclei subjected to an external magnetic field is perhaps best obtained from the ko phenomenological equations of B l o c h H e suggests that the equation of motion of the nuclear magnetization for an essemble of spins be given by = l f M x H - t % . J . ^ , .-^(M z - MQ) ( 3 - 9 ) 2 T 2 ^ The first term gives the motion of free spins, while the last terms give the motion due to relaxation. Using this equation i t is now possible to predict the non-equilibrium behaviour of the macroscopic magnetization of a sample of nuclei. The sample is first placed in a large external field HQ:and .a time long compared to either Ti or T2 is allowed to elapse so that the sample comes to thermal equilibrium. Now another magnetic field H-j_ is applied to the sample in order to drive the magnetization to a non-equilibrium value. This field is usually one of the rotating components of the field H x = 2H , cos u> t linearly polarized along the x axis; the other counter-rotating component (29) of the field may be neglectedv . The effect of this new field is to rotate the macroscopic magnetization vector from its equilibrium position along the z axis to some new position. The field Hi is usually applied in the form of very brief bursts of radio frequency power at the Larmor frequency. Using an appropriately. shaped pulse i t is possible to rotate the magnetization vector right down into the x^ -y plane. The angle -0 between the vector M and the z axis is given by: 9 = 3 ^ 7 1 ( 3 - 1 0 ) where 9 is in radians and Tu is the length of the r-f pulse. The length of the r-f pulse is usually specified.in terms of the rotation angle -9. A 9 0 ° pulse then implies that 9 = 1 radians. After the application of a 9 0 ° pulse to a sample the equilibrium magnetization is left precessing in the x-y plane and i t is this macroscopic magnetization that is detected by the n.m.r. -spectrometer. Of course any angle of rotation (except 0, 2TT, etc.) hi may be used to produce a transverse component of magnetization but the signal i s a maximum a f t e r the a p p l i c a t i o n of a 9 0 ° pulse. A f t e r the r - f f i e l d -is cut o f f the transverse component of magnetization continues to precess f o r some time but f i n a l l y decays to zero f o r several reasons.. Relaxation processes immediately s t a r t to play t h e i r r o l e ; s p i n - l a t t i c e r e l a x a t i o n acts to return the magnetization to the z d i r e c t i o n , t h i s process may be measured i n terms of the time constant T]_ ; transverse r e l a x a t i o n acts to disorder the phases of the i n d i v i d u a l moments i n a random manner and i s measured i n terms of TV,-T r a n s l a t i o n a l .diffusion of the molecules containing the nuclear spins through .an inhomogeneous external magnetic f i e l d also.acts to dephase the precessing moments i n a random manner. In a p e r f e c t l y homogeneous magnetic f i e l d the l a s t of these e f f e c t s would not be observed and i t would be possible to d i r e c t l y observe the e f f e c t s of the r e l a x a t i o n mechanisms. "Of course laboratory magnetic f i e l d s - a r e not p e r f e c t l y homogeneous and i n most cases i t i s impossible to i n f e r the re l a x a t i o n times from the observation of a single decay. Each i n d i v i d u a l moment "sees" a s l i g h t l y d i f f e r e n t magnetic f i e l d because of the inhomogeneity and so each w i l l precess at a s l i g h t l y d i f f e r e n t frequency. This r e s u l t s i n a f a i r l y rapid l o s s of phase and the s i g n a l decays to zero with a l i f e t i m e of the order of ( &H)" 1, where AH i s the average inhomogeneity across the sample. This i s not, however, an i r r e v e r s i b l e phenomenon. There i s nothing random about t h i s process so long as the spins do not move about i n the inhomogeneous f i e l d . I t i s therefore possible to restore the f u l l transverse magnetization that was created by the f i r s t pulse. This can be done by applying a l 8 0 ° pulse separated from the f i r s t 9 0 ° pulse by a time T which may be much longer than ( ^ AH) . The second pulse i s followed at a time 2 T by a spontaneous rephasing of a l l of the elementary moments and the f u l l transverse magnetization i s restored, only to decay again f o r the same reason as given above. This second s i g n a l i s ( 22 ) c a l l e d a spin echo . I t may be noted that the second refocusing pulse need not be 180 f o r the echo to occur. The amplitude of the echo varies with width of the second pulse Go as sin2 3--5-2 Measurement of spin l a t t i c e r e l a x a t i o n time In the measurement of spin l a t t i c e r e l a x a t i o n time, the system i s f i r s t disturbed from equilibrium and then the regrowth of the z component of magnetization i s monitored as a function of time. Bloch's equation p r e d i c t s that the regrowth of the magnetization w i l l obey the following law: M(t) = M w (1 - e--|. ) . (3 .11) I f the experiment i s i n i t i a t e d by a 90° pulse the z component of magnetization s t a r t s from zero and grows exponentially to i t s f i n a l value M,^  . •Unfortunately i t i s impossible to monitor continuously the z component of magnetization without d i s t u r b i n g i t . What can be done i s as follows; a f t e r some time T another 90° pulse can be applied and the si g n a l r e s u l t i n g from t h i s pulse i s an accurate measure of the magnetization j u s t preceding the pulse. This pulse however destroys the z component of magnetization by r o t a t i n g i t into the x-y plane and so i t i s necessary to s t a r t the experiment over again. This procedure i s very time consuming, since an i n t e r v a l several times T^ must elapse between each measurement to l e t the, .nuclear magnetization reach i t s equilibrium value M w . It i s then p o s s i b l e by repeating the experiment many times and varying T each time to p l o t out the recovery of the magnetization to i t s f i n a l value. A value f o r T^ i s found by p l o t t i n g the In ( M ^ - M(T) ) as a function of T. The r e s u l t i n g s t r a i g h t l i n e has a slope = z2u T l Tn p r a c t i c e the simple scheme o u t l i n e d above has several defects and the system used i n taking the measurements reported here w i l l now be described. The act u a l signals measured were echoes generated by a 90° pulse c l o s e l y followed (about 2 msec) by a l80° pulse. Echoes were used rather than the d e c a y b e c a u s e o f s i g n a l t o n o i s e c o n s i d e r a t i o n s . A n a r r o w b a n d f i l t e r c a n be u s e d t o i m p r o v e t h e s i g n a l t o n o i s e r a t i o f o r an e c h o , w h i l e i t i s n o t n e a r l y as u s e f u l f o r a d e c a y b e c a u s e t h e s h a r p p u l s e . p r e c e d i n g t h e d e c a y r e q u i r e s a much b r o a d e r b a n d w i d t h t o be p a s s e d u n d i s t o r t e d t h a n does a s i m p l e e c h o . A l80° r e f o c u s i n g p u l s e was u s e d i n s t e a d o f a 90° p u l s e s i n c e t h i s g i v e s an i n c r e a s e i n s i g n a l t o n o i s e o f a f a c t o r o f 2. To i n s u r e t h a t t h e m a g n e t i z a t i o n does i n d e e d s t a r t f r o m z e r o , a l a r g e number (more t h a n 100) o f 90° p u l s e s were a p p l i e d i n a t i m e much l e s s t h a n ' T - ^ . The e x p e r i m e n t p r o c e e d s b y f i r s t a p p l y i n g t h i s b u r s t o f 90° p u l s e s , t h e n w a i t i n g a t i m e T and t h e n a p p l y i n g . a 90° p u l s e , c l o s e l y f o l l o w e d b y a l80° p u l s e . The •ampli tude o f t h e echo f o l l o w i n g t h e l80° p u l s e i s m e a s u r e d and t h i s g i v e s a v a l u e f o r t h e z component o f m a g n e t i z a t i o n j u s t b e f o r e t h e 90° p u l s e . The e x p e r i m e n t i s r e p e a t e d f o r many d i f f e r e n t v a l u e s o f T . I t s h o u l d be n o t e d however t h a t t h e s p i n s y s t e m n e e d n o t s t a r t f r o m e q u i l i b r i u m b e f o r e e a c h e x p e r i m e n t . The b u r s t o f 90° p u l s e s s e r v e s t o i n s u r e t h a t no z component o f m a g n e t i z a t i o n r e m a i n s a n d t h i s b u r s t c a n be a p p l i e d i m m e d i a t e l y f o l l o w i n g t h e e c h o . T h i s g i v e s a c o n s i d e r a b l e s a v i n g i n t i m e when d e a l i n g w i t h s a m p l e s where t h e r e l a x a t i o n t i m e i s v e r y l o n g . A v a l u e f o r Mac? i s f o u n d b y m a k i n g T = —5 , t h a t i s , a t l e a s t t e n t i m e s T j . . To o b t a i n a v a l u e f o r T]_ i t i s n e c e s s a r y t o p l o t I n (MCJJ •-.'M(T) ) as a f u n c t i o n o f T . A g a i n t h e r e s u l t i n g s t r a i g h t l i n e h a s a s l o p e = -1 . Ln e v e r y c a s e a t l e a s t f o u r T l measurements o f M(T) were a v e r a g e d f o r e a c h T and a b o u t 15 d i f f e r e n t v a l u e s o f T were u s e d . The measurement t o o k a b o u t one h o u r t o c o m p l e t e . 3-5-3 Measurement o f s e l f d i f f u s i o n c o n s t a n t . I n l i q u i d s and g a s e s t h e n a t u r a l d e c a y t i m e Tg o f n u c l e a r s i g n a l s c a n n o t o r d i n a r i l y be i n f e r r e d f r o m t h e d e c a y o f t h e s i g n a l f o l l o w i n g a 90° p u l s e . T h e s p i n echo t e c h n i q u e makes p o s s i b l e , i n p r i n c i p l e , a measurement o f T p . The ; a r t i f i c i a l d e c a y c a u s e d b y t h e inhomogeneous e x t e r n a l f i e l d c a n be e f f e c t i v e l y eliminated and the natural decay of the nuclear signal observed. | The measurement i s made by varying the time T* between the 90 and l80° pulse and measuring the amplitude of the echo at time 2T* as a function of T . This natural decay i s expected to proceed exponentially with a time constant Tg • However, f o r many samples the decay envelope i s found to be of the form exp (-K T ). Hahn was successful i n showing that t h i s i s the form of the decay to be expected - i f the decay i s caused by d i f f u s i o n of the molecules through the inhomogenous f i e l d H Q . Carr and P u r c e l l investigated t h i s e f f e c t i n more d e t a i l and they were able to show that the i n t e n s i t y of the echo at. time 2.T i s determined by be constant over the volume of the sample. D i s the s e l f d i f f u s i o n constant 2 i n u n i t s of cm / s e c The form of the decay then depends .on the r e l a t i v e magnitude of these two terms.. If the f i e l d i s very homogeneous (G - * - 0 ) and f o r samples where the d i f f u s i o n i s slow the second term may be n e g l i g i b l e and Tg can be measured d i r e c t l y . To measure D i t i s only necessary that the second term be comparable with the f i r s t . Carr and P u r c e l l have made several assumptions i n d e r i v i n g t h i s equation. It remained f o r Muller and B l o o m ^ ^ to show in. a t h e o r e t i c a l analysis that most of these r e s t r i c t i o n s could be l i f t e d . Their study provides a much firmer b a s i s f o r the use of the free precession method i n measuring s e l f d i f f u s i o n constants. As f a r as d i f f u s i o n i s concerned Muller-and Bloom f i n d the dependence of the echo amplitude on T * to be independent of the width of both the 90° and 180° pulses. T t i s therefore not necessary that the pulse widths be adjusted p r e c i s e l y . They.also f i n d that the above equation holds independent of the i n t e n s i t y and homogeneity of the r - f f i e l d and also independent of the magnitude of the constant gradient G. The effect of a non-constant gradient in the external magnetic field has been examined by Lipsicas and Bloom (31) . For a one-dimensional gradient of form G(x)•= g 1 + xg2> where the deviation from a constant gradient is small so that xgg <_< g^, they find that the rate of echo decay for diffusion is of the form M(2 r ) = % exp [- Kx D T 31 SlnhCKgP T •?) (3 .13) 1 J K2D where % = 2_ J g]_ > ^2 = _ f ° 1 §2 ^ > d = length of the sample. For 3 3 large values of 7 - , KgD T 3 >/•• 1 So that M(2 7") = MQ exp [ -K-L D T 3 J | exp [ K2D T*3J (3-1^) = ^ e x p { - [ K l - K 2 ] D T ? } ( . Thus the slope of the ln M ( 2 T ) versus T " 3 curve may s t i l l be used to measure relative values of D in spite of the fact that G is not constant. The only condition which must be satisfied is that G(x) be the same for a l l measurements, a condition normally satisfied in experiments involving a permanent magnet. The measurements of self diffusion constants reported here are a l l relative measurements, based on the known self diffusion constant of water. In a very (32) • . careful experiment, Simpson and Carr X J ' have reliably measured the" diffusion coefficient in water as a function of temperature using n.m.r. techniques. Their value of D at 25°C "(2.-13 x 10"5 cm2/sec) was used to "calibrate" the magnetic field gradient. Water was first placed in the sample coils. The coils were then moved about in the magnetic field in search of the best placement. The criterion used for placement was that the gradient should be" as small as possible and as constant as possible. The shape of the decay following .a 90° pulse may be used as a monitor during the search. Carr and Purcell^--5' have shown that the shape of the decay (for a cylindrical sample in a constant gradient)has the form: M(t) . = :MQ 2 J X (| frGdt) ( 3, 1 5) where If G d t ) i s t h e f i r s t - o r d e r B e s s e l f u n c t i o n a n d d i s t h e d i a m e t e r o f t h e s a m p l e . The w i d t h o f t h e d e c a y may t h e n he t a k e n as a measure o f t h e m a g n i t u d e o f t h e g r a d i e n t a n d t h e shape o f t h e d e c a y i s a measure -of t h e u n i f o r m i t y o f t h e g r a d i e n t . The v a l u e .of- a v e r y u n i f o r m g r a d i e n t may i n d e e d "be m e a s u r e d i n d e p e n d e n t l y o f any c a l i b r a t i o n u s i n g E q u a t i o n 3-15- T h i s i s u n d o u b t e d l y t h e most s a t i s f y i n g way t o a p p r o a c h t h e p r o b l e m e x p e r i m e n t a l l y . I f a v e r y g o o d m a g n e t : . i s a v a i l a b l e , i t i s r e l a t i v e l y e a s y t o p r o d u c t a v e r y . u n i f o r m g r a d i e n t u s i n g two c i r c u l a r t u r n s o f w i r e as o u t l i n e d b y C a r r a n d P u r c e l l , U n f o r t u n a t e l y ..the magnet made a v a i l a b l e f o r t h i s work h a s a r e a s o n a b l y l a r g e g r a d i e n t so t h a t i t w o u l d have b e e n i m p o s s i b l e t o e s t a b l i s h a known g r a d i e n t o f s u f f i c i e n t m a g n i t u d e t o make t h e - n a t u r a l g r a d i e n t n e g l i g i b l e . However , i t i s f e l t t h a t t h e c a l i b r a t i o n u s i n g w a t e r i s more t h a n a d e q u a t e f o r t h e s e m e a s u r e m e n t s . A f t e r f i n d i n g .a s u i t a b l e p l a c e i n t h e f i e l d t h e c o i l s were r i g i d l y mounted w i t h r e s p e c t t o t h e magnet . The g r a d i e n t was t h e n m e a s u r e d u s i n g a d i s t i l l e d w a t e r s a m p l e . To o b t a i n t h e g r a d i e n t , u s e i s made o f E q u a t i o n 3-12. T a k i n g t h e l o g a r i t h m o f b o t h s i d e s y i e l d s l n M ( 2 T ) - I n M q •= - _ _ _ - i J'VDT 3 (3 .16) T 2 3 F o r a l l t h e measurements r e p o r t e d h e r e t h e Tg t e r m i s s m a l l enough t o n e g l e c t . To i n v e s t i g a t e t h i s n o t e t h a t ( 3 . I 6 ) may b e w r i t t e n i n t h e f o r m . y - b = - K ' x ^ - K"x (3 .17) where x = 7"3, y .= i n M(2 T ) , b - I n M , K 1 = 2 , , K " = 2 )f 2 G 2 D . F o r w a t e r T 2 3 T g i s a b o u t 2 s e c o n d s so t h a t K ' i s a b o u t o n e . I n t h e magnet u s e d G i s a b o u t 1 g a u s s / c m a n d t h e v a l u e f o r D may be t a k e n ,as a b o u t 2 . x 10 ^ c m 2 / s e c . T a k i n g If •= 2.7 x .10* s e c " 1 g a u s s " 1 g i v e s K " o f o r d e r 10^ . T h i s t h e n g i v e s y - b = - xH - l O ^ x (3 .18) and the slope of this curve is -y = - _____ - i o u ( 3, 1 9) For a l l values of x >2 x 10"5 the slope is constant to better than 5$ and is determined by the K" term. Now an experiment is usually continued for about two time constants which means that the maximum value of K"x is about 2. The value of x then ranges from G to 2 x 10 and. i t is apparent that for 90% of this range K" will be determined to better than 5$>< The slight non linearity at the beginning of the-curve is ignored when drawing a line through the experimental points. The diffusion constant for ethane is -an order of magnitude greater than that for water/ so the Tg term may be safely neglected in the case of ethane. The calibration consists of measuring the time "constant "(l...e. i . ) for K the diffusion process in water. If this time constant is denoted by (Tpjg^o then the self diffusion constant for the ethane sample will be given by D C 2 H 6 - ( T P V V ^ TD)G 2H6 For most of the measurements reported :here the measured values of (Tp)jj Q at 25°C are: , • . ' (TD)H20 = 1-35 +0.07 x 10"^ sec 3 (Vapor coil) * cor* ( T D ) H ^ 0 = 5^ 26 + 0.19 x-IO"5 sec^ "(Liquid coil) . These were measured several times before the experiment-and each time the sample was changed. A representative calibration curve is shown in Figure 7- It may be observed that the non :linearity at the small values of T does not significantly effect the measurement of the slope. The time constant was constant to within about 5$ so long as the sample holder was not moved. The gradient is strongly dependent on position in the magnetic field and a ECHO AMPLITUDE (VOLTS) O o r cn 8^ new calibration must be made each time the sample holder is moved. The estimated maximum random error is 5$ and the estimated maximum systematic error is also about 5$. The D measurements were taken on samples of ethane to which oxygen had been purposely, added as an impurity. This was done to improve the signal to noise ratio for these measurements. A further improvement in signal to noise was achieved using a boxcar integrator kindly loaned to the author by Walter Hardy. The oxygen concentration in these samples was estimated to be less than 1$. In making a measurement of the self diffusion constant by any .method, a means must be found to somehow "label* the molecules to enable their paths to be traced. It is important to note that in these measurements the spin (23 system is used to'"label" the molecules. -As expressed by Carr and Purcell v "To the extent that the'label,' has, a negligible effect of the diffusion process itself, a method may be said to measure truly the 'self-diffusion' constant. In the method here described a molecule is in effect labelled by the direction of the nuclear magnetic moment i t carries; a more innocuous label would be difficult to imagine.-" Chapter IV THE APPROACH TO EQUILIBRIUM The nuclear magnetic resonance signal amplitude is proportional to-the equilibrium nuclear magnetization and this is in turn proportional to the number of molecules inside the Bample coil. It i s then'possible to express the signal amplitude S by S = K T\p6 (i+.l) where K is .a constant of the systemand includes - the effective volume of the sample, n is the number of protons per molecule, ^o.is the density in molecules per cm and 6 is the fractional excess population in the lower energy state. At fixed temperature the signal amplitude is then directly proportional to the density. Tt is difficult to use Equation (k.l) to make absolute measurements of density because an evaluation of K is difficult to make accurately. K depends on the-sample geometry which may be kept fixed and the gain of the r-f amplifier which is much harder to keep fixed over a period of several days. The amplifier gain is also•difficult to measure. Equation (U.l) can however be used to measure relative density rapidly and with reasonable accuracy. Measurements were made of the signal amplitude from the vapor coil and immediately afterward of the signal amplitude f r o m the liquid coil. Since K may be considered constant for short periods of time and with fixed sample geometry this, allows,a measurement to be made of the ratio of density in the two coils. SV Ky f> y It is important to point out that this measures a macroscopic average density over the entire effective volume of the sample coil. There were slight d i f f e r e n c e s i n the sample c o i l s and t h i s means that Kj? i s not quite equal to one. The s i g n a l r a t i o can be normalized by measuring the r a t i o of signals when the sample consists e n t i r e l y of gas w e l l away from the c r i t i c a l point. This was done at 60°C To perform,a. measurement of r e l a t i v e density i t i s necessary to measure f i r s t the amplitude of s i g n a l from the vapor c o i l and then switch the */2 l i n e to the l i q u i d c o i l . Unfortunately the two c o i l s were not i d e n t i c a l even though,an attempt had been made to make then as nearly the same as possible. Since they were e l e c t r i c a l l y somewhat d i f f e r e n t t h i s meant that the spectrometer had to be retuned when switching from one c o i l to the other. When the r e l a x a t i o n time i s long t h i s i s a tedious process. 'Also care must be taken that the sample has relaxed to thermal equilibrium before measuring the s i g n a l , so that a time of at l e a s t must elapse between each measure-ment of the s i g n a l . For the pure sample these considerations contribute to degrade the accuracy of the measurement s i g n i f i c a n t l y . Measurements of the r e l a t i v e density f o r both the pure and the impure samples are presented i n Figure 8. The s o l i d l i n e shown i s the r a t i o p r edicted using the density data from the smooth curves shown i n Figure 20 (Chapter V). There i s reasonable, agreement between these p r e d i c t i o n s and the nuclear magnetic resonance measurements of the density r a t i o i n the pure sample. The density measurements constitute a convincing proof that the samples were f i l l e d to the c o r r e c t density. I f the o v e r - a l l density i n the sample tubes had been e i t h e r too high or too low the meniscus would, move e i t h e r up or down and t h i s could be detected as a rapid change i n the r e l a t i v e density as the meniscus moved through the sample c o i l . The measurements indic a t e that the density at the lower c o i l remains higher than at the upper c o i l even at temperatures i n excess of the temperature of meniscus disappearance,. •(h) This i s i n good agreement with Palmer's measurements Figure 8. Temperature dependence o f density r a t i o . 53 The dotted l i n e i n Figure 8 gives the behaviour of the density r a t i o i n the impure samples. In t h i s case the density r a t i o indicates that the c r i t i c a l temperature has been s h i f t e d to ,a lower temperature. The shape of the two curves i s s i g n i f i c a n t l y d i f f e r e n t i n d i c a t i n g that the equation of state has been a l t e r e d by the addition of oxygen. To within experimental error the two curves coincide at temperatures l e s s than about 30°C The three impure samples a l l behaved i n the same manner and a l l had an indicated c r i t i c a l temperature of about 31-9°C I t would be i n t e r e s t i n g to c a r e f u l l y examine these samples i n an adequately thermostatted bath to determine the temperature of meniscus disappearance. This was not done however. Quite e a r l y i n t h i s i n v e s t i g a t i o n i t was noticed that when the temperature was increased r a p i d l y some time was required f o r the density r a t i o to come to an equilibrium value. This e f f e c t was studied i n an impure sample. In two separate experiments the sample was allowed to remain at a constant temperature of 23 - ^ 0 °C f o r two days and then the temperature was increased suddenly to 31 ,92°C It took about 1 hour to increase the temperature of the sample holder to t h i s f i n a l temperature. The temperature was then held constant (to within about 0.003°C) f o r a period of ^5 hours during which the density r a t i o was measured. The r e s u l t s of the second of these two experiments are shown i n Figure 9- The dotted l i n e i n Figure 9 i s an exponential curve with a time constant of 7-9 hours drawn f o r comparison with the data. As can be seen, a long time i s taken f o r the sample to reach equilibrium. The approach to e q u i l i b r i u m appears to be nearly exponential. The r e p r o d u c i b i l i t y between these two experiments was excellent and the measurements were within the errors indicated i n Figure 9- To check that t h i s i s not j u s t a measurement of the thermal time constant of the sample tube, an i d e n t i c a l sample tube containing water was constructed. Two thermocouples were located at the midpoints of the vapor and l i q u i d c o i l inside the sample tube. The thermal time constant of t h i s sample tube was measured by putt i n g - o 1.6 > 1.4 O I— < cn >-CO W 1.2 O 1 . 0 O X£> o x o o o o 0 1 0 1 5 2 0 TIME IN HOURS Figure 9 . Time dependence of density ratio. i t into the sample holder which was held ;at a constant temperature above the in i t i a l temperature of the sample tube and .monitoring the rise in temperature of the water. This gives a thermal time constant for the sample tube of 2-3 minutes.and no temperature difference between the two positions in the sample tube could be detected to within the resolution of the themiocouples (about 0.05°C). Two other preliminary experiments were also done; one in which the final temperature was well above the critical temperature and the other in which the final temperature was less than the critical temperature. The final temperature in the first experiment was 3^ --09°C and the density ratio went from 1.19 to 1.00 with a time constant of about k hours. 'The final temperature in the second experiment was 31-H°C and the density ratio went from 1.77 to 1.57 with a time constant of about 3 hours. These should be looked upon as preliminary experiments and would certainly need repeating to verify the time constants given, Upon cooling the system down from above the critical temperature, there was no time lag detected in the density ratio. In other words, upon cooling,-the system seems to follow the .equilibrium curve. The reason for the long time involved in the approach to equilibrium is not at a l l obvious. It appears that a long time is involved for the liquid structure to break up and for the system to become a homogeneous gas-This .appears to be an interesting effect worth investigating further. All of these measurements were on the impure sample .and the effect may well be •an impurity effect. It would be interesting to study the pure sample. -A much better experimental -approach would be to have a movable coil so that the spectrometer does not have to be retuned between measurements.. This •would.also allow studies to be made at different heights in the sample. There are several interesting experiments which could be performed. The effect of stirring the sample would be worth investigating. Palmer^ . found that s t i r r i n g destroys the density gradient and caused the sample tube to be f i l l e d with turbulent regions of inhomogeneity. I f opalescence were •o r i g i n a l l y present, i t became immediately d i s t r i b u t e d throughout the sample. T t would also be of i n t e r e s t to study the e f f e c t of d i f f e r e n t i n i t i a l temperatures and d i f f e r e n t rates of change of temperature from the i n i t i a l to the f i n a l temperature. V i s u a l observations of the sample simultaneously with the nuclear magnetic resonance experiment would be a fascinating,and valuable experimental improvement. The following chapter describes the r e s u l t s obtained f o r measurements of the s p i n - l a t t i c e r e l a x a t i o n time and the s e l f d i f f u s i o n constant. A l l of these r e s u l t s were obtained a f t e r the sample had been held at constant temperature f o r at l e a s t 1 2 hours and i n most cases 2k hours. T t i s therefore assumed that they are equilibrium values and are not a f f e c t e d by the approach to equilibrium described above. 57 Chapter V THE EXPERIMENTAL RESULTS The s p i n - l a t t i c e r e l a x a t i o n time has been measured over the e n t i r e temperature range from the melting point to above the c r i t i c a l temperature i n pure l i q u i d ethane under i t s own vapor pressure. In t h i s experiment the emphasis was on the behaviour near the c r i t i c a l point so that a d e t a i l e d study was made on both the vapor and l i q u i d phases i n the c r i t i c a l region as w e l l as the dense gas at temperatures i n excess of the c r i t i c a l temperature. Knowledge of the d i f f u s i o n constant i s .necessary f o r the t h e o r e t i c a l i n t e r p r e t a t i o n of s p i n - l a t t i c e r e l a x a t i o n r e s u l t s . I t was therefore highly desirable to make measurements of the d i f f u s i o n constant i n the f l u i d over the same temperature range. Transport properties play an important r o l e i n the e l u c i d a t i o n of the molecular theory of l i q u i d s and gases. For t h i s reason measurements of the d i f f u s i o n constant are important i n t h e i r own r i g h t , 5-1 S e l f D i f f u s i o n Results The s e l f d i f f u s i o n constant D has been measured i n the temperature range 15°C to 60°C by the method o u t l i n e d i n Chapter I I I . These r e s u l t s are presented i n Figure 10. In the l i q u i d D increases slowly w i t h increasing temperature u n t i l the temperature i s .about 1° below the c r i t i c a l temperature and then the slope changes rapidly.and the d i f f u s i o n constant increases •sharply. In the vapor D f a l l s more r a p i d l y than it'increases i n the l i q u i d . In the dense gas above the c r i t i c a l temperature D seems to increase slowly with temperature. Several samples were used to obtain these r e s u l t s . To three of these samples oxygen had been purposely added as an impurity to decrease the r e l a x a t i o n time. "This r e s u l t s i n an increased • s i g n a l to noise r a t i o and therefore an improvement i n the accuracy of the measurement. Two Pigure 10, Self diffusion in ethane. 5 9 measurements were made In a pure sample at about 2h°C and these are shown marked with a P in Figure 10. Al l of the other measurements were made on the adulterated samples. The oxygen concentration varied from sample to sample, but was estimated to be no more than 1$ maximum. The diffusion constant was reproducible from sample to sample for the three samples containing appreciable amounts of oxygen and therefore the diffusion process is not strongly dependent on oxygen concentration over the range of concentrations used. The critical temperature for the samples is however affected by the addition of even this small amount of oxygen. "This may be seen in Figure 11 where the results of the measurements are given on an expanded scale in the region of the critical temperature. It may be noted here that the diffusion constants for the liquid and vapor become equal about O.U°C below the critical temperature for pure ethane. This probably means that the equation of state for the fluid has been changed in the critical region by the addition of the impurity. To check that this is the case measurements were also made on the relative densities of the liquid and vapor phases. These measurements were presented in Chapter IV; i t may be noted here however that the density of both phases becomes equal at the same temperature that the diffusion constants become equal. Tn the pure sample the density in the vapor coil does not become equal to the density in the liquid coil until a temperature somewhat higher than the critical temperature as was described in Chapter IV. This will be discussed in more detail in connection with the T]_ results-The measurements above the critical temperature at first seemed to indicate that the diffusion constants in the two coils were not equal. The error flags shown in Figure 11 indicate maximum random error for a given In (echo amplitude) vs. t plot. They do not Indicate any systematic error from the calibration of the magnetic field gradient. 'The measurements 8.0 shown at 33-6 C were taken a f t e r the other r e s u l t s shown and a f t e r the sample had been removed and the gradient c a r e f u l l y r e c a l i b r a t e d . I t appears then that the gradient had s h i f t e d s l i g h t l y and that the two p o s i t i o n s i n the sample do indeed have equal d i f f u s i o n constants. A l l of the d i f f u s i o n r e s u l t s were obtained with increasing temperature and i n each case the sample was allowed to remain at constant temperature f o r at l e a s t 12 hours a f t e r the temperature was increased before proceeding with a measurement. 5-1.1 The e f f e c t of density The e f f e c t of density on the d i f f u s i o n constant was investigated i n the neighbourhood of the c r i t i c a l temperature by the use of samples loaded to d i f f e r e n t d e n s i t i e s . At d e n s i t i e s below the c r i t i c a l density the sample would be composed e n t i r e l y of gas with no l i q u i d present. If the sample i s loaded to a density i n excess of the c r i t i c a l density the sample becomes completely f u l l of l i q u i d at a temperature below the c r i t i c a l temperature. The d i f f u s i o n constant was measured i n four pure samples of ethane where only one phase was present at several temperatures. These r e s u l t s are given i n Figure 12 f o r temperatures of 32 and i4-0°C In t h i s curve the r e c i p r o c a l of the measured d i f f u s i o n constant i s p l o t t e d as a function of density. The unit of density, known as the "Amagat" un i t of density, i s the density of the gas at one atmosphere pressure and 0°C One Amagat then corresponds to 1-357 x 10 _3 gm/cm^., since t h i s i s the density at NTP. The density of the gas at an a r b i t r a r y pressure and temperature i s s p e c i f i e d by g i v i n g the r a t i o of the true density to the density at NTP: P e. 1.-357 x 10-3 (5.1) and pK i s then the density ""in.Amagat u n i t s " . 62 DENSITY (A MAG AT) Figure 12. The effect of density on diffusion constant for *"non c r i t i c a l l y " loaded samples of ethane. From Figure 12 i t i s immediately' apparent that the product Dp i s equal to a constant. Now i t i s w e l l known w that f o r a d i l u t e gas Dp = o< 7 (5.2) where ^ i s a numerical constant of order unity and ^ i s the v i s c o s i t y of the gas. The v i s c o s i t y of a d i l u t e gas i s independent of density and so i t i s expected f o r a d i l u t e gas that D (O - constant at f i x e d temperature. .The r e s u l t s given i n Figure 12 i n d i c a t e that t h i s i s true even f o r the dense gas i n the neighborhood of the c r i t i c a l point. I t would be very i n t e r e s t i n g to compare these r e s u l t s with measured values of the v i s c o s i t y . The slope of the s t r a i g h t l i n e s i n Figure. 12 g i v e : at 32°C D P = — = 9.28 x l O " 2 cm 2 Amagat = 1.'26 x lO"1*' poise s l ° P e sec" at 1+0°C D/o = _ i = 9.96 x 10" 2 cm 2 Amagat = 1.-35 x 10"^ poise. s l ° P e 7e~c Unfortunately there have been no measurements of v i s c o s i t y reported with which these r e s u l t s could be compared. The density of these samples was obtained-by observing the temperature at which they became composed of a single phase. This y i e l d s the density (to within about 0.001 gm/cm ) by comparing t h i s temperature with the coexistance curve (Fig.20). 5-1-2 Temperature dependence of Dp The temperature dependence of Dp i s also of i n t e r e s t . For a d i l u t e gas t h i s should give the temperature dependence of the v i s c o s i t y . In a 1. hard sphere gas ^ should be proportional to T 2. Experimentally the dependence even f o r rather d i l u t e gases tends to be s l i g h t l y steeper than ( 3 M t h i s . No .theoretical p r e d i c t i o n i s a v a i l a b l e f o r the l i q u i d state.. 6k The temperature dependence f o r Dp i s given i n F i g . 1 3 f o r l i q u i d ethane from j u s t above i t s melting point ( 89 -9°K) to the . c r i t i c a l point. The r e s u l t s of the present study are also shown f o r temperatures above 0°C ( l n T=5-6). The (oc) d i f f u s i o n measurements below 0°C are those of Gaven, Stockmayer and Waugh . The d i f f u s i o n points are taken.from t h e i r smooth curve. They p l o t In D vs. — T and note that t h i s i s not a st r a i g h t l i n e . a s would be expected i f the a c t i v a t i o n energy were constant. The density of l i q u i d ethane was obtained from two s o u r c e s > 3 7 ) _ The s t r a i g h t l i n e obtained i n Fig.13 . shows that the product Dp increases approximately as T . This i s i n sharp contrast to the T 2 dependence expected i n the d i l u t e gas- I t may also be seen from Figure 13 that t h e r e - i s excellent agreement between the r e s u l t s reported here and those of Gaven, Stockmayer and Waugh. Tn .the•absence of a rigorous theory f o r d i f f u s i o n i n l i q u i d s , there are two rough theories that have been u s e f u l i n making order of magnitude c a l c u l a t i o n s : the hydro-dynamical theory and the Eyring theory. The hydros-dynamical theory y i e l d s the Stokes-Einstein equation: D » * T , . ( 5 . 3 ) 3TT a /j ( 38 ) The Eyring rate theory attempts to explain the transport phenomena on the basis of a simple model f o r the l i q u i d state. I t i s assumed i n t h i s theory that there i s some unimolecular rate process i n terms of which the d i f f u s i o n process can be described and i t i s further assumed that i n t h i s process there i s some configuration that can be i d e n t i f i e d as the a c t l v i a t e d • state. The Eyring theory of r e a c t i o n rates 'is applied to t h i s elementary process and i t y i e l d s the same dependence on temperature and v i s c o s i t y as the Stokes-Einstein equation with a d i f f e r e n t numerical constant. The observed behaviour of the v i s c o s i t y i n most l i q u i d s i s an exponential decrease i n v i s c o s i t y with 65 L n T (°K) Figure 1-3* Temperature dependence of D p for liquid ethane. 66 (39) •temperature w . This gives i£ oc p e& (?-k) where E a i s an activation:./.energy. There i s no p r e d i c t i o n , therefore, f o r a • simple temperature dependence of D p . 'Ethane does not f i t into t h i s simple a c t i v a t i o n energy model and i t may w e l l he that ethane i s . a s p e c i a l case. I t would he i n t e r e s t i n g to check the behaviour of'Bp as a function of temperature i n other systems. It might at t h i s point be useful'to point out an observation made as l a t e as I960 by B i r d , Stewart and Lightfoot i n t h e i r book on transport phenomena^^^. They say, " I f the reader has by now concluded that l i t t l e i s known about the p r e d i c t i o n of dense gas and l i q u i d d i f f u s i v i t i e s , he i s correct. There i s an urgent need f o r experimental measurements, both f o r t h e i r own value,and f o r the development of future theories.-" The temperature dependence of I>p near the c r i t i c a l point i s given i n Figure lk. The s o l i d l i n e i ndicates the expected temperature dependence based 3 on the r e s u l t s given i n Figure 13, that i s varying simply as T . Points were obtained f o r D from the smooth curves shown i n Figures 10 and 11, For the density use was made of the experimental values of the r a t i o of l i q u i d to vapor density as a function of temperature f o r the impure samples. A constant volume sealed•sample tube has a f i x e d number of molecules n^ which are i n e i t h e r the l i q u i d . s t a t e or the vapor state so that n L + n y •= n t . (5-5) The number of molecules i n each state can be written i n terms of the density and volume n - y _ _ _ (5.6) m where m i s the mass per molecule. Using Equation (5.6) i n Equation '(5• 5) gives ( 5 . 7 ) Figure lk. .. Temperature dependence, of D^ > near the c r i t i c a l p oint. where p is the average density- in the total volume . :If the fraction of volume .occupied by the vapor is defined by f s _ I = 1 - I_i (5-8) v t v t then Equation ('5-T) becomes / o L ( l - f)..+ PYf = p (5-9) The ratio R=J^i; has been measured in the temperature range 25 to 60°C Using this ratio Equation (5-9) becomes R (1 - f) + f = - J L or p v = R ( i - P f) + f . ( 5 - a o ) Now i f the sample is critically loaded p = /©c and the meniscus disappears at the midpoint of the sample tube. Therefore, f = near the critical temperature. This then gives v R.+ 1 -and PL = 2_R_ft_ (5..U-) R+l It Is then possible to predict the liquid .-and vapor densities from the measured ratio and using the fact that the meniscus disappear-s at the middle of a critically loaded • sample tube. The motion of the meniscus may be investigated by expressing the fraction of volume occupied by vapor at temperature T-along the vapor pressure curve as f ^ f 0 + f-L (T c - T) * • (5,12) At the critical point T = Tc and f = \ , w h i l e a t t h e m e l t i n g p o i n t and f ^ 2 3 T h e r e f o r e , 1 1 o -1 1200 K (5-13") T h i s t h e n g i v e s + ( T c - T ) • ( 5 v l i 0 1200 a n d so i n t h e t e m p e r a t u r e range shown i n F i g u r e 1^ f v a r i e s b y " l e s s t h a n 2$. T h i s . o f c o u r s e depends on t h e samples b e i n g c o r r e c t l y l o a d e d t o t h e c r i t i c a l t h a t t h e samples were i n d e e d c r i t i c a l l y l o a d e d t h e y were o b s e r v e d v i s u a l l y a s t h e t e m p e r a t u r e was r a i s e d above t h e c r i t i c a l t e m p e r a t u r e . I f t h e m e n i s c u s was o b s e r v e d t o move e i t h e r up o r down t h e sample was d i s c a r d e d . V a l u e s f o r t h e l i q u i d and v a p o r d e n s i t i e s were o b t a i n e d f r o m E q u a t i o n ( 5 - l l ) a n d t h e m e a s u r e d v a l u e s o f R. - D e n s i t y v a l u e s t a k e n f r o m t h e s e smooth c u r v e s were t h e n u s e d w i t h t h e d i f f u s i o n d a t a t o c o n s t r u c t F i g u r e Ti+. R e t u r n i n g now t o a d i s c u s s i o n o f F i g u r e lk, t h e two s q u a r e p o i n t s were o b t a i n e d f r o m t h e r e c i p r o c a l s l o p e o f F i g u r e 12 w h i c h g i v e s v a l u e s f o r Dp f o r n o n - c r i t i c a l l y l o a d e d s a m p l e s . The t e m p e r a t u r e T e i n d i c a t e d i n F i g u r e l U i s t h e t e m p e r a t u r e a t w h i c h t h e d i f f u s i o n c o n s t a n t s f o r t h e l i q u i d and v a p o r become: e x p e r i m e n t a l l y e q u a l a n d t h e d e n s i t y r a t i o becomes u n i t y . Below t h e c r i t i c a l t e m p e r a t u r e i t i s c l e a r t h a t D p f o r the l i q u i d a n d v a p o r a r e s i g n i f i c a n t l y d i f f e r e n t . T h i s i s t o be e x p e c t e d s i n c e Dp i n t h e gas v a r i e s o n l y as T 2 w h i l e f o r t h e l i q u i d - i t v a r i e s as T 3 . J u s t b e l o w t h e c r i t i c a l t e m p e r a t u r e t h e r e seems t o be an anomalous r e g i o n where D p f o r b o t h - t h e l i q u i d a n d v a p o r d e c r e a s e s a n d goes t h r o u g h a minimum, t h e n i n c r e a s i n g s h a r p l y and b e c o m i n g ; e q u a l as t h e c r i t i c a l t e m p e r a t u r e i s r e a c h e d . d e n s i t y . I f p =^ p>c t h e n f b e h a v e s i n q u i t e a d i f f e r e n t manner . To c h e c k 70 Some i n s i g h t into the reason f o r the decrease i n Dp may he obtained by considering the model of non-attracting r i g i d spheres. For t h i s model, •(hi) Herschfelder, C u r t i s s and B i r d x present the following equation: D = 2.6280 x 10-3 \j T3/M cm 2/sec (5-15) P s-'d where M •= molecular weight, T = temperature i n °K, p = pressure i n atmospheres and <ST = molecular diameter i n A". Now the density of an i d e a l gas i s (5-16) n k- T where m i s the mass per molecule. This then gives 1 Dp = constant - T2 (5-17) I f the molecular weight were to increase t h i s would r e s u l t i n a decrease i n the product D p. This i s what may be expected i f the f l u i d i s made up of a large number of c l u s t e r s of molecules of various si z e s . The c l u s t e r s would be non-interacting but an i n d i v i d u a l molecule i s trapped i n a c l u s t e r and must move with i t . It would be u s e f u l to repeat the experiment using pure ethane to check whether or not t h i s i s an impurity e f f e c t . "Unfortunately the use of pure ethane would decrease the accuracy of the measurements and the e f f e c t may be obscured i n the random scatter of the measurements. A sample with shorter natural r e l a x a t i o n time would perhaps be a bett e r choice. 5-2 Spin-Lattice Relaxation Measurements The s p i n - l a t t i c e r e l a x a t i o n time has been measured over the enti r e l i q u i d range f o r pure ethane. These r e s u l t s are presented i n Figure.'15-. I t may be seen.here that T]_ increases almost exponentially as a function of r e c i p r o c a l temperature f o r low temperatures. As the temperature i s increased T-j_ goes through a maximum', and then begins to decrease w e l l below the c r i t i c a l (k2) temperature. Muller-and'Noble have also measured T-^  i n l i q u i d ethane, however t h e i r maximum t e m p e r a t u r e was 200°K. The r e s u l t s p r e s e n t e d h e r e a r e i n e x c e l l e n t agreement w i t h i n t h i s t e m p e r a t u r e r a n g e . The low t e m p e r a t u r e s y s t e m u s e d t o o b t a i n t h e s e r e s u l t s was d e s i g n e d and c o n s t r u c t e d b y F r a n k B r i d g e s and the. . a u t h o r i s i n d e b t e d t o h i m f o r h e l p i n o b t a i n e d t h e s e m e a s u r e m e n t s . R e s u l t s o b t a i n e d u s i n g t h e t e m p e r a t u r e c o n t r o l s y s t e m d e s c r i b e d i n C h a p t e r I I I a r e p r e s e n t e d I n F i g u r e 16. A l l o f t h e p o i n t s s h o w n ' h e r e a r e t h e a v e r a g e o f two o r more s e p a r a t e m e a s u r e m e n t s . T n t h e l i q u i d t h e r e l a x a t i o n t i m e i s a d e c r e a s i n g f u n c t i o n o f t e m p e r a t u r e w i t h t h e d e c r e a s e •becoming more r a p i d as t h e c r i t i c a l t e m p e r a t u r e i s a p p r o a c h e d . F o r t h e v a p o r t h e r e l a x a t i o n t i m e i s much s h o r t e r and i n c r e a s e s w i t h t e m p e r a t u r e . T h i s i n c r e a s e c o n t i n u e s u n t i l t h e v a p o r T]_ meets t h e l i q u i d T-j_ a t t h e c r i t i c a l tempera ture - . T n t h e dense gas above t h e c r i t i c a l t e m p e r a t u r e T-^ d e c r e a s e s s l o w l y w i t h i n c r e a s i n g t e m p e r a t u r e . 5-2.1 A c c u r a c y o f T^_ measurements-I t may b e u s e f u l a t t h i s ; s tage t o make some comments o n t h e a c c u r a c y w i t h w h i c h T-^ may b e m e a s u r e d . The t e c h n i q u e o f measurement i s d e s c r i b e d i n C h a p t e r I I I a n d t h e u s u a l method i s t o p l o t I n (Moo - MCT).) V S . T . T h i s y i e l d s a s t r a i g h t l i n e w i t h s l o p e e q u a l t o - 1 .. I t c a n b e seen t h e n T l t h a t Moo e n t e r s i n t o e v e r y p o i n t w h i c h i s p l o t t e d a n d - s o i t i s i m p o r t a n t t o d e t e r m i n e MQO a c c u r a t e l y . T h i s i s d i f f i c u l t t o do s i n c e t o o b t a i n - M ^ , a t i m e o f '5 t o 10 T-j_ must e l a p s e b e t w e e n • s a t u r a t i n g t h e s y s t e m a n d o b s e r v i n g t h e m a g n e t i z a t i o n . F o r a T-^ o f 30 sec i t t h e n t a k e s 2-5 t o 5 m i n u t e s t o o b t a i n one v a l u e o f M , ^ - T t i s v e r y t i m e c o n s u m i n g t o > a t t e m p t t o i m p r o v e t h e a c c u r a c y b y r e p e t i t i v e m e a s u r e m e n t s . The e f f e c t o f e r r o r s i n was i n v e s t i g a t e d u s i n g an IBM 1620 c o m p u t e r . I t was f o u n d t h a t a change i n M , ^ o f 1$ c o n t r i b u t e s a change o f 3 o r ^ t o t h e measurement o f T]_ . I t i s d i f f i c u l t t o measure M ^ t o 1^. A n - a l t e r n a t i v e method o f t r e a t i n g t h e LIQUID Figure 16. S p i n - l a t t i c e r e l a x a t i o n i n pure ethane. 7^ data i s as follows. The echo amplitudes are to he f i t t e d to an equation of the form M(T) = M « (1 - e ^ ). (5,18) I f T i s increased by an increment o( t h i s gives M(T + «) = Mao (1 -e ^ e ^  ) . .(5.19) Row using Equation (5-l8) to evaluate e ~ t h i s y i e l d s i l M(T +O() = Moo ( l - [ 1 " M(T) 1 e ^ ) L Moo J T i M(T+o() = M(T).e ^ + M „ £l - e ,. (5-'20) I f M(T + O() i s p l o t t e d as a function of M(T) t h i s again y i e l d s a s t r a i g h t l i n e with slope e~T| • So T]_ may "be evaluated without using M^ -as long as the time i s increased by-a f i x e d amount o( each measurement. This second method was the one used here. A l l of the T^ data were f i t t e d to Equation (5.20) by the method of l e a s t squares using an IBM 1620 computer. The data were also f i t by drawing a s t r a i g h t l i n e on the curve of In (M — M(T)) vs. T. The l e a s t squares f i t by computer d i d not s i g n i f i c a n t l y reduce -the scatter i n the mqasurements. This i s probably because no weighting was applied to the points and while drawing the curve by eye t h i s i s easy to do. The problem of improving the accuracy i s d i f f i c u l t because.of the low s i g n a l to noise r a t i o . Several f a c t o r s contribute to t h i s . Since T^ i s long i t i s very time consuming to use a r e p e t i t i v e technique to improve the accuracy. The time a v a i l a b l e f o r a measurement i s also l i m i t e d by the s t a b i l i t y of the apparatus. Another important consideration i s the f i l l i n g f a c t o r f o r the c o i l . Since high pressures are involved the glass container must be thick enough to contain the pressure. For these experiments the f i l l i n g f a c t o r was about 0.5, that i s , only one h a l f the volume of the c o i l was occupied by the sample. The actual volume of sample inside the c o i l was about 0.13 cm , The net r e s u l t i s that i n the temperature region shown i n Figure l6 the o v e r a l l accuracy of the T^ measurements i s only about + 5$-5-2--2 Comparison with theory. The theory o u t l i n e d i n Chapter I I w i l l now be applied to the gross features of these r e l a x a t i o n measurements. The re l a x a t i o n rate -1 i s T l c onventionally written as the sum of three c o n t r i b u t i o n s : - = R + R + R (S.2l) T x A B C \?-^) Equation (2.l6) gives the co n t r i b u t i o n due to the intramolecular dipole-dipole i n t e r a c t i o n " R ^ : R A = i. ( 5 . 2 2 ) 18. (L.76 x l O " U ) D Taking the molecular diameter a to be equal to 3-95 A. gives U-93 x IO" 6 (5.23) D —1 2 with R^ i n sec when D i s measured i n cm / s e c •The intermolecular c o n t r i b u t i o n Rg i s given by Equation (2.l8) RB = 2 r r N y u t i 2 (5...2U) 5 a D If a i s again taken equal to 3.95 & and i f N the density of spins i s written •in terms of the mass density p t h i s y i e l d s R B = 2.18 x.-lO"* p_ (5.25) D with R-g i n s e c - 1 when p i s measured,in gm/cm3and.D i n cm 2/sec. The. spin r o t a t i o n a l c o n t r i b u t i o n R^ i s given by Equation (2.35)-R C - 6.1 x:10-l^ D ( 5 _ 2 6 ) a 2 For a =3-95 A t h i s gives R c = 39-1 D (5-27) with RQ i n sec" when'D i s measured i n cm /sec. These three contributions to the re l a x a t i o n rate are shown by the dotted lines:.in Figure 17• The values f o r D were taken from Gaven, temperature values f o r D were those reported i n section 5-1* The s o l i d l i n e i n Figure 17 gives the sum of the three contributions. Also p l o t t e d f o r agreement between the a p r i o r i estimate f o r the r e l a x a t i o n rate and the experimental values i s s u r p r i s i n g l y good. At no temperature i s the disagreement more than a f a c t o r of 3 and near the c r i t i c a l temperature i t i s l e s s than a f a c t o r of 2. In view of the rather gross approximations made i n the theory the agreement must be regarded.as somewhat f o r t u i t o u s . T t does appear however that the r e l a x a t i o n i n ethane can be adequately described over a considerable range of density and temperature by a simple superposition of the three r e l a x a t i o n mechanisms. For the low temperature l i q u i d the di p o l a r intramolecular c o n t r i b u t i o n to the r e l a x a t i o n rate i s found to be too e f f e c t i v e i n producing the re l a x a t i o n . 'This i s perhaps evidence f o r the breakdown of the simple version of the Debye model of a sphere r o t a t i n g . i n a viscous mediumin which the molecular rotations are assumed to be governed by solutions to (hp) the r o t a t i o n a l d i f f u s i o n equation. Muller and Noble v ' have investigated the r e l a t i v e magnitudes of the three contributions to r e l a x a t i o n using deuterated modifications of ethane. The dependence of R^, Rg> and on the p o s i t i o n and number of deuterons on the molecule i s d i f f e r e n t and so i t i s p o s s i b l e to sort out the r e l a t i v e magnitudes of the three contributions.. In the temperature range T 0 0 ° to lUO°K where four d i f f e r e n t Stockmayer and Waugh (35) f o r the low temperature l i q u i d , while the high The 77 Figure TJ. Temperature dependence of R^ , Rg and R^. ethanes have been measured they f i n d that the p r i n c i p a l c o n t r i b u t i o n to r e l a x a t i o n comes from the intermolecular dipolar i n t e r a c t i o n . The dipplar intramolecular i n t e r a c t i o n contributes ( l l +_ 5)$; the intermolecular i n t e r -action contributes (87 + 7')$.and the spin r o t a t i o n a l i n t e r a c t i o n i s small (2 + 6)$> and may well be zero. The t h e o r e t i c a l p r e d i c t i o n f o r RQ i s c e r t a i n l y consistant with these r e s u l t s . I f i t i s assumed that the dependence on density and d i f f u s i o n constant i s correct, i t i s then po s s i b l e to f i t R^ and .Rg. at low temperatures to the experimental r e s u l t s by m u l t i p l y i n g each by an appropriate constant. At 130°K, R^ must be m u l t i p l i e d by U.88 x I O - 2 and Rg by 1.63- I f these constants are then used throughout the temperature range then the dotted l i n e i n Figure 18 shows the f i t obtained. As i s to be expected, the agreement between theory and experiment i s very good at low temperatures, but f o r high temperatures the theory with no adjustments ( s o l i d l i n e ) gives a b e t t e r f i t . I t seems l i k e l y that the theory f o r the intermolecular i n t e r a c t i o n (Rg) i s correct to within a f a c t o r of 2. However the theory f o r the di p o l a r intramolecular i n t e r a c t i o n s has a much weaker t h e o r e t i c a l foundation, r e s t i n g -as i t does on the Debye r o t a t i o n a l d i f f u s i o n model. I t seems l i k e l y that at low temperatures the predicted c o r r e l a t i o n time may be out by a f a c t o r of 10 to 20. In an i n t e r e s t i n g experiment (kk) de Wit has investigated the deuteron r e l a x a t i o n time i n C . He finds that to within the errors of measurement the r e l a x a t i o n time i s independent of temperature r i g h t through from a s o l i d to a l i q u i d :above the melting point. The i n t e r p r e t a t i o n of t h i s r e s u l t i s that the c o r r e l a t i o n time f o r molecular r e o r i e n t a t i o n i s independent of temperature. T h i s means that the c o r r e l a t i o n time i s independent of the s e l f d i f f u s i o n constant. For low temperature ethane the same r e s u l t may very w e l l be true. The contributions from the intramolecular i n t e r a c t i o n s are small at low temperatures and the experimental r e s u l t s do not rule out the p o s s i b i l i t y that they are independent of temperature. For high temperatures near the c r i t i c a l point the 79 intermolecular i n t e r a c t i o n only contributes a few per cent to the t o t a l r e l a x a t i o n and so the intramolecular i n t e r a c t i o n must become temperature dep endent. It should be pointed out that i n the case of a d i l u t e gas where the density i s much l e s s than the c r i t i c a l density, the intramolecular i n t e r a c t i o n s and RQ are both proportional to ^* _ 1. Therefore, .at low d e n s i t i e s and constant temperature both R^ and RQ must increase as the density decreases. This i s to be contrasted with the p r e d i c t i o n s of the Debye r o t a t i o n a l d i f f u s i o n model which gives R^ cx 1_ and hence a decreasing R^ with decreasing D density. Thus, even i f the r o t a t i o n a l d i f f u s i o n model holds i n the high density l i q u i d and describes c o r r e c t l y the temperature dependence of R^, there must s t i l l be some density-at which R^ goes through -a minimum value. It may w e l l be that t h i s density i s close to the c r i t i c a l density. 5.2.3 Measurements near the c r i t i c a l point. Measurements of the s p i n - l a t t i c e r e l a x a t i o n time near the c r i t i c a l point are presented i n more d e t a i l i n Figure.'.19- The s o l i d l i n e s give the experimental temperature dependence. The actual shape of these curves should not be taken too seriously, but they do represent some sort of an average. As can be seen there i s a good deal of scatter i n the data, but most of the points l i e within about +_ 5$ of the l i n e s . The shape of the T-L curve may "be compared with the shape of the coexistence curve, Figure 20. There i s a s t r i k i n g . s i m i l a r i t y i n the shape of these two curves. The density data f o r the coexistence curve was taken from two sources. For temperatures below the c r i t i c a l temperature the r e s u l t s of Mason, Waldrett, and'Maass^) .are p l o t t e d . Their estimated error i s about 1 part i n 3000 f o r the density and they give a temperature f o r meniscus disappearance of 32.23°C The density data above the c r i t i c a l temperature was taken from the r e s u l t s of Palmer^'^ , who used a Schlieren o p t i c a l system to study the Figure 2 0 . The coexistance curve f o r ethane in. the c r i t i c a l temperature region. density d i s t r i b u t i o n i n h i s samples. He estimates that there may be errors as large as 8$ i n the density at a given height. Palmer made c a r e f u l v i s u a l observations i n determining the temperature of disappearance of the meniscus. He obtained a temperature f o r ethane of 32-32 + 0.02°C This temperature i s used throughout t h i s study as the c r i t i c a l temperature. I t was therefore necessary to s l i g h t l y adjust the temperatures given by Mason, NaldrQtt and Maass. This was done by adding 0.09°C to t h e i r experimental temperatures. As may be seen i n Figure 20 the d e n s i t i e s at the two p o s i t i o n s i n the sample are not equal at the temperature of meniscus disappearance. A difference i n density i s maintained f o r nearly a degree above the temperature of meniscus disappearance. Palmer obtained the shape of the density d i s t r i b u t i o n as a function of height i n h i s v e r t i c a l samples. The d i s t r i b u t i o n s of density under conditions of r i s i n g temperature were sigmoid i n character. The values p l o t t e d i n Figure 20 are f o r the p o s i t i o n s of the vapor and l i q u i d c o i l s used i n t h i s study. Equation 3.8 gives the v a r i a t i o n of density with height i n a compressible f l u i d : a___ = - P 2(Z)K_ g (5,28) d z So i t may be seen that when the isothermal c o m p r e s s i b i l i t y KJI i s large, as i t i s known to be i n the v i c i n i t y of TQ^ 1^, .a density gradient may be set up i n the sample due to the g r a v i t a t i o n a l f i e l d . This i s the reason that a density gradient can p e r s i s t above the c r i t i c a l temperature. Returning now to Figure 19, the t h e o r e t i c a l p r e d i c t i o n s f o r T-j_ are shown by the dotted l i n e s . This i s the a p r i o r i theory corresponding to the s o l i d curve of Figure 18.using the measured values f o r the s e l f d i f f u s i o n constant. The error f l a g shown at 3k°C i s due to measurement errors i n the d i f f u s i o n constant. There i s no p r e d i c t i o n f o r a difference i n T^ i n the two c o i l s above the c r i t i c a l temperature because there i s no experimental j u s t i f i c a t i o n Qh f o r taking the d i f f u s i o n constant to be d i f f e r e n t at these two p o s i t i o n s i n the impure samples used to obtain the d i f f u s i o n r e s u l t s . "The .main experimental error i n the d i f f u s i o n constant measurements are the systematic errors involved i n c a l i b r a t i n g the f i e l d gradient. There i s also some d i f f i c u l t y i n taking into account the change i n the equation of state i n the impure samples. Even using these imperfect d i f f u s i o n r e s u l t s i t i s apparent that the theory i s capable of p r e d i c t i n g the gross shape of the curves and .the agreement i n magnitude i s very good. It i s apparent that the density has a strong e f f e c t on T , t h i s e f f e c t was investigated using samples loaded to d e n s i t i e s other than the c r i t i c a l density. Four pure samples were prepared; these are the same samples as previously described i n the d i f f u s i o n r e s u l t s . T^ was also measured f o r these samples as a function of temperature. The r e s u l t s at 32.0°C and i40.0°C are" presented i n Figure 21. The e f f e c t of density on may be investigated using the t h e o r e t i c a l r e s u l t s already discussed. If i t i s assumed that Rg=0 at the c r i t i c a l point ( i t i s at most h or 5$ of the t o t a l ) then the relax^ ation rate i s given by $ = A D + | (5-29) where A and B are constants r e l a t e d to the spin r o t a t i o n a l and dipole-dipole intramolecular i n t e r a c t i o n s r e s p e c t i v e l y . The T-j_ data from 0° to 60°C were f i t t e d to t h i s equation by considering l i q u i d , vapor and gas r e s u l t s separately. Table'IE gives the "best f i t " values obtained f o r A and B. Table I I Best f i t values f o r A and B A(cm" 2) B(cm sec ) T h e o r e t i c a l 39-1 U.93 x 10"6 Vapor 27-5 15.6 Gas hi, 2 5.19 L i q u i d 60.2 3-68 85 86 The d i f f u s i o n constant can he obtained from D/O = K (5-30) where K i s constant at f i x e d temperature. Values f o r K may be obtained from Figure 12 and are given by •K 2 G ) = 9.28 x 10~ 2 cm^ Amagat sec -2 2 K^Q0 = 9--96" x 10 cm Amagat , •sec The s o l i d l i n e i n Figure'21 displays the density dependence of T^ using the t h e o r e t i c a l values f o r A and B and the value of -K at 32°C. The dotted l i n e makes use of the "best f i t " values f o r the gas and the. value of K-at U0°C. Values jtaker. from the smooth curves i n Figure 19 are also p l o t t e d (using the density data taken from Figure 20 with temperature -as a parameter) f o r comparison. The measured values of the r e l a x a t i o n time i n the samples loaded to l e s s than the c r i t i c a l density-appears to be somewhat lower than the c r i t i c a l l y loaded samples f o r the same density. For the samples loaded to a density i n excess of the c r i t i c a l density the r e l a x a t i o n times were i d e n t i c a l to the c r i t i c a l l y loaded samples to within experimental error. Unfortunately the most dense of these samples exploded before a measurement could be performed at kO°C. I t appears that more work would be necessary to show whether the r e l a x a t i o n time i n a non c r i t i c a l l y loaded sample was r e a l l y s i g n i f i c a n t l y - l o w e r than a c r i t i c a l l y " l o a d e d • sample. I f t h i s were true i t would mean that there i s a s i g n i f i c a n t d i f f e r e n c e between the gas and the equilibrium vapor at the same density and at nearly the same temperature. This seems most u n l i k e l y . Schwartz^) measured T-|_ i n s u l f e r hexafloride near the c r i t i c a l point. His r e s u l t s are very s i m i l a r to the r e s u l t s presented i n Figure 19« They ex h i b i t the same general shape and he also observed that the r e l a x a t i o n 87 times d i d not become equal at the c r i t i c a l temperature. Tn t h i s case the difference i n rel a x a t i o n times p e r s i s t s f o r about 5°C abOve the c r i t i c a l point. Detailed density information was unavailable f o r s u l f e r hexafloride and so Schwartz was unable to explain t h i s difference-in-'T^. 5-2.4 Some possible d i s t u r b i n g e f f e c t s . Since the measured r e l a x a t i o n times are rather long a t i s important to consider whether ( l ) r e l a x a t i o n by c o l l i s i o n s with the walls of the container, or (2) rapid.exchange of molecules between the l i q u i d and gas phases play any r o l e i n the re l a x a t i o n process. This may be investigated by considering the distance a molecule can t r a v e l i n -a time the order of the relaxation time. By d e f i n i t i o n , D i s r e l a t e d to the mean square distance t r a v e l l e d by a molecule i n the x, y, and z d i r e c t i o n s i n a time t by i?)t = (7) t = U % = 2Dt. So the distance a molecule d i f f u s e s i n a re l a x a t i o n p e r i o d i s (~xF)i = (2D Ti)2«(2.x 6.x 10 - 1 + x 30)i ~ 0.2-cm, where values f o r D and Tj_ are appropriate to the c r i t i c a l point. The root mean square distance t r a v e l l e d i s roughly independent of density since D var i e s as 1 and T-j_ varies approximately as p at f i x e d temperature. This demonstrates that i t i s possible to immediately rule out any e f f e c t due to a ra p i d exchange of molecules at the in t e r f a c e . The centers of the c o i l s are about 2 cm from the meniscus and so the molecules would not have s u f f i c i e n t time to d i f f u s e into the c o i l s during a r e l a x a t i o n period. It i s not so obvious that1.-.wall e f f e c t s may be n e g l i g i b l e , since the i n t e r n a l diameter of the sample i s only k mm and i t appears that an appreciable f r a c t i o n of the molecules may c o l l i d e with the wall during a rel a x a t i o n period. To t e s t f o r po s s i b l e w a l l e f f e c t s samples with i n t e r n a l diameters of 2, 6 and 12 mm were also studied. T^ was measured i n both the l i q u i d and vapor phases at a temperature of 0°C To within the accuracy of measurement these three samples a l l gave the same relaxation times and this was in agreement with the relaxation time obtained for the k mm samples which were studied extensively. • It therefore appears that effects due to collisions with the container walls may be ruled out as a possible relaxation process. Another process which may be very disturbing in the measurement of both the diffusion constant and the relaxation time is the effect of molecular convection. Near the critical point the density varies rapidly as a function of temperature and this is just the condition needed to set up strong convection currents. The effects of convection could indeed be observed when the temperature was being changed, however convection appeared to die out rapidly as soon as the temperature reached equilibrium. It is believed that the temperature was uniform enough over the sample to rule out the possibility of convection effects. In an effort to check this a sample container was constructed having three separate chambers about 1 cm long. The bottom chamber would contain liquid, the middle chamber would be a buffer chamber and contain the meniscus, while the top chamber would contain only vapor. These chambers were separated about 1 cm and were connected with capillary tubing. Unfortunately this container exploded as the sample was-.being warmed up to room temperature .and no measurements were performed on i t . This attempt was then abandoned. It is undoubtably possible to construct sample tubes with at least constrictions in them to -reduce convection. It is probably a legitimate criticism of this work that more effort was not put into using a constrictive sample tube. 5.2.5 Impurity relaxation. The results of the spin-lattice relaxation measurements for the impure ethane samples are shown in Figure 22. Spin-lattice relaxation in mobile fluids through dissolved paramagnetic oxygen has been•discussed by Abraganr 1.5 CO A A. A A * A A 6 4 • . • • D a a ° O VAPORl • LIQUID J SAMPLE NO. I a VAPOR\ • LIQUID J A VAPOR"! A LIQUIDJ SAMPLE NO. 2 SAMPLE NO. 3 0.5 i -I I I l _ 20 30 T c T E M P E R A T U R E 40 (°C) 50 60 Figure 22. impurity controlled spin-lattice relaxation•in ethane. It i s assumed that any r e l a x a t i o n mechanism except that due to the presence of oxygen i s n e g l i g i b l e . From-.Abragam!s analysis of the general Bloembergexi, P u r c e l l and Pound treatment i = 16 TT & N P 0 2 > ( 5 . 3 2 ) 1 D a . where T^ i s the proton s p i n - l a t t i c e r e l a x a t i o n time, ~tf„ the proton gyro-magnetic r a t i o (2 .68 x lcA g a u s s - 1 s e c - 1 ) , N p the number of oxygen molecules per u n i t volume > the mean square magnetic moment of the oxygen -h-0 2 -2 molecule (6 .7 x 10 erg gauss ), D the mutual d i f f u s i o n c o e f f i c i e n t between ethane and oxygen, and a i s a distance of c l o s e s t approach between ethane and oxygen. I f these values .are used along with the Lennard-Jones parameter for the distance of c l o s e s t approach t h i s y i e l d s N p = 7 . ^ x . l 0 2 2 D ( 5 . . 3 3 ) 1 Approximating D by the experimental s e l f d i f f u s i o n c o e f f i c i e n t and using = 1 sec t h i s gives 1 Q • 3 Np = 4 . 5 x 10 molecules of oxygen/cm This may be compared with the number of ethane molecules at the c r i t i c a l density: 21 ' 3 N A = k.l x :10 molecules of ethane/cm which gives an oxygen concentration of a b o u t l $ . This i s i n good agreement with the estimate made while preparing the samples. I t i s i n t e r e s t i n g to note that i n these impure samples T-^  i n the l i q u i d i s lower than i n the gas as opposed.to the pure samples where T^ i n the l i q u i d i s longer than i n the gas. This i s not s u r p r i s i n g because- i n t h i s example of an intermolecular i n t e r a c t i o n T^ oi. D and D i s greater i n the gas than the l i q u i d . Here i t i s assumed that Np i s constant. T t i s however somewhat s u r p r i s i n g that the r e l a x a t i o n times are so nearly equal at temperatures below the c r i t i c a l point. Taking sample number one at 15 C, f o r example, the re l a x a t i o n times are only d i f f e r e n t by a f a c t o r of 1-5- I f the oxygen remained evenly d i s t r i b u t e d throughout the sample then the re l a x a t i o n times should d i f f e r by the same fa c t o r as the d i f f u s i o n constant, i . e . a f a c t o r of about 6. Since the re l a x a t i o n times are so nearly equal t h i s means that the density of oxygen molecules has a c t u a l l y increased i n the vapor portion' of the sample by a f a c t o r of about 2.5. The s o l u b i l i t y of oxygen i n l i q u i d •,ethane must decrease r a p i d l y as the temperature i s lowered from the c r i t i c a l temperature. This would be an i n t e r e s t i n g and u s e f u l method f o r studying the s o l u b i l i t y of paramagnetic impurities near the c r i t i c a l point. 5.2.6 D i l u t e ethane gas measurements. Measurements were also made on the spin l a t t i c e r e l a x a t i o n time i n d i l u t e ethane gas over a temperature range from about l80°K to 300°K. To within the errors of measurement T^ was found to be proportional to p> . These measurements were made at d e n s i t i e s of l e s s than .k Amagats using a spectro-meter and low temperature system designed and constructed by Walter Hardy •and Frank Bridges. This system was especially•adapted to the study of gases at low d e n s i t i e s . The author i s very g r a t e f u l to Mr,. Hardy and Mr. Bridges f o r t h e i r time and help i n obtaining these measurements. The r e s u l t s obtained are presented i n Figure 23 along with a measurement due to Bloom, L i p s i c a s and M u l l e r ^ ^ . Here l n T^ i s plotted-as a function of In T to IB-show the temperature dependence of T]_ . The err o r f l a g s shown give maximum T estimated er r o r and are thought to be p e s s i m i s t i c . The data in d i c a t e s that The intermolecular d i p o l a r i n t e r a c t i o n may be neglected i n the i n t e r -p r e t a t i o n of polyatomic d i l u t e gas r e s u l t s . As may be seen i n Figure 17, Rg i s nearly n e g l i g i b l e even at the c r i t i c a l density. This leaves only the intramolecular i n t e r a c t i o n s to be considered. From the t h e o r e t i c a l r e s u l t s -0.6 _ l I I I I I I I I 1 l _ 5.2 :• 5,3 5.4 5.5 5.6 5.7 Ln T (°K) Figure 23. Temperature dependence of -T, /f> f o r ethane at low densities. 93 of Chapter II i t was found that f o r the dipole-dipole i n t e r a c t i o n (Equation 2.11) T x r 2 = 1 . 7 6 x 1 0 " 1 1 s e c 2 ( 5 . . . 3 I O This r e s u l t i s v a l i d f o r a d i l u t e gas except that i t i s not expected that the model of a r o t a t i n g sphere i n a viscous medium would be applicable to a d i l u t e gas. For the spin r o t a t i o n a l i n t e r a c t i o n the r e s u l t given by (19) Equation ( 2 . 2 l ) i s shown to be v a l i d by B l i c h a r s k i when W 2 ^ « 1 (5-35) where i s the average l i f e time of the m^. states. I t i s .usually assumed that ~r* i s approximately equal to the e f f e c t i v e molecular c o r r e l a t i o n time ~fc and f o r gases of the density considered here the above in e q u a l i t y i s we l l s a t i s f i e d . The l i f e time i n a J state i s assumed to be much longer than the l i f e time i n the m,. states and t h i s means that J <1 = _ T> 1 ( 5 , 3 6 ) Ti + r c The c o n t r i b u t i o n due to the spin r o t a t i o n a l i n t e r a c t i o n w i l l then be given by Equation (2 .27) with q set equal to one. ft) 8 . 8 x 1 0 8 T 7* sr or -9 m-1 Q P P 2 (5-37) T T x l 1 1 = 1.1 x 10 y T sec' The spin r o t a t i o n a l i n t e r a c t i o n has the same symmetry properties as:spherical harmonics of order 1 while the dipole-dipole i n t e r a c t i o n involves s p h e r i c a l (47) harmonics of order 2. This means that i n the weak c o l l i s i o n " approximation • ^ = 3 T 2 ( 5 . 3 8 ) 9h The r e l a t i v e e f f e c t i v e n e s s o f t h e s e two p r o c e s s e s may h e f o u n d f r o m ( T l ) d i p - d i p ^ 2 .= 1.76 x I O - 1 1 ('A , 3 T 2 " l.l x 10-y T"1 ( 5 ' 3 9 ) w h i c h g i v e s ( T l ) d i p - d i p = k.Q x I O - 2 T . . (5-kO) \ X ' s r I n t h e - t e m p e r a t u r e r a n g e s t u d i e d h e r e t h e s p i n r o t a t i o n a l i n t e r a c t i o n i s t h e n a b o u t 10 t i m e s as e f f e c t i v e i n p r o d u c i n g r e l a x a t i o n -as i s t h e d i p o l e -d l p o l e i n t e r a c t i o n . The above t h e o r y may be u s e d t o e v a l u a t e t h e c o r r e l a t i o n t i m e s . The a n i s o t r o p i c i n t e r a c t i o n s w h i c h p r o d u c e m o l e c u l a r r e o r i e n t a t i o n a r e s h o r t range, a n d . a p o s s i b l e m o d e l i s t h a t i n w h i c h t h e o r i e n t a t i o n o f t h e m o l e c u l e i s c h a n g e d o n l y as a r e s u l t o f c o l l i s i o n s . F o r t h e d i l u t e gas t h e r a t e o f m o l e c u l a r c o l l i s i o n s i n t h e f ramework -of a h a r d s p h e r e m o d e l i s g i v e n 1 = k a.2p ^TTk T y ( 5 . 4 l ) r where a i s t h e d i a m e t e r o f t h e m o l e c u l e , p is t h e number o f m o l e c u l e s p e r cm^ , a n d m i s t h e mass o f t h e m o l e c u l e . F o r e t h a n e a = 3-95 A and m = ^-99 x I O - 2 3 gm. U s i n g t h e s e n u m e r i c a l v a l u e s f o r e t h a n e y i e l d s 1 •= 1.8U x 1 0 " 1 1 p T 2 . (5..1+ar) T E x p r e s s i n g p I n Amagats g i v e s 1 = 5..00 x 10 8 pA T ^ (5-^3) r To compare t h i s c o l l i s i o n t i m e w i t h t h e c o r r e l a t i o n t i m e f o r t h e r e l a x a t i o n •process i t i s p o s s i b l e t o w r i t e *1 = {^sv + ( ^ ) d i p - d i p = 8.8 x 10 8 TT_ + 5.68 x 1 0 1 0T 2 = (2.6 x 10 9 T + 5-7 x 1 0 1 0)T 2 (5M) 95 The m e a s u r e d v a l u e s o f T-j_ may t h e n be u s e d t o e v a l u a t e f g . T h e s e r e s u l t s a r e summarized i n T a b l e I I I . T a b l e I I I 'Summary o f d i l u t e gas . r e s u l t s Temp. ( ° k ) ( s e c / A m a g a t ) -rz P ( s e c - A m a g a t ) rp ( s e c - A m a g a t ) T/TT, 182 0.48 -12 3.9 x 10 ±cl 1.48 x l O ' 1 0 38 223 O.38 4.1 1-34 33 263 0.28 4.8 1.-23 26 297 0.25 4.8 I . I 6 24 3t4 0.23 5-1 I . I 3 22 The v a l u e s g i v e n i n T a b l e I I I show t h a t T*'is much g r e a t e r t h a n T^ , w h i l e t h e ' " w e a k c o l l i s i o n ' ' a p p r o x i m a t i o n i m p l i c i t l y - a s s u m e s t h a t T* < I'i • (47) F r e e d h a s s u g g e s t e d t h a t u n d e r t h e s e c i r c u m s t a n c e s a " s t r o n g c o l l i s i o n " a p p r o x i m a t i o n s h o u l d be more v a l i d . T h i s w o u l d i m p l y t h a t i t . i s p o s s i b l e t o d e f i n e a T e f f g i v e n b y E q u a t i o n ' (5-4l) w i t h an '3.eff > a s u c h t h a t -j-1 Teff ' t h a t i s , i n any c o l l i s i o n where t h e m o l e c u l e s a p p r o a c h t o w i t h i n a d i s t a n c e s m a l l e r t h a n a g j . f , . . a l l memory o f t h e o r i g i n a l m o l e c u l a r o r i e n t a t i o n i s l o s t . However , a c c o r d i n g t o t h i s a p p r o x i m a t i o n -— -1 37 t h e t e m p e r a t u r e dependence o f T-j_ i s T 2 i n s t e a d o f t h e o b s e r v e d T J 1 . w d e p e n d e n c e . On : the o t h e r h a n d a s i m p l e :"weak c o l l i s i o n " m o d e l a s s u m i n g t h a t t h e c o r r e l a t i o n t i m e f o r m o l e c u l a r r e o r i e n t a t i o n i s i n d e p e n d e n t o f t h e r o t a t i o n a l state (49,50)x .3 ' g i v e s a T l t e m p e r a t u r e dependence o f T 2 w h i c h i s c l o s e t o t h e . e x p e r i m e n t a l v a l u e . S i n c e t h e "weak c o l l i s i o n " a p p r o x i m a t i o n i s i n c o n r s i s t a n t w i t h t h e r e s u l t s p r e s e n t e d i n T a b l e I I I , i t seems t h a t a more • d e t a i l e d d e s c r i p t i o n o f t h e m o l e c u l a r c o l l i s i o n s i s r e q u i r e d t o e x p l a i n t h e o b s e r v e d r e s u l t s f o r d i l u t e e t h a n e g a s . 9 6 Chapter VI CONCLUSIONS AND .SUGGESTIONS FOR FURTHER WORK The work reported i n the preceding chapters demonstrates that the nuclear magnetic resonance method, permits one to study the c r i t i c a l point of gases and to gain information not e a s i l y obtained by other means. For example the s e l f d i f f u s i o n constant i n suitable compounds may be measured with r e l a t i v e ease using t h i s technique. When using other experimental methods i t i s d i f f i c u l t to make accurate measurements of t h i s important transport property. Measurements of the d i f f u s i o n constant reveal that f o r low temperatures the product- Bp f o r l i q u i d ethane v a r i e s approximately -as T J. Near the c r i t i c a l point there appears to be. anomalous behaviour i n D. For both the l i q u i d and vapor the product Bp decreases and goes through a minimum and then increases r a p i d l y as the c r i t i c a l temperature i s reached. Above the c r i t i c a l temperature Bp i n the dense gas again, increases, approximately as T-3. An explanation is. o f f e r e d which accounts f o r t h i s e f f e c t i n terms of the theory of molecular c l u s t e r i n g . These measurements were a l l made on samples to which oxygen had been purposely added as an impurity. I t i s therefore possible that t h i s i s an impurity e f f e c t . This would be a v e r y . i n t e r e s t i n g e f f e c t to investigate further. It would be worth while to repeat the experi-ment using pure ethane to eliminate the p o s s i b i l i t y of an impurity e f f e c t . Unfortunately, t h i s would be a rather d i f f i c u l t experiment because the spin-l a t t i c e r e l a x a t i o n time of pure ethane i s very long. - I t would be b e t t e r perhaps to choose a compound with a shorter natural r e l a x a t i o n time. Measurements of the s p i n - l a t t i c e r e l a x a t i o n time as a function of temperature over the e n t i r e l i q u i d range, as well as i n the vapor below the c r i t i c a l point and i n the dense gas at temperatures above the c r i t i c a l temperature, were described. These r e s u l t s can be adequately explained i n terms of a simple superposition of three r e l a x a t i o n mechanisms: the 97 intermolecular d i p o l a r i n t e r a c t i o n , the intramolecular d i p o l a r i n t e r a c t i o n and the spin r o t a t i o n a l i n t e r a c t i o n . The theory gives the r e l a x a t i o n c o n t r i b u t i o n i n terms of the t r a n s l a t i o n a l s e l f d i f f u s i o n constant. Using measured values of D the t h e o r e t i c a l values of the r e l a x a t i o n times are In reasonable agreement with the experimental values. The v a r i a t i o n i n T^ near the c r i t i c a l point i s well explained i n terms of the v a r i a t i o n i n density.and d i f f u s i o n constant. Further work on experimental measurement of the r e l a x a t i o n time i n ethane appears u n l i k e l y to be very f r u i t f u l . Since the measurements i n the d i l u t e ethane gas can not be explained i n terms of e x i s t i n g theory, more work must be done on the t h e o r e t i c a l d e s c r i p t i o n of molecular c o l l i s i o n s and the r e s u l t i n g molecular reorie n t a t i o n s . "The problem of adequately describing the molecular reorientations i s also,a perplexing one f o r l i q u i d s and dense gases. An experimental measurement of the r e l a x a t i o n time as a function of temperature f o r t o t a l l y deuterated ethane would be valuable. This i s because the deuteron r e l a x a t i o n i s l i k e l y to be dominated by the coupling of e l e c t r i c f i e l d gradients with the quadrupole moment of the deuteron. I f t h i s were the case . i t would allow -a d i r e c t measurement of the temperature dependence of the c o r r e l a t i o n time fo r molecular reorientations.. Experimentally i t has been observed that i n the neighbourhood'of the c r i t i c a l point i t i s d i f f i c u l t to obtain equilibrium conditions throughout the system. I t has been known f o r many years that equilibrium conditions are established rather slowly i n t h i s region. This i n t e r e s t i n g phenomena was investigated by measuring the r a t i o of l i q u i d to vapor density as a function of time a f t e r a r a p i d change i n temperature of the: system. The approach of the density r a t i o to i t s equilibrium value was found to vary i n a roughly.exponential fashion with a time constant of the order of several hours. The nuclear magnetic resonance technique appears to be a very u s e f u l method of i n v e s t i g a t i n g the approach to equilibrium. I f a moveable c o i l were used i t would also be possible to study the density gradients which are set up i n the earth's g r a v i t a t i o n a l f i e l d . Experiments along these l i n e s might produce much u s e f u l information about the behaviour of a pure substance i n i t s c r i t i c a l region. APPENDIX A CIRCUIT D E T A I L S OF PULSED SPECTROMETER TIME MARK GENERATOR COUNTER NO.I OUTPUTS O r —o 2 -o 3 —o 4— o 5 - o 6 —o 7— o 8— o 9— o COUNTER NO. I COUNTER NOi n i i i i i i i i i COUNTER NO. m i i i i I I i—r SWITCHES SEE FIG. 3A TO REPETITION RATE GEN, (SEE FIG. 4 A ) TI-I TU-0 Q O DT-I DT-2 DT-3 C B A F E D 9 0 Q Q Q 0 0 1 1 CHANNEL I COINCI DENCE CHANNEL H COINCI DENCE TO PULSE W l { SEE FIG. 5A ) DTH GEN. i" Fig-are -IA. False sequence sub-unit. TRIGGER 6AU6 GATING UNIVIBRATOR I2AU7 PASS CLAMP + 3 0 0 V FROM REPETITION RATE GEN. (FIG. 4A) 0 l , H FROM SEQUENCE SUB UNIT TO TRANSMITTER GATE Figure 2A. Gate and clamp circuit-. 0 - Q 0 - Q 0 -Q, 1 - C X I - o \ I -O 2 -O \ 2 -O \ 2 - 0 3 -O \ 3 -O \ 3 - 0 4 -O X>*-D 4 -p XD-*-E 4 -O 5 - 0 5 • -to 5 - 0 6 - 0 6 - 0 6 - 0 7 - 0 7 -O 7-0 8 - 0 8 -O 8 - 0 9 - 0 9 - O 9 - 0 Figure 3.A. Switch details. 103 + 250V TARGET OUTPUTS TO ABOVE SWITCH 3 4 5 6 7 8 9 ^0 25QV FROM COUNTER (FIG. IA) 250V .240K >22K 4W + 12V I 3 K < I . I K < » I K IN629 H4 30KS.33K.:«. IN38B 47 • A A A - 1 _ . I . IK RECEPTACLES FOR DCII4'S 3 VAC 82K —W I50pf .01 •047, I IK hAAA IN38B-zt" •I3K 9.IK 330 ;82K 290K; IN628 »47K IN67 2NI605 2NI605 t '47K I80K-ZERO SET TO GATE (SEE FIG. 2A) 2NI990 Figure hA. Preset counter for repetition rate. TRIGGER AMPLIFIER I2AU7 PHANTASTRON GATE I2AU7 5725 MIXER 5670 + 150V —170V FROM CH. U. GATE CHANNEL H TRIGGER AMPLIFIER I2AU7 TO TRANSMITTER CHANNEL II PHANTASTRON GATE I2AU7 5725 Figure 5A= Pulse width generator. I2AU7 +225V 1 6AU7 6L6 +2 25V I ll »22K .1 FROM WIOTH G E N £ 6 8 0 K •22K - I70V 6SN7 >22K -300V _L7 150 •22K | Z-28 + 15 1 l 1 FERRITE BEADS '680K -20pf Ipf • 330 OV 6L6 + 225V I 1 '.Olj.pl, I20pf -\\ 40pf | SET •01 ~T 5 4 7 K 5330 J _ .01 1 J L - _ J O TO SAMPLE -300V IOK ' FERRITE t BEADS. + l 2 5 0 v t±± Figure 6k. Radio frequency, transmitter. H o BIBLIOGRAPHY F i s h e r , M . E . , J . M a t h . / P h y s . 5, 9kh (196U), Hemmer, - P . C . , K a c , M. and U h l e r i b e c k , ' G . E . - , J . M a t h . P h y s . -5, 6 C (196U); see a l s o Van H o v e , L . , P h y s . R e v . 95, 2h9 (l95Uj. K r y n i c k i , K . a n d . P o w l e s , J . G ? , -Proc . P h y s . S o c . 8 3 , 0,83 (196I+), A l s o see R e f . 16 and 17 b e l o w . P a l m e r , 11.H., J . Chem. -Phys. 22, 625 (195*0-M a s o n , - S - G . , N a l d r e t t , S . N . a n d M a a s s , 0., C a n . J . -Research B , 18, 103 (19.^0), S c h w a r t z , J . , P h . D . T h e s i s , H a r v a r d U n i v e r s i t y (1958). B l o e m b e r g e n , N . , P u r e e l l , E , M . , . a n d P o u n d , R . V . . , - P h y s . Rev . ,73, 679 (19^8). Abragam, A . , " T h e P r i n c i p o l e s o f N u c l e a r M a g n e t i s m " , O x f o r d U n i v e r s i t y P r e s s , ;London ' (19.61), H u b b a r d , P . S . , P h y s . . R e v . -109, .1153 (.1958). -."Handbook o f C h e m i s t r y a n d P h y s i c s " , kkth. E d . , C h e m i c a l R u b b e r P u b l i s h i n g C o . - , C l e v e l a n d , O h i o , p . 3517 (.1962). I v a n o v , E . N . , S o v i e t P h y s . J E T P 18, -10kl (196k), Kemp, J . D . . a n d . P i t z e r , -K.-S. , " J , . Am. Chem. S o c , 59, .27.6 (19 37.). M u l l e r , B . H . , P h . D . T h e s i s , U n i v e r s i t y o f I l l i n o i s (1-954). . H u b b a r d , P . S . . , P h y s , R e v . 131, -275 (1963). .Oppenheim, I . and B l o o m , M . , C a n . J . -Phys. 39, 8^ .5 "(1961), P o w l e s , - J . G . a n d ' S m i t h , D . W . G . , • P h y s , L e t t e r s 9, 239 (196k). K r y n i c k i . , K. .and P o w l e s , J . G . , - P h y s . L e t t e r s __, 260 (1963). Townes , C - H . a n d Shawlow, A . T . , - "Microwave S p e c t r o s c o p y " , M c G r a w - H i l l Book C o . . , I n c . , New Y o r k , p.-219 (1955), ' ' ' B l i c h a r s k i , J , S . - , - A c t a . Phys . . P o l o n . 2k, .817 (1963). A n d e r s o n , C H . , P h . D . T h e s i s - , - H a r v a r d U n i v e r s i t y • (1961). . H e r t z b e r g , G . , • " I n f r a r e d and Ramam S p e c t r a * ' , D Van N o s t r a n d " C o , , I n c . , New Y o r k , p . 437 (l9i+5)-H a h n , E . L . , P h y s . R e v . .60, ;580 (1950). 107 (23) C a r r , H . Y . -.and P u r c e l l , E . M . . , - P h y s . R e v . 94, 63O (195U). (24) - S c h w a r t z , J . , R e v . - S c i . I n s t . 28, ,780 (1957). (25) C l a r k , W . G , , R e v . ' S c i . T n s t . 35, .316 (±964). (26) Waterman, ft.., P h . D . T h e s i s , U n i v e r s i t y o f B . C . (±954). (27) Lees-, -J.. . , ' M u l l e r , B . f t , • .and N o b l e , J . D . . , J . - Chem. P h y s , -3k, '341(1961). ' (28) B l o c h , F . , P h y s . -Rev. -70, 460 (1946). (29) Andrew, E . R . , " N u c l e a r M a g n e t i c R e s o n a n c e " , -Cambridge U n i v e r s i t y P r e s s , p.-10 (1955). (30) M u l l e r , "B. and B l o o m , M , , C a n . J , . P h y s . - 3 8 , 13l8 ( i960) . (31) L i p s i c a s , M. and B l o o m , M . . , -Can,' J . .Phys-. -39, 8 8 l (1961).. (32) S i m p s o n , J . H . and C a r r , H . Y . . , P h y s . R e v . I l l , 1201 (1958).. (.33) J e a n s , " J . . , " A n ' I n t r o d u c t i o n t o t h e K i n e t i c T h e o r y o f G a s e s " , U n i v e r s i t y ' . P r e s s , C a m b r i d g e , p.211 (1962). (34) i b i d , p . 170-6. (35) G a v e n , J . . V , , . , - ' S t o c k m a y e r , W . - H - , and Waugh, J . S . , J.. Chem.- -Phys . 37, .1188' (1962), ' (36) M a x w e l l , J . , - . - " D a t a Book on H y d r o c a r b o n s " , V a n No s t r a n d , -New Y o r k , (1951)- . . . (37) A m e r i c a n P e t r o l e u m ' I n s t i t u t e , - . - P r o j e c t 40, ' - " S e l e c t e d V a l u e s :of P r o p e r t i e s o f - H y d r o c a r b o n s " a n d R e l a t e d " C o m p o u n d s " , V o l . - I l l , T a b l e T d . (38) G l a s s t o n e , S . , L a i d l e r , K . - J . and E y r i n g , H . , " T h e o r y o f R a t e . P r o c e s s e s " , M c G r a w - H i l l , New Y o r k , C h a p t e r I X , (igkl). (39) F r e n k e l , - J , . , " K i n e t i c T h e o r y o f L i q u i d s " , D o v e r , 'New Y o r k , p .191, (1.955). . . . . . . •(40') B i r d , R . B . . , S t e w a r t , ' W . E . . , and L i g h t f o o t , ; E . N , . , . " T r a n s p o r t Phenomena" , J o h n W i l e y - a n d - S o n s , New Y o r k , p-515, ( i960). •(4l) H i r s c h f e l d e r , -J . .0.-, C u r t i s s , C F . , a n d " B i r d , R . B . , " M o l e c u l a r T h e o r y o f G a s e s a n d ' L i q u i d s " , J o h n W i l e y - a n d S o n s , New Y o r k , p.24, (±954)- ' (42) M u l l e r , B . f t . a n d N o b l e , J , D . . , J . Chem. Phys.. .-38, .777 (1963). (43) R e f e r e n c e 4 l , p.1112. (44) de W i t , G . , M . S c . T h e s i s , • U n i v e r s i t y , o f B . C . (1963). Reference 8., p,3 0 2 . Bloom, M., L i p s i c a s , M. and Muller, B.H., Can. J . Phys. 39) 1093 (1961). Freed,' J.H., J. Chen. Phys. -kl, 7 (.1964). Reference kl, p.9,-Bloom, M..and Oppenheim, -I.., Can. J. Phys. 41, I58O (1963). Johnson, C S. .and Waugh, J..S.., J . Chem. Phys, '35, 2020 (1961). 

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