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UBC Theses and Dissertations

Nuclear spin relaxation in gas mixtures Lalita, Krovvidi 1967

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. The.University of B r i t i s h Columbia FACULTY OF .GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF KROWIDI LALITA B.Sc. Andhra U n i v e r s i t y , W a l t a i r , I n d i a , 1956 M.Sc. Andhra U n i v e r s i t y , Waltair,. I n d i a , 1959 IN.ROOM 302, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: I. McT. Cowan M. Bloom J„ Marko. B. G. T u r r e l l R. F. Snider D. L. Will i a m s L. W. Reeves E x t e r n a l Examiner: J . S. Waugh Massachusetts I n s t i t u t e of Technology Cambridge, Mass. Research Supervisor: M. Bloom 3:30 p.m., March 28, 1967 • • NUCLEAR SPIN RELAXATION IN GAS. MIXTURES _ ABSTRACT -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 s t u d i e d i n normal H 2 as a f u n c t i o n of d e n s i t y and temperature i n the range 293 K - 700°K. The measure-ments were made i n the r e g i o n where 7 | f . The H r e s u l t s have been i n t e r p r e t e d u s i n g the Bloora-2 > Oppenheim theory i n which the t r a n s i t i o n s between d i f f e r e n t J s t a t e s were taken i n t o account. The a n a l y s i s i n d i c a t e s that the resonant t r a n s i t i o n s (1,3«-*3,1) and quasi-resonant t r a n s i t i o n s (1,203,0) and (1,44*3,2) c o n t r i b u t e s i g n i f i c a n t l y to the r e l a x -a t i o n mechanism. The a n i s o t r o p i c i n t e r m o l e c u l a r ; p o t e n t i a l between the two H 2 molecules which,depends on the o r i e n t a t i o n of both the molecules c o u l d be given by quadrupole-quadrupole i n t e r a c t i o n w h i l e the part that depends on the o r i e n t a t i o n of one o f the molecules alone was found to be adequately represented by a Lennard-Jones p o t e n t i a l . T, was measured i n H 2 - He and H 2 - C 0 2 mixtures as a f u n c t i o n of d e n s i t y and composition i n the tem-perature range. 293°K - 700°K. The a n a l y s i s i n d i c a t e s that the i n t e r a c t i o n p o t e n t i a l f o r H 2 - He c o u l d be adequately d e s c r i b e d by a Lennard-Jones p o t e n t i a l w h i l e the dominant i n t e r a c t i o n f o r H 2 - C 0 2 c o u l d be given by quadrupole-quadrupole i n t e r a c t i o n . There were i n d i c a t i o n s that the dependence of 7) jf i n H 2 - He mixture on the percentage of He i s non-- iinear_,abo.ve 150°R. . However, t h i s was not found to -be the case i n H 2 - C 0 2 mixtures. T. was a l s o measured i n CH. and CH, - He mixture » 4 4 as a f u n c t i o n of d e n s i t y and composition i n the same temperature range. The data can be f i t t e d by T*f? =AT where n takes the value of 1.5 f o r pure CH^ and 0.79 f o r CH^ gas i n f i n i t e l y d i l u t e d i n He. The a n a l y s i s based on the e x i s t i n g theory f o r polyatomic gases shows that the i n t e r m o l e c u l a r p o t e n t i a l f o r CH^ - CH^ and CH^ - He c o u l d be described by medium range p o t e n t i a l s . The r e s u l t s i n d i c a t e that the dependence of T J j ^ on the percentage of He i s not l i n e a r below 400°K. AWARDS 1956 U n i v e r s i t y F i r s t : Gold Medal 1959 U n i v e r s i t y F i r s t : Gold Medal „ 1962 Commonwealth S c h o l a r s h i p GRADUATE STUDIES , F i e l d , of Study: -Nuclear Magnetic Resonance Elementary Quantum Mechanics W. Opechows'ki Electromagnetic Theory G. M„ V o l k o f f Advanced Magnetism M. Bloom S t a t i s t i c a l Mechanics R„ B a r r i e Quantum Theory of S o l i d s W„ Opechowski NUCLEAR. SPIN RELAXATION IN GAS MIXTURES by KROVVIDI LALITA B . S c , Andhra U n i v e r s i t y , W a l t a i r , 1956 M . S c , Andhra U n i v e r s i t y , W a l t a i r , 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA MARCH, 1967 In presenting this thes i s in p a r t i a l f u l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that th..; L ib ra ry sha l l make i t f ree l y ava i l ab le for reference and s t u d y I f u r the r agree that permission for extensive copying of th i s thes is f o r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t i on of th i s thes i s for f i n a n c i a l gain sha l l not be allowed wi tliout my wr i t ten permission,. Department of Tivs. Un i ve r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date N(McX 3f )H7  ABSTRACT 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 s t u d i e d i n normal as a f u n c t i o n o f d e n s i t y and temperature i n the range 293°K - 700°K. The measurements were made i n the r e g i o n where ~T|«j> „ The Hg r e s u l t s have been i n t e r p r e t e d u s i n g the Bloom-Oppenheim t h e o r y i n which the t r a n s i t i o n s between d i f f e r -ent J s t a t e s were t a k e n i n t o a c c o u n t . The a n a l y s i s i n d i c a t e s t h a t the r e s o n a n t t r a n s i t i o n s (l,3o3?l) and q u a s i - r e s o n a n t t r a n s i t i o n s (1,23?0) and (1,*+ <—352) 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 the r e l a x a t i o n mechanism. The a n i s o t r o p i c i n t e r -m o l e c u l a r p o t e n t i a l between the two Hg m o l e c u l e s which depends on the o r i e n t a t i o n of b o t h the m o l e c u l e s c o u l d be g i v e n by qua d r u p o l e - q u a d r u p o l e i n t e r a c t i o n w h i l e the p a r t t h a t depends on the o r i e n t a t i o n o f one of the m o l e c u l e s a l o n e was found t o be a d e q u a t e l y r e p r e s e n t e d by a Lennard-Jones p o t e n t i a l , T |. was measured i n Hg - He and Hg - COg m i x t u r e s as . a f u n c t i o n o f d e n s i t y and c o m p o s i t i o n i n the temperature range 293°K - 700°K. The a n a l y s i s i n d i c a t e s t h a t the i n t e r a c t i o n p o t e n t i a l f o r Hg - He c o u l d be a d e q u a t e l y d e s c r i b e d by a Lennard-Jones p o t e n t i a l w h i l e the dominant i n t e r a c t i o n f o r Hg - COg c o u l d be g i v e n by qu a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n . There were i n d i c a t i o n s t h a t the dependence of Tt Jj> i n Hg - He m i x t u r e on the perc e n t a g e of He i s n o n - l i n e a r above 150°K. However, t h i s was not found t o be the case i n Hg - COg m i x t u r e s . T, was a l s o measured i n C H ^ and^CH^ - He m i x t u r e as a f u n c t i o n of d e n s i t y and c o m p o s i t i o n i n the same temp e r a t u r e range. The d a t a can be f i t t e d by T,jpz/\- where n t a k e s the v a l u e of 1.5 f o r pure CH^ and 0.79 f o r CH^ gas i n f i n i t e l y d i l u t e d i n He. The a n a l y s i s based on the e x i s t i n g t h e o r y f o r p o l y a t o m i c gases shows t h a t the i n t e r m o l e c u l a r p o t e n t i a l f o r CH^ - CH^ and CH^ - He c o u l d be d e s c r i b e d by medium range p o t e n t i a l s . T h e . r e s u l t s i n d i c a t e t h a t the dependence of T/p on the p e r c e n t a g e of He i s not l i n e a r below ^fOO°K. i i i TABLE OF CONTENTS Page A b s t r a c t . i i l i s t of T a b l e s . v i L i s t of I l l u s t r a t i o n s . v i i Acknowledgements. x i CHAPTER I . INTRODUCTION 1 CHAPTER I I . APPARATUS AND EXPERIMENTAL TECHNIQUE 7 2.1. Measurement of S p i n - l a t t i c e 7 r e l a x a t i o n t i m e . 2.2. N.M.R. Spectrometer •9 2.2.1. G e n e r a l Remarks 9 2.2.2. Timing C i r c u i t 9 2.2.3- T r a n s m i t t e r 11 2.2.H-. Sample C o i l 12 2.2.5- R e c e i v e r 13 2.3- H i g h - P r e s s u r e System 13 2j+. H e a t e r 15 2.5. M i x i n g o f gases: D e t e r m i n a t i o n o f 17 C o n c e n t r a t i o n CHAPTER I I I THEORY 25 3.1. Hydrogen 25 3.2. Theory o f R e l a x a t i o n i n Hg 29 3.2.1. C o r r e l a t i o n f u n c t i o n s o f i n t r a - 31 m o l e c u l a r i n t e r a c t i o n s 3.3. A p p l i c a t i o n of the t h e o r y t o some 36 s p e c i a l cases 3.3.I. I n f r e q u e n t t r a n s i t i o n s between 36 d i f f e r e n t s t a t e s of J . i v Page CHAPTER IV CHAPTER V 3 . 3 . 2 . T w o - l e v e l System 37 3-H-. 1 E v a l u a t i o n of B ^ ( J , J ) i n terms of i n t e r m o l e c u l a r a n i s o t r o p i c i n t e r a c t i o n s . 39 EXPERIMENTAL RESULTS AND DISCUSSION H-7 H-.1 . G e n e r a l Remarks h7 H-.2. Hydrogen h? H-.2.1 0 R e s u l t s H-7 H-.2.2. I n t e r p r e t a t i o n 51 H 2 - He M i x t u r e 68 M-.3.1 • R e s u l t s 68 H-.3.2. I n t e r p r e t a t i o n 72 H-.M-. H 2 - COg M i x t u r e 81 H-.H-.1 . R e s u l t s 81 H-A . 2 . I n t e r p r e t a t i o n 85 EXPERIMENTAL RESULTS AND DISCUSSION (co n t d . ) 93 Methane 93 5 . 1 . 1 . R e s u l t s 93 5 .1 .2 . I n t e r p r e t a t i o n 98 5.2. CH^ - He M i x t u r e 102 5.2.1 . R e s u l t s 102 5 .2 .2 . I n t e r p r e t a t i o n 105 CHAPTER V I CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 110 APPENDIX A C i r c u i t Diagrams 11M-BIBLIOGRAPHY. 121 v LIST OF TABLES TABLE I F r a c t i o n a l p o p u l a t i o n of the I I V a l u e s of I ( p , n ) f o r d i l u t e the f i r s t quantum c o r r e p t i o n i s o t r o p i c p o t e n t i a l f o r 0.Q3 Page r o t a t i o n a l s t a t e s f o r Hg- 28 Hg gas i n c l u d i n g the 67 f o r a Lennard-Jones 4 $ 0.09 I I I (T./p) o b t a i n e d from e x t r a p o l a t i n g the e x p e r i m e n t a l 71 d a t a £q 100$ He. v i LIST OF ILLUSTRATIONS F i g u r e Page 1. B l o c k diagram of 3 0 M c . p u l s e d s p e c t r o m e t e r 10 2. Sample h o l d e r 1^ 3. H i g h p r e s s u r e system 16 k. D e n s i t y of Hg-COg m i x t u r e s as a f u n c t i o n of c o m p o s i t i o n 21 5. S i g n a l s t r e n g t h v s . p r e s s u r e f o r Hg-COg m i x t u r e a t 293°K 23 6. S i g n a l s t r e n g t h v s . p r e s s u r e f o r Hg-COg m i x t u r e a t 1+000Kt 2h 7. B l o c k diagram of the n a t u r e of c o u p l i n g between n u c l e a r s p i n s and the l a t t i c e 29 8a. T 1 v s . d e n s i t y a t 293 K i n H 2 ^8 8b. T 1 v s . d e n s i t y a t 738°K i n H 2 hQ 9. T-j/p as a f u n c t i o n of temperature i n normal Hg 50 10. T h e o r e t i c a l p l o t s of ( T 1 / p ) Q p / ( T 1 / p ) 0 Q as a f u n c t i o n of ^^/y/' as o b t a i n e d from Eq. (h.2.1 k) % H —H 11. Temperature dependence of (k.j) 2 2 as o b t a i n e d from Eq. (M-„2.1 k) and the e x p e r i m e n t a l v a l u e s of ( T 1 ^ ^ O . p / ^ 1 ^ ''o.O a n d ( T 1 ^ o ) f o r n o r m a l H 2 57 12. Comparison of e x p e r i m e n t a l v a l u e s of [ k 1 ( T ) / k 1 ( 2 0 0 ) j H 2 " H 2 w i t h the t h e o r e t i c a l v a l u e s u s i n g E q . ( ^ . 2 . l 8 ) 59 13- Comparison of the e x p e r i m e n t a l v a l u e s of T^/p w i t h the computed v a l u e s u s i n g Eq. 2.1 h) and assuming t h a t o n l y q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n i s i m p o r t a n t 61 H —H 1^. Temperature dependence of ( k Q ) 2 2 as o b t a i n e d from Eq.(H-.2.lH-) and the e x p e r i m e n t a l v a l u e s of (T^/f) i n normal Hg 62 v i i Comparison of e x p e r i m e n t a l v a l u e s of rk o(T)/k 0(200)j H2~ H2 w i t h the t h e o r e t i c a l v a l u e s u s i n g Eqs . (h.2.22) and (H-.2.2H-) vs„ d e n s i t y a t 293°K i n Hg-He m i x t u r e v s . d e n s i t y a t 738°K i n Hg-He m i x t u r e Dependence of T^/p on H e l i u m c o n c e n t r a t i o n i n Hg-He m i x t u r e H -He Temperature dependence of (k ) 2 as o b t a i n e d f r om Eq.(h.2.1h) and u s i n g the e x p e r i m e n t a l v a l u e s of (T^/f) i n Hg-He m i x t u r e T h e o r e t i c a l p l o t s of T^/f v s . $He when Eq.(h.2.1k) was n o r m a l i s e d w i t h the e x p e r i m e n t a l v a l u e s a t 32.2$ He, 77-0$ He and 100$ He Comparison of the e x t r a p o l a t e d v a l u e s of ( T . / p ) ^ w i t h the computed v a l u e s u s i n g Eq. (M-.2.1 k) 2 Comparison of the e x p e r i m e n t a l v a l u e s of [ k ( T ) / k (293)] H2~ H e w i t h the t h e o r e t i c a l v a l u e s u s i n g Eqs.(k.2.22) and (k.2.2h) Dependence of T^  on d e n s i t y a t 293°K f o r Hg-COg m i x t u r e Dependence of T^/p on d e n s i t y a t 293°K Dependence of T^/p on the pe r c e n t a g e of COg. i n Hg-COg m i x t u r e Dependence of l o g ( T ^ / p ) 1 Q Q ^ C O O N L O S T ° K f o r Hg-COg m i x t u r e ^ H —CO Temperature dependence of (k Q+Ck^) 2 2 as o b t a i n e d from Eqs.(h.2.1h) and (h.h.7.) u s i n g the e x p e r i m e n t a l v a l u e s of T^/p T h e o r e t i c a l p l o t s of T^/p v s . $C0g i n Hg-COg m i x t u r e when Eq.(h.2.1h) was n p r m a l i s e d w i t h the e x p e r i m e n t a l v a l u e s a t d i f f e r e n t c o n c e n t r a t i o n s of COg v i i i Comparison of \'k 1(T)/k 1 ( 2 9 3 ) ] H 2 " C 0 2 w i t h the computed v a l u e s u s i n g Eq . C + . 2 . l 8 ) Dependence of on d e n s i t y a t 293°K f o r CH^ Dependence of T^/n on d e n s i t y a t 293°K f o r CH^ Dependence of T 1 on d e n s i t y a t 720°K f o r CH^ Dependence of T^/p on d e n s i t y a t 720°K f o r CH^ Dependence of T^ /& on temperature f o r CH^ Log (T^ /y ) as a f u n c t i o n of logT°K f o r CH^ Comparison of [ l ( 6 , n ) T / I ( 6 , n ) 2 9 3 ] C I V C Hq- as o b t a i n e d from Eq.(5-1-10) w i t h the computed v a l u e s u s i n g the n u m e r i c a l v a l u e s of the i n t e g r a l s Dependence of on d e n s i t y f o r CH^-He m i x t u r e a t 293°K Dependence of /p on d e n s i t y f o r CH^-He m i x t u r e a t 293°K Dependence of on d e n s i t y f o r CH^-He m i x t u r e a t 730°K Dependence of T 1 /p on d e n s i t y f o r CH^-He m i x t u r e a t 730°K Dependence of T^/p on H e l i u m c o n c e n t r a t i o n i n CH^-He m i x t u r e Log (T^ /p ) 1 QO^He a s a f ' u r l c ' t i o n o f logT°K for CH^-He m i x t u r e Comparison of [ I (2 ,n) T / I (2 ,n) 2 ^ ° k \ C I V H e ^ o b t a i n e d from Eq.(5-2.1) w i t h the computed v a l u e s u s i n g the n u m e r i c a l v a l u e s of the i n t e g r a l s T r a n s m i t t e r Power s u p p l y 10Mc. c r y s t a l s t a n d a r d o s c i l l a t o r i x F i g u r e Page Ah. Wideband a m p l i f i e r 118 A5. 3 0 M c . t r i p l e r 119 A6. P u l s e sequencer 120 x ACKNOWLEDGEMENTS I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o P r o f . Myer Bloom f o r h i s guidance and c o n s t a n t encouragement th r o u g h o u t t h i s work. I would l i k e t o thank Dr. John D. Noble f o r h i s v a l u -a b l e h e l p i n the d e s i g n of the equipment and f o r many h e l p f u l d i s c u s s i o n s . I am i n d e b t e d t o Mr. Basanta S a r k a r who i n t r o d u c e d me to the computer programming and who was always a v a i l a b l e f o r a s s i s t a n c e i n w r i t i n g the programmes. My thanks a re due t o Mr. W i l l i a m M o r r i s o n , f o r con-s t r u c t i n g t he sample h o l d e r and f o r h i s ready h e l p , t o P e t e r Haas f o r d o i n g t h e diagrams i n t h i s t h e s i s , and t o Mr. John Lees f o r h i s a s s i s t a n c e i n t h i s work. The s c h o l a r s h i p p r o v i d e d by the Commonwealth S c h o l a r -s h i p Scheme i s g r a t e f u l l y acknowledged. I w i s h t o thank P r o f . Myer Bloom f o r the Res e a r c h A s s i s t a n t s h i p from h i s N.R.C. g r a n t . The f i n a n c i a l h e l p g i v e n by the P.E.O. S i s t e r h o o d i s v e r y much a p p r e c i a t e d . The R e s e a r c h was su p p o r t e d by the N a t i o n a l R e s e a r c h C o u n c i l of Canada. I w i s h t o thank the s t a f f o f the Computation Centre a t the U n i v e r s i t y o f B r i t i s h Columbia f o r t h e i r s e r v i c e s i n r u n n i n g the programmes. x i CHAPTER I INTRODUCTION The t e c h n i q u e s o f n u c l e a r magnetic resonance a r e w i d e l y t used t o s t u d y the p r o p e r t i e s of m a t t e r i n b u l k samples. The r a t e a t w h i c h the s p i n system appr'pfa'phes thermodynamic e q u a i l i b r i u m w i t h i t s s u r r o u n d i n g s , which i s c h a r a c t e r i s e d by the time con-s t a n t T| i n t h e l i t e r a t u r e , can be measured e x p e r i m e n t a l l y and can be i n t e r p r e t e d i n terms of i n t e r m o l e c u l a r f o r c e s t h a t cause the s p i n s t o r e l a x . The aim i n t h i s t h e s i s i s to e x t r a c t quan-t i t a t i v e i n f o r m a t i o n on the i n t e r m o l e c u l a r f o r c e s between d i f f e r -ent m o l e c u l e s from the s t u d y of T| i n gases. The p r i n c i p l e s of magnetic resonance w i l l be b r i e f l y d i s -c ussed here and a more complete d i s c u s s i o n can be found i n the l i t e r a t u r e The Zeeman energy of a n u c l e a r s p i n w i t h s p i n a n g u l a r momentum K\I and a magnetic moment /A - y £ I when p l a c e d i n an e x t e r n a l magnetic f i e l d H Q i s g i v e n by -> H = -/» • H0 ( 1 . 1 . ) where H Q i s t a k e n to be i n the z - d i r e c t i o n . The energy l e v e l s . o f such a system a r e g i v e n by Em - - Y * H0 mx ( 1 . 2 . ) where rn T - T, I -1, • • • > ' ^  I f t h e r e a re N ' s p i n s weakly i n t e r a c t i n g w i t h each o t h e r and i n t h e r m a l e q u i l i b r i u m w i t h the " l a t t i c e " t he f r a c t i o n a l p o p u l a t i o n s of t h e s e energy l e v e l s a r e g i v e n by 2 w h e i N e T i s the " l a t t i c e " t e m p e r a t u r e . " L a t t i c e " here i ' r e f e r s to the o t h e r degrees of freedom., of the sample i n which the s p i n s a l l o c a t e d . The n e t m a g n e t i s a t i o n of the sample i s t h e n g i v e n hy where the h i g h t e m p e r a t u r e a p p r o x i m a t i o n ^-Ltk « I has k T been made. N e g l e c t i n g r e l a x a t i o n e f f e c t s , the e q u a t i o n o f mo t i o n of t h e m a g n e t i s a t i o n M i s g i v e n by , V M x H (,1.5.) where H i s the t o t a l magnetic f i e l d a p p l i e d t o the sample. T r a n s f o r m i n g e q u a t i o n ( 1 . 5 0 t o a r e f e r e n c e frame r o t a t i n g w i t h an a n g u l a r v e l o c i t y ^ the e q u a t i o n of mo t i o n f o r M can be w r i t -t e n as v M K \ H + H J (1-6.) I f H = H0t , -- 0 uUe* To = - V H 6 t . U. t) c M i s f i x e d i n the r o t a t i n g r e f e r e n c e frame. I n the l a b o r a t o r y -+ r e f e r e n c e frame the m a g n e t i s a t i o n M p r e c e s s e s about H 0 a t a f r e q u e n c y W ; . v H e , The a n g u l a r f r e q u e n c y Y H0 i s c a l l e d "Larmor f r e q u e n c y " . The e f f e c t of an a d d i t i o n a l a l t e r n a t i n g f i e l d ^ 4 , ( 6 ) =. ZH,Cct> tot . p e r p e n d i c u l a r t o R"0 can be a n a l y s e d by b r e a k i n g the a l t e r n a t i n g f i e l d i n t o two r o t a t i n g components of a m p l i t u d e ( } one r o t a t i n g c l o c k w i s e and the o t h e r a n t i - c l o c k w i s e . S i n c e one of the two components r o t a t e s i n the same sense as the p r e c e s s i o n of the magnetic moment ;arii' the o t h e r i n t h e o p p o s i t e d i r e c t i o n , i t carybe shown t h a t the c o u n t e r r o t a t i n g component c a n be n e g l e c t e d a t res o n a n c e , T h e r e f o r e , the t o t a l magnetic f i e l d can be w r i t t e n t o a good a p p r o x i m a t i o n as H = Ha + H, 6'-" • . (1.7) where H , CO - H , [? Coi w £ + j s ^ o t l I f the x - a x i s of the r o t a t i n g r e f e r e n c e frame i s chosen t o be a l o n g H, , t h e n H ( i s f i x e d in, t h i s frame of r e f e r e n c e and the e q u a t i o n of m o t i o n of M i s g i v e n by -> * Y J (1.8) - y N A H e ^ . where u . ^ /„ co \ ? u (1.9) and t i s a u n i t v e c t o r i n the x - d i r e c t i o n i n the r o t a t i n g r e f e r e n c e fvrame. From e q u a t i o n (1.9) i t i s ob v i o u s t h a t "M p r e c e s --» ses about /-L, w i t h an a n g u l a r v e l o c i t y y H e„ . The/^pffect of y, on Hon i s predominant when GJ«-Y^ i n which case the magnetic moment p r e c e s s e s about H, w i t h an a n g u l a r v e l o c i t y Y H, • T l i i s phenomenon i s c a l l e d N u c l e a r M a g n e t i c Resonance (N.M.R.). I f H, were a'pplied o n l y f o r a d u r a t i o n o f time tu , the n the moment would p r e c e s s t h r o u g h an a n g l e Q g i v e n by 6 Y H , t u (1.10) I f t w and H, were chosen such t h a t 6 = * the p u l s e would i n v e r t the m a g n e t i s a t i o n and i s r e f e r r e d t o a a r a "l80° p u l s e " . S i m i l a r l y , i f 6 = 90° (90° p u l s e ) , the m a g n e t i s a t i o n would r o t a t e t h r o u g h 90° and i t would p r e c e s s about H Q i n the x-y p l a n e a f t e r the p u l s e . I n p r a c t i c e the sample i s p l a c e d i n a c o i l which i s p l a -ced p e r p e n d i c u l a r ' t o the m a g n e t i c - f i e l d H . An a l t e r n a t i n g field i s produced p e r p e n d i c u l a r t o .H0 by p a s s i n g r . f . c u r r e n t t h r o u g h the c o i l . The m a g n e t i s a t i o n p r e c e s s i n g i n the x-y..plane a f t e r the. a p p l i c a t i o n of a 90° p u l s e i n d u c e s an e.m.f. i n the c o i l which can be d e t e c t e d and observed and t h i s i s n o r m a l l y r e f e r r e d t o as the f r e e i n d u c t i o n decay. The p r e c e d i n g paragraphs i n d i c a t e a method of p r e p a r i n g a n o n - e q u i l i b r i u m sample. The approach t o e q u i l i b r i u m of such a 2 system can be d e s c r i b e d by the phenomenological BXochh e q u a t i o n s as g i v e n below dS = V M x " r t - M v i - M » t - + Mo - ivy, t ( l . l l ) d t T 2 T 2 - T i " where i , j and k are u n i t v e c t o r s i n the x, y and z d i r e c -t i o n s , T-L and T 2 are l o n g i t u d i n a l and t r a n s v e r s e r e l a x a t i o n t i m e s , M 0 i s the e q u i l i b r i u m m a g n e t i s a t i o n i n the s t a t i c f i e l d H 0 and H i s g i v e n by H = H 0k + 2 H 1 c o s ^ t i where H i « H 0 (1.12) The s o l u t i o n s - t o the e q u a t i o n ( l . l l ) can be w r i t t e n a.s M x ( t ) M x y ( 0 ) j^cosOt +^ 3e"t/T2 M y ( t ) = M x y ( 0 ) ^ s i n O t +^ ]e"t/T2 M z (0) - M 0 ] e'">/Ti M z ( t ) • = M 0 + ' 2- 2. where M.Xy = . 1 M x + My M x, My and Mz- can be mon i t o r e d e x p e r i m e n t a l l y and hence T i and T 2 can be o b t a i n e d . S i n c e T i i s a measure of the r a t e at which the s p i n system exchanges energy w i t h the l a t t i c e , i t i s p o s s i b l e t o r e l a t e T l t o the m o l e c u l a r p r o p e r t i e s of.vth-e gas.- The s p i n s r e l a x as a r e s u l t of f l u c t u a t i o n s i n the magnetic f i e l d s at the s i t e s of the n u c l e i . In d i a t o m i c and. p o l y a t o m i c gases the c o n t r i b u t i o n s t o t h e s e l o c a l 5 f i e l d s come from the i n t e r a c t i o n of the s p i n . w i t h . t h e r o t a t i o n a l a n g u l a r momentum of the m o l e c u l e and from.the d i p o l e moments of the o t h e r n u c l e i i n the same m o l e c u l e . The f l u c t u a t i o n s i n the s 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 s are produced by tho s e c o l l i s i o n s which r e o r i e n t the m o l e c u l e . Thus the study of s p i n - l a t t i c e r e l a x a t i o n can g i v e i n f o r m a t i o n on the i n t e r m o l e c u l a r i n t e r a c t i o n s which p r o -duce m o l e c u l a r r e o r i e n t a t i o n s . The mechanism by which the spins' r e l a x was f i r s t proposed 3 • ' 4' by Schwinger and then extended by Needier and Op.echowski ' and 5 Johnson and Waugh . They have o b t a i n e d e x p r e s s i o n s r e l a t i n g the s p i n l a t t i c e r e l a x a t i o n ' t ime t o the c o r r e l a t i o n t i m e s of the i n t r a -m o l e c u l a r i n t e r a c t i o n s . Bloom and. Oppenheim ^ have t r e a t e d the dynamics of the s y s -tem w i t h the "Constant A c c e l e r a t i o n A p p r o x i m a t i o n ! ' (C.A.A.) and obtained, e x p r e s s i o n s f o r the c o r r e l a t i o n t i m e s of the i n t r a m o l e c u -l a r i n t e r a c t i o n s i n terms.of the i n t e r m o l e . c u l a r I n t e r a c t i o n s . They have a l s o assumed t h a t t h e t r a n s i t i o n s between d i f f e r e n t J s t a t e s are n e g l i g i b l e compared w i t h the t r a n s i t i o n s between d i f f e r -ent Mj s t a t e s of the same J - m a n i f o l d . T h i s theory,was-used i n i n t e r p r e t i n g the d a t a below room t e m p e r a t u r e i n t h e i r subsequent 8 ! paper.' , ! 9 R e c e n t l y Bloom and. Oppenheim have extended t h e i r t h e o r y t o i n c l u d e the t r a n s i t i o n s - between d i f f e r e n t J s t a t e s which i s more s u i t a b l e t o i n t e r p r e t the r e s u l t s a t h i g h e r t e m p e r a t u r e s . T h i s t h e o r y has been used i n i n t e r p r e t i n g the d a t a on H2 i n t h i s - t h e s i s and- is- p r e s e n t e d i n d e t a i l i n Chapter I I I . The apparatus'.and e x p e r i m e n t a l t e c h n i q u e s t h a t were used i n o b t a i n i n g the d a t a are p r e s e n t e d i n Chapter I I . Chapter IV p r e -s.ents the r e s u l t s i n H2 and. H2 - He,. H2 ~ CO2 m i x t u r e s and t h e i r i n t e r p r e t a t i o n u s i n g the t h e o r y p r e s e n t e d .in Chapter I I I . Chapter V c o n t a i n s the r e s u l t s and i n t e r p r e t a t i o n of CH4 and. CH4 - He m i x t u r e s , In Chapter VI the r e s u l t s are, summarised and some s u g g e s t i o n s are made f o r f u r t h e r work.. The 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 e l e c t r o n i c s are p r e s e n t e d i n Appendix A. CHAPTER I T APPARATUS.AND ^EXPERIMENTAL TECHNIQUE 2. 1. Measurement of S p i n - l a t t i c e R e l a x a t i o n Time.-The measurement, of the s p i n - l a t t i c e , r e l a x a t i o n time. Tj_ w i l l be d e s c r i b e d v e r y b r i e f l y here s i n c e the t e c h n i q u e i s w e l l e s t a b -l i s h e d and i s a v a i l a b l e i n the- l i t e r a t u r e . 1 'IOJ-H' When a sample i s p l a c e d i n an e x t e r n a l magnetic f i e l d H 0, the n u c l e a r moments w i l l have a net m a g n e t i s a t i o n a l i g n e d w i t h the f i e l d under e q u i l i b r i u m c o n d i t i o n s . In o r d e r t o measure T l , ..-th"'e ' m a g n e t i s a t i o n o f . t h e s p i n system has t o be d i s t u r b e d from i t s e q u i l i b r i u m p o s i t i o n . From the d i s c u s s i o n i n Chapter I i t i s e v i d e n t t h a t the m a g n e t i s a t i o n can be r o t a t e d through any d e s i r e d a n g l e 6 from i t s e q u i l i b r i u m p o s i t i o n by the a p p l i c a t i o n of a r . f . f i e l d H]_ at the Larmor f r e q u e n c y p e r p e n d i c u l a r t o the s t a t i c , f i e l d H 0. The r . f . f i e l d must.be s u f f i c i e n t l y i n t e n s e t o r o t a t e the s p i n system through the d e s i r e d a n g l e i n a time much s h o r t e r than or T2 so t h a t the r e l a x a t i o n e f f e c t s , c a n be n e g l e c t e d d u r i n g t h i s time.. R e f e r r i n g t o a c o o r d i n a t e system whose z - a x i s i s cho-- 4 sen a l o n g H 0, a 90° p u l s e r o t a t e s the m a g n e t i s a t i o n t o the x-y plane and when the p u l s e i s removed the m a g n e t i s a t i o n p r e c e s s e s about H 0 i n the x-y p l a n e , thus i n d u c i n g an e.m.f. i n the sample c o i l . Due t o the inhomogeneity- of the magnetic f i e l d , a l l the. s p i n s do not p r e c e s s w i t h the same Larmor f r e q u e n c y . As a r e s u l t the s p i n s dephase among themselves- and- the induced s i g n a l decays i n a time - of the order, of (v&H) where AH i s the measure of inhomogeneity pf the f i e l d a c r o s s the sample. I f a 180°. p u l s e i s 8 a p p l i e d t o the system a f t e r a time X from the 90° p u l s e , the s p i n s " f a l l i n phase, a g a i n and the s i g n a l b u i l d s up t o . i t . s . maximum at a time 2Z from the 90° p u l s e and decays a g a i n f o r the same r e a s o n s . T h i s second s i g n a l i s c a l l e d " s p i n echo". In t he exp e r i m e n t s t o be d e s c r i b e d i n . t h i s t h e s i s "M was i n v e r t e d by a 180° p u l s e and the r e c o v e r y of M z was moni t o r e d from - M Q t o + M Q by a subsequent 90° - 180° p u l s e sequence. I f the time between 90° and the second 180© p u l s e s i s kept c o n s t a n t t h r o u g h o u t the exp e r i m e n t , the magnitude of the echo i s p r o p o r -t i o n a l . toi> the v a l u e of M z p r e c e d i n g the 90° pulse.. The measurements were made w i t h a 30 Mc coh e r e n t p u l s e d s p e c t r o m e t e r u s i n g phase 4, s e n s i t i v e d e t e c t i o n . The Box-Car 12 - • i n t e g r a t o r was used t o improve the s i g n a l t o n o i s e . . r a t i o . The p u l s e sequence was repeated'.-after every time i n t e r v a l T ^ 10Ti so t h a t the s p i n s r e l a x i n between the sequence. D u r i n g a measurement the time between the f i r s t 180° p u l s e and the 90° p u l s e was swept v e r y s l o w l y t o s e v e r a l t i m e s Tj_ keeping the time between the 90° .pUlse and. the 180° p u l s e c o n s t a n t . The box-car was t r i g g e r e d by the 90° p u l s e and the sampl i n g gate was a d j u s t e d t o sample the echo formed by the 90° and the..jsecond 180° p u l s e s . The. magnitude of the echo and hence the r e c o v e r y of M z was p l o t t e d on a. V a r i a n C h a r t r e c o r d e r . The time, t between the 1 f i r s t 1800 p u l s e and the 90° p u l s e was measured w i t h a H e w l e t t - P a c k a r d 524-C e l e c t r o n i c c o u n t e r and the., time, was p r i n t e d out a u t o m a t i c a l l y by a H e w l e t t - P a c k a r d 526-B d i g i t a l r e c o r d e r . An. event, marker on the.._Varian d h a r t r e c o r d e r made a mark whenever the time was p r i n t e d out by. the d i g i t a l r e c o r d e r and proper t i m e s t o thos e marks w e r e , a s s i g n e d l a t e r . was o b t a i n e d by. making t ^ l 0 T - j _ . I f A ( t ) i s the h e i g h t 9 of the s i g n a l at time t and. a t t^©9, the n i t can be seen t h a t A ( t ) = Am ( i - e _ t / T i ) The s l o p e of the s t r a i g h t l i n e l o g ( A w - A(0.). Vs. t g i v e s the v a l u e of T p 2. 2. N.M.R. Sp e c t r o m e t e r , 2.2.1. G e n e r a l Remarks. B The s p e c t r o m e t e r , o r i g i n a l l y b u i l t by John- -TP, Noble, was m o d i f i e d t o d e t e c t weak'N.M.R.• s i g n a l s . A coherent 30M C t r a n s m i t -t e r was b u i l t t o use phase s e n s i t i v e d e t e c t i o n and the t i m i n g c i r c u i t was m o d i f i e d t o make use. of the box-ca r i n t e g r a t o r . The 14 box-oar i n t e g r a t o r was c o n s t r u c t e d by Walter- N. Hardy based on 12 the d e s i g n of R.J. Blume and a d e t a i l e d a n a l y s i s - was report.ed i n h i s Ph.D. t h e s i s . A b l o c k diagram of the s p e c t r o m e t e r i s g i v e n i n f i g . ( l ) .2.2.2. T i m i n g C i r c u i t . . The t i m i n g c i r c u i t c o n s i s t e d of a c o l l e c t i o n of T e k t r o n i x wave-form and p u l s e g e n e r a t o r u n i t s . A Tek. 162 wave-form genera-t o r was used i n r e c u r r e n t mode t o p r o v i d e the f i r s t 180° p u l s e . The. sawtooth from the same -'generator . s t a r t e d r u n n i n g down at the same time and was f e d t o a m o d i f i e d Tek. 163 p u l s e g e n e r a t o r whose t r i g g e r i n g l e v e l was s e t by the sawtooth of a m o d i f i e d " u l t r a - s l o w " Tek 162 wave-form generator» The p u l s e f r o m the m o d i f i e d Tek. 163 p u l s e g e n e r a t o r was t a k e n t o a p u l s e sequencer- t o produce two p u l -ses whose s e p a r a t i o n was c o n t r o l l e d by'>the p u l s e width, o.f the i n p u t p u l s e . T h e . f i r s t 180° p u l s e and the 'secpHd p u l s e from the p u l s e sequencer were'added.together i n a mixer c i r c u i t . These p u l s e s weiE t a k e n t o a 180° p u l s e - w i d t h g e n e r a t o r whereas the f i r s t pulse--from H td t-1 o o W I—1 > o o o -id a C Q C Q o 1-3 O S K (-3 ~ 11 t h e p u l s e sequencer was t a k e n t o a 90° p u l s e - w i d t h g e n e r a t o r . These two p u l s e - w i d t h g e n e r a t o r s were i d e n t i c a l and were de s i g n e d 13 u s i n g p h a n t a s t r d h c i r c u i t s . The pulse-widths.,.could be v a r i e d from a microsecond t o about 100 m i c r o s e c o n d s . The p u l s e s were then mixed and. were used t o gate the t r a n s m i t t e r . The 900 p u l s e from the p u l s e sequencer was a l s o u s e d * t o : t r i g g e r the Box-car. 2.2.3. T r a n s m i t t e r . T h e . t r a n s m i t t e r was d e s i g n e d t o make use of p h a s e - s e n s i t i v e d e t e c t i o n s i n c e i t was known .•from .the beginning,'that t h e . s i g n a l s would be s m a l l . The p h a s e - S e n s i t i v e d e t e c t i o n r e q u i r e s a coherent p u l s e system and a r e f e r e n c e v o l t a g e at 30Mq. A t r a n s i s t o r i s e d 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 was b u i l t t o supply the r . f . at lOMc and•was s h i e l d e d by. a copper can. The output from t h i s a m p l i f i e r was t a k e n o u t . t h r o u g h la.... s h i e l d e d c a b l e t o p r o v i d e the r e f e r e n c e " voJLtage. The tri>ipler as w e l l as the 30Mc a m p l i f i e r were t r a n s i s -t p r i s e d which made'it possible....to b u i l d them I n s i d e the copper,, lean. A t r a n s i s t o r i s e d power su p p l y r e g u l a t e d by Zener d i o d e s was a l s o b u i l t i n s i d e the" can. Thus o n l y one. l e a d a t 6.3V a.c. was ta k e n i n s i d e the can t o . s u p p l y the. .power t r a n s f o r m e r . T h i s h e l p e d i n keep i n g the r i f . . . l e a k to., a., minimum when the r . f . gate was cl'o.sed. The r . f . from t h e gate was t a k e n t o a c o n v e n t i o n a l phase-15-s h i f . t i n g c i r c u i t and. a t r i p l e r . The t r i p l e r was a p u s h - p u l l a m p l i f i e r at 30Mc. The output.;, f r o m the t r i p l e r was t a k e n t o a gated power a m p l i f i e r s t a g e . t u n e d t o 30Mc. The power a m p l i f i e r produced r e a s o n a b l y r e c t a n g u l a r r . f . p u l s e s of 1200 v o l t s peak t o peak. With t h i s power output the p u l s e w i d t h s needed t o produce 90° and 180® p u l s e s were about 6 m i c r o s e c o n d s and 13 m i c r o s e c o n d s r e s p e c t i v e l y . . .The output power from t h e t r a n s m i t t e r 1 was t a k e n .to 12 t h e sample c o i l tuned t o 30Mc th r o u g h a s m a l l c a p a c i t o r of about 3 p f . 2.2.4. Sample C o i l The sample c o i l was made of a p p r o x i m a t e l y t w e l v e . t u r n s of 22 S.W.G. w i r e w i t h a s p a c i n g i n between the t u r n s e q u a l to. the di a m e t e r of the w i r e and was about 1*4" l o n g . S i n c e the sample c o i l was immersed i n the--sample ( f i g . . 2) the jenamel came o f f the. w i r e and i n t r o d u c e d i m p u r i t y i n the sample when the temperatur-e .of the sample holder-was r a i s e d above room temperatures T h i s r e s u l t e d i n a very' sharp i n c r e a s e i n Tj_ becaus.e.-. of- t h e c o l l i s i o n s of H2 m o l e c u l e s w i t h h e a v i e r i m p u r i t y m o l e c u l e s . T h e r e f o r e , the i n s u l a t i o n on the w i r e was c o m p l e t e l y s t r i p p e d o f f and the..wi;re'r was t h o r o u g h l y c l e a n e d b e f o r e use. Also,, thev c o i l -was pre-heated t o the h i g h e s t t:emperaturs a l o n g w i t h the samp l e - h o l d e r and the gases were pumped1 out. When the s e p r e c a u t i o n s were t a k e n no t r a c e o f . i m p u r i t y was n o t i c e d and the... r e s u l t s were r e p r o d u c i b l e t o w i t h i n A. t h i n - w i l l e d g l a s s tube was f i t t e d t i g h t l y i n s i d e the p r e s s u r e v e s s e l t o i n s u l a t e the c o i l from the w a l l of the v e s s e l . In o r d e r t o get an i d e a l 180^ p u l s e i t i s n e c e s s a r y t o have a l l the sample i n s i d e the c o i l . So ' the c o i l was- wound t o f i t almost e x a c t l y i n s i d e the g l a s s tube so' t h a t most of the sample was i n s i d e the c o i l . The r . f . l e a d was j u s t the e x t e n s i o n of one end. of the c o i l and was t a k e n t h r o u g h a s t a i n l e s s s t e e l tube t o p l u g A i n f i g . 2 where i t was s e a l e d a g a i n s t p r e s s u r e w i t h t e f l o n washers at room t e m p e r a t u r e . The o t h e r end of the c o i l was f a s t e n e d t o the s t a i n -l e s s , s t e e l p l u g by means of a screw. G l a s s was used f o r i n s u l a t i o n wherever i t was n e c e s s a r y . The r . f . l e a d was connected t o a t u n i n g c a p a c i t o r t h r o u g h a 12 p . f . c a p a c i t o r . 2.2.5. R e c e i v e r . The r e c e i v e r was a L.E.L. a m p l i f i e r model 1.F.21.B.S. This i s a wide band a m p l i f i e r w i t h a maximum g a i n of 100 db and a band w i d t h of 2Mc. c e n t e r e d around. 30Mc. A 1N295 diode d e t e c t o r was s u p p l i e d w i t h t h i s and was used i n the measurements. O r d i n a r y d i o d e d e t e c t i o n i s not l i n e a r over the e n t i r e r e g i o n and the non-l i n e a r i t y becomes v e r y i m p o r t a n t w h i l e d e a l i n g w i t h s m a l l s i g n a l s . " Phase-Goher-'e:Mitt> d e t e c t i o n was used which a l l o w e d operati.on i n t h e l i n e a r r e g i o n of the d i o d e . The 30Mc r e f e r e n c e v o l t a g e , was added io the s i g n a l a t a stage i n the r e c e i v e r where the s t a g g e r tuned t r i p ' - " l e t s t a g e s were coupled t o g e t h e r . The di o d e d e t e c t s the "sum" of the r e f e r e n c e v o l t a g e and the s i g n a l . The s i g n a l was kept below about 1/10 of t h e r e f e r e n c e v o l t a g e t o a v o i d d i s t o r t i o n . 2.3. H i g h - P r e s s u r e System. The h i g h - p r e s s u r e a p p a r a t u s used i n the ex p e r i m e n t s i s shown i n f i g . 2. I t c o n s i s t s of a t h i c k w a l l e d Be-Cu v e s s e l (O.D. 17/16" and I.D. 3/8") and a s t a i n l e s s s t e e l p l u g s e a l e d i n t h e v e s s e l by 16 17 means of Bridggmann's arrangement of p a c k i n g washers. ' • ' The c e n t r a l washer was annealed copper i n s t e a d of l e a d . Copper, when ann e a l e d , i s q u i t e s o f t and hence makes a good s e a l . A l s o , s i n c e the t h e r m a l e x p a n s i o n of copper i s the same as t h a t of Be-Cu, the •se a l , when made at room t e m p e r a t u r e , m a i n t a i n s i t s e l f a t h i g h e r t e m p e r a t u r e s . The o t h e r two washers were made of Be-Cu i n s t e a d of Ev e r d u r ^ . The o u t s i d e edge of t h e washer D]_ was t a p e r e d t o the i n s i d e t o f i t on t o t h e t a p e r of t h e s t a i n l e s s s t e e l p l u g . T h i s p r o v i d e d more area of c o n t a c t between the s t a i n l e s s s t e e l p l u g and the Be-Cu washer and hence the p r e s s u r e was a p p l i e d u n i f o r m l y on R . F . L E A D WITH riC, 2. SAMP!,.(". HOLDCR 15 the c e n t r a l washer when J:'h.e s t a i n l e s s s t e e l p l u g was t i g h t e n e d t o make..the s e a l . The r . f . le ;ad was s e a l e d a g a i n s t p r e s s u r e at room tempera-t u r e i n p l u g A w i t h t e f l o n washers and t h e n brought i n t o , t h e h i g h p r e s s u r e r e g i o n t h r o u g h a l4" s t a i n l e s s i s t e e l tube.. The. s t a i n l e s s s t e e l t u b e ' w a s , p r e s s u r e s e a l e d i n s i d e , the p l u g U s i n g t h e same t e c h n i q u e as t h a t of s t a n d a r d h i g h p r e s s u r e equipment and was q u i t e s a t i s f a c t o r y t h r o u g h o u t ;the temperature range,, The schematic of h i g h p r e s s u r e system i s g i v e n i n f i g . 3.; A l l .the h i g h p r e s s u r e c o n n e c t o r s were, purchased from A u t o c l a v e j Eng.. I n c . and were de s i g n e d t o w i t h s t a n d p r e s s u r e s up t o 30,000 p s i . P.S I. 2.4. H e a t e r . The h e a t e r was wound on the h i g h - p r e s s u r e v e s s e l using, a t h i n mica sheet t o i n s u l a t e i t from the v e s s e l . To a v o i d m u l t i -l a y e r w i n d i n g , a commercial 600 w a t t s s p i r a l element was used i n s t e a d of a s t r a i g h t nichrome w i r e . I t was wound n o n - i n d u c t i v e l y w i t h a spacing, i n between the w i n d i n g s a p p r o x i m a t e l y e q u a l to the d i a m e t e r of the element. '"A l i q u i d p o r c e l a i n c a l l e d " S a u x e i s e n " was used t o keep the w i n d i n g s i n place.. The h i g h - p r e s s u r e v e s s e l a l o n g w i i h the h e a t e r was kept i n s i d e a vacuum j a c k e t t o keep the r a d i a t i o n and c o n v e c t i o n c u r r e n t l o s s e s t o a minimum. The l e a d s from the h e a t e r were t a k e n o u t s i d e the vacuum j a c k e t t h r o u g h Kovar s e a l s . The h e a t e r c u r r e n t was r e g u l a t e d by a v a r i a c . About 150 w a t t s of power, was s u p p l i e d t o m a i n t a i n the sample h o l d e r at 750° K. The t e m p e r a t u r e was measured w i t h a c h r o m e l - a l l u m e l thermo-» TO PLUG A TO GAS TANK TO GAS CYLINDER FIG. 3- HIGH PRESSURE SYSTEM c o u p l e . The " H o t - J u n c t i o n ' 1 of the thermo-couple was p l a c e d i n the h o l e p r o v i d e d f o r i t i n the v e s s e l . The l e a d s were t a k e n o u t s i d e the vacuum j a c k e t through KoVar s e a l s . The thermal E.M.F. was;Measured w i t h a "Honey-Well" p o t e n t i o m e t e r u s i n g a n u l l d e t e c t o r . No s p e c i a l e f f o r t was made t o c o n t r o l the temp-e r a t u r e a u t o m a t i c a l l y because or}oe'' the c u r r e n t was a d j u s t e d the t e m p e r a t u r e was s t e a d y to w i t h i n a degree a f t e r r e a c h i n g the e q u i l i b r i u m . Whenever the temperature was changed a t l e a s t 8 or 10 hours was a l l o w e d b e f o r e . t a k i n g the measurements t o make sure t h a t the sample was a t e q u i l i b r i u m w i t h the v e s s e l . The t e m p e r a t u r e measurements were a c c u r a t e to w i t h i n - 0.5%- • 2 . 5 - M i x i n g of Gases; D e t e r m i n a t i o n of C o n c e n t r a t i o n . The gases t h a t were used i n the e x p e r i m e n t s were of r e s e a r c h grade and were o b t a i n e d from Matheson Company. The m i x t u r e of two gases was p r e p a r e d i n a t h i r d c y l i n d e r by l e t -t i n g i n one component a t lower p r e s s u r e f i r s t and t h e n the o t h e r component. The t o t a l p r e s s u r e was, noted b e f o r e and a f t e r the i n t r o d u c t i o n . • o f the second component. Knowing the p a r t i a l p r e s s u r e s of the i n d i v i d u a l components and the t e m p e r a t u r e , the d e n s i t y o f each .component gas i n i d e a l amagat u n i t s can be 18 found from the e q u a t i o n of s t a t e d a t a and hence the composi-t i o n of the m i x t u r e can be d e t e r m i n e d . I d e a l Amagat i s the  number d e n s i t y of, the gas a t N.T.P. The c a l c u l a t e d v a l u e of the c o n c e n t r a t i o n can be v e r i -f i e d e x p e r i m e n t a l l y by comparing the. h e i g h t of the echo of the m i x t u r e w i t h t h a t of pure gas c o n t a i n i n g p r o t o n s under i n d e n t i -c a l c o n d i t i o n s . I n the case of hydrogen and i t s m i x t u r e s the h e i g h t of the echo has t o be c o r r e c t e d f o r the e f f e c t s of T2 and d i f f u s i o n . The h e i g h t of the echo A(2, T ) , where f i s the time between 90° and 180° p u l s e s , i s g i v e n by A (zz) -- A f o ) ex ip \_-^/TZ - k ( z r ( ] (2.5.1.) where k = 1 Y'G^'D • 12 f • D «? '/./> andJ° i s the d e n s i t y of gas i n amagat u n i t s , where an "amagat" i s Z . &3 * ID9 ynate6u,les / C m 3 N e g l e c t i n g t h e d i f f u s i o n e f f e c t s A(zr) - A(b) ex|> zz j r x ^ (2.5.2.) I f the number of s p i n s per m o l e c u l e i s denoted by n^-, t h e n ^ . A t e ) o c M ( o ) wAere M <V> -- / y ) j Y * Jfr*C) (2.5.3.) 3 K T o r A fa) z C f where C i s a c o n s t a n t (2.5.H-.) E q u a t i o n (2.5«2.) becomes A(zz)= c j> exf> C - ^ ^ / T L ) (2.5.5.) S i n c e i n the h i g h d e n s i t y r e g i o n T)Jp z jf c f = Afer) e*j> (zr/r,) (Cf) .... ( f c ? ^ (2.5.6.) T h e r e f o r e v 1-^ / m i x t u r e gives- the f r a c t i o n a l c o m p o s i t i o n of the Y-m i x t u r e . v •- " J^z. I n the case of CH^ ahd i t s : m i x t u r e s the c o r r e c t i o n i s v e r y s m a l l s i n c e T-^  i s much l o n g e r t h a n 2*Zi which i s a p p r o x i -m a t e l y a m i l l i - s e c o n d i n a l l the measurements. For an i d e a l gas the d e n s i t y i n amagats of the m i x t u r e a-t-.any i n t e r m e d i a t e p r e s s u r e P i s g i v e n by 1 9 f - P ( i n atmospheres) ( 2 . 5 - 7 0 The CO2 - H2 m i x t u r e p r e s e n t s a d i f f e r e n t problem. The dependence of (Cf )yfj^j^JJ£ ° n t o t a l p r e s s u r e i s l i n e a r a t h i g h t e m p e r a t u r e s , but not a t room temperature and t h i s e f f e c t i s more prominent a t h i g h e r c o n c e n t r a t i o n s of COg. T h i s i s due to the f a c t t h a t room temperature i s not h i g h enough t o d e s c r i b e Ike the e q u a t i o n of s t a t e of COg b y ^ i d e a l gas law. From the equa-t i o n of s t a t e d a t a f o r CO2 i t can be seen t h a t the d e n s i t y has i n c r e a s e d from 9 - 5 to ^ 7 - 5 amagats when the p r e s s u r e has i n -c r e a s e d , from 10 atmospheres t o ^0 atmospheres a t room tempera-t u r e i . e . the d e n s i t y a,t -^0 atmospheres i s about 25% h i g h e r t h a n what the i d e a l gas: law p r e d i c t s . S i m i l a r study a t ^fOOOK shows -that the d e n s i t y a t ^0 atmospheres i s about 6% h i g h e r t h a n the d e n s i t y o b t a i n e d by i d e a l gaa.'.law and a t h i g h e r t e m p e r a t u r e s the i d e a l gas e q u a t i o n i s found .to be a good a p p r o x i m a t i o n to w i t h i n , 2%. As a r e s u l t of t h i s , comparison of s i g n a l s t r e n g t h s a t room temperature .doe's not g i v e t h e f r a c -t i o n a l c o m p o s i t i o n of the m i x t u r e nor- c o u l d i t be d e t e r m i n e d by n o t i n g the p r e s s u r e s b e f o r e and a f t e r the i n t r o d u c t i o n of the second component when the p r e s s u r e s i n v o l v e d a r e h i g h ( i . e ^ > 1 0 atm.). Hence a d i f f e r e n t p r o c e d u r e was adopted t o d e t e r -mine th e f r a c t i o n a l c o m p o s i t i o n as w e l l as the d e n s i t y a t any o t h e r p r e s s u r e w h i c h i s d e s c r i b e d below. JLet the -pressure of CO2 i n t r o d u c e d i n t o the c y l i n d e r be ?2_ and l e t H g b e added to t h i s t o b r i n g the t o t a l p r e s s u r e t o P. The gases were l e f t t o g e t h e r f o r about 2h hours t o mix ' ^ y - . S i f ^ s i i 6 r i - . i - - : ' ' ; ' " a y i f ' i l • The s i g n a l s t r e n g t h was c a l i b r a t e d i n u n i t s of p r o t o n • d e n s i t y by n o t i n g the s i g n a l s t r e n g t h s a t v a r i o u s p r e s s u r e s f o r Hg. The sample h o l d e r was evacuated and the m i x t u r e was l e t i n t o the sample h o l d e r a t p r e s s u r e P. Keeping the g a i n of the s p e a t f o m e t e r the same, the s i g n a l s t r e n g t h of the m i x t u r e was noted a t p r e s s u r e P as w e l l " a s a t s e v e r a l o t h e r p r e s s u r e s between 0 and P. The s t r e n g t h of thfe s i g n a l a t p r e s s u r e P g i v e s the d e n s i t y of hydrogen i n amaga,tte. The d e n s i t y of COg was o b t a i n e d from the e q u a t i o n of s t a t e d a t a f o r COg knowing P-^  and te m p e r a t u r e . I f i s the d e n s i t y o f COg and 9^ i s the d e n s i t y of Hg, / ( fc +- f ^ g i v e s the f r a c t i o n a l compo-s i t i o n of the m i x t u r e . The d e n s i t y of the m i x t u r e a t any i n t e r -m ediate p r e s s u r e P was o b t a i n e d from the s i g n a l s t r e n g t h S . of the m i x t u r e a t p r e s s u r e P". S" was c o n v e r t e d t o the d e n s i t y of Hg, , and s i n c e the r a t i o jjL0 / i s a c o n s t a n t K, j°Co z. ^  fH . The d e n s i t y of the m i x t u r e a t p r e s s u r e P^  was o b t a i n e d as f P + P ^ F i g . h shows the d e n s i t y of the m i x t u r e as a f u n c t i o n of c o m p o s i t i o n a t d i f f e r e n t p r e s s u r e s . At h i g h e r c o n c e n t r a t i o n s of COg the d e n s i t y i s p r o p o r t i o n a l t o the pe r c e n t a g e of COg but a t l o w e r c o n c e n t r a t i o n s i t i s n o n - l i n e a r . The v a l u e s e x t r a -p o l a t e d t o 1 0 0 $ COg a r e i n agreement w i t h the v a l u e s o b t a i n e d f r om the e q u a t i o n of s t a t e d a t a f o r COg w i t h i n e x p e r i m e n t a l e r r o r s . The d e n s i t y of the m i x t u r e was. o b t a i n e d e x p e r i m e n t a l l y as a f u n c t i o n of p r e s s u r e a t o t h e r t e m p e r a t u r e s of i n t e r e s t a l s o . At 3 5 0 ° K i t was found t h a t the d e n s i t y o b t a i n e d from the i d e a l gas e q u a t i o n d i f f e r s from the e x p e r i m e n t a l v a l u e by 5% O E x p e r i m e n t a l V a l u e s A D e n s i t y of H ? o b t a i n e d from Eq. of S t a t e d a t a Percentage of CO h. D e n s i t y of H ?-CO p m i x t u r e as a f u n c t i o n of c o m p o s i t i o n . a t about 60 atmospheres and was l e s s t h a n t h a t a t l o w e r tempe: t u r e s . The s i g n a l s t r e n g t h v e r s u s p r e s s u r e i s shown i n f i g . 5 f o r 5 5 $ COg and 79-h% COg at: room t e m p e r a t u r e , whereas f i g . 6 shows the s i m i l a r p l o t a t M-00°K f o r 5 5 $ C0„, M - 5 $ H 9 m i x t u r e . 23 8.0 200 ^00 600 800 1000 P r e s s u r e i n p . s . i . F I G. 6. SIGNAL STRENGTH v s . PRESSURE FOR E0 - C0o MIXTURE d * AT ^00°K. CHAPTER I I I THEORY 25 3 . 1 . HYDROGEN The i n t e r n a l degrees of freedom of a d i a t o m i c m o l e c u l e such as Hg a r e : 1. E l e c t r o n i c ( o r b i t a l and s p i n ) 2. V i b r a t i o n a l 3. R o t a t i o n a l s t a t e s of the n u c l e i h. N u c l e a r s p i n s t a t e s . , . rL « „, u „ ^ ^ i r s t excited Staie of the i i n c e "the v i b r a t i o n a l energy of the m o l e c u l e i s of the o r d e r of 6000°K and s i n c e ^ e i e c i : r o n i c > ^ v i b ' t h e d e § r e e s o f freedom c o r r e s p o n d i n g t o the e l e c t r o n i c and v i b r a t i o n a l s t a t e s of the m o l e c u l e can be c o n s i d e r e d t o be f r o z e n even up t o 1000°K. As a r e s u l t the Hg m o l e c u l e can be c o n s i d e r e d as a r i g i d r o t a t o r w i t h an e x t r a degree o f freedom c o r r e s p o n d i n g t o the s p i n s t a t e of the m o l e c u l e . Hence the s t a t e of a hydrogen m o l e c u l e can be d e s c r i b e d by the r o t a t i o n a l quantum numbers J , nij and the p r o t o n s p i n quantum numbers I , m-j-. S i n c e the p r o t o n s a r e f e r m i o n s , the t o t a l w a v e - f u n c t i o n of Hg must be a n t i s y m m e t r i c a l i n the two atoms w i t h r e s p e c t t o p e r m u t a t i o n of the two p r o t o n s . The v i b r a t i o n a l and e l e c t r o n i c p a r t s of the w a v e - f u n c t i o n a r e s y m m e t r i c a l i n t h e ground s t a t e w hich i s the o n l y s t a t q t h a t i s o c c u p i e d . The r o t a t i o n a l w a v e - f u n c t i o n i s s y m m e t r i c a l f o r even J and a n t i s y m m e t r i c a l f o r odd J . The s p i n w a v e - f u n c t i o n i s symmetric f o r 1=1 and a n t i -symmetric f o r 1=0. I n o r d e r t o have the t o t a l w a v e - f u n c t i o n t o be a n t i s y m m e t r i c 1=0 s h o u l d be a s s o c i a t e d w i t h even J and 1=1 s h o u l d be a s s o c i a t e d w i t h odd J . These two m o d i f i c a t i o n s a r e c a l l e d p a r a - and o r t h o - hydrogen r e s p e c t i v e l y . F o r Hg a t t h e r m a l e q u i l i b r i u m the d i s t r i b u t i o n of the m o l e c u l e s among the J s t a t e s i s g i v e n by Boltzmann's d i s t r i -b u t i o n f u n c t i o n P7 « ^ e x ^ - E j / l j . r ) (3.1.1.) where P j i s the f r a c t i o n o f the t o t a l number of m o l e c u l e s t h a t a r e i n J s t a t e , ^ i s the degeneracy of the s t a t e and T i s the tempe r a t u r e of the system % ( 2 I + i)(zJ + \) = ( 2 J + 0 F o r J even i . e . 1=0 = 5(l,T+i) F o r J odd i . e . 1=1 E J = = T(T+i)k 6n (3.1.2.) and where I = Moment of I n e r t i a of the m o l e c u l e k = Boltzmann's C o n s t a n t 0R = 85-3°K f o r H 2 The e q u i l i b r i u m ' r a t i o of o r t h o - t o p a r a - hydrogen i s At h i g h t e m p e r a t u r e s where &•« T, n. —> 3 ? and a t low te m p e r a t u r e s as T.->0,/t->0. At room t e m p e r a t u r e , s i n c e T i s w e l l above 0^ the r a t i o i s c l o s e t o 3• As the temperature i s lo w e r e d the above e q u a t i o n p r e d i c t s t h a t a l l the m o l e c u l e s w i l l be c o n v e r t e d t o p a r a - hydrogen. S i n c e the t r a n s i t i o n from o r t h o - t o p a r a - I n v o l v e s changing t o t a l s p i n a n g u l a r momentum i n the m o l e c u l e , the t r a n s i t i o n p r o b a b i l i t y i s v e r y s m a l l and 27 the e q u i l i b r i u m w i t h r e s p e c t t o o r t h o - p a r a r a t i o cannot be a c h i e v e d i n a s h o r t t i m e . T h e r e f o r e , Hg must be t r e a t e d as a m i x t u r e of two s e p a r a t e gases. P a r a - Hg w i t h 1=0 g i v e s no N.M.R. s i g n a l whereas o r t h o - hydrogen behaves l i k e a 1=1 system. N m o l e c u l e s of Hg at:.room temperature g i v e a s i g n a l p r o p o r t i o n a l t o 3 A NKI+ 1 ) = 3/2 N. The f r a c t i o n a l d i s t r i b u t i o n of the m o l e c u l e s among d i f f e r e n t J s t a t e s i s g i v e n i n Table 1 f o r ( i ) 100$ o r t h o -hydrogen and ( i i ) 100$ p a r a - hydrogen. TEMP IN °K J = 1 7 7 . 5 1.0 1 0 0 . 0 0 . 9 9 9 5 2 0 0 . 0 0 . 9 6 5 3 0 0 . 0 0 . 8 7 9 4-00.0 O.778 5 0 0 . o 0 . 6 8 8 6 0 0 . 0 0 . 6 1 3 7 0 0 . 0 0 . 5 5 0 100$ o r t h o - H, J = 3 0 . 0 0 0 5 0 . 0 3 5 0 . 1 1 9 0.214-0 . 2 9 0 0.3H-3 0 . 3 7 9 J = 5 J = 7 0.002 0.008 0.022 0.04-2 0.067 o.ooi 0.66k 3o<LL 100$ p a r a - H, J = 0 J = 2 J = k J = 6 0 . 9 9 3 0 . 0 0 7 0 . 9 7 1 0 . 0 2 9 0.72 0.28 0 . 5 1 7 - 0.4-68 • 0 . 0 1 5 0 . 3 9 7 0 . 5 6 2 0 . 0 5 0 . 3 2 2 0 . 5 8 0 0 . 0 9 6 0 . 0 0 3 0 . 2 7 1 0 . 5 7 8 0.14-2 0 . 0 0 9 0.234- 0.564- 0.184- 0 . 0 1 8 p : Czf+i) ex.f> f- J(j--H^)gs--3/-rj i&u£jo _ __ Tab l e I . F r a c t i o n a l p o p u l a t i o n of the r o t a t i o n a l s t a t e s f o r Hg, rv> CO 3.2. Theory o f R e l a x a t i o n i n E"2 The h a m i l t o n i a n of the s p i n r o t a t i o n a l and d i p o l a r i n t e r a c t i o n of the H^ ' m o l e c u l e can be w r i t t e n as (3.2.1.) where L-^ and a r e random f u n c t i o n s of the l a t t i c e o p e r a t o r s and S l m and S^m a r e the o p e r a t o r s a c t i n g on the n u c l e a r s p i n v a r i a b l e s . L L /ti 5 ) 0 = [2. I t .'Ar JL , .1 (3-2.2.) " 2o I n a gas where t h e r e a re f r e q u e n t c o l l i s i o n s , the r o t a t i o n a l anguLac?; momentum v e c t o r J undergoes changes due to the a n i s o t r o p i c i n t e r m o l e c u l a r f o r c e s a c t i n g , on the m o l e c u l e d u r i n g the c o l l i s i o n . T h i s . c a u s e s the magnetic f i e l d s a t the s i t e s of the n u c l e i t o f l u c t u a t e t h e r e b y p r o d u c i n g n u c l e a r s p i n t r a n s i t i o n s which"' b r i n g the s p i n system t o e q u i l i b r i u m . Thus the m o l e c u l a r c o l l i s i o n s e s t a b l i s h e q u i l i b r i u m among the J l e v e l s v e r y r a p i d l y whereas the n u c l e a r s p i n s .are r e l a x e d s l o w l y t h r o u g h a weak c o u p l i n g t o J as shown i n f i g . 7« STRONG "LATTICE" t COUPLING MOLECULAR » ZEEMAN ENERGY WEAK *~ COUPLING"^ NUCLEAR ZEEMAN ENERGY FIG. 7. Nature of C o u p l i n g between N u c l e a r S p i n s and the. " l a t t i c e " 30 •• The " L a t t i c e " here r e f e r s to a l l degrees of freedom t h a t are a v a i l a b l e t o a m o l e c u l e except the one c o r r e s p o n d i n g 4 t o the n u c l e a r s p i n s . I t remains a t c o n s t a n t t e m p e r a t u r e T even a f t e r exchanging energy w i t h the n u c l e a r s p i n system because of i t s l a r g e t h e r m a l c a p a c i t y . From the g e n e r a l t h e o r y o f r e l a x a t i o n i t can be shown t h a t (Abragam 1) x - _ ~ \ ( » r ) * ^ vVr f I + , y f Jzir«r) + 4 I T L ( J ( 3 . 2 . 3 . ) where where H' - 27 gauss i s the s p i n r o t a t i o n a l c o u p l i n g c o n s t a n t H" = 3*+ gauss i s the d i p o l a r c o u p l i n g c o n s t a n t . I = -§- i n the case of hydrogen = J e^Q.JOdL/r (3-2.4,) - «o The bar r e p r e s e n t s the ensemble average. I n Schwinger's ijiodel ~^  a l l the m o l e c u l e s were assumed to be i n the l o w e s t r o t a t i o n a l s t a t e ( J = l ) and the m o l e c u l e was a l l o w e d to make t r a n s i t i o n s between the t h r e e m.j s t a t e s . I t was f u r t h e r assumed i n h i s model t h a t a l l the c o r r e l a t i o n f u n c t i o n s decay w i t h the same time c o n s t a n t V 0 > ^ =" ^^70) ( 3 . 2 . 6 . ) k N e e d i e r and Opechowski (1961) suggested t h a t more a c c u r a t e e x p r e s s i o n f o r c o r r e l a t i o n f u n c t i o n would be ; Ljc^) - LJo)Ll(» c ' - ^ e ' V r i • (3 .2.7 .) Johnson and "Waugh (1962) p o i n t e d out t h a t i f the f r e -quency o f e f f e c t i v e c o l l i s i o n s i s much l a r g e r t h a n A a n d i f o n l y J = l s t a t e i s p o p u l a t e d , t h e n Gg^CO c a n he c h a r a c t e r i s e d by two c o r r e l a t i o n times o n l y , one f o r L-^M (m = -1, 0, +1) and the o t h e r f o r (m = -2, -1,0). The e x p r e s s i o n f o r 1 can t h e h be w r i t t e n as T l r r t -j (3.2.8.) •+ 4 r " "1 p r o v i d i n g t h a t a l l the m o l e c u l e s a r e i n t h e i r l o w e s t r o t a t i o n a l CM) , : s t a t e A .This a s s u m p t i o n s t a r t s t o break down s e r i o u s l y a t room temp e r a t u r e where the p o p u l a t i o n of J=3 s t a t e i s about 12$. Bloom ( p r i v a t e communication) has proposed t o e v a l u a t e the c o r r e l a t i o n f u n c t i o n s t a k i n g i n t o a ccount the t r a n s i t i o n s between d i f f e r e n t J s t a t e s w h i c h w i l l be d e s c r i b e d i n d e t a i l i n the f o l l o w i n g s e c t i o n s . 3-2.1. C o r r e l a t i o n F u n c t i o n s of I n t r a - m o l e c U l a r I n t e r a c t i o n s . From e q u a t i o n s (3.2.3.),, (3-2.1+.) and (3.2.5-) and the d i s c u s s i o n t h a t f o l l o w e d , i t i s c l e a r t h a t the c o r r e l a t i o n f u n c t i o n s have t o be e v a l u a t e d i n o r d e r t o 1 o b t a i n an e x p l i c i t e x p r e s s i o n f o r T^. I n the' f o l l o w i n g d i s c u s s i o n the Q t r e a t m e n t of Bloom and Oppenheim ^ i s f o l l o w e d c l o s e l y . The o p e r a t o r L-^ Q has non-zero m a t r i x elements o n l y f o r the case J = J Q , where JQ and J a r e the i n i t i a l and f i n a l r o t a t i o n a l s t a t e s of the m o l e c u l e , whereas the o p e r a t o r L^Q has non-zero . m a t r i x elements f o r the cases J=J and J=J -2. Thfe l a t t e r o o " o s c i l l a t e s f o r f r e e m o l e c u l e a t an a n g u l a r f r e q u e n c y jj^ w h i c h i s always g r e a t e r t h a n 1 0 l L f s e c - 1 f o r h-o r t h o - Hg. N e e d i e r and' Opechowski have shown t h a t the con-t r i b u t i o n from the n o n - d i a g ; D n a l terms i s n e g l i g i b l e compared w i t h the c o n t r i b u t i o n from the d i a g o n a l terms as l o n g as the 1 3 - 1 c o l l i s i o n f r e q u e n c i e s a r e s m a l l e r t h a n 10 J sec . As the 12 -1 c o l l i s i o n f r e q u e n c y i n Hg gas does not exceed 10 sec even a t s e v e r a l hundred atmospheres a t room t e m p e r a t u r e , t h e m a t r i x elements between d i f f e r e n t J s t a t e s can b e . n e g l e c t e d . T h e r e f o r e , the i m p o r t a n t non-zero m a t r i x elements o f a r e g i v e n by <JM/ L, 0|TM> [ f / r f ) LC (J.I.J; M j 0 ) ( 3 < 2 > 9 > ) where C ( J , l , J , M , o ) i s a Cl e b s c h - G o r d o n c o e f f i c i e n t as d e f i n e d by Rose 1 9 and i , ( r ) = ^ ^ 0 (3.2.10.) (Zj-i)(zrfh) (3.2.11) The c o r r e l a t i o n f u n c t i o n of L. /t) may be w r i t t e n as 0 % x / (3.2.12) y (3.2.13) where ¥(J,M,t/J ,M ,o) i s t h e p r o b a b i l i t y t h a t the m o l e c u l e i s i n the r o t a t i o n a l s t a t e J,M a t time t g i v e n t h a t i t i s i n the s t a t e J 0>M 0 a t t=0, and 1 (ZT+I) ex/.[- Ej/i<r] T 0 M 0 ~ Z7.fl To . / . _ A ~ ^ i . i (3.2.1H-) i s tfie p r o b a b i l i t y t h a t a m o l e c u l e i s i n the s t a t e J^,M of energy E T f o r a system a t temp e r a t u r e T. W(J,M,t/J ,M ,o) v o s a t i s f i e s the master e q u a t i o n W fa-"> t /j 0,.o) 1 A nj rW) W f 7 ' M U / T 0 M . . 0 ) (3.2.15) where A(J,M,JJ',M'') i s the t r a n s i t i o n p r o b a b i l i t y per u n i t time /. v f o r the t r a n s i t i o n J'.'M —> JM Fo r J = J' , M = M:/ D e f i n i n g the t e n s o r p o l a r i s a t i o n of o r d e r 1 i n the s t a t e  J f o r m o l e c u l e s i n i t i a l l y i n the s t a t e J0->M0 as ^ l ^ y ' ^ C ( U 7 > ^ U (3'2'17) and u s i n g e q u a t i o n (3.2.15), i t may be seen t h a t the f o l l o w i n g d i f f e r e n t i a l e q u a t i o n i s s a t i s f i e d Weak C o l l i s i o n A p p r o x i m a t i o n . I t i s assumed t h a t the a n i s o t r o p i c i n t e r m o l e c u l a r i n t e r -a c t i o n s a r e "weak" so t h a t f i r s t o r d e r • p e r t u r b a t i o n t h e o r y may be used t o c a l c u l a t e the t r a n s i t i o n p r o b a b i l i t y A(JM,J'Mi' ). I f each term i n the a n i s o t r o p i c i n t e r a c t i o n h a m i l t o n i a n can ; be e x p r e s s e d as a p r o d u c t of a " l a t t i c e " o p e r a t o r and a r o t a t i o n a l o p e r a t o r y (si) , where W o ) i s a s p h e r i c a l harmonic o f o r d e r \ 'KM. V*. the m a t r i x element between s t a t e s a.y,d. T'M' g i v e n hy <™ly*M\*'M'y-S , [(lp^fc(T'*T;M»)C(T'^;Oo) (3-2.19) S i n c e the t r a n s i t i o n p r o b a b i l i t y i s p r o p o r t i o n a l t o the square of the m a t r i x element between the i n i t i a l and f i n a l s t a t e s , ijlie c o n t r i b u t i o n of t h i s term t o the t r a n s i t i o n p r o -b a b i l i t y i s 2. A ^ M y M > Q ; w [ a / A x x M - M ' ) ] ( 3 > 2 > 2 0 ) Q ^ J j ) c o n t a i n s a l l the i n f o r m a t i o n about the a n i s o t r o p i c i n t e r m o l e c u l a r i n t e r a c t i o n s and w i l l be e v a l u a t e d l a t e r . U s i n g e q u a t i o n (3.2.20) and e q u a t i o n s (6.23a) and (6.23b) of Rose, i t can be shown t h a t £ C ( ^ J ; K o ) A x f T M ; T W ; C ( T W ' M O ) B £ A ( ^ ' ) ' (3.2.21) B.xC^')- q/^nfO L^'X^O] U(JJ'JJ!KI\ (3.2.22) * £ £ *T"+' n r-r*-r'\ ~ h77j» IT^T ) where /^ (xx'xj. A i s a Racah c o e f f i c i e n t . U s i n g e q u a t i o n s (3.2.18), (3.2.21) and (3.2.22), i t can be shown t h a t the t e n s o r p o l a r i s a t i o n s a t i s f i e s the f o l l o w i n g e q u a t i o n KA"y-^')Kr^ J-" (3.2.23) where B/rJ) = £ %X(J^') (3.2.2M-) From e q u a t i o n (3.2.17) the i n i t i a l c o n d i t i o n s a re o b t a i n e d as ™ J j r # * 0 ) : C f t ^ s ^ ) s « ° (3.2.25) and the s o l u t i o n s t o the e q u a t i o n (3.2.23) are g i v e n by The number o^ terms i n the sum i s e q u a l t o the number of e q u a t i o n s (3.2.15) used i n s o l v i n g e q u a t i o n (3.2/2-3) w h i c h i s the same as the number o f r o t a t i o n a l s t a t e s t h a t a r e s i g n i f i -c a n t l y p o p u l a t e d . From the i n i t i a l c o n d i t i o n s i t f o l l o w s ' t h a t o ^ i l'* C (7jjo.l%,0) (3.2.27) where X-r;r a r e independent of t i m e . The c o r r e l a t i o n f u n c t i o n G^0^) d e f i n e d by e q u a t i o n (3-2.13) can now be w r i t t e n u s i n g e q u a t i o n s (3-2.17), (3-2.26) and (3-2.27) as below. Jo 2. (3.2.28) S I N C E ^ | ^ C R J ^ ^ V ) ] = I7 + / (3-2.29) a n d pj„ ~- 7T FJ (3.2.30) e q u a t i o n (3-2.28) can be w r i t t e n as W > ° ^ e * * ' ( 3 . 2 . 3 i ) where ^ '/2 c Jo /z ^ _ (3.2.32) JJo I t may be noted t h a t the c o r r e l a t i o n f u n c t i o n can always be w r i t t e n as a sum or* e x p o n e n t i a l f u n c t i o n s o f time i n the weak c o l l i s i o n a p p r o x i m a t i o n . The F o u r i e r t r a n s f o r m of the c o r r e l a t i o n f u n c t i o n i s g i v e n by 00 . Ju (»)- j e Qu it - OO C <X s i n c e i t f o l l o w s from e q u a t i o n (3.2.7) t h a t Jim(ui^ ~~ JU ( W ~ (3.2.3^ ) 36 the e x p r e s s i o n f o r 1 c a n be w r i t t e n a s , u s i n g e q u a t i o n (3-2.3) 2. ,,2. : T l + E - v V l i P ; R + (T.)£ (3 .2 .35) From the above e x p r e s s i o n , i t may be seen t h a t - i - = £ P Jo where (I/T) i s the r e l a x a t i o n r a t e f o r an ensemble of molecu-l e s which a re a l l i n i t i a l l y i n the s t a t e J 3-3- A p p l i c a t i o n of the Theory t o some S p e c i a l Cases: 3.3-1- I n f r e q u e n t T r a n s i t i o n s , between S t a t e s of D i f f e r e n t J . When the t r a n s i t i o n s between d i f f e r e n t J s t a t e s o ccur much l e s s f r e q u e n t l y t h a n . t h e t r a n s i t i o n s between d i f f e r e n t M s t a t e s w i t h i n a J m a n i f o l d , i t can be shown from e q u a t i o n s (3 .2 .20) , (3-2.22) and (3 .2 .2*0. t h a t J') 8 / J 7 ) ^ , (3-3-1.) and the incjex can be r e p l a c e d by J Q From e q u a t i o n s (3-2 .23) , (3-2 .26 ) , (3-2.27) and ( 3 . 3 . 1 . ) , i t may be shown t h a t Cr0 -L V " ^ J $ U ^ ] tj* Z / (3-3-2.) and - ~ BJt(yo.Jo) (3-3.3. ) T h e r e f o r e z T r° " Urh^M**) ~ 1 y i ( 3 . 3 . 5 . ) where At low te m p e r a t u r e s i n Hg, , P J q = ^ and e q u a t i o n (3.3-5') i s the same r e s u l t as o r i g i n a l l y d e r i v e d by Schwinger. n E q u a t i o n ( 3 - 3 • 5 • ) was a l s o d e r i v e d by Bloom and Oppenheim ' a s a g e n e r a l i s a t i o n o f S c h w i n g e r 1 s r e s u l t f o r the case i n which t r a n s i t i o n s between d i f f e r e n t r o t a t i o n a l s t a t e s a r e n e g l i g i b l e . 3-3.2. T w o - l e v e l System. I n t h i s case i t w i l l be assumed t h a t o n l y two of the r o t a t i o n a l s t a t e s a r e a p p r e c i a b l y p o p u l a t e d so t h a t o n l y two of the e q u a t i o n s (3.2.23) need t o be c o n s i d e r e d t o e v a l u a t e T-^  w i t h r e a s o n a b l e a c c u r a c y . F o r Hg t h i s s i t u a t i o n can be a c h i e v e d below M-50°K s i n c e a t t h i s temperature the p o p u l a t i o n of J=5 s t a t e i s o n l y about 1.38$. The e q u a t i o n s t o be s o l v e d f o r the t w o - l e v e l sys-tem a r e J.£W 0 Z J'%** *2&K (3.3.60 where J can tak e the v a l u e s of J-, and J 0 . 0 1 2 S o l v i n g t h e s e two e q u a t i o n s the r a t e c o n s t a n t s A ^ and the c o e f f i c i e n t s C can be o b t a i n e d where «( = 1,2 and J o ~ J l ' J2* *- J L //z (3.3.7.) J, A / 2 - A„ J A,,-A „ - / x (3-3.8.) J' A^- A z / L i ^ J A £ i-A £, and C_ and C- are obtained from C T and C T r e s p e c t i v e l y , by permuting 1 and 2. By s u b s t i t u t i n g e q u a t i o n s (3•3•7•) and (3-3.8.) i n t o e q u a t i o n (3.2.33), JZ(co) can be e x p r e s s e d i n terms of B ( J , J 7 ) . I n t h e s h o r t c o r r e l a t i o n time l i m i t i . e . A£^> ^ f o r a t w o - l e v e l system ± — t ^ i = ^ - ^ - (3.3-9-) U s i n g e q u a t i o n (3-3-8.) the above e x p r e s s i o n can be w r i t t e n as A e / A ^ A £ z and by symmetry * ± fo\ '/z. — + ^ A I (3-3-11) A „ A ^ A £ L A£2_ S u b s t i t u t i n g i n t o e q u a t i o n (3-2.33.), £^^ °) c a n ^ e e x p r e s s e d as I n the l i m i t o f no. changes i n J B£ ( J1' J2 ) = B £ ( J 2 , J 1 ) = 0 T h e r e f o r e (3-3-12) 3 9 and e q u a t i o n ( 3 . 3 - 1 2 ) reduces t o -- hX % ^ z?+ PJMJ^ J ( 3 - 3 - 1 3 ) E q u a t i o n ( 3 . 3 . 1 3 ) was f i r s t d e r i v e d by Johnson and Waugh y f o r the s p e c i a l case d e s c r i b e d above. / 3 .h. E v a l u a t i o n o f B ^ . C J i J ) i n terms of I n t e r m o l e c u l a r A n i s o t r o p i c I n t e r a c t i o n s . The a n i s o t r o p i c i n t e r m o l e c u l a r p o t e n t i a l between two m o l e c u l e s , l a b e l l e d 1 and 2 r e s p e c t i v e l y , i s assumed t o con-s i s t of two terms where and ft rf ^ 2 H > y ^ ; ) v > ' o (3A.3.) (?£) and & (nS) are f u n c t i o n s of the s e p a r a t i o n of the c e n t r e s of mass o f 1 and 2 , P^(CaS6^ i s the Legendre p o l y n o m i a l o f o r d e r 2 of the angl e 6$, which the m o l e c u l a r a x i s o f m o l e c u l e 1 makes w i t h / i , y a n d V 2^^^"2.) a r e 3 e < J ° n d o r d e r s p h e r i c a l harmonics and .Q-'f and rt' are the o r i e n t a t i o n s of the symmetry axes o f m o l e c u l e s 1 and 2 , r e s p e c t i v e l y , w i t h r e s p e c t t o ^ l . The (k, a r e c o n s t a n t s . j - j i s the most g e n e r a l p o t e n t i a l i f m o l e c u l e 1 i s an o r t h o - Hg m o l e c u l e r e s t r i c t e d t o J=T.state, and m o l e c u l e 2 has no r o t a t i o n a l degrees of freedom such as He or p a r a - Hg r e s -ho t r i c t e d t o i t s 'ground s t a t e ( J = 0 ) . - H R depends on the o r i e n t a -t i o n s o f b o t h the m o l e c u l e s and i s . t h e most g e n e r a l form i f b o t h 1 and 2 . m o l e c u l e s are o r t h o - H 2 i n the ground r o t a t i o n a l s t a t e ( i . e . J = l ) a n d ; i f t h e r e are no t r a n s i t i o n s between d i f f e r -ent r o t a t i o n a l s t a t e s . These c o n d i t i o n s are r e a l i s e d a t -s u f f i c i e n t l y low t e m p e r a t u r e s f o r H 2 when- the p o p u l a t i o n of J=3 s t a t e i s n e g l i g i b l e . The number of terms t h a t a r e r e q u i r e d t o s p e c i f y the most g e n e r a l form of J - j a t h i g h e r t e m p e r a t u r e s depends on the s t a t e of l a r g e s t J w h i c h c o n t r i b u t e s a p p r e c i a b l y t o the r e l a x a t i o n r a t e . However, as the a d d i t i o n a l terms w i l l o n l y i n c r e a s e the number of parameters and c o m p l i c a t e the a n a l y s i s too much to o b t a i n any i n f o r m a t i o n on the i n t e r m o l e c u -l a r p o t e n t i a l s , i t w i l l be assumed t h a t MR and J ~ t R are g i v e n by the e q u a t i o n s : (^  A . 2.) and (3- 1+-3-) r e s p e c t i v e l y . I t w i l l f u r -t h e r be assumed t h a t t h e r e i s no i n t e r f e r e n c e between the con-/ o> 12.) t t r i b u t i o n s f r om Ji^and JiR t o Q^JjJ) and hence Qfe?) may be w r i t t e n as ^ ( ) ^ QJ^T')-- q2.ttr')+ G^for') G A A . ) I n o r d e r to o b t a i n the Q fa?) i t i s n e c e s s a r y to e v a l u a t e the t r a n s i t i o n p r o b a b i l i t y per u n i t time A M , T'M ' j o f m o l e c u l e 1. A ^ ^ ) i s o b t a i n e d a f t e r a v e r a g i n g over the e q u i l i b r i u m ensemble of m o l e c u l e 2, the averages b e i n g t a k e n over a l l p o s i t i o n , momentum and a n g u l a r momentum s t a t e s o f m o l e c u l e 2. The t r a n s i t i o n p r o b a b i l i t y per u n i t time of a m o l e c u l e / / 1 between the s t a t e s J,M and J' ,M i s g i v e n by 0 ° . h _ oo / p r o v i d i n g t h a t the a n i s o t r o p i c i n t e r a c t i o n between the molecu-l e s i s q u i t e s m a l l so t h a t p e r t u r b a t i o n t h e o r y can be us e d , and t h a t the energy d i f f e r e n c e between the i n i t i a l and f i n a l r o t a -t i o n a l s t a t e s Jf\Q , i s much s m a l l e r t h a n kT. The bar r e p r e -s e n t s an ensemble a v e r a g e , the average b e i n g t a k e n over a l l t h e r e l a t i v e p o s i t i o n s and momenta of m o l e c u l e s 1 and 2. T r a n s f o r m i n g J -| t o l a b o r a t o r y r e f e r e n c e frame P zCco5 e') V * £ yjxL,) V * f . n . ) O A . 6 . ) where XI, and j f i . a r e the o r i e n t a t i o n s of m o l e c u l e 1 and v e c t o r 7t r e s p e c t i v e l y , i n the space f i x e d c o o r d i n a t e system i n -> w h i c h the z - a x i s i s a l o n g H .. • S u b s t i t u t i n g e q u a t i o n (3-H-.6.) i n t o (3-H-.5.) and u s i n g (3.2.20) Q ^ - - T ( i ^ ) ^ (3A.7-) where e» . , and; — 0 0 k(t)-. ^ ^fo)y 2 < _rrz f t))^o)y 2 wC^) ( 3 A - 9 i ) i s the c o r r e l a t i o n f u n c t i o n of •& V„ ^ -O.) which i s i n d e -pendent of ft f o r a gas. The c o r r e l a t i o n time k (t) decays i n a time of the o r d e r ,of average time of a s i n g l e c o l l i s i o n between a p a i r of m o l e c u l e s i n a d i l u t e gas. r ^ ~ Jita/M)'2- = % (3A.io) where i s the approximate range of the i n t e r m o l e c u l a r i n t e r -a c t i o n s , ytc i s the reduced mass o f m o l e c u l e s 1 and 2 and V Q i s the r.m.s. v a l u e of t h e i r r e l a t i v e v e l o c i t y . S i n c e , r , i t can be con c l u d e d t h a t v e r y l i t t l e energy i s exchanged between r o t a t i o n a l and t r a n s l a t i o n a l degrees o f freedom i f The o n l y non-zero terms f o r Q^-rf) c o r r e s p o n d t o J'"=J and J""=J'±2. F o r Hg m o l e c u l e <$1(7*?±Z)« Q i f^O (3A.12) s i n c e , a t 700°K ~ 8 The c o r r e l a t i o n f u n c t i o n s g i v e n by e q u a t i o n (3A.9<) .were c a l c u l a t e d by Bloom and Oppenheim ^ u s i n g the " c o n s t a n t " a c c e l e r a t i o n a p p r o x i m a t i o n " . U s i n g the r e s u l t s of t h e i r t h e o r y . (i) (P. 866) j /o) can be w r i t t e n as y (o) - po7(l*pM) I to (3A.13) where i s the number d e n s i t y of m o l e c u l e s of type 2 and o o where ft, i s the c h a r a c t e r i s t i c l e n g t h f o r the s p h e r i c a l l y symmetric i n t e r m o l e c u l a r p o t e n t i a l between m o l e c u l e s 1 and 2, T-fl/a, > 9 (x.) i s the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r t h e p a i r of m o l e c u l e s under c o n s i d e r a t i o n , "J"^ ± i s the B e s s e l f u n c t i o n of ^^JL^ a n ^ A ^ s ^ e reduced mass. S u b s t i t u t i n g (3A.13) i n (3-H-.7-), can be w r i t t e n as E v a l u a t i n g the Racah c o e f f i c i e n t s i n e q u a t i o n (3.2.22) . tl) , V 3 f^y^) o b t a i n e d as f o r A - * ^ - , ) ( * » 3 ) ft.) The c o n t r i b u t i o n of 3~\^ t o the B^C?,?) w i l l now be e v a l u a t e d . I t i s assumed t h a t the dominant i n t e r a c t i o n i s the qu a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n f o r w h i c h the v a l u e s of C\, i n e q u a t i o n (3.M-.3-) are = ^ (3 A.18) Making the t r a n s f o r m a t i o n t o the l a b o r a t o r y r e f e r e n c e frame . 'A ^ M (3 i+ 19) (2.) T / J_J . produces t r a n s i t i o n s of m o l e c u l e 1 from the s t a t e J"M'.' t o the s t a t e J,M w h i l e m o l e c u l e 2 undergoes a s i m u l t a n e o u s t r a n s i t i o n from the s t a t e ",M'^  " t o s t a t e J 7 7 ,My/;. Denoting the t r a n s i t i o n p r o b a b i l i t y f o r t h i s p r o c e s s by A((7M/7,M,)(J"V"JT"W") ^ f r ^ T ' f ^ i s o b t a i n e d as A % i , xW ) = £ £_ £ L. A (frn/rW) ( :T"M", T V")) . N { J 7 » T ' " M * M'" V / (3 A.20) where . x . fc.) 2. . , A R ^ , , w j r / ' „ ' > ' " M ' " ) ] = 1 / 77) ^  r^ »»')Lc^ '^^ ')] ^ W T y r . _ » / = ^ T " ^ ' ) -f ^ j ' - ^ . ) ( 3 A . 2 2 ) and ?our • 1 to / \ i s t h e F o u r i e r transform, of £ fey w h i c h i s g i v e n by U s i n g e q u a t i o n (3.2.19) t o o b t a i n the m a t r i x elements i n e q u a t i o n (3.1+.21) and summing over W\ ^ M and M and d e f i n i n g £j frj'J by e q u a t i o n ( 3 . 2 . 2 0 ) , Q ( ;r J ) /// 12. i s o b t a i n e d as /OX -1 (3A.20 X a l s o decays i n a time of the o r d e r o f £" T h e r e f o r e , (2.) 1 '/ 1 I/ Q^Jj') i s non-zero f o r J' =J and J " - J + 2 and J = T and y.'" - J ''t 2 . . Up t o the o r d e r of 500°K o n l y J = l and 3 s t a t e s f o r o r t h b - Hg and J=0, 2 and h s t a t e s f o r p a r a - Hg need t o be . 1 iij //> . c o n s i d e r e d , The f o l l o w i n g t r a n s i t i o n s ( J", CT <£—> J", J" ) g i v e a p p r e c i a b l e c o n t r i b u t i o n t o Q F ° r o r t h o - o r t h o c o l l i s i o n s O > u ,„ -0 f o r (1,3 4r-> 3,1) t r a n s i t i o n s and f o r a l l o t h e r t r a n s i t i o n s ^7y>y"jitFeotZ ^ ^ a t r o o m temperature. F o r o r t h o - p a r a c o l l i s i o n s HTTW" ~ ^  o n l y f ° r (? J > 7 , x " ^ . F o r the t r a n s i t i o n s ( l , 2 < - * 3 , 0 ) and ( l , 4 - < — * 3 , 2 ) ^'j^Joa 7'~l 20 a t room t e m p e r a t u r e . A c c o r d i n g t o Van Kranendonk's c a l c u l a -t i o n of j (LS) v e r s u s W f o r a q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n , jVw) i s l a r g e f o r W "^ a n c ^ r a p i d l y d e c r e a s e s f o r h i g h e r v a l u e s of 0^ . T h e r e f o r e , the above-mentioned t r a n s i -t i o n s have t o be t a k e n i n t o account f o r . o r t h o - p a r a i n t e r a c t i o n s . U s i n g e q u a t i o n s (3.H-.2G) and (3.H-.21) the r a t i o of the t r a n s i t i o n , r a t e s ^ V — a n d 3"') I' o r m o l e c u l e 1 i s o b t a i n e d as l , y ( j j ' W r / ) S i n c e the Boltzmann f a c t o r i s l o s t i n the l a t t e r c a s e , Oppenheim suggested a p r e s c r i p t i o n t o t r e a t t h e s e c o l l i s i o n s w h i c h was d i s c u s s e d i n more d e t a i l i n a s e p a r a t e paper by 9 Bloom and Oppenheim . The p r e s c r i p t i o n i s t o m u l t i p l y the upward t r a n s i t i o n s and downward t r a n s i t i o n s of m o l e c u l e 1 by 6^  and £ ^ r e s p e c t i v e l y where and e,-- i ('+ e ^ " - T V " J (3.4.27) € ^ ± ( , t e + ^ ^ V V ) (3.4.28) The t r a n s i t i o n s ( 1 , 3 < — ^ 3 , 1 ) a n d (x,J//«-» ^ 3 " " ) w i l l be r e f e r r e d t o as r e s o n a n t t r a n s i t i o n s caused by r e s o n a n t c o l l i -s i o n s w h i l e the t r a n s i t i o n s of the type ( 1 , 2 ^ — ^ 3 50) and ( 1 , M - ^ - ^ 3 j 2 ) w i l l be c a l l e d as q u a s i - r e s o n a n t t r a n s i t i o n s caused by q u a s i - r e s o n a n t c o l l i s i o n s i n the r e s t of H i s t h e s i s . H-6 A l l the o t h e r t r a n s i t i o n s f o r which A t d , n u. ^ O are 77 7 j'" termed as non-resonant t r a n s i t i o n s . CHAPTER IV EXPERIMENTAL RESULTS AND DISCUSSION H-.1.' GENERAL REMARKS The s p i n - l a t t i c e r e l a x a t i o n time T^  was measured i n normal Hg and i t s m i x t u r e s w i t h He. and COg as a f u n c t i o n of d e n s i t y and tempe r a t u r e from 293°K up to 700°K. The d e n s i t i e s used were s u f f i c i e n t l y h i g h t o be away from T^  minimum and y e t low enough f o r t he t h r e e body c o l l i s i o n s t o be n e g l i g i b l e . Under these c o n d i t i o n s the t h e o r y p r e d i c t s t h a t T^  i s p r o p o r t i o n a l t o the d e n s i t y . The h i g h e s t p r e s s u r e s used i n the exp e r i m e n t s were of the o r d e r of 2000 p . s . i . The experiment a t any temperature was s t a r t e d w i t h the h i g h e s t p r e s s u r e a v a i l a b l e and T^  was measured a t d i f f e r e n t p r e s s u r e s r e d u c i n g the p r e s s u r e i n c o n v e n i e n t s t e p s . The l o w e s t p r e s s u r e reached a t any temperature was determined by 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 . At room temperature t h i s was about 100 p . s . i . (p*7 amagats) whereas a t 738°K i t was about 600 p . s . i . (w15 amagats). M-.2. HYDROGEN H-.2.1 . R e s u l t s . T y p i c a l p l o t s of T^  v e r s u s d e n s i t y are shown i n f i g u r e s 8a and 8b f o r normal Hg a t room temperature and a t 738°K r e s p e c t i v e l y . The most p r o b a b l e v a l u e of T^  a t any p r e s s u r e and temperature was o b t a i n e d by f i t t i n g the s t r a i g h t l i n e t h r o u g h the d a t a log[A(o°) - A ( t j ] v e r s u s t by eye and the e r r o r on t h i s v a l u e was o b t a i n e d by drawing the s t r a i g h t l i n e t h r o u g h the h8 1 0 . 0 20 kO 60 80 D e n s i t y i n Amagats 100 FIG. 8a . T 1 v s . D e n s i t y a t 293 K i n normal H 2 3-0 o CD 2.0 • H 1.0 FIG. 8b.. 50 extreme p o i n t s . However, a l e a s t squares f i t was c a r r i e d out i n a few s e l e c t e d cases on an I.B.M. 70M-0 computer and the s l o p e thus o b t a i n e d agreed w i t h t h a t of the b e s t f i t by eye w i t h i n 5%. The s o l i d l i n e i n f i g u r e s 8a and 8b r e p r e s e n t s the l e a s t squares f i t and the e r r o r s r e p r e s e n t e d as + <5 c o r r e s p o n d t o the c o n d i t i o n t h a t t h e r e i s a 5$ p r o b a b i l i t y t h a t the c o r r e c t e x p e r i m e n t a l v a l u e s be o u t s i d e the l i m i t s g i v e n by the e r r o r + g . Knowing the temperature and p r e s s u r e , the number d e n s i t y 1M-of the gas was o b t a i n e d from the t a b l e s g i v e n by Wooley e t a l . As t h e s e t a b l e s were g i v e n o n l y up t o 600°K, Wooley has suggested ( p r i v a t e communication) t h a t the e s t i m a t e of d e n s i t y a t h i g h e r t e m p e r a t u r e s as a f u n c t i o n of p r e s s u r e can be o b t a i n e d t o a good a p p r o x i m a t i o n from the f o l l o w i n g e q u a t i o n P V / K T - ex-jb \_B? + C f ^ J where -3/2 ~2-and C - oooif.782 T ' - o - o ^ o f 3 T The computations were c a r r i e d out on an I.B.M. 70M-0 computer and the d e n s i t y was o b t a i n e d as a f u n c t i o n of p r e s s u r e i n the temp e r a t u r e range. 500°K - 1200°K. I n f i g u r e 9 T-|/p i s p l o t t e d as f u n c t i o n of temperature f r o m 293°K up t o 738°K a l o n g w i t h the. d a t a a v a i l a b l e below room tempe r a t u r e from p r e v i o u s work. The d a t a a t room temperature from the p r e s e n t work may be 21 compared w i t h the p r e v i o u s work of W i l l i a m s (1962), Johnson and Waugh ^ (1962), L i p s i c a s 2 2 (1962), Armstrong 2 3 (1966) and R i e h l (1966). L i p s i c a s o b t a i n e d the .value of T^/p as 0.125 m.sec/Amagat which i s 25$ h i g h e r t h a n the v a l u e - p 05 a \ o 0 • H 16 ,1M-.12 .10 .08 06 \ A \ »A A \ A A A x"-• R i e h l A L i p s i c a s x W i l l i a m s n Johnson and Waugh o P r e s e n t Work ,0h 1 1 0 100 200 600 300 hoo 500 Temperature i n °K FIG. 9. T^/p as a f u n c t i o n of temperature i n normal R"2 700 800 M I o o b t a i n e d i n the p r e s e n t work. The v a l u e o b t a i n e d i n the p r e -s e n t work i s 0.100 m.sec/Amagat and a g r e e s , w i t h i n e x p e r i m e n t a l e r r o r s , w i t h t h a t of W i l l i a m s (0.100 mf sec/Amagat), Johnson and Waugh (0.105 m.sec/Amagat), Armstrong (0.106 m.sec/Amagat) and R i e h l (0.105 m.sec/Amagat). L i p s i c a s ' d a t a d i f f e r s s i g n i -f i c a n t l y from W i l l i a m s ' and R i e h l ' s between 120°K and 293°K a l s o , though i t agrees v e r y w e l l w i t h t h e i r d a t a below 100°K. On e x a m i n a t i o n of L i p s i c a s ' d a t a as p r e s e n t e d i n h i s t h e s i s i t appears t h a t the d i s c r e p a n c y i s due t o the h i g h d e n s i t i e s t h a t were used i n h i s measurements. Presumably,the e f f e c t of t h r e e body c o l l i s i o n s i n c r e a s e d the apparent v a l u e of "T| j ? * I n f a c t , Armstrong a n a l y s e d the dependence of T-^  on _f f o r h i g h d e n s i t i e s assuming t h a t T£ i s g i v e n by T, = Af + B f 2 " 2. and h i s work i n d i c a t e s t h a t the c o n t r i b u t i o n from f term t o T-^  i s about 10$ i f $ i s of the o r d e r of 100 amagats. i Though the a b s o l u t e v a l u e s d i f f e r , the g e n e r a l t r e n d of the curve i s the. same i n a l l the ca s e s . TjJj> d e c r e a s e s v e r y s h a r p l y w i t h i n c r e a s e of temperature from low tem p e r a t u r e s up t o 150°K and remains c o n s t a n t between 150°K and 293*°K. As the temp e r a t u r e i s i n c r e a s e d above room temperature T, jp d e c r e a s e s g r a d u a l l y t o a v a l u e o f 0.602 m.sec/Amagat a t 738°K. *+.2.2. I n t e r p r e t a t i o n . I n t h i s s e c t i o n , .the t h e o r y d i s c u s s e d i n the p r e v i o u s chap-ter w i l l be used t o e x t r a c t i n f o r m a t i o n on the i n t e r -m o l e c u l a r a n i s o t r o p i c i n t e r a c t i o n s from the e x p e r i m e n t a l d a t a d'f temperature dependence of T-^ /p i n H 2, assuming t h a t the i s o t r o p i c p a r t of the i n t e r m o l e c u l a r p o t e n t i a l i s g i v e n by a Lennard-Jones p o t e n t i a l . The d i s c u s s i o n w i l l be c o n f i n e d t o a two l e v e l system even though the f r a c t i o n a l p o p u l a t i o n of the J=5 s t a t e i s 0.03 a t 500°K and 0.07 a t 700°K and t h i s assump-t i o n i s not s t r i c t l y v a l i d a t thes e t e m p e r a t u r e s . I n o r d e r t o i n t e r p r e t the r e s i i l t s i t i s n e c e s s a r y t o w r i t e down the e x p r e s s i o n f o r 1 e x p l i c i t l y . From e q u a t i o n s T l (3.2.22), (3.3-12) and (3.2.3.) i t may be seen t h a t 1 i s T l e s s e n t i a l l y a f u n c t i o n o f o ^ f x r ) and the Racah c o e f f i c i e n t s jo, , U s i n g the e q u a t i o n s (3-H-.15) and (3.lt.2n-) QA*?) and • 12.) • Qz(:r/J) c a n be w r i t t e n as f o l l o w s : 5- ( Z T - ^ r t i ) 1 l o > (4.2.1.) where , \u (,) [ °> ~~ ^ ~ 0+.2.3.) (V'l ''IT' ^ (h-2M-) z \(ir-0(2T+i) / (4,2.5.) where <^  ^ r e p r e s e n t s the ensemble average. 12.) / W h i l e e v a l u a t i n g the i t i s assumed t h a t o n l y t h o s e c o l l i s i o n s i n which t o t a l r o t a t i o n a l a n g u l a r momentum i s cons e r v e d a re i m p o r t a n t . As the m o l e c u l e 2 c o u l d e i t h e r be an o r t h o - or p a r a - Hg m o l e c u l e , QJftj) I s a f u n c t i o n of the r o t a t i o n a l a n g u l a r momentum quantum numbers of bo t h o r t h o - and p a r a - Hg. The t r e a t m e n t of thes e two typ e s of c o l l i s i o n s was 53 g i v e n i n S e c t i o n 3A. of Chapter I I I . T r e a t i n g the r e s o n a n t and q u a s i - r e s o n a n t c o l l i s i o n s as d e s c r i b e d i n the p r e v i o u s c h a p t e r and assuming t h a t non-resonant c o l l i s i o n s do not con-U") / s. (2.) v (*•), N t r i b u t e s i g n i f i c a n t l y t o ( n ' ) > (j+z^J) o^<L a r e g i v e n by . N r / x «,-/ya_ / >i v w * P T*/'- FXfr,) V'M ( ^ - 2 . 6 . ) where . / T J - A / T N ^ F- JJ-^0-15 f o r normal H p A. 2.9-) and £ and £ are g i v e n by e q u a t i o n s (3 A.27) and (3 A.28). S u b s t i t u t i n g e q u a t i o n s A.2.1.), A.2.2.), A.2.6.) and A.2.7.) i n t o e q u a t i o n (3.2.22), u s i n g the r e l a t i o n g i v e n by e q u a t i o n (3AA.) and e v a l u a t i n g the Racah c o e f f i c i e n t s , B^fajl') a r e o b t a i n e d as f o l l o w s : ^ ^ h X where J 5 i s the d e n s i t y i n i d e a l amagats and 5M-15O TT S u b s t i t u t i n g e q u a t i o n s (h.2.10) and (H - , 2 . 1 1 ) i n the e q u a t i o n ( 3 . 3 - 1 2 ) t o o b t a i n J(<>) and 7z(o) and u s i n g e q u a t i o n ( 3 . 2 . 3 - ) 9IT ^S o b t a i n e d a s u 'V1 L ib ft"? + J»s>+f H fe+ K> I i ft t i l l I t may be noted from the above e x p r e s s i o n t h a t f J T i s a f u n c t i o n of ( V 0 ^ ) , ' ^ J and F. I n v iew of the g e n e r a l theory; now a v a i l a b l e i t was f e l t n e c e s s a r y to r e i n t e r p r e t the da t a a v a i l a b l e below room tempera-t u r e from o t h e r work. These d a t a had been i n t e r p r e t e d p r e -v i o u s l y n e g l e c t i n g the t r a n s i t i o n s between s t a t e s of d i f f e r e n t J , a r e s t r i c t i o n which has been removed by the new t h e o r y . The d a t a of T;jj> o b t a i n e d by L i p s i c a s and H a r t l a n d 2 ^ as a f u n c t i o n o f o r t h o - H 2 and R i e h l ' s 2 h r and W i l l i a m s ' 2 1 d a t a of T, j f i n normal Hg w i l l be used to o b t a i n (7)'} a n d ^ j ^ below room t e m p e r a t u r e . The v a l u e of Tf Jj> due t o o r t h o - p a r a c o l l i s i o n s a l o n e c o r r e s p o n d s t o the c o n d i t i o n t h a t F=0 and t h a t due t o o r t h o -o r t h o c o l l i s i o n s a l one i s o b t a i n e d by e q u a t i n g F t o 1. From e q u a t i o n (h.2.lk) i t can be seen t h a t Cfyp) //^ -N) depends ' 'op f /oo o n l y on fyj^ ) ^  . (r^l?\P/(T'/f)0D was C a l c u l a t e d f o r d i f f e r e n t v a l u e s of (^ /y, ) on an I.B.M. 70^-0 computer and f i g . 10. shows the t h e o r e t i c a l p l o t s of (T t J p ) Q 1 > j ( j l / f ) as a f u n c t i o n of ^ vJ, J yj't YZ'H2- a t te m p e r a t u r e s 77.8°K, 100°K, 200°K and 300°K. U s i n g t h e s e t h e o r e t i c a l c u r v e s , the v a l u e o f (V0 / ^i^i^z. c o r r e s p o n d i n g t o the e x p e r i m e n t a l v a l u e of (T'/i)of>j(TJf) was ob-t a i n e d a t d i f f e r e n t t e m p e r a t u r e s . S u b s t i t u t i n g t h e s e v a l u e s of M o / 7 ' ) i n t o e q u a t i o n (h.2.lh) and u s i n g the e x p e r i m e n t a l v a l u e s o f Tijf f o r n o r m a l ' H 2 ( V J was o b t a i n e d a t d i f f e r e n t t e m p e r a t u r e s . D e f i n i n g [K,j as (V) ) can be e x p r e s s e d , u s i n g the e q u a t i o n (h.2.h) as where _P = 2.69 X 1019cm."3, an " I d e a l Amagat". F i g . 11 shows 2 2 as a f u n c t i o n o f temperature when the q u a s i - r e s o n a n t c o l l i s i o n s a r e i n c l u d e d and a l s o when th e y a r e not i n c l u d e d . I t can be seen from the f i g u r e and the f o l l o w i n g d i s c u s s i o n t h a t the q u a s i - r e s o n a n t terms p r o b a b l y c o n t r i b u t e s i g n i f i c a n t l y even a t 200 K though the p o p u l a t i o n o f J=3 s t a t e i s o n l y 3% a t t h i s t e m p e r a t u r e . I f ' t h e i s o t r o p i c p a r t of the i n t e r m o l e c u l a r p o t e n t i a l i s assumed t o be g i v e n by a Lennard-Jones p o t e n t i a l f o r r e a s o n s 8 d i s c u s s e d by Bloom, Oppenheim e t a l i n t h e i r paper , the temp-e r a t u r e dependence of k-^  i s determined by the r a d i a l dependence of the a n i s o t r o p i c p a r t of the i n t e r m o l e c u l a r p o t e n t i a l . ON o 2.0 1 .0 0 O I n c l u d i n g the q u a s i r e s o n a n t terms • E x c l u d i n g " " " " 0 100 200 300 Temperature i n °K F I G . 1 1 . Temperature dependence of ( k 1 ) H 2 " H 2 as o b t a i n e d from Eq. (^ .2 .1^) and t h e e x p e r i m e n t a l v a l u e s of ( T 1 Z P ) Q p/^^P^Q Q a n d (T^/p) f o r normal E^, Assuming t h a t the a n i s o t r o p i c p o t e n t i a l i s g i v e n by a s i m p l e power law (<v) <r - — ( H - . 2 . 1 7 ) /I M 2" X (fG can be ex p r e s s e d as /^"^ , Ml-,, ' N (4,2.18) U s i n g the v a l u e s of the i n t e g r a l s i 7 /^") g i v e n by Bloom, 8 /, A - * -Oppenheim e t a l , the t h e o r e t i c a l v a l u e s of [k,) can be com^ -puted a t d i f f e r e n t t e m p e r a t u r e s by n o r m a l i s i n g the t h e o r e t i c a l O v a l u e t o the e x p e r i m e n t a l v a l u e a t 200 K, to o b t a i n CO^ • r i / -i H i " w 2 -F i g . 12 shows | K,(r)Ik(iod)J as a f u n c t i o n of temperature a l o n g w i t h the t h e o r e t i c a l p l o t s f o r d i f f e r e n t v a l u e s of n. F i g . 12 a l s o shows the e x p e r i m e n t a l v a l u e s of [k/r) / k( 2.oo)J when the q u a s i - r e s o n a n t terms a re not i n c l u d e d . I t can be seen from the f i g u r e t h a t n=5 gives, a r e a s o n a b l e f i t i n the h i g h tempera-i t u r e r e g i o n w h i c h i s c o n s i s t e n t w i t h q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n . The disagreement i n the low temperature r e g i o n i s b e l i e v e d t o be-due- to- the f a c t t h a t t h e quantum e f f e c t s a r e not t a k e n i n t o account a d e q u a t e l y i n the t h e o r y . The quadrupole moment o f the Hg m o l e c u l e can be o b t a i n e d by comparing the e x p e r i m e n t a l v a l u e a t 200°K w i t h the t h e o r e t i -c a l v a l u e . The v a l u e o b t a i n e d i n the p r e s e n t work i s (o-5l±0-o^ 10~ 2^ e.s.u. as compared t o the t h e o r e t i c a l v a l u e of 0.65*-10" 2 6 e.s.u. Having e s t a b l i s h e d t h a t i s g i v e n by qu a d r u p o l e -q u a d r u p o l e i n t e r a c t i o n , -the v a l u e s o f ^ jj a t d i f f e r e n t tempera-t u r e s were o b t a i n e d from the f o l l o w i n g e q u a t i o n O I n c l u d i n g the q u a s i r e s o n a n t terms • E x c l u d i n g " " " » CM I CM o o CM \ EH 1 .6 1 .k 1 .2 1 .0 0.8 0.6 oA 0.2 n = 5 0 0 100 200 300 Temperature i n K FIG . .12.. Comparison of e x p e r i m e n t a l v a l u e s of |^k1 ( T ) / k 1 (200)JH2"H2 w i t h the t h e o r e t i c a l v a l u e s u s i n g Eq. (•+.2.1 8) . The c o n t r i b u t i o n from q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n a l o n e to^jjp i s o b t a i n e d from e q u a t i o n (4-.2.14-) by p u t t i n g Y = o w h i c h i s shown i n f i g . 13 a l o n g w i t h the e x p e r i m e n t a l 0 'values of TJ jp , I t can be seen from the f i g u r e t h a t the con-t r i b u t i o n from q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n i s q u i t e s i g -n i f i c a n t t hroughout the temperature range. E v a l u a t i o n of E q u a t i o n (h.2.±h) can be w r i t t e n as a p o l y n o m i a l of o r d e r h i n X where X i s d e f i n e d by e q u a t i o n (H-,2.12) and the s o l u t i o n s of t h i s p o l y n o m i a l a r e found by B a i r s t o w e ' s method, on an I.B.M. JOhO computer a t the U n i v e r s i t y of B r i t i s h Columbia c o m p u t a t i o n c e n t r e . A l l the f o u r r o o t s were r e a l and o n l y one o f them was p o s i t i v e . Choosing the p o s i t i v e r o o t , (pf\'^zHz can be e v a l u a t e d knowing ^ ' j ^ A a n d C. D e f i n i n g H ~H >)z z can be e x p r e s s e d as 5 * ? -F i g . lh shows k Q as a f u n c t i o n o f t e m p e r a t u r e . The e r r o r b a r s r e p r e s e n t - 5$ e r r o r on the v a l u e of Tijp • A - 10$ e r r o r on the v a l u e o f pj( / i n t r o d u c e d the same e r r o r i n the v a l u e of (ko)1 1 a s shown by the e r r o r b a r s Th the f i g u r e . I f a s i m p l e power law i s assumed t o d e s c r i b e the a n i s o -t r o p i c potential,(«0) i s g i v e n by U s i n g the n u m e r i c a l v a l u e s of the i n t e g r a l s X (Zj rj), cJ^ can be o b t a i n e d by e q u a t i n g the e x p e r i m e n t a l v a l u e of ko t o - p a ho a <«: \ o CD i — E H .12 10 ,08 .06 .oh ,02 0 0 o o o o o o E x p e r i m e n t a l V a l u e s C o n t r i b u t i o n of Quadrupole-Quadrupole i n t e r a c t i o n t o T^/p 100 200 300 hOO 500 600 700 800 Temperature i n K FI G . 13* Comparison of the e x p e r i m e n t a l v a l u e s of T^/p w i t h the computed v a l u e s u s i n g Eq.(h.2.1h) and assuming t h a t . o n l y q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n i s i m p o r t a n t . ON 6.0 OJ i OJ M i \ OJ i o C D w J -OJ O X 5-0 4,0 3.0 2.0 1 .0 0 6 0 0 6 0 100 200 300 4-00 500 600 700 Temperature i n K FI G . J\h. Temperature dependence of (k Q) H2 H2 as o b t a i n e d from Eq. (h.2.1 h) and the e x p e r i m e n t a l v a l u e s of (T^/p) i n normal Hg. I V ) 63 o r i / i i ^ t ^ 3 -t h e t h e o r e t i c a l v a l u e a t 200 K f o r any n. Thus ^*»v>U faJ can be o b t a i n e d as a f u n c t i o n of temperature f o r d i f f e r e n t v a l u e s of n. F i g . 15 snows the e x p e r i m e n t a l v a l u e s of ^Ik« fao)^^3' a l o n g w i t h the t h e o r e t i c a l p l o t s f o r d i f f e r e n t v a l u e s of n. When a s i m p l e power law i s assumed to e x p r e s s the e x p e r i m e n t a l v a l u e s a r e b e s t f i t t e d f o r n=l5- However, the 8 t h e o r e t i c a l s t u d i e s of the form o f the i n t e r m o l e c u l a r p o t e n t i a l between Hg m o l e c u l e s which depends bn the o r i e n t a t i o n o f one o f the m o l e c u l e s a l o n e i n d i c a t e t h a t the s i m p l e form can b e t t e r be r e p r e s e n t e d i n terms of an a t t r a c t i v e p a r t ' p r o p o r t i o n a l t o and-.a . r e p u l s i v e p a r t p r o p o r t i o n a l t o ' / / i 1 - I n t h i s case J(r (az) i s g i v e n ; by * ! $ + « i f (^ .2.23) x r The v a l u e of OJ? was computed t h e o r e t i c a l l y ^ and i s g i v e n t b be (J^ - _/. g 5 x Io'2' Mc'. K n l s t h e n S i y e n : ; "byvr:; 60^ . i s e v a l u a t e d by e q u a t i n g t h i s e x p r e s s i o n t o the e x p e r i -m e n t a l v a l u e o f ^ a t 200°K and u s i n g t h e t a b l e s I I I and IV g i v e n by Bloom, Oppenheim e t a l to^.evaluate v,*')• (k0} *~ *~ a t any o t h e r temperature i s o b t a i n e d from e q u a t i o n (h.2.2h) u s i n g t h e v a l u e s o f T ( 2 . ^ ^ ' ) c o r r e s p o n d i n g t o t h a t p a r t i c u l a r t e m p e r a t u r e . \_^»(T^}k (2.00)^^ a s a ^ u n c t i o n of temperature i s shown i n f i g . 15 f o r d i f f e r e n t v a l u e s of r/'< I t can be seen from the f i g u r e t h a t the agreement between the e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s i s b e s t f o r n''-13. As the v a l u e s of the i n t e g r a l s f o r 1(2,12) and 1(2,6,12) a r e not p r o v i d e d i n the CM I CM W O o CM EH 7 . o h 6.0 5.0 H-.O 3.0 2.0 1 .0 0 t I I (n,n') = (6,12) n = 15 O E x p e r i m e n t a l P o i n t s 0 100 200 300 M-00 500 600 700 800 o Temperature i n K FIG. 15". Comparison of e x p e r i m e n t a l v a l u e s o f ^ k-Q(T)/kQ(200 )J H2'~H2 w i t h the t h e o r e t i c a l v a l u e s u s i n g Eqs.(>+.2.22) and (h.2.2h) -r t a b l e s , the t h e o r e t i c a l curve f o r n 'fc12 was o b t a i n e d u s i n g t h e i n t e r p o l a t e d v a l u e s of 1(2,12) and 1(2,6,12) and the p l o t t h u s o b t a i n e d was found t o be almo s t i d e n t i c a l w i t h t h a t of n'u=13. The d e p a r t u r e of the e x p e r i m e n t a l v a l u e s from the t h e o r e t i c a l v a l u e s above 500°K i s p r o b a b l y due t o the f a c t t h a t t he a n a l y s i s was c a r r i e d out on the ass u m p t i o n t h a t J=5 s t a t e can be n e g l e c t e d . At 500°K the p o p u l a t i o n o f J=5 s t a t e i about 2% and a t 700°K i t i s about 6.7$ and the e f f e c t o f J=5 s t a t e has t o be t a k e n i n t o a c c o u n t . Based on the a n a l y s i s of (1) t w o - l e v e l system 3-J R f o r Hg can be r e p r e s e n t e d by K'-1***"'*^^-"^)'1-^)6^ P^Cc*©') A.2.25) and the i n t e r m o l e c u l a r p o t e n t i a l Vfi) can be w r i t t e n f o r an o r t h o - p a r a p a i r as V ^ K t o ' W j + A ^ / o - ' f w . (\f] PJC* 6 ' ) (4,2 . 2 6 ) where e q u a t i o n s (H-,2.23) and (3.^.2.) were used and the i s o t r o -p i c p a r t of the i n t e r m o l e c u l a r p o t e n t i a l was g i v e n by Lennard-Jones p o t e n t i a l . I f i t i s assumed t h a t the a n i s o t r o p i c part* is' also:'.given* 'by-Lennard-Jones p o t e n t i a l , the t h e o r e t i c a l v a l u e s agree v e r y w e l l w i t h the e x p e r i m e n t a l v a l u e s up t o 500°K and the v a l u e o b t a i n e d f o r C J ^ i s &-)g,= - 2>-3 x /o' Z /$*c '. The i n t e r m o l e c u l a r p o t e n t i a l can be w r i t t e n f o r t h i s case as V W = ' f e L ^ W % f j [ ' + 0 - ' 7 3 l ^ e ' ) ] (4.2.27) The v a l u e s o f the i n t e g r a l s T({>H) and TCr]^,yi'} were g i v e n o n l y up t o = 0.09 i n the t a b l e s r e f e r r e d t o i n the p r e -c e d i n g paragraphs w h i c h c o r r e s p o n d s t o a p p r o x i m a t e l y H-00°K f o r Hg. However, i t was found t h a t l o g X ( f> ^ v s . l o g ( p>€) as w e l l as l o g T.(p)y\ly\) vs l o g ($6) a r e l i n g a j r and t h e v a l u e s o f the i n t e g r a l s a t h i g h e r t e m p e r a t u r e s were o b t a i n e d from the e x t r a p o l a t e d v a l u e s of thes e p l o t s . The a c t u a l c o m p u t a t i o n of t h e s e i n t e g r a l s was c a r r i e d out by L i p s i c a s ( p r i v a t e commu-n i c a t i o n ) e x t e n d i n g the range of down t o 0.Q1-* These v a l u e s agree w i t h the e x t r a p o l a t e d v a l u e s . The a c t u a l ! v a l u e s a r e p r e s e n t e d i n Table I I f o r 0.03 < f>€. ^  0.09.. 67 n 15 17 19 21 I ?± ; ! 0.03 O.O1* 0 . 0 5 0.06 0 .08 0.09 2 \ 0.139^ 0.1276 0.1190 0.1123 0.1027 0.0991 i h i O . 0 9 A O.O836 0.0799 0.0753 0,0672 0.06>+8 i i 2 j 0.16^+6 0.lh2k 0.1267 ; 0.11^9 O.O986 0.0926 h j 0.1125 0.0972 0.0863 O.O783 0.0671 0.063 2 I 0.2516 0.2067 0.1762 0.15^1 0.12M+ 0.1139 i 1+ ; 0.1763 0.lkk6 0.1231 6.1076 0.0867 0.0793 2 1.8029 1.3151 1.0131 0.8116 0.5626 OA813 h ! 1.3098 0.953^ 0.7332 0.5867 O.H-061 0.3^73 2 ".! 3.957 2,8028 2.105^ 1.6^89 1.096W 0.920 h j 2.896 2.0^55 1.5336 1.1996 0.7965 0.668 j 2 I 9.0533 6.2H-86 ^*5916 3-52^6 2.2569 1.8586 j h j 6.6630 5851 3-3617 2.5773 1.6h77 1-3565 ( 2 I 2 1 A 1 3 1 A . M + 8 1 0 A 1 5 7.85^2 ^.859 3.9322 h 15.833 10.65 7.6580 5.7678 3.5622 2.881** Table I I . V a l u e s of I(p,n) f q r d i l u t e H 2 gas i n c l u d i n g the I f i r s t quantum c o r r e c t i o n f o r a Lennard-Jones i s o t r o p i c p o t e n t i a l . H-.3. H„ a n d He M i x t u r e . M-.3.1. R e s u l t s . The p r o t o n s p i n r e l a x a t i o n time was measured i n Hg and He m i x t u r e s as a f u n c t i o n of d e n s i t y and c o m p o s i t i o n from room temp e r a t u r e up t o 700°K. The d a t a were t a k e n a t room tempera-t u r e f o r f i v e c o n c e n t r a t i o n s of the m i x t u r e but the temperature dependence was s t u d i e d f o r o n l y two m i x t u r e s , one w i t h 32 .2$ He and the o t h e r w i t h 77.0% He. T y p i c a l p l o t s of ^ v s . j 5 are shown i n f i g . 16a f o r 32'.2% He and 77-0% He m i x t u r e s a t room temperature whereas f i g . 16b shows the p l o t f o r 77-0% He m i x t u r e a t 700°K, which r e p r e s e n t s the w o r s t s i g n a l t o n o i s e c o n d i t i o n s . The s o l i d l i n e i n b o t h cases r e p r e s e n t s the l e a s t square f i t f o r the d a t a . F i g . 17 shows Ttjo as a f u n c t i o n o f p e r c e n t a g e of He p r e s e n t i n the m i x t u r e a t d i f f e r e n t t e m p e r a t u r e s . W i t h i n e x p e r i m e n t a l e r r o r s and j u d g i n g from the d a t a a v a i l a b l e , the dependence of T,jf> on the c o n c e n t r a t i o n o f He was found to be l i n e a r a t h i g h e r t e m p e r a t u r e s but a l l the d a t a a t room temp e r a t u r e cannot be f i t t e d by a s t r a i g h t l i n e . However, when a s t r a i g h t l i n e i s drawn t h r o u g h the d a t a as shown i n f i g . 17, i t i s d i f f i c u l t t o say whether the d e v i a t i o n of the d a t a from the l i n e a r f i t i s s i g n i f i c a n t or n o t . I f i t i s s i g -n i f i c a n t the e x t r a p o l a t i o n of t o 100$ He i s perhaps n ot j u s t i f i e d . The t h e o r e t i c a l i n v e s t i g a t i o n gave more i n s i g h t t o t h i s problem which w i l l be d i s c u s s e d i n the f o l l o w i n g s e c t i o n . Assuming t h a t the e x t r a p o l a t i o n i s v a l i d , the v a l u e s o f T' jj> f o r 100$ He are g i v e n i n Table I I I f o r d i f f e r e n t temp-69 10.0 8.0 i-o <D w 6.0 • H *f.0 2.0 O 3 2 . 2 $ He 9 77.0$ He 20 >+0 60 D e n s i t y i n Amagats 80 100 FIG. 16a. T 1 v e r s u s D e n s i t y a t 293 K i n Hg-He m i x t u r e , .11 • R i e h l P e r centage of He F I G . 17. Dependence of T^/j=> on Helium c o n c e n t r a t i o n i n Hg-He m i x t u r e . o e r a t u r e s . TEMP. V W ^ C / A M 0 ^ ^ ) 293°K 0 . 0 6 4 - 5 4-00°K 0.0635 500°K 0.0615 600°K 0.06 700°K 0.059 750°K 0.058 T a b l e I I I . fal^jii -Up o b t a i n e d from e x t r a p o l a t i n g the e x p e r i -2 m e n t a l d a t a t o 100$ He. 72 M-.3.2. I n t e r p r e t a t i o n . I f f and Q . r e p r e s e n t the number of H p and He m o l e c u l e s per u n i t volume r e s p e c t i v e l y , then the d e n s i t y of the combined system i s g i v e n by ? r -- ? H e (4.3.1 .) and i f j? and j? are the d e n s i t i e s of o r t h o - and p a r a - Hg r e s -p e c t i v e l y , t h e n V - p » + ( l K 3 - 2 - ) The f r a c t i o n a l d e n s i t i e s f o r the m i x t u r e can be d e f i n e d as P - tl* P - . p & »e . f T > V f r > h z J 0+.3.3O and f o r normal Hg ~90~(^'lli) ? For t h i s m i x t u r e t h e r e a re t h r e e t y p e s of i n t e r a c t i o n s t h a t have t o be c o n s i d e r e d : ( i ) the i n t e r a c t i o n between o r t h o -o r t h o m o l e c u l e s , ( i i ) the i n t e r a c t i o n between o r t h o - p a r a m o l e c u l e s , ( i i i ) t he i n t e r a c t i o n between o r t h o - He atoms. The f i r s t two ty-pes o f . i n t e r a c t i o n s were d i s c u s s e d i n the p r e -I v i o u s s e c t i o n and t h e i n t e r a c t i o n between o r t h o - Hg and He m o l e c u l e s w i l l be d i s c u s s e d below. Hg - He I n t e r a c t i o n : -T h i s i n t e r a c t i o n depends on the o r i e n t a t i o n of m o l e c u l e 1 a l o n e and hence can be r e p r e s e n t e d by the h a m i l t o n i a n jj^ . As d i s c u s s e d i n the p r e v i o u s c h a p t e r , because of t h e l a r g e energy d i f f e r e n c e between r o t a t i o n a l s t a t e s , the t r a n s i t i o n s produced by t h i s i n t e r a c t i o n a r e thos e w i t h no change i n the r o t a t i o n a l a n g u l a r momentum of the m o l e c u l e , i . e . A J = 0. U) Thus t h i s c o n t r i b u t e s o n l y t o £j . U s i n g e q u a t i o n (3.1+.l5) t h i s can be w r i t t e n as where , i/o ,n i • t (X,t M- and I (2) are e v a l u a t e d u s i n g a p p r o p r i a t e parameters n t e r a c t i o n s ^ 26 / 1 A - « e f o r the Hg - He i n t e r a c t i o n s and ( v\Q) i s d e f i n e d as where j> i s the d e n s i t y i n i d e a l amagats. U s i n g e q u a t i o n s (3 A.15), (3A. 16) and (3.*+.17), the con-t r i b u t i o n t o B'^-^.J) and g^^3") d u e t o Hg - He i n t e r a c t i o n s can be w r i t t e n as r "J -1 = --T?He^'>  f (zr-Mn+j) (^ .3.7.) 5- «ev., (zT-^fa+jf (*f.3>-SO E v a l u a t i n g e q u a t i o n s A ^ ^ O and (^ .3.8.) f o r J = l and J=3 and addi n g t o the c o n t r i b u t i o n s from Hg - Hg i n t e r a c t i o n s , B 1 ( l , l ) , B (3,3), B g ( l , l ) and Bg(3,3) f o r the m i x t u r e can s t i l l be g i v e n by e q u a t i o n s (h.2.10) and (k.2.11) i f X i s d e f i n e d as Th e r e f o r e j 5 / ^ f o r any c o n c e n t r a t i o n of He i s g i v e n by e q u a t i o n (^ .2. 1*+) where X i s d e f i n e d by e q u a t i o n 3 - 9•) • E q u a t i o n (k.2.lh) was s o l v e d f o r X as d e s c r i b e d i n the p r e v i o u s s e c t i o n u s i n g the e x p e r i m e n t a l v a l u e of iTijp f o r the m i x t u r e a t any c o m p o s i t i o n . Having o b t a i n e d X, fil^2^ c a n be o£|ained from, e q u a t i o n (H-.3.9.) where '•and ) * *" a r e o b t a i n e d from the a n a l y s i s of Hg r e s u l t s ; . I t may be noted here t h a t when the e x t r a p o l a t e d v a l u e of T, J j> f o r 100$: He i s used the hydrogen parameters need not be used and t h e r e f o r e the e r r o r due t o the u n c e r t a i n t i e s i n hydrogen parameters can be e l i m i n a t e d . The v a l u e of (V0 ) was o b t a i n e d from t h e ' e x p e r i m e n t a l v a l u e s of ".^iff f o r 32.2$ He 77.0% He and 100$ He from room tem p e r a t u r e up t o 700°K, w h i l e the v a l u e s below room temperature/were o b t a i n e d from R i e h l ' s d a t a f o r 60$ :and 100$ He- (/Q i s o b t a i n e d from e q u a t i o n (k.3.6.) and i s shown i n f i g . 18 as a f u n c t i o n o f te m p e r a t u r e . The t h e o r e t i c a l p l o t s of "7//j> as a f u n c t i o n of pe r c e n t a g e of He were o b t a i n e d f o r t h e s e t h r e e s l i g h t l y d i f f e r e n t v a l u e s of (>l| )^* W e b o t h a t room temperature and a t 700°K. F i g . 19 shows these p l o t s a l o n g w i t h the e x p e r i m e n t a l d a t a . As can be seen from the f i g u r e the t h e o r e t i c a l p l o t i n which (*)'0} o b t a i n e d from 77.0$ He m i x t u r e was used g i v e s b e s t f i t t o the e x p e r i m e n t a l r e s u l t s . A l s o the t h e o r e t i c a l p l o t s i n d i c a t e t h a t the depend-ence of T , jf> on the c o n c e n t r a t i o n of He i s not l i n e a r . The v a l u e of Tijf f o r 100$ He p r e d i c t e d by the t h e o r y i s about 15$ s m a l l e r t h a n the e x t r a p o l a t e d v a l u e a t room temperature and a t 700°K the v a l u e i s o n l y about 3$ or h% s m a l l e r . At s u f f i c i e n t l y l ow t e m p e r a t u r e s when a l l the m o l e c u l e s a re i n t h e i r l o w e s t r o t a t i o n a l s t a t e one would not expect t h i s n o n - l i n e a r depend-ence of T^f on the per c e n t a g e of He as t h i s i s a m a n i f e s t a t i o n o f t r a n s i t i o n s between d i f f e r e n t J s t a t e s . From the above con-75 •15 'u J -C\J o 10 EH 5 o o • From ( T 1 / p ) r o o ^ H e ° " <V*>33.2*He A o A o A o A o A o 100 200 300 4-00 500 600 700 Temperature i n K TT — H P FIG. 18. Temperature dependence of (k ) 2 as o b t a i n e d f r om E q . (4,2.14-) and u s i n g ,the e x p e r i m e n t a l •values of (T^/p) i n Hg-He; m i x t u r e . P e r c e n t a g e of He FIG. 19. T h e o r e t i c a l p l o t s of T^/f v e r s u s $He when Eq.(k.2.1k) was n o r m a l i s e d w i t h the e x p e r i m e n t a l v a l u e s a t 32.2$ He, 77.0$ He and 100$ He. 77 s i d e r a t i o n s i t can be s a i d t h a t the dependence of T, jf> on the c o n c e n t r a t i o n o f He i s - n o t l i n e a r above 150°K and the e x t e n t o f n o n - l i n e a r i t y i s s m a l l e r a t h i g h e r t e m p e r a t u r e s . At lo w e r c o n c e n t r a t i o n s o f He, Hg - Hg c o l l i s i o n s p l a y an i m p o r t a n t r o l e i n r e l a x a t i o n mechanism and hence the v a l u e of v ) 0 ) o b t a i n e d f r o m 32 .2$ He m i x t u r e i n c l u d e s the e r r o r s of Hg par a m e t e r s . Under t h e s e '.circumstances the v a l u e of {r}0j as o b t a i n e d from 77-0$ He m i x t u r e i s perhaps a good compromise as i s a l s o i n d i -c a t e d by f i g . 19-, F i g . 20 shows the computed v a l u e s of T,Jj> f o r 100$ He u s i n g e q u a t i o n s (*+•.3-9-) and (^f.2.1*0 when n o r m a l i s e d a t 32 .2$ He and 77-0$ He from room;temperature up t o 700°K, a l o n g w i t h t h e e x p e r i m e n t a l v a l u e s ad- o b t a i n e d from e x t r a p o l a t i n g the d a t a t o 100$ He. Below room temperature e q u a t i o n (*+. 2 . 1*0 was n o r -m a l i s e d w i t h R i e h l 1 s d a t a f o r 60$ He to compute the v a l u e of Tt/f f o r 100$ He. F i g . 21 shows (_ k„CO / k6Ca93)j a s a f u n c t i o n of temp e r a t u r e a l o n g w i t h the v a r i o u s t h e o r e t i c a l p l o t s . The t h e o r e t i c a l p l o t s were o b t a i n e d as d e s c r i b e d i n the case of Hg assuming t h a t l^*6*x) i s g i v e n by andv.also by „, \r f a x ) , _n + ^ X? X. where co6 = - H/ (0 The p l o t s were shown f o r the case n=19 when IxOu) i s g i v e n by e q u a t i o n C+.3.IO) and f o r n' =13 and 15 when 4 r J f a x ) i s g i v e n by e q u a t i o n (^-.3.11). I t was seen i n the case of Hg / / 4 • R i e h l o P r e s e n t Work Computed V a l u e s A N o r m a l i s e d a t 77.0$ He ® "• » 33.2$ He ^ » " 60.0$ He 0 100 200 300 4-00 500 600 700 800 Temperature i n K 20. Comparison of the e x t r a p o l a t e d v a l u e s of ( T . j / p ) H _ H e w i t h the computed v a l u e s u s i n g Eq.(h.2.1h) 2 5.0 h.o 3-0 2.0_ 1 .0 — 0 ( n , n ' ) = (6,15) ) = (6,13) 0 100 200 300 H-oo 500 600 700 Temperature i n K FIG.-21. Comparison of the e x p e r i m e n t a l v a l u e s o f j \ o t T)/k o ( 2 9 3 f ) H 2 ~ H e w i t h the t h e o r e t i c a l v a l u e s u s i n g Eqs. {h.2.22) and ( 4 - . 2 . 2 H - ) t h a t the . t h e o r e t i c a l curve o b t a i n e d f o r the case ii =12 i s almost i d e n t i c a l w i t h n / = 1 3 . I t can be- seen from the f i g u r e t h a t the e x p e r i m e n t a l d a t a are a d e q u a t e l y r e p r e s e n t e d when fe-"Vax) - (H- .3.12) and the i n t e r m o l e c u l a r p o t e n t i a l f o r Hg - He p a i r can be w r i t t e n as R i e h l has a l s o o b t a i n e d an e x p r e s s i o n f o r the i n t e r m o l e c u l a r p o t e n t i a l between Hg and He m o l e c u l e s u s i n g s c a t t e r i n g t h e o r y t o eiValuate the c o l l i s i o n c r o s s - s e c t i o n s . He assumed t h a t the i s o t r o p i c and a n i s o t r o p i c p a r t s of the i n t e r m o l e c u l a r p o t e n t i a l s have the same dependence on / i and t h a t t h i s depend-ence i s g i v e n by [A €JCIS (• kA.) _ QtL^ where A, k and B are con-s t a n t s . He a l s o computed the v a l u e s o f T, jp u s i n g t h e Bloom-6 7 8 Oppenheim t h e o r y ' a s w e l l as s c a t t e r i n g t h e o r y , assuming t h a t t h e a n i s o t r o p i c p a r t i s g i v e n by and the i s o t r o p i c p a r t i s g i v e n by Lennard-Jones p o t e n t i a l . F o r the b e s t f i t of h i s d a t a below room t e m p e r a t u r e , he o b t a i n e d £ = 0 . 2 8 from s c a t t e r i n g t h e o r y and S = O.lU- from Bloom-Oppenheim. t h e o r y when the t h e o r e t i c a l v a l u e was norimVlis'ed t o the e x p e r i m e n t a l v a l u e a t 100°K. T h i s i s perhaps due to the f a c t t h a t the Bloom-Oppenheim t h e o r y does not ta k e i n t o account the quantum e f f e c t s . I t was found t h a t when the t h e o r e t i c a l v a l u e u s i n g the Bloom-Oppenheim t h e o r y was n o r m a l i s e d t o the e x p e r i m e n t a l v a l u e a t 300°K or above, u s i n g t h e 77-6% He sample, the v a l u e o f g o b t a i n e d was 0.173 i n s t e a d of 0.1*+. I f R i e h l 1 s t h e o r y was f i t t e d t o the same d a t a a t 300°K, i t appears t h a t the v a l u e o f £ so o b t a i n e d would be l e s s t h a n 0.28 by a t l e a s t 10$. T h i s may i n d i c a t e t h a t the quantum e f f e c t s n e g l e c t e d i n the Bloom-Oppenheim t h e o r y may not be as i m p o r t a n t above room temperature as a t lower t e m p e r a t u r e s . M-.H-. H 2 - C 0 2 M i x t u r e . ^ . H - . I . R e s u l t s . T, was measured i n Hg - COg m i x t u r e f o r t h r e e c o n c e n t r a -t i o n s of C 0 2 a t room -temperature as a f u n c t i o n of d e n s i t y . The temperature dependence of Tjj^ was s t u d i e d f o r two of these m i x t u r e s where the t o t a l number d e n s i t y of the gas was determined as d e s c r i b e d i n Chapter I I . F i g . 22 shows the dependence o f T, o n ^ a t room tempera-t u r e f o r 16.k% C 0 2 , 55-0$ C 0 2 and 77-5% COg. I t can be seen from the f i g u r e t h a t T ( -does n ot depend l i n e a r l y on d e n s i t y f o r h i g h e r c o n c e n t r a t i o n s of COg. ^"ijp v s . P a t room tempera-t u r e i s shown i n f i g . 23 and the e x t r a p o l a t e d v a l u e of ~H jj> t o ' f = 0 g i v e s ^ijy i n d i l u t e gas. The same proced u r e was used t o o b t a i n the v a l u e of Tfy m d i l u t e gas a t h i g h e r t e m p e r a t u r e s . F i g . 2h shows Ti as a f u n c t i o n of pe r c e n t a g e of COg. from room temp e r a t u r e up t o 500°K. W i t h i n e x p e r i m e n t a l e r r o r s TiJ'p was found t o depend l i n e a r l y on the c o n c e n t r a t i o n of COg. Above 55P°K, the r e c o v e r y of m a g n e t i s a t i o n was found t o be a sum of two e x p o n e n t i a l s and the r a t i o of the time con-D e n s i t y i n Amagats FIG.22. Dependence of T 1 on d e n s i t y a t 2 9 3 K f o r H 2-C0 2 m i x t u r e . 0 10 20 30 h0 50 60 70 80 90 100 D e n s i t y i n Amagats FIG. 23. Dependence of T^/p on d e n s i t y a t 293 K. Percentage of COg FIG. 2h. Dependence of....T1 /p on the p e r c e n t a g e ' o f COg i n Hg-COg m i x t u r e . s t a n t s was a p p r o x i m a t e l y 2.5 a t 600°K. The r e a s o n f o r t h i s was su s p e c t e d t o be Hg and COg r e a c t i n g c h e m i c a l l y t o g i v e . ' r i s e t o CO and HgO and. the two time c o n s t a n t s c o u l d be due t o the r e l a -x a t i o n of p r o t o n s i n Hg and HgO. T h i s ^ r e a c t i o n was s t u d i e d by Graven and Long 2' ? from 800°C up t o 1100°C i n q u a r t z v e s s e l s . A c c o r d i n g t o t h e i r r e s u l t s the e x t e n t of r e a c t i o n v a r i e d froJn 0.5$ t o 2.0%, depending on the r e a c t a n t c o n c e n t r a t i o n , a t 900°C and 0.53 second r e a c t i o n t i m e . The t o t a l p r e s s u r e used by them was an atmosphere or l e s s . I n . t h e p r e s e n t case the e f f e c t was n o t i c e a b l e f o r 95-0% COg m i x t u r e even a t 600°K. The p r e s s u r e s used were from 15 t o 65 atmospheres. The material of the sample h o l d e r , Be-Cu, c o u l d be a c t i n g . a s a c a t a l y s t t o the r e a c t i o n and the h i g h p r e s s u r e s i n v o l v e d c o u l d be the r e a s o n f o r the r e a c t i o n t o tak e p l a c e a t much lo w e r t e m p e r a t u r e s t h a n a n t i c i -p a t e d . I t may be no t e d t h a t the e f f e c t c o u l d n o t be observed i f t h e measurement was made w i t h i n the f i r s t hour a f t e r the gas b e i n g i n t r o d u c e d i n t o the pr e h e a t e d sample h o l d e r and the e f f e c t was al m o s t complete a f t e r 8 or 10 h o u r s . Because of t h i s c o m p l i c a t i o n the temperature dependence f o r Hg - COg m i x t u r e s was s t u d i e d o n l y up t o 50d°K. The e x t r a p o l a t e d v a l u e f o r 100$ COg g i v e s the c o n t r i b u -t i o n due t o H p - COp c o l l i s i o n s a l o n e and l o g (Tilp) v s . l o g " 7 " K i s shown i n f i g . 25- The d e v i a t i o n from l i n e a r dependence o f logfVe) v s . l o g T a t H-50°K and 500°K i s p r o b a b l y due to the e f f e c t of c h e m i c a l r e a c t i o n d i s c u s s e d above. M-.H-.2. I n t e r p r e t a t i o n . The i n t e r a c t i o n between o r t h o - H 0 and CO 0 m o l e c u l e s can 00 OA be a d e q u a t e l y d e s c r i b e d by q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n s i n c e the m o l e c u l e s do not have a permanent d i p o l e moment. The h a m i l t o n i a n f o r t h i s i n t e r a c t i o n i s g i v e n by and AA 01.) I n. where g | i s the quadrupole moment of o r t h o - Hg m o l e c u l e and i s the quadrupole moment o f COg m o l e c u l e . I t i s assumed t h a t t h i s i n t e r a c t i o n cannot produce t r a n s i t i o n s between d i f f e r e n t J s t a t e s of the o r t h o - m o l e c u l e . S i n c e the COg mole-c u l e has a h i g h moment o f i n e r t i a compared w i t h Hg, many of i t s r o t a t i o n a l s t a t e s a re p o p u l a t e d . C o l l i s i o n s w i t h Hg m o l e c u l e s w i l l produce t r a n s i t i o n s between d i f f e r e n t J s t a t e s f o r COg m o l e c u l e . When t h e s e t r a n s i t i o n s a r e a l l o w e d the v a l u e of C f o r COg becomes 5 / 2 . The c o n t r i b u t i o n s t o B, ( J , J ) and B g ( J , J ) due t o Hg - COg i n t e r a c t i o n s • c a n be w r i t t e n as 6, A.2.) A A . 3 J and X f o r Hg - COg m i x t u r e can be d e f i n e d as ' cm) X where I2.5-* »zCoz (AAA.) A A . 5 - ) (ID =y^'^ ,.,3 where 0-.tju. are e v a l u a t e d f o r RV, - COg i n t e r a c t i o n s „ E v a l u a t i n g e q u a t i o n s (H-.H-.2.) and (H-.H-.3-) f o r J = l and J=3 and ad d i n g t o the c o n t r i b u t i o n s froni" Hg - Hg i n t e r a c t i o n s B^C^J) and Bg(jJ?'') f o r Hg - COg m i x t u r e a re g i v e n by e q u a t i o n s (H-.2.10) and (4-.2.11) where X i s d e f i n e d by e q u a t i o n (H-.H-.H-.). ?JT( f ° r any--composition of the m i x t u r e i s g i v e n by e q u a t i o n (H-.2.1H-) whi c h c a n be s o l v e d f o r X on t h e computer as d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s . From-"the v a l u e of X thus o b t a i n e d ^ co^ °an be o b t a i n e d f r o m e q u a t i o n (H-.H-.H-.) knowing • U s i n g e q u a t i o n s ( H - . H - . 5 - ) H - CO ' ^~ and (H,H-.6.), (k0+C-k,Y Z can be w r i t t e n as -he r e P , = fT / p ; 5 A F i g . 26 r e p r e s e n t s ( f r 0 * " ^ - 4 ( ) as a f u n c t i o n of temp-erature'. The t h e o r e t i c a l p l o t s of 7~, v s . per c e n t a g e of COg are shown i n f i g . 27 f o r d i f f e r e n t v a l u e s of XCo as o b t a i n e d f r om the a n a l y s i s of the d a t a i n d i f f e r e n t m i x t u r e s a t room tempe r a t u r e and a t 500°K. The e x p e r i m e n t a l d a t a f i t s b e s t the p l o t s i n which X C o o b t a i n e d from h i g h e r c o n c e n t r a t i o n s COg were used. I n t h i s the t h e o r e t i c a l p l o t s a l s o c o n f i r m the e x p e r i m e n t a l o b s e r v a t i o n t h a t T/jp i s p r o p o r t i o n a l t o the c o n c e n t r a t i o n of C0 o. 10.0 5.0 U 5 } 0 200 - 300 . ;.,. ,. ' . koo 500 ""•"Temperature i n °K FIG . 26. Temperature dependence of (k +Ck 1 ) 2'" 2 as o b t a i n e d from Eqs. (U-.2.1H-) and (4-.H-.7) u s i n g the e x p e r i m e n t a l v a l u e s of T^  /p co P e r c e n t a g e of CO^ FIG. 2 7 . T h e o r e t i c a l p l o t s of T^/f v s . #C0 2 i n H 2 - C 0 2 m i x t u r e when Eq. (A.2.1 h) wa-s n o r m a l i s e d w i t h the e x p e r i m e n t a l v a l u e s a t d i f f e r e n t c o n c e n t r a t i o n s of CO The f a c t t h a t (k0+ C k^j2 does n ot v a r y v e r y much w i t h temperature i n d i c a t e s t h a t the c o n t r i b u t i o n s from s h o r t range i n t e r a c t i o n s t o k0 'k, are v e r y s m a l l . S i n c e the i n f o r m a -t i o n i s a v a i l a b l e o n l y on (^t+c<k^ and not on k0 and kf s e p a r a t e l y , i t i s assumed t h a t C fc,» "fc0 a s a f i r s t approxima-t i o n . The t h e o r e t i c a l v a l u e of -fc, i s o b t a i n e d from e q u a t i o n (4-. 2.18) with.n= 5 and Q i s o b t a i n e d by n o r m a l i s i n g the t h e o r -e t i c a l and e x p e r i m e n t a l v a l u e s a t 293°K. F i g . 28 shows the t h e o r e t i c a l p l o t s o f as a f u n c t i o n of tempe r a t u r e a l o n g w i t h the e x p e r i m e n t a l v a l u e s . The agreement between the t h e o r e t i c a l and e x p e r i m e n t a l p l o t s i n d i c a t e t h a t the dominant c o n t r i b u t i o n t o the a n i s o t r o p i c i n t e r a c t i o n s between EL, and COg m o l e c u l e s i s qu a d r u p o l e - q u a d r u p o l e i n t e r a c -t i o n . The quadrupole moment was o b t a i n e d by n o r m a l i s i n g the t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s a t 293°K and the v a l u e + 26 o b t a i n e d i s (M-.83 - 0 .3) X 10" e.s.u. T h i s v a l u e i s compar-a b l e t o the v a l u e s o b t a i n e d from o t h e r e x p e r i m e n t s . 1 A CM O O I CM W m CM E H 1 .2 1 .0 100 200 300 Temperature i n °K M-00 500 FIG. 28. Comparison of [k 1(T)/k 1(293)] H2 C 02 w i t h the computed v a l u e s u s i n g Eq. A.2.18). CHAPTER V EXPERIMENTAL RESULTS AND DISCUSSION ( c o n t ' d ) 5 . 1 . METHANE 5 . 1 . 1 . R e s u l t s The 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 time T-^  was measured i n methane as a f u n c t i o n of d e n s i t y from room temperature up t o 700°K. Knowing the p r e s s u r e , the d e n s i t y a t any temperature was o b t a i n e d from the c o m p r e s s i b i l i t y t a b l e s g i v e n by "American 28 P e t r o l e u m I n s t i t u t e " . F i g . 29a shows T-^  as a f u n c t i o n of den-s i t y up t o 100 amagats a t room t e m p e r a t u r e . I t can be seen f r o m the graph t j r a t the dependence o f T^ on d e n s i t y i s l i n e a r i n the low d e n s i t y r e g i o n wherfeas i t i s n o n - l i n e a r a t h i g h e r d e n s i t i e s due t o the e f f e c t o f t h r e e body c o l l i s i o n s . The v a l u e of T/ jj> i n the d i l u t e gas i s o b t a i n e d by p l o t t i n g T/ jf as a f u n c t i o n of d e n s i t y and • ' e x t r a p o l a t i n g i t t o ze r o d e n s i t y . F i g . 29b shows ^Jf^S* a t room t e m p e r a t u r e . F i g s . 30a and 30b show s i m i l a r p l o t s f o r CH^ a t 720°!K and the same procedure i s adopted a t a l l o t h e r t e m p e r a t u r e s . The v a l u e of Ti jf obtained- a"t. room temperature (20.2 m.sec/Amagat) I s i n agr e e -pQ .^ ment w i t h the v a l u e s o b t a i n e d by B r i d g e s y (21.0 - 5$ m.sec/ Amagat). L i p s i c a s , Bloom and M u l l e r 3 0 (20.0 m.sec/Amagat) 31 and Johnson and Waugh (23.0 m.sec/Amagat). F i g . 31 shows T7Jj> as a f u n c t i o n of temperature a l o n g w i t h the low tempera-t u r e d a t a a v a i l a b l e from o t h e r work whereas l o g (r'j^j as a f u n c t i o n of l o g T °K i s shown i n f i g . 3 2 / The d a t a can be 20 I h0 60 D e n s i t y i n Amagats 86 100 FIG. 29a. Dependence of T 1 on d e n s i t y , a t 293 K f o r CH^. co - P 03 cC O cn 22.0 U 20.0 b* 18.0 h 16.0 L 20 kO 60 80 D e n s i t y i n Amagats 100 FIG. 29b. Dependence of T^/p on d e n s i t y a t 293°K f o r CH^. 5 10 15 20 D e n s i t y i n Amagats FIG . 30a. Dependence of T 1 on d e n s i t y a t 720 K f o r CH^. 7-0 25 -in -p cd uo cc • H 6.0 5.oL k.o 0 -O- I 5 25 , 1 0 15 20 D e n s i t y i n Amagats FIG . 30b. Dependence of T ^ / j 3 on d e n s i t y a t 720°K f o r CH^. 100 200 300 4-00 500 600 700 —-8G0 Temperature i n °K F I G . 31. Dependence of T^/f on temperature f o r CH^. OA f i t t e d by a s t r a i g h t l i n e w i t h a s l o p e e q u a l t o - 1.5 i n d i c a t -— 1 5 i n g t h a t Tfjj, i s r o u g h l y p r o p o r t i o n a l t o T ' 7 . T h i s i s i n on + agreement w i t h the s l o p e s o b t a i n e d by B r i d g e s 7 (-l.1+5 - 5%) , Johnson and Waugh (-l.Lf2) and T r a p p e n i e r s e t a i ' ° (-1.55 - 0.03). 5.1.2. I n t e r p r e t a t i o n . Due t o the h i g h Moment o f I n e r t i a , the' r o t a t i o n a l s t a t e s of the CH^ m o l e c u l e £T~§ c l o s e l y spaced and hence q u i t e a l a r g e number of r o t a t i o n a l s t a t e s a re a p p r e c i a b l y p o p u l a t e d even a t low t e m p e r a t u r e s . As a r e s u l t , the t h e o r y developed i n Chapter I I I cannot be d i r e c t l y a p p l i e d t o t h i s system because of the l a r g e number, of equations, o f (3-2.23) t h a t have t o be r e t a i n e d t o g i v e a c o r r e c t r e p r e s e n t a t i o n o f T-^ . T h e r e f o r e , f u r t h e r a p p r o x i m a t i o n s w i l l be made t o the g e n e r a l r e s u l t s o b t a i n e d i n Chapter I I I .in o r d e r t o get some i n f o r m a t i o n about a n i s o t r o p i c i n t e r m o l e c u l a r i n t e r a c t i o n s . Assuming t h a t the i n t r a m o l e c u l a r i n t e r a c t i o n s t h a t a r e r e s p o n s i b l e f o r the r e l a x a t i o n mechanism a r e g i v e n by 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 s a l o n e , the r e l a x a t i o n r a t e can be w r i t t e n i n the s h o r t c o r r e l a t i o n time l i m i t , u s i n g e q u a t i o n (3.2.35), as _L . h. y V f £ PT T0(jof^ £ £l<L. (5-1.1.) r " 3 T0 J° « A l o ( ' Assuming t h a t A,, i s a s l o w l y v a r y i n g f u n c t i o n of o( i t can be r e p l a c e d by i t s average v a l u e , <^  /N^N and can be c " ' t a k e n o u t s i d e the summation. S i n c e ^ c ' ^ - / e q u a t i o n (5>1.1.) can be w r i t t e n as / 2- „A n l 2 -jj- j Y Jo fatO < A ( > * (5.1.2I) where the bar r e p r e s e n t s the ensemble average. U s i n g t h e h i g h temperature r e s u l t J ( T T O O C . - T (5.1-3.) (5-iA.) the r e l a x a t i o n r a t e can he w r i t t e n as T j < A ( > I f the t r a n s i t i o n s between d i f f e r e n t J s t a t e s a r e i n f r e q u e n t compared w i t h the t r a n s i t i o n s among d i f f e r e n t nij l e v e l s of the salne J - m a n i f o l d , '/(Aj^ can be i d e n t i f i e d w'j|.th the c o r r e -l a t i o n time t£ of the ' s ' p i n - r o t a t i o n a l i h t e r a c t i b h . 'Under such c o n d i t i o n s . ^ « T rc ) (5.1.5.) S i n c e e x p e r i m e n t a l r e s u l t s i n d i c a t e t h a t X oc r 1 ' 5 (5-1.6.) V i t can be con c l u d e d t h a t '" zc <* T ^ 5 (5.1.7.) 6 7 Bloom and Oppenheim have e v a l u a t e d TC f o r the case of d i a t o m i c m o l e c u l e s i n terms of i n t e r m o l e c u l a r a n i s o t r o p i c i n t e r a c t i o n s u s i n g the " c o n s t a n t a c c e l e r a t i o n a p p r o x i m a t i o n " and th e y o b t a i n f o r ' t h e case where no r o t a t i o n a l t r a n s i t i o n s take • p l a c e , a r e s u l t o f the form ' /^A/.N T ( V I ^ (5.1.8.) where A ( J ) i s a f u n c t i o n of the r o t a t i o n a l s t a t e J , (f) i s d e f i n e d by e q u a t i o n (3.H-.14-) and / i i s the reduced mass. The a s s u m p t i o n t h a t does not s t r o n g l y depend on J i s v a l i d o n l y i f A ( J ) does not s t r o n g l y depend on J . The i n t e g r a l s J^(f) are e v a l u a t e d by Bloom, Oppenheim e t a l ^  f o r the cases where the i s o t r o p i c i n t e r m o l e c u l a r p o t e n t i a l i s g i v e n by ha r d sphere p o t e n t i a l and Lennard-Jones p o t e n t i a l . F o r the h a r d sphere model the i n t e g r a l s a r e independent of temperature and hence 2^. << T , F u r t h e r t h i s model p r e d i c t s t h a t T/If 1~/S f o r a 1 1 system^ l i k e CH^ - X. I n f a c t , the 1 5 T , y b e h a v i o u r i s not observed f o r the case when X i s He as can be seen from the f o l l o w i n g s e c t i o n . T h e r e f o r e , the temper-a t u r e dependence of JP w i l l be c o n s i d e r e d t a k i n g a more r e a l i s t i c form of the i s o t r o p i c p a r t of the i n t e r m o l e c u l a r p o t e n t i a l s namely the Lennard-Jones p o t e n t i a l , and assuming t h a t the a n i s o t r o p i c p a r t i s g i v e n by a s i m p l e power law (?\a.*) - / ^ ( 5 . 1 .9 . ) Then : lft * T 5 / 7 l ( ^ ) (5 .1 .10) where (<y) (y) 2. , From the e x p e r i m e n t a l v a l u e s of Ti/p> a t d i f f e r e n t temp-r " -e r a t u r e s the temperature dependence of ^jL(p^ / x(f>^) J i s o b t a i n e d from e q u a t i o n (5.1 .J-0) and i s shown i n f i g . 33 [ C H "~ C rT v s . t e m perature as o b t a i n e d from the t a b l e s g i v e n by Bloom, Oppenheim et a l f o r n=3, 5 and 7- I t may be w o r t h m e n t i o n -i n g h ere t h a t the r a t i o of the i n t e g r a l s a t d i f f e r e n t tempera-t u r e s i s independent of the parameter p. I f the o c t o p o l e -o c t o p o l e i n t e r a c t i o n i s assumed to be the dominant i n t e r a c t i o n between the two methane m o l e c u l e s , t h e n n s h o u l d take the v a l u e of 7- But as i t can be seen from the f i g . 33 5 the p l o t w i t h n=3 f i t s b e s t w i t h the e x p e r i m e n t a l r e s u l t s . However, t h i s cannot be t a k e n too s e r i o u s l y as the e q u a t i o n (5-1.8. ) i s o b t a i n e d assuming t h a t p e r t u r b a t i o n t h e o r y can be used to 2.0 1 .0 0 o T h e o r e t i c a l P l o t s E x p e r i m e n t a l V a l u e s P r e s e n t Work B r i d g e s •••• O n = 3 0 100 200 300 hoo 500 600 700 800 Temperature i n FIG- 33» Comparison of [ i ( 6 , n ) T / I ( 6 , n ) 2 ^ C \ C \ as o b t a i n e d from E q . (5-1 .10) w i t h the computed v a l u e s u s i n g the n u m e r i c a l v a l u e s of the integrals„ o 102 e v a l u a t e the t r a n s i t i o n p r o b a b i l i t i e s between the r o t a t i o n a l s t a t e s of the m o l e c u l e which i s not t r u e f o r t h i s c a s e . A d d i t i o n a l temperature dependence c o u l d a l s o be coming from. A ( J ) . Hence a more r i g o r o u s t h e o r y i s n e c e s s a r y t o u n d e r s t a n d t h e s e r e s u l t s . 5.2. CH^ - He M i x t u r e . 5-2.1. R e s u l t s . T-^  was measured i n CH^ - He m i x t u r e as a f u n c t i o n o f d e n s i t y f o r f i v e d i f f e r e n t c o m p o s i t i o n s of the m i x t u r e a t room temp e r a t u r e and the temperature dependence of m i x t u r e s w i t h 30.0$ He, 76.0% He and 88.2$ He was s t u d i e d from room tempera-t u r e up t o 700°K. T y p i c a l p l o t s of T-^  vs f and ^  vs are shown, i n f i g s . 3^ +a and 3*+h r e s p e c t i v e l y f o r the m i x t u r e s w i t h 88.2$ He a t room temperature whereas f i g . 35a and f i g . 3.5b show s i m i l a r p l o t s f o r the same m i x t u r e a t 730°K. The dependence of T, jf> on j 3 was l i n e a r w i t h i n e x p e r i m e n t a l e r r o r s and the v a l u e e x t r a p o l a t e d t o i n f i n i t e d i l u t i o n g i v ^ s the v a l u e of ~T\\f f o r d i l u t e gas which w i l l be r e f e r r e d t o as Tj \f i n the f o l l o w i n g . Log (Tj/f) v s . l o g T'% was found t o be a s t r a i g h t l i n e f o r a l l t he m i x t u r e s and the s l o p e of the s t r a i g h t l i n e d e c r e a s e d w i t h t h e i n c r e a s e of the He c o n c e n t r a t i o n i n the m i x t u r e and the v a l u e s a r e g i v e n below: 30.0$ He' - 1.28 - 5$ 76.0$ He -1.08 » 5$ 88.2$ He - 0.82 - 5$ 103 1000 11 „7fo CH^ «3% He kO 60 80 D e n s i t y i n Amagats FIG . 3*+a. Dependence of T 1 on d e n s i t y f o r CH^-He m i x t u r e a t 293°K. 9.0 - p 03 03 ^ 8.0 o CD m ,0 EH 6.0 T 20 0 -Q O ho . 60 • 80 D e n s i t y I n Amagats • 100 FIG. 3H>D. Dependence of T^ /p on d e n s i t y f o r CH^-He m i x t u r e a t 293 K. D e n s i t y i n Amagats FIG . 35b. Dependence of T^/p on d e n s i t y f o r CH^-He m i x t u r e a t 730°K. - The dependence of 77j$> on the p e r c e n t a g e of He a t d i f f e r -ent t e m p e r a t u r e s i s shown i n f i g . 36.. At room temperature the dependence of Tjp on the p e r c e n t a g e of He shows s l i g h t non-l i n e a r i t y f o r s m a l l e r p e r c e n t a g e s of He. I n o t h e r words when the d a t a of the. m i x t u r e s i s f i t t e d by a s t r a i g h t l i n e and e x t r a p o l a t e d t o 100$ CH^, the v a l u e o b t a i n e d i s about 15$ s m a l l e r t h a n the a c t u a l measured v a l u e . There a r e not enough d a t a f o r low c o n c e n t r a t i o n s of He i n the p r e s e n t work t o e s t a b -l i s h t h i s e f f e c t d e f i n i t e l y . The e f f e c t .disappeared w i t h i n c r e a s e of temperature and i s t o t a l l y absent above 4-00°K. T h e r e f o r e , i t i s perhaps n e c e s s a r y t o do an e x p e r i m e n t a t 0 ' around 200 K measuring yt i n s e v e r a l m i x t u r e s w i t h He concen-t r a t i o n s between 0 and 25$. T h i s i s o n l y a matter of c u r i o s i t y and does not a f f e c t v t h e purpose of the experiment s i n c e the v a l u e s o f i n t e r e s t are those e x t r a p o l a t e d to 100$ He. Thej?'• v a l u e of T, jf> e x t r a p o l a t e d t o 100$ He givers the c o n t r i b u t i o n due t o CH^ _ - He c o l l i s i o n s a l o n e . Log ( r / j f ) C H _ W £ vs l o g T i s a s t r a i g h t l i n e w i t h a s l o p e ^ ~o~& and i s shown i n f i g . 37-5-2.2. I n t e r p r e t a t i o n . I n t h e absence of a d e t a i l e d m o l e c u l a r t h e o r y f o r the p o l y a t o m i c gases '^he a n a l y s i s of t h e s e r e s u l t s w i l l be c a r r i e d out on s i m i l a r l i n e s to t h a t p r e s e n t e d i n S e c t i o n (5-1.2. ) U s i n g the r e s u l t g i v e n by e q u a t i o n ( 5-l-£j.) and remembering t h a t i t i s d e r i v e d f o r the case of d i a t o m i c m o l e c u l e s arid f o r the case of no t r a n s i t i o n s among d i f f a r e n j t J s t a t e s , Jp can be w r i t t e n as • T'iP °C T X ( ^ n ) ( 5 . 2 .1 . ) o 108 U s i n g the e x p e r i m e n t a l r e s u l t T>lf « r " 0 ' 8 (5.2.2.) the t e m p e r a t u r e dependence of XflVi) i s o b t a i n e d as l(f>,rf) .oc T'01 (5.2.3.) U s i n g e q u a t i o n (5-2.1.) and e x p e r i m e n t a l v a l u e o f fril?\H-fe the v a l u e of KI) i s o b t a i n e d a t d i f f e r e n t t e m p e r a t u r e s ^ Ch " He and [_T(P,")T I ~£ (h^zti] i s s n o w n i n fiS'« 33 "as a f u n c t i o n of t e m p e r a t u r e . The t h e o r e t i c a l p l o t s of the r a t i o as a f u n c t i o n Q o f t e m p e r a t u r e a r e o b t a i n e d from the t a b l e s u s i n g ,, - £ £, ' f o r the cases n=7 and 9« n=7 g i v e s b e t t e r agreement w i t h the e x p e r i m e n t a l r e s u l t s s u g g e s t i n g t h a t the a n i s o t r o p i c i n t e r m o l e c u l a r p o t e n t i a l v a r i e s as Ijn? • However, t h i s r e s u l t s h o u l d be c o n s i d e r e d as a q u a l i t a t i v e i n d i c a t i o n t h a t the i n t e r a c t i o n i s k medium range i n t e r a c t i o n . 0 n - 9 n - 7 O o o O E x p e r i m e n t a l P o i n t s T h e o r e t i c a l P l o t s 200 300 4-00 500 600 700 Temperature i n K FIG. 38. Comparison of [^I ( 2,n) T/I ( 2,n) 2 9 3o K] C H l T H e as o b t a i n e d from Eq.{5-2.1 ) w i t h the computed values, u s i n g the n u m e r i c a l v a l u e s of the i n t e g r a l s . CHAPTER VI CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK The p r i n c i p a l i n t e r e s t - i n the work p r e s e n t e d i n the pre ce d i n g c h a p t e r s i s t o o b t a i n q u a n t i t a t i v e i n f o r m a t i o n on the a n i s o t r o p i c i n t e r m o l e c u l a r p o t e n t i a l s from measurements of the 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 t i m e . The r e s u l t s a t tempera-t u r e s below room temperature were i n t e r p r e t e d by Bloom, 8 Oppenheim e t a l u s i n g a t h e o r y which assumed t h a t the t r a n s i t i o n s between d i f f e r e n t J: s t a t e s a re much l e s s f r e q u e n t t h a n the t r a n s i t i o n s among m.j ^ l e v e l s of the same J - m a n i f o l d . S i n c e t h e n the t h e o r y has been g e n e r a l i s e d by Bloom and Oppen-heim ( t o be p u b l i s h e d ) t o i n c l u d e the t r a n s i t i o n s between J s t a t e s and i s p r e s e n t e d i n Chapter I I I of t h i s t h e s i s . I t was assumed i n the a n a l y s i s of the d a t a t h a t ( i ) o n l y two of the r o t a t i o n a l s t a t e s a r e s i g n i f i c a n t l y p o p u l a t e d . T h i s a s s u m p t i o n i s v a l i d i n the case of Hg f o r t e m p e r a t u r e s below 500°K. ( i i ) t he t o t a l r o t a t i o n a l a n g u l a r momentum i s c o n s e r v e d . F o r Hg o n l y the; o r t h o - Hg m o l e c u l e s c o n t r i b u t e t o the N.M.R. s i g n a l . When the c o l l i d i n g m o l e c u l e i s a l s o an o r t h o - Hg the c o l l i s i o n s i n which a n g u l a r momentum i s conserved are: r e s o n a n t and when i t i s a p a r a - Hg the c o l l i s i o n s a r e ''quasi-resonant". The e v a l u a t i o n of the c o n t r i b u t i o n t o Q fjj'^ from t h e s e twq_ ty p e s of c o l l i s i o n s was d i s c u s s e d i n Chapter IV. The analyses o f the d a t a a t 200°K'indicates... tha.t- the, " q u a s i - r e s o n a n t " c o l l i s i o n s - c o n t r i b u t e about 30% t o the. e f f e c t of q u a d r u p o l e -q u a d r u p o l e i n t e r a c t i o n even though t h e p o p u l a t i o n o f J=3 s t a t e i s o n l y about 3$ a t t h i s t e m p e r a t u r e . I t i s a l s o seen t h a t the c o n t r i b u t i o n 6\f qu a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n t o "H/jp i s about 50% a t 700°K and g r e a t e r t h a n t h a t below 700°K. As t h i s a n a l y s i s give's, a r e a s o n a b l y good, v a l u e f o r the quadrupole moment of the E^ m o l e c u l e i t can be t e n t a t i v e l y c o n c l u d e d t h a t the qua&;L-resonant terms can, be t a k e n i n t o account i n the s i m p l manner as d e s c r i b e d i n t h i s t h e s i s and the o t h e r non-resonant terms a r e i n s i g n i f l e a n t . I f Gj. =. - l-o * lo /Sec • the la ) a n a l y s i s i n d i c a t e s t h a t 3^ can be a d e q u a t e l y d e s c r i b e d by an a t t r a c t i v e p a r t p r o p o r t i o n a l t o 'jrt and a r e p u l s i v e p a r t p r o -p o r t i o n a l t o '//I2"- However, i t was found t h a t the d a t a can a l s o be f i t t e d by a Lennard-Jones p o t e n t i a l i n which case the v a l u e (0 12 -1 o b t a i n e d f o r Ofe i s - 3«3 X 10 s e c . S i n c e L i p s i c a s ' r e s u l t s i n normal H 2 are about 25% h i g h e t h a n the v a l u e s o b t a i n e d by o t h e r i n v e s t i g a t o r s i t i s p o s s i b l e t h a t tjiis v a l u e s of ( T ' l f ) 0 0 a n d (""' /^)0.p a r e a l s o h i g h e r i n the tem p e r a t u r e r e g i o n 150°K - 300°K. As t h i s i n f o r m a t i o n i s o f fun d a m e n t a l i n t e r e s t i t would be w o r t h w h i l e r e p e a t i n g t h e s e measurements and e x t e n d i n g the temperature range. Such a st u d y I ! ': I would be a b l e t o e s t a b l i s h c o n c l u s i v e l y whether o r not the i n -t e r a c t i o n between two E^ m o l e c u l e s which depends on the o r i e n -t a t i p n of b o t h the m o l e c u l e s i s a d e q u a t e l y r e p r e s e n t e d by qu a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n . ! F o r the E^ - He m i x t u r e the t h e o r y p r e d i c t s t h a t the dependence of T/j> on J 3 ^ i s n o n - l i n e a r . Though the d a t a a t room, temperature i n d i c a t e d t h i s n o n - l i n e a r i t y , i t i s not enough t o e s t a b l i s h t h i s e f f e c t , d e f i n i t e l y . As t h i s e f f e c t i s 112 a m a n i f e s t a t i o n of the t r a n s i t i o n s of the Hg m o l e c u l e between d i f f e r e n t J s t a t e s , one c o u l d not expect t h i s e f f e c t t o be p r e -s e n t below 100°K where a l l the o r t h o - m o l e c u l e s a r e e s s e n t i a l l y i n J = l s t a t e . The t h e o r e t i c a l p l o t s i n d i c a t e t h a t the d e v i a t i o n f r o m l i n e a r i t y becomes s m a l l e r a t h i g h e r t e m p e r a t u r e s . T h e r e f o r e room temperature i s perhaps the most s u i t a b l e t e m p e r a t u r e t o l o o k f o r t h i s e f f e c t . The a n a l y s i s of the d a t a i n d i c a t e s t h a t the i n t e r m o l e c u l a r a n i s o t r o p i c , i n t e r a c t i o n between Hg - He m o l e c u l e s can be a d e q u a t e l y r e p r e s e n t e d by Lennar'd-Jones p o t e n t i a l . F o r the Hg - COg m i x t u r e s the ex p e r i m e n t s were c a r r i e d out o n l y up t o 500°K because c h e m i c a l r e a c t i o n between the two gases p r e v e n t e d g o i n g t o h i g h e r t e m p e r a t u r e s . The number den-s i t y o f the m i x t u r e was determined e x p e r i m e n t a l l y as .d e s c r i b e d i n C h apter I I . The a n a l y s i s of the r e s u l t s i n d i c a t e t h a t the dominant i n t e r a c t i o n between Hg and COg m o l e c u l e s can be r e p r e -s e n t e d by q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n . I n view of the c h e m i c a l r e a c t i o n Hg - N g l m i x t u r e would be a b e t t e r system i n whic h t o s t u d y the q u a d r u p o l e - q u a d r u p o l e i n t e r a c t i o n between u n l i k e m o l e c u l e s a t h i g h t e m p e r a t u r e s . F o r CH^ q u i t e a l a r g e number o f r o t a t i o n a l s t a t e s a r e p o p u l a t e d even near room temperature and hence the t h e o r y f o r t w o - l e v e l system c o u l d n o t be use d . I n the absence of a r i g o r o u s t h e o r y f o r p o l y a t o m i c gases, the r e s u l t o b t a i n e d f o r d i a t o m i c m o l e c u l e s w i t h the assumption" ' O f no t r a n s i t i o n s between J s t a t e s were used t o o b t a i n some i n f o r m a t i o n about the i n t e r -m o l e c u l a r p o t e n t i a l s , ^i/y was found t o be p r o p o r t i o n a i l t o -1 5 T f o r CH^ which would be the case i f the c o n t r i b u t i o n s (y) f r o m X(P) t o Tiff i s temperature independent. I f the i s o t r o p i c 113 p a r t of the i n t e r m o l e c u l a r p o t e n t i a l i s assumed t o be g i v e n by h a r d sphere p o t e n t i a l t h e n the i n t e g r a l s £• (f) are independent of -1-5 t e m p e r a t u r e . T h i s model p r e d i c t s t h a t T'i//€"r f o r o t h e r systems l i k e CH^,- X as w e l l . F o r the case, when X i s He i t was found -0-8 t h a t "7//p°c T T h e r e f o r e , a more r e a l i s t i c p o t e n t i a l was assumed f o r the i s o t r o p i c p a r t of the i n t e r m o l e c u l a r p o t e n t i a l , namely the Lennard-Jones p o t e n t i a l . The a n a l y s i s of the tempera- . t u r e dependence of Tijp suggests t h a t the a n i s o t r o p i c i n t e r -m o l e c u l a r p o t e n t i a l i s p r o p o r t i o n a l t o 'j^ and '//{? f o r CH^ - CH^ p a i r and CH^ - He p a i r r e s p e c t i v e l y . However, the s e r e s u l t s s h o u l d be c o n s i d e r e d more as q u a l i t a t i v e t h a n as q u a n t i t a t i v e and the a n i s o t r o p i c i n t e r j n b l e c u l a r i n t e r a c t i o n can be d e s c r i b e d as a medium range i n t e r a c t i o n . As T/^j, vs J?We i n CH^ - He m i x t u r e s appears t o be non-l i n e a r f o r s m a l l e r c o n c e n t r a t i o n s of He a t room t e m p e r a t u r e , i t may be i n t e r e s t i n g t o i n v e s t i g a t e t h i s e f f e c t . S i n c e t h i s n o n - l i n e a r i t y seems t o i n c r e a s e w i t h d e c r e a s e of t e m p e r a t u r e i t may be e a s i e r t o s t u d y t h i s a t around 200°K. However t o under-s t a n d any of t h e r e s u l t s i n p o l y a t o m i c gases the m o l e c u l a r t h e o r y has t o be improved. APPENDIX A CIRCUIT DIAGRAMS 10 Mc. O s c i -l l a t o r 1 Wideband A m p l i f i e r Power Supply 7T Phase S h i f t e r -xOJLr ± < . 6.3V AC T r i p l e r -^ V Reference Output FIG. A1. TRANSMITTER. Power O T o —5» T r i p l e r 5s> A m p l i f i e r Sample C o i l v n o +20V • 3 V . AC 3 HAMMOND TRANSFORMER 167D i 1N5H-0 or BY100 •AAAA-5K -AAAAr-1 .5K - ^ v w v 1 . 5 K 1 OOAf 200V -o+6V IN3036 1N753 -H>k}-1 NI 5 2 7 1N753 -6V FIG. A2. POWER SUPPLY. 1 5K 2-6 . 5 jah >3-9K r b l O O 3lmh p f fRFC =f 68pT +20V Supp l y -° Output t o 2.2K Load 1 2Mh FIG. A3. 10Mc. CRYSTAL STANDARD OSCILLATOR. smnnn- -6V -o +6V FIG. kk. WIDEBAND AMPLIFIER. CO 2N502 0.01 I 0.01 jj£ ~ J^f 6V FIG. A5. 30Mc. TRIPLER. To 90 P u l s e w i d t h Generator 4 To 180 P u l s e w i d t h To Box-car To X G e n e r a t o r FIG. A6. PULSE SEQUENCER. BIBLIOGRAPHY 121 A. Abragam: F. B l o c h : N. Bloembergen, E.M. P u r c e l l & R.V. Pound G. T. N e e d i e r & W. Opechowski: C S . Johnson & J.S. Waugh M. Bloom & I . Oppenheim: M. Bloom & I.- Oppenheim: M. Bloom, I . Oppenheim, M. L i p s i c a s , : '• C.G. Wade & C F . Y a r n e l l M. Bloom & I . Oppenheim: E. L. Hah^i: H.Y. C a r r & E.M. P u r c e l l : R. J . Blume: J.D. N o b l e : W.;N. Hardy:, W.G. C l a r k : P.W. Bridgmann: The P r i n c i p l e s of N u c l e a r Magnetism, Oxford U n i v e r s i t y P r e s s , 19/61. Phys. Rev. 2 Q , 6^0, (19^ 6), Phys. Rev. 211 679, (19W , Can. J . Phys., J6, 870, (1961), J . Chem. Phys., 36, 2266, (1962), Can. J . Phys., 22v (1961) Can. J . Phys., *+l, 1580, (1963), J , Chem. Phys., V i , 1036, (1965), To be p u b l i s h e d . "Advances i n Chemical P h y s i c s " , a volume on I n t e r m o l e c u l a r F o r c e s , e d i t e d by J.D. H i j r s c h f e l d e r ( I n t e r s c i e n c e ) Phys. Rev., 80, 580,! (1950), Phys. Rev., e}30, (195^ ), Rev. S c i . I n s t r , 2^, 1016, (1961), Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia (1965), i Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h C olumbia, (1965),' Rev. S c i . I n s t r . , 316, (196*0, "The P h y s i c s of, H i g h P r e s s u r e " , G. B e l l and Soris L i d . I London, 19^ 9, 1 2 2 1 7 . 1 8 . 1 9 -2 0 . 2 1 . 2 2 . 2 3 -2 4 , 25": 2 6 . 2 7 . 2 8 . 2 9 -3 0 . 3 1 -3 2 , G.B. Benedek: H.W. Wooley, R.B. S c o t t & F.G. Brickwedde: M.E. Rose: "Magnetic Resonance at* High. P r e s s u r e " Number 24-, I n t e r s c i e n c . e T r a c t s on P h y s i c s and Astronomy. J o u r n a l of Research of the N a t i o n a l Bureau of S t a n d a r d s , Research paper RP 1 9 3 2 , ]J+, 3 7 9 , (19H-8), "Elementary Theory o f A n g u l a r Momentum", John W i l e y and Sons, I n c . J . Van Kranendonk: ' Can. J . Phys. kl, 4-33, ( 1 9 6 3 ) , D.LI. W i l l i a m s : Can. J . Phys., ^+0, 1 0 2 7 , ( 1 9 6 2 ) , M. L i p s i c a s & M. Bloom: Armstrong: J.W. R i e h l : M. L i p s i c a s & A. H a r t l a n d : J.O. H i r s c h f e l d e r , C F . C u r t i s s & R.B. B i r d : Can. J . Phys., 1 2 , 8 8 1 , ( 1 9 6 1 ) , P r i v a t e Communication. Ph.D. T h e s i s , M a s s a c h u s e t t s I n s t i t u t e of Technology ( 1 9 6 6 ) , Phys. Rev., 1 ^ 1 , 1 1 8 7 , ( 1 9 6 3 ) , " M o l e c u l a r Theory of Gases and L i q u i d s " John W i l e y and Sons. W.M. Graven & F . J . Long : F. B r i d g e s , M. Bloom, M.- L i p s i c a s & B.H. M u l l e r : C S . Johnson & J.S. Waugh : J . Am. Chem. S o c , 26, 2 6 0 2 , ( 1 9 5 ^ - ) , American 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 4-4-, V o l . I l l , Table I . M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h C o l umbia, 196k. Can. J . Phys, 3^, 1 0 9 3 , ( 1 9 6 1 ) , J . Chem. Phys., 2 0 2 0 , ( 1 9 6 1 ) , N.J. T r a p p e n i e r s , C.J. G e r r i t s m a & P.H. O o s t i n g : P h y s i c a , 3 1 , 2 0 2 , ( 1 9 6 5 ) 

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