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UBC Theses and Dissertations

Nuclear spin relaxation in dilute gases Dorothy, Robert Glenn 1967

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The  U n i v e r s i t y 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 ROBERT GLENN DOROTHY B.Sc,  Iowa S t a t e U n i v e r s i t y , 1962  M . S c , U n i v e r s i t y of Wyoming, 1964 THURSDAY, JUNE 15, 1967 AT 3:30 Pr.M. IN ROOM 301, PHYSICS (HENNINGS) BUILDING COMMITTEE IN CHARGE Chairman:  J . R. Adams  M. Bloom  D. L . W i l l i a m s  R. F. Snider  J . M. M c M i l l a n  F. W. Dalby .. E x t e r n a l  Examiner;.  F . S; Hubbard-  Department of P h y s i c s U n i v e r s i t y . o f North C a r o l i n a Chapel H i l l ,  North C a r o l i n a  . U.S.A. Research S u p e r v i s o r :  M. Bloom  NUCLEAR SPIN RELAXATION IN DILUTE GASES . ABSTRACT The s p i n r e l a x a t i o n time, T^, has been measured a t low d e n s i t i e s i n normal H  a t 77°K, 196°K, 298°K and  2  392°K as a f u n c t i o n of d e n s i t y u s i n g  a 96 mHz  pulsed  N^MIR. spectrometer and a T^ minimum o b t a i n e d . at 77°K, where o n l y  data  the J=l r o t a t i o n a l s t a t e i s p o p u l a t e d ,  i s f i t by the c o n v e n t i o n a l  t h e o r i e s , but the r e s u l t s at  h i g h e r temperatures are not e x p l a i n e d t i o n of the Bloom-Oppenheim was  The  a l s o measured  a mixture o f 54.57, He  in H  ( f>)  by the g e n e r a l i z a -  theory.  as a f u n c t i o n of d e n s i t y f o r 2  a t 298°K i n an e f f o r t to  i n v e s t i g a t e the r o l e of t r a n s i t i o n s between  J states i n  the r e l a x a t i o n p r o c e s s . The s p i n r e l a x a t i o n time HD as a f u n c t i o n of d e n s i t y minimum a t 196 K and 298°K. t,  was  a l s o measured i n  i n the r e g i o n  of the T^  S i n c e the most r e c e n t  has not been extended to a system of s e v e r a l J. l e v e l s the r e s u l t s are r a t h e r T, was i density  a l s o measured  i n the r e g i o n  i n CH, as a f u n c t i o n of 4  of the  minimum a t 196°K and coupling  obtained..  From these r e s u l t s i t i s c o n c l u d e d t h a t of T^ as a f u n c t i o n of d e n s i t y theories  populated  inconclusive.  298°K a n d - i n f o r m a t i o n about the r o t a t i o n a l constants  theory  i s very useful i n t e s t i n g  of r e l a x a t i o n and o b t a i n i n g  r o t a t i o n a l coupling  measurements  information  on the  constants f o r polyatomic molecules.  GRADUATE STUDIES Field  of Study:  N u c l e a r Magnetic  Resonance  Quantum Theory of S o l i d s  R.R. H a e r i n g  Statistical  L. deSobrino  Advanced Seminar  Theory of M a t t e r  Magnetism  C. F. Schwerdtfeger  i n NMR  Noise i n P h y s i c a l  D. L. W i l l i a m s Systems  R.E. Burgess  Elementary Quantum Mechanics  G.M.  Statistical  R.  Mechanics  Volkoff  Barrie  PUBLICATIONS P r e l i m i n a r y Study o f the O p t i c a l C o n s t a n t s of S i l v e r Cadmium A l l o y s Near 300 mu, R.G. Dorothy and D.W. U.S.A.E.C. Research and Development  Report No. Is-863  Lynch,  NUCLEAR SPIN RELAXATION IN DILUTE GASES  by  ROBERT GLENN DOROTHY  BoSe. Iowa State U n i v e r s i t y , 1962 MoSeo U n i v e r s i t y of Wyoming, 1964  THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of Physics  We accept t h i s t h e s i s - as conforming to the r e q u i r e d standard  THE  UNIVERSITY OF BRITISH- COLUMBIA June 1967  In  presenting  for  an  that  advanced  thesis  shall  I further  agree  for scholarly  Department  o r by  publication  without  thesis  degree  ths Library  study.  or  this  my  at  in partial  the U n i v e r s i t y  make  i t freely  that  may  be  his representatives.  written  thesis  granted  Department Columbia  by  requirements  Columbia,  t h e Head  shall  of  of  this  my  that not  agree  and  copying  It i s understood gain  I  f o r reference  f o r extensive  for financial  permission.  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  of British  available  permission  purposes  of this  f u l f i l m e n t o f the  be  copying allowed  ABSTRACT  The t i e s  s p i n  i n  r e l a x a t i o n  normal  d e n s i t y  H  using'  a  2  a t 77°K,  96 mHz  The d a t a  p o p u l a t e d ,  i s  f i t by  temperatures  B l o o m - O p p e n h e i m ^ T^  was a l s o  54.5%  He  i n  t r a n s i t i o n s  o f  measured  between  Since'  t h e 'm o s t  s e v e r a l :  J  as- a  the  a  p o p u l a t e d  -  o f  i n  time  t i o n f o r  t h e o r y  J 'l e v e l s  measured  r o t a t i o n a l  c o u p l i n g  From  r e s u l t s  o f  d e n s i t y  a n d o b t a i n i n g p o l y a t o m i c  o n l y  e f f o r t i n  K  S  t h e J = l  by  a t  a s  l o w  a  d e n s i -  f u n c t i o n  a n d a  T-^  b u t  o f  minimum  r o t a t i o n a l  t h e o r i e s ,  s t a t e  t h e r e s u l t s  t h e g e n e r a l i z a t i o n  o f ' d e n s i t y  t o  t h e r e l a x a t i o n  o f  o f  i s a t  t h e  measured  h a s n o t been  i n :C H ^ a s a t ' 1 9 6  a  :  a n d 2 9 8 ° K  c o n s t a n t s  o b t a i n e d .  i t i s  ±s  ;  concluded-  v e r y  i n f o r m a t i o n  m o l e c u l e s .  C  u s e f u l o n  i n  a t  a r e r a t h e r '  K  1  t h a t  m i x t u r e  t h e r o l e  i n  extended  f u n c t i o n  a  o f  o f  p r o c e s s .  t h e T^_ m i n i m u m  t h e r e s u l t s '  f o r  i n v e s t i g a t e  T-^ w a s a l s o  1  1  f u n c t i o n  a n  s t a t e s  t h e T^ minimum  t h e s e  a n d 3 9 2  s p e c t r o m e t e r  f u n c t i o n  t h e r e g i o n  r e c e n t  TJL w a s a l s o  r e g i o n  i n  where  measured  t h e o r y .  a t . 2 9 8 ° K  d e n s i t y  N.M.R.  a r e n o t explained"  :  -  298°K  t h e c o n v e n t i o n a l  - The spin' r e l a x a t i o n t i o n  h a s been  196°K,  a t ' 77°K,  -  T-^,  p u l s e d  :  o b t a i n e d .  h i g h e r  t i m e ,  o f  ;  HD a s  a  2 9 8 ° K .  system  o f  i n c o n c l u s i v e . d e n s i t y  i n t h e  a n d i n f o r m a t i o n  measurements  t e s t i n g  t h e r o t a t i o n a l  f u n c -  a n d  1 9 6 ® K t o  a  t h e o r i e s c o u p l i n g  about  o f o f  T-^ a s r e l a x a -  c o n s t a n t s  TABLE  OF  CONTENTS  Page  Abstract L i s t  o f  Tables  L i s t  o f  I l l u s t r a t i o n s  Acknowledgements Chapter  I  Introduction  Chapter  I I  Experimental  1 Procedure  and Equipment  8  2 . 1 .  Apparatus  9  2 . 1 . 1 .  9 6  mHz  9  2 , l s 2 »  Low-Noise  2 . 1 . 3 .  9 6  Amplifier  mHz  mHz  9 6  Coherent  Amplifier  1 5  Reference Generator  1 9  2 . L 4 ,  Timing  Apparatus  2 1  2 J  Boxcar  Integrator.  2 1  o5 .  2 . 1 . 6 .  Matching  Network  2 . 1 . 7 .  Video  2.1.8b  Magnet  2 . 1 . 9 .  Time  2 . 1 . 1 0 .  Strip  2 . l o l l .  Gas Handling  2 3  /Amplifier and Power  Interval Chart  2 6  Supply  2 6  Unit  2 6  Recorder  2 7  System  and Gas Sample  2 . 2 .  Measurements  2 . 2 . 1 .  Temperature  2 . 2 . 2 .  Density  Measurements  2 . 2 . 3 .  General  Comments o n E q u i p m e n t and Measurements i i i  2 7 28  Measurements  2 8  2 9  2 9  No  2.2 . 4.  Chapter  Chapter  III  IV  A c c u r a c y o f Measurements  Theory  35  3.1.  Description of H  Gas  35  3.2.  D e s c r i p t i o n o f HD Gas  37  3.3.  Relaxation i n H  46  3.4.  R e l a x a t i o n i n HD  3.5.  C a l c u l a t i o n o f B^-( J , J ) f o r HD  2  2  46 1  V  48  Experimental Results and Discussion  51  4.1.  The T j Minimum  51  4.1.1.  A t 77°K  51  4.1.2.  A t Higher Temperatures  54  4.1.3.  He - H  61  4.1.4.  The T j - Minimum  1  2  in H  2  Mixture Versus Temperature  Chapter  30  4.1.5.  The S t r o n g C o l l i s i o n  4.1.6.  Higher Order  4.2.  The  4.3.  The T  Limit  Interaction Possibility Minimum -in HD  x  Measurements  Conclusion  i n CH4  63 64  66 67 70 74  iv  LIST  Integration for  T-L  f o r  Fractional States  TABLES  and  Times  Used,  Measurements  Fractional States  Constants  OF  f o r  Population  of  the  Rotational  of  the  Rotational  H2  Population HD  v  LIST OF. ILLUSTRATIONS Figure  Page No.  2 .1.  Block Diagram of 96 mHz  2 .2.  Coherent-Oscillator  2 .3.  Pulsed Spectrometer  10  and-Gate  11  25 mHz  A m p l i f i e r - and Doubler  12  2 .4. - -  96 mHz  Gated- B u f f e r A m p l i f i e r s  13  2 .5-. • •  F i n a l 96 mHz A m p l i f i e r  14  2• 6 <  96 mHz  Cascode-Input Stage  16  2 «7" o  96 mHz  Gain C o n t r o l l e d A m p l i f i e r s and Detector  17  96 mHz  Coherent Reference Generator  20  2 .8.  -  2 .9.  DC Pulse M i x e r - A m p l i f i e r  22  2v l O v  Sample C o n f i g u r a t i o n and Matching Network  25  4 .1.  T  l  versus D e n s i t y a t 77°K i n H  4 -.2. -  T  l  v e r s u s D e n s i t y a t 196°K i n H  2  55  4• 3  T  l  v e r s u s - D e n s i t y a t 298°K i n H  2  56  4 .4.  T  l  v e r s u s D e n s i t y a t 392°K i n H  2  57  4  T  l  v e r s u s - D e n s i t y a t 298°K i n a 54.5% - He - H Mixture  62  Minimum-as a F u n c t i o n of Temperature  64  versus D e n s i t y i n HD a t 196 K and 298°K  68  l versus D e n s i t y i n CH • a t 196°K and 298°K •• 4  71  :  e  53  2  2  4 -.6. 4 .?. 4.8 .  T  l  T  l  3  -- • :  "  T  vi  ACKNOWLEDGEMENTS  P wish- to express my s i n c e r e g r a t i t u d e to" P r o f e s s o r Meyer Bloom f o r h i s guidance, encouragement" and f i n a n c i a l  assis-  tance p r o v i d e d by h i s N.R.C. g r a n t . I would l i k e t o thank Dr. John Noble f o r many h e l p f u l d i s c u s s i o n s on" the equipment used i n t h i s ' e x p e r i m e n t . The use o f the- Computing  Center i s g r a t e f u l l y  acknow-  ledged . I wish to thank 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. I wish to thank Mr. Peter Haas f o r drawing the diagrams i n t h i s t h e s i s ' a n d M i s s Wendy Davis f o r t y p i n g t h i s :  thesis.  The Research was- supported by the N a t i o n a l Research C o u n c i l of Canada.  vii  1  I N T R O D U C T I O N  N u c l e a r magnetic resonance i s a well-known t e c h n i q u e for  studying  gases.  the properties  o f s o l i d s , l i q u i d s and, more r e c e n t l y ,  The l o n g i t u d i n a l r e l a x a t i o n time T-^ . i s .a measure o f t h e  time r e q u i r e d  for a disturbed  s p i n system t o a t t a i n thermodynamic  e q u i l i b r i u m with i t s surroundings.  The purpose o f t h i s t h e s i s i s  t o show t h a t a t v e r y low d e n s i t i e s t h e s p i n r e l a x a t i o n time i s v e r y s e n s i t i v e t o t h e i n t e r a c t i o n s between m o l e c u l e s and t h a t a measurement o f T-^ as a f u n c t i o n o f d e n s i t y can p r o v i d e i n f o r m a t i o n on these i n t e r a c t i o n s between- m o l e c u l e s which a r e i n e x c i t e d rotational states.  The p r i n c i p l e s o f n u c l e a r magnetism a r e d i s -  cussed i n d e t a i l i n Abragam^and w i l l o n l y be o u t l i n e d  briefly  here. The  Zeeman energy o f 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 I and magnetic moment , A<L = ~2"ri X i n an e x t e r n a l magnetic field H i s Q  - Ai-  =  7 /  (l.D  where t h e energy l e v e l s of- such a system a r e g i v e n by -  where  - 1 The  m  ~7f-k Ho  X'l,  }  (  1  >  2  )  -I  f r a c t i o n a l p o p u l a t i o n s o f t h e s e energy l e v e l s a t  e q u i l i b r i u m a r e g i v e n by I m  +r >-V>  c  /, -r  - - I  where T i s t h e temperature o f t h e g a s .  (1«3)  2 The system  of N  equilibrium  spins  as  M. =  £  r  If motion  H  H » / k T  o f t h e m a g n e t i z a t i o n - may  In ;  equation  i s fixed  the M  =  "Larmor  about  H  Q  W/fcy  1  ;  m  a  of a y  H where  reference  =  t  t  e  frequency  H  field  q  ^ £  )  cvz-tftto  the rotational  0  i s called frequency  of  applied  and a time  magnetic  varying  field  n  c  d . 7 )  °s  -i- j  us~t  as M .  H-, the  S Ih ust  J magnetic  In the rotating  equation  of motion  :  field  reference  from  this  + hi, T]  equation  frame  of M i s given  tiHo-rfl+Th/, seen  -  frame.  as the t o t a l  rMj[(Ho+f)K  c a n be  0  -h jj,(t)  sense  Hc/y =It  frequency  i f // - // t<  of the applied-alternating  along  P  i  H , L ^  i n t h e same  the x-axis  r  Ho =  angular  (1.6)  frame  and r e p r e s e n t s  static w  the part  ipl w h e r e  e  H,^)  rotating with  b  with  +^1  i f H i s considered  H  represents  rotating  i n the laboratory reference  consisting  of  (1.5)  The a n g u l a r  frequency"  Now field  ~K  0  n e g l e c t e d v "the": e q u a t i o n  ;  be w r i t t e n  xlH  7  i n the rota-ting  (^cr - ~ $ / - l  and  (i.4)  <i  frame  M  =r  m  f o r M m a y be w r i t t e n as  of motion  1^ M  o ' '  a reference  for a  1  effeets-are  ~  i s given  v*hS*X(z+1)MT «  relaxation  XI  the  f^rh  Z  hi  = where  or net magnetization  that M  precesses  by  about H y y  w i t h an angular  e  d o m i n a n t t e r m i n f/g// when Us- -  f r e q u e n c y fflcJf  when ur -  T h i s phenomenon i s n u c l e a r m a g n e t i c  -  f-}  If  the f i e l d  0  i s applied-only  are chosen such t h a t ^ w ^ f if, 0 - ^  Q -  resonance.  f o r a time t ^ . , then the ~jr/-)  When t ^  ,.~t  and  The 180° p u l s e , . a n d t h e 90° p u l s e  p  have t h e e f f e c t o f r o t a t i n g - t h e m a g n e t i z a t i o n ^ t h r o u g h 180°  be t h e  t h e p u l s e o f E-^ i s - c a l l e d , a "180° p u l s e  a "90 -pulse"-.  ;  will  i . e . , t h a t M p r e c e s s e s a b o u t H-^  moment w o u l d p r e c e s s t h r o u g h a n a n g l e  or  and t h a t  an angle o f  o r 9 0 ° , a b o u t H-j-(t) > r e s p e c t i v e l y . The m a g n e t i z a t i o n - p r e c e s s - i n g ' i n - t h e x - y - p l a n e a f t e r t h e  a p p l i c a t i o n o f a 90° p u l s e i n d u c e s a n e . m . f .  i n the c o i l  which can  b e - d e t e c t e d and observed o n - - a n - o s c i l l o s c o p e and i s r e f e r r e d a free  i n d u c t i o n decay.  the magnetization  I f a t a time  t  after  the f i r s t  t o as  pulse,  i s r o t a t e d t h r o u g h ' 180° t h e . . s p i n s t e n d t o r e p h a s  i n . t h e x - y p l a n e a t a t i m e 2L^T, r e s u l t i n g i n a n i n d u c e d e . m . f . t h e - c o i l w h i c h goes t h r o u g h - a spin  maximum v a l u e  a t time  f , called a  echo. The a p p r o a c h t o e q u i l i b r i u m o f t h e m a g n e t i z a t i o n  it  in  after  h a s b e e n d i s t u r b e d f r o m e q u i l i b r i u m c a n be d e s c r i b e d b y t h e  phenomenological  Bloch  equations  71  a-t  where  7V.  T,  and T 2 a r e 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  and w h e r e  H - tioK + A M  COSOrt  I  j H«  //,_  {  (1.10)  The s o l u t i o n s t o ( 1 . 9 ) may be w r i t t e n y  tf fe) y  -  H*y(o) y  Mi (t) = Mo +  fs/n^-f.^J  e  (l.ll)  a  CO) - Mo] e  t y T [  times  where  rf  '  /y  Tl H/t)  My  t  It to  T2  and  (6)  and  may W  be  determined  fef  ?  as  interactions  m o l e c u l a r g a s e s which- a r e  actions gible spin  between n u c l e i ^ o n  and  the  are  a function  of of  not  too  freedom of  dense, the  i.ev  relaxation  r the  the  the  responsible  7  the  nuclear  for  spins  internegli-  nuclear to  the  m o l e c u l e . T h e s e ' in tra-mo1ecu1ar  m o d u l a t e d by  the  rotation  by  of  the  d e f i n i t i o n , those c o l l i s i o n s which having  1  different  general  the  illustrate  measurement: o f ' T^  t i m e T-^ d e p e n d s on  the  Larmor  c o r r e l a t i o n ' t i m e ""fc^ f o r m o l e c u l a r  strength" of To  the  spin-spin  rotation-  numbers. The  and  thesis.  those" c o l l i s i o n s between m o l e c u l e s w h i c h c a u s e m o l e -  reorientation,  quantum  time  interest in this  cause t r a n s i t i o n s between "molecular s t a t e s al  t.  d i f f e r e n t molecules'are completely  interactions-are  m o l e c u l e and' by  observing  this relaxation  which i s of  those which couple  d e g r e e s of  spin-dependent  cular  e x p e r i m e n t a l l y by  spin^de-pendent i n t e r a c t i o n s  :  relaxation  rotational  ^  i s m a i n l y the" r e l a t i o n s h i p  the'molecular  For  -h My'J  [/ix"  can  theory of  intra-molecular by- an ;  y i e l d ' the :  spin  :  interaction.  i n f o r m a t i o n which  expression  relaxation  reorientation  spin-dependent  example t h e  frequency  for  1/T-^ i s w r i t t e n  ,1.12.,  2  i s the :  fluctuating  from  as  - I r i r K / ^ / ' ^ J ^ Ti where H ( t )  a  internal- f i e l d  described  by  the  Hamiltonian  7t and  = - ii  >  das.)  where the" c o r r e l a t i o n f u n c t i o n " of" H ( t )  <H(t)  is  = <!rTGt)l ?j(*) i  written  u.14.1  -*°  J  •  y  In o r d e r t o - s p e c i f y  two p a r a m e t e r s  1  magnitude  (1.15.)  are" needed, t h e  of the f i e l d ' which couples- the nuclear spins' t o the l a t :  t i c e a n d the" F o u r i e r  7  c o m p o n e n t o f • ^jO?) t h e " r e d u c e d - c o r r e l a t i o n  c t i o n o f H ( t ) e v a l u a t e d a t t h e Larmor  frequency".  the. f i e l d ^ | H | ^ i s u s u a l l y known f r o m o t h e r ;  A  :  m o l e c u l a r beam e x p e r i m e n t s ,  fun-  The m a g n i t u d e o f  experiments  such as  a n d i t i s o f t e n assumed t h a t  gfyf)decays  exponentially i . e .  gftj = Then  e"^^*-  (Lie.)  (1.12.)becomes (1.17.) For  this case,  t h e p l o t o f T-^ v e r s u s  c h a r a c t e r i s t i c minimum a t  goes t h r o u g h a  t *k. I w h i l e f o r iss^  I^ 77 =^ " t ^  c  If" t h i s m o d e l i s a p p l i e d t o a d i l u t e g a s i n w h i c h t h e t i m e 1  b e t w e e n c h a n g e s " i n H ( t ) i s a s s o c i a t e d " w i t h the" t i m e b e t w e e n c e r t a i n :  :  t y p e s o f m o l e c u l a r c o l l i s i o n s , tf*\f  ^ then,  i n t h e r e g i o n U/frXt 7 ? I j  d e c r e a s e s a s t h e frequency^ of - c o l l i s i o n s i n c r e a s e s , b u t in" t h e r e g i o n 6l/r^  <-<-/ j T]_ i n c r e a s e s a s t h e f r e q u e n c y  (pressure  narrowing). When t h e f o r m o f  depends  of c o l l i s i o n s increases  cjfejis  k n o w n , t h e minimum v a l u e o f 1  o n l y on t h e s t r e n g t h ' of t h e i n t r a - m o l e c u l a r  interactions  :  and o n t h e L a r m o r f r e q u e n c y .  Hardy^  measured  (T^) m i n f o r H  77°K w h e r e o n l y t h e - J = l " r o t a t i o n a l - s t a t e - o f o r t h o - H and o b t a i n e d a v a l u e o f 2 2 5 ' u s e e s .  2  2  at  i s populated,  A t t h e s a m e t i m e h e was a b l e  to^ s h o w c o n c l u s i v e l y , from" t h e d e p e n d e n c e o f "T^ onJ> n e a r that  T-^  (T-^) m i n ,  f o r the s i n g l e r o t a t i o n a l l e v e l system t h e c o r r e l a t i o n  func-  tions  are  indeed In  exponential.  this thesis,  minimum.are r e p o r t e d ciable  number' o f  results  for H2  at  -  of  measurements n e a r t h e  higher  :  t e m p e r a t u r e s where an  o r t h o - H ^ m o l e c u l e s ' are' i n an  state  (J^S);  ied .  These' m e a s u r e m e n t s were made p o s s i b l e  1  a  In  r  mHz  II.  pulse  Also  the  i n Chapter II  As  and  by  the  the  disagreement of  the  be  the  first  interpreted  such as  erature  is raised  increased  spin  discussion i s of  (1.16.).  The  and h i g h e r of  (1.17.) >  :  an  reason f o r  In states,  the  states  1  of  t r e a t i n g ' the  the  (T-^)  from  this  min  clear  in  i s that  as  the  to  ^Therefore,  1  d e c r e a s e s ' as  T  increases.  i n (T];) min T  c o n t r i b u t i o n " to-"T^ by  as  excited  had' assumed  4  different J,  be  within  correlation  exponential,  2  they  temp-  the if The  the  77°K t c room " t e m p e r a t u r e .  Oppehheim^ ^  d i f f e r e n t mj  shown t o  2  correlation  'rotational  t h a t ' no  i . e . that  the  :  the  H  H  that  transieffect  between m o l e c u l e s i s t o p r o d u c e " o n l y " t r a n s i t i o n s  approximation,  for  t o ^ / / f / * ' ^ namely  is-proportional  t i o n s ' o c c u r "between s t a t e s ' o f collisions  importance  contributions  (T-^-) • min  increased  Bloom and  the  states- populated,  experiments revealed' a-substantial-increase t e m p e r a t u r e was  of  great  exponential  rotational  r  interaction,  T-j_ i s g i v e n by  Chapter  experimentalresults  measurements o f  i n terms of  b e c a u s e one  rotation  in  of  theory.  s o o n as  function  stud-  development  :  not  were  in detail  This' d i s c u s s i o n  r  appre-  rotational  CH4  room t e m p e r a t u r e were a v a i l a b l e , ' i t became q u i t e  could  was  HD  is a critical  :  available  ;  -  the' m e a s u r e m e n t s .  here because of with  minimum' o f  spectrometer-which' i s c r e s c r i b e d  1  included  a c c u r a c y of  is  ;  1  96  at  addition,vthe T^  excited  T-^  the.same J m a n i f o l d . 1  function  but  associated  By  between  making  w i t h each J  a different correlation  of  time  this state was  7  obtained  f o r e a c h J s t a t e * : •. However>  of e x p l a i n i n g  t h e b e h a v i o u r : o f Tj_ n e a r  At this  s t a g e y Bloom and  e x a m i n e the' j u s t i f i c a t i o n of d i f f e r e n t J . cular  transitions  J = l t o t h e J=3 state".  Oppenheim^  j  were l e d t o r e -  t r a n s i t i o n s " between  1  between H^' m o l e c u l e s ,  i t turns out that  states  these  i n which J i s changed a r i s i n g  i n w h i c h one  ortho-H^ molecule  state" while another  function  i s changed.  which a r e i n i t i a l l y  i n the J - l s t a t e  each  J=3  o f t h e J = l and  states  a r e n o n - e x p o n e n t i a l . I t was non-exponential correlation  decay  the  to  the form  of  the  of  molecules  an a p p r e c i a b l e t i m e i n  so t h a t t h e i r ' c o r r e l a t i o n " f u n c t i o n s  thought  w o u l d be  that  since  to spread"the  f u n c t i o n s over a wider  i n the v a l u e of  spend  from  t h e :J=3  i n t o account,  " A collection  the  from  i s excited  i s d e - e x c i t e d from  When" c h a n g e s o f - J a r e t a k e n  the c o r r e l a t i o n  increase  (T]_) m i n .  for neglecting  for collisions  these c o l l i s i o n s  1  also" i n c a p a b l e  s h o u l d not""be- n e g l e c t e d , . t h e " m a i n ~ " c o n t r i b u t i o n t o  cross-section  J=l  1  t h e o r y was  Because of the n a t u r e of the a n i s o t r o p i c i n t e r m o l e -  interactions  1  this  ( T ^ ) min 1  range  the e f f e c t  Fourier  of  spectrum  of f r e q u e n c i e s , the  -  of  the  observed  r  with' i n c r e a s i n g  this  temperature  would  be e x p l a i n e d . • I n f a c t , ; w i t h i n the- s c o p e  of the theory a t the  time, - the experimental r e s u l t s  u n e x p l a i n e d " , even a l l o w i n g  1  an a r b i t r a r y assumption is  that  states  ;  variation  are s t i l l  of the parameters  1  o f the :  o f " t h e theory, which i s - d e s c r i b e d :  theory.  briefly  1  a s e t of master  c a s e s , t h e weak c o l l i s i o n have been t e s t e d  1  available  1  and  the experiments  Two  limiting, limits  i n this  thesis.  a d i s c u s s i o n o f them i n t e r m s o f  theory are presented  ;  i n Chapter  IV.  main  rotational  l i m i t - , and" t h e s t r o n g c o l l i s i o n  by- t h e a n a l y s i s ^ o f  The- e x p e r i m e n t a l r e s u l t s  equations.  The  i n Chapter I I I ,  t h e r a t e s - o f e h a n g e ' o f t h e p o p u l a t i o n s of" t h e  a r e d e s c r i b e d by  present  the  8  C H A P T E R  I I  EXPERIMENTAL "PROCEDURE "AND 'EQUIPMENT  The p h y s i c a l " p r i n c i p l e s u n d e r l y i n g the use of p u l s e d techniques  for' measuring T-^. are well-known and e x p l a i n e d i n d e t a i l  i n Abragamd). les  NMR  T h e r e f o r e , o n l y a s h o r t d e s c r i p t i o n o f the p r i n c i p -  of the measurement technique w i l l be g i v e n here.  used i n making the measurements w i l l ,  The equipment  however, be d e s c r i b e d i n  detail. Measurement o f : T 180°  r a d i o frequency  which were i n i t i a l l y field.  r  was accomplished  by f i r s t a p p l y i n g a  p u l s e a t the Larmor frequency  o f the s p i n s  i n ' e q u i l i b r i u m ' in" the e x t e r n a l d.c. magnetic  Then the r a t e - o f recovery o f the s p i n system t o e q u i l i b r i u m  was monitored by a p p l y i n g a 90° - 180° sampling frequency  pulses'and  sequence of r a d i o  o b s e r v i n g :the'height of the echo as the time  between the f i r s t 180° p u l s e and 90° - 180° p u l s e sequence was varied.  The - f o l l o w i n g " d i a g r a m " i l l u s t r a t e s t h i s  sequence:  The magnitude o f the echo i s a ' f u n c t i o n of the time t as d e s c r i b e d 1  by the equation  where M  Q  i s the e q u i l i b r i u m " m a g n e t i z a t i o n .  / /  R o t a t i o n "Of a l l " o f -the .spins through 1 8 0 ° o r 9 0 ° implies'*that"both:::the'-static" f i e l d quency f i e l d *H'^" are 'perfeetlythomogeneous. v  an angle of e x a c t l y and r a d i o f r e o  These are c o n d i t i o n s  t h a t cannot be s a t i s f i e d - c o m p l e t e l y but which only a f f e c t the s i z e of the s i g n a l and not the." v a l u e of T^. T  • •> • *• -The-' - appttrc^taa«f.iiScessaEfi-i^''^row±de the l a r g e s e q u e n t i a l radio" freq"uerre^pti±l5'e5^.aih^  signals' i s d e s c r i b e d i n  "the f o l l o w i n g " s e c t i o n . :  2.1.'  APPARATUS. A b l o c k diagram  ( f i g . 2 . 1 ) shows how the v a r i o u s compo-  nents' which w i l l be d e s c r i b e d s e p a r a t e l y f i t i n t o the t o t a l  2 . 1 . 1 .  -  -• 9 6  mHz PULSE" GENERATOR  (figs. 2 . 2 ,  2.3,  2 . 4 ,2 . 5 ) .  A 9 6 mHZ h i g h power, p u l s e  generator was b u i l t with" the c r i t e r i a t h a t the generated extremely  system.  p u l s e be  s t a b l e i n width and power. A 1 2 mHz  frequency  i s generated  and gated by a c r y s t a l  c o n t r o l l e d o s c i l l a t o r and gate designed by R. J . Blume^O with no m o d i f i c a t i o n s i n e i t h e r the b a s i c o s c i l l a t o r or gate.  This  1 2 mHz p u l s e i s then f e d i n t o a frequency doubler f o l l o w e d by a 2 4 mHz pentode a m p l i f i e r t o i n c r e a s e the power f o r more d o u b l i n g a c t i o n i n the subsequent stages.  efficient  S i m i l a r l y , t h i s doub-  l e r - a m p l i f i e r sequence i s repeated a t 4 8 mHz and 9 6 mHz. the" s i n g l e - e n d e d 9 6 mHz a m p l i f i e r a balanced output  From  i s obtained  by c a p a c i t a n c e c o u p l i n g and f e d i n t o a g r i d gated p u s h - p u l l amplif i e r u s i n g two  5763  beam power pentodes.  Then the p u l s e i s f u r -  12MS  jf e r ence  'i  Phase S h i f t e r -  R.F.  12Mc C r y s t a l C o n t r o l l e d  Ainpllf  O s c i l l a t o r Gating  M i x e r  Reference  N.M.R. S i g n a l  Poller  M u l t i p l i a  :  iVlc  Gated  Gate  And 7  1*6  M u l t i p l i e r  P u l s e  R'.F".  1  Mc  •Amplifier  96Mc  i e r  96  Phase C o h e r e n t D e t e c t o r  F a l s e  V  A  P u l s e A m p l i f i e r  Dawar  V i d e o A m p l i f i e r  Maanet  Boxcar Pai.se  I n t e g r a t e  Generator  Sample C o i l  Stop Input  K e w e l e t t Packard Time  S t a r t  -  I n t e r v a l  Reccrdea  U n i t  Input Boxcar  FIG-.  2  .1  Block"Diagram  of  96  mHz  Sample  P u l s e d  V a r i a n C h a r t  D i g i t a l  T r i g g e r  Spectrometer  Event Marker T r i g g e r  Recorder  12 mE2  Reference out  o—6DJS -  Cr^st^l  I0K 5W 15K W  —llllllr~rp—U^AAA^—=p H i l l III  ——V^XAAAA>  ,-I70V ?33K  22SV^ Resistors i n C a p a c i t o r s i n p f . i f >1 Capacitors i n ^ f f . i f < 1 FIG. 2.2.  7K LN307  4  =ZZ  9 0V O—  1  L y  r  90 v o l t gate  Coherent O s c i l l a t o r  2 - 5763 in p a r a l l e l  >3 9K  and Gate.  + 22511)0  ?+225VDC  5K  FIG, 2.3.  24mHz .Amplifier and Doubler.  Q. +225VDC 02 27K  5763  96mKz  T o n e x t P-P Buffer Amplifier  -O  22  O  f r o m 96rnHz Dcubler  02<1M-  RFC 25VDC  '.02X  +2 2 5 V D C  — O  96mHz  DC P u l s e i n  O-  -AA/W—I  O  -T70VDC 5763  25«  FIG.  2.4.  Gated  Buffer  Amplifier  o +-120  VDC Z L  n^mfdTo sample  coil  36 mHz + 75 0 VDC O 02 rJ  J  L  1/2 8 2 9B  ::0 mfd  L  1/2 829E 01 RFC  RFC  22  o  -VW\A  6  From P-P Buffer Amplifier  FIG«  2.5.  F i n a l 96 mHz Araplif i e r  -o DC p u l s e i n  4.7K  10K  O - 170 VDC :25K  t h e r a m p l i f i e d i n a n o t h e r g r i d " g a t e d ' 96 mlfe p u s h - p u l l 1  an  effort  to overdrive  s  the f i n a l  power a m p l i f i e r i n t o a s a t u r a t e d  mode s o t h a t t h e power o u t p u t , w i l l of  small  be more s t a b l e and i n d e p e n d e n t  f l u c t u a t i o n s i n the. p r e v i o u s  circuits.  t u a t i o n s might'be a m p l i f i e d t o c r i t i c a l c a s c a d e d a m p l i f i e r s . ' The f i n a l single the  noise  during  is-raised-to plied  the o f f time.  of  this  During  so many  the pulse  to eliminate  the d.c. g r i d of-r.f.  voltage i s sup-  +1200 v . d . c . on t h e p l a t e and  the necessary  final; amplifier well  t h i s power i n t o t h e sample c o i l  cussed  with  fluc-  c o n f i g u r a t i o n and c u t o f f i n  -170 v . d . c . on t h e g r i d  so t h a t w i t h  +•750 v . d . c . on t h e s c r e e n with  proportions  a b o u t -70 v . d . c . a n d a b o u t 400 v o l t s  t o the g r i d  obtained  Such s m a l l  power a m p l i f i e r c o n s i s t s o f a  829B d u a l p e n t o d e i n p u s h - p u l l  absence o f a p u l s e with  amplifier i n  two k i l o w a t t s o f power i s saturated.  The c o u p l i n g  where i t i s o f u s e w i l l  be d i s -  i n t h e s e c t i o n on c o u p l i n g .  2.1.2.  LOW-NOISE 96 mHz AMPLIFIER (figs.  recovery  2.6 and 2 . 7 ) .  A n a r r o w band, l o w - n o i s e ,  t i m e a m p l i f i e r was d e s i g n e d  and b u i l t  fast  by S. K o s k e n n o n o f  T e l e - S i g n a l E l e c t r o n i c s , V a n c o u v e r , B.C., ••to our, s p e c i f i c a t i o n s . This  a m p l i f i e r h a s a b a n d w i d t h o f 2.mHz a t t h e 3 dh  a recovery defined the  t i m e o f 3-4 m i c r o s e c o n d s , where t h e r e c o v e r y  t o be t h e t i m e a f t e r  input noise  noting  the pulse  of the r e c e i v e r could  that this value  of recovery  15-20 m i c r o s e c o n d s f o r t y p i c a l of'the  p o i n t s and  inherently smaller  waS t u r n e d be o b s e r v e d .  time i s  o f f a t which I t i s worth  t i m e compares w i t h  values of  30 mHz a m p l i f i e r s and i s a r e s u l t  capacitances  associated with  the higher  FIG.  2,6.  Cascode  Input  Stage  FIG.  2.7  Gain  Controlled  Amplifiers  and  Detector  18  frequency  circuits.  T h i s a m p l i f i e r has a n o i s e f i g u r e of 4.6 db.  A t the time t h i s a m p l i f i e r : w a s . b u i l t the " s t a t e - o f - t h e - a r t " of 1  vacuum tube a m p l i f i e r s seemed, t o s e t a b a s i c l i m i t a t i o n on minimum n o i s e f i g u r e of about 4.5 dL. n o i s e i n the f i r s t  stage'of  a t 96 mHz due mostly t o induced  the a m p l i f i e r .  grid  Very r e c e n t developments  u s i n g s o l i d s t a t e d e v i c e s can p o s s i b l y p r o v i d e lower n o i s e figures„ A  6CW4 n u v i s t o r which has a low e q u i v a l e n t n o i s e  tance and low i n p u t c a p a c i t a n c e caseode i n p u t .  resis-  i s used as the f i r s t h a l f of a.  The second h a l f of t h i s cascode c i r c u i t c o n s i s t s of  a very high mu 7 7 8 8 pentode, t r i o d e connected.  T h i s 7 7 8 8 i s per-  haps- the h i g h e s t g a i n low n o i s e tube a v a i l a b l e .  Two v a r i a b l e g a i n  stages u s i n g 7 7 8 8 pentodes are then cascoded to y i e l d most of the 90 d h . v o l t a g e g a i n s i n c e the cascode i n p u t c i r c u i t g i v e s o n l y a power g a i n w i t h v o l t a g e g a i n of u n i t y i n t h i s c i r c u i t s germanium diode  i s used to d e t e c t . t h e  able g a i n stages. i.e.  A 1N87  s i g n a l f o l l o w i n g the v a r i -  In order t o use phase coherent  detection,  d e t e c t i o n i n the l i n e a r c h a r a c t e r i s t i c r e g i o n of the diode, a  r e f e r e n c e o r marker s i g n a l i s i n s e r t e d i n the l a s t a m p l i f i e r stage„ T h i s g i v e s a r e f e r e n c e v o l t a g e which e f f e c t i v e l y b i a s e s the diode at  about 2.5 v o l t s p o s i t i v e and assures  linear detection.  L i n e a r i t y c a l i b r a t i o n curves were obtained u s i n g both the NMR  s i g n a l i n a gas and a c a l i b r a t e d s i g n a l generator,  and the  r e s u l t s i n d i c a t e t h a t the l i n e a r i t y of the a m p l i f i e r i s l i m i t e d o n l y by the v o l t a g e c h a r a c t e r i s t i c of the diode.  This f a c t  indi-  cates the need f o r a s t a b l e , low n o i s e , r e f e r e n c e s i g n a l to b i a s the diode onto t h i s l i n e a r p o r t i o n of the c h a r a c t e r i s t i c curve as d e s c r i b e d i n the next  chapter.  19 2.1.3.  96 mHz  COHERENT REFERENCE GENERATOR  ( f i g . 2.8)  A 12 mHz  s i g n a l i s o b t a i n e d from a tuned r f» 0  a m p l i f i e r c o n t a i n e d i n the o s c i l l a t o r compartment of the 96  mHz.  pulse generator.  follower  T h i s 12 mHz v o l t a g e i s f e d i n t o a cathode  which matches the impedance of the H e l i o p o t phase s h i f t e r  (450/1)  From the phase s h i f t e r the s i g n a l then goes to two cascoded 25 tuned a m p l i f i e r s pentodes. 96 mHz  mHz  The frequency i s then quadrupled to  i n two more cascoded a m p l i f i e r stages which are s a t u r a t e d to  reduce f l u c t u a t i o n s i n output v o l t a g e .  Before the r e f e r e n c e v o l -  tage i s f e d i n t o the a m p l i f i e r i t i s f i r s t c l i p p e d by a s e t of c r o s s e d 1N3600 diodes to f u r t h e r reduce f l u c t u a t i o n s and r i p p l e i n the output v o l t a g e .  Even though t h i s c l i p p i n g reduces the v o l t a g e t  fed i n t o the 96 mHz  a m p l i f i e r to about 0.5 v o l t s , the l a s t stage  of the a m p l i f i e r i s s u f f i c i e n t to i n c r e a s e the r.m.s. v a l u e of the r e f e r e n c e v o l t a g e to about 2.5 v o l t s which i s s u f f i c i e n t to i n s u r e l i n e a r d e t e c t i o n f o r s i g n a l s up to about 0.1  - 0.2  S e v e r a l s t r i n g e n t c r i t e r i a are imposed  volts.  on the  stability  of the r e f e r e n c e phase and frequency when one uses phase coherent d e t e c t i o n t o observe a spin-echo which may seconds a f t e r the 90° p u l s e .  occur 600 - 800 micro-  T h i s r e q u i r e s t h a t the phase of the  r e f e r e n c e and the r e f e r e n c e generator remain l o c k e d to a t l e a s t one t e n t h of a c y c l e f o r t h i s time to make even a 10% a c c u r a t e measurement of the amplitude of the echo.  In o t h e r words the  phase must remain c o n s t a n t to a t l e a s t one p a r t i n 10 f o r 600 800 microseconds. care was  To achieve t h i s s t a b i l i t y a l a r g e amount of  taken i n the l o c a t i o n of the tubes, s o l i d  mounting of the components was  non-microphonic  used and a t t e n t i o n was  p a i d to the  thermal s t a b i l i t y of the components i n the r e f e r e n c e generator  21 which was c o n t a i n e d i n a'spun copper can. These measures seemed t o be q u i t e adequate and no d r i f t  was ever noticed' which- c o u l d be a t t r i -  buted t o a phase or frequency change i n the r e f e r e n c e v o l t a g e .  2.1.4.  TIMING APPARATUS The g a t i n g p u l s e s a r e o b t a i n e d from T e k t r o n i x p u l s e gene-  rators suitably modified i n several instances.  The r e p e t i t i o n  r a t e of the e n t i r e p u l s e sequence i s c o n t r o l l e d by a T e k t r o n i x type 162 waveform generator running i n the r e c u r r e n t mode.  T h i s type  162 waveform generator t r i g g e r s a type 163 p u l s e generator t o prov i d e the f i r s t second is  180° p u l s e .  The f i r s t  type 162 a l s o t r i g g e r s a  type 162 waveform generator from which a sawtooth  waveform  taken to t r i g g e r a type 163 p u l s e generator to p r o v i d e a 90°  pulse. tor  The v o l t a g e on the sawtooth  t r i g g e r s i s s e t by a type 162 waveform generator which has been  m o d i f i e d t o p r o v i d e sawtooth duration. yet  a t which' the 90° p u l s e genera-  waveforms of up to about two hours'  A p u l s e i s taken from the 90° p u l s e generator to t r i g g e r  another type 162 waveform generator, the sawtooth  o f which  t r i g g e r s another type 163 p u l s e generator t o p r o v i d e the second 180° p u l s e and t o t r i g g e r a type 161 p u l s e generator which prov i d e s the p o s i t i v e and n e g a t i v e gate f o r the boxcar The  integrator.  two 180° p u l s e s and the 90° p u l s e are then f e d to  a mixer a m p l i f i e r , f i g . 2.9, which e f f e c t i v e l y i s o l a t e s the two p u l s e s o f d i f f e r e n t widths, a m p l i f i e s the v o l t a g e of the p u l s e and p r o v i d e s a low impedence output t o the v . h . f . p u l s e g e n e r a t o r .  A  schematic diagram of t h i s mixer a m p l i f i e r i s i n c l u d e d i n the appendix .  2.1.5.  BOXCAR INTEGRATOR. The boxcar i n t e g r a t o r used was the same one used by  CH Q  225  O-r  VDC  +225VDC  O-  170VDC  2  -  2 . 9 .  DC Pulse  M i x e r  A m p l i f i e r  —  5687's  lOG^X  FIG.  225VD<£>+225VDC  ~  i n  R e s i s t o r s  P a r a l l e l Separate  to  23 W. N , H a r d y a n d t h o r o u g h l y  described in'his  t u t i o n of multiturn potentiometers was  t h e s i s ) .  for single  turn  A  substi-  potentiometers  the only additional modification.  2.1.6.  MATCHING NETWORK (fig.  apparent getting very  2.10)  In the early  stages  o f t h e p r o j e c t i t became  t h a t t h e m a j o r o b s t a c l e t o be overcome was t h e p r o b l e m o f a large v.h.f. pulse  small signal out with  into  t h e sample a n d ' o b s e r v i n g t h e  a minimum o f l o s s  i n the signal  t o noise  ratio. The the  problem o f t r a n s m i t t i n g the l a r g e v . h . f . pulse  sample was s o l v e d w i t h  coupling This  t o t h e tuned  circuit  the necessary  i n power.  proved  t o be a p r o b l e m w i t h o u t , s u c h i n this  circuit  E x t r a c t i o n of the small signal  and'the v.h.f.  time  only  The d i f f i -  a match' t o t h e v-.h.f.  amplifier  effort  impedence i n order  that the  (3-4 m i c r o s e c o n d s )  6CW4 i n p u t t u b e s t u r d y enough t h a t no i s o l a t i o n  work, c a p a c i t a n c e  small  f r o m t h e sample  somewhat by t h e f a c t  f r o m s a t u r a t i o n was f a s t ; e n o u g h  amplifier  impe-  t o n o i s e r a t i o - and s a t u r a t i o n r e c o v e r y  of the  f r o m t h e h i g h power w h . f . p u l s e " was n e c e s s a r y .  Many d i f f e r e n t m a t c h i n g c i r c u i t s  an  the high  solution.  t o t h e 50^A. i n p u t o f t h e v . h . f .  T h i s p r o b l e m was a l l e v i a t e d  recovery  coil.  i t was a l s o n e c e s s a r y 'to m a t c h t h e h i g h  t o o b t a i n an optimum s i g n a l time.  an e a s y  c a s e was t h a t w h i l e m a i n t a i n i n g  pulse generator, tuned  matching-of  t o t h e 50S\* t r a n s m i s s i o n l i n e w i t h  losses  culty  transformer  c i r c u i t w h i c h i n c l u d e s t h e sample  arrangement achieved  dence tuned  the use o f a capacitance  into  transformer,  t o match t h e . t u n e d  including  a p i network,  and a u t o t r a n s f o r m e r ,  circuit  t o t h e 5O.A.  L  net-  were t r i e d i n  coaxial  cable.  However, the s i m p l e s t arrangement of j u s t using' a 6.8  pf„  capaci-  t o r to tap d i r e c t l y from the tuned c i r c u i t to the t r a n s m i s s i o n l i n e w i t h no attempt a t impedence matching gave a signal, to n o i s e r a t i o which was  a t l e a s t e q u i v a l e n t to any of the more complicated match-  i n g arrangements and without many of the problems encountered matching networks..  Matching networks i n v o l v i n g a h i g h Q  with  coil  one of the elements had.a tendency to r i n g a f t e r the p u l s e  as  was  switched o f f , as w e l l as being cumbersome to tune p r o p e r l y at temperature.  low  A c a p a c i t a n c e transformer matching network proved  to  be d i f f i c u l t . t o tune s i n c e the c i r c u i t i n t e r a c t e d w i t h the c a p a c i tance c o i l . .Autotransformers 96 mHz  proved  to be very i n e f f i c i e n t a t  w i t h the r i b b o n sample c o i l c o n f i g u r a t i o n used. A d i r e c t i o n a l c o u p l i n g d e v i c e , although not t r i e d s i n c e the  l o s s e s quoted by most manufacturers  were considered, to be  than o b t a i n e d by the s i m p l e s t c i r c u i t used, should be as a p o s s i b i l i t y a g a i n . 'permit  The-recent  higher  considered  advances i n c i r c u i t r y  may  l o c a t i n g an u l t r a - s m a l l i n t e g r a t e d c i r c u i t p r e - a m p l i f i e r  of low n o i s e d i r e c t l y a t the sample c o i l . .  With s u i t a b l e s w i t c h i n g  arrangements to p r o t e c t the d e v i c e from the h i g h power p u l s e  this  c o u l d e l i m i n a t e e n t i r e l y the problem of matching t c a t r a n s m i s s i o n l i n e and the i n h e r e n t l o s s e s i n matching. A sample c o i l c o n s i s t i n g of f i v e turns of 3 m i l copper r i b bon,  3 mm.  wide wound around the o u t s i d e of a 9 mm.  gdlve the maximum volume f o r a g i v e n inductance inductance and  was  s i n c e both the  self  the mutual inductance of such t h i n r i b b o n i s much  l e s s than an e q u i v a l e n t l e n g t h of round w i r e . a t 96 mHz  \J  g l a s s tube  A Q  of about  400  u s u a l l y o b t a i n e d w i t h such a c o i l .  Tuning of the sample c o i l c i r c u i t i n the t a i l of the dewar  25  k —  ~  —  1 FIG.  2.10  J  1/8"  Sample'Configuration  and  Matching  Network  26 was accomplished by u s i n g p i s t o n trimmer v a r i a b l e c a p a c i t o r s con7  s t r u c t e d o f b r a s s w i t h pyrex tube i n s u l a t i o n " which" were f a b r i c a t e d i n the  U.B.G,  P h y s i c s Department shop.  T e f l o n rods then extended  from these c a p a c i t o r s so t h a t the c i r c u i t by r o t a t i n g the T e f l o n r o d s .  could be e a s i l y tuned b  T h i s tuning a t low temperatures was  necessary s i n c e the c i r c u i t parameters v a r i e d w i t h 1  temperature  causing the c i r c u i t tuning t o v a r y w i t h temperature.  2.1.7.  VIDEO AMPLIFIER A f t e r d e t e c t i o n the s i g n a l was f u r t h e r a m p l i f i e d  using  both s e c t i o n s of the T e k t r o n i x 1A1 d u a l p l u g - i n a m p l i f i e r i n a T e k t r o n i x type 547 o s c i l l o s c o p e b e f o r e being f e d to the boxcar integrator.  2.1.8.  MAGNET AND POWER SUPPLY A Magnion f i f t e e n i n c h electromagnet w i t h p o l e caps  tapered t o 6.25 inches with a 1.75 i n c h gap was used f o r t h i s experiment.  With the  Jj( -shim d e v i c e w i t h which t h i s magnet  was equipped a p r o t o n i n d u c t i o n t a i l of about 100 microsecond width c o r r e s p o n d i n g to a f i e l d at  inhomogeniety of two p a r t s i n 10--'  22.5 k i l o g a u s s was o b t a i n e d i n a 5 c c . sample.  The power  supply was a Magnion HS-1365 power supply w i t h only c u r r e n t r e g u l a t i o n u s i n g mercury quate f i e l d  2.1.9.  c e l l s as the v o l t a g e r e f e r e n c e .  Ade-  s t a b i l i t y was o b t a i n e d from t h i s power supply.  TIME INTERVAL UNIT A Hewlett-Packard time i n t e r v a l u n i t i n a type 5245L  frequency measure  counter  the time  equipped interval  with  a digital  between  print-out  was  the inversion-pulse  used  and  to  sampling  sequence.  2.1.10v  STRIP  The Varian ker  output  triggered  GAS  further  from  Care  was  change even of  the e f f e c t . o f  though  research  0.01%  with  interval  the purity  impurities  temperature  mar-  unit.  b y W.  The hydrogen'used  as u l t r a - p u r e  to guard  an e v e n t  t h e same o n e u s e d  t h e measurements  frequently  recorded i n a  (99.99%) since,  i s certainly  used  was a n d no  a t the low negligible.  a t low temperatures against  N.  to  ortho-para conversion  the" c o n v e r s i o n t i m e i s  o f weeks.  The as  during  Company  was  SAMPLE  was  to increase  a t the lowest  the order  GAS  i n his thesis. (10)  was made  t h e sample  equipped  o f the time  S Y S T E M AND  the Matheson  taken  integrator  recorder  gas h a n d l i n g system  attempt  densities  the boxcar  chart  HANDLING  and d e s c r i b e d  obtained  from  by t h e p r i n t - o u t  This Hardy  RECORDER.  G 1 1 As t r i p  type  2.1.11.  CHART  m e t h a n e was  grade  also  (99.99%)  o b t a i n e d from  a n d no f u r t h e r  the Matheson  purification  was  Company per-  formed . Use pared  b y K.  determined Use  L a l i t a d ) 1  and  o f such  small  of a since  54.5% h e l i u m  range  i n hydrogen  t h e c o m p o s i t i o n had been  accurately  a mixture  density  certainly  was made  measured  a t moderate  provided a comparison  f o r the low d e n s i t y  a psychologically  desirable  region  mixture  accurately densities.  f o r T-^ o v e r  T^ m e a s u r e m e n t s  attribute  pre-  when  a  which i s  performing  28 such s m a l l s i g n a l measurements i n high n o i s e backgrounds.  Agree-  5  ment was o b t a i n e d w i t h the r e s u l t s ' o f L a l i t a i n ' t h e d e n s i t y :  i n which the measurements overlapped.  The' HD sample was  region  obtained  from Merck, Sharp and Dohme of Canada L t d . at" 98v5% ••isotopic p u r i t y i n a f i v e l i t e r g l a s s b o t t l e a t one a t m o s p h e r e .  :  This"unfortunate  packaging r e q u i r e d the use of a diaphragm'pump~as d e s c r i b e d by W. N. Hardy.  Even though- t h i s increased' c o m p l i c a t i o n ' i n t r o d u c e d  more p o s s i b l e courses of contamination, • i n d i c a t i o n s from comparisons of T^ i n f r e s h samples and much used samples were t h a t any contamination was  negligible.  2.2.  MEASUREMENTS  2.2.1.  TEMPERATURE MEASUREMENTS A very a c c u r a t e measurement of temperature was not con^  s i d e r e d to be of g r e a t importance and so" a copper-constantan thermocouple which had been checked by comparison w i t h a p l a t i n u m  resis-  tance thermometer was used•" to measure the temperature of the l i q u i d n i t r o g e n and the dry i c e - a c e t o n e . b a t h s . good q u a l i t y g l a s s - l i q u i d thermometer used to monitor temperature.  At' room temperature a  l o c a t e d a t the sample  was  Temperature f l u c t u a t i o n s d u r i n g the  course of a sequence of measurement i n the r e g i o n of a T^ minimum amounted t o no more than ± i . 0 ° K  as measured by the above methods.  Such a sequence of measurements sometimes week.  1  l a s t e d as long as one  29  2.2.2.  DENSITY MEASUREMENTS  Above two atmospheres p r e s s u r e a s i x - i n c h diameter bourdon tube guage  (vacuum to" 100' p. s v i . o r vacuum" to 300 p . s . i . )  a c c u r a t e t o — 2% was used-to measure the pressure i n the sample.  •r  (8)  From the d e n s i t y . c o r r e c t i o n - t a b l e s o f Woo l e y , S c o t t and Bnckwedde iv  it  i s e v i d e n t that"-the"dTen-s±-ty"may '%e--- detnBrmi*ne-d' f o r H ,  ;  2  a t the den-  s i t i e s used from the- i d e a l " gas- law> - given^the-pressure:, and- temperature-.  The' maxinranrcorrection~tg~-fehe~ 'd^s-±t:yat"-these" d e n s i t i e s i s v  of the order- of" 'OVOl'% whiehris" eerta±rrly n e g l i g i b l e , w i t h i n the  2%  a c c u r a c y of the' pressurer-measurementi"' -Similar l y below two atmospheres a mercury manometer was used t o measure  the p r e s s u r e of the  gas and the d e n s i t y determined from the- i d e a l gas law.  The measure-  ments of p r e s s u r e u s i n g a mercury manometer were a c c u r a t e t o b e t t e r than to. 5%. I t seems i n c o n c e i v a b l e - t h a t any important systematic e r r o r i n the dependence o f Tj" on d e n s i t y could be i n t r o d u c e d due to the density 2.2.3. •  measurements. GENERAL COMMENTS'.' ON"- EQUIPMENT AND* 'MEASUREMENT u  I t is•' thoughtvthat":the'^limiting' parameter on the s i z e of the s i g n a l which could'•'•be* -used-to. measure -T^ was imposed not by 'white n o i s e  1 ,  but" b y --low "frequency " f luetuations* i n the c i r c u i t  and/or i n the power' i i n e s . : For t h i s reason a l l o f the lowest den1  s i t y measurements  were performed i n the v e r y e a r l y morning hours when  the power was most s t a b l e .  Performing' the experiments a t t h a t time  had the a d d i t i o n a l advantage t h a t the background i n t e r f e r e n c e from such sources as motors and F-.M. s t a t i o n s was lowest and allowed the  30 measurements •coil' with the  a  to  be  subsequent  precaution" of  ments' i n t h e cent  increase  making'the  taken  for' a of  :  before  and  :  i n the'Q  1  low  of  density  the' c o i l .  (small  :  course' of  the  For eliminate the  :  than"ten  signal)  •• -  this  For  Q  hour  change  measured showed over  a  the  discarded.  identify  e r r o r s - which- a r e " o f ' f a r g r e a t e r :  measurements  concern  of  T^.  these-"-sma-Mi-v-s'igiiall'^«asirremerntrs? ' the''•mo'S't'- ' g l a r i n g ,  and  device;'  i n  distortion  h a v e 'been d i s c u s s e d sufficient  w h e r e c<  pos-  !  and  by  - -  the  boxcar integrator  (2)  W'."N.= H a r d y r  He  :  obtains  He « ^T,  and  inverse  RC  i s tne  i s given  of  the  the time  repetition constant  rate- at  effective  by  the  The  :  c h a r a e - t e r i s t i c s ' o f 'the  a  which  the  necessary which  pulse  time-'constant': of' t h e  and  is  separation boxcar  inte-  ——  RC is  t h i s ' "ca"se a~ b o x c a r i n t e g r a t o r .  c o n d i t i o n ' t o " avoid- i n t e g r a t i o n ' d i s t o r t i o n  i s the  varied  RC  -  M  7  two a  study.'-of' T p minima',' c a r e ' w a s - - t a k e n ^ ' t o  -  T  to  per  f o r a - d e n s i t y " d e p e n d e n t - s y s t e m a t i c " e r r o r i s of" c o u r s e  integration  where  fifty  RESULTS .  systematic  signal^ averaging  grator  f  ,  -  !  with  measure-  e v i d e n c e d " by  ;  random•errors"between-indivi d Era"l"  sibility  one  per^'cen-t "in t h e " v a l u e " o f  measurement^were  - ACCURACY- OF  • and  s i n g l e * T^* m e a s u r e m e n t " a s  sample  Even  a f t e r " e a c h - m e a s u r e m e n t - ; • '-""Those' "t?r a'e'e's"'which, greater  2.2.4.  the  t h e ? e q u i l i b r i u m • ' m a g n e t i z a t i o n ' - w h i c h : was  change' o f  and  shielding' around  e a r l y morning'there-seemed"-to'"be "almost: a  -in- t h e v a l u e  is  without  p r o b a b i l i t y ' o f - a power-f luctuat-ion''during" the  period  than  performed  = rate, of  the  t"  t h e " w i d t h - o f 'the  resistor-capacitor  sampling network.  gate "Much  31 less tor  than"  i n this  of about  ratios  7  actually  -zj^g  present, using  7  used this  As  7  the basis ;  7  i t c a n be than  1  shows  that  the c r i t e r i o n  check' t h a t ' n o ' i n t e g r a t i o n 5  7  fac-  some  ratios  ; :  90' a n d  a  :  t h a n " 'once  s h o u l d be  distortion  7  at' a given  ::  :  i n t h e - T^" v a l u e s w a s " o b t a i n e d  by  was  density a  factor  detectable  i n such  7  safe  by  H2" at-' r o o m " t e m p e r a t u r e .  777  of these  sys-  experiments.  two m e t h o d s  of c h e c k i n g f o r  to say that  t h e e x p e r i m e n t a l v a l u e s o f T^  by d i s t o r t i o n  chart  X)  than  seen that~ a i l of the  ,  Another the' s t r i p  (Table  f o r t h e d i f f e r e n t ' measurements'.•'- ' No  i t seems  affected  table-  "less  •and/or* 'tame" ••c^nst an't^*'- whic*"'"vai?ied'  1  tematic difference  not  table  a further  time's'  t o mean  measurementsiTt  7  sweep  tortion  for  T-^ was' o f t e n ' m e a s u r e d - m o r e  of. t e n o r more  On  taken  7  are greater  fulfilled.  was  e i g h t y " v " The- f o l l o w i n g  From y~l —  criterion  due  possible  recorder  7 7  to the'boxcar  integration  were  integrator.  s o u r c e o f an- i n t e g r a t i o n  which  dis-  has  T  a variable  from  7  this  distortion  response  7  i s  time' o r  damp-  ing . Possible using gain  the recorder  distortion with  o b s e r v e d on  recorder  was A  provided  safely further  7  so' t h a t  verified  by  eliminated  eliminated  check' on the' absence-"of of ^  measurement  with  the measurements  non-systematic" deviation."  source of systematic  as a f u n c t i o n  A  7  o f t h e a b s e n c e ' o f a" s y s t e m a t i c  error  errors  ;  than  at  2  of the  calculated  no more  was  for H  comparison  the' t h e o r e t i c a l l y o f H a r d y shows  the  distortion.  of density  T h e ' c o m p a r i s o n ' a t 77 °K 7  (noise)  " With' t h e s e p r e c a u t i o n s  as a p o s s i b l e  by  and with the  rapid'' t r a n s i e n t s  i s a c a s e with-' p r e d i c t a b l e ' r e s u l t s :  results' of this  best' t e s t  7  the recorder'.  b y a measurement  77 °K w h i c h  s o u r c e was  the'l o w e s t damping p o s s i b l e  of the amplifier' adjusted  would' be  as  7  a  i s perhaps  i n ' T-, f o r s u c h  curve —  5%  the possi-  T Amagats  See.  t r  Msec.'  <=<  X 10  6  "  Msec  Sec  ^10  3  sec  0.92  .068  20  65  2.7  3.60  24  9.7  460  1.71  ... 068  20  65  2.7  1.85  21  5.0  238  3.51  .068  20  65  2.7  1.09  21  2.9  138  6.1  .0068  20  . 75  2.7  1.19  1.8  3.2  1780  3.4  .0068  20  75-  6.3  1.18  1,8  7.4  4100  2.8  .068  20  75  5.4  1.47  18  8.0  445  4.5-  .068  20  75  3.6  1.02  18  3.7  205  2.55  0.1  20  75  4.5  1.38  27  6.2  230  0.01  20  65  0.6  1.44  3.1  1.86*  600  8.5  0.012  20  65  1.2  1.19  3.7  1.45  392  9.7  0.0l2  20  65  3.6  0.95  3.7  3.4 • >  920  0.92  0.1-  20  65  . 2.7  3.50  9.5  306  12.5  ,  -  ,  31  i  -TABLE I I n t e g r a t i o n Constants and Times used f o r T-, Measurements.  to  33  b i l i t i e s as r e c e i v e r linearity",'• i n c l u d i n g " coherent 7  arity,  detection l i n e -  long' base' l i n e ' s a t u r a t i o n recovery" times- and' p u l s e  interac-  7  t i o n s ' which may son was  -  w e l l be l o s t " i n the noise'. • -Even" though-the' compari77  7  -  used a' further*- safeguard', •'that-of -fceeping- the' N;M. R. r  71  signal  l e s s than" 0.1' v o l t s at" the* d e t e c t o r when using" a 2. 5 v o l t r.m.s. 7  7  7  coherent r e f e r e n c e signal", was" imposed. | 7  - •••-••-"-"  A check" to'insure' -that- there "were?.;ho l o n g base l i n e 7  r  7  r e c o v e r y times - which" might w e l l be"" b u r i e d i n " the n o i s e and which 7  7  -  would i n f l u e n c e the T-j_" measurements ' s y s t e m a t i c a l l y "with d e n s i t y , 7  7  c o n s i s t e d of sweeping"the pulse" sequence with' the magnetic  field  7  w e l l o f f ' resonance," f o r " s e v e r a l ' a m p l i f i e r " gain" 'settings correspond7  i n g to d i f f e r e n t d e n s i t i e s , and'observing' the s t r i p c h a r t r e c o r d e r 7  traces .' • T h i s check i n d i c a t e s ' that" there" were no l o n g term base 77  l i n e r e c o v e r y time' e f f e c t s 7  77  present.  -.- r •. •. • Perhaps" the" most' d i f f i c u l t . to'-detect -possible source 7  a s y s t e m a t i c e r r o r ' i s an ' i n t e r a c t i o n " between" the i n d i v i d u a l p u l s e s 7  7  o f the p u l s e sequence' which' m i g h t v a r y the" power o f t h e p u l s e s as a 7  7  -  f u n c t i o n ' of the' separation- between" p u l s e s and" hence' vary M ( t ) . z  M o s t l i k e i y t h i s ' i n t e r a c t i o n would- have- shown up as a; n o n - l i n e a r i t y when p l o t t i n g . ln'(M ' - M (t ) ) versus't", the" -time between p u l s e s . 7  7  0  7  z  r e p r o d u c i b l e " n o n - l i n e a r i t y was- ever observed'on- these-' p l o t s .  On  t h i s - b a s i s ' and'-with' t h e "comparison' of - the" 77°K" r e s u l t s w i t h  the  7  7  T  t h e o r e t i c a l r e s u l t s , it ".seems- most' u n l i k e l y t h a t there was 7  77  No  any  e f f e c t on T-j. from an i n t e r a c t i o n ' between p u l s e s . 7  -  7  7  . With " a l l of these-' p o s s i b l e ' sources" of e r r o r the most con7  v i n c i n g ;. irgument" that' there were' no systematic e r r o r s ' p r e s e n t i n the 7  measurement of  a s a" f u n c t i o n ' of- d e n s i t y i s c e r t a i n l y the e x c e l l e n t 7  agreement of the' r e s u l t s ' at" 77"°.-K- w i t h the' w e l l e s t a b l i s h e d theore-  34 t i c a l c u i t  curve'  since  parameters"  •  - -  r  f o r  were"'  the  -  higher*  T  random"errors  5  m a y now be  :  d e n s i t y  of " d e v i a t i o n  i n  rEOTlt^f-mwBB^phH'n'oiiiena"  s c a t t e r  r e t a i n e d than  £  two  as  i n a  15% o f  and"o f t e n  so  t h a t  a  t h e " s l o p e * -;of-- - I n - C M o " - * M ^ ( t ) ) " r  be  assumed.  as  determined'  measurements  than  changes 7  f i v e  m o s t I n  v  versus  :  extreme' no  case  was a  t h e m o s t  v a l u e  more" v a l u e s '  A t  o f  7  t h e  h i g h e s t  t h i s t h e  per" c e n t ,  T-]_.  i  10% a t  t h e  extreme  were  l o w e s t  from  g r e a t e r  a t  d e n s i t y  t h a t  l e a s t  averaged  random'  c a n  e r r o r s  t h e T]_  ~t 5 % .  d i s c u s s i o n  i t  dependences  of" T^' v e r s u s " d e n s i t y  m a y be  T]_)  were  Furthermore,  line-method" i n d i c a t e s  l i m i t s  measurement  v a l u e s  t h e  p e r -  imaginable  T y  a' given" d e n s i t y  d e n s i t i e s " measured  a r e accurate' t o  i n  T-^ a t  random error  t h e  by  o f  o f  t" ( i . e .  'limits  when" the" e x t r e m e  probable;  and  The maximum  ;  v a l i d ' measurement  From a t i c  T  the- data" points".  maximum  the  a s . power  o f^components  o r t e m p e r a t u r e ^ f l u c t u a t i o n s  w e r e ' o b t a i n e d " t»y d r a w i n g - " i n " t h e the  c i r -  examined.  ' f l u c t u a t i o n s > ~ " w h i t e ^ -nblse-'-'-v - ' v i b r a t i o n "  haps" even  no  t h e r e " were-- n o - s y s t e m a t i c " e r r o r s  • These" random* e r r o r s " l i n e  TjTOeasuTements  changed.  On" t h e ' p r e m i s e ' ' t h a t -  :  temperature'  m a y be  observed  as~ a  :  T  i n f e r r e d  r e a l  t h a t o f  e f f e c t .  any  system-  g r e a t e r  35  = '•••• C - H - A  P• T: E ' R  =  3 . 1 . -  DESCRIPTION-  Before, a  b r i e f  t o t a l  wave  of  p r o t o n s  f u n c t i o n  of  to  has  two  m o d i f i c a t i o n s : :  and  r o t a t i o n a l  t i o n  between  bidden  n u c l e a r the  s p i n s ;  order'  t u r e s  i t  t u r e s  the  i s  that  one: of  7  7  of i s  y e a r s ,  which"  thermodynamic d i s t r i b u t e d t i o n  as  to  a a  7  but  w i t h  produce  (ortho-H^)  7  g i v e s  e q u i l i b r i u m " at  among  d e s c r i b e d  t h e " J ' by  for" the  the-  an  a  s t a t e s  The  has' t o t a l  7  7  100% of  a  a t  i s  7  to  T  the a  t r a n s i -  of  f o r -  the i s  of  low  tempera-  h i g h  tempera-  75%.  F o r  1=1  h i g h l y  two' d i s t i n c t  N.-M .R. =s i g n a l .  w i t h  and  N . T . P .  A t  of  A  s t a t e  the  m o l e c u l e s p i n  2. .v  p a r a - H ^ . i  t h a t  3 . . . ,  and  ortho -H2  a c c o r d i n g  e q u a t i o n  J = l ,  c a t a l y s t s  temperature 7  n u c l e a r  s u c h ' a" p r o c e s s of  n e c e s s a r y .  hydrogen  a n d " a" s i n g l e  a i d  i s  a n t i - s y m m e t r i c  and' J=0,  7  theory,  r e q u i r e  state' r e p r e s e n t s  a-" m i x t u r e 7  gas  t h e r e f o r e ,  ' s p i n - I=0  a l m o s t  as  hydrogen  quantum' 'numbers  c o n c e n t r a t i o n  be' t r e a t e d  r e l a x a t i o n  m o l e c u l e b e  - t r i p l e t  7  the  p r o t o n s .  p a r a -  h a l f - l i f e  e q u i l i b r i u m  H'2' c a n  and  of  w h i c h  ?  n u c l e a r  between 7  two  momentum  t o t a l  The  easy  the  of  and,  hydrogen  o r t h o - H  o r t h o -  t r a n s i t i o n  7  d i s c u s s i o n  ferraions  of'  a n g u l a r  an  GAS -  a  the  permutation'  has  THEORY  t h e ' p r o p e r t i e s are  r e s p e c t  para-Hp; w h i c h  H?  b e g i n n i n g  d e s c r i p t i o n The  OF  III  ;  The g a s e s ,  H^- g a s  m o l e c u l e s  Boltzmann  r e s u l t o n l y i n are  d i s t r i b u -  36  -e*/kT  gv  where energy given  ~i s  the  1  d e g e n e r a c y - ' o-f> t h e " J  associated  $  w i t h ' a* g i v e n ' J s t a t e .  +  =  =  T  A J  and E , is- t h e J  This- degeneracy  r o t a t i o n a l g  i s J  + l.  Jeven,  = 3(AJ+/)..  U-r+i)Uz+0  L X  -  6)  T oat J x = l)  E j i s given by  where-' I _ i s t h e ' m o m e n t ' o f " i n e r t i a o value f o r  state  by  <$T  and  ;  1  85 . 3 ° K  ortho-H  2  f o r " H' .: 2  -  o f the molecule -  ' V a l u e s ' o f Pp' a t " t h e  r  a n d A„ h a s  temperatures  o f  the  interest  a r e • -- • = Ortho^H  /  TABLE  I I Para-H.  2  3  £~  0  a  0.993  0.007  0.7 2  0.28  77  1.0  200  0.965  0.035  300  0.87 9  0.119  0.002  0.517  0.468  0. 015  400  0.778  0.290  j 0.022  0.397  0.562  0.05  3 . 2 .  DESCRIPTION'  HD ments  d i f f e r s  -  R  where  mass  from  and" t h e f r a c t i o n a l  P  and  OF HD GAS-  c?^ =  H  2  J 64.26  i s t h a t  f l  'K  the" l a t e r  i n  1  that-  p o p u l a t i o n  f o rHD."  u n l i k e  7  there'  e ' er^^o j  C  T  +  ]  s t a t e s  r e q u i r e -  i s g i v e n b y  ^ ^ / k T GR/UT  A's i g n i f i c a n t H  2  d i f f e r e n c e  a n d , t h e r e f o r e ,  between HD  molecule' t h e c e n t e r  does' n o t c o r r e s p o n d  t o the"  a d i f f e r e n t  o f  g e o m e t r i c a l form  o f  t h e  i s e x p e c t e d .  o f P j f o rH D a t s o m e  d i s c u s s i o n s " are" g i v e n  TABLE  o  a r e " rib' s y m m e t r y '  o f the" J  t h e c a s e = o f ' t h e  i n t e r a c t i o n  V a l u e s  ;  T(ZJH)  o f t h e m o l e c u l e ,  a n i s o t r o p i c  2  fr*+0  o f t h e HDm o l e c u l e  c e n t e r  H  temperatures  i n -t h e f o l l o w i n g  o f i n t e r e s t  Table I I I .  I I I  3  /  23.0  0.99  0.011  77 . 0  0.69  0.30  0.01  200  0.294  0.452  0.204  .038  300  0.204  0.388  0.269  0.104  y-  .00  .035  i n  38 3.3.  RELAXATION: IN H  7  ?  As p o i n t e d o u t i n the i n t r o d u c t i o n  the n u c l e a r spins i n  d u l u t e hydrogen gas are" r e l a x e s b y the - f l u c u a t i h g 7  intra-molecular  f i e l d s and not"'by" the "dipole i n t e r a c t i o n s " between" n u c l e a r  spins  7  on d i f f e r e n t " moleculesv ~ These" f l u c t u a t i n g r e s u l t ' from' the. i n t r a - m o l e c u l a r tions.'  -  intra-molecular  fields  spin" r o t a t i o n " and d i p o l a r  interac-  The s p i n Hamiltonian' f o r a" s i n g l e " ortho-H? molecule i n a  magnetic f i e l d  7  H  Q  applied  along the  z - a x i s - i s ' g i v e n by  -  ^  '  m  (3.1.)  .III  where u/j^ and  a r e the" Larmor precession' f r e q u e n c i e s of I and J  r e s p e c t i v e l y i n the f i e l d H , • ^ =: 2.6 9 X 10 ^ sec"^- g a u s s :  - 1  Q  the proton gyromagnetic r a t i o n and H' *= 26.752 ~t . 007  gauss and  7  H" =- 33 .862  - . 015  gauss are" the" coupling' c o n s t a n t s a s s o c i a t e d  the s p i n r o t a t i o n ^nd d i p o l e 7  L^  interactions respectively.  a r e components" of' i r r e d u c i b l e ' tensors' a s s o c i a t e d 7  m  s p i n o p e r a t o r s Xa-y 1+ -  -^ly  -  J~  £  j-^ —  S*o  Lie  is  with  S^. and m  with nuclear  and molecular r o t a t i o n o p e r a t o r s  + i J~y r e s p e c t i v e l y and are d e f i n e d by 7  = (-r) E^ k  "Zd+Olj  J i ,  Lihi  S ±i A  ~ J±  =J*I  ±  +Z±  "Ii"  (3.3a)  ( - -> 3  4  39  is-" the" s p h e r i c a l harmonic of' order 2 , of the  (ll)  where  orien-  1  t a t i o n s of the"molecular a x i s . The" r e l a x a t i b n rate" can" be r e l a t e d " to the" f l u c t u a t i n g r  -  i n t r a - m o l e c u l a r f i e l d s " through the' Fourier" t r a n s f o r m of the 4  5  I/^f7  Lr£^ (~£ ) •'• The- r e l a x a t i o n rate'  l a t i o n f u n c t i o n of  corre-  1  :  c  may  be  writ-  ten  S-r, =  2^  _  +l%rfo,M  Uur)  ^Z (^)J  ( 3 . 6 . ,  0  J^(ur) = / £ W c/t where Gj^^fe) i s " a; c o r r e l a t i o n f u n c t i o n  where and  L^^,  symmetrized product of the  G Jt)  = ^  A  L„  {  w i t h the bar r e p r e s e n t i n g Needier and where Ufyis moment.  given  i n terms of  the  's'  + l Jo)lJ„ A  (t)}  >  ( 3 . 8 . ,  an ensemble- average.  1  Opechows"ki  ^" have: suggested t h a t  rV  Owz  us~ - U/^. x  the" Larmor f r e q u e n c y of - the' molecular r o t a t i o n a l  For H , 2  quency and  (o)  (-t)L^  t~k)  (3.7.)  IWj  i s about, one" f i f t h " of the" proton Larmor f r e -  t h i s r o t a t i o n a l Zeeman s p l i t t i n g cannot be neglected  shown e x p e r i m e n t a l l y  b y Hardy  5  Needier" and  ' .  K  O p e c h d w s k i ( ' have" a l s o " p o i n t e d :  L'  though the operator  AKrt  as  out" t h a t even  has- non-zero matrix" elements between p a i r s  of s t a t e s whose' angular- momenta- d i f f e r by  J - 0 or i 2 , the o s c i l l a :  t i o n ' frequency of the l a t t e r terms is- greater" than 1 0 " ^ sec ^" 1  as  -  compared with a c o l l i s i o n - f r e q u e n c y of 1 0 / 1 2 s e c ~ l f o r H 1  2  at  several  hundred" atmospheres'," andv t h e r e f o r e , ' the" c o n t r i b u t i o n of these  mat-  (2)  r i x elements to t h e r e l a x a t i o n process- i s ~ n e g l i g i b l e . :  has  r  f u r t h e r argued t h a t o n l y two  one- associated' w i t h the: L; distinct.  Hardy  7  m  :  '  v  independent" c o r r e l a t i o n f u n c t i o n s ,  and" one; associated: w i t h the I>  Am  are  In other words", the c o r r e l a t i o n ' f u n c t i o n G ^ ( t ) may m  written G^ (t) » m  G^ (t)e Q  l m f t  5 . t  be  40 . A" theory- -which' -enabies'-the- r e l a x a t i o n ^ rate" to .'be ted" in- terms of these two" c o r r e l a t i o n t i m e s ' f o r ' the than one  calcula-  case when more  J" s t a t e i s populated' i s the-' r e c e n t Bioom-Oppenhelm theory'^  Using t h i s t h e o r y ah" attempt' -is' made t o e x p l a i n 7  7  results; for H  and  r  2  :  the  1  experimental  HD.  , .... . . -only' a" brief"ioutiine'which- gives"'some;-of•• the' e s s e n t i a l r e s u l t s ' w i l l be" presehted---here* and- the- reader' is" r e f e r r e d paper" b y Bloom' and  :  to  the  bppehheiirr^ *) • for- the" d i s cus s i o n .  To r e l a t e the'two" c o r r e l a t i o n , functions'; a s s o c i a t e d w i t h the- L  l  m  and  L  x  to the" dynamics"of the 'system' Bloom and  m  begin' b y w r i t i n g  Oppenheim  a master' equation r e l a t i n g the p r o b a b i l i t y  of  t r a n s i t i o n s w i t h the t r a n s i t i o n rates' f o r " i n d i v i d u a l t r a n s i t i o n s . From" t h i s " master equation' they show that" i t i s p o s s i b l e the c o r r e l a t i o n f u n c t i o n  CrjiofeJ "in"•••term's"- of the' c o n d i t i o n a l  p r o b a b i l i t y that" a : t r a n s i t i o n - h a s taken" p l a c e . i n tensor - p o l a r i z a t i o n molecules-which are in the' s t a t e J ;  to w r i t e  ^ Y l §-j  f / j ' f e ) ' o r d e r JL  :  o f  Q  0  i n i t i a l l y "in the;^-state';  CTQMQ  time t .  Next a  i s " defined" f o r those and  a t a time t ;  as  where (/^Y J*/// ~t /^?t)Af<>/ @)  ;  !  i s the' c o n d i t i o n a l " p r o b a b i l i t y  that  0  the molecule is" i n " the', rotational"- state:.cry M " a t time t g i v e n it i s i n the the  are  state J , 0  M  : Q  a t t=0.  that  •• Bloonr and" Oppenheim show that  time rate' of change of this""tensor" p o l a r i z a t i o n may r  be  written  as  = r  If the  B  (  t r a n s i t i o n rate  ' y n ^ M p  (<J~M~/  J h Jean  ,3.9.,  be w r i t t e n  as  41 where  a- C l e b s c h « G o r d o n (J~,  J')  ancbsotropic  :  contains-the-dependence' on-the  1  i n t e r a c t i o n .  Then  where  rates"  7  1  I t " shonld''be' noted-'-'that' ''ih" a s s u m i n g r:  !  (3.11")  the" c o l l i s i o n s " have  been  theory" i s  culation  A  mation  of  could  these  t r a n s i t i o n rates.  equally  7  well  have  7  the" t r a n s i t i o n r a t e s " c o u l d "  been  7  be  state  l a t i o n  J'M'  of  -  words,  — O t h e r  ties  such  7  as  7  :  t o be  v a l i d  equal  so a  assuming  (3.13.)  to a  7  f o r a" t r a n s i t i o n  constant  times  a s s u m i n g - c o m b i n a t i o n s " o f weak" a n d s t r o n g " 7  be  noted"that  J=5- r o t a t i o n a l s t a t e 7  system  at" room  the  from popu-  temperature  c o l l i s i o n s , a t "this the  point.  popula-  i s - 0 . 0 0 2 a n d a t 392°K i s 0.008,  should-' b e  adequately  approximated  by  system.  W i t h i n ' the" weak" c o l l i s i o n ; m o d e l , solve  that  7  that" the hydrogen two' l e v e l  approxi-  methods" o f a p p r o x i m a t i n g "the'-"transition p r o b a b i l i -  - I t should the-  c a l -  state.  7  of  the set.of  co-  weak  f o rthe  c o l l i s i o n by  t r a n s i -  Clebsch-Gordon  a l t h o u g h ' d i f f i c u l t , t o " c a l c u l a t e . , " c o u l d " be' i n t r o d u c e d "  tion  the  as  the" t r a n s i t i o n "r a t e  i s " simply  t h e ' f i n a l  1  7  notation  aPjH  that  t o " JM  s t r o n g  introduced  written  AfyyijT'M')in' other  7 7  that  assumed  enough' that" f i r s t ' order' p e r t u r b a t i o n "  the  as  ::  a r e " p r o p o r t i o n a l " t o the""- s q u a r e " o f - t h e  e f f i c i e n t s  7  of the  'f^-^ -)-inay-'-be-'"written  •  ,  :  Or,  form  'a""^ca^i''"'co;ef"fl"c±e"n t : i n : , t h e  b'f " R o s e i " * " ^ ' tion  c o e f f i c i e n t and  l i n e a r - d i f f e r e n t i a l  Bloom  equations  and  Oppenheim  f o r t h e two  level  42 system  a n dshow  w r i t t e n  t h a t  t h e  r  s p e c t r a l - d e n s i t y  may b e  a s (-3.14.)  where  --klBJjnJ*) i-BjJ^J,) +  -Ave* = and" w h e r e t i o n s  t h e C  a s  C -- d The-  (\-^  t i v e l y "  7  J~  1 and  i n i t i a l  :  t h ea n i s o t r o p i c  c o n d i -  Bj^^'^i^ '^'^ 1  ^  from'  £  *-Hd  1  B/Y)  . .,  (3  •  a n d C • ^ i n 'equations  16  (  3  17 )  r e s p e c t  (3.16.) a n d  b y  p o t e n t i a l  =  f  B^CJljJT)-  2' e v e r y w h e r e  7  0  T^ t h e  A  j  - f  i s g i v e n  a J(S+  V  ^  a r eobtained'  C  a n d • J^CT)  (J)=  r  +  AJLI_±BJL(J.J,)  a n d  r  /  W 4 « W )  _  T o - c a l c u l a t e of  " h a v e " b e e n - d e t e r m i n e d ' -fr-ora "the  S  b y permuting  (3.1-7.)  ^  J  +  1  ZZ  - £  ^ 7 > #  ''are' c a l c u l a t e d ' assuming  g i v e n ^  ( J )  J"' a  form  b y ......  '  ,3.18.)  where  .......  y«KL (R)P^>&;) ll)  (3  . .; 19  and  = l*Wf _ \Y^:)Y^MY t  where the  a n d ^(K) are  c e n t e r s  c o n s t a n t s . 1  only,  1  and  7  o f mass Since"  a n d " 1/^) 2 ,  '  A  7  functions-' o n l y  o f t h e two" m o l e c u l e s  V ^ ^ i n v o l v e s i n v o l v e s  7  t h e 'c o n t r i b u t i o n s  r o t a t i o n a l "  (3  o f t h e 's e p a r a t i o n a n dw h e r e o p e r a t o r s  r o t a t i o n a l ' o p e r a t o r s  t h e  20  )  R o f  £L<^ a r e  f o r molecule  f o r both  o f t h e two" p o t e n t i a l s  --  m o l e c u l e s  t o the" t r a n s i t i o n  43 rate  f o r m o l e c u l e ' 1: going-' f r o m  -  Q.Ur'J  = Q CJ,J') 0)  +•  3  $ 'ft'to "3"M  state  <?,  F A J  '-additive,  1  i.e.  , 3 . 2 1 . )  the  r  j')  of  = ^(^)[c(J'^;oo)]y>(^  - Q l ' M  . .,  rj  '  where the'-^-&(w)  (3 22  [C(W;  <£KV) = ^  oofLP^  V->\  '" )  '  (3.23.)  d e p e n d s o n the' amount o f e n e r g y ,  between'the' r o t a t i o n a l has  -  (jr.J )  Bloom- a n d Oppenheim' o b t a i n ' the" values;-' f o r  ana  a r e  and' t r a n s l a t i o n a l  "fa'to*'  exchanged  degrees o f freedom.  It  f o r "jn&s* = o by Bloom, Oppenheim e t a l ^  been- c a l c u l a t e d  showed t h a t ^ fyfar) i s " v e r y " s e n s i t i v e -  t r o p i c intermolecular potential  who  t o t h e dependence" o f t h e  on- the- s e p a r a t i o n  aniso-  o f the' m o l e c u l e s .  The d e v e l o p m e n t of" a t h e o r y " t o i n c l u d e " r o t a t i o n a l ' ' t r a n s i t i o n s r  :  difficulties, ent to  b u t an" a t t e m p t " has" b e e n made t o c l a s s i f y  rotational nuclear  the  t h e t r a n s i t i o n s ' (l',3 #—> 3", l ) " ' c a l i e d r  o r ( J J"'t~*J  culated.  J " ) t h e u/"s a r e  A t room t e m p e r a t u r e  transitions  (1, 2 «-»3,  z e r o and j (1)  0) a n d (1, 4<—¥ 3, value  Kranendonkcalculation  transitions  lieu by  or  must be i n c l u d e d .  transitions  that  including  terms  resonant and j  —  a  y b  e  cal-  — V< } f o r  from the induced  (io) —has  suggested  that  m i g h t be i n c l u d e d " i n ' a n approximate" manner, i n  6)  and' 6r  z  J^).( ^^ , jr^mj u/  l  exactly  respectively  for  i n t h e summation o v e r J '  (3.23) d e p e n d i n g " o n ' w h e t h e r the" t r a n s i t i o n  lower J s t a t e ,  m  transi-  2)- w h i c h a r e t e r m e d  these quadrupolar  Oppenheim  o f - a method" o f o b t a i n i n g  equation  contributions  s u c h a s 4.7 i m p l i e s  •  these  their  the" v a l u e o f Oc^.-ji^yHjiir^ctil  guasi'-resonant; c o l l i s i o n s . A van  estimate  -  2  the d i f f e r -  spin"relaxation. For  tions  t r a n s i t i o n s : i n " H ' and" t o  poses  where  11  tu^jtjj" £6 1  in  i s to a higher  £, --kC<+- e  and  where  s t a t e s  E ^ a n d E^ a r e'the-energy  o f  t h e m o l e c u l e The-  f u s i n g  Sji  e q u a t i o n s  ("3 . 2 2 v )  W i t h  v a l u e s  weak  J  t h e  a p p r o x i m a t i o n  i f  4  ^  /,  from  thev_A-£ /.ff  j  t h e J  r e s p e c t i v e l y ,  i n c l u d i n g ' ' b o t h  i&J^  a n d 1/T-^ may b e w r i t t e n  c o l l i s i o n  2  b e c a l c u l a t e d  and"" " ( 3 . 2 3 . ) '  o f  (3.24.)  d i f f e r e n c e s " between  7  a n d molecule" J  - t e r m s .  c a l c u l a t e d  1  C<T,0''')' smay  q u a s i - r e s o n a n t  the  '  e q u a t i o n r e s o n a n t  a n d  i  f o r t h e t w o level"  Cj H  (3.12.) a n d  2  a y be  m  o  system  w i t h  a s  (*)z JL(3.26.)  where  i n  ^ y - -  - out.  I t  should' b e noted  g e n e r a l  d i f f i c u l t t o  the. d e t a i l s  o f  6)  two  -  QlfojJ*)  J J  e q u a t i o n  (3 =1 2.) .  Of .these' f o u r  The parameters  d e t a i l e d  where  m a t r i x  'Q'^('J~ J'?) I  a  n  d  of' o n l y "four,  -  QitT*?f)  a  "•parameters-,  o n l y  a n d £? ^ ^ ' / ^ j a r e z  ,  ,  J  !  o f  f o r a  parameters  S e a t e d t w o a r e r e l a t e d  b y i n d e p e n -  t h r o u g h  (3  M ' ) P J < M ' - A W f t ' * _  _  .  t h e probabillty'''tha't'"-'the"'m'0'-lecule'"-'is'"-''i n ''"'the -' ;  s  =  :  i  s t a t e  a r e  o n  b a l a n c e  AiCjrt;  i s  , $±&;J*)  depend  no knowledge  t h e  system' i s ' expressed''-in- t e r m s  Qx( " >)j  w i t h  'y'r"')  ^(^j*  t h e s e f u n c t i o n s  b u t , ' even  1  a n d^ ( W j - j ' . j O j " ' )  l e v e l '  dent.  model ,  and^  (u/'^.^y)  c a l c u l a t e - s i n c e  t h e assumed  j (uq.ji)  ^ '  t h a t  JM«  - 27  )  ("3.11.) a n d ( 3 . 2 7 . )  Using Gordon  coefficients" i t  i s e a s y  5  ,  show  the -properties  of  the  using' equations  arbitrary  , and" n o t i n g -  Clebsch  that  .  Furthermore, \  to  and  (3.28.)  analogous "to'  •("3.22.)  (3.23.) f o r  and  that  (3.29. )  -  Since  t y yO  i n the dilute  from" t h e s l o p e time  l i m i t ,  ledge  of  of  QiCTjJ-')**of"these  t h e T-^ v e r s u s >^  o n l y  ^  there  one value"of  following  equations  parameters  c u r v e i n  OC/~  's  proportional  i s only  one  even  i s  densi-  determined  correlation with  no  undetermined  the"i n d e x — \ express  c a n be  the short  Therefore,  to the  know-  parameter  important.  SjrjJtJJ )^ 1  terms  of  parameters  where  where  be  (3.29.)  Qi(Ji,Ji)  ^  /S^  relationship may  the  g a s , one  (i-1 (w)  the  The these  o f  i . e .  providing"that  and  each  a  of  obtained  m a  (^J-I^I)  ( 3 , 28".) from  Y"  b  e  ' obtained  obtained  (3.12.)  from  from detailed  ^(j/^J^using  the  balance.  and written" in" terms  SJIJLC'JJ.JJI')  ofQ-(jj J t  t  fusing  To the  only  from  that  j?  of  0 >  ;  )  and*  density  -  o n ^  slope  ,. T ^  f o r the.  Then  of the parameter  S^^tu,<j')  Q (CJJ)  i s  A  o f T' • v e r s u s  using  from, t h e  c a n - b e ' p l o t t e d "as a  The-application'of-this  function of  R. theory  to  t o ' e x p l a i n the " r e s u l t s - o b t a i n e d ' f o r T £ a s a r f u n c t i o n  density  f o rH  7  7  will' be discussedin -  2  IV along  • •• •  F o r HD  a more  de-  omissions-in'the-assumed  I N . HD  the spin  where' D i s t h e d e u t e r o n ;  gyrbmagnetic ratio "  -  Hamiltonian  may  be w r i t t e n a s  s p i n a n g u l a r - momentum"'vector- a n d  o f the  same  form  interaction  a s i n " t h e ? c a s e ' of'H ^;--"and: i / T ^ " -  may  f£L  7  ^  J  that  proton  i n HD  for the-spin  » ^ °  ^  i s of  rotational  (3.33.,  d i p o l a r i n t e r a c t i o n ' i s more"' c o m p l i c a t e d  unlike spins  has- s h o w n  D  be w r i t t e n a s  =  The  "2T t h e  deuteron.  7  T h e v • s p i n " r o t a t i o n " ' inte-r-ac't±o-n"-if©r---the  cule-of  with  of  model.  3. 4-v; "' -• " R E L A X A T I O N  the  Chapter  d i s c u s s i o n o f some; o f t h e p o s s i b l e  physical  the  known  attempt  tailed  p  j  R-v • T h e - ' q u a n t i t y  i s proportional to  ^^(ur)  f o r any v a l u e  !  a r e Q^(J J-)  the- h i g h  -  (  dependence  ' i n t h e s e - egua tions  1  parameters  determined fact  recapitulate :"  -  for  7  than  f o r " a m o l e c u l e "of  the case  7  like  for a  spins.  of" u n l i k e " s p i n s t h e c o u p l e d  mole-  Abragamd) equations  for  the rate'of  magnetization  change of  the-proton" m a g n e t i z a t i o n 'and  r  may- b e  deuteron  written  .„ _d^,-^A<^>-^-^(<^'M - d < * > l / T ] _  T  .± ( ^-I )-4r (<C>4^). M  =  u r  where  and "where  _  r  the  t  <T  c a n be' w r i t t e n  r  ,3.3,.,  BB  .3 .,  < 3  5  as  ^  ratio--"  (3.37.)  indicates  ^/TI  that  ^  may  be  neglected  the  contribution- o f t h e - entire  ten  p e r cent  of  compared  dipolar"  the contribution  with"  interaction  to^//J  from  •//TjJ-T  w i l l  since  be less  t h e"spin  than  rotation  interaction.  (2) Hardy deuteron ton  and p r o t o n . H i s - ' experiments-'  resonance  frequency ment on  so  ism' and" t h e  r  that  tend  concern  •  - - "Q"~  (-T^  while-perf orming  resonance-.  Then within  the  a "T^" " t y p e "  (j>/T\  O  C  /  were  important,  T-j_ f o r t h e " p r o t o n i n the accuracy  7~< " -L - [Ti /X) Isn  +  of  "magnetization",  HD  mechan-  t o be  was' n o t o b s e r v e d of only  two  the measurements  and'can  be  /dipo/cay,  I Tl  /dip*  Larmor  dipolar  consists  IfL) 77  pro-  measure-  t o "short:'circuit"'"th±s"~d±polar - c o u p l i n g r  of the  a t the proton  'Interaction'sotherthan  enhancement " o f "the" deuteron  i f t h e term -  rnciuded- saturating  b y a p p l i c a t i o n ' o f"an" r v f * f i e l d  the deuteron"  interactions"  ted  — 'experimentally"investigated'"the-'"'coupling  expecby  Hardy.  terms written  (3.38.)  of  48  CALCULATION OF B-]  3.5.  (3,3''  ) FOR HP  For HD t h e r e i s no r e f l e c t i o n s y m m e t r y interaction  for the"additional  which"arises- from" the fact" t h a t t h e  c e n t e r o f the HD" molecule  geometric  does'not correspond" to the c e n t e r o f mass  and a d d i t i o n a l torques"areexerted"'on" "the' "molecules ' d u r i n g a c o l l i s i o n due t o t h i s ' displacement-of'the' c e n t e r of" mass. The  f o l l o w i n g d i a g r a m ' i l l u s t r a t e s t h e "geometry I n v o l v e d  i n d e r i v i n g the p o t e n t i a l f o r HD  where the c e n t e r of m a s s i s displaced'"from"'-the geometric ;  :  the two HD molecules  b y a distance b . U s i n g  term i n the expansion  of the p o t e n t i a l  c e n t e r s of  t h i s diagram"the l e a d i n g which depends on the  mutual o r i e n t a t i o n of the" two molecules', may be w r i t t e n i n terms of the v e c t o r R which j o i n s " the" c e n t e r s of" mass "and the s o l i d angles _TL^  and  which are"the o r i e n t a t i o n s of molecules  1 and 2  r e s p e c t i v e l y , as, (3.39.)  where  a-^ = a_^ =' 1  and a  Transforming  77  Q  = -2  to" t h e l a b o r a t o r y r e f e r e n c e frame  ^^may  be w r i t t e n *  _ /f%)r Vf,  -  °  1  A.  (-i) c(i^jY,J<QVJpJKJ^ M  '  (3.40.  49 where  R ) c o n t a i n s the" r a d i a l dependence on R and  Sl are  -Z2./ and  7  z  the o r i e n t a t i o n s of" molecules 1 a n d " 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 a space f i x e d " a x i s a l i g n e d along HQ. i n t e r a c t i o n i s ^ J - +- \  The' s e l e c t i o n "rule f o r t h i s  so - t h a t a l l ' c o l l i s i o n s which change mj  are c o l l i s i o n s w h i c h a l s o ' c h a n g e J . The e f f e c t of l ^ ^ i s  t o produce a t r a n s i t i o n o f molecule  1 from the s t a t e (J'M' -tro- the-•• start*--"J^M"-where | ^ j / - / j ' - J * / r j j " - j " / = 1 1  As f o r H  2  (equationlV'. 20 in~ Bloom and'-Oppen-helm's paper  the  )(j M jJ ^  t r a n s i t i o n rate" f o r th±s process^ i s ~ denoted by -  )  ?  where J  J  "  »  (3.41.)  and where  , *\Cf'r1 'lr ' .«l>-)\r 'M'")l* IM  ^tyuTjji,  juj'")  .  ,  (3.42.)  i s ' the' Fourier" t r a n s f o r m of It ^ •) (t) which i s g i v e n 3  by  then  A (jhl 'j J 'M ') • C2)  may be w r i t t e n  and' from the- d e f i n i t i o n o f ''QYqj'J( 3 .11.)  the Q y(jr J J may be 1  written  V  ? A  W  / ^ W * ' '  ,  J  " ' ' ^ ^ )  ( 3  '  4 5  '  )  50 -  A t t h i s p o i n t " " i t s h o u l d be n o t e d t h a t HD g a s c a n be - c o n r  sidered only  as" a  two" l e v e l  system, • a t s u f f i c i e n t l y  low"temperature,  t h e J - l , 2 s t a t e s ' c o n t r i b u t i n g ' t o " the" " t e n s o r " p o l a r i z a t i o n .  therefore,  the d e f i n i t i o n s of — a n d  £^obtained  for H  2  with And,  using  t h e ' weak" c o l l i s i o n " a p p r o x i m a t i o n " a r e - " v a l i d " for" HD. • B u t , t h e J=0 state  does  7  gjf  enter  ( J, J )  i n t o the''""^Y'(^'J'^'a'S"'"'can''''be"""S'e'en'"'f"rom from- (3.12.) a s ~J7I » u*j T~ i  -  i  •!  (3.46.)  where t h e summation o v e r ' 3""" i n c l u d e s ' t h e J=0 i s zero" s i n e e " t h e " C l e b s c h " G o r d o n i . e . ' . ^ J =-i In  the; c a s e J " / » J  With  of  1  Ty f r o m  _A-^.^  where  s t a t e a n d where  coefficient  (sff^^f zero.  1  xtfutO^J'^Jj* obtain  writing  :  and  cf^.  ^}  (<J,J'Jmay be  <//(ss:rJ';  these;. r e s u l t s U^ iw)may 0  M)  ' '>  (3  - Us~  D  u s i n g ' (3.15.),(3.16.)  '  T  l may be  and where and  (3.17.) .  47  be obtained""and" u s e d t o  (3. 28 .) , (3 .31.)and-(3r3"4".")"  Cj-^  written  E x p l i c i t l y i n - terms  written  and-  may  be c a l c u l a t e d  51  •"  • • C H A -P- T E- R  IV  EXPERIMENTAL" 'RESULT S' ANBrDTSGUS SI ON 7  4.I.  " THE' T-[ MINIMUM IN H  4 . 1 . 1 V A T  9  77°K.  Hardy ^  measured T in''H^''"^^--'"^'- ^ u s i n g "a''-3'0"-mHz, N.M.R. • 0  1  spectrometer over a p r e s s u r e range of 0.05 atmosphere t o 2.0 atmospheres and observed a T-^ minimum.  With the a i d of T^ measure-  ments" Hardy was' able' to" p l a c e l i m i t s on the r a t i o o f ' the c o r r e l a t i o n times' a s s o c i a t e d w i t h • t h e d i p o l e - d i p o l e " " a n d s p i n - r o t a t i o n 7  r  con-  t r i b u t i o n s r e s p e c t i v e l y of 0.6 i) ^  I'/ O .  v a l u e o f T^ a t the minimum 'denoted b y " ( T ^ ) m i n a n d  From the  the"dependence  of T y o n ' d e n s i t y near'the minimum,"'he- was- able t o show t h a t the T  s u g g e s t i o n of Needier and' O p e c h o w s k i ^  t h a t the molecular r o t a -  t i o n a l s p l i t t i n g must a l s o be taken "into"" account "wa's" c o r r e c t . Increased s e n s i t i v i t y i s expected a t 96'mHz-due t o the f a c t t h a t the s i g n a l to" n o i s e r a t i o i s expected" t o I n c r e a s e as the 7  square o f the" magnetic" f i e l d ' ( o r frequency) .-•--in addition", the con7  d i t i o n f o r : a T^ minimum" (W^fdfc.i) p r e d i c t s that* the" minimum' w i l l occur -  at a' h i g h e r density'."  The" motivation' f o r the" experiments r e p o r t e d  here was' t h a t more a c c u r a t e T-^ measurements might be o b t a i n e d , ..thus d e c r e a s i n g the l i m i t s on' JI-\-L/IA:JI  and" the T-^ minimum might a l s o be  s t u d i e d as a f u n c t i o n " of" temperature.  7  52 The  experimental " r e s u l t s - a t 77 °K u s i n g the 96 mHz 7  meter are' shown" i n f i g .  4 . I v along with" the* best" f i t  :  curve based on Hardy's work a t a" 30 mHz 7  spectro-  theoretical  f o r comparison.  At 96  mHz  the minimum occurs a t a d e n s i t y of 2.0 :amagats and has a v a l u e of 7  600 ^ s e c as compared" with": a" d e n s i t y o f 0.6" amagat and minimum v a l u e of  / &6 p s e c as r e p o r t e d b y Hardy a t 30" mHz;.' To" w i t h i n t h e 7  of the data'these values" check" w e l l with" t h e expected r  The  accuracy  ratio  accuracy* of the data i s " about the same as the accuracy  o b t a i n e d by Hardy a t 77°K" even' though the s i g n a l to n o i s e r a t i o expected  to i n c r e a s e as a f u n c t i o n o f " f r e q u e n c y .  was  T h i s i s mainly a  r e s u l t of the i n h e r e n t l y h i g h e r i n p u t n o i s e of the a m p l i f i e r a t 96 mHz  as compared w i t h 30 mHz  cussed i n Chapter mal  due  to induced g r i d n o i s e as  I I . " At 77°K> Hardy was  a b l e t o observe  n o i s e o r i g i n a t i n g i n the sample c o i l but t h i s was  a t 96 mHz.  the t h e r -  not p o s s i b l e  However, a t room-''tempera't^r'e>""'it'"waB''pos'si'b'Ie"''tO'  the thermal n o i s e a t 96 mHz  :  ,  ,  !  :  dis-  observe  a n d " i t i s at' t h i s " h i g h e r t e m p e r a t u r e  and above t h a t the use" of higher f r e q u e n c i e s e x h i b i t e d ' i t s s u p e r i o r ity .  (4.1.)  and from (3.11.) f o r ^ = 2 (4.2.)  where (4.3.) (4.4.)  54 where the  may be c a l c u l a t e d " using- the" r e s u l t s of Bloom e t a l  and  /^ r  j"(J tlL n  T n  where the a v e r a g e ' i s  - ry <r(>r+iy ; \  (4.5.)  taken over a l l r o t a t i o n a l s t a t e s of the mole-  c u l e 2 which c o l l i d e with t h e o r t h o - H  2  molecule 1.  ^Hx^  i  s  t  h  e  19 d e n s i t y of  in- " i d e a l ' amagats""' which' i s d e f i n e d as 2.69 X 10 -  molecules/cm . I t i s possible- to write 3  (4.1.) i n terms of a s i n g l e  parameter  and o b t a i n t h i s parameter from t h e * h i g h - d e n s i t y  slope of  plotted  as a f u n c t i o n of d e n s i t y . In' other- words, the' o n l y i n f o r m a t i o n about ^fy(0)  necessary i s t h a t ^ &(h)  In t h i s manner the l a t e d from minimum". /3j  (liO/B  Qj^C"A'/J""  i s ' a l i n e a r "function of d e n s i t y . i s defined- and  may be e a s i l y c a l c u -  (3.3.6) as a' function- o f d e n s i t y i n the region; of the T, But,-as" noted e a r l i e r , the- r a t i o : obtained  t  (hi)  from  cannot' be- v e r i f i e d from" t h i s experiment.  t h i s measurement: of the" T v minimum" a t a single'temperature  (4.2.) f o r Similarly, i s com-  p l e t e l y i n s e n s i t i v e to the- value' o f the* r a t i o of Therefore,  the' measurements of Tj_ a s a f u n c t i o n of d e n s i t y  p r o v i d e a c h e c k of the" absence of s y s t e m a t i c " e r r o r s i n the apparatus :  by t h e i r e x c e l l e n t comparison with' the r e s u l t s o f Hardy, but y i e l d no a d d i t i o n a l i n f o r m a t i o n b y themselves.  4.1.2. '  HIGHER  TEMPERATURES  F i g s . 4.2, 4.3 and 4.4 show p l o t s o f the" measurements of  M.o  s  h 2 . 0  c  - 1 . 0 - 0 . 8  - 0 . 6  - 0 . 4  0.4  0.6  0.8  1.0  2.0  4.0  i  __L  - 7-imagats FIG.  4.2,  ?2 v e r s u s  Density  at  196°K  in  H  2  6-iO  I  8.010.0  L  I  20.0  FIG.  4.4.  T  r  versus Density a t 392°K i n Ho  58 T  as a f u n c t i o n of - density- i n H  a t 196*K,298°K  7  1  tively.  2  and 3 92°K r e s p e c -  The populations- of -the J=3 s t a t e at' these' temperatures are 7  0.0321, 0.121 and 0.21y r e s p e c t i v e l y . A t the- f r a c t i o n a l p o p u l a t i o n s o f " the  :  3<=5  the h i g h e s t  state i s o n l y  temperature  0.008 and can be  safely neglected. To attempt t o -explain-: these" data>:-(3 . 26v;) . f o r Tj^ i n the T  /g\ two l e v e l " system as -derived "by- Blaom and "Oppenheim" 7  r  was f i r s t  used. To i n v e s t i g a t e the p o s s i b i l i t y o f v a r y i n g the r a t i o Q^^Z)/  BjtCjj')'*  C?x( / / / / i t  C$,0 ~%Q  3,  = -ly Q*M  (3,3)  as  (2.0  0, (iJ)  B, (><*> =  (4.7.)  Q.O.V  8, bii) fc)(%)k  0J3<>)  s  and  X  the data by  i s i n s t r u c t i v e to w r i t e the  Q^J^'H  i n terms of the  S, (1,0 -  fitting  B (h0-  - J f i U h O 0  x  (4.8.)  2S/  Using- d e t a i l e d balance  A{jMjJ'/i'J&*'  =  the r a t i o between Q (1>30 7  2  And from  A/J'#';7-/4)P*M and?:Q^i3,1) i s determined t o be  (3.29.) i t can be seen t h a t the r a t i o  . -QTctt)  (3.27.)  l?^^+^/& 0L^'^ L  given  (4.10.)  _i,  /  59  Then, d e f i n i n g the r a t i o s o t S Hi-  and /?=  the  C<J~, J ) S may be w r i t t e n i n terms of  equations f o r the  which may be determined 7  from the h i g h d e n s i t y r e s u l t s , and the  parameter R, which i s d i r e c t l y r e l a t e d t o the r a t i o J.^  8,0,0  -=  &)/^^(0)  B, 0,3)  ' ''  ^0.0(^)0^.)^  R  (4.i2.)  (2. 36.), 2.37.) and 2.38.) i t can be seen  jLfa^ can be w r i t t e n  terms of o n l y R.  <4 11  --$,0,1)1%  •••-•••From equations that  as  '<?i(hi)Lk+h«*-l  Q^hOthHi)** 3,(1,0  hi)  l  i n terms of Q (1,1) and R and  in  2  A l l o w i n g t h e r a t i o R t o v a r y corresponds  i n the  p h y s i c a l model to "•allowing 'an u n c e r t a i n t y i n t h e r e l a t i v e s t r e n g t h of the quadrupole-quadrupole " i n t e r a c t i o n as compared with the interaction. With"••the--aid" of the' d i g i t a l " computer, the r a t i o R was 77  allowed to vary from zero t o 100.0 i n increments increments  of .1 to 10 and  of 10 t h e r e a f t e r . T h e r e s u l t i s t h a t the" v a l u e of the  minimum i s changed "by "no • more;"than -12-% o v e r " t h i s range of R 7  which i s not s u f f i c i e n t 'to f i t 7  The  the data.  l i m i t i n g cases R=0 and R=100 are p l o t t e d i n f i g .  4.3 (14)  as a dashed l i n e and a: dash dot" line" r e s p e c t i v e l y .  Using  value of  to a value of  ;  •which corresponds  Lalita's  R=0.181 the s o l i d curves i n f i g s . 4.2, 4.3 and 4.4 are o b t a i n e d .  60 Intuitively, a  f i t of  for  i t can  the  so  possibility  :  minimum i s t o o b t a i n  a g i v e n J2,  shorter  be seen that' t h e - o n l y  that  a large part  c o r r e l a t i o n t i m e and,  two of  for  limiting  (4.11.) and For  c a s e s R=0  the  therefore,  'and-R > / l  (4.12.) t h a t  instance,  f o r the  the  case  i t can  -X^^'x  contribution w i l l  contribute  mainly  be' s e e n u s i n g  are  f o r the  have  at  a  For  the  equations  somewhat d i f f e r e n t .  R=0  M, -- - eM-^Q^ and  -A-^  greatly different  much l o w e r d e n s i t y than' a t t a i n e d . i n " t h e s e 'measurements. two  obtaining  - -  (4  13  }  c a s e R 77 1  (4.14.)  where t h e JLJ.) further  may  be  g r e a t l y d i f f e r e n t ' - f r o m the  c a l c u l a t i o n of  case. " C a l c u l a t i o n of  the  Cj  i s necessary  t  t h e C * ^ shows t h a t ;  _Aje,  ^  so  that  to evaluate  this  a  e v e n t h o u g h t h e _/LJ2<*.may  x  Jo  differ  g r e a t l y , the  c o n t r i b u t i o n to' t h e  m i n i m i z e d ' b y a" s m a l l  It  __/lje.^ tionship the  should  value  be  using  1  £Cj*^  good f i t o f  the  correlation  times d i f f e r  of magnitude.  out  that,  w i t h one  by  varying  the  '  arbitrary  Cj  i t i s possible  Such a f i t r e q u i r e s from the  other  two  that by  two  of  of  rela-  only a  to  very  the  more t h a n an  O t h e r t h e o r i e s w i t h more u n d e t e r m i n e d  the  value  subject to o b t a i n  is  of  a r b i t r a r y manner" w i t h o u t u s i n g "any  between" them and  data.  Cassociated  pointed  i n a completely  relationship  of  spin relaxation rate  four order  parameters,  a  such as the theory of Needier' and Opechowski' allow' such v a r i a t i o n of the v a r i o u s c o r r e l a t i o n " times and i t might be s a i d t h a t these t h e o r i e s f i t the data.  But the " i n t r o d u c t i o n o f b a s i c r e l a t i o n s h i p s 7  (e. g. d e t a i l e d balance) - between "some of 'the' c o r r e l a t i o n times does :  77  not' a l l o w such' a f i t ' of t h e  4.1:.3v-  " He - H  77  2  7  data.  MIXTURE  In an e f f o r t to' i n v e s t i g a t e the" r o l e of the t r a n s i t i o n s between J s t a t e s i n the r e l a x a t i o n p r o c e s s , T^ was measured as a f u n c t i o n of d e n s i t y a t 2 9 8 °K f o r a mixture of 5 4 . 5 % Helium i n Hydrogen and a minimum type" curve o b t a i n e d , f i g . 4 . 5 . The pared w i t h  c o l l i s i o n cross-section  2.93°A  He-ortho-H  2  for  <y f o r  He i s 2 . 5 7 ° A as com-  and' the form of the  interaction for a  7  c o l l i s i o n i s the same as f o r a p a r a - H  collision.But,  the  (J=0) - ortho-H2  2  i n t e r a c t i o n which i s mainly r e s p o n s i b l e f o r  inducing J t r a n s i t i o n s i s absent.  T h e r e f o r e , a v a l u e of the T j  minimum which approaches the same v a l u e as p r e d i c t e d f o r the two 7  7  1  l e v e l system w i t h no allowed t r a n s i t i o n s ' would be expected 7  i f the  J t r a n s i t i o n s are g r e a t l y - a f f e c t i n g the r e l a x a t i o n p r o c e s s . Qualitatively,  i t may be seen from f i g .  4 . 5 t h a t the v a l u e  of the T ^ minimum i s * indeed:, lower t h a n the value".obtained f o r pure 7  H  ; 2  a t 2 9 8 ° K , and approaches the v a l u e expected -  forH  2  assuming t h a t  there a r e no J t r a n s i t i o n s , which i s i n d i c a t e d on t h e graph. 7  attempt  t o f i t t h e data u s i n g the v a l u e of -J/.JP zOi  f o r pure H -r  i n the same  7  manner as f o r pure H 2  7 2  i s plotted- i n f i g .  4 . 5 . as a s o l i d  t h i s t h e o r e t i c a l curve does "not f i t " t h e  L i n e a r l y e x t r a p o l a t i n g these 5 4 . 5 % He i n H  An  line.  As  data. 2  r e s u l t s to  the l i m i t i n g case of 1 0 0 % He seems to i n d i c a t e t h a t the theory  FIGo  4.5.;  T ± versus  D e n s i t y - a t 298°K  in a  54.5%  He  i n H>  Mixture.  63 C/,3j  would f i t t h i s case f o r which  4.1.4.  THE Fig.  T-j MINIMUM "VERSUS 'TEMPERATURE 4.6  shows a ^ p l o t of the experimental v a l u e s  T^ minimum as a" f u n c t i o n of "temperature along w i t h the e r r o r s i n the v a l u e of the minimum.  of  the  estimated  The'solid" l i n e " i n d i c a t e s the  termperature dependence .of t h e T ^ minimum' p r e d i c t e d from the Bloom:  Oppenheim t h e o r y "as obtained" f rom-'the-solid" l i n e c u r v e s - i n  figs.  -  4.1,  4. 2, 4.3  and" 4.4 .  mental values: of t h e 7  7  It" can be r e a d i l y observed t h a t the r  :  T^" minimum -increase" as a "function of -  ture w h i l e the t h e o r e t i c a l v a l u e s ' o b t a i n e d theory  -  experi-  tempera-  from the Bloom-Oppenheim  decrease with temperature.  4 .1-. 5 .'  -:  STRONG COLLISION". LIMIT.  ••••••••• I t i s i n s t r u c t i v e ? to- examine the opposed to the weak c o l l i s i o n  strong c o l l i s i o n  l i m i t ' which' was " p r e v i o u s l y  i n the Chapter I I w i t h t h e ' i n t r o d u c t i o n o#  limit  considered  perturbation-theory.  In t h i s l i m i t the assumption i s " made t h a t the r a t e s are d i r e c t l y p r o p o r t i o n a l to the" p o p u l a t i o n  of the  transition final  state  Aterf'^ ^')  '*  1  for  JM  at J ' M'  y ^ j j j b  and  (4.15.) /  I  F i r s t c o n s i d e r i n g the case -  J ' equation 11.38  i n Bloom  (6)  and  Oppenheim's p a p e r  v  -' may  With the r e l a t i o n  be  P_ j t  written  ~—Lp^  the second summation i n  as  (3.32.) may be w r i t t e n  Then"the""on"l'y 'non-zerp'""contrxbu't±Gn"''to" T  the g e n e r a l i z e d  p o l a r i z a t i o n in' t h i s s t r o n g c o l l i s i o n l i m i t must' come from the case J = J ' - . F o r t h i s case equation" I T . 38' may be w r i t t e n  *  O h c o l l e c t i n g terms" t h i s  \ —j  (4.18.)  yields \\\  ,  ,  e  r  .  (4.19.)  T h i s equation i s e a s i l y s o l v e d t o g i v e :  ^4^  j  =™s*»p> ~** e  (4  - °-' 2  At t h i s point' i t becomes e v i d e n t t h a t t h i s s t r o n g c o l l i s i o n l i m i t w i l l " y i e l d ' a c o r r e l a t i o n f u n c t i o n which c o n s i s t s of a s i n g l e e x p o n e n t i a l ' . A t 273°K t h i s c o u l d h a v e ' o n l y a" 13% e f f e c t on the 7  v a l u e of (T^) min ascompared' w i t h the" s i n g l e l e v e l v a l u e of the T^ minimum', which' would tend to- reduce" the " v a l u e o f the minimum s i n c e both J states' would' c o n t r i b u t e "with" the same c o r r e l a t i o n  time.  :  The  f a c t t h a t n e i t h e r the weak" c o l l i s i o n a p p r o x i m a t i o n ;  nor t h i s  s t r o n g c o l l i s i o n approximation' can e x p l a i n the data does not r u l e out the" p o s s i b i l i t y that' some i n t e r m e d i a t e s t r e n g t h c o l l i s i o n x i m a t i o n , o r combination  appro-  of" weak" and" strong' c o l l i s i o n s which i s be-  yond the scope of t h i s t h e s i s to" i n v e s t i g a t e i n d e t a i l , might e x p l a i n the' d a t a .  4.1.6. -  HIGHER ORDER-INTERACTION From the e q u a t i o n f o r  POSSIBILITY  —/L^«< i t can be seen t h a t  i s of about the same o r d e r o f magnitude as QS^d,!) the p o s s i b i l i t y of o b t a i n i n g -greatly d i f f e r e n t duction  of a j^ (Jl/)i\\.\.e.x action T  states  there e x i s t s  -A-j^^'s , The i n t r o -  term which has o n l y m a t r i x elements -  m  between the m  ~ Bp  iff^(>i)/3^/J  f o r J=3 and between the"two s t a t e s o f d i f f e r -  ent J and no m a t r i x elements between m  s t a t e s f o r J - l was a p o s s i J  b i l i t y which might have f i t i n t o the p h y s i c a l model and which was calculated. The f i r s t assumption t r i e d e m p i r i c a l l y f o r t h i s a d d i t i o n a l A= 4  i n t e r a c t i o n was - to assumethat' t h i s i n t e r a c t i o n added o n l y a  correction  to the / j ^ C3, j) term, where the r a t i o  B^i^j)/B^CJ^T)  i s g i v e n by  B^'frf) _  sLyj(s-n) - MM)  -Q  or s p e c i f i c a l l y f o r ^ = 4  - C a l c u l a t i n g  T^ as a f u n c t i o n  d i f f e r e n t v a l u e s of 0^(3,3) the  i  n  of y f o r the a d d i t i o n of  jB^fajJ showed  u n i t s o f Qfe^to  that  v a l u e of the T^ minimum was not changed by more than 13% f o r  t h i s assumption which was a g a i n not s u f f i c i e n t t o f i t the d a t a . Since the assumption that Qy(jfj')-6 and r  p o s s i b i l i t y of' f i t t i n g t r i e d was t h a t  would not, i n g e n e r a l ,  d i d not l e a d to a  the data the next" q u a l i t a t i v e assumption  <jj^(j J')$0 t  (3,3)^-0  a n d ^ Q«.(3,3) = 0.  be t r u e ,  This  but a t t h i s p o i n t o n l y the p o s s i b i -  l i t y o f a q u a l i t a t i v e e f f e c t on the s p e c t r a l d e n s i t y ined.  C a l c u l a t i o n of ;Ty as a"• function'-of  of d i f f e r e n t v a l u e s of  '^(/^J  assumption  to  i s being exam-  density'for  B^C^T)  and  the a d d i t i o n in  u n i t s of Q (_l,3)  showed t h a t the v a l u e of" the  x  minimum was  not  changed by more than 13% for* any v a l u e of the" a d d i t i o n a l " term between 0 and  100  4.2.  Q (1,3). 2  HP RESULTS Fig.  4.6  s i t y between 0.8  i s a p l o t o f T y measurements a s a f u n c t i o n of den-  and  1  :  6 amagats f o r HD a t 298°K and  196°K.  As f o r the case of hydrogen , the e x t e n s i o n of the Bloom-  Oppenheim theory to i n c l u d e t r a n s i t i o n s between J s t a t e s was used to attempt  to f i t the r e s u l t s .  CTjjO'^  the  may  From" equations  first  (3.80) and  be w r i t t e n  where i t should be noted t h a t  Bjzj(j j)  i  t  r o t a t i o n and d i p o l e i n t e r a c t i o n . "  s  the same f o r "both the  E v a l u a t i o n of the  (p^(o~ J)  a f u n c t i o n of d e n s i t y were- obtained"by (3.12.) to o b t a i n the  (3.15.),  (3.16.) and  spin yields  t  With the a i d of a d i g i t a l computer the v a l u e s of T^  in  (3.81)  as  u s i n g these v a l u e s of  then- o b t a i n i n g the  and Cj  0  from  (3.17.) and e v e n t u a l l y u s i n g these v a l u e s of  and C-j^ i n equation  (3.44.) .  high d e n s i t y slope of T-j_ versus  The  term^  and,  r  (0)is  determined  by  t h e r e f o r e , there are no  the  parameters to be v a r i e d to o b t a i n a f i t of the d a t a . The  solid  l i n e s i n f i g . 4 . 6 p l o t s the v a l u e s of T^  ^  as c a l c u l a t e d i n ' t h i s manner"neglecting  J=3  s t a t e which a t 298°K"is about 9.0%,  versus  the p o p u l a t i o n of the  but o n l y 3.0%  a t 196°K  0  Even w i t h , o r b e c a u s e of t h i s poor assumption, an e x c e l l e n t f i t to ;  w i t h i n i 5% the data" i s obtained- u s i n g the" Bioom=Oppenhelm theory. To a c t u a l l y v e r i f y " that" t h i s - theory- i s c o r r e c t f o r HD,  thetheory  ;  would have to be extended" to a t h r e e l e v e l system.  In other words,  the r e s u l t i n g three l i n e a r " dependent d i f f e r e n t i a l equations  descri-  bing the time r a t e of change" of the" g e n e r a l i z e d p o l a r i z a t i o n would ahve to be s o l v e d f o r t h e a n d conditions.  The  Cj^  u s i n g the a p p r o p r i a t e  initial  low p o i n t on" t h e . g r a p h i n d i c a t e s the value of the  T^ minimum p r e d i c t e d i n c l u d i n g the J=3  s t a t e and assuming a s i n g l e  c o r r e l a t i o n time f o r a l l three r o t a t i o n a l  states.  In t h i s d i s c u s s i o n of HD" the assumption has been made t h a t the l/^ and(/jf^ terms, which are the dominant i n t e r a c t i o n s f o r 1  -  are n e g l i g i b l e i n comparison to  I/A f o r HD.  H, 2  - Arguments f o r t h i s  as sump t i o n ' a r e : 1.  A c a l c u l a t i o n of  &Jf*(j' J ') and  Bjt^J? ) 1  t  from the h i g h d e n s i t y slope of T^ versus d e n s i t y f o r HD  i/0) l^f  assuming t h a t  [/^^ are the dominant" t e r m s - p r e d i c t s the v a l u e s of :  and  ^-^Ify  ^^^^fo)  which are a f a c t o r of ten g r e a t e r than the v a l u e s obtained i n the 7  same manner" f o r  H^. /1  2  A comparison of the measurements by L a l i t a 7  H"* w i t h the measurements of 2  v  "7/,  o \  '  o f i n  i n HD as performed by-Hardy ^  shows  t h a t " ^ decreases more than ten times" f a s t e r with i n c r e a s i n g temperat u r e a t h i g h temperature i n HD,  than in" H . 2  This implies that a  d i f f e r e n t i n t e r a c t i o n , with a d i f f e r e n t " temperature dependence i s  \  70  dominant' i n ' HD: than i n H j . These' arguments" do not' r u l e "out' the' p o s s i b i l i t y and t ^ ^ d o p l a y a r o l e v i n " HD, action,  4 .3.  y  l/fif*^"  i•••  t h a t l/^f ^ 1  but only argue t h a t a d i f f e r e n t i n t e r -  , i s " dominant.  MEASUREMENTS: IN CH^ was measured i n methane a t 2S8"°K and 196°K over a  density  range of 0.1 to 7.0 amagats a t 96 mHz"and a d e v i a t i o n  linearity  which appears t o be the approach t o the T^ minimum was  observed. -  F  from  o  F i g . 4.8  r  shows these d a t a .  CH4 the d i p o l a r i n t e r a c t i o n c o n t r i b u t e s  l e s s than  3% to the r e l a x a t i o n r a t e and may be s a f e l y n e g l e c t e d i n comparison w i t h the s p i n ^ - r o t a t i o n a l i n t e r a c t i o n . r e l a x a t i o n r a t e may  c  A  ~  are c o u p l i n g u  spin-rotational  be written" f o r a s p h e r i c a l top m o l e c u l e ( ) * 1 4  .  where  The  CC^  C) d  ^  ,  s  -  C  ,  C  „  .  ( ^  (4.24.)  ~  (4.25.)  constants,  1 (t>v)  and where  —  ^-^1  _  (4.26.)  and  Bloom  ( p r i v a t e communication)  i n t h e : e x p r e s s i o n f o r 1/T^ the  same term in; 1/T^ The ^  has ""shown the second term  i n a d i l u t e gas to be d i f f e r e n t from  i n a l i q u i d by a f a c t o r o f  2/5.  ^ denote an average over an e q u i l i b r i u m  ensemble  andmay" be approximated by  •  '<ctfj>0>  =  " ' - ^ l -  6  (4.28.)  l5  72  i n the h i g h J l i m i t , where I The q u a n t i t y , tj  cule.  -  i n t e r a c t i o n s and" ^  Q  i s the moment of i n e r t i a of the mole-  is-the* correlation" time"for spin-rotation  i s ' a d e q u a t e l y approximated by  ^ where  ^  (4.30.)  i s the c o r r e l a t i o n - time f o r  the s p h e r i c a l  Y^—^j\Jjj  har-  monies o f o r d e r 2 a s s o c i a t e d ' with" the" o r i e n t a t i o n j v ° f a v e c t o r -  f i x e d to  the molecule. -  The inequalityy".-••(Bloom' and Oppehhelm (^M ) ;  - •may  -  r  X,  - J f i  (4.31.)  be used to determine the l i m i t s on the r a t i o s J*^* -  i s no d e t a i l e d t h e o r y of molecular r e o r i e n t a t i o n f o r -  The c o u p l i n g constants  and  andCj  CH^.-  can,be determined,  p r i n c i p l e , " u s i n g the molecular beam technique. q u i t e d i f f i c u l t to measure C ^  s i n c e there  In p r a c t i c e , i t i s  , although the measurement  of the average c o u p l i n g constant i s s t r a i g h t f o r w a r d .  Anderson and  Rqmsey (16) have e v a l u a t e d these" c o u p l i n g c o n s t a n t s f o r C H 4 and 7  I.  7  C,t - 11, </ ± / C /<t/ ec S  j  in  C  Z -3, & + 3, X  ±  find  kc/sec  However, they are n o t a b i e to r i g o r o u s l y exclude the a l t e r n a t i v e :  -  -  assignments. II.  C„  - ~/7**/±  1>Lkc/*ec  j  C±  r  +  2,*kc/sec  A f i t of the experimental data f o r T y i n d i c a t e s t h a t the a l t e r n a t i v e assignments f o r Cjj and Cj_ may  be r i g o r o u s l y excluded.  s o l i d l i n e and the dashed l i n e i n f i g . T y data a t 298°K and  4.8.  The  which seem to f i t the -  196°K were obtained by assuming t h e v a l u e s of  73  where the two l i n e s i n d i c a t e the uncertainty" i n Tv a s s o c i a t e d w i t h the u n c e r t a i n t y in" these v a l u e s of C ^ and Cj,- . in f i g .  The lowest  4.8. i s a p l o t of the T-^ v a l u e s obtained using, the ;  curve  alterna-  t i v e v a l u e s of C^j and Cj_ assuming" the same value f o r the r a t i o ^ j ^ ; ^ as above.  I t should be noted" that' the v a l u e s of T^ are very  i n s e n s i t i v e to the v a l u e o f the r a t i o o f ^ :  i n the range 2 ^  ^ 4.  From t h i s d i s c u s s i o n - i t " may be" r e a d i l y seen t h a t the measurement of T^ as a f u n c t i o n of d e n s i t y i n the r e g i o n of the T^ :  minimum may be used t o e l i m i n a t e any ambiguity and  and furthermore,  v a l u e s of C  ted by Anderson and Ramsey  a  i n the v a l u e s of C//  of the same a c c u r a c y as r e p o r -  may be o b t a i n e d .  Recently i t h a s b e e n reported(17) 7  t h a t the ambiguity  i n the  c o u p l i n g c o n s t a n t s has been e l i m i n a t e d showing t h a t the' f i r s t a s s i g n ment was indeed  c o r r e c t u s i n g molecular beam techniques and a more  a c c u r a t e v a l u e o f C ~obtained. I t should be noted i n p a s s i n g t h a t the" h i g h d e n s i t y s l o p e s of T^ v e r s u s d e n s i t y are i n good agreement w i t h the v a l u e :  by B r i d g e s ( ) . 1 8  obtained  74  C H' A P T E R  V  ; CONCLUSION  The  project" discussed i n t h i s t h e s i s involved  a study, d e s i g n and /  firstly  c o n s t r u c t i o n of a high' f r e q u e n c y , s e n s i t i v e , p u l s e  \  spectrometer. frequency  was  The  i n c r e a s e i n the s e n s i t i v i t y with an i n c r e a s e i n  not r e a l i z e d due  to such e f f e c t s as" induced  more inhomogeneous r . f . magnetic f i e l d , critical  grid  noise,  s m a l l e r samples and more  matching problems i n h e r e n t at the higher f r e q u e n c i e s .  with the 96 mHz  But,  spectrometer, the measurements of t h e - T - L minimum  c o u l d be performed at" higher-temperatures s i n c e the" Tj_ minimum s h i f t e d to higher d e n s i t i e s ' w i t h the" higher frequency  and  was  since at  the higher' temperatures the temperature' dependent' "white n o i s e " of 7  the tuned c i r c u i t became of the same order as the induced  grid  noise. Ty was  measured' i n H  2  a t 77°K, 196°K, 298°K and  392°K  and  t h e Bloom-Oppenheim theory was"used i n ' an u n s u c c e s s f u l attempt to :  f i t the data a t the higher" temperatures where the p o p u l a t i o n of J=3  s t a t e i s not n e g l i g i b l e . '  T h i s attempt to f i t the data' made use  of the' most g e n e r a l form of' the theory 1  i n which'no knowledge of  r a d i a l i n t e g r a l s i s assumed with t h e - r e s u l t t h a t f o r no v a l u e s the r a t i o The  conld  the  a' f i t ; of the T"^ minimum' be  l i m i t i n g case of strong i n t e r a c t i o n s was  the r e s u l t t h a t no f i t c o u l d be obtained  of  obtained.  also discussed  in this limit.  the  The  with possi-  75  bility  t h a t a h i g h e r order- i n t e r a c t i o n ' of the form' Vy^ (had  been  -  n e g l e c t e d was i n v e s t i g a t e d " assuming' a wide range of r e l a t i v e s t r e n g t h s 7  of t h i s i n t e r a c t i o n as compared w i t h the 7  Y-x^C-^^) i n t e r a c t i o n s w i t h  the r e s u l t " that" there' was" no p o s s i b i l i t y of' o b t a i n i n g a f i t of the -  ;  Ty minimum a t the" higher" temperatures -".  by i n c l u d i n g - t h i s ' - i n t e r a c t i o n . 7  was a l s o measured i n t h e region: of t h e r T ^ •minimum f o r -  a mixture'of  7  54.5%: He::in H  with' the" r e s u l t ' t h a t the.value of T  : 2  1  minimum approached the'value expected i f no A J t r a n s i t i o n s were 7  allowed, but s t i l l " c o u l d not' be" f i t " using" the most g e n e r a l form of (6)  the Bloom-Oppenheim. A p l o t of the" experimental' (T^') min was presented along w i t h the v a l u e s o b t a i n e d from the Bloom -Oppenheim theory w i t h the i  r e s u l t ' t h a t the"temperature of  dependence of the experimental  (T-j_) min i n c r e a s e with' temperature  whereas the t h e o r y p r e d i c t s  -  a decrease w i t h  values  temperature.  To summarize the'H"2" r e s u l t s , " i t " i s obvious t h a t the Tj_ measurements i n the region; o f the T-^ minimum cannot 7  .... ..  be f i t by the  /gX  theory of Bloom and Oppenheim • e v e n ' i n ' the most"general  form,  assuming e i t h e r a weak i n t e r a c t i o n l i m i t o r a s t r o n g i n t e r a c t i o n l i m i t o r with the i n c l u s i o n of a" h i g h e r o r d e r ' " i n t e r a c t i o n . ;  This  f a i l u r e seems to imply" that" the- e f f e c t o f the" t r a n s i t i o n s between 7  J s t a t e s has not been'taken*into" account c o r r e c t l y f o r H -  T  mum o b t a i n e d .  2 >  was measured in- HD at" 196°K and 298°K and a T^ m i n i 7  Even though the two" l e v e l " t h e o r y was probably not 77  ;  v a l i d , t h i s two l e v e l theory was used in' an attempt  to f i t the  theory w i t h the r e s u l t t h a t the" d i s c r e p a n c y between the t h e o r e t i 7  7  c a l v a l u e s and t h e e x p e r i m e n t a l values" i s not as g r e a t as f o r H . 7  2  But, to p r o p e r l y check' the' theory f o r HD the theory should be 7  extended  7  t o i n c l u d e the' three" l e v e l system". 7  7  Since" t h i s has not  been done, the r e s u l t s are" r a t h e r 1  was' measured i n CH^ of the.  inconclusive. a t 196°K and  298 °K i n the  region  minimum and" i t was" shown t h a t such a measurement y i e l d s  enough' information" about'the" r o t a t i o n a l " coupling' constants Cj_  and  Cjj  con-  to eliminate" an ambiguity which" e x i s t s "when these c o u p l i n g -  s t a n t s are measured' b y molecular- beanr t e c h n i q u e s . This" experiment" has "shown" that" a measurement of T.^ as a f u n c t i o n " of" density" i n " the-regionL of" the" T y minimum f o r a d i l u t e d i a t o m i c gas  i s most: u s e f u l ' i n t e s t i n g ' t h e o r i e s "of s p i n - r e l a x a t i o n  i n d i a t o m i c gases.  Andy for" gases "composed of" polyatomic molecules  a measurement of  i n the' region:: of "".the"  minimum"-: has "been shown  to be' useful" in' evaluating'the""rotational'" coupling'  constants.  77 BIBLIOGRAPHY  I.  A. Abragam  •'The- P r i n c i p l e s of Nuclear Magnetism", Oxford' U n i v e r s i t y Press, 1961.  2.  W.N.  Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1964, Can. J\ Phys., 44, 265, (1966).  3.  M. Bloom and I . Oppenheim  Hardy  Can. J . Phys., 39, 845, (1961).  4. • M. Bloom and I. Oppenheim  Can. J . Phys-., 41, 1580, (1963).  5.  M. Bloom I. Oppenheim M. L i p s i c a s C.G. Wade and C.F. Y a r n e l l  J . Chem. Phys., 43, 1036 (1965).  6.  M. 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O z i e r Ashok Khosia and N.F. Ramsey  B u l l .  Amer. Phys. Soc.  18. F. Bridges  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  5 0 9 ,  Spring  (1967).  Columbia.  

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