UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Deuteron magnetic resonance properties in deuterated modifications of methane De Wit, Gerald Aloysius 1962

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1963_A6_7_D3 D3.pdf [ 2.42MB ]
Metadata
JSON: 831-1.0085822.json
JSON-LD: 831-1.0085822-ld.json
RDF/XML (Pretty): 831-1.0085822-rdf.xml
RDF/JSON: 831-1.0085822-rdf.json
Turtle: 831-1.0085822-turtle.txt
N-Triples: 831-1.0085822-rdf-ntriples.txt
Original Record: 831-1.0085822-source.json
Full Text
831-1.0085822-fulltext.txt
Citation
831-1.0085822.ris

Full Text

DEUTERON MAGNETIC RESONANCE PROPERTIES IN DEUTERATED MODIFICATIONS OF  METHANE by  B.Sc.  GERHARDUS ALOYSIUS de WIT The U n i v e r s i t y o f B r i t i s h Columbia, 1961  A T H E S I S SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming r e q u i r e d standard  THE  t o the  UNIVERSITY OF BRITISH COLUMBIA May, 1962  In presenting  this thesis in p a r t i a l fulfilment of  the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall, make i t freely available for reference  and study.  I further agree that per-  mission for extensive copying of this thesis for scholarly purposes may  be granted by the Head of my Department or by  his representatives.  It i s understood that copying, or publi-  cation of this thesis for f i n a n c i a l gain shall not be allowed without my written permission.  Department of  Ph  ^  c  The University of B r i t i s h Columbia,. Vancouver 8, Canada. Date  M'a^j,  3  ^  /<&6a  - i i -  ABSTRACT  Relaxation  properties  CD4 and CD3H were s t u d i e d 105°K - 57*K.  of the deuteron s p i n system i n  i n the temperature range from  These r e s u l t s show that the  intra-molecular  quadrupole i n t e r a c t i o n s dominate and the r e l a x a t i o n o c c u r s through the m o l e c u l a r r e o r i e n t a t i o n s  o f the molecule.  The deuteron s p i n - l a t t i c e r e l a x a t i o n times are approximately temperature independent, except f o r a contribution melting point  small  from the magnetic d i p o l a r i n t e r a c t i o n s near the i n CD3H.  From t h i s data i t i s concluded t h a t  the r e o r i e n t a t l o n a l c o r r e l a t i o n time i s temperature independent. The deuteron T^ shows the same temperature dependence as the proton T j 2  the  the deuteron T% can be accounted f o r on  b a s i s o f magnetic d i p o l a r i n t e r a c t i o n s a l o n e .  ACKNOWLEDGEMENT  My s i n c e r e s t a p p r e c i a t i o n i s due Dr. M. Bloom f o r h i s constant  interest  and advice  throughout the course o f  t h i s work. I a l s o wish t o express my a p p r e c i a t i o n t o Dr. D, L. Williams  f o r the c o n t r i b u t i o n o f h i s time and equipment  i n the l a s t phase o f the experiments,, T h i s r e s e a r c h has been supported f i n a n c i a l l y by the N a t i o n a l Research C o u n c i l through r e s e a r c h grants Bloom and the award o f a N a t i o n a l Research C o u n c i l  t o Dr. M. Studentship.  - ill  -  TABLE OF CONTENTS Page ABSTRACT  i i  LIST OF ILLUSTRATIONS  iv  ACKNOWLEDGMENT  . v  CHAPTER I  INTRODUCTION  1  CHAPTER 2  EXPERIMENTAL APPARATUS AND TECHNIQUES  5  CHAPTER 3  THEORY  14  CHAPTER 4  RESULTS AND DISCUSSION  27  APPENDIX  THE PROTON RELAXATION DATA AT 30 MCS.  38  REFERENCES  46  - iv LIST OF ILLUSTRATIONS Page F i g u r e 1.  Diagram o f the dewar and sample  F i g u r e 2.  Schematic diagram o f the t r a n s m i t t e r  F i g u r e 3.  Modified receiver c i r c u i t  Figure 4.  Deuteron T^ o f CD4 versus temperature  28  F i g u r e 5.  Deuteron T j o f CDgH v e r s u s temperature  29  F i g u r e 6.  Deuteron T^ i n CD^ versus the r e c i p r o c a l  F i g u r e 7.  arrangement  diagram  6 9 11  of temperature  33  Proton T^ i n CD^H v e r s u s temperature  35  CHAPTER 1 INTRODUCTION  Nuclear magnetic resonance (N.M.R.) p r o v i d e s on the p r o p e r t i e s o f matter. f r e q u e n c i e s g i v e s a measure of  information  Measurement o f N.M.R. the time-averaged  local  magnetic f i e l d s a t the n u c l e a r s i t e s , while o b s e r v a t i o n s on the approach t o thermal e q u i l i b r i u m o f the n u c l e a r s p i n system ( s p i n r e l a x a t i o n ) g i v e i n f o r m a t i o n on the f l u c t u a t i o n s o f the local fields.  T h i s t h e s i s i s o n l y concerned with s p i n  r e l a x a t i o n , as a p p l i e d t o l i q u i d and s o l i d methane. Let  M. k be the macroscopic magnetization  vector f o r a  s p i n system i n e q u i l i b r i u m with i t s surroundings at a temperature T i n an e x t e r n a l f i e l d H„k.  (the l a t t i c e )  If, initially,  the s p i n system i s not i n e q u i l i b r i u m with the l a t t i c e , and has a magnetization  MIPI =  M ^ L ->- M U>) k  i t will  4  approach e q u i l i b r i u m a c c o r d i n g to the f o l l o w i n g e q u a t i o n , |M») =  where  cx>„ = y « H  SM^Co) [ cos u.t I  -t- s^n tOo-fc j j  i s the Larmor frequency  o f the n u c l e i  having  gyromagnetic r a t i o y , F ( t ) and G ( t ) are the t r a n s v e r s e and longitudinal relaxation functions respectively. F(0) - G(0) » 1 and F ( t ) , G ( t )  0 as t -*• <*> .  2 O f t e n , the approach t o e q u i l i b r i u m those c a s e s , the r e l a x a t i o n  functions  s p e c i f i e d by t h e i r time c o n s t a n t s . spin" relaxation  i s exponential.  are completely  The t r a n s v e r s e o r " s p i n -  time i s u s u a l l y denoted by T  l o n g i t u d i n a l or " s p i n - l a t t i c e " r e l a x a t i o n and  T  2  have been p r e v i o u s l y  (see Appendix) i n CH , CH3D, C H D 4  2  2  In  2  and the  time by T^.  measured f o r protons  and CHD3 between 110°K and  5 5 ° K i n an attempt t o understand the b a s i c  relaxation  mechanisms i n methane through the dependence o f T j and T the  isotopes  o f hydrogen.  There i s a s t r o n g  isotope  2  on  effect  because the deuteron has a magnetic moment about 1/7 of the proton magnetic moment, thereby producing weaker  local  f i e l d s and hence being f a r l e s s e f f e c t i v e than protons i n c a u s i n g s p i n r e l a x a t i o n o f n e i g h b o r i n g protons. The  three b a s i c c o n t r i b u t i o n s  (1) i n t r a - m o l e c u l a r dipolar  t o the l o c a l  dipolar interactions,  i n t e r a c t i o n s , (3) the i n f l u e n c e  fields,  (2) i n t e r - m o l e c u l a r  of the r o t a t i o n a l  magnetic moments of the molecules, have d i f f e r e n t dependences on n, the number of deuterons i n the C H _ D 4  n  n  molecule.  The  proton s p i n r e l a x a t i o n study was unable t o p r o v i d e a unique s o l u t i o n t o the r e l a t i v e c o n t r i b u t i o n s  of ( 1 ) , (2) and (3) t o  the r e l a x a t i o n r a t e 1/T^, though i t d i d g i v e upper and lower l i m i t s f o r each o f them.  The s i m p l e s t  possible  interpretation,  namely t h a t mechanism (2) predominates over the whole temperature range, i s i n c o n f l i c t w i t h the s i m p l e s t  theories  which p r e d i c t t h a t mechanism (1) should be more important  than  - 3 (2) i n the l i q u i d  (90°K - 110°K) and the s o l i d below about  65°K. The r e s u l t s r e p o r t e d i n t h i s t h e s i s a r e p a r t o f a program designed t o f u r t h e r c l a r i f y the r e l a x a t i o n i n methane. in CD  4  T j and T 2 have been measured f o r the deuterons  and CHD  3  between 105°K and 57°K.  deuteron magnetic 4.3  problem  Because o f the s m a l l e r  moment t h i s study has been c a r r i e d out a t  megacycles per second, as compared with 30mc/sec.'for the  p r e v i o u s work on p r o t o n s .  To r e l a t e these measurements more  c l e a r l y t o the proton T^  measurements, the proton T^ i n CHD  was  3  a l s o s t u d i e d a t 4.3 roc/sec. The T^ and T.^ measurements were performed  using standard  p u l s e techniques which a r e d e s c r i b e d b r i e f l y i n Chapter 2, along w i t h other experimental a s p e c t s of t h i s work. A new r e l a x a t i o n mechanism i s Introduced f o r deuterons s i n c e the deuteron s p i n i s 1 (the proton s p i n i s 1/2). deuterons have quadrupole e f f e c t s produced  moments so t h a t , i n a d d i t i o n t o  by f l u c t u a t i n g magnetic  c o n t r i b u t i o n from  The  f i e l d s , there i s a  (4) the f l u c t u a t i n g e l e c t r i c f i e l d g r a d i e n t  a s s o c i a t e d with the r o t a t i o n o f the molecules.  The r e l a t i o n -  s h i p s between the proton and deuteron T j and  as a f u n c t i o n  of n a r e d i s c u s s e d under Theory  i n Chapter 3.  mechanism (4) i s c l o s e l y analogous both mechanisms a r e governed  I t i s seen  that  t o mechanism (1) i n s o f a r as  by molecular r o t a t i o n .  The experimental r e s u l t s a r e presented i n Chapter 4„ I t i s demonstrated  c l e a r l y t h a t as f a r as the deuteron T, i s  - 4 concerned, mechanism (4) i s dominant over the e n t i r e range s t u d i e d f o r CD^ and CHDg.  temperature  In p r i n c i p l e , these measure-  ments should l e a d to an a c c u r a t e c a l c u l a t i o n o f the c o n t r i b u t i o n of mechanism (1) t o 1/T^ f o r protons.  However, the quadrupole  c o u p l i n g c o n s t a n t , i . e . the magnitude o f the e l e c t r i c g r a d i e n t a t the deuteron, must be known f i r s t . are suggested t o o b t a i n t h i s On the other hand i s governed  field  Experiments  parameter.  f o r the deuterons i n s o l i d  by (2) as i s the case f o r the p r o t o n s .  methane  The s e l f -  c o n s i s t e n c y between these measurements and the proton T measurements i s a l s o d i s c u s s e d i n Chapter 4. The p r e v i o u s l y r e p o r t e d measurements on T^ and T^ f o r protons i n CH4_ D Q  this laboratory. Appendix  n  have been performed by Bloom and Sandhu of R e p r i n t s of t h e i r papers a r e g i v e n i n the  f o r ready r e f e r e n c e .  - 5 CHAPTER 2 EXPERIMENTAL APPARATUS AND  TECHNIQUES  TEMPERATURE CONTROL: The temperature c o n t r o l s e c t i o n of the apparatus  was  designed t o cover a range o f temperatures above the b o i l i n g p o i n t of the c o o l a n t used.  E i t h e r l i q u i d oxygen or  n i t r o g e n was used i n a l l the experiments. coverage from 77°K - 105°K was  obtained.  A continuous Moreover,  pumping on the l i q u i d n i t r o g e n , with a Hyvac no.7, was  liquid  by the range  extended down to 57°K. F i g u r e (1) shows s c h e m a t i c a l l y the dewars and the sample  arrangement.  There are t h r e e dewars, two g l a s s dewars and a  metal dewar.  The procedure c o n s i s t e d e s s e n t i a l l y of c o o l i n g  the  whole system by f i l l i n g  the outer dewar w i t h c o o l a n t ,  then e v a c u a t i n g the i n n e r g l a s s dewar to 0.1 but l e a v i n g a few cm.  micron of p r e s s u r e ,  of p r e s s u r e i n s i d e the metal dewar.  h e a t i n g r a t e o f the sample was  The  governed by the net heat input  by c o n d u c t i o n along the rods i n t o the dewar, the heat s u p p l i e d by the heater above the sample, was  and by the heat output, which  mainly caused by r a d i a t i o n to the c o o l a n t .  To o b t a i n  temperatures both the o u t e r and inner dewars were f i l l e d  lower with  l i q u i d n i t r o g e n , then the p r e s s u r e i n the i n n e r g l a s s dewar  was  reduced by pumping.  by  The temperature was  roughly c o n t r o l l e d  - 6 -  F i g u r e 1.  Diagram of the dewar and sample arrangement.  m o n i t o r i n g the vapor p r e s s u r e i n s i d e the inner dewar u s i n g a mercury manometer.  The temperature was measured by means o f  a platinum r e s i s t a n c e thermometer p r e v i o u s l y c a l i b r a t e d u s i n g the oxygen and n i t r o g e n b o i l i n g p o i n t s and the i c e p o i n t . The r e s i s t a n c e was measured w i t h a Rubicon M u e l l e r B r i d g e . The a c c u r a c y of the temperature measurements was 0.2°K. the p u r i t y of the l i q u i d n i t r o g e n was  Since  not known, the vapour  p r e s s u r e r e a d i n g s were not used to measure the temperature, but they d i d agree roughly with the p u b l i s h e d vapour p r e s s u r e 2 v e r s u s temperature curves f o r n i t r o g e n . Because of the poor thermal c o n d u c t i v i t y of g l a s s , p r e c a u t i o n s were taken to a v o i d thermal g r a d i e n t s , and t o ensure c o r r e c t temperature measurements. p r e s s u r e i n s i d e the metal dewar was  For these reasons, the kept at a few c e n t i m e t e r s ,  and copper s p r i n g f i n g e r s were used t o p r o v i d e as l a r g e an area o f c o n t a c t with the g l a s s as p o s s i b l e .  The f a c t that the  m e l t i n g p o i n t t r a n s i t i o n temperature, as noted from the experimental r e s u l t s , agrees t o w i t h i n a t l e a s t  .5°K w i t h that  given i n the l i t e r a t u r e ^ , shows that the temperature was  s u f f i c i e n t l y accurate f o r t h i s  control  experiment.  SAMPLE PREPARATION: The samples were prepared a c c o r d i n g to the procedure 4  developed by H. Sandhu, J . Lees, and M. Bloom .  The problem i s  to c o m p l e t e l y remove any paramagnetic i m p u r i t i e s , which are  - 8 normally present i n commercially  prepared methane samples.  The  by a g e t t e r i n g techniquej a  purification  i s accomplished  d i f f u s e l a y e r of misch metal at  i s d e p o s i t e d on a s p h e r i c a l  the top of the sample tube ( f i g u r e 1 ) .  The  bulb  procedure  is  as f o l l o w s , a tungsten c o i l c o n t a i n i n g some p i e c e s of misch metal  i s s e a l e d i n t o the g l a s s bulb.  evacuated  to 1 micron  of p r e s s u r e .  The whole assembly i s In order to o b t a i n a  d i f f u s e l a y e r of the d e p o s i t , the sample tube i s f i l l e d Argon up to about 2mm.  of p r e s s u r e .  The  with  reason f o r t h i s s t e p  i s that a d i f f u s e d e p o s i t has a l a r g e r a b s o r p t i o n c a p a c i t y . The g e t t e r i s f l a s h e d , the Argon i s pumped out, the methane i s i n t r o d u c e d , and  the sample i s s e a l e d o f f .  The  slight  enlarge-  ment a t the end of the sample tube, around which the sample coil  fits,  i s to i n c r e a s e the sample volume and hence to  improve the s i g n a l to n o i s e r a t i o .  ELECTRONICS: The e l e c t r o n i c s i s t h a t of a standard pulsed N.M.R. spectrometer.  I t c o n s i s t s of a t i m i n g s e c t i o n , which  g a t i n g p u l s e s of d e f i n i t e width and at d e f i n i t e times;  generates a  t r a n s m i t t e r s e c t i o n , which produces sharp p u l s e s of r . f . power; a sample c i r c u i t and a r e c e i v e r .  The  timing s e c t i o n i s b u i l t  up from three T e k t r o n i x u n i t s , two p u l s e generators and one sawtooth generator  (type 162).  The  (type  163)  sawtooths produced  by the sawtooth generator, e i t h e r s i n g l y or r e p e t i t i v e l y ,  are  - l o used t o t r i g g e r the pulse g e n e r a t o r s a t predetermined on the sawtooth. jitter  times  T h i s t i m i n g c i r c u i t works v e r y w e l l , the  i s l e s s than 0.5%.  The t r a n s m i t t e r , see f i g u r e 2,(on  p r e v i o u s page) i s b a s i c a l l y a gated H a r t l e y o s c i l l a t o r f o l l o w e d by b u f f e r s t a g e , and a power a m p l i f i e r . and f a l l  The r i s e  times o f the edges o f the r . f . p u l s e s a r e  approximately 1-2 microsec; the peak-to-peak v o l t a g e a c r o s s the sample c i r c u i t  i s 800 v o l t s .  The r e c e i v e r i s a h i g h g a i n  wide-band I . F . I , r e c e i v e r type SPC.-230.  The r e c o v e r y  c h a r a c t e r i s t i c s , as w e l l as i t s s t a b i l i t y a g a i n s t s e l f o s c i l l a t i o n were very poor; consequently a number o f changes were made.  The schematic i n f i g u r e 3 shows the m o d i f i e d c i r c u i t .  The s i g n a l - t o - n o i s e r a t i o f o r the deuteron resonance was approximately 100:1 f o r maximum s i g n a l ; the n o n - l i n e a r i t y of the a m p l i f i e r was found t o be n e g l i g i b l e f o r the s i g n a l  amplitudes  o b t a i n e d and no c o r r e c t i o n s were found to be necessary.  THE MEASUREMENT OF RELAXATION TIMES: T^ and T^ measurements were accomplished by s t a n d a r d N.M.R. t e c h n i q u e s .  F o r l o n g T^ (4 s e c . and up) the sample  i s s a t u r a t e d w i t h r . f . a t time t«0 by a t r a i n of c l o s e l y spaced 90° p u l s e s ; t h i s produces the i n i t i a l n o n - e q u i l i b r i u m c o n d i t i o n M„(0) » M„(0) » M„(0) » 0. x y z  Then a t a d e f i n i t e time t  l a t e r a s i n g l e 90° pulse i s a p p l i e d ; i . e . i s r o t a t e d i n t o the x-y plane by the p u l s e .  M ^ ( * ) - , M [ i - e'* '] /T  0  The amplitude A ( t )  F i g u r e 3,  Modified receiver c i r c u i t  diagram.  12 of  the I n d u c t i o n t a i l  f o l l o w i n g t h i s 90° p u l s e i s r e c o r d e d .  I t can be 3hown from the d e f i n i t i o n of A(M » R«,[' - e' T,J y  of  •  t  n  u  s  T  c  a  n  b  e  amplitude measurements a t d i f f e r e n t  that  o b t a i n e d from a s e r i e s times t .  A s t o p watch  was used f o r t i m i n g ; the i n d u c t i o n t a i l s were r e c o r d e d on f i l m w i t h a Du Mont Scope camera type 2620.  For s h o r t e r  s p i n - l a t t i c e r e l a x a t i o n times a 90°- 90° p u l s e sequence used. in  was  Again the amplitude a f t e r the second p u l s e s h o u l d vary  the same manner.  p u l s e s was  In t h i s case, the time i n t e r v a l between  measured u s i n g a Hewlett-Packard frequency meter  model 524C with a time i n t e r v a l p l u g - i n u n i t model 526E. the  average, the r e p r o d u c i b i l i t y of the deuteron  measurements was measuring  approximately 5%;  t h i s was  a r r i v e d at by  a number of times a t the same temperature.  a l a r g e source of e r r o r i s i n t r o d u c e d , i f the sample (the  On  But  itself  l a t t i c e ) has not yet a t t a i n e d thermal e q u i l i b r i u m a t  the  time of measurement.  (an  hour or so) i f the sample has been c y c l e d  melting point.  The p r o t o n  T h i s time c o u l d  long  through the  measurements had a lower a c c u r a c y  because of a lower s i g n a l to n o i s e r a t i o . r e l a x a t i o n was  be q u i t e  The  spin-spin  measured u s i n g a 90° - 180° p u l s e sequence.  For  two such p u l s e s s e p a r a t e d by a time T a s p i n echo appears a t 2T w i t h an amplitude A(2T) - A e x p Q  In  <  s e e  Hahn ). 5  the. l i q u i d s t a t e the decay of the i n d u c t i o n t a i l i s  governed by the d i s t r i b u t i o n of v a l u e s of the main magnetic  - 13 field  over the sample.  f i e l d s inside  In the s o l i d the l o c a l magnetic  the sample due t o the d i s t r i b u t i o n o f the  n u c l e a r magnetic moments can be much l a r g e r inhomogeneity.  than the magnet  Under those c o n d i t i o n s the d i s t r i b u t i o n s i s  approximately Gaussian and i t can be shown that t a i l has the form chapter). <|co^  A(t) = A(o) e x [ - i  By f i t t i n g  P  the i n d u c t i o n  can be o b t a i n e d .  <u>*> -t-] 2  tail  the i n d u c t i o n (  s e e  next  the second moment  14 CHAPTER 3 THEORY  In t h i s s e c t i o n the theory r e l e v a n t w i l l be d i s c u s s e d . spin species,  t o t h i s experiment  F o r a s p i n system c o n s i s t i n g o f two  one o f which has I - 1 / 2 and the o t h e r I > l / 2 and  a quadrupole moment Q, the g e n e r a l H a m i l t o n i a n i n c l u d i n g the quadrupole and s p i n - r o t a t i o n i n t e r a c t i o n s , i s :  * H. Z  Ijj  + V  H„ I  I,  (1)  + X fl 1 J.'. J , i -  +  21 A I j . . J j . j' ~ ~  where primed symbols r e f e r t o the s p i n s p e c i e s w i t h 1=1/2. The f i r s t  two terms a r e the Zeeman energy terms;  the next  three terms a r e magnetic d i p o l e i n t e r a c t i o n terms; the f i f t h r e p r e s e n t s the quadrupole energy; the l a s t the s p i n - r o t a t i o n i n t e r a c t i o n terms  two terms r e p r e s e n t  - 15 where  ^1=  sc.  -  ±  c  <  f  e  ^  e o  Yt Y* ^  3  Q»c =  e Q  >9^  ^  ( ^ l t 3  - I^J  E(ar-<)  I U i - .) E  0  =  \  &  2  Etn ~  +  _!_  (3 c ^ i  !  9 - t]  2.  s  ^  sen 9  cc s &  S-~ ^ c  where the symmetry a x i s o f the molecule i s d e f i n e d by <9 and  <^> w i t h r e s p e c t t o the main magnetic  field  The  physical  relaxation Puree11 and  p i c t u r e u n d e r l y i n g the d e s c r i p t i o n  phenomena was Pound .  the  f i r s t proposed by Bloerabergen,  The  l o c a l f i e l d s a c t i n g on  of  Fourier  the  spectrum o f the  nucleus may  frequency c o r r e s p o n d i n g to the  time-dependent  iiave components a t  energy d i f f e r e n c e  a p a i r of l e v e l s of the n u c l e a r s p i n system; there  between  exists  then a f i n i t e p r o b a b i l i t y of a s p i n t r a n s i t i o n being induced by  the  fluctuating f i e l d .  t r a n s i t i o n i s p r o v i d e d by i n thermal e q u i l i b r i u m , the  The the  lattice.  a p p r o p r i a t e Boltzman f a c t o r s and  equilibrium To  with the  for this  Since the  lattice is  these t r a n s i t i o n s are weighted  t r a n s i t i o n s i s to b r i n g  the e f f e c t of  by  these  the n u c l e a r s p i n system i n t o thermal  lattice.  be a b l e to d e s c r i b e the above i d e a s mathematically  a correlation function by d e f i n i t i o n  G (t) of a f u n c t i o n  G(r) = <( W  ensemble average.  The  f">**)]>  where < > denotes an made that  i n o t h e r words, one  depend upon the o r i g i n i n time.  tion function  f ( t ) i s introduced;  assumption i s u s u a l l y  distribution i s stationary, not  energy r e q u i r e d  depends o n l y on  x .  which does  In t h i s case the For  the  most p h y s i c a l  correlaquantities  a c e r t a i n amount of c o r r e l a t i o n e x i s t s between the v a l u e s of a physical G(x. t  quantity within  a s m a l l i n t e r v a l of time, i . e .  ) i s i n g e n e r a l non-zero f o r a f i n i t e range of v a l u e s of .  The  l e n g t h of time i n which some c o r r e l a t i o n  i s c a l l e d the c o r r e l a t i o n time; f o r example, t d e f i n e d by  G (? ) c  G  C?i  c  persists  could  be  - 17 The  -  above H a m i l t o n i a n d e s c r i b e s  four  contributions  to changes i n the l o c a l magnetic f i e l d each with a lation function.  The  d i p o l e - d i p o l e i n t e r a c t i o n s can F• =  expressed as a sum  of terms i n v o l v i n g  where k « 0,  the a^ are c o n s t a n t s and  1, 2,  s p h e r i c a l harmonics.  1/Tj  and  1/T  on the motions  the  s p i n s t a t e s and  <^ f V )  g i v e an  =h  t+  I t i s convenient  (where  A/V  a c o r r e l a t i o n time of  the order of the average time f o r a p p r e c i a b l e t r • C  molecule  correlation function  i s a constant)which has  fc  intra-molecular  r e o r i e n t a t l o n a l motions of the  intra-molecular  which  i *C ^))>, which depend  of the n e i g h b o r i n g s p i n s .  The  are  2  hence i n the e x p r e s s i o n s f o r  to s e p a r a t e t h i s i n t e r a c t i o n i n t o i n t e r - and contributions.  y ^  t  f o r the t r a n s i t i o n p r o b a b i l i t y  are of the form  2  be  °- w VakCe.f)  k  Thus the c o r r e l a t i o n f u n c t i o n s  appear i n the e x p r e s s i o n between nuclear  corre-  reorientations  For a s p i n t r a n s i t i o n i n v o l v i n g an energy change  a e - tf co be w r i t t e n  t  the i n t r a m o l e c u l a r  c o n t r i b u t i o n to T^  f  can  thus  i n terms of the s p e c t r a l d e n s i t i e s  C") =  G  C-c) e.  dx  (3)  J- CO  The  complete e x p r e s s i o n  to 1/Tj  i s (Abragam  7  f o r the i n t r a - m o l e c u l a r  page  291)  contribution  - IS -  (jj  =x  . t f v t d * . ] - o — > '  (4)  where i and k s p i n s are s i m i l a r , but the 1 s p i n s a r e d i s ..i \ v ; : similar. The  inter-molecular FV .?)  because the  8  c o n t r i b u t i o n i s more c o m p l i c a t e d ,  o f n e i g h b o r i n g s p i n s are determined  by the o r i e n t a t i o n s as w e l l as the r e l a t i v e s p a t i a l of the m o l e c u l e s . general  locations  Consequently the s p e c t r a l d e n s i t y  i sin  a f u n c t i o n o f two c o r r e l a t i o n times, one a s s o c i a t e d  with m o l e c u l a r r e o r i e n t a t i o n and the other w i t h t r a n s l a t i o n a l motion.  I f the assumption i s made t h a t the r e o r i e n t a t i o n s  occur much f a s t e r than the t r a n s l a t i o n a l motions, then the s p e c t r a l d e n s i t y and the c o r r e l a t i o n f u n c t i o n can be w r i t t e n as the sum o f two independent terms. Hamiltonian  H,  perturbation  i s a f u n c t i o n o f the v e c t o r  between the c e n t e r s r_c ~ (re,  The  &c ,<fc)  (fi,©,4>)  o f the molecules, and the v e c t o r s  where 1=1, 2, s p e c i f y i n g the p o s i t i o n s o f  each o f the two i n t e r a c t i n g n u c l e i with r e s p e c t of the two molecules. VI, =  £=  Mr  +  Now w r i t e Vi  t  t o the c e n t e r s  the i d e n t i t y (5)  - 19 where  <^  r e p r e s e n t s an average over a l l o r i e n t a -  t l o n s o f each o f the molecules.  The time dependence o f K^.  i s p r i m a r i l y due to r e o r i e n t a t i o n s  o f the molecules and /H  t  depends o n l y on the r e l a t i v e p o s i t i o n s o f the two n u c l e i , so t h a t i t s time dependence i s due t o t r a n s l a t i o n a l motions. Now the c o r r e l a t i o n f u n c t i o n has the form from e q u a t i o n (5)  <H,(t) M,ct**)> = < u„cu K<t+0 + <H (tj H£^)> t  The l a s t  two terms average t o z e r o , i f the t r a n s l a t i o n a l  and r o t a t i o n a l motions a r e u n c o r r e l a t e d , i . e .  The s p e c t r a l d e n s i t y  one o f which time,  J S-^ B  J 6°) 8t  %  &  a  The c o n t r i b u t i o n (4) w i t h  w i l l be the sum o f two terms  involves  the t r a n s l a t i o n a l c o r r e l a t i o n  f u n c t i o n o f the r o t a t i o n a l c o r r e l a t i o n time. t o 1/T^ w i l l have the same form as e q u a t i o n  J^o) substituted  f o r J~*c<w .  - 20 -  As methane i s a molecular c r y s t a l , that the e l e c t r i c f i e l d the neighbors  i t w i l l be assumed  g r a d i e n t a t the n u c l e a r s i t e due t o  i s negligible.  Consequently  r e l a x a t i o n mechanism can be expressed  the quadrupole  i n terms o f molecular  r e o r i e n t a t i o n s o n l y ; the f i e l d g r a d i e n t being f i x e d w i t h r e s p e c t to the molecular body a x i s .  The H a m i l t o n i a n ,  e q u a t i o n ( 1 ) , shows t h a t the quadrupole expressed functions for  i n terms of the y ~ v . 2  <^y ~>J y X . ^ ^ ) > +  2  a  i n t e r a c t i o n can be  Thus the c o r r e l a t i o n  i n v o l v e d a r e the same as those  the i n t r a - m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n e q u a t i o n (2).  The s p i n l a t t i c e r e l a x a t i o n time f o r 1=1 and f o r an a x i a l l y symmetric f i e l d  i n s i d e the molecule  (Abragam  where eQ i s the n u c l e a r e l e c t r i c quadrupole the e l e c t r i c f i e l d  page 314}  moment and q i s  g r a d i e n t a t the nucleus.  The H a m i l t o n i a n d e s c r i b i n g the i n t e r a c t i o n between the s p i n and the r o t a t i o n a l magnetic moments i s  21  X » -  R t. J  •  At *!* 1  where A i s a c o n s t a n t .  * i - l M -  - - J*). L  The r o t a t i o n a l quantum s t a t e m o f T  the molecule changes due t o c o l l i s i o n s . changes i n the l o c a l magnetic f i e l d F o r . t h i s type of p e r t u r b a t i o n the c o n t r i b u t i o n  ' 'i  The r e s u l t i n g  induce s p i n t r a n s i t i o n s .  Abragam, page 309, shows  -  where  (9)  1  and  The  that  G "Cc) = c  A* < J <?) J-Ct)> +  complete e x p r e s s i o n f o r 1/Tj i s the sum o f the c o n t r i -  b u t i o n s g i v e n by equations ( 4 ) , ( 7 ) , ( 8 ) and ( 9 ) . This expression i s c o r r e c t only  i f the s p i n  of the coupled s p i n system i s independent o f time. condition  temperature If this  i s not s a t i s f i e d the r e l a x a t i o n f u n c t i o n w i l l not  be d e s c r i b e d  by a s i n g l e r e l a x a t i o n time, t h e r e would be two  e x p o n e n t i a l terms.  In the experimental r e s u l t s r e p o r t e d  here,  the r e l a x a t i o n f u n c t i o n was always a s i n g l e e x p o n e n t i a l w i t h i n experimental e r r o r . If exponential c o r r e l a t i o n functions the s p e c t r a l d e n s i t y For  has the f o l l o w i n g form  a r e assumed, then J"(o) <x , — £ * •+- (O*  t r a n s l a t i o n a l and r o t a t i o n a l d i f f u s i o n i t can be shown  - 22 that and  J * i J .* Jf ^ 6 : 1 : 4 . c  9)  1  The  -  above expressions  f o r the  i n t r a - m o l e c u l a r d i p o l a r c o n t r i b u t i o n s to the  r e l a x a t i o n times can bo s i m p l i f i e d f o r the case t h i s l i m i t J can be approximated by For the i n t r a - m o l e c u l a r equation  (4), _1_ -r  Joe  inter-  spin-lattice cotr »/.  In  c  _r  i n t e r a c t i o n s alone the  expression,  becomes =  3  L(  «?  I  ^-  *  FI  w  3,  Upon e v a l u a t i o n of the expressions t h a t I.6A1, = B,. K  ^>  fl„  For a CD. 4—n  JL  =  ft  • (4-)  a n (  -  Where the f u n c t i o n s  for  and  one  finds  H„ molecule n  J  [ ( ' - a)  ^  £ ^  ^ 3  -  i.t a ]  S  B  ( 1 0 )  depends upon the r o t a t i o n a l c o r r e l a t i o n  time, the gyromagnetic r a t i o s , l a t t i c e parameters, e t c . For the i n t e r - m o l e c u l a r d i p o l e - d i p o l e i n t e r a c t i o n the r e l a t i v e c o n t r i b u t i o n s of the two same; except the a b s o l u t e functions RB and  and  value  s p i n s p e c i e s w i l l be  the  i s determined by d i f f e r e n t  RQ, which are dependent on the r o t a t i o n a l  t r a n s l a t i o n a l c o r r e l a t i o n times r e s p e c t i v e l y ft)  =  Id-?)  +• ' -  6  5-j  («*  < > n  Johnson and Waugh  have suggested that  the r e l a x a t i o n  r a t e due t o the s p i n r o t a t i o n i n t e r a c t i o n i s p r o p o r t i o n a l to  <j~( J+i))DC T I „ 5 i f t h i s i s v a l i d the dependence o f the  s p i n r o t a t i o n mechanism on n takes the form  (12)  where R~ w i l l be a f u n c t i o n  o f the c o r r e l a t i o n time f o r  changes i n the mj v a l u e s . The quadrupole r e l a x a t i o n  mechanism o f the deuteron  s p i n system i s independent o f the i s o t o p i c c o n s t i t u t i o n o f the molecule, i n so f a r as the m o l e c u l a r wave f u n c t i o n s remain unchanged.  I t can be d e s c r i b e d by a f u n c t i o n  Rg, which  i s dependent on the r o t a t i o n a l c o r r e l a t i o n time. I t s h o u l d be remarked  that R , R , R A  g  c  and Rg a r e  s t r i c t l y speaking f u n c t i o n s o f n too, because the c o r r e l a t i o n times a r e f u n c t i o n s o f the mass o f the molecule through the diffusion coefficients. The complete e x p r e s s i o n f o r ( l / T j )  from (10), (11)  and (12)  (13)  In the i n t e r p r e t a t i o n some o f the r e s u l t s o f the theory  - 24 of the i n f l u e n c e of motion of the s p i n s on l i n e width are utilized order.  (Abragam Chapter 10), thus a few comments are i n The development  of the a d l a b a t i c case  co f 0  c  » (  by  Abragam i s a s e m i - c l a s s i c a l one, where the matrix elements of the s p i n - s p i n i n t e r a c t i o n s are assumed to be random f u n c t i o n s o f time. bution of l o c a l  At each i n s t a n t the m i c r o s c o p i c  fields  throughout the sample  be o f a s t a t i o n a r y c h a r a c t e r rigid  lattice  (assumed  distri-  i s assumed to  and i s the same as that f o r a  t o be Gaussian)„  but the l o c a l  field  at each p o i n t i s assumed to f l u c t u a t e a t a r a t e d e s c r i b e d the c o r r e l a t i o n f u n c t i o n  GJt) =  <>(t) «o(ttt)). = <6o*> ^Jp) ,  where  ^Lo > i s the second moment of the r i g i d l a t t i c e  line.  Abragam o b t a i n s  a  transform  the f i n a l e x p r e s s i o n  of the a b s o r p t i o n  line  by  resonance  f o r the F o u r i e r  shape  (14)  Without making any e x p l i c i t  assumptions about g ^ ( f ) the  f o l l o w i n g extreme cases can be 1) Long c o r r e l a t i o n time  considered: limit:  The c o r r e l a t i o n time i s so long that <6->> t£ a  i . e . f o r t < -t G(t)  =  »  /  can approximate g ( r)« 1, thus /  UJ  e ~°* C  e x p j ^ - o . </co-> t J 2  (15)  - 25  -  In other words, the shape of the  transverse  r e l a x a t i o n f u n c t i o n i s Gaussian, 2) Short c o r r e l a t i o n time l i m i t : If t it  c  ^  i s so s h o r t that  '  f°r  t. »  x  t  i s p e r m i s s i b l e to w r i t e r* (t-X)  \ *  cLT,  <^>  t  the d e f i n i t i o n of . tri i s of the order of t  and  The  ^  J_ =  c  CO f 5 JSC?** = O-O as used here .  t x'  c  -  Then  <to > <  (16)  x  above formulae h o l d f o r any c o r r e l a t i o n f u n c t i o n , i n  p a r t i c u l a r f o r the cases of Brownian motion and In those  self-diffusion  i n s t a n c e s f o r which the s e l f - d i f f u s i o n mechanism i s  dominant, the  expression  i s o b t a i n e d u s i n g the f a c t t h a t f o r s e l f - d i f f u s i o n t' = t  e /«T Ett  a  where E  0  In the n o n - a d i a b a t i c u> t »i e a  i s c a l l e d the a c t i v a t i o n energy. approximation the  restriction  i s removed, i t can be shown t h a t there e x i s t s a  unique s p i n - s p i n r e l a x a t i o n time, i . e . the a b s o r p t i o n  curve  i s L o r e n t z i a n ; the r e s u l t of the d e r i v a t i o n i s (Abragam p.292)  - 26 -  18)  If  «kt »/ c  o n l y the j(°)(0) term i s important and e q u a t i o n  (18) g i v e s the same r e s u l t as e q u a t i o n (16).  - 27 CHAPTER 4 RESULTS AND  DISCUSSION  Measurements of s p i n - l a t t i c e T^ and s p i n - s p i n relaxation  times of the deuteron resonance i n CD^  T  2  and CD3H,  and T j o f the proton resonance i n CD3H were c a r r i e d out a t 4.3 mc/sec. as a f u n c t i o n 105°K to 57°K.  of temperature  i n the range  The r e s u l t s a r e shown i n f i g u r e s  from  4-7.  Upon comparison of the proton measurements a t 30  mcs  (see Appendix) w i t h the deuteron measurements some o b s e r v a t i o n s can be made.  F i r s t of a l l , the s t r o n g  temperature dependence e x h i b i t e d  i n the proton  measure-  ments i s completely absent i n the deuteron measurements f o r both CD4 and CD3H (see f i g u r e s 4 and 5 ) .  Moreover,  the  v a l u e s o f T j i n most of the temperature range are s h o r t e r  for  the deuteron resonance, whereas they are expected to be l o n g e r i f the r e l a x a t i o n actions, the  i s due to magnetic  dipolar  inter-  e s p e c i a l l y i n the l i q u i d s t a t e 90°K - 105°K.  l i q u i d , oo %«i 0  9  thus  J %) L  = J^^j  - J^(aw )  In a  n  d  r  from e q u a t i o n (7)  These o b s e r v a t i o n s a l l o w one t o conclude that e i t h e r  the  c o u p l i n g between the n u c l e i i s of a d i f f e r e n t nature or the random motions of the molecules are of an e n t i r e l y d i f f e r e n t  Melting I ' o f  A1 i 1 1  50  4  ~ 89.8 °K  1 1  70  80  90  Temperature F i g u r e 4.  CD  Point  Deuteron  o f CD  4  °K  versus temperature,  100  110  Melting Point  _J  70  i  I  80  i  I  Temperature F i g u r e 5„  Deuteron  i  90  I  100  °K  of CD H versus temperature. 3  i  I  110  - 30 character,  which would e f f e c t changes i n the s p e c t r a l  d e n s i t i e s governing the r e l a x a t i o n  times.  The second of the two a l t e r n a t i v e s can be r u l e d o u t , because  the temperature  CDgH and the deuteron T 30  racs.  dependences o f the proton 2  in  i n CD^ are s i m i l a r t o those a t  In the range o f 105°K - 90°K the sample i s a l i q u i d ,  below 90°K i t i s a s o l i d but the behaviour o f the proton T j and Tg i n d i c a t e that  t r a n s l a t i o n a l motions a s s o c i a t e d  with  s e l f - d i f f u s i o n a r e important down t o approximately 65°K 9 (Waugh J. D e s p i t e t h i s l a r g e change i n the m o l e c u l a r  mobility,  there i s no i n d i c a t i o n o f i t i n the deuteron T^ measurements f o r CD  4  and i s only s l i g h t l y n o t i c e a b l e  therefore  concluded that  intra-molecular  the s p i n - l a t t i c e r e l a x a t i o n .  f o r CDgHj i t i s i n t e r a c t i o n s dominate  The proton r e l a x a t i o n measure-  ments a t 30 mcs. i n d i c a t e that the magnetic  intra-molecular  r e l a x a t i o n mechanism i s very i n e f f i c i e n t ; i t i s suggested the molecules a r e undergoing r a p i d r e o r i e n t a t i o n s Sandhu). magnetic  As the deuteron magnetic moment, the magnetic  that  (Bloom and  moment i s 1/7 o f the proton  intra-molecular  i n t e r a c t i o n s are  expected t o be even l e s s e f f i c i e n t f o r the deuteron s p i n  system  and can thus be d i s r e g a r d e d . The i n t e r a c t i o n o f the deuteron quadrupole moment with the e l e c t r i c f i e l d g r a d i e n t  i s suggested as a p o s s i b l e  molecular i n t e r a c t i o n t o account f o r the above r e s u l t s . contribution  intraThe  t o the r e l a x a t i o n i s through the modulation o f  - 31 the e l e c t r i c f i e l d by the m o l e c u l a r r e o r i e n t a t i o n s . temperature independence  The  of the deuteron T^'s shows t h a t the  c o r r e l a t i o n time f o r m o l e c u l a r r e o r i e n t a t i o n i s independent of  temperature. The T^ measurements f o r CD  4  can be e x p l a i n e d i f the  above model i s used, and i f the r e o r i e n t a t i o n s * o c c u r a t a s u f f i c i e n t l y h i g h frequency.  Under these c i r c u m s t a n c e s the  c o n t r i b u t i o n to the l i n e shape from the quadrupole  interaction  w i l l have a L o r e n t z i a n c h a r a c t e r w i t h a long decay c o n s t a n t , e q u a t i o n (18) J L =» <o>*> %  c  , t h i s i s the behaviour c h a r a c t e r i s t i c  of extreme m o t i o n a l narrowing.  The e f f e c t of the quadrupole  i n t e r a c t i o n i s unobservable, because the i n d u c t i o n decay i s dominated  by the i n t e r - m o l e c u l a r d i p o l a r i n t e r a c t i o n s ; the  e f f e c t would o n l y be o b s e r v a b l e f a r out i n the i n d u c t i o n  tail,  where i t i s i n s e p a r a b l e from the n o i s e . I f the above c o n s i d e r a t i o n s are c o r r e c t , the v a l u e s of the deuteron T proton T  2  2  should be c a l c u l a b l e from the v a l u e s of the  a t 30 mcs.  The e x p r e s s i o n f o r T  2  f o r a system o f  i d e n t i c a l s p i n s i s , equation (18)  2 In the range o f temperatures j u s t below the 15.P. the s p e c t r a l d e n s i t i e s are dominated approximated by  J ^ j=  by the s e l f - d i f f u s i o n and can be CLZ %<•_  where  a.;a, tx^= 6-. i -. * ;  The v a l u e s of . t i n t h i s r e g i o n can be o b t a i n e d from the v a l u e s c  - 32 o f the p r o t o n T A oc T  2  -< "> '^ co  e  ^  a  co•r.C^z.S T(^)oc term  t  j  and  2  f  Q  "•  r  -  the l i n e shape g i v e n by e q u a t i o n t>T^  , thus  _J  can be w r i t t e n as  Using these v a l u e s f o r t  i n the range of temperatures  , one f i n d s t h a t  t  90°K - 70°K, thus  and e q u a t i o n (19) i s dominated by the  J°(o)  and  The same approximations resonance.  are a l s o c o r r e c t f o r the proton  Using the v a l u e s o f the proton T  2  a t 30  one can p r e d i c t those f o r the deuteron resonance ( T ^  The  (16)  »  YP  r  P  u  P  t i )  mcs.,  (T ) 2  D  (TJ  p r e d i c t e d v a l u e s are shown i n f i g u r e 6;  the agreement  between the p r e d i c t e d and e x p e r i m e n t a l v a l u e s i s r e a s o n a b l e ; the d i f f e r e n c e i n a c t i v a t i o n e n e r g i e s accounts f o r the s y s t e m a t i c d e v i a t i o n of the s l o p e s of the two above r e s u l t i m p l i e s t h a t the quadrupole e f f e c t on  lines.  The  i n t e r a c t i o n has  no  T . 2  In order t o o b t a i n s i g n i f i c a n t r e s u l t s i t i s of the utmost importance,  t h a t a l l paramagnetic  have been removed. 4  In the paper by H.  M. Bloom  i t i s shown t h a t an almost  oxygen i m p u r i t i e s  Sandhu, J . Lees and  temperature  independent  r e s u l t i s o b t a i n e d , i f the oxygen i m p u r i t y c o n c e n t r a t i o n i s 1% or more, but e f f e c t s of the magnetic i n t e r a c t i o n s w i t h  - 33 -  1000  F i g u r e 6.  Deuteron T i n CD temperature. g  4  versus the r e c i p r o c a l of  -  34  -  other methane molecules s t a r t to be observed concentration.  f o r lower  When the r e l a x a t i o n e f f e c t s are dominated  by the oxygen i m p u r i t i e s  should e q u a l T „ 2  A value f o r  T j of 34 m i l l i seconds i s o b t a i n e d f o r an oxygen c o n c e n t r a t i o n of  1%.  C o n v i n c i n g proof t h a t the CD  sample used was  4  free  from i m p u r i t i e s i s the f a c t t h a t the v a l u e s of ( T ) 2  can  D  be  p r e d i c t e d from the v a l u e s f o r ( T ) , as i s d i s c u s s e d above. 2  p  The CDgH must a l s o be f r e e from oxygen i m p u r i t i e s , the v a l u e s o f ( T j ) a t 4.3 p  temperature  mcs.  show the c h a r a c t e r i s t i c  dependence of i n t e r - m o l e c u l a r d i p o l a r  modulated by s e l f - d i f f u s i o n , The a c t i v a t i o n energy  because  (see  interactions  below).  f o r the s e l f - d i f f u s i o n  process  as o b t a i n e d from the ( T ) p temperature dependence i s 3.5  kcal.  2  per mole f o r CD , 4  as compared with 3.2  kcal/roole f o r  o b t a i n e d from the proton measurements at 30 i n c r e a s e i n the a c t i v a t i o n energy  can p o s s i b l y be  t o the 25% i n c r e a s e i n the mass of the The  (T^Jp r e s u l t s f o r CD^H  show some i n t e r e s t i n g f e a t u r e s .  rocs.  a t 4.3  CH  The  4  10%  attributed  molecule. mcs.,  see f i g u r e  7,  They can be c a l c u l a t e d  approximately u s i n g e q u a t i o n (7) and the ( T ^ ) r e s u l t s a t p  30 mcs.,  as before i t i s assumed t h a t  the p r e d i c t e d value of ( T j ) p a t 4.3 2.1 mcs.  x 10""  2  ( T j ) p as 30 mcs.,  mcs.  T(H «- —±—  Then  i s approximately  i . e . the (T-^p r e s u l t s a t  should show the same dependence on temperature,  ( T j ) p r e s u l t s a t 30 mcs.  .  4.3  as do  the  However, the p r e d i c t e d r e s u l t s  are s m a l l e r than the experimental v a l u e s by as much as a f a c t o r  -  35  -  i  0J\  1  I  80  .  I  90  Te mp era ture F i g u r e 7.  Proton T, i n CD~H  °K  versus  temperature.  - 36 of t h r e e a t the h i g h e s t temperatures, though the lower temperatures.  they agree a t  The assumption o f a s i n g l e  correla-  t i o n time f o r s e l f - d i f f u s i o n , which l e a d s t o c o t ^ s f o r 0  t  4.3 mcs. below the m e l t i n g p o i n t , may be an o v e r s i m p l i f i c a t i o n . Perhaps a more d e t a i l e d model, T o r r e y ' s m o d e l ^  for,example,  s h o u l d be employed, but t h i s has not y e t been attempted. The d i p i n the deuteron  r e s u l t s near the m e l t i n g  p o i n t f o r CD3H can be e x p l a i n e d q u a l i t a t i v e l y , i f the d i p o l a r i n t e r a c t i o n s a r e taken i n t o account.  The c o n t r i b u t i o n to 1/T^  due t o d i p o l a r i n t e r a c t i o n s can be o b t a i n e d u s i n g e q u a t i o n (7) and the T^ r e s u l t s f o r CH^ a t 30 mcs.  The same a p p r o x i -  mations r e g a r d i n g J(w) w i l l be made as above. For example a t 89° K, the c o n t r i b u t i o n t o T j due t o d i p o l a r i n t e r a c t i o n i s approximately 24 s e c . f o r CT> H, i f the quadru3  pole c o n t r i b u t i o n i s assumed t o be 10 s e c . , then the observed T j s h o u l d be approximately 7.1 s e c . as compared with an e x p e r i m e n t a l v a l u e of 6.0 s e c . be improved  Again t h i s r e s u l t may p o s s i b l y  through the use o f a more a c c u r a t e s p e c t r a l d e n s i t y .  The quadrupole c o n t r i b u t i o n t o the r e l a x a t i o n o f the s p i n system can o n l y be i n t e r p r e t e d , i f somehow a v a l u e f o r the quadrupole c o u p l i n g c o n s t a n t can be o b t a i n e d .  One  p o s s i b i l i t y i s from the l i n e shape o f a p o l y - c r y s t a l l i n e sample a t helium temperatures.  I f the quadrupole c o u p l i n g i s s u f f i -  c i e n t l y s t r o n g and i f the i n t e r n a l motions a r e completely quenched, then the a b s o r p t i o n l i n e shape w i l l e x h i b i t two  0  - 37 peaks.  The d i s t a n c e between the peaks i s a f u n c t i o n of the  coupling constant.  Another p o s s i b i l i t y  i s to c a l c u l a t e the  f i e l d g r a d i e n t from Hartree-Fock approximate for  the molecule  wave f u n c t i o n s  (e.g. K r a u s e * * ) .  To o b t a i n the a b s o l u t e magnitude* of the i n t e r - m o l e c u l a r c o n t r i b u t i o n to T^,  the f o l l o w i n g experiment  i s suggested.  I f the s p i n l a t t i c e r e l a x a t i o n times of mixtures of CH CD  4  a r e measured, then the term  w i l l be r e p l a c e d by  [(«-*)  0  (R  x} ( R  B  B  + R^)  -t- R )  and  4  i n e q u a t i o n (13 J where x i s the  c  c o n c e n t r a t i o n of CD4 molecules and A i s a c o n s t a n t which d e s c r i b e s the r e l a t i v e s t r e n g t h of the i n t e r - m o l e c u l a r a c t i o n s of the two types of molecules.  I f the r e l a x a t i o n  i s measured f o r a number of c o n c e n t r a t i o n s a t each then ( R  B  + R^) can be e v a l u a t e d as a f u n c t i o n of  In c o n j u n c t i o n with the p r o t o n r e s u l t s a t 30 mcs,  1/T,.  time  temperature,  temperature. i t may  p o s s i b l e to p r o v i d e a unique s o l u t i o n to the r e l a t i v e c o n t r i b u t i o n s to the r e l a x a t i o n r a t e  inter-  be  38  -  APPENDIX  Proton R e l a x a t i o n Data a t 30 i)  mcs:  "Proton S p i n - L a t t i c e R e l a x a t i o n i n Pure Methane and i t s Deuterated M o d i f i c a t i o n s . " M. Bloom and H. S. Sandhu Can. Jo Phys. 40, 289 (1962).  ii)  "N.M.R. Line-shape S t u d i e s i n Methane Using P u l s e Techniques." M. Bloom and H. S. Sandhu Can. J . Phys. 40, 292 (1962).  -39-  NOTES PROTON  SPIN-LATTICE ITS  RELAXATION  DEUTERATED  M.  BLOOM f  IN PURE M E T H A N E A N D  MODIFICATIONS*  AND H.  S.  SANDHU  We report here measurements of the proton spin-lattice relaxation time T\ using pulse techniques (Hahn 1950) in liquid and solid C H , C H 3 D , C H 2 D 2 , and C H D between 57° K and 110° K . By varying the isotopic form of methane in this way it is possible to obtain information on the nature of the interactions contributing to spin-lattice relaxation. Reliable measurements were made possible by the use of a purification technique reported previously (Sandhu, Lees, and Bloom 1960). When this technique is used, 7\ in liquid and solid C H is found to be about 1000 times longer than previously reported (Thomas, Alpert, and Torrey 1950), presumably because of 0 contamination of the samples used previously. The detailed interpretation of T\ by Tomita (1953) must therefore be reconsidered. 4  3  4  2  The interactions governing 7\ for spin \ nuclei in polyatomic molecules such as methane are: (A) intramolecular dipolar interactions, (B) intermolecular dipolar interactions, (C) A\ . J type interactions between the nuclear spin I and the rotational angular momentum of the molecules J . There has been some evidence recently that mechanism (C) is probably predominant for fluorine nuclei in systems such as liquid C H F (Gutowsky, Lawrenson, and Shimonura 1961). Recently Johnson and Waugh (1961) have suggested that it is also important for liquid C H . They estimate a contribution to T\~ of approximately 0.02 sec at its normal boiling point. In making this suggestion Johnson and Waugh have surmised that the relaxation rate due to mechanism (C) is proportional to (J(J+l)) S? TI , where Tis the absolute temperature and Io is the moment of inertia of the molecule. If we accept this assumption, T\ in the molecule CH „„£> has the following dependence on n, the number of deuterons, at constant temperature 3  4  x  -1  r  0  4  n  (1)  whereR , R , and R are the contributions to 2 \ for n = 0 due to mechanisms A, B, and C respectively. In writing equation (1) we neglect the changes in the correlation functions appearing in R , R , R due to the changes in n. In the conventional theories of T\ (Abragam 1961, pp. 300 and 302), R and i? would depend on the diffusion coefficient and if the diffusion coefficient were proportional to M~ , R and R would be multiplied by [l + ( « / 1 6 ) ] , having at most an influence of 9% for C H D . - 1  A  B  c  A  B  c  K  B  m  1/2  A  B  3  *Research supported by National Research Council of Canada. fAlfred P. Sloan Foundation Fellow. Canadian Journal of Physics. Volume 40 (1962)  -kc-  C A N A D I A N JOURNAL O F PHYSICS. VOL. 40, 1962  • i  If (1/Ti) is fitted to a linear function of n at each temperature as implied by (1), one obtains only two parameters from the equations. a  (2a)  Qr) =  (2b)  dJ\J^)  R +R +R ,  fl  A  =  -°-  B  32i?  C  A-0.24i? +0.25i? . B  c  Although one cannot obtain a unique solution to equations (2), the fact that R , R b i and R must all be positive enables us to establish upper and lower bounds for R , R , and R . A  C  A  B  C  The Liquids A reasonable fit of the experimental data given in Fig. 1 over the entire liquid range is obtained for  h(rl  = -°- fe)o 2  w i t h i n±  1  0  %  -  200-  f _j  ,  S0 K m  ,  60'K  !  )  70'K 80'K  Tempera  90°K  ture  , 100'K  , //O'K  FIG. 1. Plot of T\ versus temperature for the proton resonances in liquid|and solid CH<, CH D, C H D , and C H D between 56° K and 110° K. 3  2  2  3  -lfl-  NOTES  Ra, Rn > 0 give the following approximate limits for R : c  If the correction factor [l + (w/16)] = l + 0 . 0 3 « + . . . mentioned above for R and R is applied, the limits are changed to 1/2  A  B  More precise values can be obtained by performing double resonance experiments on the proton-deuteron systems, which we plan to carry out. We may conclude, however, that Johnson and Waugh are correct in predicting that R 7* 0 but the experimental upper limit on R is at least a factor of 2 lower than their predicted value, which represents a theoretical upper limit assuming no quenching of the rotational angular momentum. c  c  The Solids Here we obtain (4) which provides the limits  The simplest interpretation of the results in both the liquids and solids is that R = R = 0, i.e. that the contributions to proton spin-lattice relaxation due to all intramolecular interactions is negligible compared with the intermolecular dipolar interactions. The conventional theory for R and R in the liquid (Abragam 1961) would predict that they are of comparable orders of magnitude. This possibility is not excluded by our results. In the solid, it is expected that mechanism (B) is predominant in the region just below the melting point since a plot of In 7\ versus l/T gives the same activation energy, 3.2 kcal/mole, as obtained from line-width measurements (Bloom and Sandhu 1961). The line width is predominantly due to intermolecular interactions. If mechanism (B) is predominant at lower temperatures ( < 7 0 ° K ) , the mechanism is associated with dipolar intermolecular interactions modulated A  c  A  B  -1+2-  C A N A D I A N J O U R N A L O F PHYSICS. V O L . 40, 1962  MT  by the reorientations of the molecules. Using the rigid-lattice line widths (Bloom and Sandhu 1961), the correlation times for molecular reorientation required to give these results are found to be reasonable ( = 10 seconds). However, the conventional theory would predict that for such reorientational motions R should be several times larger than R - It may be that mechanism (A) is sensitive to the influence of the crystalline electric fields on the rotational states of the molecules (Toniita 1953). -12  A  B  ABRAGAM, A . 1961. The principles of nuclear magnetism (Oxford University Press). BLOOM, M . and SANDHU, H . S. 1961. Can. J . Phys. 40. This issue. GUTOWSKY, H. S., LAWRENSON, I. J., and SHIMONURA, K.  1961.  Phys. Rev. Letters, 6,  349.  HAHN, E . ' L .  1950.  Phys. Rev. 80, 580.  JOHNSON, C. S., JR. and WAUGH, J . S. SANDHU, H . S., L E E S , J . , and  1961. J . Chem. Phys. To be published.  BLOOM, M .  1960.  Can.  THOMAS, J . T., ALPERT, N. L., and TORREY, H . C.  TOMITA, K. 1953. Phys. Rev. 89, 429.  J . Chem. 38,  1950.  493.  J . Chem. Phys. 18, 1511.  RECEIVED OCTOBER 31, 1961. DEPARTMENT OF PHYSICS, UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER 8, B . C .  N.M.R. LINE-SHAPE STUDIES IN M E T H A N E  M.  BLOOM+  A N D H.  S.  USING PULSE  TECHNIQUES*  SANDHU  When diamagnetic solids are cooled to sufficiently low temperatures so that very little translational motion of the molecules is taking place, the nuclear magnetic resonance absorption as a function of frequency is usually independent of temperature. This rigid-lattice line shape is often closely approximated by a Gaussian function, i.e. /(co) ~ exp[(co — co ) /2cop], where coo = yH is the Larmor frequency of the nuclear spins in the external field Ho and cop is the second moment of the line. The free induction signal observed in a pulse experiment (Hahn 1950) is proportional to the "relaxation function" G(t) which is the Fourier transform of I(w), i.e. G(t) ~ exp[ —cop</2] for a Gaussian line shape (Abragam 1961, p. 114). With the onset of rapid molecular motion the observable line shape or relaxation function changes to a Lorentzian form, G(t) ~ exp[ — Wpr /], where T is the correlation time for changes in local fields due to translational motions with copTc 1. Under very general assumptions the relaxation function for all times is predicted to be (Abragam 1961, p. 439) 2  0  0  2  c  c  (1)  G{t) ~exp[-eo^c{exp(-*/T )-l+*/T }]. e  *Research supported by National Research Council of Canada. fAlfred P. Sloan Foundation Fellow. Canadian Journal of Physics. Volume 40 (1962)  c  NOTES  MB  The general form of G(t) in (1) has not been extensively used in evaluating experimental results. Normally, in order to obtain T as a function of temperature in a system where diffusion takes place, one works in a region where r is so short that G(t) is only studied for / » r , so that the Lorentzian form is obtained. When the motion has slowed down to the point that the "rigidlattice" line shape is obtained, it is usually assumed that no further information on t can be obtained. Abragam (1961, p. 456) has pointed out that it may be fruitful to use the general form for G(t) in (1) to interpret experimental results. In fact, we have found in studying the proton resonance in C H , C H D , C H D , and C H D that the use of the pulse technique enables the direct stud)- of (1) over a range of T not usually accessible to absorption methods. When T in such systems becomes of the order of 10~ seconds or longer, the absorption signal gives the temperature-independent, rigid-lattice line shape. In the pulse experiment, G(t) is Gaussian for t <K r , enabling one to study the (temperature-independent) second moment. However, the proton induction signals in such systems are so large that one can also study G(t) for t » r . Here, the line shape is indeed found to be Lorentzian. The corresponding observation in absorption experiments is that is Lorentzian for (u-ajn) Tc « 1 . However, the observation of a Lorentzian pip near the center of an absorption line is difficult since one must subtract the large Gaussian contribution to I(oi). In the pulse experiment, the Gaussian portion is allowed to die away, leaving the Lorentzian tail to be studied separately. The results are shown in Fig. 1 where T» is plotted versus temperature. The crosses show T evaluated from signals observed at / « T , fitting the curve G{t) ~ exp[ — t /2Tl]. The circles are obtained from spin echo experiments at higher temperatures where T is short. The squares are obtained from free precession signals for t X> r fitting G(t) ~ exp[ — t/T->]. The results show that for the molecules CHi- D , the correlation times accurately follow activation energy curves over the entire temperature range. c  c  c  c  4  3  2  2  3  c  5  c  c  c  !  2  (  2  c  c  n  (2)  n  (r ), = e  (r ) exp[-EJRT}. 0  n  The activation energies E are found to be 3.2 kcal/mole independent of n, in disagreement with previous measurements for C H (Waugh 1957). Professor Waugh's estimates are based on the line-width data of Thomas, Alpert, and Torrey (1950). The lack of agreement between our activation energies and those obtained from the line-width data is probably due to the change in line shape as the temperature is changed. This is not taken into account in the line-width data. Steady state experiments on methane should be repeated to check whether or not this conjecture is true. If it is true, many of the published activation energies derived from line-width measurements may have to be re-examined. The second moments are, in units of 10 sec , 4.8, 3.95, 3.1, and 1.4 for n = 0, 1, 2, and 3 respectively, corresponding to the ratios 1:0.8:0.63:0.28. a  4  9  -2  C A N A D I A N J O U R N A L O F PHYSICS. V O L . 40, 1962  FIG. 1. Plot of Ti versus the reciprocal of temperature for solid C H , C H D , C H > D , and C H D between 56° K and 90° K . As indicated in the figure and discussed in the text, two different types of time constants are plotted, one associated with the approximately Gaussian shape of the relaxation function at short times and the other with the approximately Lorentzian shape at long times. 4  S  2  3  These ratios are to be compared with the ratios predicted for intermolecular interactions (Bloom and Sandhu 1961; Abragam 1961) (3)  (4)n = ( c 4 ) ( l - 0 . 2 4 » ) 0  which predicts 1:0.76:0.52:0.28. In using (3), we neglect isotopic changes in intermolecular separations and assume that the molecules are undergoing such rapid reorientations that the intramolecular interactions do not contribute to the observable second moments.  -U5-  NOTES  MB  If one assumes a uniform distribution of protons or deuterons on a sphere of radius given by the C - H distance in C H , and if one uses the known crystal structure of C H (see, for example, James and Keenan 1959), the predicted second moments agree with the experimental values within a few per cent. We wish to thank Professor J. S. Waugh for some helpful comments. 4  4  ABRAGAM, A . 1961. The principles of nuclear magnetism (Oxford University Press). BLOOM, M . and SANDHU, H . S. 1961. Can. J . Phys. 40. This issue. HAHN, E . L.  1950.  Phys. Rev. 80,  580.  JAMES, H . M . and KEENAN, T. A . 1959. THOMAS, J . T., ALPERT, N . L., and  WAUGH, J . S.  1957.  J . Chem. Phys. 31, 12.  TORREY, H . C.  J . Chem. Phys. 26, 966.  RECEIVED OCTOBER 31, 1961. DEPARTMENT OF PHYSICS, UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER 8, B . C .  1950.  J . Chem. Phys. 18,  1511.  - 46 -  REFERENCES  1  M. Bloom and H. S. Sandhu: Can. J . Phys. 40, 289-295 (1962).  2  G. T. Armstrong:  3  G. T. Armstrong, F. G. Brickwedde, and R. B. S c o t t :  J . Research N.B.S. 53, 263 (1953).  J . Research N.B.S. 55, 39 (1955). 4  H. Sandhu, J . Lees and M. Bloom: Can. J . Chem. 38, 493 (1960).  5  E . L. Hahn: Phys. Rev. 80, 580 (1950).  6  N. Bloembergen,  E. P u r c e l l , R. Pound: Phys. Rev. 73, 679,  (1948). 7  A. Abragam:  The P r i n c i p l e s  o f Nuclear  Magnetism.  8  C. Johnson and J . Waugh:  9  J . Waugh:  10  H. C. T o r r e y : Phys. Rev. 92, 962 (1953).  11  M. Krause:  J . Chem. Phys. 35, 2020 (1961).  J . Chem. Phys. 26, 966 (1957).  J . Chem. Phys. 38, 564 (1963).  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085822/manifest

Comment

Related Items