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Many-quantum transitions in the conduction electron spin system of lithium metal. Koss, Terry A. 1968

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(i)  .MANY-QUANTUM T R A N S I T I O N S IN THE ELECTRON S P I N SYSTEM OF L I T H I U M  CONDUCTION METAL  by  TERRY B . S c ,  University 1966  A THESIS.-.SUBMITTED THE  KOSS  IN  of  Washington,  PARTIAL  FULFILMENT  OF  REQUIREMENTS FOR THE DEGREE OF MASTER in  the  OF  SCIENCE  department  of  PHYSICS  We  accept  this  thesis  as  conforming  THE  to  the  UNIVERSITY  required  OF B R I T I S H  July  1968  standard.  COLUMBIA  In  presenting  for  an  that  advanced  thesis  shall  I further  agree  for scholarly  Department  or by  publication  without  thesis  degree  the Library  Study.  or  this  my  Department  h its  of  make  i t freely  that  may  thesis  Physics  1968  Columbia  the  granted  by  requirements  Columbia,  t h e Head  shall  and  copying  It i s understood  gain  I agree  for reference  for extensive  for financial  permission.  J u l y 24,  be  of  of British  available  permission  representatives.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date  fulfilment  at the U n i v e r s i t y  purposes  of this  written  in partial  of  of  this  my  that  n o t be  copying  allowed  (ii)  ABSTRACT  Resonance a b s o r p t i o n s p e c t r a have been o b s e r v e d  which  may be i n t e r p r e t e d a s many-quantum t r a n s i t i o n s where t h e a x i s o f q u a n t i z a t i o n i s a l o n g the e f f e c t i v e magnetic in  the r o t a t i n g frame.  By t h i s d e s c r i p t i o n ,  field  resonances  w h i c h r e q u i r e d u p t o f i v e q u a n t a were o b s e r v e d .  The s p i n  s y s t e m u s e d was t h a t o f c o n d u c t i o n e l e c t r o n s o f l i t h i u m metal i n neutron i r r a d i a t e d  lithium fluoride crystals.  An  a n a l y s i s of the experimental r e s u l t s using a modified Bloch -7  e q u a t i o n under the assumption t h a t r = 7^7^1.5x10 v  and  t h a t the s p i n system r e l a x e s toward the i n s t a n t a n e o u s  field  i s presented.  A b r i e f o u t l i n e of the concept of s p i n  temperature i s included. the  sec.  A comparison  i s made b e t w e e n some o f  p r e d i c t i o n s o f t h e s p i n t e m p e r a t u r e concept and t h e s i m p l e  Bloch  theory used  "x  i n analyzing the experimental data.  (iii) TABLE  OF  CONTENTS  ABSTRACT TABLE LIST  i i  OF CONTENTS OF  i i i  FIGURES  .  . . .iv,  ACKNOWLEDGEMENT  vi  INTRODUCTION  1  THEORY  8  APPARATUS  13  .  EXPERIMENTAL  P R O C E D U R E AND  THE MEASUREMENT  RESULTS  OF THE RELEVANT  MAGNETIC F I E L D S  23 47  APPENDIX  52  BIBLIOGRAPPIY  57  (iv) LIST OF FIGURES Figure  Page  1.  2  Fields i n the Lab. and Rotating Frames  ,2. Block Diagram of Experimental Apparatus  17  3.  Spectrometer C i r c u i t  18  4.  TE104 Mode Cavity  19  5.  Klystron Frequency S t a b i l i z e r  20  6.  Variable Coupling Device  21  7.  Block Diagram of Klystron Power Supply  22  8.  Preliminary Derivative Traces at 7.25 and 23 MHz  23  9.  Qualitative Effect on Derivative of Increasing H,Through Cavity Coupling 10. Qualitative Effect on Derivative Traces of Increasing H ^  24 25  11. The Effect of Increasing H on the Relative Peak Strength at 2.6 MHz  29.  12. The Effect of Increasing H^on the Relative Peak Strength at 3.0 MHz  30  13. Plot of  31  vi  vs In of Relative Peak Strength  14. Theoretical Derivative Traces at d i f f e r e n t T'S  34  15. The Theoretical Effect of Increasing H,While the Other Parameters are held constant  35, 36  16. The t h e o r e t i c a l effect of Increasing H $ while the Other Parameters are held constant  37,38 and  17. Comparison of Theory and Experiment for two frequencies  40  r  (v) LIST O F FIGURES (continued)  Figure  '  Page  18. Comparison of Theory and Experiment f o r D i f f e r e n t V a l u e s of H ^ a n d II,  41  19. The N=l resonance L i n e Run f o r , Four V a l u e s of  43  20. Comparison o f D i f f e r e n t P r e d i c t i o n s of the L i n e Width as Measured i n the L a b o r a t o r y Frame  44  21. S e p a r a t i o n i n the Lab. Frarae of the Two N=l Resonance L i n e s as a F u n c t i o n of 9  46  22. C a l i b r a t i o n Data to F i n d the Magnetic F i e l d Produced by the RF Spectrosieter c o i l  49  (vi)  ACKNOWLEDGEMENT  The  e x p e r i m e n t s d e s c r i b e d h e r e i n were m o t i v a t e d  by  t h e p r e v i o u s work o f E r i c E n g a t o whom I am g r a t e f u l f o r t h e c o n s t r u c t i o n o f most o f t h e n e c e s s a r y Valuable  equipment.  a s s i s t e n c e was g i v e n b y Bob P a r s o n s a n d P a t  R y a l l who w r o t e a n d r e f i n e d t h e c o m p u t e r p r o g r a m u s e d i n the a n a l y s i s . S p e c i a l t h a n k s i s g i v e n t o D r . S. A l e x a n d e r  who g a v e  invaluable assistence i n the presentation of the theory. The  p r o j e c t was g u i d e d  Schwerdtfeger  without  and s u p p o r t e d  b y Dzv C h a r l e s  whose a s s i s t e n c e a n d p a t i e n c e t h e  e x p e r i m e n t would n e v e r have been c o m p l e t e d . The  r e s e a r c h was s u p p o r t e d  through  C o u n c i l o p e r a t i n g g r a n t , NRC-A-2226.  a N a t i o n a l Research  -1-  1. INTRODUCTION  The  c l a s s i c a l equation  d e s c r i b i n g the motion o f a f r e e  m a g n e t i c d i p o l e w i t h a d i p o l e moment y. i n a s t a t i c j £  f i e l d H. i s  =  ^  ^  x  ( ) M  where y i s d e f i n e d b y t h e r e l a t i o n [i=yJ y i scalled  momentum o f t h e d i p o l e . This equation coordinate  magnetic  and J i s t h e a n g u l a r  t h e gyroruagnetic  ratio.  i s g e n e r a l l y s o l v e d by n o t i n g t h a t i n a  system r o t a t i n g a t an a n g u l a r  the a x i s which i s p a r a l l e l  frequency  t o H the equation 0  u around  of motion o f  t h e d i p o l e moment becomes  The at  m a g n e t i c moment i s t h u s u=-yl\.  This solution  the l a b o r a t o r y frame. t h e quantum t h e o r y  s t a t i o n a r y i n a frarae r o t a t i n g  then  i seasily  transformed  back t o  Likewise  i t i s e a s y t o show t h a t i n  the equation  f o r the expection value o f  t h e m a g n e t i c niOEient f o r a n i s o l a t e d s p i n i s g i v e n b y i < * > which i s j u s t  =  <M.>-x  yff  0  the c l a s s i c a l equation.  I /. 3) I na d d i t i o n i f the  spins are non-interacting the expectation value of the t o t a l magnetization  o b e y s t h e same e q u a t i o n .  The r o t a t i n g  reference  f r a t a e m e n t i o n e d a b o v e a s s u m e s s p e c i a l i m p o r t a n c e when a r o t a t i n g magnetic f i e l d perpendicular is a static  o f f r e q u e n c y y i and t o the s t a t i c  field  H  e f  field  magnitude H , i s a p p l i e d H . e  I n t h i s case  there  i n t h e r o t a t i n g frame and an a n a l y s i s  s i m i l a r t o t h a t a b o v e shows t h a t t h e m a g n e t i z a t i o n s h o u l d p r e c e s s about  viewed i n the r o t a t i n g frame.  Thus i t  i s seen t h a t t h e r o t a t i n g frame i s a u s e f u l d e v i c e f o r v i s u a l i z i n g the resonance p r o c e s s . of  The  angular frequency  t h e p r e c e s s i o n o f t h e s p i n s i n the r o t a t i n g frame  i s given  AH.  C«0  LAB  H* -  Fmne  Hi  cos  by w e f f = y H e f f . in  the  H  =  B  frame.  i s o b v i o u s f r o m t h e a b o v e d i s c u s s i o n t h a t when ( H - / 0  effective field  i s j u s t H,and a s t h e m a g n e t i z a t i o n  p r e c e s s e s about this f i e l d i s zero.  8  F i g u r e 1 shows t h e m a g n e t i c f i e l d s a s v i e w e d  t h e l a b frame and t h e r o t a t i n g It  Hi  i t s time average i n the z  direction  So f a r i n t h e d i s c u s s i o n a l l i n t e r a c t i o n s o f t h e  s p i n s w i t h the l a t t i c e o r t h e m s e l v e s has been n e g l e c t e d . r e a l systems there are s p i n - l a t t i c e the  i n t e r a c t i o n s which  In  cause  s p i n system t o tend toward the thermal e q u i l i b r i u m v a l u e  )=0  it  v/ould h a v e i n t h e a b s e n c e o f a r o t a t i n g f i e l d .  thus  two  competing processes,  magnetization  to process  r e l a x a t i o n processes H .  e  s h o r t e r than  and  e ?  of very s t r o n g  causing  the  lattice  to relax  spin-lattice  the s p i n Larmor p r e c e s s i o n p e r i o d t h e r e  g i v e n by  magnetization  the u s u a l e x p r e s s i o n f o r the  thermal  of a s p i n system i n a  Larmor p r e c e s s i o n p e r i o d the magnitude of the e  f  c a n be much l a r g e r .  by d e f i n i n g a new remains v a l i d  temperature  static  / / 0 / v  such  the  magnetization  T h i s s i t u a t i o n c a n be T  described 1.4  that equation  f o r the g i v e n m a g n e t i z a t i o n .  I n any  case i t  i s p o s s i b l e t o cause a resonance i n the r o t a t i n g frame applying a perturbing f i e l d dicular  T  should  the o t h e r hand r i s comparable t o o r l a r g e r than  along H j  toward  to c h a r a c t e r i s t i c r e l a x a t i o n times  e q u i l i b r i u m of the m a g n e t i z a t i o n 1 magnetic f i e l d .  on  are  the s p i n  i n the r o t a t i n g f r a m e , a n e t component of  along H £  If  5  c a u s i n g the m a g n e t i z a t i o n  coupling corresponding  be,  about H ^  However e v e n i n t h e c a s e  0  the r o t a t i n g f i e l d  There  of the c o r r e c t frequency  by  perpen-  t o H Qf. . 2 f  R e d f i e l d developed of resonance experiments  an e s s e n t i a l l y  thermodynamic  i n the r o t a t i n g frame.  One  description of  the 3  a s s u m p t i o n s of the t h e o r y as o u t l i n e d is  t h a t the s p i n - s p i n i n t e r a c t i o n  lattice  interaction.  have a t e m p e r a t u r e  i n Miss Franz's  thesis  i s l a r g e compared t o the  T h i s i m p l i e s t h a t the s p i n system  d i f f e r e n t from t h a t of the l a t t i c e .  can The  spin  -4-  s t a r t i n g p o i n t f o r t h i s t h e o r y i s the assumption o f a system H a m i l t o n i a n o f t h e f o r m H=H2 +llss where H z i s t h e Zeeman i n t e r a c t i o n e n e r g y a n d R is  the spin-spin coupling  ss  Note t h a t t h i s H a m i l t o n i a n i g n o r e s any s p i n - l a t t i c e The H a m i l t o n i a n i s t r a n s f o r m e d t o t h e r o t a t i n g f r a m e two t r a n s f o r m a t i o n  H*'= e'*  Gly  HV  t > G l  through  (I**)  *  t a k e s t h e system i n t o a frame r o t a t i n g a t a n g u l a r  frequency u around the z a x i s . of  interaction.  operators.  !  The f i r s t  energy.  The second a l i g n s t h e a x i s  quantization along the e f f e c t i v e f i e l d  axis.  The r e s u l t i n g  Hamiltonian i s :  *Z $ A ,T*T+ 0i*rZ^Xn*  *»H*'--v*ffc :iy  iKf  w  *  Jv3  *Oinpr^  '4 ( 3 Cos ©-/) 1  Ej*e =f  A  l  e> c o s e Sj,/  #  T h e r e a r e two p o i n t s o f i m m e d i a t e  interest.  z a t i o n o f a s p i n system i n a magnetic f i e l d  The m a g n e t i -  defines a spin  temperature based on C u r i e ' s l a w .  I f a s p i n system i s i n i t i a l l y  i n e q u i l i b r i u m with a s t a t i c  a t some t e m p e r a t u r e T a n d  if  the s t a t i c  field  field  c a n be r e d u c e d w i t h o u t c h a n g i n g t h e  The c o u p l i n g c o e f f i c i e n t s ^ ^ a n d tfjtf a r e d e f i n e d b y the s p i n - s p i n H a m i l t o n i a n i n t h e l a b frame. The c o n s t a n t s Aj\i a n d B J K a r e d e f i n e d i n t e r m s o f these by the r e l a t i o n s , /Ci  ^ '/a  ^c  e s  *a; -l) K  ,  * - %  &  (IW*-*!*-!)  F o r a c o m p l e t e d i s c u s s i o n , a r e f e r e n c e i s Abragam pp. 5 4 6 - 5 4 8 .  -5-  m a g n e t i z a t i o n we s a y t h a t t h e t e m p e r a t u r e i s l o w e r t h a n It  before.  h a p p e n s t h a t i f t h e r e i s no s p i n - l a t t i c e r e l a x a t i o n t h e  total  s t a t i c m a g n e t i z a t i o n c a n be made t o p r e c e s s a b o u t t h e  much s m a l l e r adiabatic lattice  i n t h e r o t a t i n g f r a m e , a p r o c e s s known a s  f a s t passage.  T h i s means t h a t t h e r a t i o o f t h e u  t o s p i n t e m p e r a t u r e i s o f t h e o r d e r o f m a g n i t u d e ~f— . 4  An e x a c t c a l c u l a t i o n  shows:  where H ^ i s t h e l o c a l  field  nzis  i n the r o t a t i n g frame.  From  C u r i e ' s Law t h e m a g n e t i z a t i o n i n t h e r o t a t i n g f r a m e i s fl  The  *  H t u j l  I  1.7)  second p o i n t i s that the resonant l i n e width i n the 5  r o t a t i n g frame The  s h o u l d be d e p e n d e n t  (3cos^-l)  dependency i s such t h a t t h e l i n e w i d t h s h o u l d narrow as  ( 3 c o s ^ - l ) goes t o z e r o o r  54.7°.  as a t e s t f o r t h e a p p l i c a b i l i t y  The  initial  This e f f e c t should serve  of the spin  theory i n t h i s form t o i n d i v i d u a l  temperature  systems.  experiments of R e d f i e l d detected large  dispersion signals at r f field to  on t h e t e r m  strengths higher than  ensure the s a t u r a t i o n o f the absorption s i g n a l . 23 6  s y s t e m was t h e n u c l e i o f Na what he t e r m e d frequency f i e l d  i n NaCl.  He a l s o  needed The s p i n  observed  " r o t a r y s a t u r a t i o n " by a p p l y i n g an a u d i o perpendicular to H ^ c  f  in"the rotating  frame.  When t h e n u c l e a r s p i n s y s t e m was a t r e s o n a n c e a d e c r e a s e i n the  dispersion derivative  frequency  satisfied  s i g n a l was o b s e r v e d when t h e a u d i o  t h e c o n d i t i o n u=y8.iff .  In this  -6-  experiment  the H , f i e l d  was  7  s t r o n g enough t o e n s u r e  saturation.  S i n c e t h e r e s u l t s o f t h e s e e x p e r i m e n t s c o u l d n o t be factorily  e x p l a i n e d by t h e u s u a l p h e n o m e m o l o g i c a l B l o c h  equations Redfield  i n t r o d u c e d the concept  of s p i n  S u b s e q u e n t l y R e d f i e l d ' s t h e o r y has been a p p l i e d to  satis-  many n u c l e a r r e s o n a n c e  temperature.  successfully  experiments, e s p e c i a l l y  those  i n v o l v i n g h i g h e n o u g h H, f i e l d s t o c a u s e s a t u r a t i o n . 8 Enga a p p l i e d t h e s e i d e a s t o e l e c t r o n s p i n resonance. the  energy  field  i s about  a typical the  o f i n t e r a c t i o n o f an e l e c t r o n s p i n w i t h a  analogous  magnetic  1 0 0 0 times g r e a t e r than the i n t e r a c t i o n  n u c l e a r s p i n i n t h e same f i e l d , fields  used  o r d e r s of magnitude l a r g e r  As  of  the frequency of  i n Enga's experiment t h a n t h o s e u s e d by  were t h r e e Redfield.  T h i s f a c t e n a b l e d Enga t o use a m a r g i n a l o s c i l l a t o r o p e r a t i n g in  the megacycle r e g i o n to d i r e c t l y  magnetization along H ^ e  radical  $  .  The  monitor changes of  s p i n system  resonance  properties.  l i n e w i d t h of about  four gauss.  were c o n f i n e d t o f i e l d s  free  Dpph h a s a u s u a l e . s . r . Therefore h i s experiments  (He$* ) i n e x c e s s o f t h r e e  This corresponds to a radio frequency  to  the  i n D p p h , a n o r g a n i c compound w i t h w e l l known p a r a -  magnetic  ten  u s e d was  megacycles.  resonance  ( r f ) of approximately  Y/hen he a p p l i e d an r f f i e l d  the microwave f i e l d when|H ^[ £  p e r p e n d i c u l a r t o Ho he T h i s has 'k  two  gauss.  a l o n g H„in observed  solutions and  this  addition  a  as: is satisfied for,  -7-  T h e s e two l i n e s were o b s e r v e d w i t h The  t h e e x p e c t e d symmetry.  l i n e f o r h i g h e s t H w a s a b s o r p t i v e a n d t h a t f o r l o w e s t H^was e  emissive.  T h i s would c o r r e s p o n d t o t h e f a c t  t h a t t h e magnet-  i z a t i o n s h o u l d be p a r a l l e l a n d a n t i p a r a l l e l t o t h e H f i e l d 0  direction  i n t h e two c a s e s r e s p e c t i v e l y .  t h e s e two l i n e s a n i n t e r e s t i n g c o r r e s p o n d i n g t o H £$ =H». e  In addition  c e n t e r l i n e was o b s e r v e d  A s t h i s c e n t e r l i n e h a d no r e a d y  interpretation  i n terms o f the t h e o r y proposed  to i n v e s t i g a t e  this rotating  s p i n system.  i t was d e c i d e d  frame e x p e r i m e n t u s i n g a d i f f e r e n t  Neutron i r r a d i a t e d  seemed a g o o d s y s t e m b e c a u s e 9  lithium  fluoride  crystals  v e r y narrow c o n d u c t i o n e . s . r .  l i n e s had been o b s e r v e d and t h e c r y s t a l s temperature.  to  had a h i g h m e l t i n g  Dpph c r y s t a l s w o u l d m e l t u n d e r  a b s o r p t i o n o f power f r o m t h e k l y s t r o n  the strong  used i n t h i s  experiment  a n d i t was p r o p o s e d t h a t t h e m e l t i n g was i n some way r e s p o n sible  f o r t h e anomalous c e n t e r l i n e .  -8-  2. THEORY  The L i F samples used had a peak to peak d e r i v a t i v e l i n e w i d t h of a p p r o x i m a t e l y .35 gauss.  e.s.r.  I t i s w e l l known that  the peak to peak d e r i v a t i v e width i s r e l a t e d t o the h a l f power resonance curve width by  ^^ff/i'  The c h a r a c t e r i s t i c  spin-  l a t t i c e r e l a x a t i o n time can be found from the h a l f width of the U s i n g the u n c e r t a i n t y p r i n c i p l e At&T-ii  a b s o r p t i o n peak.  i n t e r p r e t i n g kT as the c h a r a c t e r i s t i c s p i n to the l a t t i c e  and  time f o r r e l a x a t i o n of a  gives:  where p. i s the magnetic d i p o l e moment of the e l e c t r o n . Thus:  T'/a  *  TiL  _ -  1  <*  'A2. 7 I'll YJ£Z^~ I'ls'hJjZ—  ( ^ *  where r i s i n seconds and H i s i n gauss.  "2-)  These e x p r e s s i o n s -7  g i v e a r e l a x a t i o n time of approximately 1.9x10  seconds f o r  the measured peak to peak width of the sample.  T h i s means  t h a t the s p i n l a t t i c e r e l a x a t i o n time i s s h o r t e r than the Larmor p r e c e s s i o n time.  T h i s immediately throws s e r i o u s doubt  upon the v a l i d i t y of u s i n g the c o n c l u s i o n s of the s p i n temperature theory to d e s c r i b e the r e s u l t s of t h i s  experiment.  The s i m p l e s t and most standard approach f o r most resonance experiments has been to s t a r t with the phenomenalogical e q u a t i o n s of B l o c h .  Bloch's equations are: ~ ( ^ - ^ }  'p.3)  where M i s the m a g n e t i z a t i o n , H i s the t o t a l i n s t a n t a n e o u s  -9-  7 field  -1  and ^=1.76x10 oe  the e l e c t r o n .  sec.  When one  s a m p l e i s s u c h t h a t 7~J to  the  -1  l  i s the gyroraagnetic r a t i o  has l a r g e p e r t u r b i n g f i e l d s =r  a  , the Bloch  and  are  the  modified  7  microwave f i e l d  dicular  and  form, JT  The  equations  of  to H . e  i s linearly  The  oscillating  radio frequency f i e l d  i s also linearly  oscillating, equations  interpreting  e.s.r. results.  low f i e l d  detected along H j$ e  i s parallel ce* ut  t h u s . H=  T h i s form of the B l o c h  and i s p e r p e n to  + k(lU  *KfCo$jn)  has been used b e f o r e 10,11  i n the r o t a t i n g  The  Ho  in  resonance i s  f r a m e s o we must  transform  t h i s e q u a t i o n i n t o t h e r o t a t i n g f r a m e by t a k i n g H a s (H^-^-jfr and H x as c H, . R e l a x a t i o n i s s t i l l a s s u m e d t o w a r d s H b e c a u s e 2  e  of the s t r o n g s p i n l a t t i c e c o u p l i n g . is:  ~~-  i y(ATKfBfH,-^+*H,7)  This equation  The  equation  t o be  solved  -^ft-* l&(HtiH,fCesut+i(ll,t>9sJlty]'^  ^  9  ^  c a n n o t be s o l v e d e x a c t l y and s i n c e t h e m a g n e t ic  ization  i s p e r i o d i c i t i s e f f a c a c i o u s t o use a F o u r i e r ft  Setting  sr X  S1  h  v/here o b v i o u s l y  e *  /7n -  a n d s u b s t i t u t i n g 2.6 ( LhJl-A)  M  n  -  n  J  l  (<2.0  t  rl-h i n t o 2.5  g i v e s the i t e r i t i v e  8 (/?*»-! * / ? , ) + Cn =0 h t  where t h e m a t r i c e s A and B a r e d e f i n e d s o  a /T  n  r.  y /?"„ x  and where IL,^ =  expansion;  (5.?) that  (a. ?; £ W, */? (w„ -  eq.  -10-  To u s e t h e s e e q u a t i o n s t h e y must be w r i t t e n o u t e x p l i c i t l y . I g i v e enough h e r e s o t h a t a l l t h e o t h e r s a r e a p p a r e n t . are c l e a r l y  6n e q u a t i o n s a s t h e r e i s a r e a l  and i m a g i n a r y  component o f t h e m a g n e t i z a t i o n j e.g.,  The  3. ^  10. II  18.  equations are:  R  ^ W ' / i e  fl,1*+si7fi,**  ri,**-jiT  •ayr%Ai.  ^rfon**  Ai,*  r  +*  ,[/r  -yr^*' ^»o ,  * t^t/iMF  r  +v ** > rH  {  •=  ^ . r / l / S In t h e above e q u a t i o n s  ^  =• (^'''¥'),  .  0  There  -11-  This suggests  s o l v i n g t h e 6n x 6n m a t r i x e q u a t i o n AM=C  w h e r e A i s t h e m a t r i x d e f i n e d by magnetization  and  the c o e f f i c i e n t s of  M i s a 6n d i m e n s i o n a l  the  vector representing  the components of m a g n e t i z a t i o n , C i s a n o t h e r  column v e c t o r  r e p r e s e n t i n g the inhomogeneous terms i n the e q u a t i o n s . a r e o n l y t h r e e non  There  ,C , and C . C and C are 1 5 11 1 11 three o r d e r s of magnitude s m a l l e r than C and i t p r o v e d t o 5 _make no n o t i c e a b l e d i f f e r e n c e i n t h e r e s u l t s t o n e g l e c t t h e s e t e r m s and  zero terms C  r e t a i n only C  .  This i s reasonable  s i n c e H. ? ^ H, °*  H  n.  5 The  m a t r i x e q u a t i o n was  computer.  Terminating  s o l v e d n u m e r i c a l l y on t h e s e r i e s a t n=5  f o r a l l the parameters except one  one  m a c h i n e r u n t o o k 10 m i n u t e s .  f o r our cisely  n e e d s a s we  and  an  w h i c h assumed f i v e T h i s was  found  needed o n l y t o determine  values  values,  t o be  sufficient  M o r more p r e v  r  terms served  7040  chosing fixed  the d e r i v a t i v e of M * w i t h r e s p e c t t o H„.  t h e n=6  IBM  Including  t o change the c a l c u l a t e d d e r i v a t i v e  at  4  n=5 by o n l y one p a r t i n 10, r e g a r d l e s s o f t h e f a c t t h a t t h e M c o m p o n e n t s were n o t n e g l i g i b l e c o m p a r e d t o t h e M components. 6 5 One i n t e r e s t i n g p o i n t o f t h e e x p e r i m e n t i s t h e e f f e c t on t h e l i n e w i d t h when i n c r e a s i n g t h e a n g l e H . 6  I t was  noted  the f u n c t i o n  becomes s m a l l e r i f t h e s p i n t e m p e r a t u r e  apparent  The  and  i n the i n t r o d u c t i o n t h a t the l i n e w i d t h i n  the r o t a t i n g frame s h o u l d narrow as  f o r the case.  b e t w e e n Keff  p r e d i c t i o n of our  hypothesis  theory i s not  3cos#-l i s valid immediately  b u t when a c o n s t a n t T i s u s e d i n t h e e q u a t i o n s  the  -12-  l i n e w i d t h s h o u l d be c o n s t a n t  i n the r o t a t i n g frame.  r e l a t i o n of the l i n e w i d t h along  The  t o t h a t i n t h e l a b frame  i s s i m p l e a s l o n g a s t h e l i n e w i d t h i s s m a l l c o m p a r e d t o H<?# . The  r e l a t i o n f o l l o w s f r o m e q u a t i o n 1.10.  AWc  *  -K ^ l ^ .  S o  A  / / w  ,  A(H  .^  So a s s u m i n g t h e l i n e w i d t h i s a c o n s t a n t  (0, 30)  }  t h e dependence o f t h e  m e a s u r e d l i n e w i d t h i n t h e l a b f r a m e s h o u l d be ^Ull  .  It is  cos e unfortunate  t h a t t h e h i g h e s t v a l u e s o f H, o b t a i n a b l e were o n l y  a r o u n d one g a u s s w h i c h i s n o t l a r g e e n o u g h t o t e s t t h i s diction accurately.  F o r l a r g e angles,  n o t much l a r g e r t h a n  the l i n e w i d t h .  matched t h e e x p e r i m e n t a l  one g a u s s w h i c h i s  S i n c e t h e computer  l i n e s f o r a l l cases  r u n f o r a s e r i e s o f a n g l e s w i t h H ^ ^ ^ I O Hy. given along with a plot of  pre-  a n d AH-/  X  observed  results  i t was  This plot i s  (3003*0-1)  i n figure  -IS.  -13-  3.  The  APPARATUS  a p p a r a t u s c o n s i s t s o f a h i g h power 34 GHz  t y p e 8TFK2 k l y s t r o n c a p a b l e o f 10 w a t t s o u t p u t , a n  Elliot isolator,  l o a d , and a s s o c i a t e d m i c r o w a v e e q u i p m e n t i n c l u d i n g a TE104 retangular cavity, power m e t e r , with f i e l d  a bolometer  a field  and H e w l e t t P a c k a r d M o d e l 430C  controlled  9.5  i n c h M a g n i o n magnet a l o n g  m o d u l a t i o n e q u i p m e n t , two m a r g i n a l o s c i l l a t o r s ,  a  Hewlett Packard frequency counter, a l o c k - i n detector. Chart r e c o r d e r and The  the a u x i l i a r y equipment needed f o r  a r r a n g e m e n t i s shown i n t h e b l o c k d i a g r a m  stabilization.  (Figure 2).  The  k l y s t r o n i s w a t e r c o o l e d p r o v i d i n g enough s t a b i l i t y so  that  it  an  c o u l d be r u n s a t i s f a c t o r i l y w i t h o u t a n a . f . c . a f t e r  initial  warm up p e r i o d o f h a l f an h o u r .  The  klystron could  n o t be s w e p t t h r o u g h a n e n t i r e mode and d i s p l a y e d on  the  o s c i l l i s c o p e a s t h i s i n v o l v e d m o d u l a t i n g t h e beam v o l t a g e s u p p l y by a b o u t n o t u s e d and  150 v o l t s .  In p r a c t i c e  t h e k l y s t r o n was  then the a . f . c .  allowed to d r i f t  accomplished  f r e q u e n c y was  with  obtained.  fine  The  F i r s t a rough  tuned u n t i l  This  s e t t i n g of  the  While  The  Then  t h e maximum s i g n a l s t r e n g t h  f r e q u e n c y c o u l d be f o u n d  the wavemeter. 13  Enga's t h e s i s .  steps.  This  o b t a i n e d t h r o u g h the c a l i b r a t e d wavemeter.  t h e k l y s t r o n was was  i n two  was  slightly.  meant r e t u n i n g t h e k l y s t r o n t o t h e c a v i t y f r e q u e n t l y . was  :  t o about  ten  coupler i s described i n detail  the s t r e n g t h of the f i e l d  t h r o u g h t h e c o u p l i n g t h e r e i s no c a l i b r a t i o n a n d  c a n be the  MHz in  controlled  settings  -14-  are not reproducible. described  i n Enga's t h e s i s and a l t h o u g h  deteriorated was  The c a v i t y u s e d was t h e same a s t h e s i l v e r p l a t i n g had  t h e Q was n o t s e r i o u s l y a f f e c t e d .  a l s o cooled through  an e x t e r n a l copper  the s i d e o f the c a v i t y .  The c a v i t y  tube a f f i x e d t o  D e t a i l s o f the c a v i t y and t e f l o n  s a m p l e h o l d e r s a r e shown i n f i g u r e 4. The s a m p l e c o i l depended on t h e f r e q u e n c y  d e s i r e d b u t c o i l s made o f 1 0 0  t u r n s o f no. 46 enameled c o p p e r MHz.  wire o s c i l l a t e d a t about  three  The o t h e r m i c r o w a v e e q u i p m e n t i s d e s c r i b e d i n d e t a i l  i n Enga's t h e s i s .  A schematic  o f t h e k l y s t r o n power  and  i t s connections  The  power s u p p l y i s n o i s y a n d d e t e c t i o n o f e . s . r . b y  difficult.  from  t h e microwave system  An a d d i t i o n a l d i f f i c u l t y  the k l y s t r o n frequency  t o the c a v i t y resonant  frequency  counter  i n u n i t was u s e d t o d e t e r m i n e e . s . r . resonance frequency  i nH . 0  a l o n g lie$f  The  t o match  frequency.  w i t h a 50-100 MHz p l u g  the exact frequencies f o r the  and t h e p r o t o n  resonance  Using the p l u g i n u n i t f r e q u e n c i e s both  a b o v e a n d b e l o w 50 MHz c a n be d e t e r m i n e d s e l e c t o r knob.  would  was t h a t t h e k l y s t r o n  mode c o u l d n o t be d i s p l a y e d t h u s m a k i n g i t d i f f i c u l t  A Hewlett Packard  supply  t o t h e k l y s t r o n i s shown i n f i g u r e 7.  m o n i t o r i n g power a b s o r b e d be  size  by c h a n g i n g t h e  T h i s a v o i d s t h e n e c e s s i t y o f two c o u n t e r s .  marginal o s c i l l a t o r  i s a s l i g h t l y m o d i f i e d form 14 of t h e one d e s c r i b e d b y B e n e d e k a n d K u s h i d a and V o l k o f f 15 (  et  al.  The 6 J 6 t u b e  the rEsonant  operates as a p u s h - p u l l o s c i l l a t o r  frequency  determined  by t h e s a m p l e c o i l a n d  with  -15-  butterfly  tuning capacitors.  i t o r s are adjusted  The  f o r marginal  g r i d - p l a t e feedback  oscillation.  energy i s absorbed from the sample c o i l  At  resonance,  changing i t s Q  hence c h a n g i n g the c u r r e n t d e l i v e r e d t o the c i r c u i t plate supply. modulation  This signal  c o i l s at w  lock-in detector. recorder.  i s m o d u l a t e d by  and  This  The  i s used to monitor  T h i s amounts t o o v e r  d.c.  m e t e r on  via  the  Helmholtz  i n t u r n i s connected to a  Packard  chart  It  provides frequency  .1 v o l t a t t h e f r e q u e n c i e s  the f r o n t of the m a r g i n a l  by  the  grid  on  wave f o r m .  s m a l l g r i d c a p a c i t o r s can  The  the p o s i t i v e h a l f of  u s e d t o a d j u s t t h e g r i d v o l t a g e and strength.  The  unmodified  b e t w e e n a b o u t 5 and  60 MHz.  was  coil  o f a b o u t 100  t u r n s and  grid  t r i m m e r o f 47  uencies  added t o the g r i d  A p a r a l l e l capacitance MHz.  MHz  i t was  To  o f 100  frequencies  capacitors.  o f 3 t o 3.5  MHz  was  With a with  could  pf would g i v e  limited  t o a few  t o use  two  spectrometers,  one  the be  freqthe  hundred  o b t a i n a l l d e s i r e d f r e q u e n c i e s f r o m 2.3  necessary  be  operating  However a t s u c h f r e q u e n c i e s  t u n i n g range of the o s c i l l a t o r kilocycles.  of  a capacitor in parallel  pf a frequency  clown t o 2.3  capable  the  field  For o p e r a t i o n at lower  extra capacitance  obtained.  h e n c e t h e d.c.  o s c i l l a t o r was  used.  oscillator  c u r r e n t a c r o s s a 51 K r e s i s t o r f i g u r e 3.  the  m o d e l HP5245 L  e s s e n t i a l l y measures the v o l t a g e developed  See  the  a m p l i f i e r connected to the f r e q u e n c y .  enough g a i n t o d r i v e the H e w l e t t counter.  the  and  i s fed v i a a c o a x i a l cable to  A t r a n s i s t o r i z e d audio  plate circuit  capac-  to 4  modified  -16-  anci one u n m o d i f i e d , a n d two s p e c t r o m e t e r t o s i g n a l c o i l lengths. of  When o p e r a t i n g p r o p e r l y t h e s i g n a l t o n o i s e  the o s c i l l a t o r  LiF.  was o v e r 100 f o r t h e n = l r e s o n a n c e  The p h y s i c a l c o n s t r u c t i o n o f t h e o s c i l l a t o r  i n Enga's t h e s i s , and a s c h e m a t i c diagram  cavity.  ratio line i n  i s discussed  i s g i v e n i n f i g u r e 3.  F i g u r e 6 c o n t a i n diagrams o f the c a v i t y c o u p l i n g and a T E 0 1 1 mode c y l i n d r i c a l  cable  mechanism  T h i s c a v i t y was n o t  u s e d i n t h e p r e s e n t e x p e r i m e n t s b u t was u s e d b y E n g a i n h i s o r i g i n a l experiments and so i s i n c l u d e d f o r completeness.  -17-  BLOCK  DIAGRAM  OF  THE  EXPERIMENTAL  APPARATUS  COUNTER  LOCK-IN AMPLIFIER  CHART RECORDER  MARGINAL OSCILLATOR  MARGINAL OSCILLATOR  4 0 0 HZ OSCILLATOR  BALANCED LINES ...  AUDIO A M ?  V  /  /  TE :  KLYSTRON POWER SUP.  9.5 BOLOMETER.  POWER METER  DIRECTIONAL COUPLER (lOdb)  r.  Vs  2  KLYSTRON AFC  --\7L. f  ATT  FIGURE  104  ^ S A M P L E HOLDER / MODULATION COILS  / / MAGNION MAGNET  3 4 GHZ CAVITY  WAVEMETER  3 PORT CIRCULATOR  XTAL  -18-  FIG.  3.  -19-  lev ion  Sample  Spectrometer Coup!u-iCj  HoltScr  Coil  Silver* Pjqttncj  Hols-  .071 Diet. W/G  Fiance.  Soldered  S f l m pie.  onto  Water  Cooliriq  VIEW CUTAWAY  SIDE  Slot  for  introa'ucinj  $urrevs\<lm^ metal to  r  Jl  I*  •)—  1  _rn_  T  "I  rcac'iinecC  ilr\i$!tnes$  VIEW  TOP  .07}  .QQ'f  -.100  "\nc  Oia.  >.02S"  OIO  -#<S0  LUC ITS CAVITY  FIG.  PLASTIC !VAS  V-  FORM  WHICH  S A M P L E . B0.B51N  ELECTROPLATED  t£  0/'}  sample.  iM0D£  CAVITY  (RECTAWG-ULAR)  O n II  (TcRON)  cuTavwy  VKW  saowm  FIG 6. ' VARIABLE  ggass jtsms  COUPLING  in' vn-zz  DEVICE  vilo  -22BLOCK DIAGRAM OF KLYSTRON SUPPLY current operated r e l a y EIIT SUPPLY r n  cavity  collector  connected toghter i f a . f . c . not used  heater  FIGURE 7  -23•4. EXPERIMENTAL PROCEDURE AND RESULTS  The e x p e r i m e n t s w e r e c a r r i e d o u t a t room t e m p e r a t u r e i n all  cases.  The s t a t i c m a g n e t i c  f i e l d H was modulated  Hz. and a l o c k - i n d e t e c t i o n system were o b t a i n e d by s e t t i n g  was e m p l o y e d .  the k l y s t r o n at a f i x e d  corresponding t o t h e c a v i t y resonance oscillator  f o r the frequency which  then s l o w l y sweeping  a t 400  D  The s i g n a l s frequency  and s e t t i n g t h e m a r g i n a l  gave t h e d e s i r e d E  H^via the calibrated  Sif  and  sweep power s u p p l y .  T y p i c a l sweep s p e e d s were o f t h e o r d e r o f a f e w g a u s s p e r minute. recorder.  The d e r i v a t i v e s i g n a l s were d i s p l a y e d on a c h a r t P r e l i m i n a r y r u n s were f i r s t  g r e a t e r t h a n 6 MHz.  F o r these runs l i n e s s i m i l a r t o those  f o u n d b y E n g a were o b s e r v e d . taken a t these frequencies.  FIG.  9  made a t f r e q u e n c i e s  F i g u r e 8 shows t y p i c a l  traces  -24Fig.  9 shows t h e q u a l i t a t i v e e f f e c t o f i n c r e a s i n g  i n c r e a s i n g the c o u p l i n g t o the c a v i t y . 3.58 MHz  through  A l l t h e s e r u n s were a t  a n d a p p r o x i m a t e l y 10 t o 14|iA o f  which i n d i c a t e s the r f f i e l d  H,through  o s c i l l a t o r meter r e a d i n g  s t r e n g t h . . H , i n c r e a s e s from (a)  ( d ) b u t no a c c u r a t e q u a n t i t a t i v e v a l u e s f o r H w e r e  obtained f o r these  (  runs.  -25-  In f i g . 1 0 a l l the s i g n a l s were o b t a i n e d 2.6 M H z .  a t a frequency  The c u r r e n t r e a d i n g r e f l e c t s the r e l a t i v e  of H ^ b u t H , i s not c a l i b r a t e d .  Figure  ( b ) , however, has the  l a r g e s t value o f H . .  (c)  1 2 (iA  FIGURE 10  (  strength  d  )  2  0 jiA  -26-  The  magnetic f i e l d  i n a r y runs but r u n s i t was  apparent that at these s t r e n g t h was  the u s u a l f i r s t  treatment  the e v o l u t i o n of the  is  s o l v e d e x a c t l y by  H .  Obviously  If H^is  =  a linearly  C  V,  oscillator  frequencies  s m a l l compared t o  oscillating  field  i s rotating 2Kcos;jt  COSCO<  -  0  l a r g e compared t o K i t can (  be  one  be  toward e f f e c t i n g a  t r e a t e d as a r o t a t i n g f i e l d .  c o n d i t i o n f o r r e s o n a n c e i s p o s s i b l e and  when U = i ^ a n d if f  frequency.  2iriJ  vf  corresponds  i s almost always used.  low  was  In t h i s  resonance.  oscillating  this  precession field  linearly for  strengths.  t o n e g l e c t one  only  occurs  oscillating  experiment the  high r f f i e l d  longer proper  components of the f i e l d .  one  comparable i n s t r e n g t h to  f r e q u e n c i e s o f H r f and  c a s e i t i s no  fields.  Doing t h i s ,  t o the Larmor  In a c t u a l p r a c t i c e a l i n e a r l y  oscillating rf field  can  shown t h a t o n l y  d  may  about  StJVb/t  H,  H e n c e when H j i s s m a l l c o m p a r e d t o I I a l i n e a r l y field  for  coiaponents:  o p p o s i t e l y r o t a t i n g magnetic  r o t a t i n g component i s i m p o r t a n t  rf  He^.  of resonance the e q u a t i o n  perpendicular r f f i e l d  d e c o m p o s e d i n t o two  HL  prelim-  i n subsequent  s e p a r a t i n g H i n t o the v e c t o r  i m p l i e s t h a t the  be  i n these  magnetization,  This 0  not c a l i b r a t e d  from the c a l i b r a t i o n o b t a i n e d  the microwave f i e l d In  was  of the  In  this  rotating  I n f a c t an a n a l y s i s o f t h e  fields  -27-  pi^esent i n t h e d o u b l y r o t a t i n g frame the  axis defined  by  n^^ ), 5  shows t h a t t h e r e  c o n d i t i o n s t h a n t h e one g i v e n  above.  shows t h a t f o r a r f m a g n e t i c f i e l d in  a l l of the coordinate  U=WJ  6  where  GJ  0  .  are other  w i t h non z e r o  components  d i r e c t i o n s resonant conditions e x i s t f o r frequency i n the r o t a t i n g  The c h a n c e s o f o b s e r v i n g  these  higher  resonances i s g r e a t l y enhanced as t h e r f f i e l d  becomes c o m p a r a b l e t o t h e s t a t i c  field.  resonances f o re l e c t r o n s p i n resonance. to achieve  an  large relative  oscillator  "these  higher  to H ^j , either require e  s m a l l by u s i n g a l o w e r r f f r e q u e n c y .  order  T h e r e a r e tv/o ways  produce a l a r g e o s c i l l a t i n g  because t h e s e n s i t i v i t y  strength  Thus t h e r o t a t i n g  f r a m e p o s s i b l y a f f o r d s t h e o n l y means t o s t u d y  the  resonance 16  Y/inter's a n a l y s i s  i s t h e Larmor p r e c e s s i o n  frame about order  ( t h e one r o t a t i n g a b o u t  field  The l a t t e r  of the o s c i l l a t o r  that  o r make Heyy i s better  i s greatest  f o r low  f i e l d strengths. F o r t h e s e r e a s o n s i t was d e c i d e d t o make r u n s at as low as p r a c t i c a l r f f r e q u e n c i e s . A perturbation 17 calculation  shows t h a t i f t h e e f f e c t i s s m a l l t h e n=2  resonance s i g n a l strength the  n=3 r e s o n a n c e s h o u l d  n=l  resonance strength Figure  should  is  i s proportional to  while  Of c o u r s e t h e  .  11 shows t h e e f f e c t o f i n c r e a s i n g I I j w h i l e v  the frequency being  the other  2.6 MHz.  12 shows t h e same e x p e r i m e n t b u t w i t h f ^ = 3 . 0  obvious from these  order  ri  be p r o p o r t i o n a l t o H£.  parameters are hold constant, Figure  be p r o p o r t i o n a l t o ll  MHz.  It  r e s u l t s that the strength of the higher  r e s o n a n c e s i s h i g h l y s e n s i t i v e t o t h e s t r e n g t h o f II .  -28-  A l s o f o r t h e same r f f i e l d order  strength the strength of the higher  resonances i n c r e a s e s as  a s d e c r e a s i n g H^^. . e f f e c t of H  T i  decreases.  T h i s i s t h e same  A semi-quantitative d e s c r i p t i o n of the  s t r e n g t h on t h e r e l a t i v e  signal  made b y n o t i n g t h a t f o r a L o r e n t z i a n l i n e area under the a b s o r p t i o n curve  s t r e n g t h s was  shape t h e i n t e g r a t e d  i s p r o p o r t i o n a l t o t h e peak  h e i g h t and t h e d e r i v a t i v e peak h e i g h t a s s u m i n g t h a t t h e l i n e width  i s constant.  T h i s i s a good a p p r o x i m a t i o n  of the magnetic f i e l d s used i n t h i s shows t h e r e s u l t s o f t h i s 3.0 MHz.  measurement y  of H ^the r a t i o Y  on t h e o s c i l l a t o r  run at  s t r e n g t h s o f from 18  meter.  o f peak h e i g h t 2 - t o - l ,  F o r each value  3-to-l,  O n l y a t t h e two h i g h e s t v a l u e s o f H  be made f o r t h e r a t i o  F i g u r e 13  f o ra typical  S i x t r a c e s were made f o r H ^ f i e l d  7 t o 3 0 |iA a s r e a d  measured.  experiment.  f o r the ranges  o f p e a k 4 t o peak 1.  a n d 4 - t o - l was t j  could  The peak  measurements heights  c o u l d n o t be c o m p a r e d b e t w e e n one v a l u e o f I-L , and. a n o t h e r because the s e n s i t i v i t y  of the o s c i l l a t o r  function of the o s c i l l a t o r the  l o gl o gplot  i s some unknown  signal strength.  From t h e s l o p e o f  i t i s seen t h a t :  i. Within the l i m i t s  This  of error  then,  i s what t h e p e r t u r b a t i o n t h e o r y f o r low H * w o u l d p r e d i c t ,  although  v  perturbation theory  i s definitely  not a p p l i c a b l e as  -3  (c) 30 FIGURE  12  -32-  The n e x t for of  three f i g u r e s  the equations "77?''' . o Ho  time  r.  lines.  (14-16J  developed  show n u m e r i c a l s o l u t i o n s  i n the theory.  They a r e p l o t s  F i g u r e 14 shows t h e e f f e c t o f c h a n g i n g  S h o r t e n i n g T has the e f f e c t o f broadening However a s w i l l  due t o T a n d t h a t due t o  o t h e r e f f e c t s must be d i s t i n g u i s h e d i f t h e p r o p e r i n t h i s manner.  due t o t h e f a c t  It will  d i f f e r e n t from u n i t y .  broadening reson-  o n l y when c o s 0 i s s i g n i f i c a n t l y  Therefore  v a l u e o f T s h o u l d be l i m i t e d  to  be s e e n t h a t t h e  r i s t o be  t h a t we a r e o b s e r v i n g a r o t a t i n g f r a m e  ance i n the l a b i s i m p o r t a n t  and  the resonance  be shown o t h e r p a r a m e t e r s c a n h a v e  s i r a i l a r e f f e c t s and t h e b r o a d e n i n g  found  the r e l a x a t i o n  the search f o r the c o r r e c t  to frequencies higher  t h a n 3.5  MHz  l o w f i e l d s t r e n g t h s . The r e s u l t i s t h a t t h e b e s t r was f o u n d -7 -7 be 1.5x10 seconds. T h i s i s c o m p a r a b l e t o 1.9x10 calculated  f r o m t h e p e a k t o peak d e r i v a t i v e  l i n e width. -7  c o u l d be due t o t h e f a c t t h a t t h e 1.9x10 d e r i v e d from the u n s a t u r a t e d l o w power k l y s t r o n .  discrepency  second f i g u r e  was  resonance l i n e o b t a i n e d from a  A l l t h e s i g n a l s were o b t a i n e d i n t h e  r o t a t i n g frame w i t h H high f i e l d  The  r f  a s the p e r t u r b i n g f i e l d .  strengths of H  The k l y s t r o n power w i l l  r f  s h o u l d induce  not i t s e l f  The r a t h e r  significant  produce  saturation.  significant  s a t u r a t i o n s i n c e a l l o f t h e s i g n a l s were o b t a i n e d a t l e a s t 2 MHz  o r about. 3 h a l f w i d t h s f r o m t h e k l y s t r o n r e s o n a n t 19 The h a l f w i d t h a t r e s o n a n c e i s g i v e n b y : w h e r e ^/£is t h e u n s a t u r a t e d -7 and  a T o f 1.5x10  linewidth.  For the y of the e l e c t r o n  s e c . a n d H ^.of .5 g a u s s , r  frequency.  ^ ' - j  f£y  -33-  1.9 The o b s e r v e d d i f f e r e n c e , 1 . 5 g i v e s o n l y Ay =/*2fA'/ z  argument e x p l a i n s t h e e f f e c t Figure other  , but the  x  qualitatively.  15 shows t h e e f f e c t o f i n c r e a s i n g H , w h i l e t h e  parameters are held constant,  shows t h e e f f e c t o f i n c r e a s i n g H ,.  and f i n a l l y ,  f i g u r e 16  I n some r e s p e c t  the e f f e c t s  o f i n c r e a s i n g H , o r H ^ a r e q u i t e s i m i l a r and t h e d i f f e r e n c e s r  rather subtle. be s i m i l a r .  I t i s q u i t e e a s y t o s e e why  O n l y t h e component o f Yi  rf  effective  i n causing  H  From f i g u r e 1, s i n e = H , / H  t f  sin0.  resonant  perpendicular  absorption. e  c  kept constant.  , so the p e r t u r b i n g rf  e  H,effects  absorptive  experimental  H  ri  H ,  and t h i s  component  i s kept constant.  resonance l i n e s .  Of  H ^ h a s no s u c h  effect.  17 and 18 show a c o m p a r i s o n o f t h e t h e o r e t i c a l a n d curves  f o r some t y p i c a l r u n s .  the t h e o r e t i c a l curves t o the experiment.  while  however t h e r e i s  t h e a n g l e e and t h u s t h e s e p a r a t i o n o f t h e  and e m i s s i v e  Figures  fit  along  accounts f o r the d i f f e r e n c e s i n the lineshape  o b s e r v e d e v e n when t h e p r o d u c t  in  field  due t o H, and II,. ^  w o u l d be i n d i s t i n g u i s h a b l e . E x p e r i m e n t a l l y a l w a y s a component o f H ^ i n t h e x-y p l a n e  is  which i s  T h u s i f t h e o n l y component o f H ^ w e r e  t h e z a x i s t h e e f f e c t s on t h e peak h e i g h t  course  to H  T h i s component i s  H /H # w h e r e H ^ i s d e t e r m i n e d b y u  is actually  probably  the e f f e c t s should  the other  The  parameters  have been c h o s e n t o g i v e the b e s t  Figure  17 shows r u n s a t two  parameters are held constant.  i s c l e a r l y a change i n r e l a t i v e peak h e i g h t s .  frequencies  The m a i n e f f e c t The  parameters  THE  9=  3.S  FIGURE  THEORETICAL  rfc  14  E F F E C T OF  CHANGING  T  -39-  used f o r the The  FIGURE 16  (c)  theoretical  plots  experimental  Hrf=1.0g are g i v e n beside  parameters agree w i t h i n  F i g u r e 18 shows t h r e e c u r v e s t h e same f r e q u e n c y  (  traces,  i.e.,where  and U n v a r i e d .  Again  broader  and  show l e s s  the e x p e r i m e n t a l  d e t a i l than  the  p o s s i b l e e x p l a n a t i o n i s t h a t the treatment with neutron  i r r a d i a t i o n has  of l i t h i u m metal at  34GHz i s  field  also.  the s i z e  of L i F  The  over  I f the above e s t i m a t e of s i z e  i s actually  say  appear A  crystals  skin  the l i t h i u m  the  ones.  of producing 20  T h i s means t h a t t h e  s t r e n g t h c o u l d v a r y by 3 0 %  lp. i n d i a m e t e r .  varies  the e f f e c t  of about l j i i n diameter.  1 micron  curves  theoretical  the  values.  i s t h a t near the c e n t e r of  (H ~ ^ ) = 0 ,  error.  approximately  e x p e r i m e n t a l v a l u e s agree w e l l w i t h the t h e o r e t i c a l A s l i g h t discrepancy  figures".  experimental  w h i c h were r u n a t  (2.6MHz) w h i l e H  the  platelets depth  microwave particle  i s wrong  of  and  . l>i i n d i a m e t e r , t h e f i e l d s t r e n g t h 8 by o n l y 1 p a r t i n 10 • A c c e p t i n g t h e v a l u e q u o t e d  THEORY  FIGURE  17  EXPERIMENT  THEORY  f=2.60MHz -6 X.=lxl0 -7 r=I.5xl0 sec,  Hi  gauss  A4 gauss  FIGURE  IS  = .40g  Kri =.83g H i =.60g  H =1.03g H, =.75g r i  a b o v e o f l p may a c c o u n t  f o r m i n o r d i s c r e p a n c i e s o f o u r theox'y  and  b e c a u s e n e a r H~u/y-0  experimental  curves  a v a r i a t i o n i n Hj  c a u s e s a p p r e c i a b l e s h i f t i n g o f a b s o r p t i o n peak p o s i t i o n s w h i c h w o u l d f o r a n i n h o m o g e n e o u s H, a c r o s s t h e s a m p l e l e a d t o a broadening  of the l i n e s r e l a t i v e  t o those  o b t a i n e d by t h e  theory assuming a constant II, over  the ' p a i - t i c l e  While  be l i n e  i ti s true that there should  the p o i n t where t h e k l y s t r o n Larmor f r e q u e n c y ,  H —~-0 0  t  i s resonant  size.  broadening  near  with the e l e c t r o n  due t o t h e s a t u r a t i o n o f t h e s p i n  f  o  s y s t e m w i t h t h e k l y s t r o n p o w e r , t h i s e f f e c t s h o u l d be i n c o r porated  i n our Bloch equation  f o r m u l a t i o n a n d l e a d t o no  d i s c r e p a n c i e s 'between t h e t h e o r e t i c a l a n d e x p e r i m e n t a l Figure  19 s h o w s t h e n = l r e s o n a n c e l i n e r u n a t a b o u t  MHz a n d f o r f o u r v a l u e s o f H . be  changed enough t o observe  and  curves.  p e a k p o s i t i o n o n H,.-  E x p e r i m e n t a l l y II , c o u l d n o t  the e f f e c t of the l i n e  shape  This i s simply accomplished  the computer program however.  At the frequency  with  chosen  i s  a b o u t 4.5 g a u s s a n d v i s i n g v a l u e s o f H , o f l , 2 , 3 , a n d 4 g a u s s , angles that  of  u p t o CO d e g r e e s c a n be o b t a i n e d .  t h e l i n e w i d t h and peak p o s i t i o n  d e r i v a t i v e zero) a r e both 20 shows f o u r c u r v e s .  I t i s noticed  (determined  by t h e  s t r o n g l y d e p e n d e n t on Q .  The c u r v e  denoted "computer" i s the  peak t o peak l i n e w i d t h a s a f u n c t i o n o f H , d e t e r m i n e d the computer p l o t . expected  Figure  The s e c o n d shows t h e p e a k t o peak  from width  on t h e b a s i s o f a c o n s t a n t 'width i n t h e r o t a t i n g  The  third  curve  i s fitted  t o natch  is:  Y~5/8X~.£0 w h e r e Y i s t h e v a l u e  the second curve.  frame.  I t s 'formula  f o r a p o i n t on c u r v e  COMPARISON OF D I F F E R E N T PREDICTIONS OF THE L I N E WIDTH AS MEASURED I N THE LAB-. FRAME  30 Fz&.  2 0  3o  10  SO  €0  70  t h r e e and  X i s a v a l u e on c u r v e two. The f o u r t h c u r v e i s a . 2 "'. . (ScosO^l} w h i c h shows t h e d e p e n d e n c y w h i c h one cos o  p l o t of  would expect While  i f t h e R e d f i e l d t h e o r y were s t r i c t l y a p p l i c a b l e .  the v a l u e s o f H , used i n the experiments'were not  enough t o c o m p l e t e l y  determine  t h i s dependency, f i g u r e  large 9 shows  t h a t t h e l i n e w i d t h d i d i n c r e a s e r o u g h l y as t h e a s s u m p t i o n a constant width From f i g u r e  i n the r o t a t i o n frame i n d i c a t e d  i t would.  1 t h e s e p a r a t i o n o f t h e two  resonance  n-1 *  lines this  i n t h e l a b f r a m e s h o u l d be H ^ i s the formula  results  i t i s i n t e r e s t i n g t o see  20  f o r cose=  on  The  .975,  The  .895,  .745,  and  experimental  .460  and  times cos6.  t h e s e p a r a t i o n a t 0-0.  1.10  from this i s  The  separation  i s a valid  one  However f o r s m a l l the c o n c l u s i o n i s  t o c a l c u l a t e H ,.  Figure  comparison. line  s e p a r a t i o n i s a l s o a f f e c t e d by  Siegart effect.  The  the B l o c h  e x p r e s s i o n f o r the p o s i t i o n of  r e s o n a n c e i n t e r m s o f t h e p o s i t i o n where r e s o n a n c e occur  Since  s e p a r a t i o n i s computed  v a l u e s o f G t h e a g r e e m e n t i s good and  shows . t h i s  - H , ).  the t h e o r e t i c a l p l o t does not f o l l o w BcosG  e x a c t l y where B-  that equation  (?Jf  '/z  i f i t holds f o r  c o i a p a r e d w i t h t h e s e p a r a t i o n a t 0=0 of the l i n e s  cose=(II  >  used t o o b t a i n H , from the  the t h e o r e t i c a l p l o t s . figure  of  n e g l e c t i n g the e f f e c t  the would  i s g i v e n 21 by_.  •---•(' Taking  H  r  *  =11  I T  a s h i f t o f a b o u t 6%. and  another  are  as a t y n i c a l The  example  u  ~ 7€  ^°  This i s  u n c e r t a i n t i e s i n H b e t w e e n one  greater than  r  this  c a t i o n of the B l o c h - S i e g a r t e f f e c t  s o no e x p e r i m e n t a l was  made.  run  verifi-  21  SEPARATION I N THE L A B . FRAME OF THE TWO N = l RESONANCE L I N E S AS A FUNCTION OF 0.  .?  .g  .?  ( a ) AS FOUND FROM THE THEORY  DEVELOPED  I N THE T E X T , AND ( b )  .6  ,i  C 0 3 9 FOR COMPARISON  -475.  THE MEASUREMENT  It  was  magnetic ,H,  is  necessary  fields  d i f f i c u l t  degree  of  accuracy  independent is  to  reflected Then  the  power  and  the  the  from  the  it  f i e l d  dissipated  of  would  in  the  The  can  be  be  of  power and  f i e l d  the in  equation and  to  the  the  any  relating  of some  the  make  way  to  an do  the  cavity.  cavity  at  to  cavity,  Q of  the  and  best  the  strength  cavity,  measurement  curves  to  various  determined  The  H,.  FIELDS  the  desireable  distribution  Q an  calibrate  experimental  cavity  approximate  definition  It  incident  the  MAGNETIC  experiment.  measurement  measure  the  measure the  from  direct  knowing  From  in  However  power  calculate  to  experimentally.  1.10.  equation  this  used  OF THE RELEVANT  one  can  given  the  magnetic  Q,  point. the  f i e l d  22 distribution  in  The  c r i t i c a l  measurement  and  reflected  attempted ment The  with  meter main  jected about  to 10  couplers factor  but  the  cavity  power a  to  watts  were 4 o f 10.  is  the  Hewlett  a  is  few  of  thus  Packard  that  power. to  found.  a  measure These  the  reflections  the  and  made  this  could  while  klystron  the  from  the  two  lOdb  incident the  incident  power  bolometer  Therefore reduce  of  measurements  bolometer  d i f f i c u l t i e s  milliwatts  u t i l i z e d The  be  cavity.  experimental  d i f f i c u l t y over  can  were  measure-  inaccurate.  not  be  sub-  produced  directional power  couplings  by  and  a  - 4 8 -  discontinuities  i n the guide  i m p o s s i b l e t o determine  s y s t e m made i t d i f f i c u l t  i f not  t h e d i f f e r e n c e o f t h e i n c i d e n t and  r e f l e c t e d power t o t h e c a v i t y w i t h s u f f i c i e n t  accuracy.  A l s o i t was n o t p o s s i b l e t o l o c k t h e k l y s t r o n t o t h e c a v i t y a n d .as t h e k l y s t r o n d r i f t s s l i g h t l y f r o m t h e c a v i t y frequency  the f i e l d  significantly.  resonant  strength i n the c a v i t y can f l u x u a t e  I t i s f o r these reasons  t h a t no a c c u r a t e  e x p e r i m e n t a l v a l u e c a n be g i v e n t o t h e H , f i e l d s w h i c h i s independent of the l i n e  s e p a r a t i o n measurements.  The  r e s u l t s o f t h e power m e t e r m e a s u r e m e n t was t h a t a maximum of 1 t o 2 watts The  o f power v/ere d i s s i p a t e d  l o a d e d Q was m e a s u r e d t o be 1600 a n d a c a l c u l a t i o n  t h a t t h e maximum f i e l d times  strength i n the c a v i t y  the average f i e l d .  probably  experiments  f  f i e l d s measured i n  w e r e a b o u t one g a u s s .  This i s  due t o t h e f a c t t h a t t h e s a m p l e g r e a t l y r e d u c e s t h e  Q a n d most o f t h e power i s d i s s i p a t e d the cavity.  field  i s about 1 . 4 3  T h i s i s i n agreement w i t h t h e  v a l u e s o b t a i n e d by E n g a b u t t h e l a r g e s t H the present  shows  These v a l u e s g i v e w i t h our e q u a t i o n  a b o v e a maximum H = 3 . 2 g a u s s .  in  i n the c a v i t y .  i n t h e sample and not .  From e q u a t i o n 5.1 i t i s s e e n t h a t t h e a v e r a g e  s t r e n g t h i s p r o p o r t i o n a l t o t h e square r o o t of Q f o r  a g i v e n power  dissipated.  A determination of the f i e l d indirectly.  In the spectrometer  strength H  was made  c i r c u i t t h e r e a r e two 51 K  r e s i s t o r s which a r e i n s e r i e s and p l a c e d i n p a r a l l e l spectrometer  coil.  w i t h the  Hence t h e v o l t a g e a c r o s s t h e r e s i s t o r s  i n d i c a t e s the voltage across  the c o i l .  made t o m e a s u r e t h e v o l t a g e a c r o s s  I f a n - a t t e m p t was  the c o i l  d i r e c t l y with  t h e o s c i l l i s c o p e t h e o s c i l l a t o r was i n t e r r u p t e d and c e a s e d t o oscillate.  However t h e v o l t a g e c o u l d be m e a s u r e d a c r o s s one  o f t h e r e s i s t o r s on a f a s t r i s e t i m e o s c i l l i s c o p e . seemed t o be a p u r e s i n e wave w i t h v e r y content ular to  harmonic  over the s t r e n g t h s and f r e q u e n c i e s used.  interest  i s that the voltage varied  the g r i d c u r r e n t as read  22 i s a p l o t on  little  of the induced  the o s c i l l a t o r meter.  operating a t approximately  FIG.  The f o r m  i n direct  Of p a r t i c proportion  on t h e o s c i l l a t o r m e t e r . coil  Figure  voltage vs the current  T h i s d a t a was f o r a 100 t u r n  read coil  3.5 MHz.  « 1 3 VOLTAGE ACROSS C O I L I N VOLTS  H  S~  -50-  Assuming t h a t the sample i s c o n f i n e d the f i e l d  p r o d u c e d by t h e c o i l  t o a p o i n t on t h e a x i s o f  and t h a t t h e r e a r e n e g l i g i b l e  s h i e l d i n g e f f e c t s due t o t h e c a v i t y , t h e f i e l d s a m p l e c a n be c a l c u l a t e d k n o w i n g t h e v o l t a g e i t and the f r e q u e n c y o f the o s c i l l a t o r .  strength a t the  induced  across  A sketch of the  calculation follows.  6=£3 c o n . 2 where B i s i n Webers p e r m e t e r a i s i n m e t e r s and I i s i n amps, z i s t h e d i s t a n c e o f the sample from the c e n t e r o f the c o i l .  Then  Where B i s i n g a u s s E i s i n v o l t s and f i s i n m e g a c y c l e s .  Using  t h i s c a l c u l a t i o n and f i g u r e  frequency  22 i t i s s e e n t h a t f o r a  o f 3.5 MHz, H ^ c o u l d be v a r i e d f r o m a b o u t v  .4 t o  1.3 g a u s s when t h e g r i d c u r r e n t r e a d i n g v a r i e d f r o m 7 t o 3 0 |iA.  This corresponds  as f i t t e d  c l o s e l y with the experimental  by t h e c o m p u t e r p r o g r a m t o o u r t h e o r y .  magnetic f i e l d  H w a s s e t by a f i e l d  d r i v i n g t h e M a g n i o n magnet. on  i m a t e l y 100 g a u s s . was f o u n d on  However t h e f i e l d  t h e power s u p p l y was f o u n d  supply  dialed i n  t o be m i s c a l i b r a t e d by a p p r o x -  A l s o t h e sweep r a t e q u o t e d on t h e power s u p p l y  t o be i n e r r o r .  whether  The l a r g e  r e g u l a t e d power  n  results  In f a c t  t h e sweep r a t e d e p e n d e d  t h e sweep was u p o r down.  the w i d t h of the s i g n a l s .  T h i s was r e a l i z e d  from  S i n c e i t was d e s i r e d t o know t h e  s e p a r a t i o n o f t h e r e s o n a n c e s a c c u r a t e l y t h e sweep h a d t o be calibrated. separate  The way t h i s was done a c c u r a t e l y was t o u s e a  marginal  o s c i l l a t o r monitoring  a g l y c e r i n e ^ sample. MHz  This o s c i l l a t o r  the magnetization i n  was o p e r a t e d  a t a b o u t 55  f o r t h e p r o t o n r e s o n a n c e i n a 12 k i l o g a u s s f i e l d .  r a t e c o u l d be f o u n d  to three s i g n i f i c a n t figures  The sweep  t h i s way.  At  t h e 5 g a u s s p e r m i n u t e s e t t i n g t h e sweep u p was 3.49 g a u s s p e r m i n u t e w h i l e t h e r a t e down was 5.04 g a u s s p e r m i n u t e .  Actually  t h e r a t e was c a l i b r a t e d on t h e c h a r t p a p e r s o t h a t no e r r o r s owing t o a m i s c a l i b r a t i o n o f t h e c h a r t speed c o u l d e n t e r .  -52APPENDIX  T h i s a p p e n d i x i n c l u d e s a c o m p l e t e s t a t e m e n t o f t h e comp u t e r program  used t o s o l v e e q u a t i o n s ( 2 - 1 1 ) ~ ( 2 ~ 2 9 ) f o r Mf and  d / d t Mf" .' B e l o w 1  1  i s a brief  appear i n t h e program the  d e s c r i p t i o n o f t h e symbols  that follows.  which  I n l i n e one M r e f e r s t o  matrix of the c o e f f i c i e n t s of the magnetization.  The  d i m e n s i o n 50 i s w r i t t e n h e r e a s i t w a s n ' t a p p a r e n t i n t h e b e g i n n i n g t h a t a 3 6 x 3 6 w o u l d be a d e q u a t e . program  The m a j o r i t y o f t h e  i s the reading i n of the matrix elements.  A l l state-  ments o f t h e form M(a,b) r e f e r t o s p e c i f i c e l e m e n t s .  The  n u m b e r s w h i c h must be s u p p l i e d f o r t h e e q u a t i o n s a r e : N, t h e d i m e n s i o n o f t h e m a t r i x , w h i c h i n t h i s c a s e i s 3 6 , A, t h e maximum v a l u e o f (E -y 0  frequency f i e l d , field  ) , B, t h e a n g u l a r f r e q u e n c y o f t h e r a d i o  C l , the spin l a t t i c e  s t r e n g t h o f t h e microwave  field  r e l a x a t i o n t i m e , D, t h e R^, G, t h e g y r o m a g n e t i c  r a t i o o f t h e e l e c t r o n , GM, t h e r a d i o f r e q u e n c y f i e l d DA, t h e i n c r e m e n t i n ( H - y) e  strength,  w h i c h was t a k e n t o be .1 g a u s s ,  Dend i s t h e e n d o f t h e sv/eep r a n g e o f (H ~y) a  w h i c h was -2  g a u s s , GA i s t h e e l e c t i ' o n i c m a g n e t i c s u s c e p t i b i l i t y o f -6 l i t h i u m w h i c h I c h o s e t o be 1 x 1 0 t h e P a u l i p a r a m a g n e t i s m result.  Experimental evidence i n d i c a t e s t h i s i s low,the 23  t r u e v a l u e b e i n g two t o t h r e e t i a e s a s l a r g e . as d i s c u s s e d e a r l i e r  However  t h i s enters only as a s c a l i n g  factor  and hence o n l y t h e r e l a t i v e v a l u e i s o f i n t e r e s t h e r e . a c t u a l s o l u t i o n of the matrix i s c a r r i e d out i n a library  subroutine.  The d e r i v a t i v e  The  computer  i s o b t a i n e d by t a k i n g  -53-  d i f f e r e n c e s of M  f o r successive values o f ( H ~ u r 0  has the e f f e c t of s h i f t i n g gauss t o the r i g h t .  The  ),  This  t h e c e n t e r o f t h e ; t r a c e .05  program  includes instructions to  have t h e o u t p u t p l o t t e d , though i t does n o t i n c l u d e a s c a l i n g r o u t i n e and t h e a x e s must be a d j u s t e d t o f i t t h e particular  output.  .:,  -54-  ''ftJTIME 10 . : $PAGE 50 $ I B F I C MAIN DIMENSION M ( 5 0 , 5 0 ) , 0 ( 5 0 ) C A L L PLOTS REAL M NSOS =10 YMIN^2.5E-3 XMIN=-6. YL=10. XL=3. DY=.5E-3 READ 1 0 , N, A, B, C I , D 10 FORMAT ( 1 5 , 4D15.8) READ 2 0 , G, GM, DA, DEND 20 FORMAT ( 4 E 1 5 . 8 ) READ 2 1 , GA, G A I 21 FORMAT ( 2 E 1 5 . 5 ) PRINT 101,N,A,B,C1,D,G,GM,DA,DEND,GA,GAI DAD=A 101 F0RMAT(I5,5E16.8/5E16.8) NLL=(DEND-DAD J/DA+1.5 C A L L GRID (NLL,DAD,DEND,.5,101,-20.,20.,4.} DO 113 11=1,5 C A L L A X I S (0.,0.,6HX AXIS,-6,XL,0.,XMIH,DX) C A L L A X I S (0.,0.,6HY AXIS,6,YL,90.,YMIN,DY) A=DAD 100 DO 3 0 1=1,N DO 3 0 J=1,N M(I,J)=0.0 30 CONTINUE DO 4 0 1=1,N M(I,I)=1.0 40 CONTINUE M(7,3)=-1.0*B*C1 M(9,10)=M(7,8) M(11,12)=M(7,8) M(13,14)=M(7,8)*2.0 M(15,16)=M(13,14) M(17,18)=M(13,14) M(19,20)=M(7 8)*3.0 M(21,22)=M(19,20) M(23,24)=M(19,20) M(25,26)=M(7,8)*4.0 M(27,28)=M(25,26) M(29,30)=M(25,26) M(31,32)=M(7,S)*4.0 M(33,34)=M(31,32) M(35,36)=M(31,32) f  -55-  FACT=-1.0*G*C1*A NCUE=0 NST=-1 70 DO 50 1=1,N NCUE=NCUE+1 I F ( N C U E . L T . 3 ) GOTO 5 1 I F ( N C U E . L T . 6 ) GOTO 50 NCUE=0 ' GOTO 50 51 J=I-i-2 M(I,J)=FACT 50 CONTINUE IF(NST.GT.O) GOTO 60 NST=10 NCUE=4 FACT=1.G*C1*G*D GOTO 70 60 DO 80 1=1,N DO 80 K = l , 2 J=I+K 80 M ( J , I ) = M( I , J ) M(3,7}=2.0*C1*G*GM M(9,13)=M(3,7)/2.0 M(10,14)=M(9,13) M(15,19)=M(9,13) M(16,20)=M(9,13) M(22,26)=M(9,13> M(1,9)=-1.0*M(3,7) M(7,15)=M(l,9)/2.0 M(8,16)=M(7,15) M(13,21)=M(8,16) M ( 1 4 , 2 2 ) =41 ( 1 3 , 2 1 ) M(19,27)=M(14,22) M(20,28)=M(19,27j M(7,3)=M(20,28) . M(8,4)=M(7,3) M(13,9)=M(8,4) M(14,10)=M(13,9) M(19,15)-M(14,10) M(20,16)=M(19,15) M(25,21)=M(20,16) M(26,22)=M(25,21) M(9,1)=~1.0*M(26,22) M(10,2)=M(9,1J M(15,7)=M(10,2) M(16,S)=M(15,7) M(21,13)=M(16,8) M(22,14)=M(21,13) M(27,19)=M(22,14)  -56-  .  90  200  78  112 113 99  M(28,20)=M(27,19) M(25,33)=M(7,15) M(26,34)=M(25,33) M(31,27)=M(26,34) M(32,28)=M(31,27) M(27,31)=-1.0*M(32,28) M(28,32)=M(27,31) M(33,25)=M(28,32) M(34,26)=M(33,25) DO 90 1=1,N C(I)==0.0 C(2)=GAI*D C(5)=1.2E-2 C(6)=GAI*A C(11)«GA*GM IF(NSOS.GT.O) GO TO 200 NS0S=10 C A L L MATOUT (M,N,50) C A L L VECOUT (C,N) CONTINUE C A L L SOLTN(H,C,N,50,DET) CC=(C(12)-CC)*10. PRINT 78,A,C(12),CC,DET FORMAT ( 1 P 4 E 1 6 . 6 ) C A L L POINT (A,CC*l.E-!-4,II) XX=(A-XMIN)/DX YY=(CC-YMIN)/DY C A L L SYMBOL ( X X , Y Y , . 0 7 , 3 , 0 . , - 1 ) CC=C(12) A=A+DA. I F (A.GT.DEND) GO TO 112 GO TO 100 D=D+.25 C A L L PLOT (.1. 5*XL, 0. , -3 ) CONTINUE C A L L PLOTND C A L L OUTPUT STOP END  -57BIBLIOGRAPHY  1.  P a k e , G. , P a r a m a g n e t i c R e s o n a n c e New  York.  W.A.  Benjamin Inc.  1962  2.  R e d f i e l d , A.,  3.  F r a n z , J . , D o c t o r a l T h e s i s , U. o f I l l i n o i s , 1965 (unpublished) A b r a g a m , A., The P r i n c i p l e s o f N u c l e a r M a g n e t i s m (Oxford U n i v e r s i t y Press. L o n d o n , 1961") ~~  4-5.  P h y s . Rev.  P h y s . Rev.  9 8 , 1787  6-7.  R e d f i e l d , A.,  8.  E n g a , E., M a s t e r s T h e s i s , U,B C., 1966 ( u n p u b l i s h e d )  9.  R y t e r , Ch., P h y s . Rev. L e t t e r s ,  10.  G a r s t e n s , M.A.  and K a p l a n , J . I . , P h y s . Rev.  11.  W h i t f i e l d , G.,  and R e d f i e l d , A.G.,  12.  K o s s , T.A.,  (1955)  e  A l e x a n d e r , S.,  Can. J . P h y s . 13.  E n g a , E., op. c i t .  14.  Benedek,  15.  V o l k o f f , G.,  G.B.  3 0 , 270  P e t c h , H.,  (1960) 9 9 , 459  P h y s . Rev.  Ann.  (1957)  1968)  P h y s . Rev.  and S m e l l i e , D.,  de P h y s . 4, 745 a n d K o s s , T.A.,  118, 46  (1960)  Can. J . P h y s .  (1959) P h y s . Rev.  (1968)  19.  A b r a g a m , A.,  op. c i t .  20.  R y t e r , Ch., op. c i t .  21.  Abragam A.,  22.  P o r t i s , A.M.,  23.  S c h u m a c h e r , R.T. and S l i c h t e r , C., 1 0 1 , 58 ( 1 9 5 6 )  op. c i t . P h y s . Rev.  (1955)  106, 918  (1952)  S c h w e r d t f e g e r , C.F. 259  (August  10,  and S c h w e r d t f e g e r , C.F.  and K u s h i d a , T.,  1 6 - 1 7 . W i n t e r , J.M., 18.  9 8 , 1787  (1955)  9 1 , 1071  (1953) Phys.  Rev.  166,  

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