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A theoretical investigation of the nuclear resonance spectrum of B¹¹ in Kernite Rau, Jayaseetha Nittoor Sreenivasa 1958

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A THEORETICAL INVESTIGATION  OP THE NUCLEAR  RESONANCE SPECTRUM OF B 1 1 IN KERNITE  by JAYASEETHA NITTOOR SREENIVASA RAU  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the  Department of  Physica  We accept t h i s  t h e s i s as  conforming to  standard r e q u i r e d from candidates f o r  the the  degree o f MASTER OF SCIENCE  Members of the Department of THE UNIVERSITY OF BRITISH COLUMBIA May,  1958  Physics  ABSTRACT In t h i s t h e s i s a t h e o r e t i c a l study Is made of the resonance s p e c t r a  of the B 1 1 n u c l e i i n a s i n g l e  c r y s t a l of  K e r n i t e (Na2B40>7.4H20) whose, quadrupole moments i n t e r a c t w i t h the e l e c t r i c f i e l d g r a d i e n t at the n u c l e a r s i t e s i n the c r y s t a l i n the presence o f an e x t e r n a l magnetic f i e l d when the magnetic and quadrupole i n t e r a c t i o n e n e r g i e s are  comparable.  Using the r e s u l t s o f the h i g h f i e l d work o f Waterman and V o l k o f f ,  energy l e v e l s and t r a n s i t i o n frequencies have been  c a l c u l a t e d n u m e r i c a l l y u s i n g the U n i v e r s i t y o f B r i t i s h Columbia ALWAC-III-E d i g i t a l computer over a range of magnetic f i e l d s the r e g i o n i n t e r m e d i a t e  in  between the l i m i t i n g cases of pure  quadrupole energy l e v e l s s l i g h t l y p e r t u r b e d by magnetic i n t e r a c t i o n s and Zeeman energy l e v e l s s l i g h t l y p e r t u r b e d b y quadrupole i n t e r a c t i o n s .  The magnetic f i e l d was taken t o  be o r i e n t e d i n the p l a n e p e r p e n d i c u l a r to the t w o - f o l d symmetry a x i s o f the c r y s t a l and the c a l c u l a t i o n s were c a r r i e d out f o r d i f f e r e n t  o r i e n t a t i o n s o f the e x t e r n a l magnetic  field  i n steps of 3 0 ° . A g e n e r a l t h e o r y has been developed f o r the c a l c u l a t i o n of  signal intensities  i n the case of c r o s s e d c o i l  spectrometer  ( i n d u c t i o n method) and has been a p p l i e d to the p r e s e n t The s i g n a l i n t e n s i t i e s spectrometers  case.  f o r a b s o r p t i o n type  are p r o p o r t i o n a l ^ to the t r a n s i t i o n p r o b a b i l i t i e s ,  and r e s u l t s are g i v e n f o r the case of K e r n i t e which enables one to compare the expected s i g n a l i n t e n s i t i e s spectrometers w i t h those o f i n d u c t i o n  of  absorption  spectrometers.  In presenting the  thesis in partial fulfilment  r e q u i r e m e n t s f o r an  of B r i t i s h it  this  freely  advanced degree a t the  Columbia, I agree that  the  a v a i l a b l e f o r r e f e r e n c e and  agree t h a t p e r m i s s i o n f o r extensive t h e s i s f o r s c h o l a r l y p u r p o s e s may o f my  D e p a r t m e n t o r by  stood that financial  be  s h a l l not  be  study.  I  copying of  Department 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 &, Canada.  Columbia,  make  further this  g r a n t e d by  the  Head  I t i s underthesis  a l l o w e d w i t h o u t my  permission.  Date  shall  copying or p u b l i c a t i o n of t h i s gain  University  Library  his representative.  of  for  written  ii TABLE OP CONTENTS PAGE ACKNOWLEDGEMENTS  iv  ABSTRACT  v  CHAPTER I - INTRODUCTION  1  CHAPTER I I  7  CHAPTER I I I  - INFORMATION ON KERNITE - GENERAL HAMILTONIAN AND CASE OF KERNITE  10  CHAPTER IV - NUMERICAL RESULTS FOR ENERGY LEVELS AND TRANSITION FREQUENCIES  14  CHAPTER V - TRANSITION PROBABILITIES AND SIGNAL  CHAPTER VI  INTENSITIES  18  CONCLUSIONS  31  REFERENCES  32  TABLE 1 - D i r e c t i o n c o s i n e s of e l e c t r i c f i e l d g r a d i e n t p r i n c i p a l axes from h i g h f i e l d r e s u l t s of Waterman and V o l k o f f f o r E and F s i t e s .  13  TABLE I I - Asymmetry parameter ^ and quadrupole c o u p l i n g constant jeQ^>zz/h| from high f i e l d r e s u l t s .  15  TABLE I I I -  R e l a t i v e s i g n a l I n t e n s i t i e s i n the case where H Q a r * d H^ both are assumed to be p a r a l l e l to Y a x i s f o r a b s o r p t i o n type s p e c t r o m e t e r .  TABLE IV - R e l a t i v e s i g n a l i n t e n s i t i e s i n the case where H i i s assumed to be p a r a l l e l to Z a x i s and H 0 p a r a l l e l to Y a x i s , f o r a b s o r p t i o n type spectrometer. TABLE V - R e l a t i v e s i g n a l i n t e n s i t i e s i n the case where H i and H Q are assumed to be p a r a l l e l to Y a x i s and p i c k - u p c o i l p a r a l l e l to Z a x i s , f o r i n d u c t i o n type spectrometer.  28  29  30  iii  LIST OP ILLUSTRATIONS PACING PAGE Pig.  1 - A model of the u n i t  c e l l of K e r n i t e  Pig.  2 - Energy l e v e l s f o r  Pig.  3 - T r a n s i t i o n frequencies for ^ = 0 E-sites  Pig.  4 - Energy l e v e l s f o r  Pig.  5 - Energy l e v e l s  for  Pig.  6 - Energy l e v e l s f o r  Pig.  ^=0  ^  for E-sites for  8 14 14  = 3 0 ° for E-sites  14  y=  6 0 ° for E - s i t e s  14  Y3  9 0 ° for E-sites  14  7 - T r a n s i t i o n frequencies for f = 9 0 ° for E-sites  14  Pig.  8 - Energy l e v e l f o r f =  1 2 0 ° for E - s i t e s  14  Pig.  9 - Energy l e v e l f o r  1 5 0 ° for E - s i t e s  14  0 for P-sites  14  y=  P i g . 1 0 - Energy l e v e l f o r ^ =  P i g . 11 - T r a n s i t i o n f r e q u e n c i e s f o r ^ = 0 P-sites P i g . 1 2 - Energy l e v e l s f o r Y =  for  9 0 ° for F-sites  P i g . 1 3 - T r a n s i t i o n frequenctas f o r for P-sites  = 90°  P i g . 1 4 - Energy l e v e l s f o r E - s i t e s i n the case where the magnetic f i e l d i s a p p l i e d along the z p r i n c i p a l a x i s Pig.  15- Energy l e v e l f o r P - s i t e s i n the case where the magnetic f i e l d i s a p p l i e d along the z p r i n c i p a l a x i s  14 14 14  16  16  ACKNOWLEDGEMENTS  I would l i k e to express my s i n c e r e  thanks  to  D r . G . M . V o l k o f f f o r suggesting the problem and I am indebted to D r . M. Bloom f o r g u i d i n g t h i s r e s e a r c h and for  t a k i n g a constant  interest  i n the work.  T h i s work was made p o s s i b l e by the use of  the  e l e c t r o n i c computer at the U n i v e r s i t y of B r i t i s h Columbia. I gratefully  acknowledge the a s s i s t a n c e and  g i v e n by the s t a f f to thank D r . J .  of the computing c e n t r e .  cooperation I should l i k e  M. D a n i e l s and M r . J . R . H . Dempster  for  a c q u a i n t i n g me w i t h programming. F i n a l l y I would l i k e to thank the members of Department who through l e c t u r e s i n c r e a s e d my knowledge o f  and d i s c u s s i o n s  physics.  have  the  CHAPTER 1 INTRODUCTION  The p r o p e r t i e s were f i r s t  o f angular momentum and magnetic moment  a s c r i b e d to the nucleus by P a u l !  (1924) i n order t o  account f o r the h y p e r f i n e s t r u c t u r e o f atomic A  magnetic d i p o l e experiences  an inhomogeneous magnetic f i e l d .  spectra,  a f o r c e when p l a c e d i n  Atoms or molecules which  possess a magnetic moment are t h e r e f o r e d e f l e c t e d on p a s s i n g through such a f i e l d . i n the c l a s s i c  T h i s technique was used w i t h g r e a t  success  atomic beam experiments o f S t e r n and G e r l a c h t o  prove e x p e r i m e n t a l l y t h a t the measurable values of the components o f an atomic magnetic moment do not take on a continuous range of values;  i n s t e a d they form a d i s c r e t e set  c o r r e s p o n d i n g t o the  space q u a n t i z a t i o n o f the atom i n the magnetic f i e l d .  Prom the  magnitude o f the d e f l e c t i o n o f the beam the atomic magnetic moment can be e v a l u a t e d . The beam method was p o w e r f u l l y improved by R a b i and his  colleagues  by the i n t r o d u c t i o n o f the resonance method.  a d d i t i o n to the steady magnetic f i e l d *  In  the molecules o f the beam  are s u b j e c t e d to e l e c t r o m a g n e t i c r a d i a t i o n o f such a frequency as to induce t r a n s i t i o n s between t h e i r q u a n t i z e d energy l e v e l s by a process o f a b s o r p t i o n or s t i m u l a t e d e m i s s i o n o f quanta o f If  the n u c l e a r s p i n i s  magnetic f i e l d  I,  each energy l e v e l i s  energy.  s p l i t by the steady  i n t o 2 l + 1 e q u a l l y spaced sub l e v e l s .  If  the  2  maximum measurable  component o f the n u c l e a r magnetic moment Is  the s e p a r a t i o n between the lowest and the h i g h e s t sub l e v e l s 2 y H H i n a steady magnetic f i e l d H . s u c c e s s i v e sub l e v e l s  i s yHE/l.  is  The s e p a r a t i o n between  The frequency of the e l e c t r o  magnetic r a d i a t i o n whose quanta can e x c i t e t r a n s i t i o n s these sub l e v e l s i s thus equal to  y<H/lfi.  resonant exchange o f energy i n a g i v e n f i e l d  between  The frequency of t h i s i s found  e x p e r i m e n t a l l y by the sharp r e d u c t i o n i n the number of molecules r e a c h i n g the d e t e c t o r . The resonant exchange o f energy between the  21+1  energy l e v e l s o f a n u c l e a r magnetic moment i n a magnetic f i e l d  is  not r e s t r i c t e d t o matter i n the form o f molecular beams, but a l s o occurs i n o r d i n a r y s o l i d s , The f i r s t  l i q u i d s or g a s e s .  successful  n u c l e a r magnetic resonance  experiments u s i n g b u l k m a t e r i a l were c a r r i e d out i n d e p e n d e n t l y at the end o f 1945 by two g r o u p s .  B l o c h , Hansen and Packard (1)  measured i n d u c t i o n s i g n a l s from t r a n s i t i o n s between p r o t o n l e v e l s i n w a t e r , w h i l e P u r c e l l , T o r r e y and Pound (2) d e t e c t e d the a b s o r p t i o n of energy from a resonant e l e c t r o n i c c i r c u i t by p r o t o n t r a n s i t i o n i n p a r a f f i n wax. N u c l e i of s p i n I g r e a t e r quadrupole moments•  than l / 2 possess e l e c t r i c  The quadrupole moment i s a measure o f the  departure o f the n u c l e a r charge d i s t r i b u t i o n from s p h e r i c a l symmetry.  D e v i a t i o n s from an i n t e r v a l r u l e i n the spectrum of  Europium, observed i n 1935 were e x p l a i n e d by C a simir ( 3 ) , who suggested that the nucleus had a n o n - s p e r i c a l charge d i s t r i b u t i o n which would l e a d t o a quadrupole moment.  The i n t e r a c t i o n energy  3  of  the nucleus w i t h the e l e c t r o s t a t i c  f i e l d g r a d i e n t produced by  the surrounding e l e c t r o n c l o u d i s d i f f e r e n t orientations  for  different  of the n u c l e u s .  In many substances electrostatic  field  quadrupole e f f e c t are concerned. many c r y s t a l s ,  (e.g.  cubic crystals)  the  g r a d i e n t averages out to z e r o , so t h a t  can be n e g l e c t e d so f a r as the energy  the  levels  However, i n the case o f some m o l e c u l e s , and i n it  is possible  f o r a nucleus to be p l a c e d a t  p o s i t i o n where the e l e c t r o s t a t i c  a  f i e l d g r a d i e n t does not average  to z e r o , and the r e s u l t i n g i n t e r a c t i o n w i l l change the  energy  l e v e l s o f the n u c l e u s • In 1948 Bloembergen (4) r e p o r t e d e f f e c t s Interactions resonances  w i t h the d e u t e r o n .  which were s p l i t  interaction i n single  Pound (5)  into several  crystal,  of  quadrupole  found n u c l e a r magnetic  l i n e s by the quadrupole  i n 1951 Dehmelt and Kruger  (6)  found resonance due to t r a n s i t i o n s b e t w e e n n u c l e a r quadrupole energy l e v e l s , u s i n g the techniques o f n u c l e a r magnetic but w i t h no a p p l i e d magnetic  5  resonance  field.  I f we s u b j e c t a c r y s t a l i n which there i s  an e l e c t r i c  quadrupole i n t e r a c t i o n to an e x t e r n a l magnetic f i e l d , l e v e l s w i l l be p e r t u r b e d , and the s i n g l e  the  Zeeman  Zeeman l i n e w i l l be  split  i n t o many components. At one extreme we have the case where the magnetic I n t e r a c t i o n o f each d i p o l e w i t h an e x t e r n a l magnetic produces  Zeeman l e v e l s  separated b y e n e r g i e s  field  quite large  compared  to the s p l i t t i n g due to quadrupole i n t e r a c t i o n o f the nucleus w i t h  4  its is  surroundings.  In these i n s t a n c e s ,  taken as a p e r t u r b a t i o n on the  Kikuehi  the quadrupole i n t e r a c t i o n  Zeeman e f f e c t .  Carr and  (7) have obtained an e x p r e s s i o n f o r the t r a n s i t i o n  frequencies  t o the f i r s t  o r d e r i n the r a t i o  e  Q^zz.  pound ( 5 )  /tHo extended I t  to the t h i r d order i n case o f a x i a l l y symmetric  field,  and t h i s t h e o r y has been f u r t h e r extended t o the case o f asymmetric f i e l d  gradients  by Bersohn (8)  and V o l k o f f  et a l  At the other extreme, Dehmelt and Kruger (10) Investigated  (9).  have  t h e o r e t i c a l l y and e x p e r i m e n t a l l y the pure quadrupole  case and have o b t a i n e d i n f o r m a t i o n on the quadrupole c o u p l i n g constant  i n case of a x i a l l y symmetric c r y s t a l s .  The t h e o r y was  then extended to non symmetric c r y s t a l l i n e f i e l d g r a d i e n t by Kruger  (11)  i n which case the s p l i t t i n g s  depend on b o t h the  quadrupole c o u p l i n g constant and the degree of asymmetry of field.  Bersohn ('80  the  g i v e s a g e n e r a l e x p r e s s i o n f o r the p e r t u r b a t i o n  i n terms of an a x i a l asymmetry parameter up t o the f o u r t h o r d e r . The i n t r o d u c t i o n o f a weak e x t e r n a l magnetic f i e l d be t r e a t e d as a p e r t u r b a t i o n on the pure quadrupole Kruger (11)  obtained a f i r s t  magnetic f i e l d  discussed  the  strength. is  increased  p e r t u r b a t i o n t h e o r y breaks down where the quadrupole comparable w i t h magnetic I n t e r a c t i o n .  Investigations  levels•  order expansion i n terms of  As the e x t e r n a l magnetic f i e l d  energy i s  can  the Interaction  Experimental  on K e r n i t e have been made at the two extreme  above.  Waterman and V o l k o f f  resonance a b s o r p t i o n spectrum o f  B ^  (13)  cases  have I n v e s t i g a t e d  nuclei i n a single  the  crystal  5  o f K e r n i t e ( N a 2 B 4 0 7 , 4HgG) p l a c e d i n a l a r g e uniform e x t e r n a l magnetic f i e l d  and have d i s c u s s e d  p e r t u r b a t i o n o f the n u c l e a r  their result  i n terms o f  Zeeman energy l e v e l s by the  I n t e r a c t i o n o f the n u c l e a r e l e c t r i c quadrupole moment w i t h the crystalline electrostatic  f i e l d gradient.  I t was n e c e s s a r y to  go t o second order p e r t u r b a t i o n t h e o r y to get agreement experiment.  The work o f Hearing and V o l k o f f  with  (14) g i v e s i n f o r m a t i o n  about the resonance a b s o r p t i o n spectrum of the same n u c l e i  In  terms o f p e r t u r b a t i o n o f the n u c l e a r e l e c t r i c quadrupole energy l e v e l s by the magnetic i n t e r a c t i o n .  Not a l l the e x p e r i m e n t a l  r e s u l t s at low f i e l d are c o n s i s t e n t w i t h the h i g h f i e l d The p r e s e n t  experiment.  c a l c u l a t i o n s have been c a r r i e d out along the  same l i n e s as has been done p r e v i o u s l y f o r the case o f Spodumene (15 and 16)  In which ease the two extreme r e g i o n s a r e  linked  together and a complete knowledge o f the energy l e v e l s , f r e q u e n c i e s and i n t e n s i t i e s o f t r a n s i t i o n s  are o b t a i n e d by a  d i r e c t n u m e r i c a l s o l u t i o n o f the s e c u l a r d e t e r m i n a n t . work on Spodumene r e p r e s e n t s the f i r s t  the  The above  system i n which the n u c l e a r  resonance spectrum extending from the low f i e l d  to the h i g h  field  r e g i o n through the r e g i o n o f i n t e r m e d i a t e f i e l d s has been s t u d i e d c o m p l e t e l y b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y . complete agreement o f the experimental v a l u e s f r e q u e n c i e s w i t h the t h e o r e t i c a l c a l c u l a t i o n s .  There was  •  o f the t r a n s i t i o n The s i g n a l  i n t e n s i t i e s w i l l be d i s c u s s e d below. In t h i s t h e s i s we c a l c u l a t e the energy l e v e l s and t r a n s i t i o n frequencies  f o r B1"*" i n K e r n i t e over an i n t e r m e d i a t e  range of values o f magnetic f i e l d  showing the g r a d u a l  transition  6  from a pure quadrupole spectrum to the o t h e r extreme of a Zeeman l i n e s p l i t  i n t o a number o f c l o s e l y spaced  In a d d i t i o n , c a l c u l a t i o n s relative signal Intensities  spectrometers.  That the two types  measure d i f f e r e n t r e l a t i v e field  components.  have been done to p r e d i c t  as the magnetic f i e l d  b o t h a b s o r p t i o n and i n d u c t i o n ( e . g .  single  crossed  Is  varied for  c o i l ) types  of spectrometers  signal intensities  of  should  as a f u n c t i o n of  In systems i n v o l v i n g quadrupole i n t e r a c t i o n s arose out o f  Robinson's experiments on Spodumene, mentioned above, which were done u s i n g a c r o s s e d c o i l s p e c t r o m e t e r . developed t o e x p l a i n R o b i n s o n ' s r e s u l t s  A t h e o r y has been on i n t e n s i t i e s  (15,  17).  In Chapter V , a g e n e r a l i z a t i o n o f t h i s t h e o r y has been d e v e l o p e d , which Is a p p l i c a b l e to the case o f K e r n i t e .  It  is  felt  that  these c a l c u l a t i o n s w i l l a i d the experimenter i n choosing the type o f spectrometer  suitable  f o r i n v e s t i g a t i n g the  spectrum  in Kernlte. In Chapter I I o f t h i s t h e s i s the c r y s t a l s t r u c t u r e K e r n l t e and the h i g h f i e l d r e s u l t s discussed.  of Waterman and V o l k o f f  We d i s c u s s i n Chapter I I I  d i a g o n a l l z e d i n order to c a l c u l a t e i n K e r n l t e , while the r e s u l t s g i v e n i n Chapter I V .  of  are-  the H a m i l t o n i a n to be  energy l e v e l s  of B"^ n u c l e i  of numerical c a l c u l a t i o n s  are  CHAPTER  II  INFORMATION ON KERNITE  (a)  Crystallographlo The K e r n i t e C r y s t a l , Nag B4 O7, 4H 2 0 i s m o n o c l i n l c and  contains  four formula u n i t s ,  unit c e l l .  X-ray studies  and t h e r e f o r e  (18)  s i x t e e n - B ^ n u c l e i per  g i v e the i n f o r m a t i o n t h a t  starting  w i t h one formula u n i t a second may be o b t a i n e d by i n v e r s i o n i n a symmetry c e n t r e , w h i l e the other two may be obtained from t h e first  two b y a r o t a t i o n about a t w o - f o l d symmetry X c r y s t a l  I n v e r s i o n i n a symmetry centre guarantees that the sites consist electric  of eight p a i r s ,  f i e l d gradient  and magnitude•#  axis.  sixteen  the members o f each p a i r having  tensors i d e n t i c a l both i n o r i e n t a t i o n  The e x i s t e n c e of a t w o f o l d symmetry a x i s  further  guarantees t h a t the e i g h t p a i r s break up i n t o four groups of two pairs each.  One p a i r o f each group may d i f f e r  from the  other  p a i r o f the same group o n l y by the o r i e n t a t i o n o f the p r i n c i p a l axes o f the e l e c t r i c  f i e l d gradient  t e n s o r s , which are  symmetrically  i n c l i n e d w i t h r e s p e c t to the twofold r o t a t i o n a x i s o f the  crystal.  Thus among the s i x t e e n boron s i t e s i n a u n i t c e l l we may expect to f i n d a t most four n u m e r i c a l l y d i f f e r e n t w i t h each o f these t e n s o r s repeated orientation.  #  f i e l d gradient  i n a possibly  tensors,  different  I f the c r y s t a l symmetry a x i s i s h e l d e i t h e r  A more p r e c i s e d e s c r i p t i o n of the e l e c t r i c tensor i s postponed u n t i l Chapter I I I .  field  gradient  F i g u r e 1:  A three dimensional model o f the u n i t c e l l o f K e r n i t e ( N a 2 B 4 0 7 , 4H20). The brown b a l l s are b o r o n , the white are oxygen, the yellow a r e sodium and the b l u e are water. The symoB t r y a x i s Is v e r t i c a l .  F a c i n g page 8  8  p e r p e n d i c u l a r or p a r a l l e l to the magnetic f i e l d the two S y m m e t r i c a l l y i n c l i n e d t e n s o r s are  e q u i v a l e n t , and there  at most o n l y four n o n - e q u i v a l e n t B1-*- s i t e s . orientations B^  For a l l o t h e r  o f the c r y s t a l t h e r e may be e i g h t  s i t e s , p r o v i d e d none o f the tensor  the symmetry a x i s o f the c r y s t a l .  are  non-equivalent  axes are p a r a l l e l  to  Because of the e x p e r i m e n t a l  s i m p l i f i c a t i o n r e s u l t i n g from s u i t a b l e  choice of magnetic  field  d i r e c t i o n s as d e s c r i b e d above, the c a l c u l a t i o n s r e p o r t e d I n t h i s t h e s i s are c o n f i n e d t o o r i e n t a t i o n s field  o f the e x t e r n a l magnetic  i n the plane p e r p e n d i c u l a r to the X symmetry a x i s  (the  YZ p l a n e ) . F i g u r e 1 shows a photograph of a model of the K e r n l t e unit  cell.  (b)  High f i e l d r e s u l t s The work of H . H . Waterman and V o l k o f f  l a b o r a t o r y has g i v e n a complete a n a l y s i s  o f the  coupling tensors f o r B H n u c l e i i n K e r n l t e .  (13)  of s i n g l e  insofar  as  this  quadrupole  The above work  was an e x t e n s i o n o f the study o f n u c l e a r resonance spectra  at  absorption  c r y s t a l s i n h i g h e x t e r n a l magnetic f i e l d s ,  i t t r e a t s a spectrum c o m p l i c a t e d by the  existence  of n o n - e q u i v a l e n t s i t e s f o r n u c l e i o f the same s p e c i e s . a d d i t i o n to l e a d i n g to i n f o r m a t i o n r e g a r d i n g the  orientation  of the p r i n c i p a l axes o f the e l e c t r i c f i e l d g r a d i e n t relative  to the c r y s t a l l o g r a p h i e  In  tensors  a x e s , a study o f the  dependence  on c r y s t a l o r i e n t a t i o n i n the e x t e r n a l magnetic f i e l d of Zeeman resonances  the  p e r t u r b e d by the quadrupole i n t e r a c t i o n of  the n u c l e i , g i v e s the value o f the quadrupole c o u p l i n g  constant  -  9  and of the asymmetry parameter Hamiltonian. (13)  the  quadrupole  These quantitative r e s u l t s o f Waterman and V o l k o f f  are t a b u l a t e d  the s t a r t i n g  which c h a r a c t e r i s e  i n the next s e c t i o n and these r e s u l t s form  p o i n t of the p r e s e n t  investigation.  I t was found from the experiments  o f H . H . Waterman  on K e r n i t e t h a t two o f the f o u r n o n - e q u i v a l e n t are d e s i g n a t e d by C and D have s m a l l e r constants and f i r s t f o r these s i t e s .  quadrupole  coupling  order p e r t u r b a t i o n t h e o r y i s good enough  On the other hand f o r the other two s i t e s  which are r e f e r r e d perturbation is  s i t e s which  to as E and P , the e l e c t r i c  quadrupole  strong enough to r e q u i r e second order  to be c o n s i d e r e d . Waterman and V o l k o f f s h a l l be i n t e r e s t e d  The s i t e s r e f e r r e d  to as C and D by  w i l l not be f u r t h e r i n calculations  t r a n s i t i o n frequencies  effects  discussed here.  of energy l e v e l s  f o r the E and P s i t e s  and  i n a region of  magnetic f i e l d s t r e n g t h s i n which p e r t u r b a t i o n t h e o r y i s applicable.  Over a l a r g e p a r t  #•  not  o f t h i s r e g i o n second order  p e r t u r b a t i o n t h e o r y would be adequate t o c a l c u l a t e frequencies  We  transition  f o r C and D s i t e s .  These remarks p e r t a i n to e x t e r n a l magnetic order of 7000 gauss or h i g h e r .  fields  of  the  CHAPTER  III  GENERAL HAMILTONIAN AND THE CASE OF KERNITE  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 n u c l e a r e l e c t r i c quadrupole moment Q and the f i e l d gradient  at the n u c l e a r s i t e / a n d  the n u c l e a r magnetic moment field  electric  the I n t e r a c t i o n between  and an e x t e r n a l l y a p p l i e d magnetic  H £ , most c o n v e n i e n t l y expressed f o r our purposes i n the  p r i n c i p a l a x i s c o o r d i n a t e system of the e l e c t r i c f i e l d i s g i v e n by  (13).  where x , y , z are the p r i n c i p a l axes of where <p i s l y and I z is  the e l e c t r o s t a t i c  ^ij  = ~^<j) / 3 x i 5 x j ,  p o t e n t i a l at the n u c l e a r s i t e ,  are the component of the n u c l e a r s p i n operator  the asymmetry  £,  -= Ix, ^\  parameter  and a , b , c are the d i r e c t i o n cosines  of Hg r e l a t i v e to  p r i n c i p a l axes o f the e l e c t r i c f i e l d g r a d i e n t I f we s e t  i n which I_ i s d i a g o n a l ,  Hamiltonian m a t r i x to be d i a g o n a l i z e d i s  the  tensor.  I = 3/2 f o r B 1 1 , and d i v i d e p . ] by  t h e n , i n the r e p r e s e n t a t i o n  form  gradient,  eq #zz  the  of the f o l l o w i n g  /  y  A  h* g  h  B  g  i  o  i * C h  g  Since we are i n t e r e s t e d here i n a n u m e r i c a l calculation for a specific quantities  i n terms  the K e r n l t e  system, we s h a l l express the  above  of parameters convenient f o r the study of  system.  As shown by Waterman and V o l k o f f i n the p r e v i o u s  c h a p t e r , the  resonance  (13)  and  discussed  spectrum i n K e r n l t e  s i m p l i f i e s c o n s i d e r a b l y because of symmetry c o n s i d e r a t i o n s i f one o r i e n t s H 0 i n the YZ plane of the c r y s t a l . Here Y and Z are two o f the c r y s t a l symmetry axes (not to be confused  with  p r i n c i p a l axes) and are m u t u a l l y p e r p e n d i c u l a r . In the case o f Spodumene s t u d i e d by Robinson the 27  four A l  s i t e s , which are a l l r e l a t e d by i n v e r s i o n have  e l e c t r i c f i e l d gradients values,  which not o n l y have the same n u m e r i c a l  but which are a l s o o r i e n t e d i n an i d e n t i c a l way w i t h  r e s p e c t t o the c r y s t a l a x e s . the resonance  Thus from the p o i n t o f view of 27  spectrum of these n u c l e i , a l l the A l  the u n i t c e l l ,  and t h e r e f o r e  i n the whole c r y s t a l ,  completely e q u i v a l e n t to one a n o t h e r . possible single  T h i s Is  the  sites are simplest  k i n d o f spectrum due to one k i n d of nucleus  crystal.  Moreover, the e x i s t e n c e  in a  of a two f o l d  symmetry a x i s i n a d d i t i o n t o the two kinds of symmetry requires  in  centres  t h a t the aluminum s i t e s l i e on these symmetry a x e s ,  and t h a t one p r i n c i p a l a x i s o f the e l e c t r i c f i e l d  gradient  12  tensor  at  the s i t e s  of the two n u c l e i c o i n c i d e s w i t h the symmetry  axis.  On account o f the above s t r u c t u r e  o f Spodumene the d i r e c t i o n  of the e x t e r n a l magnetic f i e l d can be so chosen as t o g i v e r i s e a r e a l H a m i l t o n i a n to be d i a g o n a l i z e d to o b t a i n energy On the other hand, as we have a l r e a d y c r y s t a l structure  of Kernite i s  levels.  seen i n Chapter I I ,  such that  it  is  to  the  convenient to  choose the d i r e c t i o n o f the e x t e r n a l magnetic f i e l d i n the YZ plane which would not r e s u l t different  B-^ s i t e s .  i n r e a l Hamiltonians f o r  Thus the present  calculations  the  are more  g e n e r a l i n so f a r  as  they d e a l w i t h H a m i l t o n i a n s w i t h complex  elements.  As f a r  as  the e i g e n values are  difference  as  concerned t h e r e i s no  b o t h complex and r e a l Hamiltonians g i v e r e a l  but the e i g e n f u n c t i o n s  f o r a complex H a m i l t o n i a n are a l l  These c o n s i d e r a t i o n s  make the p r e s e n t  than i n the p r e v i o u s  case o f  If HQ is  be w r i t t e n as f o l l o w s  more c o m p l i c a t e d  w i t h the Y a x i s and i n the YZ  i n the above H a m i l t o n i a n m a t r i x may  i n units  o f eQ<J>zz.  A = (-3/2) c p + 3 B = (-1/2) c p - 3  where  complex.  Spodumene.  at an angle  p l a n e , then the q u a n t i t i e s  calculations  values,  C =  (1/2)  c  p  -3  D =  (3/2)  c  T  + 3  I = -(a+ib)p  13  Al»  A g and A 3  a  r  e  d e f i n e d as  the d i r e c t i o n cosines  o f the  f i e l d g r a d i e n t tensor p r i n c i p a l axes x , y , z r e s p e c t i v e l y with respect  to the X - c r y s t a l a x e s ,  jH\*  ^2  a  n  d  /^3  a  r  d i r e c t i o n cosines with respect  to the Y c r y s t a l a x i s and  *^1»  cosines w i t h r e s p e c t  a n c  Z crystal  *  ~^3  a  r  e  dirQGti  0 1 1  e  to the  axis. The e i g e n values o f the H a m i l t o n i a n , E , are  i n terms o f the dimensionless A =  expressed  quantity. E / e Q fizz 12"  Prom the h i g h f i e l d work o f Waterman and V o l k o f f the values of the d i r e c t i o n c o s i n e s Table  (13)  are g i v e n by the v a l u e s  in  1. TABLE 1  Site  Axis X  Ex  F  2  1 > 2  x - .76 - .04 + (.22  Y  y -  .60 - .03  - .03).+ (.05  Z  + (.60  - .07)  X  - .05 - .03  Y  t  Z  + (.37  (.92 t t  - (.79  + .21 ±  - .01) - .07)  - .94 t  .04  + (.97 - (.01  .01  -  .02)  -  .01)  + .340 -  .005  .04)  -  (.09  -  .03)  -  (.368  ±  .005)  .02)  t  (.33  t  .02)  ± (.865  ±  .001)  To make the c a l c u l a t i o n s as values o f the d i r e c t i o n c o s i n e s  z  accurate  as p o s s i b l e  the  i n T a b l e 1 have been m o d i f i e d  w i t h i n the p e r m i s s i b l e range o f e r r o r so that  they would  the o r t h o g o n a l i t y and n o r m a l i z a t i o n c o n d i t i o n s .  satisfy  CAPTION POR FIGURES 2 to 15  In a l l o f the f o l l o w i n g graphs showing energyl e v e l s and t r a n s i t i o n f r e q u e n c i e s as a f u n c t i o n of (Figures 2 to 15 i n c l u s i v e ) , the u n i t s of energy and t r a n s i t i o n f r e q u e n c i e s are  eQ <ftzz I2h  while the u n i t s o f  i n terms of the dimensionless q u a n t i t y L P —  are expressed 8 ytH0/eQ(£zz.  One u n i t o f energy corresponds to almost 214 k c / s e c while one u n i t i n  P corresponds to a f i e l d  of about 156 g a u s s .  1  1  1  1)14  1  S) 13  i  '  1  1  1  1  1  1  1  1  1  1  1  r  CHAPTER IV NUMERICAL RESULTS FOR ENERGY LEVELS AND TRANSITION FREQUENCIES  S o l u t i o n of the H a m i l t o n i a n The computation o f a i g e n v a l u e s range o f p  and f o r s e v e r a l angles i s  over a  a tedious  e l e c t r o n i c computers have become almost  task.  Fortunately,  commonplace a i d s  s c i e n t i f i c r e s e a r c h i n the p a s t few y e a r s , diagonal'izing matrices  reasonable  to  and the work o f  can be performed i n a much s h o r t e r  time  than by hand. In March 1957, a computer ALWAC-III-E was  installed  by the U n i v e r s i t y o f B r i t i s h Columbia and has been used to c a r r y out a l l the p r e s e n t  calculations.  Diagonalization of a  m a t r i x such as £23 and d e t e r m i n a t i o n o f S i g e n v a l u e s and S i g e n functions  takes about 3 hours u s i n g a desk computer w h i l e the  machine does the same c a l c u l a t i o n s w i t h i n 3 - l / 2 m i n u t e s . be seen how u s e f u l calculations.  It  can  these computers are to speed up the  I t would have been beyond the a u t h o r ' s  courage  to undertake the c a l c u l a t i o n s r e p o r t e d here without the a i d of the computer. Energy l e v e l s  as a f u n c t i o n of e x t e r n a l magnetic  field  Ho f o r Y=  0, 3 0 ° , 6 0 ° , 90°, 120° and 150° and t r a n s i t i o n  frequencies  as a f u n c t i o n o f f i e l d  shown i n f i g u r e s  for  0 and Y'-  90° are  2, 3, 4 , 5, 6, 7 , 8 , and 9 f o r the  E-sites.  Energy l e v e l s and frequency t r a n s i t i o n s  f o r y = 0 ° and ^ j f  3  90°  as a f u n c t i o n of f i e l d figures  i n the case o f P - s i t e s are shown i n C a l c u l a t i o n s f o r ^=  10, 11, 12 and 13.  30°,60°,  1 2 0 ° and 1 5 0 ° have been c a r r i e d out but are not r e p o r t e d h e r e . The values f o r asymmetry parameters c o u p l i n g constants  &fa n from the h i g h f i e l d r e s u l t s (13).  e  f  z  or  and quadrupole  b o t h E and P s i t e s are  of Waterman and V o l k o f f  These are g i v e n i n Table I I below and are  w i t h the pure quadrupole resonance f r e q u e n c i e s by Hearing  taken  on K e r n i t e  consistent  ( H 0 = 0)  measured  (14).  TABLE I I Site  Asymmetry parameter  Quadrupole c o u p l i n g constant eq z z / h  E  0.163  -  0.101  2.563 - 0.007 M c / s e c .  P  0.116 t  0.010  2.567 - 0.010 M c / s e c .  As might be expected the energy l e v e l s each other i n any o f the c r y s t a l o r i e n t a t i o n s as of Spodumene.  In the case of c r y s t a l s  do not c r o s s i n the case  which have one of the  p r i n c i p a l axes c o i n c i d i n g with the c r y s t a l a x i s and the e x t e r n a l magnetic f i e l d Ho t a k e n along t h i s a x i s , the energy levels  cross each o t h e r .  Let us c o n s i d e r such an example.  Let the n u c l e i under c o n s i d e r a t i o n have s p i n 3/2 as ease of B 1 1 .  Then i n the H a m i l t o n i a n m a t r i x £2 jf the  i n the elements  h , h t , i , i * v a n i s h and we are l e f t w i t h the f o l l o w i n g e x p r e s s i o n f o r the s e c u l a r  determinant  16 A  0  g  0  0  B  0  g  g  0  c  0  0  g  0  D  We can break up the determinant i n t o two 2 x 2  determinants  r e a r r a n g i n g rows and columns so t h a t the order  i n s t e a d o f 1,  3, 4 i s  1,  3, 2,  4.  The energy l e v e l s f a l l  c l a s s L s t a t e s have mixtures  o f s p i n 3/2  selection rule different  and c l a s s M  Due to the  m = - 1 o n l y t r a n s i t i o n between l e v e l s  c l a s s e s can occur and these energy l e v e l s  each o t h e r .  To i l l u s t r a t e  t h i s , Figures  of  can c r o s s  14 and 15 g i v e  the  energy l e v e l s f o r H Q along the z p r i n c i p a l a x i s o f one o f E and F s i t e s r e s p e c t i v e l y . calculations  It  should be noted that  the  these  would be d i f f i c u l t t o v e r i f y e x p e r i m e n t a l l y as  remaining 3 s i t e s  2,  i n t o two c l a s s e s ;  and - l / 2  s t a t e s have mixtures o f s p i n l / 2 and - 3 / 2 .  by  (out o f E ^ , E 2 , Fj_, F2) would each g i v e  the  a  complicated spectrum. In the p r e s e n t c r y s t a l as d i s c u s s e d i s not p o s s i b l e elements  case where no such o r i e n t a t i o n o f  above i s p o s s i b l e  for a l l B  to have c l a s s L and M s t a t e s .  11  sites,  The o f f  the It  diagonal  o f the Zeeman H a m i l t o n i a n l e a d to m i x i n g o f a l l  spin  s t a t e s so t h a t every e i g e n s t a t e of the H a m i l t o n i a n w i l l be a mixture of a l l f o u r p o s s i b l e  s p i n s t a t e s and the energy  no longer c r o s s one a n o t h e r .  A t o t a l of 6 transitions  possible  levels will  be  i n p r i n c i p l e , s i n c e t h e r e w i l l be a non-zero p r o b a b i l i t y  of t r a n s i t i o n between any two l e v e l s .  17  The system of numbering the energy s t a t e s and transitions levels  between them i s shown i n f i g u r e  are numbered i n order of d e c r e a s i n g  t r a n s i t i o n between i and j l e v e l s  is  2.  The energy  energy and the  i n d i c a t e d by " ^ i j .  convention i s f o l l o w e d f o r a l l graphs of t r a n s i t i o n We can see from f i g u r e  11 where  frequency f o r the F - s i t e between l e v e l s gradually  to about  900 K. c/s  frequencies.  = 0 , the  transition  2 and 3 decreases  at f1 = 2.5 where i t - g o e s t h r o u g h  a minimum and then we n o t i c e a g r a d u a l i n c r e a s e , figure  This  while i n  12 whereof = 9 0 ° the t r a n s i t i o n frequency o f "^23 has a  minimum o f 250 K. c/s transitions  at  p =3.2.  show c h a r a c t e r i s t i c  S i m i l a r l y , many other  minima or maxima whose  and p o s i t i o n v a r y w i t h o r i e n t a t i o n o f the magnetic of these maxima and minima are  field.  Moat  i n a frequency r e g i o n r e l a t i v e l y  easy to a t t a i n e x p e r i m e n t a l l y and should be observable.  frequency  experimentally  CHAPTER V TRANSITION PROBABILITIES AND SIGNAL INTENSITIES  Theory Nuclear resonance t r a n s i t i o n s  are u s u a l l y caused by  a r a d i o frequency magnetic f i e l d produced by r . f . coil. of  The resonance c o n d i t i o n i s  current i n a  o f t e n d e t e c t e d by a b s o r p t i o n  energy from the c o i l by the n u c l e a r s p i n s .  In s u c h a c a s e ,  the s i g n a l i n t e n s i t y i s p r o p o r t i o n a l to the t r a n s i t i o n p r o b a b i l i t y , a s i d e from v a r i a t i o n s i n s i g n a l i n t e n s i t y due t o variations time,  i n other q u a n t i t i e s  such as  line width, relaxation  etc. In an i n d u c t i o n type spectrometer a s i g n a l v o l t a g e  produced by magnetic i n d u c t i o n due t o the r . f .  is  nuclear  m a g n e t i s a t i o n induced by the r . f . . magnetic f i e l d which produces transitions.  The s i g n a l v o l t a g e  In an i n d u c t i o n  spectrometer  w i l l not n e c e s s a r i l y be s i m p l y r e l a t e d t o the t r a n s i t i o n p r o b a b i l i t y as for  i n the case of a b s o r p t i o n spectrometer, except  the case o f a nucleus which experiences  i n t e r a c t i o n and t h e r e f o r e has 2 1 + 1  e q u a l l y spaced l e v e l s  to i n t e r a c t i o n w i t h an e x t e r n a l magnetic For  o n l y a magnetic due  field.  t r a n s i t i o n s between s t a t e s which cannot be  expressed as e i g e n s t a t e s o f the operator I z  f o r any c o o r d i n a t e  system the s i g n a l i n t e n s i t y may not be the same f o r the two types o f s p e c t r o m e t e r s .  R e c e n t l y a t h e o r y has been developed  which enables one to c a l c u l a t e  how s i g n a l i n t e n s i t i e s  for  resonance between any p a i r o f s t a t e s may be r e l a t e d t o the properties  o f these s t a t e s f o r e i t h e r a b s o r p t i o n or i n d u c t i o n  spectrometers  (15, 1 7 ) .  In order t o apply t h i s theory t o  K e r n i t e , I t must be g e n e r a l i z e d t o the case o f a complex H a m i l t o n i a n , as the theory was o r i g i n a l l y developed f o r r e a l Hamiltonians i n order to e x p l a i n the s i g n a l observed i n Spodumene.  intensities  G e n e r a l i z a t i o n o f t h i s theory t o compl  Hamiltonian i s d e s c r i b e d below and i s then a p p l i e d t o the calculation of signal intensities  In K e r n i t e .  D e r i v a t i o n o f the M o d i f i e d B l o c h Equations Consider the H a m i l t o n i a n o f the nucleus o f s p i n I , which i n t e r a c t s We s p l i t  w i t h a r b i t r a r y e l e c t r i c and magnetic  i t i n t o two p a r t s , one H Q which i s  time-independent  and another V which i s time dependent and assume t h a t functions  and e i g e n values  by equations  1 and 2  ,  fields.  o f H 0 are known.  eigen  Here H Q i s g i v e n  while V w i l l be the H a m i l t o n i a n  d e s c r i b i n g i n t e r a c t i o n between the n u c l e a r magnetic moment and the r . f . magnetic f i e l d which produces H0 f l  =  E i Y i ,  E f 0  2  = E  2  f  2  transitions.  ^ c .  In g e n e r a l there w i l l be 21 + 1 energy l e v e l s , but we confine our a t t e n t i o n t o two o f them, 1 and 2 which are assumed t o be non-degenerate w i t h r e s p e c t t o any other of the 21+1  states. The o n l y important m a t r i x elements  assumed t o be those of d i a g o n a l elements  o f V are  connecting states  1 and 2. present  Such would be the case to f i r s t  order.  In the  case V i s assumed to i n v o l v e o n l y magnetic  interactions  and we may w r i t e V = Ytl  £ I U H U + I V H V + I W H W "]  =TfiI..  where H/1 i s the gyromagnetic r a t i o and I , i components o f the angular momentum operator  v  H  [5]  and I w are  I i n an a r b i t r a r y  coordinate system where axes are denoted by u , v , and w r e s p e c t i v e l y , w h i l e Hu» H v and H w are the components o f the time magnetic f i e l d H i n t h i s coordinate system.  The m a t r i x  elements between the s t a t e s 1 and 2 are assumed t o be known and we d e f i n e  (tlVlu/^2) - F ' • ( f 2 V l u / f  < f l'Vl v /t 2 )  l ) 5  ( fgVlv/fl)  = iS  < tl*/lw/Y2) = T  p* a  *iS*  [ 6 a ]  [6b]  ( YgVlw/ti) " T *  P , S and T can be e a s i l y found i f the e i g e n f u n c t i o n s  M "^g  are known. The s o l u t i o n o f the time dependent Schroedinger equation  f = (H Q + V) ^  r  = a  ti  where the e o e f f i o i e n t s  +  i s assumed t o be b  T2  a and b a r e , i n g e n e r a l , time dependent.  We now i n t r o d u c e t h r e e v a r i a b l e s , p a r t s o f the e x p e c t a t i o n values  the "time dependent"  o f I u and I v which a r e denoted  by  and I  v  ^  r e s p e c t i v e l y and the p o p u l a t i o n  between the s t a t e s 1 and 2, which i s fjt)  (YVl /Y) = ;  =  =  (YViv/y)'"  n  =  a*a  denoted by n  Pa*b + P*ab*  u  Iv(t)  difference  [7]  iSa*b - IS*ab*  [e]  - b*b  [9]  The primes i n d i c a t e that o n l y the o f f d i a g o n a l m a t r i x of I u ,  Iv  are c o n s i d e r e d i n [7] and \&] r e s p e c t i v e l y .  i n Peynman et a l  (12)  the time dependent Schroedinger  can be expressed i n terms of the q u a n t i t i e s and £9"] as  appearing  elements As shown equation in [7] ,  follows  iF = w ' x ^ 7  d  £10]  r  dt where  = ab* + b a * r  f11]  = - i (a*b-ab*)  2  r3  = aa*-bb*  =  [12] n  [l3] —9  It may be seen that r ^ , r 2 >  r ^ are a l l r e a l and w i s a l s o a  three v e c t o r d e f i n e d by w  l  = (V12  w2 =  w  3  i(v  12  +  V21) / ^  [14]  - V 2 l ) /-B  [15]  = WG = ( E  X  - Eg) / T i  [167  where V]_ 2 and V21 are the m a t r i x elements between s t a t e s 12 and 21 and t h e y are g i v e n by  CVl'VluHu + I V H V + I w H / t )  Vie  w  =  l»  w  2'  w  [17]  2*/luHu + IVHV + I w H w /  h (  =  w  T Ti  ( P ^ + TH W + i S H v )  V21 =  (P*HU + T*HW - i S * H ) Y  r  l »  r  2  a r i  x  )  li  1  v  3»  2  [18]  ^ r 3 may t h e r e f o r e be w r i t t e n as  wi - y [ ( P + P * ) H + ( T + T * ) H + l ( S - S * ) H ]  C  ,  u  where  w  v  1  9  l  w  2  •T[i(P-P*)H +(T-T*)H -(S+S*)H l  [20]  W  5  =  [16]  u  v  w0  ri  = ^s+S^  r  =  3  w  T  u  + i ^ M * J  T  [21]  v  n  m=  P*S - PS* If  P , S and T are r e a l e q u a t i o n s [10j  reduce t o  phenomenological equations p r e v i o u s l y d e r i v e d to Spodumene r e s u l t s Furthermore, I f  (15,  _ 2 , + n n - n are  T2  T2  added to the three components o f e q u a t i o n [10]  * T dr2 dt  e x p l a i n the  17).  the damping terms - _ 1 , -  the f o l l o w i n g equations  the  are  =  =  Tx respectively,  obtained  ****  - ^  wv  -  1  W l  " ^ r  3  [24a]  - V2 T  2  r24b-| L  dn dT  w  lr2  "  W2I>  1  +  [24e]  -°i  These three equations are the a n a l o g s , f o r an a r b i t r a r y two l e v e l system of the phenomenological equations may be seen to reduce to them f o r I - l / 2 , m a g n e t i z a t i o n o f the s p i n system i s Y  is  the e x p e c t a t i o n value o f  o f B l o c h and  i f the macroscopic  identified with  I,  where  I.  S o l u t i o n s of the M o d i f i e d B i o c h Equations We now s o l v e the M o d i f i e d B i o c h E q u a t i o n t  1  7  l  with  damping terras f o r the case o f a l i n e a r l y o s c i l l a t i n g r - f  field  2H-j_ cos wt where H-^ i s o r i e n t e d i n the d i r e c t i o n o f the u a x i s . Hu = 2 H | cos wt  [25a]  Hv = 0  125b]  *w =  [25c]  0  We w i l l not go through the s o l u t i o n o f the r e s u l t i n g  equations  i n d e t a i l , but w i l l o u t l i n e the p r o c e d u r e , and g i v e the  results.  We s o l v e f o r the c a s e o f "slow passage" i n the sense d e f i n e d by B l o c h ; that i s , we assume that any frequency or  field  modulation used has a s u f f i c i e n t l y slow time v a r i a t i o n t h a t n a t t a i n s a q u a s i - s t a t i c v a l u e at any time and we can put n = 0.  Then t a k i n g n as  constant,  equations  C24a] and  w i t h damping terms may be converted t o second order equations particular  in r i , rg.  The slow passage s o l u t i o n s  s o l u t i o n o f these e q u a t i o n s .  which depend l i n e a r l y on n , are s u b s t i t u t e d equation.  The r e s u l t a n t  differential  are  The values  24b  the of r ^ ,  i n t o the  e q u a t i o n f o r n c o n t a i n s some  rg  third constant  terms p l u s terms p r o p o r t i o n a l to s i n 2 wt and cos 2 wt. time dependent terms are d i s c a r d e d . f o r the B l o c h equations  These  The p r o c e d u r e , i f f o l l o w e d  f o r magnetic resonance,  corresponds  to  s p l i t t i n g the l i n e a r o s c i l l a t i n g f i e l d i n t o two components r o t a t i n g i n opposite  senses and n e g l e c t i n g the component which  r o t a t e s i n the sense opposite vector.  to the n u c l e a r  magnetization  The n e g l e c t e d component o n l y produces  small nutations  o f the m a g n e t i z a t i o n at the angular frequency 2w. we c o u l d , i n p r i n c i p l e , s p l i t effective  and i n e f f e c t i v e  Here a l s o ,  our l i n e a r l y p o l a r i s e d f i e l d  components, except  that  into  these  components, i n s t e a d o f b e i n g s i m p l y c i r c u l a r l y p o l a r i s e d , would be e l l i p t i e a l l y p o l a r i s e d . effective  The c r i t e r i o n f o r choosing the  component becomes a b i t more t r i c k y a n d , s o , we adopt  the above p r o c e d u r e .  The f i n a l r e s u l t s  are:  rn  =  7&> T2 D 2  ( A WTvjK+L) cos wt + (K- AWT 2 L) s i n wt  L 26]  r2  =  Tto T2 D ~2~  ( Aw'T 2 L-K) cos wt + (L+AWT K) s i n wt  [27]  where D i s  2  the s a t u r a t i o n D  factor  = 1 + K2+L2  T l  T  2  +  (AW)2T22  AW = W0 - w and  K = 2YH1(P+P*) L a'2VH1i(p-p*)  In a c r o s s e d  c o i l spectrometer,  u s u a l l y o r i e n t e d p e r p e n d i c u l a r to H i .  a pick-up c o i l  An o s c i l l a t o r y  is  voltage  25  of  angular frequency w Is  induced i n the c o i l due to induced  n u c l e a r m a g n e t i z a t i o n caused by the a p p l i e d r - f f i e l d . the p i c k - u p c o i l i s  Suppose  assumed to be along the v a x i s , then the  induced n u c l e a r m a g n e t i z a t i o n along the a x i s of the c o i l p r o p o r t i o n a l to T ^ ^ . v  calculate  I  v  ^  It  is  therefore  o n l y necessary to  i n order to compare the induced v o l t a g e and  hence the r e l a t i v e s i g n a l i n t e n s i t y f o r such a spectrometer.  is  Using [8]  , [ll]  , [12]  crossed-coil  , [13] ,[26]  and  [27]  we o b t a i n ,  TS>v  "HpT2 D  +  j ^ A w T 2 i ( S P * - S * P ) + (SP*+S*P) J- cos wt  «4wT2(SP*+S'""P) + i(SP**-S*'P)  ^  s i n wt J  Unlike the case of the a b « s g r p t i o n spectrometer voltage  the  [28] signal  and hence the s i g n a l i n t e n s i t y i s not p r o p o r t i o n a l  to the square of the m a t r i x element  (= PP*'  ),  except f o r  the  case of magnetic i n t e r a c t i o n s a l o n e , i n which P = S. To i n t e r p r e t e q u a t i o n [28]  , i t should be noted  t h a t the terms i n v o l v i n g A W T 2 would g i v e r i s e to a c h a r a c t e r 1 s t I e d i s p e r s i o n s i g n a l , w h i l e the terms not involving signal.  AwT 2 would by themselves  g i v e an a b s o r p t i o n  I t can be c l e a r l y s e e n , i f S and P are r e a l  type  the  term p r o p o r t i o n a l to cos wt w i l l have a pure a b s o r p t i o n type signal  (out o f phase component) and term p r o p o r t i o n to s i n wt  ( i n phase component) w i l l j u s t have a pure d i s p e r s i o n signal.  T h i s i s the ease i n o r d i n a r y n u c l e a r  type  magnetic  resonance where no quadrupole i n t e r a c t i o n s are p r e s e n t .  The  s t r i k i n g f e a t u r e i n t r o d u c e d by complex P and S i s that b o t h  i n phase and out o f phase components now should c o n s i s t  of a  mixture of a b s o r p t i o n and d i s p e r s i o n type s i g n a l s and hence the l i n e shape can i n g e n e r a l be more complex than f o r pure a b s o r p t i o n or pure d i s p e r s i o n . shape might be  In f a c t  an asymmetric l i n e  expected.  In the case i n which H]_ i s taken p a r a l l e l to u , can be shown t h a t I u v  ' i s p r o p o r t i o n a l to P P * and t h i s would  g i v e the same s i g n a l i n t e n s i t i e s spectrometer.  it  as the a b s o r p t i o n type  T h e r e f o r e a spectrometer which d e t e c t s  the  n u c l e a r resonance by i n d u c t i o n but which o n l y employs the same c o i l f o r d e t e c t i n g the n u c l e a r resonance as  i s used f o r  producing the r . f .  intensities  as an a b s o r p t i o n  f i e l d gives  the same r e l a t i v e  spectrometer.  The c a l c u l a t i o n s f o r the s i g n a l i n t e n s i t i e s  for  a b s o r p t i o n method have been c a r r i e d out f o r 2 s p e c i f i c for E - s i t e s .  In the f i r s t  case both  and H 0 are taken  p a r a l l e l to c r y s t a l Y a x i s .  In the second case H i Is  p a r a l l e l to Z a x i s and H G i s  taken p a r a l l e l to Y a x i s .  results  of these c a l c u l a t i o n s are g i v e n i n Tables I I I  From these r e s u l t s ,  i t may be seen that  some of the  would g i v e very much weaker a b s o r p t i o n s i g n a l s In g e n e r a l , as  cases  i n the case of Spodumene (16),  taken The and I V .  transitions  than o t h e r s . strong v a r i a t i o n  i n the r e l a t i v e i n t e n s i t y of a l i n e as the e x t e r n a l magnetic field  is  v a r i e d , occurs i n the r e g i o n s near maxima or minima  i n the t r a n s i t i o n  frequencies.  The values of P, S and T have been c a l c u l a t e d *y = 0 ° to ^  for  = 1 5 0 ° i n steps of 3 0 ° as a f u n c t i o n o f the  e x t e r n a l magnetic f i e l d H Q f o r both E and F s i t e s and they  are a v a i l a b l e from D r . M. Bloom, U . B . C . signal intensities straightforward  The c a l c u l a t i o n of  making use of these above v a l u e s  is  quite  f o r any o r i e n t a t i o n o f the r - f f i e l d .  As i t  i s d i f f i c u l t to f o r e s e e a l l the experimental r e q u i r e m e n t s , may not be p o s s i b l e present  to get a l l the i n f o r m a t i o n from the  c a l c u l a t i o n s , but one can c a l c u l a t e  difficulty  it  them without much  on ALWAC-III-E computer at U . B . C .  The programme  w r i t t e n f o r the c a l c u l a t i o n of m a t r i x elements and P , S and T are a v a i l a b l e from D r . Bloom. The r e s u l t s  for signal i n t e n s i t i e s  for induction  type spectrometer  have been g i v e n i n Table V f o r  E-sites.  calculation  In t h i s  = 0 for  the  was assumed t© be along t h e  c r y s t a l Y a x i s and the p i c k - u p c o i l along the c r y s t a l Z a x i s , e.g.  i n terms of the p r e v i o u s d i s c u s s i o n , the c r y s t a l Y and Z  axes are e q u i v a l e n t to the u and v axes r e s p e c t i v e l y . t h a t here i t  i s necessary  to c a l c u l a t e  two q u a n t i t i e s  corresponding to the r e a l and Imaginary p a r t s of SP* (see  e q u a t i o n 28)  Note  respectively  i n Order to c o m p l e t e l y d e s c r i b e  the  intensities. I t should be s t a t e d that  these c a l c u l a t i o n s  o n l y p r e s e n t e d here as sample c a l c u l a t i o n s . calculations requirements. variations  are  Detailed  should be made to s u i t the experimental In p a r t i c u l a r ,  i n r e g i o n s where  are p r e d i c t e d , the c a l c u l a t i o n s  u s i n g smaller I n t e r v a l s  of p  sharp  should be made  than chosen here i n order to  get a d e t a i l e d p r e d i c t i o n of the s i g n a l  intensities.  TABLE I I I Square of the m a t r i x elements of the p e r t u r b i n g o p e r a t o r V ( i n a r b i t r a r y u n i t s ) f o r E - s i t e s as a f u n c t i o n o f p i n the case where H-^ and H Q are b o t h p a r a l l e l to Y a x i s f o r a l l s i x possible transitions  r  ^ 1 2  A s  ^14  ^23  ^ 2 4  Q  .0005  .0624  .0083  .0680  .0624  .0083  1  .0001  .0409  .0165  .0416  .0675  .0053  2  .0004  .0336  .0071  .1373  .0920  .0034  2.5  .0018  .0299  .0123  .3408  .1039  .0015  3  .0095  .0198  .0046  .9265  .0834  .0179  3.5  .0207  .0297  .0037  .3513  .0400  .1552  4  .0225  .0030  .0030  .1392  .0271  .2077  4.5  .0220  .0024  .0023  .0523  .0220  .2560  5  .0212  .0345  .0017  .0245  .0200  .2970  5.5  .0204  .0020  .0012  .0122  .0180  .3184  6  .0195  .0020  .0009  .0068  .0190  .3138  8  .0170  .0017  .0002  .0019  .0084  .1238  10  .0150  .0013  0  .0040  .0075  .0632  TABLE IV Square o f the m a t r i x elements of the p e r t u r b i n g o p e r a t o r V ( i n a r b i t r a r y u n i t s ) f o r E - s i t e s as a f u n c t i o n o f p i n the case where HQ i s p a r a l l e l to Y a x i s and % p a r a l l e l to Z a x i s for a l l s i x possible t r a n s i t i o n s  r  \*  \z  A  4  0  .0002  .0104  .0001  .0587  .0002  .0001  1  .0002  .0122  .0010  .0026  .0043  .1216  2  .0005  .0139  .0006  .0071  .0009  .1877  2.5  .0014  .0137  .0005  .0193  .0360  .3219  3  .0065  .0093  .0004  .1128  .4348  .8160  3.5  .0138  .0022  .0004  .0027  .0560  .5286  4  .0159  .0010  .0003  .0028  .0287  .3874  4.5  .0168  .0006  .0002  .0063  .0009  .3759  5  .0173  .0005  .0002  .0064  .0033  .4429  5.5  .0177  .0004  .0001  .0092  .0046  .5678  6  .0183  .0004  .0001  .0132  .0448  .6826  8  .0193  .0003  0  .0227  .0023  .1362  10  .0205  .0002  0  .0140  .0012  .0660  TABLE V Product o f the m a t r i x elements of the p e r t u r b i n g o p e r a t i o n V ( i n a r b i t r a r y u n i t s ) f o r E - s i t e s as a f u n c t i o n o f p i n the case where H Q i s p a r a l l e l to Y a x i s , p i c k - u p c o i l p a r a l l e l to Z a x i s and r . f c o i l p a r a l l e l to Y a x i s f o r a l l s i x p o s s i b l e transitions. Qf) = 2 i ( S * p - S P ' » ) 7 £ = 2(SP'*+S*P)  12  p  13  C9  to  0  0  1  .0002  2  14  0)  23  H  0  -.0255  e  -.0009  0  .0007  -.0448  .0004  -.0066  .0004  -.0008  .0028  -.0432  .0004  -.0040  2.5  .0016  -.0028  .0004  -.0276  .0006  -.0040  3  .0046  -.0150  .0026  3.5  .0006  -.0330  .0028  -.0270 -Q0160  .0004 .0002  4  .0036  -.0372  .0002  .0034  4.5  .0074  -.0378  .0008  5  .0030  -.0040  5.5  .0108  6  -.0003  24  34 to  to  0 .0012  -.0630  0  -.0255  0  SL -.0009  -.0210  -.0024  -.0340  -.0626  .0032  .0198  .0454  .0216  -.0080  -.1620  .0226  .1202  .0408  .0148  -.0028 -.0022  -.0682  -.6428  .1180  .3620  .0266  .2^00  .0128  .0608  -.0444  .0826 -.0346 -.5718  -.0008  -.0018  .0044  .0396  -.0128  .0106  .0116 -.5672  .0022  .0002  -.0014  -.0036  .0294  -.0090  .0162 -.6201  .0040  .0076  0  -.0010  .0052  .0242  -.0018 ;,0026  -.0160  .0294 -.7248  -.0366  .0004  -.0018  0  S.0008  .0078  .0196  .0040  .0178  .0450 -.8692  .0094  -.0366  0  -.0014  0  -.0004  .0154  .0262  .0022  .2178 -.9236  8  ..0136  -.0342  .0002  ...0014  0  0  .0004 .0132  -.0015  .0010  -.0086  .3444  10  .0134  -.0320  .0002  -..0010  0  0  • 0142  -.0020  -.0026  -.0054  .1070 -.0722  0  .0386 -.0340  .2360  o  CHAPTER VI CONCLUSIONS  Prom the t h e o r e t i c a l study of i n d u c t i o n and a b s o r p t i o n s p e c t r a of  n u c l e i i n a single c r y s t a l of  K e r n i t e , we can say that i t would be i n t e r e s t i n g to do both types o f experiments results.  to i n v e s t i g a t e  the t h e o r e t i c a l  An e x p e r i m e n t a l cheek on the t h e o r e t i c a l  c a l c u l a t i o n s presented here would r e p r e s e n t  the most  d e t a i l e d t e s t yet on the g e n e r a l H a m i l t o n i a n d e s c r i b i n g quadrupole i n t e r a c t i o n s Induction spectra,  in crystals.  In the case of  i t would be i n t e r e s t i n g to study the  shape of the l i n e as the t h e o r e t i c a l I n v e s t i g a t i o n show asymmetry i n the l i n e  shape.  32  REFERENCES  1)  B l o c h , P . , Hansen, W. W. and P a c k a r d , M . E . , P h y s . Rev. 69, 127, (1946)  2)  P u r c e l l , E . M . , T o r r e y , H . C . and Pound, R . V . , Phys. Rev. 69, 37, (1946)  3)  Casimer, H . B . G . ,  4)  Bloembergen, N . , Nuclear Magnetic Resonance, Martinus The Hague (1948)  5)  Pound, R . V . , P h y s . Rev. 79, 685 (1950)  6)  Dehmelt, H . G . and Kruger, H . , Z . P h y s i k , 129, 401 (1950)  7)  Carrand K i k u c h i , Phy. Rev. 78, 1470 (1950)  8)  Bersohn, R . , J .  9)  V o l k o f f , G . M . , P e t c h , H . E . and S m e l l i e , 30, 270 (1952)  10)  Dehmelt, H . G . and K r u g e r , H . Z . , Z e i t s c h r i f t 130, 385 (1951)  11)  Kruger,  12)  Peynman, R . P . , Vernon, P . L . , J r .  P h y s i c a 2, 719 (1935) Nijhoff,  Chem. Phys. 20, 1505 (1952) D . W . L . , Can. J . fur  Phys.  Physik,  z . P . P h s . 130, 371,(1951) and H e l l w o r t h , R . W . ,  J o u r . App. Phys. 28, 49 (1957) 13)  Waterman, H . H . and V o l k o f f , G . M . , Can. J .  P h y s . 33, 156 (1955)  14)  R . R . Haering and G . M . V o l k o f f , Can. J .  15)  L . B . Robinson, P h . D. T h e s i s , Vancouver, B . C . (1957)  16)  Lamarche, G . and V o l k o f f , G . M . , Can. J .  17)  M. Bloom, L . B . R o b i n s o n , G . M . V 0 l k o f f , C a l c u l a t i o n s of f r e q u e n c i e s and r e l a t i v e i n t e n s i t i e s o f n u c l e a r s p i n resonance l i n e s i n c r y s t a l s . To be p u b l i s h e d i n Can. J . Phys *  18)  P o r t o l e s , L . , E u c l i d e s 5,599 (1945); E s t u d . G e o l . g , 3(1947); E s t u d . G e o l . 7, 21 (1948). Minder W . , Z . K r i s t a l l o g r . A 92, 301 (1935J  Phys. 34, 577-585 (1956)  U n i v e r s i t y of B r i t i s h Columbia, Phys. 31,1010 (1953)  

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