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The direct detection of double quantum coherence Legros, Mark Anthony 1981

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THE DIRECT DETECTION OF DOUBLE QUANTUM COHERENCE by MARK ANTHONY LEGROS B.Sc.,Massey U n i v e r s i t y ,  1981  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE  STUDIES  D e p a r t m e n t Of P h y s i c s  We a c c e p t t h i s to  t h e s i s as conforming  the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1981  ©  Mark A n t h o n y L e G r o s ,  1981  In  presenting  requirements  this f o r an  Columbia,  I  available  for  permission  agree  or  her  of  October  the  shall  reference  and  study.  I  extensive granted  by  this thesis written  copying the It for  Head of is  1984  this thesis my  Columbia  gain  the  of  British  it  freely  agree for  Department  understood  permission.  make  further  financial  Physics  8,  of  of  University  Library  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date:  fulfilment  the  w i t h o u t my  Department of  partial  advanced degree at  representatives.  publication allowed  be  in  that  for  p u r p o s e s may  thesis  that  that  scholarly  or  by  copying  shall  not  his or be  ii  Abstract  When and  spin-1  a  an e l e c t r i c  field  states  describing  spaced  levels.  term  between  i t ' s orientation  with  When t h e n o n s e c u l a r  occur;  theory  observation to provide  transitions  the  spectroscopy  splittings, information obtainable which  another  and  long  other  prevents  the  f o r the  unequally  techniques  probabilities. with l z ,  a s s o c i a t e d with  i s presented.  with  doing  resonance  when  spectroscopy  The r e s u l t s  large  relaxation m=2  from  curiosity"  show t h a t  quadrupole times.  experiments  a n d t h e major  being  i s their  of  i s practical for  (>500Khz)  the  the  The i n t e n t i o n i s  the  techniques  experiments  of a " S c i e n t i f i c  transitions  m=2.  spin-spin by  terms  d i p o l e t r a n s i t i o n can  forbidden t r a n s i t i o n  systems  available  dipole  transition  magnetic  a n d examples o f  of t h i s  field  i s t h e sum o f a Zeeman  i s a valuable procedure.  in  three  w h i c h do n o t commute  transitions  from  catagory  finite  those  i n which  m=2  insights  nuclei  have  and e x p e r i m e n t a l  such  spin-1  m=1  account  of  (which  of  diagram  term) i s t r u n c a t e d t o i n c l u d e o n l y  terms,  a transition  The  level  t h e Zeeman t e r m , m a g n e t i c  with  into  i n a l a r g e magnetic  consist  I f the hamiltonian  levels  taken  i s placed  g r a d i e n t , t h e energy  and a q u a d r u p o l a r  w h i c h commute  are  nucleus  The is  feature  r e l e g a t e d t o the simplicity.  iii  T a b l e of C o n t e n t s Abstract L i s t of Tables L i s t of F i g u r e s Acknowledgement  i i iv v vi  Chapter I INTRODUCTION  1  Chapter II BACKGROUND  3  A.  THE FIRST ORDER QUADRUPOLE  B.  NONSECULAR EFFECTS  SPECTRUM  5 17  Chapter I I I HAMILTONIANS  ;  18  A.  PERTURBATION THEORY AND THE F I C T I T I O U S SPIN  1/2  21  B.  F I C T I T I O U S SPIN-1/2  24  C.  PREPARATION OF DOUBLE QUANTUM COHERENCE  30  D. TWO HARD 90 DEGREE PULSES  30  E . THE SOFT DOUBLE QUANTUM PULSE  31  F. THE EVOLUTION OF DOUBLE QUANTUM COHERENCE  37  G. THE SECOND ORDER  41  SHIFT  C h a p t e r IV EXPERIMENTAL  44  A.  PROBE  45  B.  COILS  45  C.  SAMPLES  47  D. COMPARISON BETWEEN THE EXCITATION METHODS  56  E. APPLICATIONS  58  F. FURTHER APPLICATIONS  59  APPENDIX A - ANGULAR MOMENTUM BIBLIOGRAPHY  62 61  iv  List  of  T e n s o r s and Axioms  Tables  I.  Basis  of L i o u v i l l e  II.  A n g u l a r momentum c h a n g e s d u r i n g transition  a  |1>  Space to  11  |-1> 67  V  List  1. The  first  order  of  Figures  quadrupole  spectrum  6  f o r a spin-1  nucleus  8  2.  Powder p a t t e r n  3.  R e l a t i o n s h i p between  4.  Eigenstates  i n the  the  p r i m e d and  rotating  the  frame  unprimed  frames 28 32  5. T r a n s i t i o n  probabilities  35  6.  The  order  43  7.  Coils  8.  Satellite  9.  Transverse  second  shift  powder p a t t e r n  ...46 lines excitation  10.  Longitudinal  11.  Rotation  excitation  pattern  49 51 53 54  vi  Acknowledgement  I would l i k e inventing the  c e n t r a l idea  cohabitants  enjoyable Pratum with  the  t o thank my  of Myer's  work p l a c e . ,Dr.  supervisor,  "Room  also give  A l e x a n d e r Mackay  100"  for  Bloom  for  work  providing  an  s p e c i a l m e n t i o n t o Dr.Tom  and Edward S t e r n i n  t h e t e c h n i c a l a s p e c t s o f my  fashion.  Myer  f o r t h e t h e s i s , and I w i s h t o thank  lab,  I must  Prof.  in  a  for helping  patient  me  friendly  1  I.  This relevant  thesis to  reports a novel  nuclei  interaction  with  which  local  state  techniques  high  of  resolution  which  is  field  an  resonance  technique,  electric  quadrupole  gradients.  experiments  of  a  normally  The method i s  collectively  techniques".  i s the observation  hamiltonian  magnetic  experience  electric  a contribution to a class "Solid  INTRODUCTION  The a i m o f t h e s e  small  masked  termed,  part by  of  a  a  much  spin  larger  interact ion. The  experimental  procedures  those  normally  method  involves transitions  The  ideas  Before  a  radically  which a r e o n l y  to  the usual  partially methods,  of a s p e c i a l  beginning  a  detailed  description  o f d o u b l e quantum c o h e r e n c e "  few n o t a t i o n a l p o i n t s  large  this z  axis.  called,  i s termed The  plane  field,  I  longitudinal which  have  lies  designated  normal F o u r i e r t r a n s f o r m satellite  the  allowed.  i n that the  initial  state,  NMR  as  of the " D i r e c t  those  "not i n t h e  h e r e a r e done  and the d i r e c t i o n and d e s i g n a t e d perpendicular  t h e t r a n s v e r s e , o r x, y , p l a n e .  nitrogen-14,  of  i t may be w i s e t o c l a r i f y  f o r the b e n e f i t of  e x t e r n a l magnetic  field  from  by t h e t r o u b l e s o m e p a r t o f t h e h a m i l t o n i a n .  business'.' The r e s o n a n c e e x p e r i m e n t s r e p o r t e d a  different  s t a t e NMR, a s t h e h e a r t  i n v o l v e s the p r e p a r a t i o n  i s unaffected  detection  insolid  are similar  experiment which  employed  are  parallel to  the  laboratory  t o the f i e l d i s  F o r the experiments  the doublet satellite  i s used t o d i s t i n g u s h these  using  lines,  on  which a r i s e s i n  lines.  The  which occur  f r e q u e n c i e s Wo+Wq a n d Wo-Wq, from t h e Am=2 t r a n s i t i o n  term  a t the  a t Wo i n a  2  spin-1  system.  3  II.  The  interaction  between t h e e l e c t r i c  n u c l e u s and the e l e c t r i c nucleus  i s governed  absence  of  nuclear  electric  invariant the  any  present  the  rotations,  .  the  electric  This  technology  molecular  spatially  processes.  of  advances  years  much  powder,  the  applied  NMR o f n u c l e i  ordered  major  was  on s t r u c t u r e dictated simple  by  in electronics  be  vastly  study  of  and d y n a m i c s . the  existing  and c o m p u t i n g ,  increased  the  range  studied.  having  h a s been p a i d a  t o systems  distribution  common  distribution  which  cannot  give  fields,  the  s y s t e m s be u n d e r s t o o d  most  of e l e c t r i c  tool  H i s t o r i c a l l y the f i r s t  attention  crystallites  is  and  o f p u l s e d NMR,  of systems which c o u l d  The  field  systems  information  subsequent  orientations.  I f , as i n  on t h e r e l a t i v e  why  in solids  that  of  the  to molecular  and  reason  In t h e  rotationally  invariant  gradient  main  and t h e r e q u i r e m e n t  consisting  orientation  the  was  recent  isotropic  field  the  describing  magnetic  information  of  two.  is  external  longer  system  The  nature In  no  interaction  with the i n t r o d u c t i o n and  , a large  of  crystals, giving  first.  the hamiltonian  of a  q u a d r u p o l e moments h a s become a p r e c i s e  investigation  i n i t i a l choice  site  o f rank  interaction  is  is  of the q u a d r u p o l e  single  NMR  fact  the  of the m o l e c u l e as a whole.  of the e l e c t r i c  anisotropic use  quadrupole  field  at  interaction  fields,  hamiltonian  This  possessing for  by a t e n s o r  applied  q u a d r u p o l e moment  gradient  and t h e s p e c t r u m c o n t a i n s  orientations field  field  w.r.t r o t a t i o n  case of high  BACKGROUND  is  that  information  b u t t h e r e s u l t i n g powder  of o f an  on  the  pattern  4  s p e c t r u m has  features which  magnitude and The  to  symmetry p r o p e r t i e s of  most  nuclei,  fruitful  probably  Three  nitrogen-14,  and  easiest  study.  easy  to  study  technique NMR.  2)  coupling  over  The  good  the  produced  whole when  quadrupolar  interaction  negligible  during  example  spectrum  i s the  of  frequencies exhibits  to  experiments,  the of  i s the in a  pulse  frequency  duration s t a t e of  i s the  must  be  simple  Zeeman  of  echo  a  the  nucleus with  the  90  with  echo,  of  the  if  the  L i m i t a t i o n s on  of  the  system  is  state.  A  power  over  the  the  echo NMR  most  important  which  prepares  largest the  be the  bandwidth  magnetisation. the  to  the  of  wide  degree pulse  to  conditions  appreciably  For  small  effect  transform  which  a  interaction  sequence;  fourier  dipole  c o n d i t i o n can  means t h e  a  field  essential  nonequilibrium  varies  transverse  compared.  far  have  initial  evolution  quadrupolar  standard  by  are  initial  distortion.  of  must  uniform  preparation  the  deuterium,  large magnetic  and  the  dependent as  a  features  the  e x c i t e d , the  simplest  are is  spin-1  what makes a  two  excitation  such  system this  be  the  t h e s i s i s concerned  is large, this on  on  the  systems a c c e s s i b l e to high  quadrupole  the  frequency  parameter  the  which deuterium  A of  interaction  isotopes  of  of  performed  spectra are  hamiltonian  to noise  ratio  been  possess  spectrum.  the  interaction.  have  this  These  quadrupole  good  as  quadrupole  signal  the  statement  must  constant.  obtaining  , of  range of  nucleus  determination  spin-1  A concise  the  the  resulting  common  i s necessary,  The  moment.  the  lithium-6  to widen  1)  studies  because  interpret.  to  enable  90  The  length  quadrupole degree  pulse  5  degree pulse discussed  in  microsecond 300 of  KHZ  the  pulse  practically and  solid  constants ratio  i f not  time  this  version  first  order  the  true high  of  commuting  with  levels  Iz  magnetic  the  with  with  single  total  the  examined suitable  is  coupling  The  best  difficult are  90  approximately  large quadrupole  crystals  quadrupole field  have  to  coupling  study  using  available.  hamiltonian  hamiltonian  been as  shown  the  eigenstates  2-dimensional  rotation  angular  momentum, and  the  s p i n system w i l l  between  Selection the  be  I can  expanded  for be  A  /V  y\  be  i n terms o f ,  terms  terms a f f e c t as  will group.  [H,lz]=0  Because  the  derived  by  of  angular  numbers  magnetic  Iz,I+,I-,which  the  transform  quantum  e i g e n s t a t e s , can  /\  only  z p r o j e c t i o n of  good  rules  truncated  t o r o t a t i o n s about  the  the  a  in f i g . ( l ) .  field,  and  is  i n which  r e t a i n e d , these  is invariant  eigenstates.  transitions that  nitrogen  this  gyromagnetic  of d e u t e r i u m .  hamiltonian  under  momentum, of the  that  achieve  in f i r s t order  truncated  irreducibly this  be  its  two  width  within  i s much l e s s  and  a  The  fall  be  spectra  i t s r a n g e of q u a d r u p o l e  compounds  unless  allows  compounds can  megahertz,  combined  this  spectra  will  deuterium  distortion.  Nitrogen-14  half  can  For  and  FIRST ORDER QUADRUPOLE SPECTRUM  The  main  little  quadrupole  many  makes n i t r o g e n  c o n v e n t i o n a l NMR;  the  with  s t u d i e s as  one  section.  a l l deuterium  approximately  nature  a t t a i n a b l e , and  methods.  extends over  constants  THE  recorded  NMR  5 microseconds,  energy  experimental  s t a t e NMR  is  experimental  is readily  most  degree p u l s e  A.  of an  a l l deuterium  conventional  for  are  wide t o be  range, by  time  for  dipole noting  transform  6  1 - The  Figure  First  order  axially  first  quadrupole  symmetric  order quadrupole  spectrum  electric  field  for  spectrum  spin-1  a  nucleus  i n an  gradient.  -> 1  <1  |0>  t  1<  ^ ,  -—•  .  —-  "  ii  i  Wo  Do  2.  C J a  L O G L  (Jo  (JJO A  -  A  H= -Dolz. + (J<aTz.z.  7 irreducibly m=-1, a  under  respectively.  symmetry  group  eigenstates. possess  between  and  differing  a periodicity  the  z direction,  same amount  the  from  is m=0  and  by  consequently the pure  a direct  has  If  the  equal  states  existed,  t o o wide to  m=-1  the energy systems  for traditional transition  interaction  and  explained  a rotation t h e m=1  for  experimental  axis  those  transverse magnetisation  perpendicular  The  a  but  as  by t h e order  dependent  energy  the  the  then  each  first  order  applied  field  g r a d i e n t , the f i r s t  SiNfpi^i) oj^  to  first  powder  between  ^ fig(2).  between t h e  spectroscopy  order  If a direct  m=1  and  c o u l d be s t u d i e d .  i n which t h i s i s just  the  not  proceedure  coherence  m=-1  i n w h i c h t h e powder p a t t e r n s p e c t r a a r e  f o r cases  of  system  states are shifted  i s u n a f f e c t e d by t h e f i r s t  features  the  The s e c u l a r h a m i l t o n i a n  is  field  the  i s phase  o f an o r i e n t a t i o n sample  i s also  labelling  that  splitting.  difference  t h e powder p a t t e r n s p e c t r u m Some  to  i f there  on t h e a n g l e  by  m=1, L=1  2-dimensional  the  implies  |m|=1,0.  z a x i s of the e l e c t r i c  way t o m o n i t o r  also  probability,  powder p a t t e r n i s w e i g h t e d  with useful  only  Zeeman  result  depends s o l e l y  the  rule  of 2 0 about  state.  orientation spectrum  nothing  a net magnetisation  has  of  adds  L=1 m=0,L=1  i n the transverse plane  i t commutes  The s e l e c t i o n  levels  spectrum  Reflection  o p e r a t i o n but  rotation  can  t h e symmetry g r o u p a s ,  familiar  i s t o prepare and monitor  in with  m=1  quadrupole  approximation  a single  spectra  order  The  i s valid  line.  fig.(2) NMR.  the system  need The  in a  the decay of t h i s  to  be  standard state  of  8  Figure  2 - Powder p a t t e r n  77=0 p o w d e r  -200  f o r a spin-1  pattern  0 Frequency  200 (kHz)  nucleus  9  magnetisation. the  The  i s the a u t o c o r r e l a t i o n function f o r  transverse magnetisation.  autocorrelation components tensor. often  the  The  phase  description  relates  A  via  the  economical  features  essential  fashion.  reviewed  The  variables.  of  to  Liouville  first  For  sums o f p r o d u c t s  that  nuclei  an  to  so  decision isolated  I  is  the  of  interaction  an  choice  independent  under  the three dimensional  components  resonance i s because  tensors;  t e n s o r , a second rank  each  relevant  consist  of a l l  Ix,Iy,Iz.  For  operators  w h i c h c a n be  tensor  transforms  rotation  group,  tensor.  P\  3  irreducibly  .We  tensor,  The c h o i c e o f  t e n s o r s c a n be made by r e q u i r i n g  i n each b a s i s t r a n s f o r m  the  spin-1  of a  rank a n t i s y m m e t r i c  symmetric  matrix on  of  these  i t  i n a unique  concentrate  s p i n system  irreducibly  w i t h i n each of these  of  theoretical  t o the d e s c r i p t i o n  three  basis  method  dealt with  will  into  second  i n the  f o r m a l i s i m and t h e d e n s i t y  specific  there a r e nine  traceless  line  the  common  of the fundamental observables  an i n v a r i a n t  which  to the  mechanics,  be  partitioned  a  in  from e a c h  the  quantum  physics  the theory  transverse  the z a x i s .  i n many a r t i c l e s ,  of  nucleus.  and  scheme  spectrum  hamiltonians;  formalisim  this  susceptibility  most u s e f u l f o r m a l i s i m t o d e s c r i b e m a g n e t i c  enables  obtain  the  of  the F o u r i e r transform a r e  i s subtracted  and  to  magnetic  frequency  detection,  transform  equivalent  the observed  a t Wo a b o u t  Liouville  spin-1  Fourier  temperature  Wo,  of s e c u l a r  rotating  is  obtained  frequency, This  The  i n a reduced  sensitive  frame  high  spectra  represented  spectrum.  the  function  of  excitation  are  result  the  under a s u b g r o u p  10  of  P\3  , such as  or C o o v  Coo  •  T  n  e  subgroup chosen to d e f i n e  the b a s i s i s t a y l o r e d to the symmetry of the h a m i l t o n i a n dealt  with.  can  ignored and  be  rotation,  During  hence  the r . f . the  the  secular  Coo  o r  evolution  states  r e p r e s e n t a t i o n s of  a  r  a n <  operator  belonging e  quadrupole  Coov  p u l s e s the quadrupole  mixed.  n o t  hamiltonian  s t a t e s belonging  3  to  is  essentially  evolution  to d i f f e r e n t (1)  groups are not coupled.  the  b a s i s systems i n terms of the, fundumental  and  symmetry  groups  presentation  used  of  to  label  invariant  shows  the  symmetry i s  the  basis  subspaces  explanation. external  properties For an  dimensional tensor  that  operations. described  the  which  is  beside they  The  an  states.  under  group C o o y  This  is  r e p r e s e n t a t i o n s , the p a r t n e r s are e n c l o s e d  in brackets.  The  p,^  is  by  their  with  an  in the z d i r e c t i o n ) the has  1-  appearance of more than  one  group  and  representation  label,  merely  i n the same way  under the group our  spin  r o t a t i n g frame, so t h a t there  i s no  contains  group  of  r e q u i r e some  v  i s the symmetry group of  Zeeman i n t e r a c t i o n  hamiltonian  momentum).  transform,  C<*  interacting  albelian  The  each  i n a resonant  remnant of the if  ( d e f i n e d to be  representations.  component  indicates  as  r  and  C ^  i s o l a t e d s p i n system  magnetic f i e l d ,  symmetry i s C o o  under  form  observables,  s t r a i g h t f o r w a r d , but c a t a g o r i s a t i o n of the components transformation  under  representations  of these two  a  irreducible  the p r e v a i l i n g  Table  be  interaction  different  During  to  ( r e f l e c t i o n symmetry i s broken  terms  linear  nonabelian for  a  and  in has  particular  the  angular  2-dimensional representation  11  Table  I - B a s i s T e n s o r s a n d Axioms o f L i o u v i l l e  B a s i s T e n s o r s grouped  i n terms  of i r r e d u c i b l e  Space  r e p r e s e n t a t i o n s of  R3  Cartesian  Sprier i c a l A  A  2  ^  T « =1 ( I +) A  A  A  Txi = - l ( I *  A  , A  A  A  /V  Txy = l ( I x l y + I y l x )  A  + Txz = 1. ( I z l * + I x l .  fxO= 1 ( 3 1 2  f f V l - l ( ± z ± - - r ± - i  E  2 Z  =JL  (3iz-A)  ) tyz: = J_ ( l ^ I y + T I z ) y  Ta-A  = J_ I 2  A  A  TV-y* = 1 ( I x + I y )  A  A  A  I* A  I x  A  A  I-= J x - Z X y  A  Xz.  1 2  Table-1 Tensors  grouped  subgroups  continued  i n terms o f i r r e d u c i b l e  representations of  o f R3.  'oo  v  Tao  T»-i  I-  (Ix.Iy)  (T  y 2  (Txy,Tx'Y)  T*  ,Tx ) 2  13  Table-1 Axioms  of  Liouville  (1) L i o u v i l l e operators (2)  A  {  |Ai>  quantum  supposition  of  (3) A s c a l a r (4) L i n e a r by  all  A  A  .  space  space is a linear  mechanical  product  . A  .A  spanned  by a s e t  of  is  represented  by a  linear  acting of  with  as  <Ai|Aj>=Tra(AiAj).  on t h e s p a c e  form an a l g e b r a  operators  the  A .  Ai  of  form  spanned  Ai*Aj;  where  A .  and  A]  being  any o p e r a t o r s  from  space.  (5) T h e e v o l u t i o n I £Ho>>=  state  is defined  ,  Ai*Aj|Ak>=|AiAjAk>, Liouville  space  operators  operators  . A  vector  }.  sums a n d p r o d u c t s A  continued  £cuco) iAi> t  where H i s  problem  is  specified  by an  , and the dynamical  the hamiltonian  of  the  system.  initial  equation:  condition  14  As  an  example  l e t us  now  solve  the  following i n i t i a l  value  A  problem,;  If the  t i m e t>0 and  a  i f the  s t a t e at system  secular  simplified  by  this  coincides  Formally  the  define  problem  and  A  more  tractable  eigenfunctions under  the  e l e m e n t s of operators  and  that  laboratory  solution  be  can  operator  L vanish  unless  transform  can  at  larmor  eigenfunctions shows  the  the the  x axis  axis  L  found  must  at  one  transform  of  L,  i e . Coov  it  is  a  matrix  the  same  under  if  '  a n  ^  (0(t)=  be  a ( I ^  of L w i l l  be  of  eigenfunctions  that  row  the and  of  -i.U)a+ b ( I x - i T \ , z ) e  matrix  the  7\=Tyz.  form  same We  alx+bTyz,  corresponding  written: +  that  between  + iTya  + i T y z ) e  of  irreducibly  are:  s o l u t i o n can  be  t=0.  notes  element  r e p r e s e n t a t i o n ; h e n c e <A|L|lx> = 0 u n l e s s the  hamiltonian  problem  that x  at  as:  H=u)aT.  the  state  Zeeman  rotating  lp-[Hp-pH]  I x  Our  the  i s expressed  group  some a l g e b r a  eigenvalues  frame  i s the  The  r o t a t i n g frame so  with  symmetry  which  irreducible know  of  the  a large  hamiltonian.  t r a n s f e r r i n g to a  and  i s I x , what  i n t e r a c t s with  quadrupole  frequency, frame  t i m e t=0  (2)  15  The the  state  of  components  hamiltonian  secular  are in  a high  terms  of  of  second  We h a v e the  state  Ix  determined theorem are  by  for  such  element  of  seen  the  symmetry  tensor  from  equation  and  m  operator  of  P\  3  the  by  listing  and  if  of  basis  bilinear of  If  evolution  density  (2)  that  a  the  initial can  operators.  in  the  the  process  operator  vector  spin  be The  angular  magnetisation  tells  us  Tyz  initially  at  a  to  constant.  that  if  the  on  the  nuclear  same r o w will  of  The  Wigner  nuclear  states  the  matrix  coordinates,  the  same  be p r o p o r t i o n a l  in  frequency  good quantum numbers,  the  Tyz,  a system  state  coupling  acting  as  given  the  state  are  to  ,  is  into  operators  be  polarization.  quadrupole  L  can  8-dimensional space.  number  initial  according  representation  of  rank  tensor  transforming  an  simple  small  precesses  that  in  hamiltonian  the.  any  a  time  subspaces,  sufficiently  quadrupole  state  degree  any  disjoint  momenta a n d c o u p l e s a  system at a vector  into  conditions expressed  of  has  partitions  the  and  irreducible to  those  of  /\ Tyz  between  with  these  electric the  to  for  evolution the  dance  electric the  problem  such  field  quadrupole  the  same L a n d m.  the  y-z  component  operator.  nuclear  moment  of  can  ensemble is  with  is  moment  a given  quadrupole  initially  The  characteristics  operator  nuclear  labelled  quadrupole  electric The  states  state  the  have of  that  gradient  the  at  interaction,  magnetic  the  nucleus  and the  a  the  nuclear  charge  in  a  of  for the  distribution. visualization;  s t a t e "ix,  dipole  moment  produces  nuclear  value  measure  following is  operator  expectation  is  nuclear  nuclei  a net  The  of  An  dance  and  the  exists.  a torque  due  changes  so  16  that  the  e n s e m b l e d e v e l o p s an e l e c t r i c  so t h e o s c i l l a t i o n dipole  moment  monitoring sample  by  i s put i n t o the across  magnetic  dipole  that  the  other  second  frame.  the  coil,  quadrupole  capacitor.  rank  states  having  correspond  different  Tzz i s a s t a t i c  electric  the  sample  oriented  moment  between  quadrupole induce  the nuclear  moment  the  quadrupole  orientations quadrupole  w.r.t.  charge  distribution,  precesses into precise  usually  of  Txx-yy  a t t w i c e t h e Larmor  and  nature  frequency.  of the n u c l e a r charge  interest  to  the  state  Tyz,  and  the  from  A  frequencies associated earlier we must  oscillating  but  We have seen how t o o b s e r v e t h e o s c i l l a t i o n  into  of  spectroscopist,  states  are.  lifetimes  the  distribution i s  and  this  find  w i t h Txy  and  Txx-yy,  m=1  and  transition  m=-1  peak t o m=-1  states.  but  of  the  deduce t h e  as  remarked  cannot  be seen  energy  The f i r s t  of the r . f .  corresponding  coupling  m=1  appears.  processes  i s c o m p l i c a t e d because  to  field a  difference  method u s e d t o  was a c o n t i n u o u s wave method  ( 1 ) ) where t h e s t r e n g t h absorption  nonequi1ibrium  can  signal  a n o t h e r way o f o b s e r v i n g t h e m=2  the  observe t h i s  we  the  A  i f f o r some r e a s o n t h e t r a n s v e r s e  between  an  The  the l a b  frequencies  al  shows  i s unobservably small. to  a  A  The  Ix  and  i f the  will  quadrupole  ; and  oscillating  similarly  Comparison  to the c a p a c i t o r  an  around  moment a n d t h e e l e c t r i c  A  not  coil  the c e n t e r of a s u i t a b l y  coupling  distribution  a  induced i n the  oscillating  voltage  We c a n c o u p l e t o  wrapping  the voltage  capacitor  Txy  continues.  q u a d r u p o l e moment  (Yatsiv e t .  i s increased  two  photon  until  process  The t h e o r y o f l i n e s h a p e f o r s u c h the exact s t r e n g t h  of the r . f .  17  field is  i s an  no  longer  techniques and  important  to  manipulated  linear  the  problem  they  generally  by  s p e c t r u m , and  response  regime.  spectroscopy  a transverse f i e l d  that  recently  developed,  methods a r e  monitoring  require  system  More  have been  experimental  of  the  multiple  the  quantum  system  a t t h e Larmor  well  can  be  frequency.  NONSECULAR EFFECTS  The  discussion  principally  with  the  section  theory  some  very  Inclusion  to  The  to describe spin as  the  magnetisation of  this  the quadrupole  terms.  reasonable  transverse  up  point  nonsecular  experimental  spin  at  i n these  included Larmor  and  one  effects  The  from  principal  oscillates the z  very  examined  magnetisation  at  and  or  an  this  direction.  of  t h e Larmor  usually a  larmor  of  secular  would  spin  of d o u b l e  this  magnetisation  frequency,  and  lead  to  than  are the  nonsecular oscillating  quantum  is  a  polarized  The an  state  as  effects  longitudinally occur.  is  dynamics f o r  frequencies other  in  have  frequency.  nonsecular  result  in  experiments  experiments  effects  thesis  initial  features twice  detects  observable  in  at  shown  manipulated  the  theory  the  does e x p e r i m e n t s  frequency,  magnetisation,  If  is  concerned  be  of  i n most NMR  system  oscillating  reward.  as w i l l  approximation  dynamics  terms  been  hamiltonian w i l l  g r e a t d e a l of c o m p l i c a t i o n i n t h e  little  has  s e c u l a r h a m i l t o n i a n s , and  nonsecular  hamiltonian  a  i n the  t h e p h i l o s o p h i e s and  transitions,  B.  the  of m u l t i p l e quantum  although  suited  in  parameter  are  coherence. that  it  polarized  in  18  III. The  electric  interaction and  the  quadrupole  interaction  between t h e e l e c t r i c  electric  interaction  HAMILTONIANS  quadrupole  may be e x p r e s s e d  field  i s a second  tensor  gradient at the nucleus,  moment  of  the  nucleus.  i n terms of s p h e r i c a l  Wa = 2 j A x B x  rank  The  tensors as:  Q  m--2  pCx  where t h e  A° =  are spin operators  eo. ( 5 ± » - ± a > i ) -• 1(2.1-1)  eqf6 x* -  rY* =  d e f i n e d t o be:  - - A i=  e a ^ ( L L t i t t )  t  1(21-1)4  (D  1(21-1)4  The  §>*  nucleus  are operators  and they  a r e g i v e n by:  Ba^lfO^tivV),  Vij  are  function. to able  B*=AVzz.  operators  electrostatic  w h i c h a c t on e l e c t r o n s s u r r o u n d i n g t h e  which  potential  • -  fi~a = l  yield when  The h a m i l t o n i a n  given  second  applied  Abragam  (2).  electric  field  the is  in this  If  the p r i n c i p a l  we  choose  gradient  (e.f.g.)  an  to  way  can  -<_>  of  the  electron  wave  approximation  and j u s t i f i c a t i o n be  f o r being found  in  a x i s system of the  describe  both  spin  (1) becomes:  e c| Q c^fl - i ( i - n ) - i - i n ( T * + T - n 4 1(21-1)1 A J z  to  the i n t e r a c t i o n  c o o r d i n a t e s and t h e e . f . g . , fta =  derivatives  i n eq.(5)  the true quadrupole hamiltonian, to express  (X/xx-Vyy t a i V x y )  V TI= £ I  \Ai-v  ¥ y  (f)  19  In  the  presence  hamiltonian  of  a large  external  magnetic  field,  H,  the  total  becomes: .# = - y * H - I + & a  We z  now w i s h axis  of  .(which  to  the  is  transform  system  defined  principal  axis  rotation  described  given  in  terms  is to  system  of  the  parallel lie  is  by  coordinate  to  along  the  the  obtained  the  spin  spin  Euler  laboratory  + j 3 e ' * s i w( B) + n e~* ( i i ^ s c g ) ) e  The  1  z  purpose  of  interaction frames. axes  of  around axis  The  this  two  Euler  the  coordinate  the  z axis  with  the of  lab the  discussed  of in  systems. an a n g l e  hamiltonian  e*. a  by  rotations  c< ,  for is  the  but  it  purpose  important  later  *-n - n ee llc<  to  of  for  section.  i w  system  the  (3); lab  the  via  a  then  (8)  system  is:  6 sI Ni ( N (BB))/|±i J +- CcOoSs((BB ))\jee * \ j f i - i X l  ^i-cos(B)jV''j f i a  diagonalize of  the  aligns  to  the  not  I  a  hamiltonian  z is  x and  y  symmetry  the  will  some e x p e r i m e n t s  the  which  not  change  simplicity  The  align  molecular is  Zeeman  molecular  rotation,  rotation  does  the  l a b and  required  final  This  If  >llf  are  The  field  + ne*siN(B)|cosip)-ij e jf»i  +tie'  is  the  •V 1  misalignment  x and y a x e s .  s p a c i n g and value  1,r  transformation  from a g e n e r a l  Only  operation level  2  1  that  z axis).  lab  to  so  magnetic  <X , |3 , V  refered  s i N ( f i ) | c o s j B ) - i |e* e  siNcpjcos(f3)-ne  the  angles  + |-3e"'*snvi(p)Cos (Bi+ne^siNtp^i+cosiBije* +p e  external  from  coordinates  system  energy  set  a=0.  and w i l l is  be now  20  reexpressed  in  systems  each  as  terms  of  basis  two  has  different  merit  spin  for  operator  certain  basis  types  of  calculat ions:  = - Yf.H l a + U)a I j J T (3COS (B) - 1 + TI SlN^tB) COS*( X)) Tzi. A  + (3 -C5T Sil\J(B) C 0 5 ( B ) - f 5 : n 5iN((3)C05(p.)C05CA^) T X H - i J T T l 5IIV/(|3) 5|I\J(A*) T « , z + j 3 il3C  51  NAB)  ^l-COS(B)j CQ5(Ay) + X n ^-t-COS(fi)j A  t T l A  2  C06(Xy)j  + | V X n | l + C05(p)j 6lMqY)-^njJ.-COS(|3)j Sl^(SLnjTxy j a  ^ - - Y h H I2 -r W  Q  Z  "fV-y  l  (±o)  r j J J 3O0S*(B)-1) +_L5iW*p) COS( A 2 0 j ( 3 l i - I ( I t l )  + | 3 € l N ( B ) C05(6)-r_5lN((3)C05((3)CO5CA!r)j ( ± ( ± + + X - ) + ( 1 + + X - ) X z ) 2  + i n si*/rp) s i M ^ y ) ( ± z ( i * -X-) +  + 17L 51 W( R V) / 5  1  (X*-x-)x«)  ) - C O S " ^ ) j f±I - ±-) ) 4  ©  21  A.  P E R T U R B A T I O N THEORY AND T H E F I C T I T I O U S  The  most  efficient  terms,  is  and  determine  to  #  matrix  =-U)o±  2  which  elements  gives  due t o  calculation  of  I  will  the desired  the  the  governing  To do t h e c a l c u l a t i o n  the hamiltonian a  see e f f e c t s  t o do a p e r t u r b a t i o n  transitions. of  way t o  SPIN 1 / 2  nonsecular  eigenstates,  magnetic  use t h e s i m p l e s t  +  FOR  4^  Ye = i J a S l W ^ p )  fcCEi-rii)  +  A = OV *  SpiN-l  form  effects:  + _k(3±* -1(1+1) + A ( I z ( T + ± - ) + ( X - + - X ) ± , ; +  £j«=3e*qo.  dipole  smfft)coses)  (_?)  4 W-Ho +H '  U/RiTlNG and  the zero  by  | >, +  theory  order  |0>,  a  eigenfunctions  |— >•  Using  the eigenfunctions  to  i>'- lm) +£__HK__J K>  k. Em~E\K.  and  H o = - u ) o i z + 6 J ( 3 ± ^ - I ( X + l))  t h e Em a r e e i g e n v a l u e s  of  are eigenfunctions  standard  nondegenerate  first  order  kpm  HK-m = < k l H ' l o i )  Ho.  i n H'  of  given  perturbation  are given  The r e s u l t i n g  Ho  by:  eigenfunctions  are: l+)'--l+>+  1 _ A IO) + ^ej~> (CL)«+ 3 bJ ) 0)o a  10>'= 10> I->'=  Although  the  unchanged  in  dependent  l-> ~  A f X A I +>' +- _ _ _ A _ J - > (k>° + 3 U » ) (LJo-30Ja)  A I T A I0> — I +> (30Ja-CJo) U>  energy first  energy  (yh  difference  order,  between  the levels  difference  acquire  i n second  |+>' an  order.  and  |~~>'  is  orientationally The  perturbation  22  expansion  f o r the  eigenvalues i s : Yt  +  The  result  of  difference  the  between  iH^ml*  second the  |+>'  |c^m  order and  (f§)  calculation  |—>'  of  the  levels i s :  EC ~ E - = A cJo -i-U)* SlN*(p) (CQ5*(6) + JL SlN*(ft)) *  ^3  where and  the  z  Let dipole the  between  the  symmetry a x i s of  the  e.f.g.  direction. us  now  calculate  radiation  |+>'  (l6)  4  U)o  i s the angle  energy  |—>'  at  a  the  matrix  frequency  elements  2Wo,  as  this  for  magnetic  i s resonant  with  transition:  <+LX«l-/= -  SW*(p)  _  ZDo  (g)  <+IIx|->'=JA)aS»IVl(p) COS(B) As  the m a t r i x  elements c o r r e s p o n d i n g  vanish,  i t i s p o s s i b l e to  |+>'  |—>'  and  direction, system |+>'  and  detected Let  using  and  is  and  |—>'  radiation  oscillating  in  a  I—>',  state then  polarized us  polarization,  induce  at  i n the  By  z  Ix  the  do  the  the  states  x  or  same r e a s o n i n g  phase c o h e r e n c e magnetisation  not  z  i f the between  would  be  direction. the from  is complicated;  case a general  unless  been p r e p a r e d  using a transverse f i e l d  the  longitudinal  induced  between  in either  a net  on  e v o l u t i o n of  coherence  2Wo.  oscillating  concentrate as  polarized  x and  transition  transitions  w h i c h has  an  to t h i s  at  magnetisation  the 2Wo. we  of  longitudinal  state initial  of state  |+>' has  When c o n s i d e r i n g can  ignore  the  23  presence signal r.f. and  of the middle and  is  field. was  .(4). this  level  not c o u p l e d t o the o t h e r l e v e l s  This approximation  first  applied  h a s been  t o magnetic  effective  two  level  The n e t  considered  as  a  observables,  which  commutation  relationships.  described  anisotropic effective Fictitious  result  two  level  can  be  as t h e r o t a t i o n  and  system  gyromagnetic  applied  fields.  The  that  system to  have  state  of a vector  in  i s equivalent  ratio,  common  the  with  of a  to a  form o f years  by Bloom e t .  spin-1/2  system  three  may  be  fundamental  angular the  al  name f o r  i s the " F i c t i t i o u s  chosen  the mathematics  ).  is  The  by t h i s  known f o r many  resonance  ( s e e a l s o Feynman e t . a l . ( 5 )  approximation".  space,  |0>* a s i t d o e s n o t c o n t r i b u t e  momentum  system  three  c a n be  dimensional  t o a s p i n - 1 / 2 w i t h an  e x p e r i e n c i n g some c o m b i n a t i o n o f  I will  spin-1/2 approximation  now s y s t e m a t i c a l l y f o r our system.  develop the  24  B.  FICTITIOUS  By  SPIN-1/2  assuming  frame, d i f f e r and  that  the  form:  that  only  the molecular  the e . f . g .  is axially  a  | —>  symmetric,  [ ( - i ( 3 i i - ± ( I + l ) j  3  has  and  the  by a r o t a t i o n o f 90 d e g r e e s a b o u t  6 a = - U ) o I * +U)  which  frame  *  '  the hamiltonian  +3.(±l+±-}  ]  representation  i n t h e |+>,  3  •(Ja  fro.--  4  3 Lk  4  1  J l 3  LJ« /I  _TL  takes  @)  basis:  •Do  the x a x i s ,  4  the f o l l o w i n g matrix  Do  laboratory  @  1.  1  (Ja  |0>,  25  The  hamiltonian  can  be  diagonalized  by  a  similarity  transformation:  T  S  K  =  [\2  1  a  XL*  b  8=f 1 + 1  ( U . -h-Joo +  J1 ")' X  XL  IT  XL  2.  If  we  total  also  apply  hamiltonian  a longitudinal field  oscillating  +  (Ji COS(  ZLdot)  e  Y e - K S - K S ( ( J o A > V i L * ' )((Jb--Jub*+ JL 7  coherence the  follows.  between  the  density matrix  following  c a n be  I f the s t a t e s of i n t e r e s t  only  involve  highest  and lowest as  matrices: Ix=l  o  A  A  -r A  i  (A!)  and  may be e x p r e s s e d  X* =.1  )  w i t h /\jL0o+ s\?~  The t e r m s e a n d d a r e s m a l l compared i n what  the  i n the primed b a s i s i s :  -•wTTsiF  ignored  a t 2Wo,  i=  l I I  a  energy  eigenvectors,  combination  of the  26  We  can  with  remove components o f the  above m a t r i c e s ,  to  the  in  terms o f  As  e v o l u t i o n of  the  the  1/2,  down by  the  as:  + Qx( Ix  have be  hamiltonian  angular  magnetic  ar\d  as  a  field,  s t r e n g t h W1.  The  not can  commute  contribute be  written  commutation  ficticious  and  solution  c o n d i t i o n of  (?<t>= X-r  which  momentum  visualised  initial  matrix  components do  operators  i n s p e c t i o n f o r an i-%,y,«  these The  static of  hamiltonian  system.  s y s t e m may  field  (pco)= 1+ ^ d i l i i  as  'operators  experieYicing a  oscillating  If  primed  primed  relationships  the  the  the  a  spin-  transverse  may  be  written  form:  ( a* COS( 2 0 , C l y SlW( AU.  t))  la  CO'bCXUo-L) -r Xi SlW^U.'-fc)) +-( Oy COS(xU y t ) 5 l W ( A . U . t ) ) a  i  x(iy cosdu't)  +  the  s t a t e of  initial  equilibrium  x  a'={ijf+i?  siiMUcJo't)).  state  the  i n the  system absence  i s taken of  time  @  to  be  varying  the  thermal  fields  we  may  terms  of  write:  Trae FOR  BU»«1  Trae- *6  Oi^flBAlJo  The  primed  laboratory  (A?)  frame angular  operators momentum  can  be  expressed  operators  by  in  applying  the  27  transformat ion:  T Ij T  ,  TIxT  TlyT  =-lTV  = L)° X* A or  TV-y  +• IL  ej;  A  = OJo TV-y* - II. Xz A U '  The  T L T  y  AO,'  relationship  represented  between  as  problem a f t e r  the  two  in- fig.(3).  the  systems The  longitudinal  can  be  solution  pulse  conveniently  of  the  i s represented  evolution  as  a  vector  A  precessing the  Iz  about  axis  be  Iz'  results  magnetisation can  the  axis. in  oscillating  represented  frequency  of  the  rotating  frame  at  i n an  applied A Iy'  a  The  linearly 2Wo'.  r.f.  i s not  It  field  pure  double  frame  interaction  represented typical  the  system  quantum  in  values  interaction length  can  the to  picture  an  idea  density  for  of  matrix  the after  the  laboratory  a  can  of  put  of  Iz'  the  to  primed when  i n some  expected.  longitudinal  pulse  T is: (?<:T) = fi( I - A j x _ . ( C 0 5 ( Z U ) . teJ) l a + S l N U U i t f e T ) ± y ) )  the frame  rotation now  the  values  state  signals a  at  that  special  We  pulse  rotates  succinctly,  frame.  the  noted  i s not  on  longitudinal  be  from a  More  transformation  laboratory  get  to  transformed  coherence.  picture  should  only  t h i s motion  during  frame w h i c h  proportional  be  of  polarized  Evolution  interaction  p u r e d o u b l e quantum c o h e r e n c e , and r.f.  projection  @>  The of  28  Figure  The  primed  reflections  3 - R e l a t i o n s h i p between t h e p r i m e d and unprimed frames  frame  can be o b t a i n e d  i n t h e I x , Tx-y,  Txy by: g'TfliyPViL \  the  from the l a b o r a t o r y  plane followed  frame  by a r o t a t i o n  by  about  29  The  maximum p r o j e c t i o n  of  oscillating  iiimox) = zn  of  90  a  experimentally  attainable  magnetisation, after  degree  compared  90 d e g r e e p u l s e  a  factor  principal  |0>  the  magnitude  state  Larmor  '  will  into of  normalization  F  of  the  m a g n i t u d e of  magnetisation  frequency,  is  the  induced,  reduced  orientation  of  by the  done  the  are  Because  preserves  the  |~>',  and  these  can  be  the  the  admix  admixtures  affect  interest  axis.  and  the  disregarded.  between the  the  the  relationship  two  excitation  i s invariant  change  equation  i s a more the  via  If  hamiltonian,  frames  between  experiment  lab z  not  of  same but  transformation  relationship  do  wave f u n c t i o n ,  laboratory  t h e ..result "of any  c r y s t a l about  but  which  element  essentially  the  phase  operators  f o r t h e most g e n e r a l  primed  rotation.  and  which  is  results  e'gq  perturbed  calculation  the  for  41(71-1)  the percent  of  The  arbitrary  r  matrix  few  response,  o  contain  |+>'  amounts t o a  between  o f W1.  microseconds  the  axes:  hamiltonian  the  values  at  %(do~15o"  the  (9),  80  is  transverse  4  the  X  t o the  SI = 5Sll\A(3)  and  £iq)  pulse  the  is:  (Jo = A T T A / 6-q x 10* r o d s" ,  and  1  length  the  SIM(3.(JiXeT)  Da = 3 e * q Q - = 3.TT A 3, x IO* r o d s"  If  z magnetisation  to  general frames and  the  rotations  30  C.  PREPARATION OF DOUBLE QUANTUM  Preparation achieved  in  Two h a r d Method now  of a s t a t e of double  three  ways.  (3) A s o f t  (1) h a s been d i s c u s s e d the  quantum c o h e r e n c e  (1) A l o n g i t u d i n a l  90 d e g r e e p u l s e s .  describe  COHERENCE  other  pulse  double  i n the previous  methods  can  a t 2Wo.  quantum  be (2)  pulse.  s e c t i o n and I w i l l  in detail  (see Pines  et.al.  (6)). D.  TWO HARD 90 DEGREE PULSES  In  this  transverse applied  pulse  evolution  regime,  which  of the r . f .  splitting. quadrupole treated  It  is  field  phase o f t h e e x c i t a t i o n .  form, degree  equation  with  <o(o)^(I-&(JoIi) , and a pulse  about  examined  means  that  i s much l a r g e r  during  a s an i n s t a n t a n e o u s  is  a t the larmor  legitimate  interaction  Liouville  ,T,  the  i n the transverse plane  strength  the  case  to  about  hamiltonian  pulse  along  Larmor  frequency  frame  which  t h e +x a x i s .  m o t i o n c a n be s i m p l y  time  integrated:  a  pulse  may  be  an a x i s s p e c i f i e d by i s a solution  t o the  condition  of the  consisting  and then  a n d we c a n s o l v e t h e p r o b l e m  lose their  quadrupole  of  a  90  another  90  The p u l s e s a r e a p p l i e d a t t h e  r o t a t e s a t Wo a r o u n d  frame t h e p u l s e s  the  and the  f o l l o w e d by e v o l u t i o n f o r t i m e  under a s e c u l a r q u a d r u p o l e h a m i l t o n i a n ,  degree  the e x c i t a t i o n i s  than  initial  the x axis,  hard  i g n o r e e v o l u t i o n due t o t h e  What we seek an  the  frequency,  the p u l s e s , and rotation  in  the z a x i s .  i n an  interaction  In t h i s  rotating  dependence and t h e e q u a t i o n s of  31  =  C O S ( < J T ) ± +SlN(UaT) Txy))  fl(l+UW  a  @)  a  Kt  So  t h e s y s t e m c a n be t a k e n  state  of pure double  evolution  quantum c o h e r e n c e ;  of t h e system a f t e r  coherence  is  prepared  essentially  via a  differences E.  from a s t a t e o f  THE SOFT DOUBLE QUANTUM  dynamics  is  diagonalised and  t h e Larmor  under  exactly  analytically  quadrupole  assumptions  by a s o f t but  the pulse  frequency,  -^-.-  a  The  quantum  pulse;  there  are  important  i n t h e s e c t i o n on p o w d e r s .  PULSE double the  quantum p u l s e conditions  soluble  as  the  for arbitrary  frequency.  that  c J a T _  to  t h e same a s when t h e s y s t e m h a s been  which a r e d i s c u s s e d  straigtforward,  provided  order  the p r e p a r a t i o n of double  longitudinal  Preparation  Zeeman  The  listed  below  hamiltonian  values  problem  i s n o t q u i t e so  is  of  the  can  r.f.  be field  s o l v e d under t h e  i s a p p l i e d i n the transverse plane at  and t h a t  the  quadrupole  hamiltonian  is  secular: 6---WoI» +  After  making  a  rotating  is  g i v e n by:  fcfcL(3±i--l(±+l))-XCJ COS(0,t)±x  the rotating frame,  wave a p p r o x i m a t i o n  the hamiltonian  presence,  f o rthe i n t e r a c t i o n  and t r a n s f e r r i n g t o  becomes t i m e  6i = U a ( 3 ± £ - I ( ± + l ) ) - U i l *  Eigenstates  @  A  i n d e p e n d e n t .and  -(34)  p i c t u r e hamiltonian  and absence of t h e r . f .  field  are given  i n both  the  in fig(4).  32  Figure A)  Shows  the  a b s e n c e o f an a p p l i e d r . f .  level  the  4 - Eigenstates eigenstates  i n the r o t a t i n g  of the r o t a t i n g field.  B)  d i a g r a m i n t h e p r e s e n c e o f an r . f .  frame  frame h a m i l t o n i a n i n Shows  field.  (x3a  ~J—  a.  3  3  6Jo.-h  3  S  6  _ I 0 > 3>  1.  6= A A #=10(3 T z z  energy  (See page  l">  l+ >  the  ( c J a + ecoi") A  31)  33  The  interaction  picture  e i g e n s t a t e s a r e e x p r e s s e d i n terms of  a n g u l a r momentum s t a t e s q u a n t i z e d a l o n g t h e z a x i s a s :  ii>=A(i+>-n-> + e i o > ) e  '  l  ,  iz> =_i_(i +>—i-))q  1  fx 3>= B(n-> + i->-^io>)e 6=  \/ (Jo" +9  -H  0)1*"  A = (*-t-eA)~*  l<J3  U)a  ,  '  t  ,  1  f  dynamical problem  initial  state  can  be  state  absence  of the a p p l i e d  |3>:  , Ui = - u)g - 1 V  -6Jz»(Ja  i s now  initial  convenient  (Ja. - y/ bJo +  E>=(x+J3 ]^  Ux - - (Ja -t-1 / i J a + 9 t J *  The  3=  solved,  determined  and by  a  i n t e r m s o f |1>, |2>, |3>.  to  field  are  have t h e s e s t a t e s  I+> = AI±> + 1_IX>+BI3>  |0>,  +  expanded l-? =  any  |—>  i n terms  flll>-J_+  and  i n the i t is  o f |1>,  |2>,  B|3>  fX  lO> = Aei±> + B : B l 3 >  state  from  T=0 e x p a n s i o n o f t h e  ft  From an i n i t i a l  evolution  The e i g e n s t a t e s  | >,  ,  (Jg + 0 U&  (36)  Iz=\+><+1-><—  | t h e system  i s described  at  time T by: QLt.)- I a ( f t C05[ U,~Uz)t  + Q COS (  + "CC fxy  -t-2.3 Z SIN((Ji-[Ji)t)-t-^ fx (eA COS((J .-(J )t + 5& COS(6Ji-iJv^  (2ft5"5lN(U3-U,)t  from  tedious  algebra;  defined  the  3  y  (9 A*S I W( UJa- lj,)-t 1- 3 B*Si M ( U i - U $ 2  :i  Z  —  Passage  - U) ) € J + fiT I  X  ®  J  1  —  initial  state  expansion  of  to ^(+)  on p a g e ( 9 ) c a n be p e r f o r m e d  eqn. in  (37) i s s i m p l e b u t  cartesian  by e x p a n d i n g  operators  t h e fundamental  34  observables fairly  in  the  |+>,  complicated for  considerably  and t h i s  arbitrary  W1  w h e n W1<<Wq o r W1>>Wq.  i s nothing  evolution  | 0 > , |—> b a s i s .  under  more  the  than  r.f  and  Wq,  but  (37)  is  simplifies  F o r W1>>Wq:  the hard  field  The s o l u t i o n  pulse  is treated  approximation  where  as a r o t a t i o n .  When  W1«Wq:  and t h e system p r e c e s s e s quantum  coherence,  coherence W1/Wq.  between  the  a  function  the  probability  quickly state the  fact  but  there  tensors  of  of  time  t o be  the  in  in time  than  of  the  of  the  above  limits  t o be i n  other  rank  pulse  soft  |0> r e m a i n s  small |0>  to  we s e e  much  more  t o be i n t h e  rank  tensors  of  states  case  develops  the  from  we  in a state  |0>  In  with  prepared  h a s some s e c o n d  probability  double  involving  quantum coherence  second  of  states  the probability  of double state  to a state  |0> a n d |—> v a n i s h e s  state  value,  Iz  For the hard  evolution.  transition from  or  system  the  t o be i n t h e s t a t e  than  a  fig(5).  the i n i t i a l  during  |+>  the p r o b a b i l i t i e s  i s no c o n v e r s i o n  order  greater  and  The p r e s e n c e  that  probability lowest  of  understanding  and t o a l a r g e r |—>.  amplitudes  evolution  as  order  |0>  and examine  p(o)=|+><+|  of  the  To g e t a b e t e r  consider  from a s t a t e  i s due t o  components,  to first pulse even |+>  rank  limit  the  though  the  is  much  |+> t o | - > ( i n  fact  one h a s t o go t o t h e n e x t  to see a t r a n s i t i o n  from  |+> t o  |—> a s s u c h a  35  Figure These a  equations  spin-1  5 -  Transition  describe  initially  probabilities  the evolution  in a state  |+>,  of  the p r o b a b i l i t i e s  t o be i n t h e s t a t e s  |+>,  for |0>,  hr>.  \f> = £ C i ft) I i >  i= t, o , -  i  IC*<T)I* =  3 + e x  ± cos((j.t) + i c o s ( a u . + ) 8  ICo(t) l*= 1 ( 1 - C 0 S ( A ( J . + ) )  4  IC-W  I = 3 - l C O S ( U i t ) + J-C0S(2(J.+)  a .a  e  |C+(t)l = 0 - 4 6 + 0 - 4 4 C O S ( 0 - 5 t ) + 0 0 3 C 0 5 ( 3 + ) + O - 0 7 C O 5 ( 3 - 5 i ) ICo't) i = a i l x  (l-COS(3t))  |C-(+) l*= 0 - 4 - 6 - 0 - 4 4 C O S ( 0 - 5 t ) +• 0 - 0 3 C 0 S ( 3 t ) - O-O ? C O S (3-5 t )  36  Figure Graphs  of  the  e q u a t i o n s on t h e quadrupole compared the  r.f.  transition previous  interaction  interaction.  probabilities  page.  or  to the quadrupole Zeeman  5 continued  a  A)  large  Shows r.f.  interaction.  interaction  is  l  c =  the  by  case  Zeeman  to  the  the  of  no  interaction  B) Shows t h e c a s e  comparable  a= lC*W| , b = l c . o ( 0 | \  described  where  quadrupole  \C-(+)r*  A  0.0  1.2  2.4  -  3.6  Time  4.8  6.0  37  process never  is  processes time.  photon  one  must a d d  frequency  of  |0>  the the  zero  of p r o c e s s e s  of  to  destructively.  |+>  the  |—>  processes The  cancelled F.  THE  the  development a  of  state  longitudinal which  matrix is  longitudinal  the  elements  the  symmetric £ = -y  Hit  as  to  in  multiphoton  state  i s not  to the  at  photon  commensurate  even  t o be  from  the  the  to  the  state  connecting  though  i n the  |+>  and  |0>  process  phase,  any  r . f . , thus  of  |+>  |—>  such  state.  is exactly  |-r>.  magnetisation  2Wo,  at  hamiltonian of  formalisim I w i l l  by  in this  l z between  |+>'  o f e q n . ( 1 0 ) t o an describes  from  the  As  parts m=1 way and  now a  system  we  of  the  does  prepared  hamiltonian are  little  '.  (£d)  the  are c o n s i d e r i n g  o r m=-1  |->  look at  The  ignored. to  effect  hamiltonian  axially  symmetric  e.f.g.,  essence  of  axially  the  non  well:  + UaJJJOCOS^pJ-ljfzz  (j(o)oc I + a f xy  two  amplitude  differing  calculation case  given  to the amplitude the  many  |0>  highly  |-r> t r a n s i t i o n  quantum c o h e r e n c e .  states  a specialization  but  10>  to  relative  going  density matrix  magnetisation  of  are  in  DOUBLE QUANTUM COHERENCE  of d o u b l e  couple  Truncation  the  from  EVOLUTION OF  Using  in  passage  considered  i n any  10>  the  For  in  t o be  applied r . f .  changes  by  difference by  t o be  c o n t r i b u t i o n s add  are mediated  phase  or  contributing  interfere  amplitude  c o n t r i b u t i o n s from  the  |+>  level  The  interactions  t o compute a m p l i t u d e s  either  to  process).  because the  and  The  phase sum  two  develops  nonlinear  with  a  + J S i N Z f B ) ( C O S ( 2 « x ) f x ' V + SlW(2<x)f xy) "  38  Successive which It  commutators o f H w i t h  Liouville  subspace  i sconvenient  which  will  the  suffice  The d y n a m i c a l s u b s p a c e  this  /\+-  transformation  -A A  y\ ft  /Kit  (>(t)'=  solution  Q, (+) l z + C\x(t)  QM) = Tra(X*  shows  t h e dynamics. frame i n  of. a b o u t t h e  A.  l z , Txy,  3_ (3 C O S  Tx-y,  invariant  assumes t h e f o r m :  f2"  +CU(o)Tx -y 1  of t h e L i o u v i l l e  i sleft  (6)-l)T^z+_3_Txy  2.  ^lo) = O i ( o ) I z  general  A  and t h e h a m i l t o n i a n  LN  The  to describe  a r e r o t a t e d by an a n g l e A  under  condition  t o do t h e c a l c u l a t i o n s i n a r e f e r e n c e  the spin coordinates  z axis.  initial  +-Qs(o)Txy  1  •<Q>  equation i s :  f xy + Q (t)f x'-f' 3  (3(t))- ( J o ( UoX-Q^Q)-tSlQUo))  lc^S(.1uU)  + f Q, (o)/1 - U)o  + JL-Qii^ SiM(2(Jot) V OJ^iL " 1  a  ( J o W  - f l - L  ((Jo'i-iL )/  I  tJo  1  X  + iI  1  J  VcJoSit 1  cuct)  = a»co) cos(ii^'t)-VuL±JL-(a'W i -  (J°  .  \ - .a  (J°  CKM  swdu'-t)  --(4.51)—•  If  we l i m i t  the i n i t i a l  c o n d i t i o n t o t h e form:  (D (O) = Qxlo)  TV-y 1  + Ch(o)  Txy A  and  consider  only  the expectation  value  A  o f l z " =Iz we  find:  39  (Jo fl-C0SUa't)) +  - Q. (t)^ Qi(o)Jl  (u)oVii ) z  During  initial where  are  treated  the  spin  as having  coordinate  for a crystallite  l  powder  sample, a l l  secular hamiltonians. after  preparation  The A i  is  Txy,  s y s t e m i s t h e same f o r a l l i .  The  w i t h an  the  orientation  f r a m e d e f i n e d by t h e E u l e r a n g l e  with the f i n a l  +xi  of a  s t a t e of the i t h c r y s t a l l i t e  hamiltonian lab  VLJO  v  the transverse e x c i t a t i o n  crystallites  XL CU(o) SiW(iU't)  c<  w.r.t.  (we a r e o n l y  concerned  r o t a t i o n a b o u t t h e l a b z a x i s , p g . (I *?))  is  given  i n terms of the i t h s p i n c o o r d i n a t e s a s :  6-  Z  = r\*(<*0|^-yH±k + - L J a ( ( 3 C O S ( p ) - l ) T i z + 3. SIf\J ((3) T x y ) " Z  2  ftz  frfi)  @  A.  and  the  expectation  v a l u e o f I z f o r t h e w h o l e powder  i s given  by:  I f we assume t h e d i s t r i b u t i o n  < I* >- J Tra (  t o be  continuous:  R i W T*yR*(«)e  e  p((*)  i s the p r o b a b i l i t y d i s t r i b u t i o n  p(e*0  i s symmetric about  FUU) I a ) p<*) do*.  f u n c t i o n f o r o<.  ,  (g) and  if  zero:  IT  0 = J ( R a ( - < ) T x y fta(«0) pc*)cU  (g)  -Tr  We  see  that  signal w i l l  in  vanish  the for  case  o f a powder e q n . ( 4 7 ) h o l d s  excitation  by  either  a  soft  and t h e double  40  quantum  pulse  excitation as  the  or  two  i s used then the phase f a c t o r  initial  condition  c r y s t a l l i t e orientation excitation-response, powder.  hard 90 degree p u l s e s .  in  i s of  If l o n g i t u d i n a l no  consequence,  f o r each c r y s t a l l i t e depends on the such  a  way  as  to  maintain  the,  phase r e l a t i o n s h i p constant over the whole  41  G.  THE  SECOND ORDER SHIFT  If  the  equivalent  sample  is  a  nuclei,  the  s p e c t r u m of  the  m=2  transition  be  attributed  processes,  to  some  section.  For  quadrupole e.f.g.,  the  the  duration  the  intensity  e.f.g.  case per  be  is  a  symmetry a x i s and  the  given  of  static  |3  an  , the  way  can  one  axial by  the  i n t e n s i t y of gyromagnetic  frequency  most c o n v e n i e n t  function  with  broadened  effective  a  line  experimental  powder  The  with  relaxation  i n the  shift.  the  with  The  this  which e x p e r i e n c e s  on  spins  as  spin-spin  second order  excitation. is  of  inhomogeneously  i s dependent of  w i d t h of  isotropic  unit c e l l ,  magnetically  associated  The  described  an  with  frequencies  variety  of  dependent  number  of  wide  spectrum  powder p a t t e r n  crystal  single line.  which w i l l  nucleus  the  ratio,  the  a of  orientationally the  is a  perfect  and  to  the  express  a n g l e between  the  field:.  o  The not a  degree only  of  of  the  uniform  of  the  applied  initial  to achieve. effective  excitation a crystallite  This  r.f.  condition  is  due  gyromagnetic  the  external  W1T,  of  the  static  a  excitation.  the Ye  small  field,  but  and  The  •  angular For  is  dependence  a given  d e p e n d e n c e on the  is a  function  also orientation,  for a l l c r y s t a l l i t e s  to  ratio  s p e c t r u m a l s o has  field  receives  strength  impossibility  hence  impossible of  sample t h e the  shape  strength  duration of  the  of  product,  achieving  a  42  uniform  initial  c o n d i t i o n over the whole spectrum a l s o has some  important consequences as i t means the F o u r i e r transform free  induction  decay  temperature magnetic pattern that  i s given it  has  i s no  susceptibility.  in f i g ( 6 ) ; sharp  equivalent The  to  theoretical  the high powder  the main p o i n t about the spectrum i s  edges  c o n s i d e r a t i o n s , a r e the only accurately.  longer  of the  which,  features  due which  to can  experimental be  measured  43  Figure  The  frequency  degrees this  6 - The  is  second  normalized  t o remove t h e f i e l d diagram  intensity  order s h i f t  Ti=0,  the  to  powder  the second o r d e r s h i f t  dependence  of  nonaxially  symmetric  spread over a wider  range  pattern  the  spectrum. spectrum  of f r e q u e n c y .  1.300 1.114  -  0.928  -  '175 0.742 C Q)  0.557 0.571  -  0.185 0.000 0.00  0.42  0.84  1.26  Normalized frequency  at  90 For has  44  IV.  The  equipment  multinuclear  of  low  must  Wo,  2Wo,  must  is the  of  these  stable.  output  components a t  of  Wo,  the  the  at  the  sensitive  detector,  can  lead  done a t  at the  although  the  highest  the  m a g n e t i c moment population in  the  Larmor  is  any  to  d i f f e r e n c e between  good  increase  double field  of  spectrometer  s i g n a l s are  small  signal  noise  to achieve by  must be  to  The  of  l e v e l s and  of to the  with  the Bo,  phase quantum  should  Bo,  be  because transition both  voltage Bo.  this  the  experiments  Wo  remove  at  single  field,  the  to  signal of  Wo,  required  doubling  input  ratio  linearly  the  filtered  coherent  proportional the  r.f.  must  r e l a t i o n s h i p s between,  external  gyromagnetic  the  capable  the  , the  detection  of  If  The  reference  frequency.  inversely  detection circuit,  a transmitter  frequency  doubler  value  effective  way  of  spectrometer  2Wo  at  standard  ratio  power.  to get  2Wo  ,  is a  a transverse  phase  because  present  processes  by  only  the  frequency  and  frequencies.  The  create  the  r.f.  averaged  means t h a t  to  The  and  signal detected  coherently  is  experiments  gyromagnetic  i s small  excited  b o t h of  be  the  The  1kw  least  This  frequency.  t o do  detection c i r c u i t r y ,  and  have t o be  stability  any  at  provide  spectra. and  noise  coherence  frequency  and  magnetisation  delivering  quantum  required  spectrometer.  longitudinal have,  EXPERIMENTAL  the  induced  45  A.  PROBE The  probe  unsophisticated series being  tuned  the  LC by  B.  with  no  of  rotation  circuit  is  is  a  matching c a p a c i t o r , matching  coil  dependence  The  here  geometry. the  about  with a r e s o l u t i o n  of  signals an  1  In  order  to  t h e p r o b e has  axis perpendicular  a to  degree.  COILS The  coil  of  two  one  excitation.  both  geometry basic For  transverse  excitation  i t was  sample  about  transverse  with  inside  filling  an  and  of  factor  sources.  To  the c o i l s  had  for  the  experiments  depending  on  the  nature  excitation  along  detection.  the A  a saddle  coil  external f i e l d  saddle c o i l  of  the  with  its  was  used  is inferior  to  a  of  the  a x i s p e r p e n d i c u l a r t o the e x t e r n a l f i e l d .  For  because  and  used.  i t allowed  longitudinal The  easy  was  the  saddle c o i l  the and  t o be  s o l e n o i d to decrease arcing  improve  saddle  the  the c a p a c i t i v e  d u r i n g the  separated  a  excitation  the d e t e c t i o n c o i l The  by  a s s o c i a t e d with  at  rotation  detection a crossed  transverse  solenoid.  factor,  for  the  was  filling  is  and  prevent  probe c i r c u i t r y  7) u s e d  in a l l f e a t u r e s except  used  was  solenoidal,  the  and  excitation  arrangement  coaxially  types  aligned  coil  reason  (figure  longitudinal  axis  solenoidal  on  reported  standards.  a d j u s t i n g the  which a l l o w s  the a p p l i e d f i e l d  experiments  NMR  circuit  angular  goniometer,  for  by modern  performed  study  used  coil  coil  coil  s h o u l d be  was  mounted mounted  detection  circuit  coupling to  noise  intense transverse pulse,  least  2mm  of  teflon.  the d e t e c t i o n i s almost  The  46  Figure A)  Shows  from of  the  the  the  two  coil  solenoid  longitudinal  field.  B)  by  7 -  arrangement a teflon  Coils with  the  shroud.  B)  excitation-detection  saddle Shows  coil  coil the  w.r.t  separated orientation  the  external  47  as  simple  as  inclusion build  that  of  a high  between fields  associated  a  matching  Q detection  the  coils  produced  by  especially  when a c c o u n t  high  C.  It i s very  as  the  The  to  use  is  a  of  l o s s e s , even  cross of  the  the  easy  to  the  points  configuration  i s such  i n v a r i a b l y pushed past  be  though  intrinsically  experiment  to  coupling  These  coil  the  difficult  capacitive  orthogonal.  taken  nature  should  are  with  poor  that  the  their  limit  and  in  chosing  replace.  SAMPLES i s u s e f u l to  sample  to  indication The  nuclear with  of  the  types  signal  of  ratio  a high  requires  that  the  maintained  as  longitudinal  protons),  only  ratio.  a  long  and  give  A  and  small and  the ,  fraction  fraction  constant  nuclei,  as  we  of  as  the  intrinsic  increases  linearly  need a  sample  s i g n a l means we  the  need t o  r e l a x a t i o n t i m e be  dipolar  coupling,  second order  (in  those  quadrupole  banding  systems in  of  be the  spectrum  mechanisims  shift  to  Phase  should  narrow power  broadening  have  short.  t o accommodate major  with  s i g n a l average  narrow  the  good  effective  p o s s i b l e , to allow  The  a  a  technique.  ensemble d o u b l e quantum c o h e r e n c e  s i g n a l and  the  these  small  small  This  spin l a t t i c e  excitation.  heteronuclear  is  interaction.  the  as  s y s t e m s amenable t o t h i s  interest  d e n s i t y of  coherence w i t h i n  the  of  considered  theory,  quadrupole coupling  large quadrupole  achieve  points  the  is  gyromagnetic  the  review  demonstrate  gyromagnetic  of  two  decision  components  It  a  circuit  power components a r e  these  excitation,  capacitor.  the  any  to n o i s e .  the  causes c o n s i d e r a b l e  prejudice  signal  with  are  containing imperfect  48  crystals.  The  magnetic  coupling,  are  coherence  dephases  coherence the  interactions,  especially  under  at  damaging  twice  the  quadrupole  obtained  on  available  as a  nitrite  the  double  rate  best  nitrite,.  NaN02,  single  crystal.  large unit  The  cell  The  constant  G QQ- =5.8flhz, n  parameter  n=0.4.  The  lattice  crystal,  and t h e o s c i l l a t i n g  power  excitation  obliterates  the s i g n a l  Piezoelectric various echo  induces  ringing  the  nitrogen  satellite  lines  of  orientation  the  habit  made  echo  were  inhomogeneously  ranged  from  the  isa  an  one  asymmetry  time  i s 1.5  ferroelectric  produced  by t h e h i g h  ringing, after  the  pulse  outside  which pulse.  by H u g h e s ( 7 ) and  suggested.  I u s e d an  spacing  being  the r i n g i n g .  The  and a r o u g h  determination  with respect  t o the c r y s t a l  as t h e goniometer  or  resolution.  broadened  broadening  4 t o 50  the  has  An a c c u r a t e d e t e r m i n a t i o n o f t h e r o t a t i o n  reproducibility  structure,  formed  were s t u d i e d  p a t t e r n s was n o t a t t e m p t e d required  crystal  studied  the problem  of the e . f . g .  (Fig.(8)).  field  i n NaN02 h a s been  ensure  solid  relaxation  f o r 300 m i c r o s e c o n d s  technique to observe the s i g n a l , to  i s an i o n i c  piezoelectric  methods o f o v e r c o m i n g  chosen  been  and  Sodium N i t r i t e electric  have  nucleus experiences a  coupling  a t room t e m p e r a t u r e .  was  a b u n d a n c e and a  results  quadrupole  seconds  quantum  Nitrogen-14  which  and the n i t r o g e n  spin  quantum  of the s i n g l e  of a h i g h n a t u r a l  interaction.  sodium  per  as d i p o l e - d i p o l e  an inhomogeneous Zeeman t e r m .  n u c l e u s of c h o i c e because  large  as  such  was  kilohertz.  by  d i d not possess the  The an  satellite imperfect  orientationally  lines crystal  dependent  and  49 A i  3 0 us  1  B  50 khz  Figure A)  Shows  sequence. zero  the  f.i.d.  B) The F o u r i e r  of time taken  8 - Satellite of  a  lines  satellite line  transform  t o be t h e e c h o  using  o f t h e above  peak.  echo  a Hahn  echo  with  the  50  Although  complete  orientation finding  of  the  satellite  rotation  the  z axis  position  lines,  diagrams  of  the  were  not  e.f.g.  measured  was  the  determined  by  w h i c h gave t h e maximum s e p a r a t i o n of  consistant with  the  known v a l u e s  Q  for  the  C\Qh  and  Ti.  The  determined of  by  s t r e n g t h ^ of finding  the m a g n e t i s a t i o n  dual c o i l  effective  A  To  homogeneous  factor  on  amplitude  was  the 1.5  d e p e n d e n c e of After  of  was  is  relaxation  fig(9),  of  signal  alternately  memory.  successive  The  double  added and  from  With  the  was  8.5  the  larger in  time  signal  to  of  the  a n o r m a l Hahn time  t h e r e was  due  echo  was  the  echo  of  some  were  angular  was  400  l e n g t h of p u l s e  The  into  double  double  where  with  ^  subtracted  quantum c o h e r e n c e  90  from  precesses  -  transfers  quantum  coherence  longitudinal  by  p n  which  degrees, the at  and  signal,  s u b t r a c t r o u t i n e , i n which  is shifted  the The  pulse approximation,  resulting  add  the  kilohertz.  a transverse f i e l d  soft  finished  o f f t h e magic a n g l e ,  Zeeman o r d e r  pulses  line.  examined  relaxation  doublet  u s i n g an  rotation  satellite  investigations  The  =80us.  averaged  phase  Wo.  degree  was  time.  with  u s i n g the  T=  was  the  degrees  irradiated  frequency  by  decay  gained  processes  preliminary  computed  given  a  .field  I could achieve  was  of  the  the quadrupole  then  a satellite  m i l l i s e c o n d s , although  t h e maximum amount of be  ratio  mounted a few  =25KHZ, and  can  1.4  satellite,  this  the  was  separation  of  establish  r.f.  r e q u i r e d f o r a 90  s h o r t e s t time  (non-refocusable)  performed  system  the  gyromagnetic  isolation.  time  transverse  a s s o c i a t e d with  arrangement  microseconds.  crystal  the  the  and  the the  acquisition twice  the  51  Figure A)  Shows  double  the  quantum  9 - Transverse  longitudinal pulse.  f.i.d.  excitation following  a  transverse  52  interaction by  f r e q u e n c y o f t h e Zeeman h a m i l t o n i a n , hence a change  of  phase  90  degrees a t the Larmor  of  t h e d o u b l e quantum c o h e r e n c e  by  180  longitudinal  tuned  t o 33.6fthz, and  tesla  (fig.(10)).  scanning  over  minimum  be  quadrupole  Do  angle  orientations  has  the and  coupling  the  patterns  found  to  the this  perpendicular  z axis p a r a l l e l .  by This  require  a  spectrometer range.  remembered  that  constant i s  When  the  crystal  known  o f 'iz it  is  f r e q u e n c y t o a v a l u e where t h e and  crystal. have  to  then The  been  of the c r y s t a l  the e f f e c t i v e  be  was  where t h e m a t r i x e l e m e n t  the  search second  plotted  fig(11).  One  rotation  E x p r e s s i o n s f o r the  gyromagnetic  ratio  when  for order  for has axis,  two the z the  resonance r\.^0  are  by: - 3 U)o  +  Uo.  3.4-  " 3 . S I N ( B ) ( 4 - C O S ^ ) -f±) + T l C O S ( K ) 5 l N \ 6 ) ( 4 C O S ( ( 3 ) - l ) A  1  J3Siw (:B)+ncosttir)(s^ a  J  LJo  + n ( z - 4-cos\xp)(cos\$) 3  and  ratio, i s large,  rotating  rotation  of the e . f . g .  frequency  an  can  coil  frequency range.  over be  a saddle  produced  200khz,  probe  i t must  gyromagnetic by  field  shift  unchanged  at  shift  orthogonal  given  phase  magnitisation  using  frequency  the spectrometer  resonance  other  the  remain  set  r.f.  order  I f the quadrupole  to  applied  approximately  of  oriented  corresponding  axis  second  f o r a resonance  vanishes.  the  longitudinal  was  resonance  enough,  characteristics  better  The  retuning  searching  pulse  the v a l u e of  the  i s narrow  may  and  the  degrees.  The  range  frequency changes  -ifj  i  53  Figure The  longitudinal  saddle  coil  with  1 0 - Longitudinal  signal  excitation  has been e x c i t e d  the o r i e n t a t i o n  given  and d e t e c t e d  i n f i g - 7 b on page  using 44.  a  54  34.015  33.860  30  0  60  90  Rotation angle y (°)  Figure  Rotation  pattern  perpendicular  with  to the  11  - Rotation  the  rotation  z  axis axis.  pattern  of  the  e.f.g.  oriented  55  33.875  33.850 N 33.825 O c  CD 3 33.800  cr o  33.775  33.750  1  1  J  1  0 - 4 0  1  1  80  «160  J  120  Rotation angle (3 (°)  Figure Rotation rotation  pattern axis  with  the  11  continued  z axis  of  the  e.f.g.  parallel  to  the  56  The  m a g n i t u d e of  be  a  complex  Euler  tfe  number.  angle  and  excitation resonance  frequency but  components of  the  orthogonal  that  a  D.  on of  no  When the  be  an  powder  duration  of  an  quantum  coherence  the  coupling  constant, the  ratio  optimal  oriented  only  and  linear  of Wq  t o Wo  of  i n the which  there  are  as  i t i s more  different  many  a l l five three  when  (2) and  the  by  the  Zeeman i s the  the  e.f.g.  is  a  is  a  comparison  of  the  pulse  order  induced  r e q u i r e d to into  double  quadrupole  interaction,  a  hence  parameter.  solenoidal  solenoid offers  the  coil best  rotation.  difficult  which  results  Limitations  important  require  excitation  factors  pulse.  the T  Zeeman  arrangement  sample d o e s not  as  are  find  p r o p o r t i o n a l to the  the  three  with  d u r a t i o n of  p e r f o r m a n c e and  the  ratio  method w h i c h  determined  excitation  z direction,  of  the  points concerning  longitudinal  the  which  for  p a t t e r n s about  excitation  inversely  crystals  longitudinal  to  orientation  the  EXCITATION METHODS  The  experimental  single  used  even when z a x i s of  along  For  be  Important  amount  is  the  gyromagnetic  can  s t u d i e d are  appreciable  f u n c t i o n of  expressions  i f rotation  i s the  quantum c o h e r e n c e .  transfer  The  the  be  is a  will  field.  signal  s y s t e m s w h i c h can  is  (1)  =<1|IZ|^1>,  in  the  f o r any  observed  samples  observable  the  n f O  effective  vanish  external  factor  effect  a r e measured.  COMPARISON BETWEEN THE  double  it  has  e.f.g tensor  can  t o the  For in  and  d o e s not  signal  parallel  phase  in principal  axes  Ye  that  it  #e  in general,  The  experiment.  complicated,  are  a p p e a r s as  to  characterize  methods s h o u l d involve  be  used,  spectrometer  57  capabilities explain  and  sample  the strengths  comparisons The  only  hard  typical  of a g e n e r a l pulse  excitation  method  excitation  microseconds,  and  the  spin  range of p u l s e  sequences.  angle.  The  has  is  the s o f t  splitting  becomes  too  is  that  and,  as  coil  system.  quadrupole larger coil.  in  the  The  A  single  sample  filling  of the hard  constants  range of c r y s t a l  is  splittings.  f o r angles  longitudinal  coupling  advantages as there the  case  close  pulse  >1Mhz,  that  pulse  t o a wide the  to  the  If  the  system  the  i s no t r a n s v e r s e factor  c a n be  f o r an splitting  longitudinal  pulse  of the s o f t  pulse  magic  becomes  angle  i t  practical  works o v e r  tremendous s i g n a l coil  magic  method, r e q u i r e s a two  >500khz;  has  with  b r e a k s down and i f t h e  o r i e n t a t i o n s and r e q u i r e s o n l y  coil  first  t h e power  useful  c l o s e t o the  pulse  10-20  method t o s y s t e m s  The main d i s a d v a n t a g e  i t c a n n o t be u s e d  be  well within  angles  approximation  large,  becomes more e f f e c t i v e .  make  because the  can  is  be  this  quantum  range o f q u a d r u p o l e pulse  to  or  intermediate too small  good  sequence  limiting  interactions double  i sto  and  s y s t e m c a n be s u b j e c t e d  splitting  soft  method  is  The d i s a d v a n t a g e  spectrum of the e x c i t a t i o n , small quadrupole  each  approach  nature.  the  quadrupole  The b e s t  and w e a k n e s s e s o f  for  order  time  properties.  coupling  optimised.  a  at  a much single  to noise  i n noise  and  58  E.  APPLICATIONS I  must now  experiments, the  same  compared  was  order  observe such  an  divides  attempt  quadrupole  into  powders  quadrupole  dominate  the  pulse,  resonance  methods w h i c h c a n  yield  primary  the  attainable  make  only  systems  to  experiment.  A l l i s not  determination asymmetry  easily  spectroscopic the but  and  technique  more the  i t may  be  used  relaxation  single  crystals  samples For  The  is  of  this  the  to those  this  with  size  of  interaction  v a l u e s of  finding  I t must be  will  r.f.  thus  it  field of a  is  spectrum  these  coupling  remarked  that  can  these  attained  longitudinal'  pulse  by  t h e power  some l i m i t e d  90  only  in  lead  constant  accurately  t o get  to  again  one  as t h e powder p a t t e r n s p e c t r u m  quadrupole  and  the  spectrum,  lineshape i s determined  lattice For  the  parameter.  more  because  of  edges  was  discussion  f o r t h e power s p e c t r u m  portion  lost  the  spectrum,  excitation  quadrupole  whole  a  of  aim  order  The  >500khz.  impossible  observe  central  first  of  limits  order  the  these  i n the absence  The the  for  crystals.  interactions.  it  possess d i s t i n c t i v e  by  method  this  second  pulse to cover  possible  single  constants  the magnetic  motivation  interactions.  and  coupling  interaction  spin  these  interaction.  powders and  longitudinal,  pulse,  the u s e f u l n e s s of  t o view  dipole-dipole  For  are  The  on  f e a t u r e s normally obscured as  degree  to other  information.  experiments first  make some s t a t e m e n t s  does to  and  a the  quantities  by  NQR.  fares spectrum  As  a  poorly, of  the  i n f o r m a t i o n on  the  rate. the p r o s p e c t s are a l i t t l e  brighter,  as  59  conventional isolation of  the  the  NMR  has  to  when  the  splitting  satellite  spectrometer  focus  transition  on  the  quadrupole  is large.  frequencies  b a n d w i d t h , and  The  angular  v a r i e s over  to produce  excitation  d e t e c t i o n method makes i t u n n e c e s s a r y  order  of  the  shift  pattern  characterised, parameter,  imperfect  crystal  frequency  of a  broadened  the  only  experiments with  method,  they probe  be  capable by  and  found. of  In  in this  to  the  Am=2  the  NMR  t h e s i s are  little  and  l i e parallel  to  easy  t o do  for  the  as  of  the  an  exact  of  not  is  small think is  not  but for  the  samples  longitudinal  turning z  well  such  which  NQR,  the  a  case  I do  or  more t h a n  second  asymmetry  character  summary  the  transition  giving information  interactions,  require  and  In  the  know  the  to determine  whereas t h e  order  to  interactions  measured.  conventional  described  a conventional F.  second  large quadrupole  pulse  line,  times  sample has  constant  i t i s often d i f f i c u l t  in  available  be  dependence  longitudinal  out  I f the  magnetic  can  The  mapping  easy.  coupling  shifts  satellite  is  and  very  on  i n t e r a c t i o n s can  technique  already  is  quadrupole  chemical  retuning.  lines  information  anisotropic  magnetic  satellite  in  rotation patterns  needs  frequency  lines  many  spectrometer and  frequent  doublet  the  coil  of  direction.  FURTHER APPLICATIONS  There are thesis  which  protons  near  broadening decoupling  some e x t e n s i o n s deserve  to  the  can  be  the  of  mention.  nitrogen  by  ideas  Most  nucleus  prohibitive.  protons  the  The  irradiating  and  presented  nitrogen  in  this  samples  have  heteronuclear  p r o b l e m can  be  them w i t h  resonant  a  dipole  overcome  by  r.f.  60  field  many t i m e s  would  be  inside  of a s a d d l e c o i l  larmor  may  spin are and  transverse  solenoidal which  The  proton  coil  to  twice  i s tuned  best  setup  mounted on t h e the  nitrogen  frequency.  The it  a  the s t r e n g t h of the c o u p l i n g .  ideas are also a p p l i c a b l e to nuclei be  fruitful  3/2-nuclei limited for  such  t o those  m>2  the p a r t i a l l y  to search  f o r unconventional  a s L i 7 , C135, Na23. due t o a s e c o n d  the e f f e c t i v e  of h i g h e r  rank  gyromagnetic  forbidden t r a n s i t i o n s  scale  The  resonances i n  admixture  nonsecular ratios as  s p i n , and  hamiltonian  a s s o c i a t e d with  VkJ&). n  , where n  i s an i n t e g e r g r e a t e r t h a n  1.  t h i s means t h e s a m p l e s would  to  of  Wq  have v e r y  methods.  large values  to  warrant  terms  study  by  have these  61  BIBLIOGRAPHY 1) S . Y a t s i v ,  Phys.  Rev.  n_3  2) A.Abragam, The P r i n c i p l e s P r e s s , London. (1961) 3) G o l d s t e i n , C l a s s i c a l Pub. (1959)  , 1522.  (1959).  of N u c l e a r M a g n e t i s m .  M e c h a n i c s (page 107).  4) M.Bloom, L.B.Robinson< (1958).  Oxford  Addison  Univ.  Wesley  and G . M . V o l k o f f , C a n . J . P h y s 36  1286  5) R.P.Feynman, F . L . V e r n o n , and R . W . H e l l w o r t h , J . A p p l . P h y s 28 , 49  (1957).  6) S.Vega,  and A . P i n e s , J.Chem.Phys., 66 5624  7) D.G.Hughes and Lakshman (1984).  Pandy,  (1977).  J . M a g n e t i c Resonance  5_6 428  62  APPENDIX A ~ ANGULAR MOMENTUM  Whenever point  a  o f v i e w o f quantum m e c h a n i c s ,  approximations For  a  concerning  completely  invariant  angular  result  momentum,  w.r.t.  conservation  strictly  introduction the w o r l d the are  original  the  partition,  variables  they  in laws,  the can  over the  approximation  angular  momentum  two s e t s ,  the  one s e t  the other  system.  and some o f t h e symmetry  The  operations  operations  unless  transformation  of  of they the  I f the i n t r o d u c t i o n of v a r i a b l e s  world be  complicates  excluded  and  some p r e s c r i b e d s t a t e . hamiltonian  excludes  a  small  the  replaced This  problem by t h e i r  replacement  by a number means a l o s s o f presence  means t h a t t h e e n e r g y  depends on t h e c o u p l i n g b e i n g  groups  the system and the r e s t of  f o r a l l p o s s i b l e motions.  which  be  These  The symmetry  into  outside  f o r example t h e v e r y  the o u t s i d e world be a c o n s t a n t  the  energy.  of a simultaneous  of  value  conservation  not  this  rest  will  of the magnitude of the  of  c o u p l i n g between  and t h e o u t s i d e w o r l d .  expectation  with  world  make  translations.  partition  s e t a r e no l o n g e r  unnecessarily,  of  the  performed as p a r t  system from  a  upsets  hamiltonian  on v a r i a b l e s o f t h e c l o s e d s y s t e m ,  with  of  the  projection  laws n a t u r a l l y  contains operations deals  system  and t h e i n t e r n a l  invariably  s t a t e and t h e h a m i l t o n i a n .  i n the conservation z  i s s t u d i e d from t h e  must  r o t a t i o n s and time  the  any a x i s ,  system one  the i n i t i a l  isolated  to arbitrary  invariances  and  complex many b o d i e d  full  of  coupling  of t h e system The a c c u r a c y  dynamical  compared  a  to  of  will an  description the  internal  63  hamiltonian, is  not  a n d on t h e r e s t  measurably  interest.  With  conservation  affected  these  of  a  momentum  longitudinal  experiment nuclear  spins  and  is  as  nonsecular  r.f.  conserved  field,  consider  the  a p p l i e d t o our s p i n  quadrupole  field.  If  t o be  hamiltonian  we  imagine  composed  i s taken  an  of t h e  from an  invariant  a  transverse  to a rotation  coordinates  about  the nonsecular  considered with  until  (a l o n g i t u d i n a l This  where t h e field  of both  at  initial  axis).  Wo.  consists  The  (conserving  the  hamiltonian  of t h e  hamiltonian and  is field  c o n s e r v a t i o n law  of the world  regains  striking  the p r o j e c t i o n of  To r e g a i n t h i s  enough o f t h e r e s t  angular  c a n c a r r y no  i s t h e most  system  case  the f o l l o w i n g  field  of  the spin coordinates  the z d i r e c t i o n  momentum on t h i s  t h e z component  the z a x i s ) .  f r o m n o r m a l NMR,  and  angular  not  momentum w . r . t .  difference spins  motions o f the system of  energy  oscillating  the  i n a s t a t e which  i n mind, l e t us  I z t o a s t a t e - I z ; we s e e t h a t  angular  begin  a  being  A.  momentum  for  and  the  i n w h i c h t h e s y s t e m , assumed  A  state  by  statements  system, which e x p e r i e n c e s and  of the world  axial  must  be  symmetry.  We  hamiltonian:  na  H(xtal) object  with  because small of an  i s an a p p r o x i m a t i o n  which  treats  s i x degrees of freedom.  the motions  i n w h i c h we w i l l  crystal  This approximation  of the  other  a r e r a p i d enough t o f o l l o w t h e c h a n g e s ,  equilibrium.  If  we  ignore  as  translation  and  an  i s valid  be i n t e r e s t e d i n v o l v e  energy changes, and t h e f r e q u e n c i e s  freedom  the  very  degrees  maintaining  consider  the  64  crystal  to  be  unrestrained  f u n c t i o n s c h a r a c t e r i s e d by momentum. parallel  The  eigenstates  A  of H ( x t a l )  system  main m a g n e t i c  c o n s i s t s of v e r y  for  field.  closely  The  has  energy  level  spaced  levels  i s , the  moment  i^.«(Jo  w  e  c  a  n  of a  s  s  u  inertia m  e  the  A  the  crystal  to  angular  momentum i f t h e q u a n t i z a t i o n a x i s  degrees energy  of level  levels  spin  field.  The  is  spacing.  longitudinal to  Wolz+WqTzz  freedom  lattice  by  at  the  A  has  the 1  ,  orbital  o -  \ xx  nucleus  we  and  are  invariance,(due rotation  must  be  freedom  except  are I will  eigenfunctions  -x  B  The and  -t A the  x  6 f\  the  ) ,  where  states  electrons that  coupling  and  to l i e along of  the  the  limit  the  are  the tensor  spin  rotational  interested  In  good  in  discussion absence  of  products  of  we  The  perturbation  iX  are  a simultaneous spin)  The  only  H(xtal). g>"  the  are  operators and  spin  is  one  H(xtal)  between d e g r e e s of transformation may  as  on  the  second  rank  involving  i n i t ' s immediate exact  use  operators  acting  transform  true perturbation  assuming to the  z axis;  A —X  momentum  tensors.  particular  2Wo  spacing  l a r g e r than  we  A  and  r  angular  spherical  however  C(R  the  s t a t e s of  i s chosen  level  much  o f , Wolz+WqTzz, and  form,  energy:  spherical.  again  such a p e r t u r b a t i o n .  x  diagram  with  about  be  once  energy  Once a g a i n  coupling  eigenfunctions  are  very  excitation  coupled  axis  A  levels  main m a g n e t i c  its z  ©  crystal  energy  the  of  of  be  angular  H(xtal)  3"(r+i) - o~=o,i,--I  will  good quantum numbers w . r . t .  coordinate  to the  of H ( x t a l )  the  of  has  a'  vicinity, rotational  freedom  the  a l l degrees  of  t h e Wigner  theorem  to  65  replace same  the e l e c t r o n  rotational  the product points  operator  properties.  f u n c t i o n s as (1)  arise.  w i t h an  operator  Applying  p e r t u r b a t i o n theory  zero  order  functions  ( 2 ) The e n e r g y  difference  coupled  is  good a p p r o x i m a t i o n ,  difference functions  a  the  The p e r t u r b a t i o n c a n n o t c o u p l e  levels.  to  possessing  very  of the s p i n of i n t e r e s t  degrees  between  of freedom a l o n e .  using  following degenerate  states equal  the  which  are  t o the energy The z e r o  order  a r e g i v e n by:  l l > - | + > ) ^ m  I-l>=|-> f « which t o f i r s t I 1>'= l+>/m +  order C  |->  C  in  ( Q f i m A  +  EL+ - E-  becomes:  b/'mVz + C ^ L l + 6  f n A  +  )  i-i> = i->/£ - c + i> (-f fk-x + g y>i*-i + h fLx + i _  The  coefficients  specific from  L  a state  induced  a-j arise  r.f.  in  z  calculating  m a n d may be z e r o  |1>'  to a state  momentum  the  from  and  by a l o n g i t u d i n a l  angular  field  [-1>',  field.  which o c c u r .  a torque  projection  of  with L=L'+2  Table  angular  momentum  experiment  orientation,  a  results A  and  real  hence t h e s t a t e  for  transition m=m'-2,  system  is  i s d r i v e n by  i s e x e r t e d on t h e l a t t i c e .  the  defined  j  ( 2 ) shows t h e c h a n g e s i n  As t h e s p i n  change  For  the  i n some c a s e s .  a c c o m p a n i e d by a c o m p e n s a t i n g lattice.  +j  E-  of  The  the spin  i n the angular  system i s  momentum  the c r y s t a l  of the l a t t i c e  change  of  has a w e l l  consists  of  66  Inclusion elements is  of t h e l a t t i c e of a s p i n  unnecessary  states  o p e r a t o r and  except  does  little  to a f f e c t  such a comprehensive  f o r a n g u l a r momentum  the m a t r i x treatment  considerations.  67  Table  IL>=  II - Angular  l+> i->  l-l>  VV-A  f i  momentum c h a n g e s d u r i n g a transition  + C |-l>^Q|//m-gg  (fft:  + Cy  -rbfn'  *  9  f z  * y z >  m' + C(m'- 2) E*-E-  -i-t- C  m-2. -+ Con' E*-E-  1-  C E*-E-  -f-ClJ/m'  * « ^  |-1>  +  to  e J^m'  |1>  j  K£  )  

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