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Electron spin resonance studies of NO₂ trapped in inert matrices at 4°K Hutchinson, Douglas Allen 1960

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ELECTRON SPIN RESONANCE STUDIES OP N0 TRAPPED IN INERT MATRICES AT 4°K 2  by  Douglas Allen Hutchinson B.Sc, University of British Columbia, 1958  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of CHEMISTRY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1960  In presenting the  this  r e q u i r e m e n t s f o r an  thesis  in partial  advanced degree a t  of B r i t i s h Columbia, I agree that it  freely  agree that for  available  the  f o r r e f e r e n c e and  permission f o r extensive  s c h o l a r l y p u r p o s e s may  D e p a r t m e n t o r by  be  gain  shall  not  be  a l l o w e d w i t h o u t my  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 $, C a n a d a .  Columbia,  University s h a l l make  study.  I  copying of  his representatives.  copying or p u b l i c a t i o n of t h i s  the  of  Library  g r a n t e d by  that  fulfilment  the  further this  Head o f  thesis my  I t i s understood  thesis for written  financial  permission.  ABSTRACT A 9 K Mc e l e c t r o n s p i n resonance spectrometer has been constructed and has been used i n conjunction w i t h a s p e c i a l l y designed l i q u i d helium dewar* P r o v i s i o n was made f o r condensing f r e e r a d i c a l s from the gas phase and f o r generating r a d i c a l s by the i r r a d i a t i o n o f low temperature d e p o s i t s . Experiments were performed i n which monomeric NO  was trapped a t 4°K  i n a s e r i e s o f d i f f e r e n t m a t r i c e s . An attempt i s made t o e x p l a i n the observed s p e c t r a l l i n e s i n terms of magnetic i n t e r a c t i o n s due t o n u c l e a r spins and e l e c t r o s t a t i c i n t e r a c t i o n s due t o neighbouring matrix p a r t i c l e s . The equation which g i v e s the e l e c t r o n s p i n resonance s p e c t r a l l i n e s as a f u n c t i o n o f magnetic f i e l d i s B =  h_v  +  A Mj  Thus an e s r spectrum i s c h a r a c t e r i z e d by two numbers, the h y p e r f i n e s p l i t t i n g constant A and the g v a l u e . When a f r e e r a d i c a l i s trapped i n an i n e r t matrix, s h i f t s occur i n the hyperfine s p l i t t i n g constant and i n the g v a l u e . A d r i a n developed a semiq u a n t i t a t i v e theory t o e x p l a i n these e f f e c t s f o r hydrogen atoms trapped i n i n e r t m a t r i c e s . T h i s theory considers two important i n t e r a c t i o n s between the f r e e r a d i c a l and matrix p a r t i c l e . The van der Waals i n t e r a c t i o n leads t o a negative s h i f t i n A, while o v e r l a p e f f e c t s l e a d t o a p o s i t i v e s h i f t i n A and a negative s h i f t i n g v a l u e . E l e c t r o n s p i n resonance s p e c t r a were obtained f o r n i t r o g e n d i o x i d e trapped i n argon,methane and n i t r o g e n a t 4°K. Attempts were made t o employ the ideas o f Adrian's theory t o e x p l a i n the e s r s p e c t r a obtained from these systems. P a r t i a l success was achieved  i n e x p l a i n i n g the form o f the s p e c t r a  obtained* The divergences between the p r e d i c t i o n s o f the theory and the experimental r e s u l t s leads t o a questioning o f the assumptions made and the approximations used i n developing the theory. ABSTRACT APPROVED  - iii  -  TABLE OF CONTENTS Page L i s t of Figures  v  CHAPTER I. BASIC THEORY OF ELECTRON SPIN RESONANCE  1  A. Elementary Theory o f Angular Momentum  1  B. Resonance Phenomenon For Free E l e c t r o n In Uniform Magnetic Field %  8  C. E l e c t r o n o f Angular Momentum N u c l e i o f Angular Momenta  In I n t e r a c t i o n w i t h N  clcii cfi.v' '  (i2^  13  D. Weissman's Theory o f Hyperfine I n t e r a c t i o n s f o r alT  Electron  17  E . Mc C o r n e l l ' s Theory o f Hyperfine I n t e r a c t i o n s f o r a l T Electron  27  F. Adrian's Theory o f M a t r i x E f f e c t s on E.S.R. S p e c t r a o f Trapped Hydrogen Atoms  30  CHAPTER I I . EXPERIMENTAL  40  A. E l e c t r o n Spin Resonance Spectrometer  40  B. L i q u i d Helium Dewar  40  C. Techniques  42  CHAPTER I I I . RESULTS AND DISCUSSION  47  A. Introductory Remarks on NOg  47  B. Nature of Trapping S i t e s  47  C. NO^A  49  Spectrum  D. P o s s i b i l i t y o f I n t e r a c t i o n between Unpaired E l e c t r o n and Magnetic N u c l e i o f M a t r i x P a r t i c l e  51  E . L i m i t a t i o n s o f Approach Used In D i s c u s s i n g Spectra  53  F. Suggestions f o r Future Experiments  54  BIBLIOGRAPHY  62 - iv-  L i s t of F i g u r e s Fig. 2 - 1  Block Diagram of e s r Spectrometer  44  Fig, 2 - 2  Block Diagram of Microwave System  45  Fig. 2 - 3  L i q u i d Helium Dewar  46  Fig, 3 - 1  ESE Spectrum o f N(> i n Argon M a t r i x at 4°K  57  Fig, 3 - 2  ESR Spectrum o f N(> i n Methane M a t r i x at 4°K  58  Fig. 3 - 3  ESR Spectrum of N 0  59  Fig, 3 — 4  A v a i l a b l e Trapping S i t e s i n Face - Centred  Fig. 3 - 5  2  2  2  i n Nitrogen M a t r i x a t 4°K  Cubic L a t t i c e  60  ESR Spectrum P r e d i c t e d by Theory  61  —v—  ACMOWLEDGEMENTS F i r s t I wish t o express my a p p r e c i a t i o n t o Dr. J. B. Farmer f o r h i s encouragement and guidance during t h i s work. Without h i s patience and understanding much o f the work could not have been accomplished. Secondly I want t o thank Mr. Rudolf Muelchen f o r the c o n s t r u c t i o n o f the l i q u i d helium dewar, f o r teaching me t h e technique o f t r a n s f e r r i n g l i q u i d helium and f o r h i s w i l l i n g n e s s t o g i v e a s s i s t a n c e throughout the l a s t two y e a r s . A l s o I am g r a t e f u l t o P r o f e s s o r C, A, Mc Dowell f o r many i n f o r m a t i v e d i s c u s s i o n s , D r . R. F. Snider f o r h i s l u c i d explanation o f time dependent p e r t u r b a t i o n theory and i t s a p p l i c a t i o n t o t r a n s i t i o n s induced by magnetic d i p o l e r a d i a t i o n . I wish t o thank Dr. J. A. R. Coope f o r the h e l p he has given me i n g a i n i n g some understanding o f Quantum Mechanics, both i n h i s l e c t u r e s and i n private discussions. I thank the s t a f f o f the Chemistry Department workshops , Mr. L. Stewart and Mr. E. Giesen i n the e l e c t r o n i c s shop, Mr. A. J. Hawkins, the glassblower and Mr. F. Sawford and Mr. J. Cattermole i n the mechanical shop. The N a t i o n a l Research C o u n c i l of Canada gave generous  financial  a s s i s t a n c e f o r the c o n s t r u c t i o n of the esr spectrometer. For t h i s I wish t o thank them. A l s o  I express  my g r a t i t u d e f o r the bursary awarded t o me  and f o r the studentship i n the f o l l o w i n g year  - vi -  (1959-60).  (1958-59)  I . BASIC THEORY OP ELECTRON SPIN RESONANCE  A.  Elementary Theory o f Angular Momentum ( l , 2 ) The Heisenberg Uncertainty  and ^  representing  observables  P  P r i n c i p l e s t a t e s t h a t i f two operators'! and  Q  have the commutation r e l a t i o n s h i p  c then the product o f the root mean square d e v i a t i o n s 4P^Q  obeys the f o l l o w i n g  inequality:  Wh6re  AP r C P AG? - CQ -Q "fe - ~  - Los x /o-*  tec  7  By any experiment t h a t can be conceived  i t i s always p o s s i b l e t o show t h a t  the same components of the p o s i t i o n and l i n e a r momentum v e c t o r s obey the relationship  From t h i s we i n f e r that the operators the commutation r e l a t i o n s h i p  X^cvL i.e.  if,  « a  -  %  6  0  representing these observables s a t i s f y  - 2 However an operator r e p r e s e n t i n g a g i v e n component o f t h e p o s i t i o n v e c t o r w i l l commute w i t h an operator r e p r e s e n t i n g any other component o f t h e l i n e a r momentum v e c t o r *  i... Also  rr„  y \  =  =  o  etc.  t h e operators representing t h e components o f the p o s i t i o n v e c t o r commute  among themselves as do t h e operators representing t h e components o f t h e momentum v e c t o r *  -ft  C l a s s i c a l l y the angular momentum - X  ^  where 1^  l _ i s d e f i n e d by the f o l l o w i n g equation:  -»»  = R X P and P  are t h e p o s i t i o n and l i n e a r momentum v e c t o r s * W r i t i n g t h i s  equation out i n component form we o b t a i n : L L L  x y z  = =  y p z  -  z p y  z p " * x p *x. z  rs x p y  -  y p x  17  -We w r i t e t h e operators representing t h e components o f angular momentum i n the form  4c  u --  -  y  -Iff,  - 3 Commutation  R e l a t i o n s h i p s D e f i n i n g Angular  L e t us now compute t h e commutator  O^X  Momentum •  y 1_  S i m i l a r l y i i can be shown t h a t :  i*v-i\  =[un. =o  The above commutation r e l a t i o n s h i p s which we have j u s t obtained are s a t i s f i e d by the o r b i t a l angular momentum operator  if  which we d e f i n e d by:  - t  We c a l l t h i s t h e " o r b i t a l angular momentum" s i n c e t h e operator £~ represents the angular momentum o f an e l e c t r o n due t o i t s motion i n an atom or molecule* We now make t h e f o l l o w i n g g e n e r a l i z a t i o n : Any angular momentum observable rules:  _  J i  s  represented by an operator _  t | S a t i s f y i n g t h e commutation  - 4 T h i s g e n e r a l i z a t i o n i s necessary because an elementary p a r t i c l e l i k e an e l e c t r o n has an i n t r i n s i c angular momentum as w e l l as angular momentum a r i s i n g from i t s e x t e r n a l motion. We see t h a t each o f the components o f t h e angular momentum commutes w i t h the t o t a l angular momentum but t h e components do n o t commute among themselves* Thus t h e operator representing the square o f t h e t o t a l angular momentum  ^ and  the operator r e p r e s e n t i n g any one o f t h e components.say ^ ^ f o r m s a s e t o f commuting operators r e p r e s e n t i n g a s e t o f compatible observables.Thus  a state  of angular momentum i s s p e c i f i e d by the t o t a l angular momentum J " and one component J  and t h i s s t a t e i s represented by the k e t ) J  3  ^ - r ^ .where J  r e f e r s t o t h e t o t a l angular momentum and Mj r e f e r s t o t h e z-component o f t h e angular momentum. The p o s s i b l e v a l u e s which we can observe f o r t h e z-component o f the angular momentum are obtained from t h e eigenvalue equation:  P o s i t i v e and Negative S h i f t Operators. We now d e f i n e two new operators:  which we r e f e r t o as t h e p o s i t i v e and negative s h i f t operators.Let us now evaluate t h e commutator  -  Next we study the a c t i o n o f ^ a n d ^  ^j|J M >=  Becall  )  5  -  on t h e s t a t e  H , - t i IO"_,Mj>  7  ^ J ,  Of)  We thus see t h a t ^ I j ^ M ^ s a t i s f i e s an eigenvalue equation l i k e (*) and hence we can w r i t e  Where  C  i s some constant y e t t o be determined.  S i m i l a r l y i t can be shown t h a t  f  \7  D  » » ^ J - ' >  Thus we see t h a t the a c t i o n o f ^ of angular momentum by"fc  on a s t a t e i s t o r a i s e the z-component  and the a c t i o n o f ^**on a s t a t e i s t o lower the eigenvalue  of t h e z-component o f angular momentum by"K . Since | J j ^ c a n n o t  exceed  \ J Vthe fore-mentioned processes must terminate.  Hence we impose t h e f o l l o w i n g two c o n d i t i o n s : (1)  "3 a number  (2)  3  a number  such t h a t MJ  such t h a t  fy*  1 7j  ^" \ J  }  } Kj* >  =  O  -  O  Note t h a t throughout t h i s d i s c u s s i o n we have r e f e r r e d t o s t a t e s o f the same t o t a l angular momentum. Eigenvalues o f & . We are now i n a p o s i t i o n t o evaluate the eigenvalues o f Q .  S i m i l a r l y we can show t h a t  - 6Now l e t us study t h e a c t i o n o f  since  ^  |  ^  Q  state { J > M j  n  O  shown t h a t  A l s o i t can be  J * | J , H , S  f o r the s t a t e l y M ^ ^ t h e f o l l o w i n g i s t r u e :  »  M/£M,«  - I )  **  U , H / >  (B)  Now the s t a t e s I X , V I a n d ) J , M o h a v e the same t o t a l angular momentum,hence the eigenvalues  o f equations (A) and (B) are equal.  i.e.  ( Mj + 1 ) -  Mj ( M j * l  T h i s i s s a t i s f i e d i f and only i f -M^ or  Mj = J  =  eigenvalues  of ^  of ^  =- J  and M^- = - J  We hare seen from the a c t i o n o f ij* and ^  vary by u n i t s  )  , i.e. i f  on eigenstates o f ^  t h a t the eigenvalues  i s an eigenvalue so are (Mj +  1 )Kand  (Mj -  . Thus we see t h a t the p o s s i b l e values f o r M a r e :  - J , - J + l , - J + 2 ,  . J - 1 , J  Such a scheme can be s a t i s f i e d i f and only i f J i s e i t h e r i n t e g r a l or h a l f i n t e g r a l . So a f t e r a l l t h i s d i s c u s s i o n we obtain the r e s u l t s :  Now a l l we have l e f t t o do i s t o evaluate the constants  C and D,  1  - 7-  <W  since  ^  $T iw>  = < a*"*'"** ^  and ^ y a r e Hermitean ( i . e .  I f X, «,>  self-adjoint )  s i n c e we represent a s t a t e by a v e c t o r o f norm u n i t y i n a H i l b e r t space • I. 2> J  Hence  or Q  2 ,  ])  :? •  £ 7 I  =  >/ JiJ+l)  1)  -  M y ( M, -  - Mj  l>}  ~») 4i g ' *  3  '  i s an a r b i t r a r y phase f a c t o r which we do n o t need t o consider i n most  discussions.Hence we w i l l n e g l e c t i t i n the f o l l o w i n g d i s c u s s i o n . S i m i l a r l y we can show t h a t  Matrix  Elements  o f Angular Momentum  Operators.  Ih the r e p r e s e n t a t i o n i n which we use t h e set o f eigenvectors as a b a s i s f o r our H i l b e r t space,the v a r i o u s angular momentum operators have the f o l l o w i n g m a t r i x elements.  (r)  ,  -  Vnx*rt-M,'K-<> *  S„,  - 8-  B.  Resonance Phenomenon For Free Electron In Uniform Magnetic Field BQ An electric charge q possessing angular momentum L has a magnetic  dipole moment \K, given by: - f\ — ' where  L.  (Gaussian units)  m = mass of charge c = velocity of light  = 3 x 1 0 ^ cm sec *  g = 1 for classical systems and orbital angular momentum, g = 2 for the intrinsic angular momentum of an electron For an electron of charge - e we obtain  where  S is the intrinsic angular momentum of the electron which we quite often  refer to as spin.The energy of a magnetic dipole y- in a magnetic f i e l d B given by:  0  is  '  For convenience we define the z - Axis in the field direction,i.e. which simplifies the above equation to: W  =  8 Tm^  B  S z  o  The Hamiltonian operator is of the form  04= 3 -aIf we define the Bohr Magneton ^-^7 "the following equation  we then have  We obtain the available energy levels from the eigenvalue equation:  B^ = (0,0,B j  - 9 -  =  U^S,  but  M - f c l * , S , M s  3  >  hence F o r an e l e c t r o n  S = J  i n a magnetic f i e l d  B  M Q  = i £  so we have two energy l e v e l s f o r a f r e e e l e c t r o n  •  the d i f f e r e n c e between these two l e v e l s being given by:  A p p l i c a t i o n of electromagnetic r a d i a t i o n whose quanta have energy  4E.  should induce a t r a n s i t i o n between the two energy l e v e l s a v a i l a b l e t o the e l e c t r o n , i . e . i f the e l e c t r o n i s i n the lower energy l e v e l , i t should  absorb  a quantum of r a d i a t i o n and be e x c i t e d t o the higher energy level.The r e q u i r e d YQ  frequency  f o r "this t r a n s i t i o n i s g i v e n by the Planck  formula:  u P r e c e s s i o n of E x p e c t a t i o n Value of Spin About the Magnetic F i e l d . From the Schrodinger  we  equation  o b t a i n the expression f o r the time dependence of the expectation value of  any p h y s i c a l observable A :  i < A ) : i .  <Z*4>1  >  F o r the case of a f r e e e l e c t r o n i n a magnetic f i e l d of the form  the Hamiltonian i s  - 10 -  We can now  *U Now  compute the time dependence of the expectation value of the s p i n :  *  •*  -*»  -ft  we have t o solve the system of equations  a  / c  \  <Sx> = ^  S-y>  c  .  o  $ * C*s S  fc  w  h  e  r  e  ^>  d  -  S^slt  w>."k  Si*  s a t i s f i e s equations ( l ) and  (2)  <S > r Const 2  T h i s corresponds t o a vector<"s>=  «S>,^),<S> )  v  ^  processing about the a p p l i e d x y z magnetic f i e l d i n the manner i l l u s t r a t e d i n f i g u r e 1-1 . The p r e c e s s i o n frequency Q  i s g i v e n by:  ^  ^ >  =  2n  B  ,  „ 2.7T  g ^ 3  0  i,  T h i s i s what we r e f e r t o as the Larmor Precession.This i s e x a c t l y equal t o the frequency of electromagnetic r a d i a t i o n r e q u i r e d to cause a t r a n s i t i o n between the two energy l e v e l s a v a i l a b l e t o the e l e c t r o n . A p p l i c a t i o n of C i r c u l a r l y P o l a r i z e d Electromagnetic R a d i a t i o n t o a Free E l e c t r o n i n a Steady Magnetic F i e l d  BQ  T h i s problem i s r e a d i l y t r e a t e d by time dependent p e r t u r b a t i o n theory.(3,4) Consider the Hamiltonian of the form  (o)  AA  where the eigenstates and eigenvalues o f the  eigenstates and eigenvalues of  are known,and we wish t o determine  .That i s , we know the s o l u t i o n o f the  eigenvalue problem: and we wish t o o b t a i n the s o l u t i o n of the problem:  Writing 1 i n  the form  j , £ -i w  and applying the Schrodinger equation  (, l t .  we o b t a i n the r e s u l t  V H  -E„)'*  Assuming t h a t i n i t i a l l y the system i s i n the ground s t a t e | K > , w e o b t a i n the  approximation  c  j  ^  e  iQ"^  ^  Now l e t us apply t h i s r e s u l t t o an e l e c t r o n i n a steady magnetic f i e l d to which we apply a c i r c u l a r l y p o l a r i z e d f i e l d  where  Let  %* l  -  ££±  ^  us l a b e l the lower s t a t e  B  = (0,0,BQ)  B ^ ( cos(«>t , sin<0t , 0 ) .  0  - 1 ^ and l a b e l the upper statel<*-> - \ \ \ }  Then i f before we apply the c i r c u l a r l y p o l a r i z e d r a d i a t i o n the e l e c t r o n i s i n the  lower state,we o b t a i n f o r the c o e f f i c i e n t  c  u  a f t e r a time  t :  - 12 -  ,2 I c ^ l i s the p r o b a b i l i t y of a t r a n s i t i o n from the lower state t o the  where  upper s t a t e .  .  2-fc  .  ,  u  x  ,  u  y i  Define the gyromagnetic r a t i o  ~  ^  -  ^  -  172 J  ^*  ^  In a t r a n s i t i o n from the ground state t o the upper state only the second term i s of importance,hence we o b t a i n : j^£^,-E --fc %*> ) * ft  <*<u. —  —  ^ujt  1  Z  h ^ - k  7 x  The p r o b a b i l i t y of a t r a n s i t i o n from the ground s t a t e t o the e x c i t e d state i s g i v e n by:  For  r  »C j  1  ;  g,» ) |»  ^  f  €  0"  *  A  1  ^  ^  -J  small times we see t h a t t h i s t r a n s i t i o n p r o b a b i l i t y i s quadratic i n time  where by i n t u i t i o n we would expect i t t o depend l i n e a r l y on time.However so f a r we have been t a l k i n g about monochromatic r a d i a t i o n which i s only an i d e a l i z e d concept. So l e t us replace  B^^  by  B^(v)  thus i n t r o d u c i n g a frequency  d i s t r i b u t i o n . So now our t r a n s i t i o n p r o b a b i l i t y i s replaced by :  Now  E  u  -  = h v  0  °*  OS  " « - " )* E  k  - 13 -  The f u n c t i o n  s i n *fl" ( VQ .- v )  h  ( V - V: )  2  behaves approximately l i k e a d e l t a f u n c t i o n ,  2  Q  i . e . i t i s p r a c t i c a l l y zero unless  fi*  Now a p p l y i n g the r e s u l t  for  M  = |  v = v  J i ^ ^ s  Q  TL  , so we make the f o l l o w i n g approximation:  w  e  obtain  p  s  M_ = - £  T^ff^*  ^>  f  M  7f £ 2  1*,M *|  so Thus we o b t a i n a t r a n s i t i o n p r o b a b i l i t y which i s l i n e a r i n time.Also note t h a t the t r a n s i t i o n p r o b a b i l i t y i s q u a d r a t i c i n the square o f the amplitude o f the magnetic f i e l d whose frequency i s equal t o the Larmor f r e q u e n c y . P i n a l l y i t i s i n t e r e s t i n g t o see t h a t the t r a n s i t i o n p r o b a b i l i t y depends on t h e square o f the gyromagnetic  ratio.  —* C.  E l e c t r o n o f Angular Momentum / j . i n I n t e r a c t i o n w i t h ""ft* ~ A *  Momenta pi\  c£y  N  n u c l e i o f Angular  - * • Of . H  T h i s problem w i l l be t r e a t e d by time-independent p e r t u r b a t i o n t h e o r y . ( 5 , 6 ) In t h i s case we w r i t e the Hamiltonian i n the form  where we know the eigenvalues and e i g e n s t a t e s o f term and i t does not depend on time.  A - /  i s the p e r t u r b a t i o n  . We i n s e r t t h i s constant so t h a t  we can separate out zero o r d e r , f i r s t order,second order e t c . terms i n the c a l c u l a t i o n s . The problem i s t o determine t h e eigenstates and eigenvalues o f 99  *  - 14 -  i.e.  94 |  we want t o solve  when we know t h e s o l u t i o n o f  we w r i t e  lKvT>  - J  U  M  -  djd^  E*» \ ' U . ^ -  E  ^  ^>  By using the r e l a t i o n s h i p s  and  we a r r i v e a t the r e s u l t s :  s  Now l e t us apply t h i s theory t o an e l e c t r o n i n t e r a c t i n g w i t h N n u c l e i . For such a system the Hamiltonian i s o f the form:  i s t h e Hamiltonian d e s c r i b i n g the coulombic i n t e r a c t i o n s between the electron; and the n u c l e i and a l s o d e s c r i b i n g the k i n e t i c energy o f the e l e c t r o n , . &°  -r^J£  describes  the energy of the e l e c t r o n i n the magnetic f i e l d BQ.  $1*6 % f*-y£Jt-$(l2rfli)describes  the i s o t r o p i c i n t e r a c t i o n o f the e l e c t r o n  w i t h the i - t h nucleus. (7) T h i s i s the Fermi contact term and i s a consequence o f the D i r a c theory o f the e l e c t r o n . ^ of t h e e l e c t r o n and ^ nucleus.  i s the operator representing  i s the operator representing  the p o s i t i o n  the p o s i t i o n of the i - t h  - 15 - ^  ^Vctfr- ^ d e s c r i b e s the i n t e r a c t i o n of the i - t h nucleus w i t h the magnetic Hi t  field B  Q  .  N  So we w r i t e  which has the eigenstates  w i t h the  ^  eigenvalues  t =»  and the p e r t u r b a t i o n term i s :  44'°c  iff life f e  ^ J * , ^.i.jSfM?)  The f i r s t order approximation t o the energy of the system i s g i v e n  by:  T h i s expression reduces t o :  N  r-  *»  *0 \SHs>TTll >4_.> :>  With a b i t naive c a l c u l a t i o n we  Now  get:  1  a l l we have t o do i s evaluate ^ i S ^ i e ^ c ^ k ^ T h i s i s most conveniently  done i n p o s i t i o n r e p r e s e n t a t i o n .  Now  v=-»  8 ( * - ? v ) » ? > =  S C ^ - ^ J  - 16 -  integrate with respect t o r '  integrate with respect t o  ^^Cc*-) is result Let  r  t h e wave f u n c t i o n of the e l e c t r o n a t the i - t h nucleus.We now have the  s+tT  M f  ^  s  1^ C%)l *  us d e f i n e the h y p e r f i n e s p l i t t i n g constant f o r the i - t h nucleus by the  following r e l a t i o n  We then o b t a i n  £ c>  =  ^, ^  ^  ^  ^  x  So t h e t o t a l energy o f our system t o f i r s t order i s :  Let  us now apply t h i s formula i n d i s c u s s i n g the spectrum o f the methyl r a d i c a l . (8)  Eaeh hydrogen nucleus has a s p i n  I = -J and each hydrogen atom w i l l have the  same h y p e r f i n e s p l i t t i n g constant.So our formula takes t h e form:  define  'SI  ,then we g e t  Since the t r a n s i t i o n p r o b a b i l i t y f o r magnetic d i p o l e r a d i a t i o n depends upon the square of the gyromagnetic r a t i o , we can apply the s e l e c t i o n r u l e s s i n c e the gyromagnetic r a t i o  AMg=t/ A^zo  the e l e c t r o n i s 1836 times as great as the  gyromagnetic r a t i o of the nucleus.Applying these s e l e c t i o n r u l e s we o b t a i n  or w r i t i n g i n terms o f  fa  :  - 17 -  Now  Mj. has the values  - 3/2  , - 1/2  , l / 2 , 3/2  degeneracy when Mj takes on the values of form shown i n f i g u r e 1-2  . Since there i s a t h r e e f o l d  and \ we o b t a i n a spectrum of the  • We o b t a i n a f o u r - l i n e spectrum where the i n t e n s i t y of  the l i n e s i s i n the r a t i o  1:3:3:1 and there i s a constant separation of  A  between the l i n e s .  D.  Weissman's Theory of Hyperfine I n t e r a c t i o n s f o r aTT  Electron.(9)  In the previous s e c t i o n we obtained the f o l l o w i n g expression f o r the h y p e r f i n e s p l i t t i n g constant:  A- = ^  ftp.  t  W^)) 2  For many r a d i c a l s the odd e l e c t r o n i s aTC-electron.But  f o r an e l e c t r o n i n a  H - s t a t e there e x i s t s a nodal plane where the wave f u n c t i o n vanishes.Now i f the nucleus whose i n t e r a c t i o n w i t h the e l e c t r o n we wish t o study l i e s i n t h i s plane, the above expression would p r e d i c t no hyperfine s t r u c t u r e * I n aromatic  hydrocarbon  r a d i c a l s the odd e l e c t r o n i s i n a IT - s t a t e and the r i n g protons l i e i n the node of the wave f u n c t i o n of the odd e l e c t r o n * l e t hyperfine s t r u c t u r e from these r i n g protons has been obtained i n the e . s . r . s p e c t r a of a great nUmber of aromatic radicals*(10,11,12) T h i s apparent paradox l i e s i n the f a c t t h a t we have used a v e r y crude model f o r our f r e e r a d i c a l . The Hamiltonian f o r the r a d i c a l i s of the  f  o  m  '  <H = 4 t  +  21  u i j  +  i  j J j (  where UCj  rt» $0  u i j , 0 + t - I . u n r , s )  «-,*•  <  Is i ^ e k i n e t i c energy operator r e  P  r e s e n  * s " t ^ i n t e r a c t i o n between e l e c t r o n j and nucleus Y e  1^(J^R) represents the i n t e r a c t i o n between e l e c t r o n s j and k  ( ^ ( ^ 3 ) represents the i n t e r a c t i o n between n u c l e i  y  and  £  T h i s problem i s solved by a p e r t u r b a t i o n treatment where we w r i t e  - 18 -  The eigenstates of ^  ' are the states which we considered available  to the electron in our previous discussion.  We place the N electrons of  the free radical in the £(N + l ) lowest energy states. The £(N - l ) lowest states have two electrons with spins opposed while the j(N + l ) state contains the odd electron.  Thus we have a configuration  like that illustrated in figure 1-3 for the zero-order solution.  4__  W_H  -4-Hr  4-fc  £=?  4—t— 4—t— 1  C0V\-fi<^U.r4.-VvO« "for-  3 1  h4 - e.le.cVDwS.  I  '  In our f i r s t order solutions we admix with this configuration, configurations where an electron is excited. the odd electron in a  state.  Some of these configurations have  Thus in the true state of a free radical  the odd electron has some of the characteristics of a  state. Since a  state does not have any nodal plane, hyperfine interactions will occur. Weissman's treatment of the problem is as follows.  In considering our free  radical let us consider only the upper two energy levels and think of the rest as forming a complete subshell.  - 19 We can write for our zero order solution:  i<r cti,y> 9  I «-g e o »  lo»cuy>  \<r o»uY>  i <r & u v > u  %  I(r  i  &u)>  %  Let ) 0 * ^ ^ be the state of ()* character immediately below the I L state and let \ 0 * b e the state of 0 * character immediately above theTC state. In our f i r s t order solution we admix excited states with our zero order solution.  The states we choose to admix with  are subject  to the following restrictions: (1) M S  i.e. z-corapound of spin angular momentum must be £ = z  (2) The symmetry of the excited states must be the same as that for \ 3 k ( f ; 2 ? ) ^ >  (3)  S = -J, i.e. the total spin angular momentum must be equal to  The following three states satisfy the above three restrictions.  Other states  are also allowable but their contribution is very small.  \ O^tftO^  t T T O C O  toevio^  ITT<2C%»  l<X «fu» 6  ITT(W>  - 20 -  N3*  i T  k ° >  2  '  :  0  "  ^<r«c(co>  iQfcoKi^  i(TA^to>  l<r fca»  10^0^  1 i^rfto>  ^  I \ir«eco>  On combining three spins  =  = 2  \<r*«a*v>  A  l v  A  ^ >  \Ct*(tf>  l«ipaY>  »o- <»»  »<V"^>  A  nr^»  = £ vre obtain a quartet state  3 (S = — ) and two doublet states (S = £ ) . The three states which have IL =  \  are:  From the previous restrictions the quartet state i s of no use to us so we are l e f t with the two doublet states:  - 21  Our f i r s t order solution i s then of the forms  Now l e t us consider the Hamiltonian for our radical i n a magnetic f i e l d *  For  sake of simplicity we w i l l consider an atomic radical or a molecular radical i n which only one nucleus has a nuclear spin.  ^  ^  The Hamiltonian i s of the forms  i s that part of the Hamiltonian which describes the Coulombic interaction  between the particles comprising the molecule*  f\ v  ^  ^ *Bfe  describes the interactions of the electrons with the magnetic field describes the interactions of the nucleus with the magnetic field.  - 22  describes the isotropic hyperfine interaction between the electrons and nucleus. We treat this problem by perturbation theory and write  ^  ^ I 59xJ^ ^ f. .y s#) C =i  For the zero order Hamiltonian we have the eigenstate  and eigen-value  The f i r s t order correction to the energy i s :  Before us remains the task of evaluating this bracket:  -V V 3 ^ . U V \ C v > ^ * y  S*v»a>l  S«66vaA  Ov-^Cv-  +ev»v,S  v I i - 23 -  £o  wow  we  Wave  "Kc  jofe  ©4  +Ur«e  evaUfchn^  tv»'a.ck« +  <<r «(x)) <r dvt)\ n  < * l  9  <«**o)l <^C»I <r w^)| f  <ir*iA. <ir«»a)| <tr«u*)l  l(r ^(3)>  KM>  *5  -L  in  5!  3  A*srtU4*si«3 ***  &  \M > 3  <<T fclril « T | ¥ U > |  <7 QQ)|  <!TolU)l  <U^O)l  B  *  (  <1Tofo)|  M t , ) >  WMfo  * * Q  tt)>  ,g  **  rt>  lTTdW> | T d U ) >  i•  - 24 -  <<r cttO. B  <cr o/(i)| &  <:<r ^)l <cr ?W» < ^ ^ l 6  <TT<*UM  l(T -«iO>  \<r o^>  i(T o/^>  l^ o<lO>  \<T °Ux)>  \^ ^(Ci)>  c  2  <Ae*  -  A  <Aet  f t  A  , T T M O ) >  <TT^)I  g  A  \<r fcW>  B  A  l<r £Ci)> A  \r po)> A  \TTrflO> OTolU)>  K<H«V> i n ^ >  1^^>  <n«»«J)|  - 25 Before preceed let us note the following three selection rules: \ 6"*^ l l O  (i) \ and  8  X  9  members of an orthonormal set of states  are orthogonal and normalized  (ii) If we take a term from the expansion of the bra determinant,say for example  .this can only combine with terms of the  expansion of the three ket determinants of the form | *» «1«)^\ $(0>>| "«AJ) > ,  ,  ,,,  >  This reduces the above expression from 108 brackets to 36 brackets. ( i i i ) A term from the expansion of the bra determinant in which electron k is in  state can only combine with terms of the expansion of the three ket  determinants in which electron k  is in a "FX state.  Of the 36 terms remaining from the previous selection rule,only 12 remain.  la figure 1-4 below the electron configurations of the three Spates 13^(1,2,3^  0)2-^^  |1>J5 O ^ i s illustrated.  cr - 4 —  *k  TI —4}  T -4—  A  — J —  - 4 —  Ux (w>>  ^-4-  TT —4-  a  %  — i —  ]l>ir.tW>>  By applying these selection rules we reduce equation ( XX ) to t  -  <TTc<c.)|<MuKcr py)M s  SflJ)  ve  + ^ l o l ^ ^ K i T j p ^ i ^ s r f i )  4,  &tf \  ^  S^i)  x  -  <<r f(0|<ir«iw|<r KO)U^5"(«;) -* <W  +  <<r*o>| 4 « u ) i < ^ ) l ^ 6 ( i C ) -4. A *  -  <rr^i.M<^ (iw|  +  <ir«itoi < ^ i d l < ( r ^ « ) i  -  f  £  h  8  < V ^ I <ir%uiM < ( T ( i U ) ) i 8  <ir*ol ^BQ«I  •+ ^<r pco| 9  l v  «r ^)l^* g  < ^ M \ ^iroiiij i <TTottt)l  -* A *  S&)  ^) * 4 , <n*? j  ffrift^atfC^)  %  + -  b  - K C ^ f t f J ) l ^ c » ) > Kr£«a)>\cfc «/)>  ^  •+ ^  J G S )  •+  HfO  Ut)  B  l<r MT> Ir^coV  -*A*Jfl$)  ^^i<6(H$)  Now evaluating these remaining brackets we get :  S"<£) +feXl %) J  ir*o>UV<*y  l<r f » » l « y * « ) > A  1TT<*W>  l<r ftu>> K w a >  l<T*«i*>  I T T ^ lr .Pi?»  i»sc*i) + v ^ s f i y  fc*  \Trw»>»o;<e»  A  S C ^ y -t < * i * S e ? J | t y u l >  5fi0 s i ^ )+  Wrfu>> Iqfciuft K pu)>  B  A  A  hrtf( r> 2  i<w>  l < r ^ > i<r u»V f e  Afl  |oigd»i**^> 1TTV^>)  - 27 -  (3)  < M \ < $ ( l , 2 , 3 ) \ M l ^ 0 , 2 , ? ) > |M "> - O £  l O  x  f i  X  T h i s r e s u l t i s obtained by using the technique as a p p l i e d i n the previous calculation. The f i r s t order term i n the energy i s then given by:  <  we then have  r  B  !<T > rr ^ *  I hit)  ..  A  „ -  -r$  (o) ^  / c > )  A  f*a ^ x ^> H  F o r our energy we now have the expression  Using the s e l e c t i o n r u l e s  0 1  B = Ju> + A 0  *A  m  =• O  s 'Jt  ^  I  z  n  I f , f o r example.we take  (if^  ^  ^  ' ° 0  ^  ^  '  2  Then f o r  0.5*  ~  A  ^  2.g*«tJ.s  T h i s compares q u i t e f a v o u r a b l y with the hyperfine s p l i t t i n g constants t h a t have been observed experimentally f o r a l a r g e number of aromatic f r e e r a d i c a l s . (13,14) E»  M° C o n n e l l ^ Theory of I s o t r o p i c Hyperfine I n t e r a c t i o n s f o r a "U" E l e c t r o n . (15) Weissman's theory,which we have j u s t treated,used m.o.  considerations  - 28 -  and admixed with the zero-order ground s t a t e | ^ 0 , Z , ^ ) > ~A  O^^ltf^O^tfk)))  where the three e x c i t e d s t a t e s  b>  x  0>h"#>  - H 0 <M<«>> K«<*M  For the hyperfine  A  TTU1)>)  s p l i t t i n g constant the v a l u e obtained was  9*** * t C ^ *fo)^  co))  Mc C o n n e l l , i n h i s treatment of aromatic f r e e r a d i c a l s uses the valence bond t h e o r y . He considers the Sp" ~ hydrogen atomic o r b i t a l Writing)^  1S>  and the^;£> carbon o r b i t a l .  CO 6 (."»> ^C?^4KV3^»  Mc  Connell considers the three  Note t h a t i n Mc Ponnell's treatment \f>^> electron.By  carbon h y b r i d o r b i t a l , t h e  and  states  each share t h e odd  imposing the c o n d i t i o n s :  we obtain the two l i n e a r combinations  | ({^^corresponds t o the ground s t a t e o f the C-H fragment, |($^)to an e x c i t e d s t a t e . T h i s i s the zero order s o l u t i o n n e g l e c t i n g e l e c t r o n - e l e c t r o n i n t e r a c t i o n s . By considering these i n t e r a c t i o n s , p e r t u r b a t i o n theory gives us the 1 s t order solution:  - 29 -  {XA  d e s c r i b e s e l e c t r o n - e l e c t r o n i n t e r a c t i o n s among the three e l e c t r o n s .  Me Connell then obtains the r e s u l t  C  hence  di!L  ^t-z.| —  *- - ™ ( ^ ^ p t f "  ~~  -*^~|>«g)  ^1*£_  Now c o n s i d e r i n g the hyperfine i n t e r a c t i o n  ^CO  and t a k i n g  <  $,  1  | <^  M  |  >  Mc Connell obtains f o r the s p l i t t i n g due t o a proton  ^>^* = £j  e l e c t r o n d e n s i t y f o r odd e l e c t r o n a t carbon atom  —  j  f a c t o r f o r admixture o f the e x c i t e d s t a t e  \g(o^ ^ ^-square o f H - atom wave f u n c t i o n a t proton T h i s expression reduces t o  4*»(Hj)  =  t ^ 2 o >  By f u r t h e r considerations,Mc  M<-/  $ e c  Connell and other workers(16,17,18)  have shown t h a t the hyperfine s p l i t t i n g constant i s approximately p r o p o r t i o n a l t o the square o f wave f u n c t i o n o f the odd e l e c t r o n a t the a d j o i n i n g carbon atom. T h i s i s t r u e f o r a wide range o f aromatic  a j where  -  a  radicals,i.e.  js  i s the h y p e r f i n e s p l i t t i n g constant f o r proton j J)j Q  i s the s p i n d e n s i t y a t carbon atom j i s approximately  constant. Experimentally i t i s found t h a t  - 30 -  F.  Adrian's Theory o f M a t r i x E f f e c t s on E.S.R. Spectra of Trapped Hydrogen  Atoms.(19) When a f r e e r a d i c a l i s placed i n an i n e r t matrix s e v e r a l i n t e r e s t i n g e f f e c t s can occur i n the e . s . r . spectrum. (1) S h i f t i n the h y p e r f i n e s p l i t t i n g  constant  (2) S h i f t i n the g - value (3) M u l t i p l e l i n e s i f d i f f e r e n t t r a p p i n g s i t e s are a v a i l a b l e (4) Hyperfine i n t e r a c t i o n s with magnetic n u c l e i o f matrix p a r t i c l e s can l e a d t o a d d i t i o n a l l i n e s i n the spectrum. In the matrix,there are two important  i n t e r a c t i o n s between the matrix  p a r t i c l e and the r a d i c a l . T h e f i r s t i s t h e van der Waals i n t e r a c t i o n s , i . e . induced d i p o l e - induced d i p o l e i n t e r a c t i o n s . Secondly there i s t h e "overlap" i n t e r a c t i o n , w h i c h a r i s e s from overlap between the wave f u n c t i o n s o f the matrix p a r t i c l e and the f r e e r a d i c a l . The van der Waals i n t e r a c t i o n leads t o a r e d u c t i o n i n the h y p e r f i n e s p l i t t i n g constant whereas the "overlap" i n t e r a c t i o n leads t o an increase i n the hyperfine s p l i t t i n g constant and t o a s h i f t i n the g - v a l u e . A l s o the "overlap" i n t e r a c t i o n leads t o an i n t e r a c t i o n with any magnetic n u c l e i of the matrix p a r t i c l e and hence e x t r a l i n e s i n the e . s . r . spectrum.In the f o l l o w i n g d i s c u s s i o n we s h a l l study the i n t e r a c t i o n between the r a d i c a l and the N  nearest neighbour matrix p a r t i c l e s . T h e van der Waals and "overlap" i n t e r a c t i o n s  are short range and thus we need only consider t h e nearest neighbour matrix p a r t i c l e s . B e f o r e proceeding f u r t h e r , l e t us r e c a l l t h a t f o r a Hamiltonian of the  we o b t a i n the energy eigenvalues  and the eigenstates  - 31 -  I f we add t h e p e r t u r b a t i o n term  '  MgJfeMx /f' d $  —  we o b t a i n t h e f i r s t order c o r r e c t i o n t o the e n e r g j r ^- ; _ ^Jpt^ 7  where  A  —  ^xV^C^c-^^  ^  o d de  l  e  c  t  r  o  i s i n <T  n  (^fj state)  We s h a l l now f o l l o w Adrian's example o f studying Hydrogen atoms trapped out i n an i n e r t matrix* van der Waals I n t e r a c t i o n s Let the form  us consider the f o l l o w i n g problem.The zero order Hamiltonian i s o f ^  M  ~  ^  +  ^  M  where ^4j/ i s t h e Hamiltonian f o r an i s o l a t e d hydrogen atom i n a magnetic f i e l d and  i s the Hamiltonian f o r an i s o l a t e d matrix p a r t i c l e . T h e zero order  s o l u t i o n f o r the energy i s :  E * ° * = E * * + E * ) + Mg g ^ B ^ * M  C  \  Z-^^q  -s energy o f matrix p a r t i c l e  (C) E  = energy f o r coulombic i n t e r a c t i o n between e l e c t r o n and proton of hydrogen  atom and k i n e t i c energy o f e l e c t r o n . The corresponding eigenstates are ^ ^ i o / ^  slV>, ^  l ^ o l ^  where 1»* J ^ ^ M S / H ^ ^ . H ^ o l u t i o n 1V H d ^  = s o l u t i o n f o r matrix p a r t i c l e  Now consider the p e r t u r b a t i o n term  "* 7^1^.^  f o r the hydrogen atom  =  | J s ^  ^ f * "  2  h % c o  ~  ^4 ^ ^ r  - ^ u " V - ^  o r d i j l a  *  e s  °*  centre o f mass o f the matrix p a r t i c l e . and  ^ l y f j  ^  yA-th e l e c t r o n measured from the i s the hyperfine s p l i t t i n g term  d e s c r i b e s the d i p o l e - d i p o l e i n t e r a c t i o n s .  From time independent p e r t u r b a t i o n theory T h i s leads t o the r e s u l t  {T^—  A  =  ^  Ssr  S^'* ~*^Ui|^*V^ l ^ * ' I fi\  \A  £  ^ l ^ u o l  2  l^c//'  - 32 -  To obtain a f i r s t order s h i f t i n A we want t o c o l l e c t terms t h a t are l i n e a r in  ^$(^ .Let us now s t a t e t h e two s e l e c t i o n r u l e s :  0)  '^•)<.Vh.|44VI^\VISV»D  if  l h f > i s an s-state  JA  Now i f  hence  • i s an s—state  j ^ *  *****  (  f  ^  ^  a  ^^T"  O  W & i i , * ^ * * *  •I 4 o K ^ \ ^ l  GO  r  T  <WoK^n*  unless I U ^ i s  \W >  \ v v « 4 V 0 i f \ k l > i s an s-state  I O K > °  an s-state 'vjjf'.  i  f  1  n  o  t  °* s  s t a t e  =0,i.e. the wave f u n c t i o n vanishes a t the  nucleus.  ^VtxA ^ K > \  I hi*} \ ^k^z  0  a fLO . T h i s f o l l o w s since the s e t  i s orthonormal.Using thqse s e l e c t i o n r u l e s the second order energy term reduces to: i n f | ^  "* 4-  "' '"' "i~ v  y"' 'I -  Here we have no terms l i n e a r  ,so we go t o the t h i r d order term and c o l l e c t terms l i n e a r i n  •  = 33 -  <^l<h».|^ |U >|h, > K4i  Tk,s  Now  reduces.  lex u,  e  e  +o  "ive^j*  co*ce*+  - 34  Now E  W  I*  <U \<w»J 44 e  =  *  v  \ k>  *O  so w t ob-l-;u«  5  <<U<wM<VW^ |u >lK, >  \s ho4- d ' i + ^ i c u l T -  2  w  e  e  ^.e H i a > +  bracked  <W*\<^o\<Ww*%MO\*o> v&nfclvs^ So our resul-V ts  - 35 The van der Waal8 energy i s d e f i n e d as  where  A=  £  2 <k  y  I <%1  p  ^<v  l*0/**,>>  t ^ ^ r ^ C © } ! ^  and we now have an expression  f o r t h e energy a f t e r we take i n t o account t h e  e f f e c t of the p e r t u r b a t i o n :  £ '=  £~ ° ' +  I  which e f f e c t i v e l y introduces a r e l a t i v e change i n the hyperfine s p l i t t i n g constant  Thus the van der Waals i n t e r a c t i o n reduces the hyperfine s p l i t t i n g constant.  Overlap E f f e c t s  ( E f f e c t on Hyperfine s p l i t t i n g constant A ) .  These become appreciable when the d i s t a n c e between t h e H - atom,and the matrix p a r t i c l e i s reduced t o the p o i n t where the overlap between the electronic  charge clouds i s appreciable. L e t £ ( l t « i ^ be the s e t o f enetfgy eigenstates f o r the matrix p a r t i c l e  and ^W«*^  be the ground s t a t e f o r the H - atom.When the two systems are s u f f i c i e n t l y  c l o s e together t h a t overlap e f f e c t s must be considered,we can no longer t h i n k of the hydrogen atom and matrix p a r t i c l e as two d i s t i n c t systems.The energy eigenstates of t h i s new combined system must form an orthonormal s e t o f eigenvectors since the Hamiltonian operator representing the energy observable i s Hermitffan. We obtain an approximation lk^>  t o the s o l u t i o n by o r t h o g o n a l i z i n g  t o the orthonormal s e t ^ j H ^ and renormalizing.So  but s o C j r - ^ - J k ^  and  K m  <w li i  0  >  ^  H t  -> V  rr  V»o>  -  l e t us w r i t e :  Zjl 5 \ < t e . . \ k ^  IVH;>  - 36 -  N o r * * * \\  So Set  our  aW  »tw  vr-a+e.  wUia,  i s OV>4-U Mo^m) 0  \o  H e  {l*V.>} i s ^  |Hefl«(i)>  \H ^ll)> 0  U,o/CN)>  9 *  - 37 The hyperfine s p l i t t i n g i s given  by:  which reduces t o  from which we o b t a i n  __tO  c_  N e g l e c t i n g small order terms we  where  £ -~  < £  ,n  „, .  o b t a i n the approximation  J ^  1 kj»> r  x  ;  *  ^  ^  *  "overlap i n t e g r a l "  Defining  Thus we  see t h a t we have a r e l a t i v e change i n the hyperfine s p l i t t i n g constant  g i v e n by!  _^  A  ~  V  So the overlap e f f e c t i n c r e a s e s the hyperfine s p l i t t i n g constant.  Overlap E f f e c t s  ( E f f e c t on  g - value )*  In a magnetic f i e l d the o r b i t a l angular momentum has an energy given by:  In t r a v e r s i n g magnetic f i e l d  an e l e c t r i c f i e l d B  given by  E  at a v e l o c i t y  B = - " x E c  of a p o s i t i v e charge  £ ^  =  — _i=— W<_  <£__ <^<~  v  there i s an apparent  . In the s p h e r i c a l l y symmetric s  -V XE*  field  ( "~  u = electrostatic potential p o s i t i v e charge.  of  - 38 -  The s p i n - o r b i t  i n t e r a c t i o n energy i s :  the f a c t o r o f % i s a r e l a t i v i s t i c e f f e c t known as the "Thomas Precession". The Hamiltonian f o r a s p i n possessing o r b i t a l angular momentum i s :  R e c a l l from page  ,the zero order energy f o r an i s o l a t e d H - atom i s :  To compute the s h i f t i n g - value we look f o r terms l i n e a r i n H ^ Q B  • We get  no such term from f i r s t order p e r t u r b a t i o n theory, i n f a c t  < * J < M  %.  ) H->  \O=0(20)  So we go t o second order p e r t u r b a t i o n theory and c o l l e c t the appropriate terms.  Introduce the average e x c i t a t i o n  energy i n order t o o b t a i n the above expression  i n c l o s e d form.  Since  O t ^ ^ M . l ^ , ^  |H*> = 0  Now r e t a i n i n g terms l i n e a r i n H^PQ  IVT*>> = I W O Take the outer  w  e  S * e  ^  -.  s  x^t^  p - o r b i t a l o f the matrix p a r t i c l e d i r e c t e d  as the one g i v i n g an e f f e c t i v e c o n t r i b u t i o n  to  ?  (2)  ,  towards t h e r a d i c a l  - 39 -  ^p^>is an eigenvalue o f  since  we can write  define  «  P  .  Mix,HkifcH.'Sl « *A,, t  Summing over the N nearest neighbours we g e t Hence  J  ^>f  3  > N  w  - . | |  i  )  H  t  9 £  w  so t h a t  ( » , l ,  K  f  . U . M ^  f  <  - 40 -  II.  A.  EXPERIMENTAL  E l e c t r o n S p i n Resonance Spectrometer The  spectrometer  (see f i g u r e s 2-1  the Department o f Chemistry. A 7-153  and 2-2)  was  designed and b u i l t i n  k l y s t r o n o s c i l l a t o r , i m m e r s e d i n a water  cooled o i l bath,generates 50 m i l l i w a t t s of power a t a frequency of 9 K Mc/sec. A r e c t a n g u l a r r e f l e c t i o n - t y p e c a v i t y resonates i n the T E Q J ^ arranged  m 0  ^e  and i s  i n a magic-tee bridge provided with an attenuator and a p h a s e - s h i f t e r  f o r b a l a n c i n g . A s e l e c t e d 1N23E s i l i c o n diode i s employed as a microwave d e t e c t o r . For a diode operating on the square-law p o r t i o n of i t s c h a r a c t e r i s t i c , -4 i . e . i n c i d e n t power l e s s than 10  watts,the maximum d e t e c t o r s i g n a l i s obtained  when the c a v i t y r e f l e c t s one t h i r d o f the power reaching i t . T h i s corresponds a v o l t a g e standing wave r a t i o of 2 the c a v i t y was  3.74.  The  s i z e of the coupling i r i s of  adjusted u n t i l the c o r r e c t standing wave p a t t e r n was  magnetic f i e l d of 3 k i l o g a u s s i s s u p p l i e d by a s i x - i n c h V a r i a n having a 2.5  obtained.The  electromagnet  i n c h gap and f i t t e d w i t h ring-shim pole caps. Magnetic f i e l d  modulation a t 400  c/sec i s introduced through a p a i r of a u x i l i a r y c o i l s mounted  around the cavity.The f i e l d was motor-driven  to  scanned by i n j e c t i n g a v o l t a g e d e r i v e d from a  h e l i p o t i n s e r i e s w i t h the s t a n d a r d - c e l l v o l t a g e of the magnet  power supply. The 400  c/sec resonance s i g n a l passes through a  low-noise  p r e a m p l i f i e r t o an a m p l i f i e r and a p h a s e - s e n s i t i v e demodulator,which uses a p a i r o f synchroverters.The  resonance f i n a l l y appears as a t r a c e on a r e c o r d i n g  m i l l i v o l t m e t e r . T h e d e r i v a t i v e o f s u s c e p t i b i l i t y with respect t o magnetic f i e l d i s plotted. B.  L i q u i d Helium Dewar Free r a d i c a l s , f o r m e d i n the gas phase by e l e c t r o n i m p a c t , p y r o l y s i s ,  i r r a d i a t i o n , o r chemical r e a c t i o n , c a n be condensed i n an i n e r t matrix on a surface  - 41 -  cooled t o l i q u i d helium temperature, 4 K. Reaction i n the gas phase i s prevented by conveying- the sample i n a molecular  beam t o the r e g i o n o f condensation*An  a l t e r n a t i v e technique i s t o d e p o s i t s t a b l e substances i n a matrix a t the low temperature and then produce the r a d i c a l s by i r r a d i a t i o n of the solid.The equipment was made s u f f i c i e n t l y v e r s a t i l e t o permit e i t h e r approach* The design o f the l i q u i d helium dewar i s based on t h a t published by Duerig and Mador ( 21 ) . M o d i f i c a t i o n s were made t o allow the i n c o r p o r a t i o n o f the microwave c a v i t y i n s i d e the dewar. A c y l i n d r i c a l container f o r h o l d i n g the l i q u i d helium i s surrounded by a vacuum j a c k e t which i s surrounded by a l i q u i d n i t r o g e n container which i s surrounded by a f u r t h e r vacuum j a c k e t . In f i g u r e 2-3,  A represents the c a v i t y b o l t e d t o a l i q u i d helium cooled stub B, which i s  an extension o f the helium container C. The p a r t s a t l i q u i d helium temperature are enclosed by a l i q u i d n i t r o g e n cooled s h i e l d D i n thermal contact with the n i t r o g e n container E . To minimize heat leakage t h e e x t e r n a l microwave c i r c u i t r y i s coupled t o the c a v i t y by a s e c t i o n o f f i b e r - g l a s s waveguide F, s i l v e r p l a t e d on the i n s i d e , and f u r t h e r by a matched vacuum gap G. A mica window H a c t s as a vacuum s e a l . A r e a c t o r J was devised i n which a f a s t - f l o w i n g stream o f gas could be subjected t o a microwave discharge. The gas enters by the inner tube ( 10 mm diameter ) as i n d i c a t e d and i s evacuated through the outer one ( 40 mm diameter) by means o f a f a s t pump , Welch 1397 r a t e d a t 375 l i t e r s per minute. The microwave source i s a 100 watt Raytheon Microtherm u n i t and the r a d i a t i o n i s d i r e c t e d a t the r e a c t o r by the antenna supplied with the equipment. The discharge takes place i n the inner tube o f the r e a c t o r i n the l a s t 2 cm before the t i p . U s u a l l y the substance under i n v e s t i g a t i o n i s mixed with the matrix gas i n the p r o p o r t i o n o f 1:100 .  I t i s estimated  t h a t gas molecules t r a v e r s e the discharge  - 42 -  r e g i o n i n about one m i l l i s e c o n d . A p o r t i o n of the discharge products i s p r o j e c t e d i n a molecular beam through a 500 micron hole i n a 20 micron platinum diaphragm, and through f u r t h e r holes i n the l i q u i d n i t r o g e n s h i e l d and c a v i t y w a l l . Condensation occurs on the end of a 0.1 i n c h s y n t h e t i c sapphire rod K mounted i n s i d e the c a v i t y a t l i q u i d helium temperature. In some experiments i t i s d e s i r a b l e t o i r r a d i a t e s o l i d d e p o s i t s on the sapphire r o d . L i g h t i s then shone through the mica window and along the waveguide t o the c a v i t y . A spectrophotometer t e s t showed t h a t the mica window was transparent t o wavelengths as short as 3500 X . The vacuum system r e q u i r e d f o r evacuating the dewar,pumping the r e a c t o r gas stream and p r e p a r i n g the samples was of conventional design w i t h thermocouple  and  i o n i z a t i o n gauges used f o r monitoring the pressure at appropriate p o i n t s . C.  Techniques The f o l l o w i n g procedure was used i n p r e c o o l i n g the dewar i n p r e p a r a t i o n  f o r a l i q u i d helium t r a n s f e r . The l i q u i d n i t r o g e n s h i e l d was f i l l e d and a few minutes were allowed f o r temperature e q u i l i b r i u m t o be e s t a b l i s h e d . A t t h i s p o i n t the  blow-off r a t e of n i t r o g e n through the vents was v e r y slow.The next step was  t o introduce about one cm of Hg of dry helium gas i n t o the vacuum space of the dewar. T h i s allowed heat t r a n s f e r between the helium container and the n i t r o g e n s h i e l d . The gas was l e f t i n the dewar f o r 15 - 20 minutes and then pumped out. U s u a l l y the apparatus was l e f t f o r 3 - 4  hours t o ensure t h a t the helium  container was cooled t o l i q u i d n i t r o g e n temperature by r a d i a t i o n . L i q u i d helium was t r a n s f e r r e d from the storage v e s s e l t o the experimental dewar through a vacuum j a c k e t t e d syphon. Helium gas a t a few p s i f o r c e d the l i q u i d over. In the i n i t i a l stage the r a t e of t r a n s f e r was slow so t h a t most of the c o o l i n g  was  acheived by c o l d helium gas r a t h e r than the l i q u i d . T h i s leads t o an economical use of the r e f r i g e r a n t since the l a t e n t heat of v a p o r i z a t i o n i s small, 6 c a l o r i e s per gram. The helium gas evolved i s recovered f o r p u r i f i c a t i o n and f u t u r e l i q u e f a c t i o n . D e p o s i t i o n of the sample can be observed through the mica window  - 43 -  of the dewar. Usually when the deposit just becomes v i s i b l e there i s enough to produce acceptable  spectra.  The electron spin resonance spectrometer was allowed to warm up for several hours prior to an experiment. After sample deposition the microwave system was bolted into place on the dewar. The klystron was aligned with respect to the cavity by adjustment of the tuning screw and the reflector voltage control. The bridge was balanced by successive adjustments of the attenuator and phase shifter u n t i l the detector current reached zero. An out-of-balance current of about 10 microamperes was established by adjusting the attenuator. The magnetic f i e l d was scanned to produce a spectrum and the controls of the scanning unit, modulation amplifier, signal amplifier and demodulator were set to attain the optimum resolution and signal-to-noise r a t i o .  S C A N N I N G UNIT  M A G N E T S Y S T E M  MODUl . A T O R  MICROWAVE  K L Y S T R O N  S Y S T E M  S U P P L I E S  O S C I L L A T O R  PREAMPLIFIER  R E F E F IENCE A M P L IFIER  A M P L I F I E R  S Y N C H R O N O U S D E M O D U L A T O R  R E C O R D I N G MILL I V O L T M E T E R  F I G U R E  2-1.  B L O C K  D I A G R A M  OF  E S R  S P E C T R O M E T E R  K  L Y S T R O N  F  ERRITE  I S O L A T O R  A T T E N U A T O R  in •"t  C A V I T Y  MAGIC  A T T E N U A T O R  T E E  P H A S E S H I F T E R  R E F L E C T O R  D E T E C T O R  F I G U R E  2 - 2 .  B L O C K  D I A G R A M  OF  M I C R O W A V E  S Y S T E M  F I G U R E  2 - 3 .  LIQUID  H E L I U M  D E W A R  - 47 -  III.  A.  RESULTS AND DISCUSSION  Introductory Remarks on NO,» Nitrogen d i o x i d e dimerizes according t o the f o l l o w i n g equation 2N0  =  2  N 0 2  4  At room temperature the r a t i o of NO„ : N O i n the e q u i l i b r i u m mixture i s 2 c 4 roughly 1:4  (22). At temperatures below the f r e e z i n g p o i n t , - 9.5°C, only the  dimeric form i s found. However i t i s p o s s i b l e t o f r e e z e out the room temperature e q u i l i b r i u m mixture i n an i n e r t matrix such as A,  , CH^ etc* Since an esr  spectrum has been observed by C a s t l e and Beringer (23) i n the gas phase,one would expect an e s r spectrum from the room temperature e q u i l i b r i u m trapped out a t low temperatures i n an i n e r t m a t r i x . An experiment performed a t 77°K y i e l d e d no e s r spectrum thus i n d i c a t i n g t h a t at t h i s temperature the N0 c o u l d d i f f u s e through the l a t t i c e and dimerize t o ^ 0 ^ of e l e c t r o n s whereas would be expected from N0  2  molecules  has an odd number  has an even number of e l e c t r o n s , t h u s an e s r spectrum 2  but not from N ^  The s p e c t r a f o r NO^A, 3-1, 3-2, 3-3  . N0  2  .  NO^CH^ and N0 /N 2  2  at 4°K are shown i n f i g u r e s  . One f e a t u r e common t o the three s p e c t r a i s t h a t the i n t e n s i t y  r a t i o of the three l i n e s i s not the simple r a t i o 1:1:1  expected f o r the n i t r o g e n  t r i p l e t . For the NO,,/A spectrum three groups of three l i n e s are r e a d i l y seen whereas f o r the NOg/CH^ and NOg/^  s p e c t r a there are three r a t h e r broad l i n e s  w i t h some i n d i c a t i o n o f s p l i t t i n g . The NO^A  spectrum shown i n f i g u r e 3-1 agrees  i n general f e a t u r e s w i t h t h a t reported i n the l i t e r a t u r e (24). However the present spectrum has more d e t a i l . The ideas developed i n Adrian's theory (19) e x p l a i n a number of the p r o p e r t i e s of the three spectra obtained. B.  Nature of Trapping S i t e s Argon, methane and n i t r o g e n a l l c r y s t a l l i z e i n a f a c e - centred cubic  - 48 -  l a t t i c e (26). For a face - centred cubic l a t t i c e there are three s i t e s i n which a f r e e r a d i c a l can be l o c a t e d . The f i r s t p o s s i b l e s i t e i s s u b s t i t u t i o n at a l a t t i c e p o i n t where there are '''1?/ nearest neighbours and cubic symmetry. The second s i t e i s an i n t e r s t i t i a l s i t e at the body centre of the face-centred cube. Here there are s i x nearest neighbours and octahedral symmetry. The t h i r d s i t e i s an i n t e r s t i t i a l s i t e h a l f way between a corner and the body centre of the face centred cube. Here there are f o u r nearest neighbours and t e t r a h e d r a l symmetry. The three a v a i l a b l e s i t e s are i l l u s t r a t e d i n f i g u r e 3-4  and t h e i r p r o p e r t i e s  are summarized i n the f o l l o w i n g t a b l e . SITE  Cubic  NUMBER OF NEAREST NEIGHBOURS .  (Subst.)  Octahedral  12  (Inter.)  Tetrahedral (Inter.)  DISTANCE TO NEAREST NEIGHBOUR ( SIDE OF FACE CENTRED CUBE = 1 ) $2 =  1.414  6  f = 0.500  4  = 0.433  Each s i t e w i l l have a d i f f e r e n t e f f e c t on the hyperfine s p l i t t i n g constant A  and  upon the g v a l u e . T h i s accounts f o r the s p l i t t i n g of each main l i n e i n t o three components f o r the NO^A  spectrum. Foner, Cochran, Bowers and Jen observed the  same e f f e c t f o r HI photolyzed There are two  i n A. (25)  i n t e r a c t i o n s which a f f e c t the s h i f t of the hyperfine  s p l i t t i n g constant. The van der Waals i n t e r a c t i o n s l e a d t o a decrease i n A,while the overlap e f f e c t s l e a d t o an i n c r e a s e i n A. According t o the measurements of Jen, Foner, Cochran and Bowers (24) the hyperfine s p l i t t i n g constant A i s 12?S g r e a t e r f o r the TSQ^/A  spectrum at 4°K than f o r the spectrum of HO^  i n the  gas  phase. Thus i t appears t h a t the overlap e f f e c t s are the predominant effects.The s h i f t s i n the hyperfine s p l i t t i n g constant and the g value are given by the formulae:  A  1  From the proceeding  formulae i t i s seen t h a t the s h i f t i n A and g depends upon  the squares of overlap i n t e g r a l s . I f the square of the overlap i n t e g r a l behaved l i k e an i n v e r s e square l a v on the separation R of the f r e e r a d i c a l and matrix p a r t i c l e then the r e s u l t shown i n the f o l l o w i n g t a b l e would be  SITE  R  1/R  Tetrahedral  0.433  Octahedral Cubic  obtained.  Ni  N/R  5.33  4  21.3  0.500  4.00  6  24  1.414  0.50  12  6  2  2  i I f the overlap i n t e g r a l squared behaved l i k e an i n v e r s e cube law on the s e p a r a t i o n R of the f r e e r a d i c a l and matrix p a r t i c l e then the f o l l o w i n g r e s u l t would be  obtained.  SITE  R  1/R  N  N/R  Tetrahedral  0.433  12.3  4  49.2  Octahedral  0.500  8.0  6  48.0  Cubic  1.414  0.354  12  4.2  The  3  i n v e r s e square law i s a long range e f f e c t and the inverse cube law i s a  moderate range e f f e c t . The t h i s we  overlap i n t e g r a l i s a short range e f f e c t , and from  can assume t h a t the t e t r a h e d r a l s i t e g i v e s the g r e a t e s t s h i f t to A  g while the cubic s i t e g i v e s the l e a s t s h i f t t o A and C.  NO^A  g.  Spectrum  In f i g u r e 3 - 5 (b)  3  4g>0  the f o l l o w i n g assumptions were made: (a)  (C)|M\A>2.U«JJ  &  . Although A d r i a n ^ treatment leads t o a  A>0  and  - 50 -  negative s h i f t i n g, a p o s i t i v e s h i f t i n g i s r e q u i r e d t o e x p l a i n the observed spectrum f o r NO^A. F o r A A > 0 and  e f t  A g > 0 (M  ) the equation  ^  w i l l p r e d i c t l e s s o f a s h i f t on the h i g h f i e l d s i d e ( Mj = 1 ) than f o r the low f i e l d s i d e ( Mj = - 1 )• T h i s f o l l o w s s i n c e the two e f f e c t s oppose each other f o r t h e h i g h f i e l d side and are a d d i t i v e f o r the low f i e l d s i d e . Thus the three l i n e s a r i s i n g from the three d i f f e r e n t t r a p p i n g s i t e s w i l l be c l o s e together a t the h i g h f i e l d end and w i l l be spread apart a t the low f i e l d end. F o r the case when  , the spacings between the three centre l i n e s and the three  h i g h f i e l d l i n e s are about the same. In a c t u a l p r a c t i s e the s p l i t t i n g o f each l i n e o f the n i t r o g e n t r i p l e t i n t o three l i n e s by the three d i f f e r e n t t r a p p i n g s i t e s i s n o t completely r e s o l v e d . T h i s i s a consequence o f l i n e broadening. As a r e s u l t the low f i e l d l i n e s ( Mj = - 1 ) w i l l appear broad and d i f f u s e whereas the centre l i n e s (Mj = 0) w i l l appear as a more intense and narrower l i n e . The h i g h f i e l d l i n e s ( Mj = 1 ) w i l l have a s i m i l a r appearance t o the centre l i n e s . F o r a l l three l i n e s some degree o f s p l i t t i n g occurs. The l i n e s observed i n the NO /A spectrum are broad compared t o the l i n e s observed f o r d i l u t e s o l u t i o n s o f aromatic f r e e r a d i c a l s . The l i n e s o f the spectrum o f NO  2  i n the gas phase are v e r y broad. T h i s i s b e l i e v e d t o be due t o  unresolved r o t a t i o n a l s t r u c t u r e . The n i t r o g e n - oxygen bond l e n g t h i n NO  i s 1.2 4  and the bond angle i s about 120 • The l a t t i c e constants f o r A, CH^ and  are  r e s p e c t i v e l y 5.43 X , 5 . 6 6 X and 5.89 X • From a c o n s i d e r a t i o n of the dimensions of the NO  molecule  and t h e l a t t i c e constants o f t h e matrix m a t e r i a l s , i t i s  seen t h a t the p o s s i b i l i t y o f r e s t r i c t e d r o t a t i o n e x i s t s . T h i s could w e l l be an important  c o n t r i b u t i o n t o the l i n e broadening  i n the N0 /A spectrum. Jen, Foner, 2  Cochran and Bowers (24) have a l s o suggested t h a t c r y s t a l l i n e f i e l d  effects  - 51 could lead to line broadening. D, Possibility of Interaction between Unpaired Electron and Magnetic Nuclei of Matrix Particle. For the case where overlap effects are appreciable the ground state \ H *> 0  for the free radical adjacent to a matrix particle i s obtained by orthonormalizing the ground state \U  for an isolated free radical to the set of matrix particle  states^lrvii')^ • The result obtained i s  In©/ -  >-  ^  "  - -  From this treatment are obtained the effects on the hyperfine splitting constant and the g value. If the matrix particle has a nuclear spin, interactions between the unpaired spin of the free radical and the magnetic nucleus of the matrix particle can occur. This follows since \ W^y possesses some characteristics of the C  electrons of the matrix particle. Foner, Cochran, Bowers and Jen have  observed this phenomenon for the case of hydrogen atoms trapped out in Xenon (25). If \w»f^is the outermost occupied  0* state of the matrix particle, then we  obtain the contribution to the energy  £'=  -  6 M  t  M  4  o  g^ = g value for magnetic nucleus of matrix particle = nuclear magneton for magnetic nucleus of matrix particle \*|{^Jo)|*  t ^ j "ft  = electron density of \*\)} state at magnetic nucleus of matrix particle = z - component of angular momentum of magnetic nucleus of matrix particle  An experiment with NO /CH was performed in the hope that this effect would be observed. Theoretically an esr spectrum of 45 lines would be expected. This arises from the fact that the nitrogen triplet i s split into nine lines by  - 52 -  the three different trapping sites of the matrix. These nine lines would be further split into 45 lines by the four protons of the methane molecule. However a l l that was observed were three lines roughly of the same intensity ratio as the lines in the NO^/A spectrum,although much broader. Some indication of structure was given by the shoulders on the lines and some splitting did occur. The broadening was much greater in the CH^ lattice than in the A lattice. One possible source of this broadening i s splittings of the CH^ protons which are not resolved. If the odd electron of NCv, i s not in a state of symmetry which i s the same as the symmetry in which the N nearest neighbours are arranged, the coupling between the unpaired electron of NOg and the magnetic nucleus of the matrix particle will be different for each of the N nearest neighbours. This could be one possible reason for observing a broadening rather than a splitting. Another possible source of broadening i s restricted rotation of the matrix particles and NO^ molecules. These rotations would be particularly effective in broadening i f they gave the same order of magnitude contribution to the energy of the system as the interactions between the unpaired electron and the magnetic nuclei of the matrix particles. In order to test this explanation for the further broadening of the NO^/CH^ spectrum as compared to the NO^/A spectrum, an experiment was performed with NO /N . since nitrogen also has a magnetic nucleus. Again three very broad  2 2  lines were observed and there was some indication of splitting. This gives some plausibility to the argument that unresolved splittings from the magnetic nuclei of the matrix particles gives rise to line broadening. However there are other possible explanations for the increase in broadening of the NO^CH^ and NO ^  s  P  e c t r a o v e r  t h e N0  2/  A s  P  e c t r u m  » The  - 53 -  c r y s t a l l i n e f i e l d would c e r t a i n l y be stronger f o r CH^ and the A l a t t i c e * A l s o CH N0  2  l a t t i c e s than f o r  and N« possess g r e a t e r p o l a r i z a b i l i t i e s than A, and s i n c e  has an e l e c t r i c d i p o l e moment, broadening could r e s u l t from d i p o l e - induced  d i p o l e i n t e r a c t i o n s . An i n t e r e s t i n g experiment  would be t o deposit NO^ i n a  l a t t i c e with a p o l a r i z a b i l i t y o f the order of CH^ and N  2  but with no magnetic  n u c l e i . T h i s may w e l l i n d i c a t e whether unresolved i n t e r a c t i o n s w i t h magnetic n u c l e i i s an important source o f broadening. E.  L i m i t a t i o n s o f Approach Used In D i s c u s s i n g Spectra. In the theory o f hyperfine s p l i t t i n g the zero order Hamiltonian i s o f  the form  and the p e r t u r b a t i o n term i s of the form  3  4,0.  F i r s t order p e r t u r b a t i o n theory i s used t o o b t a i n t h e energy l e v e l s and a p p l i c a t i o n of the s e l e c t i o n r u l e s lines  y i e l d s the equation f o r the s p e c t r a l  |  g =  ha  Thus an e s r spectrum  +  a m  t  i s c h a r a c t e r i z e d by two numbers; the hyperfine s p l i t t i n g  constant A and the g v a l u e . A l s o l i n e width needs t o be s p e c i f i e d i n order t o completely d e s c r i b e the spectrum. When a f r e e r a d i c a l i s deposited i n an i n e r t matrix apparent  shifts  occur i n the hyperfine s p l i t t i n g constant A and the g v a l u e . A d r i a n developed a semi-quantitative theory t o e x p l a i n these s h i f t s f o r hydrogen atoms trapped i n i n e r t matrices (19). The s h i f t s were obtained by orthonormalizing the ground s t a t e of the hydrogen atom t o the s e t o f s t a t e s o f the matrix p a r t i c l e . The s h i f t i n A was obtained by computing the f i r s t order c o n t r i b u t i o n from the Hamiltonian  - 54 -  The s h i f t i n g value was  obtained by r e t a i n i n g terms l i n e a r i n  ^g^^Q  from  the second order c o n t r i b u t i o n of the Hamiltonian  * I t was  e-tir ti M  set  g  =2  assumed t h a t the f i r s t order c o n t r i b u t i o n from t h i s Hamiltonian  vanished*  The extension of these ideas t o the NO^/matrix s p e c t r a d i d not e x p l a i n a l l the p r o p e r t i e s of these s p e c t r a  satisfactorily*  F i r s t the p o s t u l a t e  A g > 0 had t o be invoked* T h i s i s c o n t r a r y t o  the r e s u l t s of Adrian's theory which p r e d i c t s a negative s h i f t i n g v a l u e . Secondly the s y n t h e t i c spectrum f o r NO^/A w i t h the observed  ( f i g u r e 3-5)  c e r t a i n l y does not agree  spectrum i n a l l d e t a i l s ( f i g u r e 3-1 )•  In computing  the s h i f t i n g value only terms l i n e a r i n  Mg Ug  r e t a i n e d from the second order c o n t r i b u t i o n of the p e r t u r b a t i o n term 9^  were .By  n e g l e c t i n g the other terms from the second order c o n t r i b u t i o n , terms which may g i v e j u s t as l a r g e a c o n t r i b u t i o n to the energy have been discarded* A l s o i t i s q u i t e p o s s i b l e t h a t the f i r s t order term does not v a n i s h f o r the case of NO^*  If  second order terms have to be c o l l e c t e d i n order to o b t a i n a s h i f t i n g, then second order terms from the Hamiltonian ^t^j  may  g i v e e q u a l l y important  contri-  butions t o the energy. These terms have not even been considered. By c h a r a c t e r i z i n g an e s r spectrum by the two numbers A and g, many of the c o n t r i b u t i o n s t o the energy which p e r t u r b a t i o n theory( t o second order) allows are not considered* I t could w e l l be t h a t these terms are manifesting themselves i n the esr spectrum as a p o s i t i v e s h i f t i n the g v a l u e . Thus the anomalies encountered may F.  be consequences of o v e r s i m p l i f i c a t i o n s i n the treatment used*  Suggestions f o r Future  Experiments.  At present the e s r spectrometer used i n these s t u d i e s i s s t i l l i n the stage of development. F i r s t l y the frequency s t a b i l i z e r  c i r c u i t f o r the k l y s t r o n  - 55 -  has not y e t been p e r f e c t e d . I f the k l y s t r o n frequency v a r i e s during the scanning o f a spectrum, l i n e shapes are a f f e c t e d . While reproducable  s p e c t r a have been  obtained, t h i s improvement would allow determination as t o whether some small indentations i n the s p e c t r a l l i n e s are unresolved s p l i t t i n g s or noise a r i s i n g from frequency i n s t a b i l i t y . Secondly an nmr probe f o r the purpose o f measuring the magnetic f i e l d i s being developed.  Once t h i s i s completed, q u a n t i t a t i v e  measurements w i l l be able t o be made on the s p e c t r a obtained, and thus much s p e c u l a t i o n w i l l be removed. The a l k a l i metals are v e r y i n t e r e s t i n g systems t o study by e l e c t r o n s p i n resonance. The wave f u n c t i o n o f the valence e l e c t r o n i s w e l l known f o r sodium. From a knowledge o f the wave f u n c t i o n , the hyperfine s p l i t t i n g constant can be c a l c u l a t e d . The s e n s i t i v i t y of the 100 Kc spectrometer, i n the Department o f Chemistry,  soon t o be i n use  should be s u f f i c i e n t t o o b t a i n an e s r spectrum  from sodium i n the gas phase. The hyperfine s p l i t t i n g constant obtained experimentally could be compared t o the value c a l c u l a t e d from the above c onsiderations• Another i n t e r e s t i n g experiment would be t o d e p o s i t sodium i n argon. F o r Na , I = 3/2, hence a f o u r l i n e spectrum would r e s u l t , the d i s t a n c e between the outermost l i n e s ( M^ = - 3/2 and M^. = 3/2 ) i s 3 A , where A i s the h y p e r f i n e s p l i t t i n g constant. By u s i n g Adrian's theory, or refinements o f i t , a determination o f t h e s h i f t i n the hyperfine s p l i t t i n g constant could be made, and t h i s could be compared t o the experimental  result.  The next experiment planned i s t o photolyze a d e p o s i t o f CF^I/A • I t i s hoped t h a t t h i s method w i l l y i e l d the t r i f l u o r o m e t h y l r a d i c a l . An attempt was made u s i n g a 250 watt BTH lamp, however a f t e r f i v e hours o f i r r a d i a t i o n , no spectrum was obtained. A t present an AH6 lamp i s a v a i l a b l e f o r the purpose and the experiment w i l l be t r i e d again.  - 56 I t a l s o would be q u i t e i n t e r e s t i n g t o deposit the NO^/A mixture  very  s l o w l y . In a v e r y slow d e p o s i t i o n , the most ordered arrangement would be expected* i . e . NO^ molecules l o c a t e d a t s u b s t i t u t i o n a l s i t e s i n the argon l a t t i c e . T h i s would r e s u l t i n the simple three l i n e n i t r o g e n t r i p l e t with a s h i f t of A and g value from the values obtained i n the gas phase spectrum. The three experiments performed so f a r , along w i t h s i m i l a r experiments r e p o r t e d i n the l i t e r a t u r e (25) p o i n t out an i n t e r e s t i n g f i e l d o f study. Once i t i s p o s s i b l e t o make q u a n t i t a t i v e measurements, some i n t e r e s t i n g r e s u l t s f o r f r e e r a d i c a l s trapped out i n an i n e r t matrix a t low temperatures  should be obtained.  - 57 »  FIGURE 3-1. E S R S P E C T R U M OF IN A R G O N M A T R I X A T 4 ° K  NO  F I G U R E IN  3 - 2 .  M E T H A N E  E S R  S P E C T R U M  M A T R I X  A T  4°K  OF  NO.  -  59  -  - 60 -  F I G U R E  O  3-4  MATRIX A T O M , S U B S T I T U T I O N A L  •  OCTAHEDRAL SITE  $  TETRAHEDRAL  Trapping  sites in  SITE  SITE a  face centered cubic lattice.  the  substitutional and  octahedral sites are  the  tetrahedral sites is indicated.  shown.  Only  A l l  of  one  of  FIGURE 3-5. ESR SPECTRA PREDICTED BY THEORY  A. FREE N0  2  5  B. N0  TRAPPED AT CUBIC SITE  2  C. N0  2  D. N0  TRAPPED AT OCTAHEDRAL SITE  2  TRAPPED AT TETRAHEDRAL SITE  f 1 1 1  E. N0  2  F. SAME  II  II  TRAPPED AT ALL THREE SITES  AS E BUT LINE WIDTH CONSIDERED  11  II  B ,  - 62 -  BIBLIOGRAPHY 1.  Condon, E . U. and S h o r t l e y , G. H., The Theory o f Atomic Spectra. Cambridge U n i v e r s i t y Press, 1957.  2.  D i r a c , P. A. M., Quantum Mechanics. Clarendon Press, Oxford, 1958.  3.  Bohm, D., Quantum Theory. P r e n t i c e H a l l , Englewood C l i f f s , N.J., 1954.  4.  P a u l i n g , L. and Wilson, E . B., Introduction t o Quantum Mechanics. McGrawH i l l , New York, 1935.  5.  Landau,E« and L i f s c h i t z , P., Quantum Mechanics, N o n - R e l a t i v i s t i c Theory. Addison Wesley, Reading, Mass., 1957.  6.  Mandl, F., Quantum Mechanics. Butterworths ,London, 1957.  7.  Fermi, E., Z. Physik, 60 320 (1930).  8., Ingram, D. J . E., Free R a d i c a l s Studied by E l e c t r o n Spin Resonance, Butterworths, London, 1958. 9.  Weissman, S.I., J . Chem. Phys., 25 890 (1956).  10.  de Boer, E., J . Chem. Phys., 25 190 (1956).  11.  Wertz, J . E.and V i v o , J . L., J . Chem. Phys., 24 479 (1956)  12.  T u t t l e , T. R., Ward, R. L. and Weissman, S. I . , J . Chem. Phys., 25 189  0  (1956). 13.  Hoskins, R., J . Chem. Phys., 25 788 (1956).  14.  T u t t l e , T. R. and Weissman, S. I . , J . Am. Chem. S o c , 80 5342 (1958).  15.  Mc Connell, H. M., J . Chem. Phys., 24 764 (1956).  16.  Mc Connell, H. M.,and Chestnut, D. B., J . Chem. Phys. 28 107 (l§58),  17.  Mc Connell, H. M. and Robertson, R. E., J . Chem. Phys. 28 991 (1958).  18.  Mc Connell, H. M. and Dearman, H. H., J . Chem. Phys. 28 51 (1958).  19.  A d r i a n , F . J . , J . Chem. Phys., 22,972  20.  A d r i a n , F . J . , Phys. Rev., 107 488 (1957).  21.  Duerig, W.H. and Mador, I . L., Rev. S c i . I n s t r . , 23 421 (1952)  22.  Sidgwick, N. V,, Chemical Elements and t h e i r Compounds. Clarendon Press, Oxford, 1950.  (i960).  - 63 -  23o  Castle, J. G> and Beringer, R., Phys. Rev., 80 114 (1950).  24. Jen, C. K., Foner, S. N., Cochran, £. L. and Bowers, V. A., Phys.Rev., 112 1169 (1958). 25. Foner, S. N., Cochran, E. L., Bowers, V. A. and Jen, C. K., J. Chem. Phys., 32 (1960). 26. Handbook of Physics and Chemistry, 40 th edition, Chemical Rubber Publishing Co., Cleveland, 1959. 27.  Conder, D. W., Private Communications  

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