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EPR studies of aromatic nitrenes Dickinson, James Russell 1974

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EPB STUDIES OF AROMATIC NITRENES BY JAMES RUSSELL DICKINSON B.Sc, University of B r i t i s h Columbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF CHEMISTRY We accept this thesis as conforming to the required standard: THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1974 In presenting th i s thesis in par t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th is thesis for scholar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of this thesis for f inanc ia l gain sha l l not be allowed without my written permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date (UtUrttrSC If-M i Supervisor: C. A. McDowell ABSTRACT The studies presented i n t h i s thesis are i n two areas. An investigation of aromatic nitrene molecules using EPR spectroscopy was undertaken. This revealed a general tendency for these molecules to exhibit two forms when trapped in a c r y s t a l l i n e l a t t i c e . This i s q u a l i t a t i v e l y interpreted as a perturbing e f f e c t of nitrogen evolved during generation from the azide. Single c r y s t a l experiments were made on 2,4,6-tribromophenyl nitrene, which exhibited an unusually large anisotropy in the zero f i e l d s p l i t t i n g , implying a bent structure. Temperature variation of several aromatic nitrene zero f i e l d s p l i t t i n g s were made. The variation was interpreted as thermal averaging amongst states possessing di f f e r e n t zero f i e l d parameters. The nature of these states however, could not be inferred. The variation i s compared with similar studies. A computer program to least squares f i t EPR angular data to a spin Hamiltonian has been written. This programme was used to obtain a description of the angularly dependent EPR data from an aromatic nitrene, and also for a r a d i c a l species with a large hyperfine i n t e r a c t i o n . i i TABLE OF CONTENTS A b s t r a c t i Tabl e o f Contents i i L i s t of T a b l e s i v L i s t o f F i g u r e s v Acknowledgements v i i i I n t r o d u c t i o n 1 CHAPTER ONE: Theory . 6 1.1 T r i p l e t S t a t e 6 1.2 Sp i n Hamiltonian 8 1.3 T r i p l e t S t a t e EPR 13 CHAPTER TWO: Experimental 16 2.1 Apparatus ..16 2.2 P r e p a r a t i o n of Azides ..18 2.3 Sample P r e p a r a t i o n ....20 CHAPTER THREE: Matrix E f f e c t s 22 CHAPTER FOUR: S i n g l e C r y s t a l Experiments 37 4.1 2 ,4 , 6 - t r i c h l o r o p h e n y l N i t r e n e ....38 4.2 2,4,6-tribromophenyl N i t r e n e 42 i i i 4.3 C r y s t a l S t r u c t u r e Of 2 , 4 , 6 - t r i b r o m o p h e n y l A z i d e ., 50 4.4 S i n g l e C r y s t a l EPS ......51 4.5 I n t e r p r e t a t i o n .........59 CHAPTER F I V E : T e m p e r a t u r e E f f e c t s 62 5.1 R e s u l t s And D i s c u s s i o n ...........62 5.2 C o m p a r i s o n W i t h O t h e r S t u d i e s ....84 5.3 C o n c l u s i o n s 87 CHAPTER S I X : D a t a r e f i n e m e n t 88 6.1 I n t r o d u c t o r y Remarks .....88 6.2 T h e o r e t i c a l B a s i s .......90 6.3 F o r m u l a t i o n 94 6.4 P r o p e r t i e s ......97 6.5 Some I l l u s t r a t i v e E x a m p l e s .......102 i. 6.6 T e n s o r Q u a n t i t i e s .115 6.7 E r r o r M e a s u r e .............116 B i b l i o g r a p h y 1 20 A p p e n d i x ...........125 i v LIST OF TABLES 1. Data Characterizing Two State Model .73 2. Typical Nitrene Magnetic Data 103 3. Zero F i e l d S p l i t t i n g Convergence ................ 106 4 . Zero F i e l d S p l i t t i n g Tensor ......108 5. Hyper fine S p l i t t i n g Convergence .110 V LIST OF FIGURES 1. P - n i t r o p h e n y l N i t r e n e EPR S p e c t r a ; (a) Azide Host A f t e r Annealing (b) Azide Host Before Annealing (c) Glassy Matrix .....23 2. P - n i t r o p h e n y l Nitrene EPR Spectra; Azide Host Temperature Dependence. ......................... 26 3. P-chlorophenyl N i t r e n e EPR S p e c t r a ; Azide Host Temperature Dependence .........28 4. Decay K i n e t i c s ; P-chlorophenyl N i t r e n e Species (A) And (B) .........30 5. P - n i t r e n o p h e n y l a r s o n i c A c i d EPR Spectra Showing Two S p e c i e s . ............................ 32 6. M-bromophenyl N i t r e n e EPR Spectra (a) G l a s s y Matrix (b) Powdered Azide Host 36 7. 2, 4 , 6 - t r i c h l o r o p h e n y l N i t r e n e P o l y c r y s t a l l i n e EPR Spectrum 39 8. Angular Dependence Of T r a n s i t i o n F i e l d s ; 2 , 4 , 6 - t r i c h l o r o p h e n y l N i t r e n e In Parent Azide Host. .4 1 9. 2 ,4,6-tribromophenyl Nitrene P o l y c r y s t a l l i n e EPR Spectrum; (a), (b) XY F e a t u r e s ; (c) , (d) Z F e a t u r e s . 43 1 0 . 2,4,6-tribromophenyl N i t r e n e P o l y c r y s t a l l i n e Spectrum; (a) Observed (b) Simulated. 45 11. T r i p l e t Energy L e v e l s As A F u n c t i o n Of v i Applied Magnetic F i e l d Strength, (a) , (b) XY Features; (c) Z Feature. .......46 12. Frequency Response Of Nitrene EPR Transitions, (a) , (b) XY Features; (c) , (d) Z Features. 48 13. Crystal Morphology And Axis System. .....52 14. Angular Dependence Of Transition F i e l d s ; AB And AC Planes 54 15. Angular Dependence Of Transition Fields; BC Plane 55 16. Temperature Dependence Of Zero F i e l d Parameter D In Various Mitrenes. ................ 65 17. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; P-methoxypheny1 Nitrene. ............. 67 18. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; P-carboxyphenyl Nitrene. ...68 19. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; P-nitropheny1 Nitrene. .......69 20. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; 2,4,6-tribromophenyl Nitrene. ........70 21. Temperature Dependence Of Zero Fie l d S p l i t t i n g ; P-chlorophenyl Nitrene (B) ...71 22. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; P-chlorophenyl Nitrene (A). ..........72 23. Graph Of Ln{D(0) - D(T) } Vs. 103/T For P-nitrophenyl Nitrene. ..........77 v i i 24. Graph Of Ln{D{0) - D (T) } Vs. 10VT *ov A Three State Description. ........................ 79 25. Convergence Behaviour. .......................... 98 26. Convergence Behaviour. ........99 27. Axis System. ....................................105 28. Angular Behaviour Of Resonant Fields. KDA .......112 29. Powder EPR Spectrum. KDA ........................114 ACKNOWLEDGEMENTS I would l i k e to express ray g r a t i t u d e f o r the o p p o r t u n i t y of a t t e n d i n g t h i s i n s t i t u t i o n . In p a r t i c u l a r , I f i r s t wish to acknowledge the guidance of my r e s e a r c h d i r e c t o r , P r o f . C A . McDowell, both i n the t o p i c and p r e p a r a t i o n o f t h i s t h e s i s . I want to thank him, as head o f the department, f o r the manifold f a c i l i t i e s a v a i l a b l e to me. In a d d i t i o n , d i s c u s s i o n s with other f a c u l t y members which have been of a s s i s t a n c e to my s t u d i e s are g r a t e f u l l y acknowledged., S i m i l a r l y , numerous d i s s c u s s i o n s with other members o f the magnetic resonance group are acknowledged. I am p a r t i c u l a r l y indebted to Dr. J . Hebden and Dr. N.S. D a l a i . I a l s o thank Dr. D. Bendle f o r p e r s e r v e r i n g with the s t r u c t u r e d e t e r m i n a t i o n , and the e l e c t r o n i c s shop of t h i s department f o r maintenance of the spectrometers. F i n a l l y I acknowledge the f i n a n c i a l support of the N a t i o n a l Research C o u n c i l by way of a s c h o l a r s h i p . 1 INTRODUCTION The s t u d i e s to be presented i n t h i s t h e s i s may be broadly separated i n t o two d i v i s i o n s . The f i r s t of these are s t u d i e s on aromatic n i t r e n e s p e c i e s by E l e c t r o n Paramagnetic Resonance (EPR) spectroscopy. The second p o r t i o n p e r t a i n s to e f f o r t s i n d e v e l o p i n g a computer program t o a i d i n the a n a l y s i s of EPR data." While these s t u d i e s are not the f i r s t i n t h e i r r e s p e c t i v e areas, I do not expect them t o be the l a s t . The i n v e s t i g a t i o n of aromatic n i t r e n e s r e v e a l e d a r a t h e r g e n e r a l tendency f o r these molecules to e x i s t i n two forms, when trapped i n a c r y s t a l l i n e l a t t i c e . T h i s o b s e r v a t i o n was i n t e r p r e t e d as r e s u l t i n g from p e r t u r b i n g e f f e c t s o f evolved n i t r o g e n . A l a r g e temperature e f f e c t on the EPR s p e c t r a of these molecules was observed. The temperature e f f e c t was i n t e r p r e t e d with a model c o n s i s t i n g of thermal averaging amongst s t a t e s with d i f f e r e n t zero f i e l d s p l i t t i n g s . The nature of these s t a t e s c o u l d not be i n f e r r e d with c e r t a i n t y , however. A comparative study of the r e l a t i v e m e r i ts of gl a s s y and c r y s t a l l i n e host matrices was made. T h i s i m p l i e d that f u r t h e r i n t e r e s t should be centred on the c r y s t a l l i n e host matrix, and s i n g l e c r y s t a l experiments were undertaken. A computer program of c o n s i d e r a b l e g e n e r a l i t y was w r i t t e n . I t r e f i n e s s p i n Hamiltonian parameters using l e a s t squares methods and was used t o i n t e r p r e t v a r i o u s problems of / 2 i n t e r e s t i n t h i s l a b o r a t o r y . In order t o present t h i s m a t e r i a l i n context, i t i s necessary t o present a b r i e f review of p u b l i s h e d work on the n i t r e n e s . Due to the volume o f i n f o r m a t i o n which has been p u b l i s h e d on n i t r e n e s or analogues, a f u l l review i s not p r a c t i c a b l e . P u b l i s h e d s t u d i e s on n i t r e n e s can be d i v i d e d roughly i n t o two c l a s s e s . These are i n v e s t i g a t i o n s of the chemical r e a c t i v i t y of n i t r e n e s and s p e c t r o s c o p i c i n v e s t i g a t i o n s o f the n i t r e n e s . The former c l a s s o f s t u d i e s does not c o n t a i n the emphasis of the present i n v e s t i g a t i o n , so s h a l l not be followed f u r t h e r except to remark that i n v e s t i g a t i o n s i n t o the chemical r e a c t i v i t y of n i t r e n e s and the p a r t which they p l a y i n c e r t a i n r e a c t i o n s were numerous enough that by 1967 a review of t h i s f i e l d was compiled (1). From the s p e c t r o s c o p i c s t a n d p o i n t , the emphasis of i n v e s t i g a t i o n s seems t o have been on EPR spectroscopy. By no means however i s t h i s the only technigue which has been u t i l i z e d s i n c e the o p t i c a l a b s o r p t i o n s p e c t r a of s e v e r a l types of n i t r e n e s have been observed (2, <3). Numerous i n v e s t i g a t i o n s of the EPR of n i t r e n e s and carbenes have been made. Many of these have been i n c l u d e d i n a review a r t i c l e on t h i s t o p i c p u b l i s h e d i n 1971 (4). The f i r s t r e p o r t s appeared nine years e a r l i e r . EPR was used to d e t e c t ground-state t r i p l e t s p e c i e s p h e n y l n i t r e n e and a l s o 3 diphenylmethylene (5) . At the same time the EPR of s e v e r a l ground s t a t e t r i p l e t n i t r e n e s was i n v e s t i g a t e d (6). I n t e r p r e t a t i o n of the o b s e r v a t i o n s , that i s to say the magnitude of the zero f i e l d s p l i t t i n g s was not yet c e r t a i n f o r the n i t r e n e s . A s h o r t time l a t e r , EPR o b s e r v a t i o n s provided the f i r s t evidence f o r dicarbene and d i n i t r e n e s p e c i e s ( 7 ) . EPR was a l s o used to i n v e s t i g a t e the s t r u c t u r e of some s u b s t i t u t e d methylenes (8). T h i s study i n d i c a t e d t h a t the bonds to the methylene carbon were not c o l i n e a r . Using diphenylmethylene as the paramagnetic molecule, environmental e f f e c t s i n the EPR s p e c t r a of t r i p l e t s t a t e molecules were i n v e s t i g a t e d (9). I t was found that the narrowest d i s t r i b u t i o n of v a r i a t i o n s i n zero f i e l d s p l i t t i n g s occurred when the geometry of the t r i p l e t molecule approximated the host. A study of three c y c l i c methylenes by EPR gave f u r t h e r evidence t h a t bonds to the d i v a l e n t carbon atom i n these compounds were not c o l i n e a r (10). EPR of randomly o r i e n t e d t r i p l e t s t a t e molecules was a l s o i n v e s t i g a t e d . T h i s i n v e s t i g a t i o n a l s o presented s p e c t r a l s i m u l a t i o n s of the allowed t r a n s i t i o n s of the randomly o r i e n t e d molecules diphenylmethylene, and m e t h y l s u l f o n y l n i t r e n e (11). The e x i s t e n c e o f a l k y l n i t r e n e s was demonstrated by EPR methods (12). Geometric isomers of t r i p l e t naphthylmethylenes were r e s o l v e d by EPR (13). T h e o r e t i c a l accounts of the zero f i e l d s p l i t t i n g s i n t r i p l e t 4 n i t r e n e s were r e p o r t e d by the end of 1964 (14, (15). N i t r o g e n h y p e r f i n e s t r u c t u r e i n the EPR of n i t r e n e s was observed i n s i n g l e c r y s t a l s t u d i e s (16, (17, (18). EPR of t r i p l e t c y anonitrene and dicyanomethylene was detected (19). In subsequent s t u d i e s the emphasis s h i f t e d to e x p l o r a t i o n s of known t r i p l e t molecules i n s i n g l e c r y s t a l s v i a the E l e c t r o n - N u c l e a r Double-Resonance (ENDOR) technique. A l s o s t r e s s e d was c o n s t r u c t i o n of molecules i n higher m u l t i p l e t s t a t e s by i n c o r p o r a t i n g v a r i o u s paramagnetic f u n c t i o n a l groups i n t o molecules (20). A s i n g l e c r y s t a l EPR study of diphenylmethylene preceded the r e p o r t of ENDOR i n v e s t i g a t i o n s on the same system (21). Quintet ground s t a t e s of metadicarbene and metadinitrene compounds were detec t e d by EPR methods, as was a ground s e p t e t s t a t e of a t r i n i t r e n e (22, (23, (24). Further ENDOR i n v e s t i g a t i o n s of diphenylmethylene molecules i n s i n g l e c r y s t a l s were made (25 ). T r i p l e t s t a t e methylene was detected by EPR (26). An a d d i t i o n a l ground s e p t e t s t a t e s p e c i e s was c h a r a c t e r i z e d by EPR (27). One may see from t h i s b r i e f survey the trends which r e s e a r c h i n t h i s f i e l d has f o l l o w e d from i t s conception i n the e a r l y 1960»s to the present. The s t r u c t u r a l p i c t u r e of aromatic n i t r e n e s , which emerged from EPS i n v e s t i g a t i o n s i s of importance to the present study. T h i s p i c t u r e of n i t r e n e s as r e v e a l e d by the 5 z e r o f i e l d s p l i t t i n g parameters, i s one of d e l o c a l i z a t i o n of one of the e l e c t r o n s i n v o l v e d i n the t r i p l e t i n t o the p i system of the aromatic r i n g . The other e l e c t r o n comprising the t r i p l e t i s l o c a l i s e d i n a sigma o r b i t a l on the n i t r e n e n i t r o g e n atom. The experimental evidence f o r t h i s p i c t u r e i s the f a c t t h a t zero f i e l d s p l i t t i n g s i n a l k y l n i t r e n e s are l a r g e r than those of aromatic n i t r e n e s . T h i s p i c t u r e i s f u r t h e r i l l u m i n a t e d by c a l c u l a t i o n s of the s p i n d i s t r i b u t i o n i n the aromatic p i system. T h i s allows a c a l c u l a t i o n of expected zero f i e l d s p l i t t i n g s f o r these molecules, and hence a comparison with experiment. The trends f o r the s u b s t i t u t e d aromatic n i t r e n e s i n d i c a t e d that para s u b s t i t u t i o n c o n t r i b u t e s to i n c r e a s i n g d e l o c a l i z a t i o n , while meta s u b s t i t u t i o n has g e n e r a l l y l i t t l e or no e f f e c t (28). A d d i t i o n a l i n f o r m a t i o n concerning the generation of n i t r e n e s p e c i e s has been obtained. Both EPB and o p t i c a l a b s o r p t i o n methods have been used i n these s t u d i e s . The most common p r e c u r s o r to the n i t r e n e s p e c i e s by p h o t o l y s i s i s the a z i d e group. In t h i s case n i t r o g e n i s evolved during the p h o t o l y t i c decomposition. The i s o c y a n a t e group a l s o has been shown to y i e l d n i t r e n e s p e c i e s on p h o t o l y t i c decomposition (29). I t was found t h a t X - i r r a d i a t i o n of a z i d e s u b s t i t u t e d molecules would a l s o r e s u l t i n n i t r e n e formation although no s y s t e m a t i c i n v e s t i g a t i o n of t h i s aspect was undertaken. » 6 CHAPTER ONE Theory 1 A ! T r i o l e t State The importance of t r i p l e t s t a t e molecules to chemistry l i e s i n the f a c t t h a t these molecules have s u f f i c i e n t l o n g e v i t y t o take p a r t i n chemical r e a c t i o n s . The e x i s t e n c e and o p t i c a l p r o p e r t i e s of molecules i n the t r i p l e t s t a t e was i l l u m i n a t e d by Lewis and Kasha i n 1944 (30). T h i s s t a t e draws i t s name from the d i s t i n g u i s h i n g f e a t u r e of i t s s t r u c t u r e . Two o f the e l e c t r o n s c o n f i n e d to t r i p l e t molecules are i n d i f f e r e n t s p a t i a l o r b i t a l s , a l l o w i n g the e l e c t r o n s p i n s t o couple together t o give a t r i p l e t s t a t e . The energy of the t r i p l e t s t a t e f o r a molecule i s r e l a t e d to the energy of the s i n g l e t s t a t e by the e l e c t r o s t a t i c exchange i n t e r a c t i o n . E x c l u s i v e o f s p i n f u n c t i o n s , unnormed s i n g l e t and, t r i p l e t wavef u n c t i o n s belonging to the s p a t i a l c o n f i g u r a t i o n UV a r e : | S>=0(1) V(2) +D (2) V (1) |T>=0 (1)V (2)-D (2)V (1) Coulson has given the energy of these s t a t e s as: (31) 7 E(S) = Eo+(Q+J)/(1+S2) E (T)=Eo+ (Q-J)/ (1-S 2) where Q denotes the coulomb i n t e g r a l , J denotes the exchange i n t e g r a l , and S denotes the o v e r l a p i n t e g r a l . Phosphorescence i m p l i e s t h a t j i s p o s i t i v e , s i n c e the ground s t a t e i s u s u a l l y a s i n g l e t s t a t e corresponding to the c o n f i g u r a t i o n UU. The q u a l i f i c a t i o n o f the l a t t e r remark i m p l i e s that e x c e p t i o n s e x i s t . Necessary c r i t e r i a f o r the e x i s t e n c e of a ground s t a t e t r i p l e t molecule have been enumerated as f o l l o w s : (32) (1) even number of e l e c t r o n s (2) h i g h e s t energy o r b i t a l must be (nearly) degenerate (3) t h i s o r b i t a l must be p a r t l y f i l l e d . The n i t r e n e s are examples of ground s t a t e t r i p l e t molecules. I t may be remarked t h a t the paramagnetic p r o p e r t i e s of these molecules evaded d e t e c t i o n by EPB u n t i l 1958, when the lowest t r i p l e t s t a t e of the naphthalene molecule was s u c c e s s f u l l y s t u d i e d (33) . 8 -ls.2 S£in Hamiltonian The s p i n Hamiltonian i s the language used i n the d e s c r i p t i o n of magnetic resonance experiments. As i t s name i m p l i e s , i t i s an energy operator. I t i s separated i n t o v a r i o u s e n e r g e t i c i n t e r a c t i o n s . These terms are l i k e verbs; the parameters c h a r a c t e r i z i n g each term are l i k e adverbs. The s u b j e c t of the n sentence " i s always energy, and the o b j e c t s of the sentence are o f t e n i m p l i e d . These are v a r i o u s s p i n f u n c t i o n s c o n s t i t u t i n g the s t a t e of a p a r t i c u l a r s p i n system. The terms appear i n va r y i n g importance depending upon the s p e c i f i c nature of the paramagnetic system. A s p i n Hamiltonian, which may be compared to a sentence may l i k e w i s e be compound. I t i s t h e r e f o r e necessary to mention s e p a r a t e l y the terms which may appear i n the s p i n Hamiltonian f o r a s o l i d sample. The f i r s t terms to be mentioned are the Zeeman terms. These are w r i t t e n i n the form: ]_[ = B R>G«S -B• B>G»«I . . . . . . . . . . . . . . . , . . [ 1] The e l e c t r o n i c Zeeman term, which appears f i r s t i n the above equation, i s l a r g e r than the nucl e a r Zeeman term by a f a c t o r of t h r e e o r d e r s of magnitude due to r e l a t i v e s i z e s of e l e c t r o n i c and n u c l e a r magnetic moments. These terms both 9 rep r e s e n t the energy a s s o c i a t e d with the o r i e n t a t i o n o f a magnetic d i p o l e moment immersed i n a magnetic f i e l d . Both of these terms may e x h i b i t a n i s o t r o p i c behaviour. The magnetic p r o p e r t i e s of the s p e c i e s i n which the s p i n s a re c o n f i n e d are contained i n the parameters G and G*. The next type of i n t e r a c t i o n which commonly a r i s e s i s h y p e r f i n e c o u p l i n g . E l e c t r o n and nuclear s p i n s are found to be coupled and the energy a s s o c i a t e d with t h i s i n t e r a c t i o n i s w r i t t e n : ]-[ = S«A«I . . . . , . . . . . . . . . . . . . . [ 2 ] The s p i n •S' c h a r a c t e r i z e s the e l e c t r o n and s p i n ' I * r e f e r s to the nucleus; h y p e r f i n e parameters 'A * c h a r a c t e r i z e the i n t e r a c t i o n of the two s p i n s . The h y p e r f i n e i n t e r a c t i o n term i s s e p a r a b l e i n t o two c o n t r i b u t i o n s : d i p o l a r and i s o t r o p i c p a r t s . The i s o t r o p i c p a r t i s p r o p o r t i o n a l t o t h a t p o r t i o n of the s p i n d i s t r i b u t i o n which i s s p h e r i c a l l y disposed about the nuc l e a r s p i n , while the d i p o l a r c o n t r i b u t i o n a r i s e s from the remaining non s p h e r i c a l p a r t . T h i s s e p a r a t i o n enables e x t r a c t i o n of i n f o r m a t i o n about the s p a t i a l d i s p o s i t i o n of the paramagnetic e l e c t r o n about a p a r t i c u l a r nucleus. Coupling of two or more e l e c t r o n s p i n s may a l s o occur, and i n t h i s case the i n t e r a c t i o n i s c o n v e n t i o n a l l y denoted : 1 0 M = S»D«S . . . . . [ 3 ] The q u a n t i t y *D» i s known as the zero f i e l d s p l i t t i n g s i n c e no e x t e r n a l magnetic f i e l d need be a p p l i e d t o observe t h i s i n t e r a c t i o n . The magnitude o f t h i s i n t e r a c t i o n depends upon the p a r t i c u l a r molecule i n which the e l e c t r o n s p i n s are co n f i n e d . However, s p l i t t i n g s o f the order of a cm.-1 or l e s s c o n s t i t u t e the energy range i n which most zero f i e l d s p l i t t i n g s are found i n the t r i p l e t s t a t e . Analogous t o the zero f i e l d s p l i t t i n g term a r i s i n g from the c o u p l i n g of two e l e c t r o n s p i n s i s the guadrupole i n t e r a c t i o n denoted: 3-E = i - Q - i m Here, the q u a n t i t y Q denotes the energy of i n t e r a c t i o n o f a nucle a r quadrupole moment with an e l e c t r i c f i e l d g r a d i e n t . T h i s term i s not as f r e q u e n t l y encountered as previous terms s i n c e n u c l e i of s p i n 1/2 do not possess a quadrupole moment. When the i n t e r a c t i o n i s encountered, i t i s u s u a l l y of s m a l l e r magnitude than p r e v i o u s l y mentioned terms. Further mention of t h i s i n t e r a c t i o n w i l l not be made here. S i n c e the s t u d i e s i n t h i s work p e r t a i n e s p e c i a l l y to n i t r e n e s which are i n the t r i p l e t s t a t e , a f u r t h e r d i s c u s s i o n of the s p i n Hamiltonian f o r t h i s s t a t e i s warranted. The f i n e 11 s t r u c t u r e of t h i s system may be d e s c r i b e d by: ]-[ = S«D»S + BH»G»S [ 5 ] the zero f i e l d s p l i t t i n g term can be d e r i v e d from the f o l l o w i n g r e l a t i o n : (34) ]_[ = g 2 B 2 [ r z (S1«S2) - 3 (r»S1) (r»S2) ]/r~s . [ 6 ] T h i s form may be r e c o g n i s e d as the energy a s s o c i a t e d with one d i p o l e immersed i n the f i e l d of another. I t may be r e w r i t t e n i n terms of t o t a l s p i n : (35) ]-[ = 0 . 7 5 g 2 B 2 { r 2 - 3 z 2 } r - s [ S z 2 - S 2 / 3 ] + 0 . 7 5 g 2 B 2 { y 2 - x 2 } r - s [ s x 2 - S y 2 ] the c o e f f i c i e n t s of the t o t a l s p i n o p e r a t o r s are c a l l e d D and E r e s p e c t i v e l y : D = 0 . 7 5 g 2 B 2 < | ( r 2 - 3 z 2 ) / r - s | > E = 0 . 7 5 g 2B 2< | ( y 2 - x 2 ) / r - 5 | > As i n d i c a t e d these q u a n t i t i e s are e x p e c t a t i o n s over r e l a t i v e s p a t i a l p o s i t i o n s of the two e l e c t r o n s which are c o n t r o l l e d by the much higher e n e r g e t i c s of the molecular Hamiltonian. The parameter D measures the n o n s p h e r i c a l nature of the s p a t i a l d i s p o s i t i o n of the e l e c t r o n s comprising the t r i p l e t s p i n s t a t e . In an analogous manner the parameter E measures the n o n c y l i n d r i c a l nature of the s p a t i a l d i s p o s i t i o n of the same e l e c t r o n s . The q u e s t i o n of s p i n - o r b i t c o u p l i n g c o n t r i b u t i o n s t o the zero f i e l d s p l i t t i n g should a l s o be 12 mentioned. I n v e s t i g a t i o n s of t h i s problem have been c o n f i n e d to c a l c u l a t i o n s on v a r i o u s molecules. The form of t h i s c o n t r i b u t i o n has been shown t o be the same as the s p i n - s p i n i n t e r a c t i o n so t h a t i t i s not p o s s i b l e to separate the two (36). The r e s u l t s of the i n v e s t i g a t i o n s which have been made i n d i c a t e t h a t t h i s c o n t r i b u t i o n may range from n e g l i g i b l e i n c e r t a i n molecules to a s m a l l but s i g n i f i c a n t p o r t i o n i n ot h e r s . The l a r g e s t c o n t r i b u t i o n r e p o r t e d i s about f i f t e e n p ercent, f o r the NH r a d i c a l (37). The e l e c t r o n Zeeman term i n org a n i c t r i p l e t molecules i s s u b s t a n t i a l l y i s o t r o p i c and i s found very c l o s e to that of a f r e e e l e c t r o n (38). S p i n - o r b i t c o u p l i n g i n these systems i s very s m a l l . Anisotropy o f the f i n e s t r u c t u r e terms does not a r i s e i n the Zeeman i n t e r a c t i o n i n these systems. An i n v e s t i g a t i o n o f t h i s p o i n t f o r some t r i p l e t methylene molecules r e v e a l e d t h a t the assumption of a f r e e e l e c t r o n g value was v a l i d f o r the dete r m i n a t i o n of the zero f i e l d s p l i t t i n g parameters to a p r e c i s i o n of one percent (39). 1 3 ! i 3 T r i p l e t State EPS Theory d e s c r i b i n g magnetic resonance of t r i p l e t s t a t e molecules has been gi v e n thorough a t t e n t i o n i n s e v e r a l p l a c e s , and convenient accounts of the r e s u l t s are a v a i l a b l e ( 4 0 , ( 4 1 ) . Some important f e a t u r e s o f t r i p l e t s t a t e EPR s p e c t r a may be c o n v e n i e n t l y r e v e a l e d by r e p r e s e n t i n g the s p i n Hamiltonian i n the zero f i e l d b a s i s f u n c t i o n s . Apart from n o r m a l i z a t i o n , these f u n c t i o n s are the following- combinations of the high f i e l d b a s i s f u n c t i o n s T (Ms): ( 4 2 ) | Tx>=| T ( - 1 ) -T (+1) > | T y > = i | T ( - 1 ) + T ( + 1 ) > |Tz>=|T ( 0 ) > The Hamiltonian matrix has the form: X -igBHz igBHy igBHz Y -igBHx -igBHy igBHx Z From t h i s form one may recognise the a p p r o p r i a t e name of these b a s i s f u n c t i o n s . An important aspect of t h i s form comes from the f a c t t h a t p o l y c r y s t a l l i n e s p e c t r a l f e a t u r e s occur at a x i a l o r i e n t a t i o n s of the e x t e r n a l l y a p p l i e d magnetic f i e l d . T h i s being the case, i t i s enough to s o l v e the resonant c o n d i t i o n f o r H along X, Y, and Z. T h i s has been done and the r e s u l t s have been t a b u l a t e d . However, one may see some of the p r o p e r t i e s of the t r a n s i t i o n s r a t h e r d i r e c t l y . F i r s t , one of 14 the energy l e v e l s corresponding to the p o l y c r y s t a l l i n e resonances i s independent of H. ,The remaining two l e v e l s are f u n c t i o n s of H given by a 2x2 determinant. P r o v i d i n g the resonant magnetic f i e l d s t r e n g t h i s s m a l l compared with the a n i s o t r o p y of the s p i n s p i n i n t e r a c t i o n i n the p e r p e n d i c u l a r d i r e c t i o n , the e i g e n s t a t e s remain predominantly zero f i e l d s t a t e s . I f the converse c o n d i t i o n p r e v a i l s , the e i g e n s t a t e s tend to the high f i e l d b a s i s s t a t e s . There are two types of t r a n s i t i o n s which are p o s s i b l e and these are c u s t o m a r i l y l a b e l l e d by the a l t e r a t i o n of the high f i e l d l i m i t quantum number during the t r a n s i t i o n . The p o l a r i z a t i o n of these two t y p es of t r a n s i t i o n s i s q u i t e d i f f e r e n t . The p o l a r i z a t i o n of the normally allowed t r a n s i t i o n s i s p e r p e n d i c u l a r , t h a t i s Hrf J. Ho, whereas the p o l a r i z a t i o n of the f o r b i d d e n t r a n s i t i o n s i s p a r a l l e l . T h i s may be seen by f i n d i n g the matrix elements of S between the two s t a t e s which are i n v o l v e d i n the t r a n s i t i o n . However i t seems e a s i e s t to r e c a l l t h a t Si|Ti>=0 f o r i=x,y,z; the allowed t r a n s i t i o n s can h a r d l y be p a r a l l e l p o l a r i z e d . In g e n e r a l one would expect s i x t r a n s i t i o n s of the p e r p e n d i c u l a r p o l a r i z a t i o n , and another three of the p a r a l l e l p o l a r i z a t i o n ; t h i s s i t u a t i o n may not always be r e a l i z e d , s i n c e i t may not be p o s s i b l e to s a t i s f y the resonant c o n d i t i o n . The r e c o n s t r u c t i o n of a p o l y c r y s t a l l i n e spectrum 15 consists i n part, of replacing the set of molecules, randomly oriented with respect to an external magnetic f i e l d with an equivalent set of appropriate calculations. This set of calculations could be taken equivalently as representing a sum of randomly oriented set of external magnetic f i e l d s with respect to a single molecule, or more precisely, with respect to the spin-spin i n t e r a c t i o n . I t i s expressly implied that at each orientation, required calculations consist of determining the resonant condition, p a r t i c i p a t i n g states, and r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s . Further, a lineshape function should be merged with t h i s information, and the t o t a l i t y of such orientations should be summed to yield the absorption. CHAPTER TWO 2&I A££aratus The EPR measurements i n t h i s work were made on two X band spectrometers. The e s s e n t i a l d i f f e r e n c e between these two spectrometers was s e n s i t i v i t y and magnet c a p a b i l i t y and f l e x i b i l i t y . One spectrometer was a modified V a r i a n 4502 instrument, equipped with a V a r i a n mark I I f i e l d i a l and a r o t a t a b l e 12 i n c h magnet (Varian 3900). The pole gap was 2.75 i n c h . V a r i a n c a v i t i e s V4533 ( c y l i n d r i c a l ) and V4531 (rectangular) or e q u i v a l e n t were employed on t h i s spectrometer. The other-spectrometer used was a Varian model E-3 EPS spectrometer. Although l e s s powerful i n magnet design, i t was .very u s e f u l f o r o b s e r v a t i o n s of p o l y c r y s t a l l i n e samples, p r i n c i p a l l y due to i t s higher s e n s i t i v i t y . Measurements of X band microwave f r e q u e n c i e s were made with a Hewlett Packard model 5246 counter equipped with a 5256A c o n v e r t e r . Magnetic f i e l d s t r e n g t h was monitored with a 1 7 s m a l l P.M.R. setup c o n s t r u c t e d by the e l e c t r o n i c s shop of t h i s department. In the case of the E-3 spectrometer, the magnetic f i e l d s t r e n g t h r e q u i r e d was j u s t beyond the o p e r a t i o n a l range f o r which the instrument was designed, t h e r e f o r e s m a l l P.M.R. probes were c o n s t r u c t e d t o monitor the magnetic f i e l d s t r e n g t h j u s t e x t e r n a l to the c a v i t y . Temperature v a r i a t i o n o f th e samples was achieved by usin g a cooled n i t r o g e n gas flow through a gas flow dewar assemblies of c o n v e n t i o n a l design. Sample temperature was monitored by a copper-constantan thermocouple/potentiometer arrangement. One couple was pla c e d as c l o s e t o the sample as p o s s i b l e to minimize e f f e c t s of thermal g r a d i e n t s . One assembly c o u l d be s u i t a b l y c o n t r o l l e d by a V a r i a n temperature r e g u l a t i o n u n i t while v a r i a t i o n of temperature i n the other assembly was achieved simply by a l t e r i n g the gas flow, and a l l o w i n g the assembly t o come to thermal e g u i l i b r i u m . Sample tubes were c o n s t r u c t e d of high p u r i t y S i l i c a of diameter s u i t a b l e to the p a r t i c u l a r dewar assembly, t y p i c a l l y OD 3-4mm. The i n t e n t was always t o c o n t a i n as much sample as p o s s i b l e w i t h i n the c a v i t y . The u l t r a v i o l e t r a d i a t i o n source was a low pressure mercury a r c . (Bausch and Lomb model # SP-200). T h i s was equipped with a g r a t i n g monochrometer. Some samples were a l s o i r r a d i a t e d with X-rays produced by a Mat c h l e t t OEG-60 X-ray 18 tube o p e r a t i n g a t 40 kV and 30 mA. 2j.2 P r e p a r a t i o n of Azides The s t a r t i n g p o i n t s f o r the s y n t h e s i s of the a z i d e s used i n t h i s work were commercially a v a i l a b l e aromatic amines. The amine group was r e p l a c e d with the a z i d e group by standard d i a z o t i z a t i o n r e a c t i o n . The r e a c t i o n s o l v e n t c o n s i s t e d of a ( 1:1 by volume ) mixture of concentrated h y d r o c h l o r i c a c i d and water i n s t e a d of concentrated s u l f u r i c a c i d to a v o i d problems of decomposition. A d e t a i l e d example of the p r e p a r a t i o n of 1,3,5-tribromophenyl a z i d e i s given below: 20 grams of 2, 4 , 6 - t r i B r o m o a n i l i n e , r e c r y s t a l l i z e d from methanol, are suspended i n an i c e c o l d mixture of 80 ml. of water and 80 ml. of concentrated h y d r o c h l o r i c a c i d . Then d i a z o t i z a t o n i s e f f e c t e d by slow a d d i t i o n . (3/4 hour) of a s o l u t i o n of 12 grams sodium n i t r i t e i n 30 ml. of water. T h i s r e a c t i o n should be s t i r r e d i n an i c e bath. The r e s u l t i n g yellow suspension i s f i l t e r e d t o separate u n d i s s o l v e d amine from the c l e a r y e l l o u s o l u t i o n c o n t a i n i n g the diazonium i o n . To t h i s s o l u t i o n i s . s l o w l y added a cooled s o l u t i o n of 5 grams of sodium a z i d e i n 30 ml. of water. The crude s o l i d a z i d e i s 19 best recovered by a l l o w i n g the r e a c t i o n mixture to stand u n d i s t u r b e d f o r a p e r i o d of two hours, when i t w i l l f l o a t to the top of the l i q u i d . It may then be separated, washed with water, and r e c r y s t a l l i z e d from d i e t h y l ether. A good y i e l d (85%) can be obtained. Elemental a n a l y s i s f o r carbon, n i t r o g e n , and hydrogen f o r t h i s compound was: carbon 20.2 (20.4); n i t r o g e n 11.8 (11.6); hydrogen 0.6 (0.7). An a d d i t i o n a l check of the presence of the a z i d e . group i s provided by a c h a r a c t e r i s t i c i n f r a r e d a b s o r p t i o n a t 2100 cm.-*. More e x t e n s i v e examination of the chemical p u r i t y of a l l a z i d e s was not s t r e s s e d f o r three reasons: (1) the r e a c t i o n i s w e l l e s t a b l i s h e d . (2) the a z i d e s are known p r e c u r s o r s to n i t r e n e s by u l t r a v i o l e t i r r a d i a t i o n . (3) V n i t r e n e EPR s p e c t r a are d i s t i n c t . i 20 2^3 Samp_le P£§£aration Samples which were prepared i n the course of t h i s study were of t h r e e t y p e s : s i n g l e c r y s t a l , powders, and g l a s s y matrix. The g l a s s y samples were prepared i n a c o n v e n t i o n a l manner using EPS (ether, isopentane and a l c o h o l ; 1:1:3 by volume) as a matrix. Other s o l v e n t s i n v e s t i g a t e d y i e l d e d samples of i n f e r i o r q u a l i t y and were abandoned. Con c e n t r a t i o n s of these samples were adjusted i n order to o b t a i n a c l e a r g l a s s , but were not monitored . Degassing of the samples was achieved by a r o u t i n e freeze-thaw c y c l e . Powdered samples were prepared by g r i n d i n g c r y s t a l l i n e samples at room temperature i n an agate mortar . However , i f the a z i d e was a l i q u i d at room temperature, i t was f r o z e n under l i q u i d n i t r o g e n , and powdered at t h a t temperture i n a p o r c e l a i n mortar . T h e powdered sample c o u l d be t r a n s f e r r e d to a sample tube while immersed under l i q u i d n i t r o g e n . T h i s could be e a s i l y removed by pumping on a vacuum l i n e . S i n g l e c r y s t a l samples were grown from s o l u t i o n by c o n t r o l l e d e v a p o r a t i o n at room temperature. D i e t h y l ether was found to be the most u s e f u l s o l v e n t f o r t h i s purpose. As the q u a l i t y or q u a n t i t y of c r y s t a l s which were produced was not found r e p r o d u c i b l e , repeated e v a p o r a t i o n s were attempted u n t i l a s u f f i c i e n t q u a n t i t y of c r y s t a l s was obtained. The 2 1 vacuum sublimation technique was not found useful for the purpose of growing single c r y s t a l s of these azides. 22 CHAPTER THREE Matrix E f f e c t s An i n v e s t i g a t i o n of the EPR s p e c t r a of n i t r e n e s i s o l a t e d i n g l a s s y media and p o l y c r y s t a l l i n e media was made. The motive f o r t h i s i n v e s t i g a t i o n was t h a t h y p e r f i n e i n t e r a c t i o n s had not been observed i n published s p e c t r a of these s p e c i e s when i s o l a t e d i n g l a s s y hydrocarbon matrices. I t i s of course d e s i r a b l e t o r e s o l v e t h i s i n t e r a c t i o n s i n c e i t g i v e s more d e t a i l e d i n f o r m a t i o n on the d i s p o s i t i o n of the unpaired e l e c t r o n s than the zero f i e l d s p l i t t i n g . During the course of experiments i n t h i s area, temperature e f f e c t s on these s p e c i e s were noted. The f i r s t e f f e c t s which were e l u c i d a t e d are i l l u s t r a t e d i n F i g u r e 1 . These s p e c t r a are a l l due to the p a r a n i t r o p h e n y l n i t r e n e trapped i n d i f f e r e n t media under v a r i o u s temperature c o n d i t i o n s . F i g u r e 1(c) i s a t y p i c a l spectrum of the HXY M t r a n s i t i o n s which can be observed from aromatic n i t r e n e s trapped i n a g l a s s y matrix. F i g u r e s 1 (a) and (b) are p o l y c r y s t a l l i n e s p e c t r a of the same t r a n s i t i o n s from n i t r e n e s trapped i n the a z i d e p r e c u r s o r , but under s l i g h t l y d i f f e r e n t c o n d i t i o n s . A l l three s p e c t r a were recorded at low temperatures. Although the n i t r e n e trapped i n the c r y s t a l l i n e matrix i s s t a b l e a t room temperature* the 1. P - n i t r o p h o n y l N i t r e n e EPR S p e c t r a ; (a) A z i d e Host A f t e r A n n e a l i n g (b) A z i d e Host B e f o r e A n n e a l i n g (c) G l a s s y M a t r i x 24 n i t r e n e trapped i n a hydrocarbon matrix such as EPA i s not. T h i s decay of the r e a c t i v e n i t r e n e s p e c i e s can of course be e a s i l y f o l l o w e d by ge n e r a t i o n of the n i t r e n e i n a g l a s s y matrix at 77 K and subsequent h e a t i n g . As remarked i n previous experiments, the b l e a c h i n g of the n i t r e n e i n a gla s s y medium occurs w e l l below room temperature (4 3 ) . I t i s t h e r e f o r e i m p r a c t i c a l t o c o n s i d e r i n v e s t i g a t i o n s of the temperature dependence of these s p e c i e s i n a g l a s s y EPA above 140 K. The important c o n c l u s i o n suggested immediately by Fi g u r e 1 i s t h a t there i s more r e s o l u t i o n a v a i l a b l e i n the c r y s t a l l i n e medium than the g l a s s y medium. Spectrum (b) i s observed a f t e r g e n e r a t i o n by u l t r a v i o l e t l i g h t on a p o l y c r y s t a l l i n e sample at 77 K. Spectrum (a) i s observed a f t e r a l l o w i n g the sample to warm up to 300 K to anneal f o r a period of s e v e r a l hours and quenching to approximately 77 K. I r r e v e r s i b l e changes o c c u r r i n g during the warmup c y c l e f o r spectrum (a) r e v e a l the e x i s t e n c e o f two n i t r e n e s p e c i e s with very s i m i l a r EPR s p e c t r a . Although the g l a s s y spectrum does not r e v e a l h y p e r f i n e s p l i t t i n g , a comparison of i t s form and p o s i t i o n s t r o n g l y suggests that only one of these two very s i m i l a r n i t r e n e s p e c i e s d i s c e r n e d i n the p o l y c r y s t a l l i n e medium i s a c t u a l l y present i n the g l a s s y medium. C h a r a c t e r i s t i c s of these two s p e c i e s which may be disce r n e d from the s p e c t r a are: 25 p a r a n i t r o p h e n y l n i t r e n e ; T = 152 K (A) : D < D (B) (B) : D = 0.974 cm-1 E > E(B) E = 0.001 cm-1 Hfs = 17.7 gauss Hfs = 14.4 gauss The best e x p l a n a t i o n which can be drawn from these r e s u l t s i s t h a t s p e c i e s B i s the more c l o s e l y l i n e a r and t h a t s p e c i e s A i s l i k e l y to be bent, i n the case of s p e c i e s B one may note t h a t the s p i n d e n s i t y (p) p r e d i c t e d f o r the observed h y p e r f i n e s p l i t t i n g g i v e s s u b s t a n t i a l agreement with the observed D value: Dixon has determined the value of d f o r the N-H r a d i c l e (44)., Spin d e n s i t y p has been taken as the r a t i o of observed to c a l c u l a t e d h y p e r f i n e s p l i t t i n g . T h i s equation views the i n t e r a c t i o n as having c y l i n d r i c a l symmetry. Spe c i e s A does not f i t t h i s p i c t u r e . The s m a l l e r D value f o r s p e c i e s A would be i n c o n s i s t e n t with a l i n e a r s t r u c t u r e . One may examine the annealing process during which the two s p e c i e s were detected i n F i g u r e 2 . T h i s f i g u r e i l l u s t r a t e s the EPR s p e c t r a (XY t r a n s i t i o n s ) of the n i t r e n e s p e c i e s during a warming and c o o l i n g c y c l e i n which the p = 14.4/19.8 = 0.727 D ( c a l c u l a t e d ) = D (NH) • p 2 1.82 • 0.529 = 0.96 cm-1 27 i r r e v e r s i b l e change from s p e c i e s A to B oc c u r s . The temperatures of the sample are i n d i c a t e d i n the f i g u r e i n the K e l v i n s c a l e . Two s p e c t r a l a b e l l e d 300 degrees are separated c h r o n o l o g i c a l l y by approximately ten hours d u r a t i o n when f u r t h e r changes o c c u r r e d ; a l l other s p e c t r a were taken s e q u e n t i a l l y during a much s h o r t e r time p e r i o d s u f f i c i e n t o n l y to r e c o r d the s p e c t r a . I t can be seen t h a t both s p e c i e s were present i n comparable c o n c e n t r a t i o n at 77 K. The con v e r s i o n from s p e c i e s A to B was not apparent on the time s c a l e of the experiment u n t i l the temperature o f the sample was r a i s e d to approximately 250 K. Spectra recorded during the c o o l i n g process demonstrate t h a t the e f f e c t i s not r e v e r s i b l e . S i m i l a r o b s e r v a t i o n s are recorded i n Fig u r e 3 f o r parachlorophenyl n i t r e n e trapped i n parac h l o r o p h e n y l a z i d e and generated at 77 K, i n order t h a t i r r e v e r s i b l e changes i n the sample would not occur undetected. In t h i s case the a z i d e sample was f r e s h l y r e c r y s t a l l i z e d , and s u i t a b l y p r o t e c t e d from ambient l i g h t c o n d i t i o n s . Exposure t o a moderately i l l u m i n a t e d room i s s u f f i c i e n t to generate d e t e c t a b l e n i t r e n e c o n c e n t r a t i o n s i n a few minutes. As can be seen the form of the f i r s t spectrum i s c h a r a c t e r i s t i c of only one sp e c i e s , As the temperature of the sample i s r a i s e d t o approximately 200 K, a new s p e c i e s (B) a r i s e s c o n c o m i t a n t l y with the decay of 29 the f i r s t s p e c i e s (A). Both s p e c i e s are c h a r a c t e r i s e d by r e s o l v e d h y p e r f i n e s t r u c t u r e on only the highest f i e l d m u l t i p l e t . I t can be seen t h a t the n o n - c y l i n d r i c a l nature of the new s p e c i e s (B) formed from the f i r s t s p e c i e s (A) i s n o t i c e a b l y l e s s , as evidenced by the s p l i t t i n g of the XY t r a n s i t i o n s . The temperature dependence of both s p e c i e s can be f o l l o w e d on the c o o l i n g c y c l e d e p i c t e d on the same f i g u r e . The formation of s p e c i e s (B) from (A) i s e v i d e n t l y i r r e v e r s i b l e , j u s t as f o r the case of the p a r a n i t r o p h e n y l n i t r e n e . In order to i n v e s t i g a t e the formation of the s p e c i e s (B) from (A) more c l o s e l y , a sample of parachlorophenyl azide was subjected to u l t r a v i o l e t r a d i a t i o n at 77 K, generating s p e c i e s (A). The sample temperature was allowed to r i s e a b r u p t l y to 212 K where i t was s t a b i l i z e d . The decay of s p e c i e s (A) and growth of s p e c i e s (B) as a f u n c t i o n of time was monitored d i r e c t l y from the i n t e n s i t i e s of the EPR s i g n a l s . F i g u r e 4 i l l u s t r a t e s the r e s u l t s of t h i s experiment. As can be seen i n the f i g u r e , the experiment s u b s t a n t i a t e s the r e a c t i o n (A) — > (B) by the constancy of (A+B) except i n the e a r l y stages of the decay. The d e s c r i p t i o n of the observed s p e c i e s as e i t h e r of the two s p e c i e s i s thus s u b s t a n t i a l l y c o r r e c t . Discrepancy from a constant (A + B) may then be a t t r i b u t e d to d i s t r i b u t i o n s of perturbed environments which border on (A) or (B) . T h i s i s c o n s i s t e n t with the i g n a l I n t e n s i t y ( a r b . u n i t s ) (a) puv (V) saxosds OE 31 n o t i c e a b l e l a c k of r e s o l v e d h y p e r f i n e s t r u c t u r e on both m u l t i p l e t s , and i n c o n t r a s t to the re s o l v e d s t r u c t u r e found i n p a r a n i t r o p h e n y l n i t r e n e . Observations made a f t e r the f i r s t 30 minutes of the decay a f f o r d an e x p o n e n t i a l d e s c r i p t i o n with a decay constant of 0.04 m i n - 1 . F u r t h e r q u a n t i t a t i v e i n v e s t i g a t i o n s of decay k i n e t i c s were not made; however from q u a l i t a t i v e experiments, i t was found t h a t the t r a n s i t i o n from (A) t o (B) i s much a c c e l e r a t e d a t higher temperatures. F i g u r e 5 i l l u s t r a t e s EPR s p e c t r a of two n i t r e n e s p e c i e s produced by X - i r r a d i a t i o n i n p-azido p h e n y l a r s o n i c a c i d . Spectrum (a) i s recorded at room temperature. The XY t r a n s i t i o n s of the two s p e c i e s o v e r l a p s i g n i f i c a n t l y , having the appearence of n i t r o g e n h y p e r f i n e s t r u c t u r e ; however the Z t r a n s i t i o n s at higher f i e l d are r e s o l v e d and e a s i l y r e v e a l the r e l a t i v e amounts of the two s p e c i e s . Spectrum (b) was recorded a f t e r a n n e a l i n g the sample i n b o i l i n g water. The r e l a t i v e amounts of the two s p e c i e s have been interchanged by the decay of one s p e c i e s i n t o the other. The decay c h a r a t e r i s t i c s of these n i t r e n e s f a l l at d i f f e r e n t temperatures i n d i f f e r e n t h osts. The pro d u c t i o n of these two s p e c i e s of n i t r e n e s i n p o l y c r y s t a l l i n e media i s f a i r l y g e n e r a l . I t does not depend upon the s u b s t i t u e n t s of the aromatic r i n g . S p e c t r a f o r t h r e e d i f f e r e n t s u b s t i t u e n t s i n the para p o s i t i o n demonstrate t h i s . The two s p e c i e s a l s o e x i s t i n meta s u b s t i t u t e d d e r i v a t i v e s , such as 32 3 3 metanitrophenyl n i t r e n e . The e x i s t e n c e of two s p e c i e s of methylenes with very s i m i l a r EPR parameters has been noted p r e v i o u s l y (45). In t h i s case geometric isomers of 1-naphthylmethylene and 2-naphthylmethylene were r e p o r t e d . The d i f f e r e n c e between the two s p e c i e s was a t t r i b u t e d to two s t r u c t u r e s c o n s i s t i n g of bent methylene fragments i n t e r a c t i n g with an asymmetric s p i n d i s t r i b u t i o n i n the aromatic r i n g . T h i s e x p l a n a t i o n , although i n reasonable agreement with observed zero f i e l d s p l i t t i n g s , would not account f o r the s i m i l a r o b s e r v a t i o n s i n phenyl n i t r e n e s , u n l e s s there i s a l s o marked asymmetry i n the s p i n d i s t r i b u t i o n i n the aromatic p i system. The o n l y samples i n which the two s p e c i e s were not detected were 2 , 4 , 6 - T r i c h l o r o p h e n y l n i t r e n e and 2,4,6-Tribromophenyl n i t r e n e . The annealing h i s t o r y of the medium i s an important part of o b s e r v a t i o n of these s p e c i e s . EPR parameters i n d i c a t e t h a t one of the s p e c i e s i s more c y l i n d r i c a l . One of the s p e c i e s i s unstable with r e s p e c t t o an i r r e v e r s i b l e c o n v e r s i o n i n t o the o t h e r . These f a c t s are- c o n s i s t e n t with the n o t i o n t h a t s p e c i e s (A) and (B) are d i f f e r e n t i a t e d by p e r t u r b a t i o n s by the evolved n i t r o g e n (presumably molecular) which must be formed con c o m i t a n t l y with the n i t r e n e on i r r a d i a t i o n of the a z i d e p r e c u r s o r . I f the n i t r o g e n formed remains i n c l o s e p r o x i m i t y to the n i t r e n e , by v i r t u e of the c o n f i n i n g c r y s t a l l i n e s t r u c t u r e of the a z i d e host, and energy l i b e r a t e d 34 by the p h o t o l y t i c cleavage i s i n s u f f i c i e n t to destr o y t h i s s t r u c t u r e , then one might expect s i g n i f i c a n t i n t e r m o l e c u l a r i n t e r a c t i o n s to occur. The space a f f o r d e d the n i t r o g e n and the n i t r e n e i n the a z i d e l a t t i c e may be i n s u f f i c i e n t , r e s u l t i n g i n a d i s t r i b u t i o n of perturbed n i t r e n e s . The e x i s t e n c e of b a s i c a l l y two s p e c i e s would demand two d i s p o s i t i o n s of the n i t r o g e n molecule r e l a t i v e to the n i t r e n e molecule. The exact nature of these d i s p o s i t i o n s i s not known and would be expected t o be a complex f u n c t i o n of the c r y s t a l s t r u c t u r e s of these compounds, and perhaps not amenable to i n v e s t i g a t i o n . The i r r e v e r s i b i l i t y of the co n v e r s i o n from s p e c i e s (A) to (B)... may be a consequence of d i f f u s i o n of the n i t r o g e n molecule out of a c o n f i n i n g t r a p i n the c r y s t a l s t r u c t u r e which c h a r a c t e r i s e s n i t r e n e (A) . An a c t i v a t e d process might be expected f o r such a s i t u a t i o n . The e x i s t e n c e of phase t r a n s i t i o n s i n the a z i d e c r y s t a l s i s not known. Phase t r a n s i t i o n s would be expected t o be manifested by a l t e r a t i o n s i n the r a t e of c o n v e r s i o n from (A) to <E). T h i s p i c t u r e w i l l q u a l i t a t i v e l y account f o r the f e a t u r e s which c h a r a c t e r i s e the two s p e c i e s , but of course i s not amenable to c a l c u l a b l e s u b s t a n t i a t i o n . I t should be pointed out that a s i m i l a r o b s e r v a t i o n , namely an i r r e v e r s i b l e change i n the c o n c e n t r a t i o n of t r i p l e t s p e c i e s has been made p r e v i o u s l y (46 ). The r e s u l t s of t h i s i n v e s t i g a t i o n are a sequel to e a r l y 35 experiments on phenylmethylene doped i n v a r i o u s matrices (47 ). These e a r l y s t u d i e s e s t a b l i s h e d the f a c t t h a t the d i s t r i b u t i o n of zero f i e l d s p l i t t i n g s f o r methylenes i s s m a l l e s t when the geometry of the host medium most c l o s e l y matches that of the methylene . I t i s t h e r e f o r e reasonable to seek the best r e s o l u t i o n i n the a z i d e host which i s the p r e c u r s o r to the n i t r e n e . A l l samples could not^ be prepared i n a c r y s t a l l i n e form. T h i s f a c t lead t o the experiment i l l u s t r a t e d i n F i g u r e 6 . Metabromophenyl a z i d e f o r example i s an o i l y l i q u i d at standard c o n d i t i o n s . C r y s t a l s were not obtained by c o o l i n g t h i s o i l , hence the o i l was dropped i n t o l i q u i d n i t r o g e n forming g l a s s y beads. These were ground to a powder at l i q u i d n i t r o g e n temperature, and n i t r e n e s were generated by u l t r a v i o l e t i r r a d i a t i o n . The comparison of the g l a s s y spectrum i n F i g u r e 6 (a) with the spectrum from the powdered m a t e r i a l r e v e a l s that the l i n e w i d t h i s f i f t y percent narrower i n the powdered m a t e r i a l . The d i s o r d e r e d nature of the g l a s s must have been reduced j u s t by the powdering process; u n f o r t u n a t e l y t h i s d i d . n o t allow r e s o l u t i o n of any n i t r o g e n h y p e r f i n e s p l i t t i n g s , t h a t i s to say a g r e a t improvement i n the q u a l i t y of the spectrum was not obtained. 37 CHAPTER FOUR Sincjle C r y s t a l Experiments P r e v i o u s l y o b t a i n e d r e s u l t s on aromatic n i t r e n e s by EPR spectroscopy had e s t a b l i s h e d the f a c t t h a t the n o n - c y l i n d r i c a l zero f i e l d s p l i t t i n g parameter E i s u s u a l l y very s m a l l , t y p i c a l l y 0.003 cm-* or l e s s (48). The s u b s t i t u t e d s p e c i e s which had been s t u d i e d were mainly s u b s t i t u t e d i n the meta and para p o s i t i o n s . Two n i t r e n e s were s t u d i e d by E. P. R. which d i d not f a l l i n t o t h i s l a t t e r category. These were 2 , 4 , 6 - t r i c h l o r o p h e n y l n i t r e n e , and 2,4,6-tribromophenyl n i t r e n e . Random (powder) and o r i e n t e d ( s i n g l e c r y s t a l ) E. P. R. s p e c t r a of these s p e c i e s were i n v e s t i g a t e d . 38 iiiJi 2 x 4 A 6 z t r i c h l o r o r ^ h e n x l Nitrene T h i s s p e c i e s was generated by photochemical decomposition of 2,4,6-trichloropheny1 a z i d e . The a z i d e , prepared from the corresponding a n i l i n e , was c r y s t a l l i z e d from d i e t h y l ether. The q u a l i t y of the c r y s t a l s which could be o b t a i n e d was not good; long needle-shaped c r y s t a l s were produced. However, these were s u f f i c i e n t t o o b t a i n reasonable q u a l i t y EPR s i g n a l s . The powder EPR spectrum of t h i s s p e c i e s a t room temperature i s i l l u s t r a t e d i n F i g u r e 7 . The main f e a t u r e s are a well r e s o l v e d 'E* s p l i t t i n g , and r e s o l v e d n i t r o g e n h y p e r f i n e s p l i t t i n g of 18 gauss. The f i n e s t r u c t u r e of t h i s s p e c i e s may be d e s c r i b e d by: D = 27.28 GHz. (0.909 cm-*) E = 96. MHz. (0.003 cm~») g = 2.0023 ( assumed ) The observed ' D« s p l i t t i n g i s ten percent s m a l l e r than the z e r o f i e l d s p l i t t i n g s of other, u n s u b s t i t u t e d n i t r e n e s p e c i e s . T h i s c o u l d be q u a n t i t a t i v e l y understood i n terms of enhanced d e l o c a l i z a t i o n which occurs f o r example on para s u b s t i t u t i o n . However, i n view of the temperature e f f e c t s on these measurements to be d i s c u s s e d l a t e r , i t should be mentioned t h a t the comparison i s q u a l i t a t i v e r a t h e r than 4 0 q u a n t i t a t i v e . The »E' s p l i t t i n g , o f magnitude 0.003 cm - 1 i s t y p i c a l of p r e v i o u s l y i n v e s t i g a t e d aromatic n i t r e n e s . Nitrogen h y p e r f i n e s t r u c t u r e i s to be expected i n n i t r e n e s (49) . In a d d i t i o n to the r e s o l v e d m u l t i p l e t s t r u c t u r e on the high f i e l d f i n e s t r u c t u r e l i n e , there are i n d i c a t i o n s i n the lineshape of the lower f i e l d f i n e s t r u c t u r e a b s o r p t i o n o f h y p e r f i n e s p l i t t i n g . T h i s however i s not r e s o l v e d . A s i n g l e c r y s t a l specimen was examined and the r e s u l t i n g f i n e s t r u c t u r e angular v a r i a t i o n i s d e p i c t e d i n F i g u r e 8 The angular v a r i a t i o n r e v e a l s two d i s t i n g u i s h a b l e s i t e s ; i n one of the planes of o b s e r v a t i o n the e f f e c t s of a symmetry o p e r a t i o n are obvious. Hyperfine s t r u c t u r e was detected only i n a very r e s t r i c t e d r e g i o n of the angular v a r i a t i o n . T r a n s i t i o n s r e s u l t i n g from alignment of the magnetic f i e l d along the •Z' a x i s of the n i t r e n e ( i . e. the C-N bond) r e v e a l that the molecules are arrayed i n such a manner t h a t t h i s d i r e c t i o n of the n i t r e n e s i s approximately p e r p e n d i c u l a r to the needle a x i s of the c r y s t a l . Comparison with the angular v a r i a t i o n of the brominated analogue s t r o n g l y suggests t h a t those molecules must a l s o be arrayed i n s i m i l a r -f a s h i o n . 41 8. A n n u l a r Dependence Of T r a n s i t i o n F i e l d s ; 2 , 4 , 6 - t r i c h l o r o p h e n y l N i t r o n s In P a r e n t A z i d e H o s t . V / 2 4 42 its.2 2 X 4 j L ^ t r i b r o m o t o h e n y l N i t r e n e The second n i t r e n e s p e c i e s which was examined i n random and o r i e n t e d samples was 2,4,6-tribromophenyl n i t r e n e . T h i s molecule i s s t a b l e i n c r y s t a l s of i t s a z i d e p r e c u r s o r , from which i t i s produced by photochemical decomposition. The X band powder spectrum of the n i t r e n e a t room temperature i s i l l u s t r a t e d i n F i g u r e 9 . Features (a) and (b) belong to molecules a l i g n e d along X and I d i r e c t i o n s of zero f i e l d s p l i t t i n g tensor; these .two f e a t u r e s are a measure of the E zero f i e l d s p l i t t i n g parameter which i s abnormally l a r g e f o r t h i s n i t r e n e . The usual s p l i t t i n g encountered i s approximately 100 to 150 gauss, while the s p l i t t i n g i n t h i s n i t r e n e i s of the order of 1800 gauss. Features (c) and (d) of the same f i g u r e are from molecules a l i g n e d such that the Z a x i s of the zero f i e l d s p l i t t i n g tensor l i e s along the a p p l i e d magnetic f i e l d d i r e c t i o n . I t can be seen t h a t the i n t e n s i t y of these t r a n s i t i o n s i s between 5 and 10 times l e s s t h a t the i n t e n s i t y of the XY t r a n s i t i o n s . Resolved h y p e r f i n e s p l i t t i n g does not appear on any of these t r a n s i t i o n s , and the l i n e w i d t h of these t r a n s i t i o n s i s 55 gauss, a comparatively l a r g e width. The XY f e a t u r e s were simulated using a program s u p p l i e d by Dr. J . C. T a i t and the parameters; 2,4,6-tribromophenyl N i t r e n e F o l y c r y s t ' a l l i n e EPR Spectrum; (a) , (b) XY F e a t u r e s ; ( c ) # ( d ) Z F e a t u r e s . D = .939 cm-i E = .025 cm--g = 2.0023 (assumed) T h i s s i m u l a t i o n may be compared with the observed spectrum i n F i g u r e 10 . A s i m u l a t i o n of the Z t r a n s i t i o n s was not attempted due t o the f a c t that at X band these t r a n s i t i o n s are not a continuous f u n c t i o n of angular displacement. The powder (random) spectrum f e a t u r e s a r i s e from the f a c t t h a t the angular v a r i a t i o n of resonant magnetic f i e l d possesses l o c a l extrema along d i r e c t i o n s p a r a l l e l to p r i n c i p a l d i r e c t i o n s of the zero, f i e l d i n t e r a c t i o n . These f e a t u r e s may then be c o n s i d e r e d to a r i s e from molecules a l i g n e d along these d i r e c t i o n s . A c c o r d i n g l y , f i g u r e 11 d e p i c t s the energy " l e v e l s of the t r i p l e t as a f u n c t i o n of magnetic f i e l d s t r e n g t h along each of the three p r i n c i p a l d i r e c t i o n s of the zero f i e l d s p l i t t i n g . The powder spectrum f e a t u r e s of F i g u r e 9 may then be considered to correspond to the l a b e l l e d t r a n s i t i o n s d e p i c t e d as arrows i n F i g u r e 11. The h i g h e s t f i e l d Z t r a n s i t i o n s have been omitted i n order t h a t the diagram may more c l e a r l y i l l u s t r a t e the t r a n s i t i o n s o b s e rvable a t lower f i e l d s t r e n g t h s . A d d i t i o n a l proof that the assignment of the observed t r a n s i t i o n s i s c o r r e c t may be had by observing the frequency 45 2 , 4 , 6 - t r i b r o r a o p h e n y l N i t r e n e P o l / c r y s t a l l i S p e c t r u m ; (a) O b s e r v e d (b) S i m u l a t e d . 11 T r i p l e t Energy L e v e l s As A F u n c t i o n Of A p p l i e d Magnetic- F i e l d S t r e n g t h . (a),(b) XY F e a t u r e s ; (c) Z F e a t u r e . co to E n e r g y ( c m " ) 47 dependence of the powder t r a n s i t i o n s . F i g u r e 12 i l l u s t r a t e s the e f f e c t of frequency changes upon the random n i t r e n e a b s o r p t i o n s . The a b s o r p t i o n s which have a p o s i t i v e slope must connect d i v e r g e n t (as a f u n c t i o n of i n c r e a s i n g magnetic f i e l d ) energy l e v e l s , while a negative s l o p e i m p l i e s the t r a n s i t i o n s connects l e v e l s which are converging. The s l o p e s are c o n s i s t e n t with the energy l e v e l scheme of F i g u r e 11 The approximate e q u a l i t y of the a b s o l u t e value of the s l o p e s i n d i c a t e s that a l l these t r a n s i t i o n s i n v o l v e the same increment i n s p i n quantum number, which i s ±1 f o r allowed t r a n s i t i o n s . The " f o r b i d d e n " t r a n s i t i o n s of the h a l f f i e l d r e g i o n , f o r example, would be expected to s h i f t by only 1/2 t h i s amount. The a c t u a l v a r i a t i o n of the frequency i n t h i s experiment was achieved by p a r t i a l l y removing a quartz gas flow dewar assembly from a standard X band c a v i t y . By t h i s means the resonant freguency of the c a v i t y c o u l d be v a r i e d by f i v e percent. Freguency v a r i a t i o n of resonant f i e l d s t r e n g t h s should depend upon g values. The resonance c o n d i t i o n s f o r the observed powder f e a t u r e s a r e : G x * B 2 H x 2 = v 2 + v ( D - 3 E ) - 2E (D-E) Gy 2B 2Hy 2 = V 2 + v(D+ 3 E ) + 2E(D+E) G z 2 B 2 H z 2 = (D-v) 2 - E 2 D i f f e r e n t i a t i o n of these c o n d i t i o n s with r e s p e c t to resonant frequency v y i e l d s : 48 © — ____ o C ^ (b) ( d ) - 9 • i — — I — y - ( G H z ) 9.0 9.5 12. Frequency Response Of N i t r e n e EPR , „ T r a n s i t i o n s , (a) , (b) XY F e a t u r e s ; (c) , (d) Z Features. 49 Gx 2B* (dHx/dv)2 = (v+0.5 (D-3E)) 2/ ( v 2 + v(D-3E)-2E (D-3E) ) Gy2B2 (dHy/dv) 2 = (v + 0.5 (D+3E)) 2 / (v 2 +v (D+3 E) +2E {B+3 E) ) G z 2 B 2 (dHz/dv) 2 = 1/(1-E 2/(D-v) 2) Dsing t y p i c a l v a l u e s f o r n i t r e n e s : GxB(dHx/dv) = GyB (dHy/dv) = 5/4 GzB(dHz/dv) = 1 From these r e l a t i o n s one has only an approximate method to determine g v a l u e s s i n c e the e r r o r i n the d e t e r m i n a t i o n of the d e r i v a t i v e s may be c o n s i d e r a b l e . In t h i s case the s h i f t s of the magnetic resonance was of order 250 gauss and the l i n e w i d t h s are of order 55 gauss. Despite the f a c t t h a t the measurements which are d e p i c t e d i n F i g u r e 12 are l i k e l y a c c u rate to o n l y 10 gauss, the p e r c e p t i b l e tendency of the g r e a t e r magnitude of slope f o r the XY d i r e c t i o n s has not been masked. An e s t i m a t i o n o f the g v a l u e s based on these data d i d not r e s u l t i n a c o n s i s t e n t d e s c r i p t i o n of the data. One must conclude t h a t these data enabled one to say nothing s u b s t a n t i a l about the g v a l u e s , and r e c a l l the o r i g i n a l purpose f o r which the data were taken. That was to support the assignment o f the t r a n s i t i o n s . 50 4.3 C r y s t a l S t r u c t u r e of 2 X 4 x 6 - t r i b r p m g p h e n y l Azide The space group of the c r y s t a l and u n i t c e l l s p e c i f i c a t i o n s were determined a t t h i s department by Dr. H. Sherar and Dr. D. Rendle. The r e s u l t s a r e : space group: P2*/c a = 3.966 (1) angstrom b = 14.738 (2) angstrom c = 15.857(3) angstrom B = 98.37 (3)° Z = 4 The s t r u c t u r e d e t e r m i n a t i o n of t h i s c r y s t a l was made at at t h i s department by Dr. D. Rendle. The u n i t c e l l dimensions suggest a l a y e r e d s t r u c t u r e , and the a a x i s was found to c o i n c i d e with the needle a x i s o f the c r y s t a l . A p r o j e c t i o n of the u n i t c e l l c o n tents onto the c r y s t a l l o g r a p h i c be plane i s giv e n i n F i g u r e 13 . T h i s view i s the most u s e f u l s i n c e the molecules do not o v e r l a p . The molecules must l i e b a s i c a l l y "prone" i n t h i s plane due to the s m a l l u n i t c e l l ' dimension along the a a x i s . 51 it i i i S i n g l e C r y s t a l EPR An i n v e s t i g a t i o n of the magnetic resonance of the 2,4,6-tribromophenyl n i t r e n e , o r i e n t e d i n a s i n g l e c r y s t a l of 2,4,6-tribromophenyl a z i d e was made. The purpose of t h i s study was to determine d i r e c t i o n a l i n f o r m a t i o n i n h e r e n t l y l o s t i n a p o l y c r y s t a l l i n e i n v e s t i g a t i o n . S i n c e the p o l y c r y s t a l l i n e examination of t h i s n i t r e n e had r e v e a l e d t h a t t h e r e i s marked asymmetry to the zero f i e l d s p l i t t i n g , one should be able t o determine a l l t h r e e p r i n c i p a l d i r e c t i o n s of the z e r o f i e l d s p l i t t i n g , and be able to r e l a t e these to the molecule. The c r y s t a l s t r u c t u r e of the a z i d e host had not been determined at the time the EPR measurements had begun. Th e r e f o r e no p a r t i c u l a r reason e x i s t e d f o r j u d i c i o u s c h o i c e of planes of o b s e r v a t i o n o f the angular dependence of the z e r o f i e l d s p l i t t i n g term. The space group however, had been determined, and the expected two m a g n e t i c a l l y d i s t i n c t s i t e s were observed. The c r y s t a l had the shape i l l u s t r a t e d i n F i g u r e 13 . I t was glued t o a s m a l l l u c i t e cube whose f a c e s had been m i l l e d p e r p e n d i c u l a r and l a b e l l e d A,B, and C. In t h i s manner, o b s e r v a t i o n s c o u l d be taken i n three approximately p e r p e n d i c u l a r planes, by remounting the cube i n a standard notched t e f l o n sample mount. The most well 13. C r y s t a l Morphology And A x i s System. 5 3 developed f a c e s of the c r y s t a l were approximately p a r a l l e l to the AB and AC planes of the l u c i t e cube; p o l a r i z e d l i g h t e x t i n g u i s h e d at the f o l l o w i n g d i r e c t i o n s r e l a t i v e to the ABC system: AB plane -12 ± 3° AC plane 13 ± 3° BC plane 45 ± 3° T h i s i n f o r m a t i o n i s s u f f i c i e n t to d e f i n e the mounting procedure. The angular dependence of the magnetic resonance which was observed i n each of the three planes i s depicted i n F i g u r e s 14 and 15 . Both of the expected magnetic s i t e s were observed i n a l l three p l a n e s ; they are d i s t i n g u i s h e d i n the f i g u r e by c r o s s e s and a d d i t i o n symbols. The e l l i p s o i d a l f e a t u r e s at the higher magnetic f i e l d s t r e n g t h are d i s t i n c t i v e t o the z t r a n s i t i o n s of the t r i p l e t . E f f e c t s of v a n i s h i n g t r a n s i t i o n p r o b a b i l i t y f o r the lower f i e l d t r a n s i t i o n i n a neighbourhood of t h i s d i r e c t i o n i s evidenced by the l a c k of measured resonances at the lower " h a l f f i e l d " r e g i o n of the angular v a r i a t i o n . . These p l o t s are q u a n t i t a t i v e l y s i m i l a r t o those of a p r e v i o u s s i n g l e c r y s t a l examination of a n i t r e n e with s i m i l a r zero f i e l d parameters (50). I t should be remarked t h a t there are two f a i r l y obvious \ c h o i c e s f o r the planes of o b s e r v a t i o n . One i s the axes of the z e r o f i e l d s p l i t t i n g f o r a p a r t i c u l a r s i t e , and the other 14. Angular Dependence Of T r a n s i t i o n F i e l d s ; AE And AC Planes. (GAUSS) (X101 200.0 soo.o 4)0.0 sno.o _ i i i 1 eoo.o 100.0 i eoo.o soo.o 100.0 X + + + + + + • + + + + + + + + + + + + -r + + .+ +• + + + + + +x X X CT I mo 200.0 — I SOO.O (GRUSS) (X10> « 0 . 0 soo.o 700.0 I eoo.o _ i POO.O _ i X X X X X X X X X X X X X x+ X X X H X X X X X X X X -X X X + + X X X + 55 o «1 15. A n g u l a r Dependence Of T r a n s i t i o n F i e l d s ; EC P l a n e . 56 c h o i c e i s symmetry planes a r i s i n g from c r y s t a l symmetry o p e r a t i o n s . The main reason t h a t n e i t h e r of these were chosen was the experimental l a c k of a f a c i l i t y t o reproducably o r i e n t the specimen i n such o r i e n t a t i o n s . The experimental proof that e i t h e r of these c o n d i t i o n s i s s a t i s f i e d r e s t s i n the a n i s o t r o p y of the resonance which i s being observed. In the n i t r e n e s p e c i e s the main f e a t u r e i s the l a r g e a n i s o t r o p y of the f i n e s t r u c t u r e terms; while i t was p o s s i b l e to o r i e n t a sample with the Z d i r e c t i o n of one s i t e i n the plane of the r o t a t a b l e magnet with the a i d of a standard r o t a t a b l e sample mount, no guarantee of the d i r e c t i o n of the magnet i n a p e r p e n d i c u l a r p o s i t i o n could be arranged. O r i e n t a t i o n a l d i f f i c u l t y had been a n t i c i p a t e d , so t h a t s u f f i c i e n t computational f a c i l i t i e s t o o b v i a t e . the n e c e s s i t y to take data i n these planes were a v a i l a b l e . The observed data i n d i c a t e d i n F i g u r e s 14 and 15 were f i t t e d to a s p i n H a m iltonian: ]-[ = S«D«S + gBH«S with S = 1 and g = 2.0023. As the planes of o b s e r v a t i o n were only approximately p e r p e n d i c u l a r , the r e l a t i v e o r i e n t a t i o n s of the planes were v a r i e d i n a neighbourhood of two degrees from mutually p e r p e n d i c u l a r during the f i t t i n g process. A minimal e r r o r measure was found by a l l o w i n g t h i s u n c e r t a i n t y to be t r e a t e d i n t h i s manner. I t was found t h a t the o r i e n t a t i o n which minimised e r r o r f o r one of the s i t e s d i d 57 not a l s o accomplish t h i s f o r the other s i t e ; as might be expected the p r i n c i p a l values deduced f o r the two s i t e s d i d not c o i n c i d e . T h i s may i n d i c a t e t h a t there i s seme a n i s o t r o p y i n the e l e c t r o n i c Zeeman term which was not i n c l u d e d i n the d e s c r i p t i o n ; however to deduce t h i s from the s i n g l e c r y s t a l data seems a d i f f i c u l t task. J o method c o u l d be found to remedy the u n c e r t a i n t y i n the magnet o r i e n t a t i o n s f o r these o b s e r v a t i o n s . T h i s u n c e r t a i n t y i s l i k e l y of magnitude 2 to 3 degrees f o r these planes. The zero f i e l d parameters deduced f o r the two s i t e s a r e : s i t e 1 D = 0.938 cm- 1 E = 0.025 cm-* s i t e 2 D = 0.933 cm-* E = 0.025 cm-* These were e x t r a c t e d from the p r i n c i p a l values of the zero f i e l d s p l i t t i n g t e n s o r s which r e s u l t e d from the f i t t i n g process. The values are c o n s i s t e n t with the powder spectrum i n v e s t i g a t i o n . The d i r e c t i o n a l i n f o r m a t i o n which i s represented i n F i g u r e s 14 and 15 are the e i g e n v e c t o r s of the matrix D which f i t t e d the data p o i n t s due to a s i t e . T h i s i n f o r m a t i o n r e l a t e s t o the symmetry o p e r a t i o n s of the space group of the c r y s t a l P2*/c. Omitting the t r a n s l a t i o n a l p a r t s of the o p e r a t i o n s of t h i s group there remains the i d e n t i t y , a twofold r o t a t i o n , a r e f l e c t i o n plane, and the i n v e r s i o n o p e r a t i o n s . That the presence of an i n v e r s i o n o p e r a t i o n cannot be d i s c e r n e d by EPS i m p l i e s t h a t of the f o u r molecules 58 i n the u n i t c e l l , one may expect to d i s t i n g u i s h o n l y two s i t e s . These may be c o n s i d e r e d r e l a t e d by the twofold r o t a t i o n . T h i s o p e r a t i o n can be c h a r a c t e r i z e d by a r r a n g i n g the e i g e n v e c t o r s of the zero f i e l d s p l i t t i n g t e n s o r s f o r the two s i t e s as matrices U, and v such that RO = V or B = VU _ 1. I f one chooses the e i g e n v e c t o r s of U and V to d e f i n e a x i s systems of the same handedness, then B w i l l represent a proper r o t a t i o n . For a twofold r o t a t i o n , the e i g e n v a l u e s of R s h o u l d be 1, -1, and -1, with the a x i s of the r o t a t i o n belonging to 1. In t h i s case one found the o p e r a t i o n which i s d e p i c t e d i n f i g u r e 13 T h i s i s not unique. Due t o the l a c k of knowledge of the p r e c i s e t r a n s f o r m a t i o n between the a x i s system of the EPR measurements, and the c r y s t a l l o g r a p h i c axes, i t i s d i f f i c u l t to attempt a g u a n t i t a t i v e comparison. However, r e f e r r i n g a g a i n to f i g u r e 13 , one may note t h a t the e s s e n t i a l f e a t u r e s are c o n s i s t e n t . The b c r y s t a l l o g r a p h i c a x i s n e c e s s a r i l y b i s e c t s the angle between the axes of the C-N bonds belonging to two p a i r s of molecules not r e l a t e d by the i n v e r s i o n o p e r a t i o n . These l a t t e r d i r e c t i o n s would be a s s o c i a t e d with the Z a x i s of the s p i n - s p i n c o u p l i n g i n the two s i t e s , and are l a b e l l e d by one c l o s e d c i r c l e and one arrowhead i n f i g u r e 13 . The o p e r a t i o n deduced i s one p l a u s i b l e c h o i c e f o r the b a x i s . The other c h o i c e l i e s approximately 90 degrees away from the d e p i c t e d c h o i c e , s t i l l 59 l y i n g approximately i n the BC plane. Due to the l a y e r e d s t r u c t u r e , the energy l e v e l .335 must correspond with a d i r e c t i o n s u b s t a n t i a l l y p e r p e n d i c u l a r to the aromatic r i n g . 4 ,_5 I n t e r p r e t a t i o n The combined r e s u l t s of the EPR s t u d i e s on 2,4,6—tribromophenyl n i t r e n e i n d i c a t e that t h e r e i s marked n o n - c y l i n d r i c a l nature to the t r i p l e t . The s i n g l e c r y s t a l study i n d i c a t e s t h a t a d i r e c t i o n roughly p e r p e n d i c u l a r to the phenyl r i n g corresponds to the l a r g e r of the two zero f i e l d s p l i t s t a t e s , d i s t i n g u i s h e d by non zero E. The l i n e w i d t h of the a b s o r p t i o n s i s approximately 55 gauss. These f a c t s i n d i c a t e that the d i s p o s i t i o n of e l e c t r o n s about the n i t r e n e n i t r o g e n i s not c y l i n d r i c a l l y symmetric. The s t r u c t u r e may be analogous to those of bent methylene fragments (51). The n i t r e n e n i t r o g e n has two p a i r e d e l e c t r o n s i n s t e a d of the hydrogen of the methylene. The d e s c r i p t i o n of the s t r u c t u r e of these fragments i s due to H i g u c h i (52). T h i s d e s c r i b e s a h y b r i d i z e d o r b i t a l d i r e c t e d along the b i s e c t o r of the fragment bond angle ©, The second o r b i t a l comprising the t r i p l e t i s carbon 2p o r b i t a l d i r e c t e d 60 p e r p e n d i c u l a r l y . H i g u c h i c a l c u l a t e d zero f i e l d s p l i t t i n g parameters f o r methylene as a f u n c t i o n of the bond angle 6. I n t e r p o l a t i o n of the E parameter f o r 2,4,6-tribromophenyl n i t r e n e i n Hi g u c h i * s E (6) y i e l d s an estimate of 165 degrees f o r the bent s t r u c t u r e . A bent s t r u c t u r e seems a reasonable e x p l a n a t i o n of the non zero E parameter. E f f e c t s of s p i n d e n s i t y d e l o c a l i s e d i n t o the phenyl r i n g would not seem to provide a reasonable e x p l a n a t i o n s i n c e c a l c u l a t i o n s on s u b s t i t u t e d aromatic n i t r e n e s have shown t h i s e f f e c t to be s m a l l (53). Hyperfine s t r u c t u r e was not r e s o l v e d i n the 55 Gauss l i n e w i d t h . C a l c u l a t e d c o u p l i n g constants f o r the n i t r o g e n 2s and 2p o r b i t a l s are 640 and 19.8 gauss (54). The magnitude of the h y p e r f i n e i n t e r a c t i o n expected from 2p o r b i t a l s alone seems s m a l l i n comparison t o the l i n e w i d t h , p a r t i c u l a r l y so i n view of models p r e d i c t i n g l e s s than u n i t s p i n d e n s i t y i n t h i s o r b i t a l (55). However, s i n c e the h y p e r f i n e i n t e r a c t i o n was not r e s o l v e d i t i s d i f f i c u l t to make any p r e c i s e statements about the e l e c t r o n i c s t r u c t u r e . Large l i n e w i d t h s have been a s c r i b e d to matrix e f f e c t s i n these compounds (56). The d i r e c t i o n of the p o s t u l a t e d bending of the n i t r e n e fragment r e l a t i v e to the remainder of the molecule could not be e l u c i d a t e d . As the bending i s not manifest i n the 61 c h l o r i n a t e d analogue, i t seems reasonable t h a t s t e r i c i n t e r a c t i o n s between the n i t r e n e fragment and the adjacent s u b s t i t u e n t may be r e s p o n s i b l e . CHAPTER FIVE Temperature E f f e c t s 5_s.l R e s u l t s and D i s c u s s i o n I t has been known f o r over t e n years t h a t the zero f i e l d s p l i t t i n g c h a r a c t e r i s t i c of t r i p l e t s t a t e molecules i s s u b j e c t to temperature e f f e c t s . These temperature e f f e c t s have not been i n v e s t i g a t e d with as wide chemical i n t e r e s t as de t e r m i n a t i o n of zero f i e l d s p l i t t i n g s i n a v a r i e t y of molecules. However, a number of c o n s i d e r a t i o n s of temperature e f f e c t s have appeared and are b r i e f l y summarized below. H i l d e b r a n t and BcConnell c o n s i d e r the e f f e c t of t o r s i o n a l i n t r a m o l e c u l a r motion upon the zero f i e l d s p l i t t i n g (57). Using a model based on two c o a x i a l benzene anions i n r e l a t i v e t o r s i o n a l motion, they deduce the r e l a t i o n : D-idD/dT = constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ 7 ] T h i s form r e s u l t s from the f o l l o w i n g c o n s i d e r a t i o n s : the carbon atoms of the c o a x i a l benzene anions possess uniform s p i n d e n s i t y 1/6. The s p i n Hamiltonian i s S z 2 . I n c l u d i n g n earest neighbour i n t e r m o l e c u l a r carbon-carbon p a i r s o n l y : D=0.75g2B2 (6 (1/6) 2) (1-3cos20) R-3 D= (constants) (3sin2Q-2) dD/dT= (constants) 3dsin 29/dT f o r s m a l l 6 , D _ 1dD/dT=- (3/2) d (6 2) /dT but from X ray s t u d i e s d (Q 2)/dT=constant Where experiments have been c a r r i e d out at s e v e r a l temperatures, the numerical value of the e g u a l i t y [ 7 ] i s sometimes r e p o r t e d (58). Temperature e f f e c t s on t r i p l e t s t a t e molecules trapped i n p l a s t i c s were i n v e s t i g a t e d by Thomson i n 1964 (59). A f i v e percent decrease i n the zero f i e l d parameter D* over the temperature i n t e r v a l 50 K to 300 K was determined. The e x p l a n a t i o n of t h i s e f f e c t was t h a t at higher temperatures, higher v i b r a t i o n a l l e v e l s of the t r i p l e t coronene become a p p r e c i a b l y p o p u l a t e d , r e s u l t i n g i n the observed r e d u c t i o n i n D*. A study of the EPB of phosphorescent mesitylene doped on t r i m e t h y l b orazole r e v e a l e d an unusual temperature e f f e c t (60 ). D remained independent of temperature, while E decreased with i n c r e a s i n g temperature. C o n c l u s i o n s drawn reg a r d i n g t h i s e f f e c t are q u a l i t a t i v e l y s i m i l a r t o previous deductions based on o b s e r v a t i o n s of the EPR of t r i p l e t benzene (61). I t was 64 suggested that the t r i p l e t molecules may e x i s t i n t h r e e d i s t o r t e d s t r u c t u r e s . The authors f e l t however, t h a t a r a p i d t u n n e l i n g motion ( of order ten GHz.) which had been p o s t u l a t e d f o r benzene was not r e s p o n s i b l e . Rather, they f e l t t h a t the c r y s t a l f i e l d h o l d i n g the molecules i n the l a t t i c e f a v o u r s one deformation r a t h e r than the other two. Higher temperatures would be expected to produce a more uniform p o p u l a t i o n of these v i b r o n i c l e v e l s , with concomitant r e d u c t i o n of the parameter E. , A more q u a n t i t a t i v e account of t h i s temperature e f f e c t i n the mesitylene t r i p l e t was rep o r t e d i n a subsequent study (62). The form of the observed v a r i a t i o n i n the zero f i e l d parameter D f o r s e v e r a l aromatic n i t r e n e s p e c i e s i s i l l u s t r a t e d i n F i g u r e 16 . In these temperature s t u d i e s , the zero f i e l d parameters were deduced from X,Y t r a n s i t i o n s of p o l y c r y s t a l l i n e s p e c t r a , assuming G i s o t r o p i c at the f r e e e l e c t r o n value. The f i g u r e p o r t r a y s the r e d u c t i o n of the D v a l u e with i n c r e a s i n g temperature. The magnitude of t h i s e f f e c t i n these n i t r e n e s i s about four percent, q u i t e comparable to o b s e r v a t i o n s by Thomson i n phosphorescent t r i p l e t coronene (63). T h i s e f f e c t i s common to a l l of the n i t r e n e s i n v e s t i g a t e d , and t h e r e f o r e not dependent upon the nature of the r i n g s u b s t i t u e n t . The n o n - c y l i n d r i c a l parameter E on the other hand, i s an i n c r e a s i n g f u n c t i o n of temperature f o r the para s u b s t i t u t e d n i t r e n e s . 65 6 6 Figures 17 18 19 20 21 and 22 i l l u s t r a t e the v a r i a t i o n of both zero f i e l d parameters as f u n c t i o n s of temperature. -1 Experimental o b s e r v a t i o n s are i n d i c a t e d on the f i g u r e s as squares, and f o u r c a l c u l a t e d f u n c t i o n s are d i s p l a y e d as c a r v e s . The portrayed f u n c t i o n s i l l u s t r a t e the behaviour expected f o r a model c o n s i s t i n g of two s t a t e s populated a c c o r d i n g to the Boltzmann d i s t r i b u t i o n : D(T) = (D(0) + D(1) exp (-W (1)/kT) )/1+exp (-W (1)/kT) [ 8 ] As can be seen, the f o u r curves i n t e r s e c t a t two p o i n t s on the diagram. T h i s i s a consequence of the procedure used to s e l e c t v a l u e s f o r the c o n s t a n t s D (0) , D (1) , E (0) , and E(1) which c h a r a c t e r i z e the zero f i e l d parameters of these two s t a t e s . I t was decided to d e f i n e these c o n s t a n t s such t h a t the f u n c t i o n s s a t i s f i e d the two extremal o b s e r v a t i o n s . These data are presented i n Table 1 , The remaining parameter W(1) was a r b i t r a r i l y s e l e c t e d as one of the four values: 250 cm - 1, 350 cm - 1, 450 cm - 1, and 550 cm - 1. T h i s accounts f o r the f o u r i l l u s t r a t e d c u r v e s . These f o u r f u n c t i o n s may be e a s i l y d i s t i n g u i s h e d by noting t h a t the f u n c t i o n belonging to the most c l o s e l y spaced l e v e l s (250 cm - 1) . w i l l d i s p l a y the temperature e f f e c t as the onset of c u r v a t u r e a t the lowest temperature. I t can be seen t h a t the experimental p o i n t s may be contained by the f a m i l y of f u n c t i o n s ; i t may a l s o be seen t h a t t h e r e are s y s t e m a t i c d i f f e r e n c e s between o b s e r v a t i o n s 17.. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; P-methoxyphenyl N i t r e n e . 19. Temperature Dependence Of Zero F i e l d S p l i t t i n g ; P - n i t r o p h e n y l N i t r e n e . 73 Table 1. Data Characterizing Two State Model substituent: W{1) D(0) D(1) 1(0) E(1) (cm-») (GHz) (GHz) (MHz) (MHz) p-methoxy 250. 29.882 24.348 27. 438. 350. 29.774 22.318 35. 589. 4 50. 29.738 18.774 37. 852. 550. 29.726 12.930 38. 1287. 310. 29.803 23.283 33. 517. p^carboxylate 250. 29.771 24.817 -3. 328. 350. 29.591 23.434 8. 421. 450. 29.516 20.876 13. 593. 550. 29.481 16.581 16. 881. 310. 29.643 24.109 5. 376. p-nitro 250. 29.663 23.321 18. 389. 350. 29.483 21.286 29. 507. 450. 29.415 17.626 33. 721. 550. 29.386 11.524 35. 1077. 310. 29.534 22.267 26. 450. 2,4,6-tribromo 250. 29.532 23.790 808. 595. 350. 29.450 21.432 805. 507. 450. 29.426 17.340 804. 355. 550. 29.418 10.548 804. 102. 310. 29.471 22.547 806. 548. p-chloro (B) 250. 29.173 25.718 2. 430. 350. 29.110 23.925 9. 652. 450. 29.087 20.492 12. 1078. 550. 29.078 14.163 13. 1864. 310. 29.127 24.794 7. 545. p-chloro (A) 250. 28.836 24.688 1. 916. 350. 28.759 22.392 18. 1422. 450. 28.731 17.888 24. 2416. 550. 28.719 9. 364 27. 4297. 310. 28.781 23.510 13. 1176. 74 and a p a r t i c u l a r f u n c t i o n . A f u n c t i o n of the form given by equation £8] should possess an i n f l e c t i o n p o i n t f o r H (1) = 2.4kT. By v i r t u e of the absence of a n o t i c e a b l e i n f l e c t i o n i n the experimental o b s e r v a t i o n s , one may conclude t h a t W(1) > 500 cm - 1. By g r a p h i c a l l y e s t i m a t i n g the d e r i v a t i v e : dD/dT = W(1) (D (1) —D (0) )/4kT z c o s h 2 (K (1)/2kT) ....[9] from experimental data, a t two d i f f e r e n t temperatures, W{1) was estimated to be approximately 500 cm - 1. These two methods i n d i c a t e d s i m i l a r v a l u e s f o r the -energy s e p a r a t i o n , however both e s t i m a t e s are by t h e i r nature open to c o n s i d e r a b l e u n c e r t a i n t y . More fundamental i s the t a c i t assumption t h a t the experimental data are a c c u r a t e l y d e s c r i b e d by or conform to equation [ 8 ] . In f a c t i t was f o r t h i s reason t h a t F i g u r e s 17 through 22 were c o n s t r u c t e d . Apart from the g e n e r a l agreement i t can be seen t h a t a s y s t e m a t i c d e v i a t i o n between t h i s f a m i l y of f u n c t i o n s and the observed p o i n t s e x i s t s . The g e n e r a l tendency i s that at lower temperatures, the experimental p o i n t s f a l l c l o s e to a f u n c t i o n with the lower s t a t e s e p a r a t i o n , while at higher temperatures, the experimental p o i n t s tend to a f u n c t i o n c h a r a c t e r i s t i c of h i g h e r energy s e p a r a t i o n . T h i s suggests t h a t the f u n c t i o n a l form of equation [ 8 ] i s b a s i c a l l y i n accord with the o b s e r v a t i o n s , but t h a t i n t h i s temperature ranqe more than 75 j u s t two s t a t e s are a c t u a l l y i n v o l v e d . I t i s t h e r e f o r e of i n t e r e s t t o examine t h i s p o s s i b i l i t y . The a d d i t i o n of another s t a t e i n t o the Boltzmann d i s t r i b u t i o n to y i e l d the average D (T) i s easy. However, d e a l i n g with t h i s f u n c t i o n and i t s p r o p e r t i e s i s not. One c o u l d not deconvolute from a v a i l a b l e D (T) the values of D(0), D(1), D (2) e t c . which are necessary to d e f i n e t h i s f u n c t i o n . Instead of adopting a t r i a l and e r r o r approach, or f i t t i n g techniques which might be a n t i c i p a t e d t o be i n d e c i s i v e , i t was decided to compare the behaviour of a t h r e e s t a t e model f o r v a r i o u s assumed parameter values with observed behaviour of p a r a n i t r o p h e n y l n i t r e n e . Before t h i s comparison can be e f f e c t i v e l y presented, i t i s necessary to review the form of some s i m i l a r r e s u l t s o b t a i n e d by an EPR study of t r i p l e t mesitylene. The temperature dependence of E f o r t h i s molecule was presented as a graph of ln{ E (T)-E (0) } as a f u n c t i o n of 1/T. The motive f o r such a p l o t i s t h a t the energy s e p a r a t i o n between the s t a t e s of a two s t a t e system may be e x t r a c t e d from the s l o p e of such a p l o t : ln{ D(0)-D (T) } = ln{D(0)-0(1) } - W(1)/kT -exp (-W (1) /kT) + [ 10 ] The f u n c t i o n ln{D (0)-D (T) } versus 1/T i s expected t o be l i n e a r provided exp[ - W (1)/kT} « 1 . T h i s i s e x a c t l y the range of a v a i l a b l e experimental o b s e r v a t i o n s ; t h e i r s i g n a l s 76 became unobservable b e f o r e a p p r e c i a b l e c u r v a t u r e from the p a r t i t i o n f u n c t i o n term (represented by the s e r i e s expansion) became e v i d e n t . The c o n s t r u c t i o n of t h i s graph a l s o r e q u i r e s a knowledge of D(0). In f a c t the authors f i n d t h a t "a f i t of equation [10] to the l i m i t e d range of experimental data l e a v e s a f a i r amount of freedom f o r the values of W (1), but c o n s i s t e n c y with experimental o b s e r v a t i o n s was ob t a i n e d " . They f i n d W(1) = 310 cm - 1 f o r the mesitylene t r i p l e t by t h i s procedure. F i g u r e 23 i l l u s t r a t e s the same type of p l o t f o r the p-n i t r o p h e n y l n i t r e n e z e r o f i e l d s p l i t t i n g parameter D. The f u n c t i o n ln{ D (0)-D (T) } i s p l o t t e d a g a i n s t 10 3/T and a f a i r l y l i n e a r r e l a t i o n i s shown to e x i s t . The dashed l i n e corresponds t o H(1) •= 315 cm-*. The value o f D(0) was suggested by the data i n t a b l e 1. I t may be n o t i c e d that i n s t e a d of dropping o f f below a s t r a i g h t l i n e at the high temperature l i m i t as would be expected f o r data a c t u a l l y obeying equation [10] , the experimental p o i n t s tend t o r i s e above the l i n e a r p o r t i o n of the curve, f u r t h e r suggesting t h a t a two s t a t e model i s not qu i t e complete. Since the temperature range which was i n v e s t i g a t e d f o r the n i t r e n e s was higher than f o r the mesitylene case, such behaviour might w e l l go undetected i n t h a t experiment. The experimental o b s e r v a t i o n s suggested that a model 77 23. Graph Of Ln[D(0) - D (T) } Vs. 1 0 V T For P-nitrophenyl Hitrene. 1 i -I o -1 •a - r -3 © \ \ o \ 4 6 7 8 - 9 9 10/T 78 with two s t a t e s may not be complete, and t h a t perhaps other s t a t e s may be c o n t r i b u t i n g to the observed zero f i e l d s p l i t t i n g s . However, when the number of s t a t e s c o n t r i b u t i n g to the observed zero f i e l d s p l i t t i n g s was expanded, i t was not f e a s i b l e to attempt f i t t i n g the observed data. Rather, i t was decided to show t h a t behaviour s i m i l a r to the observed c o u l d be o b t a i n e d with assumed v a l u e s of the parameters i n a t h r e e s t a t e model, a c c o r d i n g l y . F i g u r e 24 d e p i c t s the f u n c t i o n ln{ D (0) -D (T) } versus 10 3/T f o r f o u r d i f f e r e n t f u n c t i o n s : D (T). Four d i f f e r e n t s e t s of parameters l i s t e d below the f i g u r e d e f i n e D(T) under a Boltzmann d i s t r i b u t i o n . The s o l i d l i n e i s v i r t u a l l y s t r a i g h t over the p l o t t e d range; t h i s may be compared with the dashed l i n e which i s more t y p i c a l of a two s t a t e d e s c r i p t i o n s i n c e W(2) was chosen much l a r g e r than kT. The asymptotic approach to a l i n e a r behaviour at low temperature can be e a s i l y seen. The dotted curve i l l u s t r a t e s t h a t i s i s p o s s i b l e to o b t a i n the upward cu r v a t u r e d i s p l a y e d by the., p a r a n i t r o p h e n y l n i t r e n e ; t h i s r e g u i r e s o n l y that the zero f i e l d s p l i t t i n g parameter c h a r a c t e r i s t i c of the h i g h e s t energy s t a t e to d i f f e r more from the ground s t a t e than does the s t a t e i n t e r m e d i a t e i n energy. There seems s u f f i c i e n t f l e x i b i l i t y i n a t h r e e s t a t e d e s c r i p t i o n to account f o r the c h a r a c t e r i s t i c s which were observed i n the n i t r e n e zero f i e l d s p l i t t i n g temperature dependence. Of course, one may not expect to a s c e r t a i n the 79 24. Graph Of Ln{D(0) - D (T) } V s . 10 V T F o r A Three S t a t e D e s c r i p t i o n . - ii 4 5 6 7 8 SO/T ^ Data f o r i l l u a t r u t e d f u n c t i o n s : W(l) W(2) D(0) D ( l ) D(2)' 314. 628. 8 3 £4.533 .1 K '.v M. • «J 314. 628. . V J . 4 83 24.533 19.56 3 314. 2000. 20.483 24.533 19.583 314. 500. 2014Q3 24.533 24.533 80 a c t u a l number of s t a t e s which c o n t r i b u t e to the observed zero f i e l d s p l i t t i n g s ; t h e r e i s an i n h e r e n t l a c k of r e s o l u t i o n i n the average property D (T) which cannot be overcome. That three s t a t e s a t l e a s t are i n v o l v e d seems to be a reasonable c o n c l u s i o n . T h i s i n based on the r e s u l t s of a two s t a t e model, and the p o s s i b l e behaviours of the three s t a t e model. The preceding treatment has e s t a b l i s h e d a model a s c r i b i n g the observed temperature dependence of the n i t r e n e z e r o f i e l d s p l i t t i n g s to a Boltzmann average of s e v e r a l s t a t e s of d i f f e r i n g zero f i e l d s p l i t t i n g s . EPR r e s u l t s suggest t h a t the lowest two s t a t e s are separated e n e r g e t i c a l l y by approximately 300 cm - 1. I t becomes reasonable t o g u e s t i o n the nature of these s t a t e s , that i s to s u b s t a n t i a t e t h e i r e x i s t e n c e . T h i s l e a d s d i r e c t l y to examining r e s u l t s from other s p e c t r o s c o p i c methods of a more s u i t a b l e energy s c a l e . I t i s d e s i r a b l e t o examine the known e n e r g e t i c s of v i b r a t i o n a l processes i n s o l i d s . For t h i s reason one examined the l i t e r a t u r e on v i b r a t i o n a l a n a l yses of benzenoid hydrocarbons. Of p a r t i c u l a r i n t e r e s t w i l l be normal modes of f r e q u e n c i e s 1000 cm - 1 or l e s s . T h i s s u b j e c t i s reviewed comprehensively i n Varsanyi's book (64). C a r e f u l i n s p e c t i o n of the r e s u l t s presented t h e r e i n r e v e a l s that w i t h i n the 200 cm - 1 to 400 cm - 1 i n t e r v a l , normal modes e x i s t f o r a l l three types of motion i n 81 the s u b s t i t u t e d benzene d e r i v a t i v e s . These three types of motion are normal modes whose v i b r a t i o n a l motion may be roughly c h a r a c t e r i z e d as s t r e t c h i n g , i n plane bending, or out of plane bending. A d i s t i n c t i o n as to which type of v i b r a t i o n a l motion may be r e s p o n s i b l e f o r the temperature e f f e c t s being d i s c u s s e d t h e r e f o r e cannot be p l a u s i b l y based upon frequency c o n s i d e r a t i o n s . I t i s r e a l i z e d that these v i b r a t i o n s are c o n s i d e r e d n u c l e a r motions wi t h i n a f i x e d e l e c t r o n i c p o t e n t i a l , and that the zero f i e l d s p l i t t i n g i s not a f u n c t i o n of n u c l e a r c o o r d i n a t e s . As such i t cannot be i n f l u e n c e d by such motion u n l e s s the e l e c t r o n i c motion i s coupled to the n u c l e a r motion r a t h e r than being independent of i t . Nevertheless, the energy range of t h i s s o r t of v i b r a t i o n i s a reasonable one to c o n s i d e r . T h i s i s not the only form of motion i n c r y s t a l l i n e s o l i d s which may s t o r e energy. In a d d i t i o n to the energy a s s o c i a t e d with the s t a t i c packing of molecules i n a l a t t i c e , s m a l l deformations from e q u i l i b r i u m p o s i t i o n s c o n s t i t u t e an energy storage mechanism. The d i s c r e t e f r e q u e n c i e s a s s o c i a t e d w i t h t h i s type of motion are c a l l e d l a t t i c e f r e q u e n c i e s . The p o t e n t i a l i n t h i s case would be i n t e r m o l e c u l a r . Frequency ranges f o r a b s o r p t i o n of energy by these modes seem to f a l l i n the r e g i o n below 200 cm-1, f o r such aromatic hydrocarbons (65). T h i s type of motion should, not be immediately d i s c a r d e d as a c a n d i d a t e r e s p o n s i b l e f o r the temperature 82 dependent e f f e c t . However, the f r e q u e n c i e s which seem to be c h a r a c t e r i s t i c of t h i s motion do tend to be s m a l l e r than the range suggested by the EPR r e s u l t s . In order to a t t r i b u t e changes i n the zero f i e l d s p l i t t i n g t o e f f e c t s of motion of t h i s type, i t would be necessary t o provide a d e s c r i p t i o n of e x i s t i n g i n t e r m o l e c u l a r f o r c e s which might modulate the geometry of the t r i p l e t molecule. Such an e n t e r p r i s e may be expected to be nebulous at best, by v i r t u e of the complexity of the s i t u a t i o n . In order to i n v e s t i g a t e f u r t h e r the s t a t e s r e s p o n s i b l e f o r the v a r i a t i o n i n the n i t r e n e zero f i e l d s p l i t t i n g , i t i s necessary to c o n s i d e r r e s u l t s from o p t i c a l spectroscopy. The e l e c t r o n i c a b s o r p t i o n s p e c t r a of n i t r e n e s p e c i e s trapped i n o r g a n i c matrices has been i n v e s t i g a t e d (66). The s p e c t r a of the n i t r e n e s are very s i m i l a r to aromatic r a d i c a l s . Two bands are r e p o r t e d , the higher energy band having the s t r o n g e r a b s o r p t i o n . Although some v i b r a t i o n a l s t r u c t u r e i s apparent i n the s p e c t r a of these s p e c i e s , i t i s b a r e l y r e s o l v e d and g e n e r a l l y of l i t t l e value i n seeking to confirm s t a t e s s p l i t by 300 cm - 1. G. P o r t e r and B. Ward have observed the a b s o r p t i o n spectrum of phenyl n i t r e n e generated during f l a s h p h o t o l y s i s of ortho s u b s t i t u t e d a n i l i n e s . They found a strong a b s o r p t i o n a t 368nm. t o be c h a r a c t e r i s t i c of the 0-0 band of an allowed e l e c t r o n i c t r a n s i t i o n , and v i b r a t i o n a l s t r u c t u r e i n the v i c i n i t y was r e s o l v e d . Presuming a c o r r e c t 83 i d e n t i f i c a t i o n of t h i s band o r i g i n , evidence of ground e l e c t r o n i c s t a t e v i b r a t i o n a l s t r u c t u r e (lower wavenumber) extends o n l y 170 cm-* lower. No a n a l y s i s of the v i b r a t i o n a l s t r u c t u r e was made however. That the a b s o r p t i o n spectrum of phenyl n i t r e n e trapped i n an o r g a n i c g l a s s shows a band extending to 400 nm. may. be a r e s u l t of an o v e r l a p p i n g a b s o r p t i o n of the a n i l i n o r a d i c a l . C o n f i r m a t i o n of s t a t e s s p l i t by a few hundred cm - 1 by published o p t i c a l a b s o r p t i o n measurements on n i t r e n e s i s t h e r e f o r e not a v a i l a b l e . While no p o s i t i v e evidence i s forthcoming, n e i t h e r i s s u b s t a n t i a l c o n t r a r y e v i d e n c e , and i t should be r e c a l l e d t h a t phenyl n i t r e n e c o u l d not be p r o p e r l y i n v e s t i g a t e d by EPR f o r want of s t a b i l i t y . Evidence has been obtained f o r an e x c i t e d v i b r a t i o n a l s t a t e 325 cm - 1 from the o r i g i n i n an u l t r a v i o l e t a b s o r p t i o n spectrum of diphenylmethylene (67). T h i s r e s u l t was obtained by o b s e r v i n g the change i n absorbance due to diphenylmethylene produced by u l t r a v i o l e t i r r a d i a t i o n of s i n g l e c r y s t a l s of d i p h e n y l e t h y l e n e doped with diphenyldiazomethane. While l i t t l e i s e x p l i c i t e l y i m p l i e d by t h i s r e s u l t concerning v i b r a t i o n a l s t a t e s i n the ground e l e c t r o n i c s t a t e , i t does show th a t v i b r a t i o n s of a p p r o p r i a t e frequency e x i s t i n an e x c i t e d s t a t e of a c l o s e l y r e l a t e d molecule. 84 5.2 Com^arison With Other S t u d i e s One may compare t h i s temperature e f f e c t with s t u d i e s by J . H. Van der Waals e t . a l . (68) . These s t u d i e s focus t h e i r a t t e n t i o n on the lower t r i p l e t s t a t e s of benzene, and a p a r t i c u l a r carbon-carbon s t r e t c h i n g mode. The reasons f o r t h i s a t t e n t i o n were experimental evidence (both o p t i c a l and magnetic resonant) t h a t the phosphorescent benzene molecules possess lower symmetry than D3h. Although " a s o l u t i o n to the complete problem i s not f e a s i b l e " , they study the c o u p l i n g of the p i e l e c t r o n s with t h i s v i b r a t i o n . The f o r m u l a t i o n i s complex, and i t would be i n a p p r o p r i a t e to reproduce i t here. However, these authors do conclude that the lowest v i b r o n i c s t a t e s i n t h i s problem are probably separated by only a few hundred cm-*. They conclude t h a t the aforementioned r e s u l t s f o r the temperature dependence of the zero f i e l d s p l i t t i n g i n mesi t y l e n e , i n d i c a t i n g a s e p a r a t i o n of approximately 300 cm-*, are i n q u a l i t a t i v e accord with t h e i r t h e o r e t i c a l model. T h r e e f o l d t o f o u r f o l d changes are known to e x i s t i n the phosphorescent l i f e t i m e of the benzene t r i p l e t i n a r i g i d g l a s s on c o o l i n g from 80 K to 4.2 K. I t was suggested t h a t the temperature dependence of t h i s q u a n t i t y may be caused by v i b r o n i c l e v e l s . o f a p p r e c i a b l y d i f f e r e n t decay r a t e s , and i f t h i s i s the case, s i m i l a r o b s e r v a t i o n s on the mesitylene system would be d e s i r e a b l e (69). Such a v a r i a t i o n i n t r i p l e t 85 s t a t e l i f e t i m e has been r e p o r t e d by Thomson f o r coronene and t r i p h e n y l e n e i n a p l a s t i c host and a s i m i l a r e x p l a n a t i o n was deduced (70). A corresponding measurement f o r the n i t r e n e s of course i s not p o s s i b l e . In a subsequent study, a l s o concerned with v i b r o n i c i n t e r a c t i o n s i n the lower e l e c t r o n i c s t a t e s of benzene, J. H. Van der Baals e t . a l . i n v e s t i g a t e d the e f f e c t of an a n i s o t r o p i c c r y s t a l f i e l d on the lowest t r i p l e t s t a t e (71). Only q u a l i t a t i v e agreement with experimental r e s u l t s was o b t a i n e d . They do conclude that a n i s o t r o p y i n a c r y s t a l f i e l d may be r e s p o n s i b l e f o r the non h e x a g o n a l i t y of the e l e c t r o n i c s t a t e . T h i s would seem a mechanism c o n s i s t e n t with the o b s e r v a t i o n t h a t the d i s t r i b u t i o n of zero f i e l d s p l i t t i n g s of diphenylmethylene t r i p l e t s i n v a r i o u s hosts i s narrowest when the host molecular geometry most c l o s e l y approximates that of the guest (72). I t a l s o seems to.be i n good accord with the o b s e r v a t i o n that the EPR l i n e w i d t h s are narrower f o r n i t r e n e s trapped i n a p o l y c r y s t a l l i n e host as opposed to a g l a s s y matrix. The l a t t e r would be expected to possess a l e s s ordered s t r u c t u r e i n the immediate v i c i n i t y of the t r i p l e t molecule, and hence d i s p l a y a l a r g e r d i s t r i b u t i o n of i n t e r m o l e c u l a r i n t e r a c t i o n s . i n S ince a s a t i s f y i n g account s u b s t i t u t e d aromatic n i t r e n e s of the zero f i e l d was obtained from s p l i t t i n g s a model i n 86 which the t r i p l e t e l e c t r o n s are of p i and sigma symmetry, one may make use of t h i s d i v i s i o n . I t may be f r u i t f u l to compare temperature e f f e c t s observed i n EPR of v a r i o u s i o n s and r a d i c a l s p e c i e s of p i symmetry. Experimental evidence of temperature e f f e c t s i n these molecules c o n s i s t s of a l t e r a t i o n s i n s u b s t i t u e n t h y p e r f i n e s p l i t t i n g constants, and a l s o s m a l l a l t e r a t i o n s i n G values. S u b s t i t u t e d benzene anions are a system which has r e c e i v e d much a t t e n t i o n (73). The EPR s p e c t r a o f monoalkyl and d i a l k y l benzene anions suggested t h a t the molecular o r b i t a l s which are degenerate i n the case of the u n s u b s t i t u t e d benzene anion were s l i g h t l y s p l i t by the i n t r o d u c t i o n of the a l k y l s u b s t i t u e n t s (74). Estimates of t h i s s p l i t t i n g are of the order of a few hundred cm - 1 (75). The s t a t e s are v i b r o n i c i n nature. Out of plane v i b r a t i o n s have been suggested as being r e s p o n s i b l e f o r temperature dependent proton h y p e r f i n e c o u p l i n g s of some aromatic r a d i c a l s (76). These temperature e f f e c t s i n v o l v e v i b r a t i o n s i n the freguency range suggested i n the present study. 8 7 5.3 Conclusions The temperature variation of the zero f i e l d s p l i t t i n g i n aromatic nitrenes has been investigated and interpreted i n terms of thermal averaging of several states of d i f f e r i n g zero f i e l d s p l i t t i n g . Similar behaviour has been noted in other t r i p l e t molecules. The nature of the vibrations which are responsible for t h i s e f f e c t could not be elucidated by t h i s study. 88 CHAPTER SIX Data refinement 6_a.l Introductory. Remarks The g e n e r a l i n t e n t of t h i s computational procedure i s to p r o v i d e a means f o r the a n a l y s i s of angular dependent EPR data. While not a l l EPR s t u d i e s f a l l i n t o t h i s category i t was f e l t t h a t a procedure capable of h a n d l i n g t h i s type of data should a l s o be a b l e to handle cases of l e s s complexity. At the p l a n n i n g stage of t h i s work previous e f f o r t i n t h i s area had been s u c c e s s f u l , and t h e r e were no problems with d e v e l o p i n g the theory behind the procedure ( 7 7 ) . In a d d i t i o n , at the time t h i s work commenced one had a v a i l a b l e a p r a c t i c a l computational t o o l of c o n s i d e r a b l e g e n e r a l i t y which co u l d simulate a paramagnetic c e n t r e . T h i s computer program i s c a l l e d " F i e l d s " and i t s major i n t e n t was to s o l v e the resonant c o n d i t i o n given the parametric d e s c r i p t i o n of the magnetic c e n t r e . That i s to say, i t would c a l c u l a t e the magnitude of magnetic f i e l d s t r e n g t h necessary to produce a resonance between s p e c i f i e d energy l e v e l s under the known experimental c o n d i t i o n s . To analyse a problem, one could 89 compare c a l c u l a t e d magnetic f i e l d s t r e n g t h s with those observed and a l t e r the d e s c r i p t i o n of the magnetic cent r e u n t i l s u f f i c i e n t agreement i s obtained. Thus, the i n t e n t of the program "L.S.F. " was t o improve t h i s l a t t e r process which the e x p e r i m e n t a l i s t f a c e s a f t e r c o l l e c t i n g r e l e v a n t data. I t should.be apparent t h a t while the o b j e c t i v e s of these two programs are d i f f e r e n t , together they should provide a potent method of approaching the problem of a n a l y s i s and d e s c r i p t i o n of EPR data. The m o t i v a t i o n f o r t h i s work l a y i n the f a c t that l i g h t c ould be shed on the a n a l y s i s a spects of the problem by t h i s method, and t h a t the emphasis i n the development of the program should l i e i n s i m i l a r i t i e s between the a l r e a d y e x i s t i n g program F i e l d s . I t can be seen t h a t the o b j e c t i v e s of the two programs are q u i t e complementary. At a l a t e r stage i n the development of t h i s program i t was r e a l i z e d t h a t the extension of the program to treatment of s i n g l e c r y s t a l Endor data should not be m a t e r i a l l y d i f f e r e n t and t h a t such a treatment might be made. The approach to the s o l u t i o n of the a n a l y t i c a l problem which i s u t i l i z e d here i s not new. B a s i c a l l y i t i s an i t e r a t i v e method. I t r e g u i r e s one to guess or p r o v i d e an i n i t i a l d e s c r i p t i o n of the data i n terms of s p i n Hamiltonian parameters, and the r e l e v a n t e x p e r i m e n t a l l y determined measurements,' With t h i s i n f o r m a t i o n the program then c a l c u l a t e s expected f e a t u r e s and compares them with those 90 which are observed, As w e l l , i t estimates the dependence of t h e s e c a l c u l a t e d f e a t u r e s on each of the parameters which may be v a r i e d i n the s p i n Hamiltonian used t o d e s c r i b e the experiment. Armed with t h i s i n f o r m a t i o n , the program should be able to improve the agreement between the c a l c u l a t e d and observed f e a t u r e s , by s u i t a b l y a l t e r i n g the i n i t i a l or p r e v a i l i n g parameters, with t h i s o u t l i n e i n mind, one may examine the t h e o r e t i c a l b a s i s of the method. 6^2 T h e o r e t i c a l B a s i s T h i s theory does not o r i g i n a t e here and i s s e t down i n i n t e r e s t of completeness (78). The energy operator i s c o n s i d e r e d i n the f o l l o w i n g form: ]-[ = P(3) M (j) + c o n s t a n t s . . . . . . . . . . . . [ 1 1 ] where the ] - [ ( j ) c o n s i s t of s p i n o p e r a t o r s , magnetic f i e l d s t r e n g t h components, and a s s o c i a t e d c o n s t a n t s , and the p (j) are the parametric d e s c r i p t i o n to be sought. The o perator P(j) ]-[ (j) may r e p r e s e n t only those p a r t s of the s p i n Hamiltonian whose parameters are t o be r e f i n e d or v a r i e d . One has many o b s e r v a t i o n s of the system ( t r a n s i t i o n s ) indexed by 91 ]-[ (m) = p(j) ]-[ (j) 1<j<J ..........,........[12] An assignment of the m»th t r a n s i t i o n to s t a t e s r and g i s made, and the f o l l o w i n g q u a n t i t i e s are c a l c u l a t e d : ]-[ (m,r) |r> = E(m,r) |r> [13] ]-[ (m,q) |q> = E(m,q) |q> ...[14] A c a l c u l a t e d resonant frequency Fcalc(m) d e f i n e d n o n - n e g a t i v e l y by Fcalc(m) = (E(m,r) - E(m,q))/h ..............,..[15] i s then to be compared with the e x p e r i m e n t a l l y observed t r a n s i t i o n frequency Fexp(m) y i e l d i n g an e r r o r measure e (m): e (m) = Fexp(m) - F c a l c (m) .................,..,,[16] Suppose t h a t a l l t h i s i n f o r m a t i o n has been e v a l u a t e d f o r a l l M t r a n s i t i o n s with the p r e v a i l i n g parametric d e s c r i p t i o n {P* (j) }. One proceeds by the method of l e a s t sguares with S = W(m)e(m)2 1<m<H .....,,.[17] being the e r r o r measure to be minimised by s o l v i n g the set of normal equations: 92 { W (m) e (m) d F c a l c (m)/dp (j) = 0 1<m<M } j=1,J<M [18] i n v o l v i n g H t r a n s i t i o n s and J parameters. Denoting d F c a l c (m)/dp (j) by K (m, j) and K transpose by K +, the normal equations become: K+We = 0 .................................. [19] One must determine s u i t a b l e increments t o the p r e v a i l i n g parameter s e t {p'(j) } such t h a t the d e s c r i p t i o n i s improved. To accomplish t h i s one d e f i n e s an increment to the j ' t h parameter: P(j) = P , + MJ) - PMJ) [20] I t would be s u i t a b l e to choose the p'+* such that the P(j) s a t i s f y : K ( m r j ) P ( j ) = e(m) 1<m<M .......[21] i . e . KP = e ..........[22] and e q u i v a l e n t l y : K+WKP = K+We ,....[23] The concept i s to i t e r a t i v e l y solve t h i s l a s t equation. As the parameter increments tend t o zero, the normal equations on the r i g h t hand s i d e of [ 2 3 ] must a l s o tend to s o l u t i o n . 93 The s e t of increments i n parameters at any p a r t i c u l a r step i n the i t e r a t i v e process i s g i v e n by: P = [K+WK]-»K+We [24] The frequency d e r i v a t i v e s are c a l c u l a t e d from changes i n the energy l e v e l s , with r e s p e c t t o v a r i a t i o n of the parameters as f o l l o w s : K(m,j) = dE(m,r)/dp (j) - dE (m, q)/dp (j) [25] D e r i v a t i v e s of the ei g e n v a l u e s are c a l c u l a t e d by: dE (m,r)/dp (j) = <r| ]-[ (j) | r > . . . . . . .[26] The e i g e n v e c t o r s of ]-[(m) are necessary to c a l c u l a t e the above matrix element. The form o f the program which was w r i t t e n changed somewhat through i t s development. In the e a r l y s t a g e s , a t t e n t i o n was centred upon c e r t a i n terms i n the s p i n Hamiltonian, namely the e l e c t r o n i c Zeeman and h y p e r f i n e c o u p l i n g terms. Later one wanted t o allow the program t c d e a l with e l e c t r o n spin^-spin c o u p l i n g and quadrupole c o u p l i n g terms. T h i s o r d e r i n g was p a r t i a l l y a consequence of the frequency of occurence of these terms i n work at t h i s l a b o r a t o r y and p a r t i a l l y due t o the magnitude of such terms. The apparatus and m a t e r i a l s a v a i l a b l e a t the commencement of t h i s work were b a s i c a l l y c o n t a i n e d i n the program F i e l d s . In 94 order to make use o f t h i s i n f o r m a t i o n an understanding of the c o n s t r u c t i o n and o r d e r i n g of t h i s program had to be achieved. The g e n e r a l i t y o f approach i m p l i c i t i n t h i s program meant t h a t t h i s task was not minimal', and i n order t h a t the f i t t i n g program would r e t a i n a g e n e r a l u t i l i t y one c o u l d not d i s c a r d the work accomplished i n the program F i e l d s , P a r t of the e f f o r t i n v o l v e d i n s e t t i n g up the f i t t i n g program l a y i n r e f o r m u l a t i n g s e c t i o n s of the program F i e l d s to conform with the l i n e a r i t y of the s p i n Hamiltonian, The f i t t i n g program must r e f l e c t t h i s l i n e a r i t y i n order to r e t a i n an ordered c o n s t r u c t i o n and hence f l e x i b i l i t y of use. 6_. 3 Formulation The work i n v o l v e d f a l l s b r o a d l y i n t o three c a t e g o r i e s . The f i r s t requirement was f o r m u l a t i o n of the energy o p e r a t o r s ]-[ {J) r which are termed s k e l e t a l matrices i n the program due to the r o l e they p l a y i n c o n s t r u c t i n g the s p i n Hamiltonian. A s s o c i a t e d with t h i s are the simple a d d i t i v e c o n s t r u c t i o n of the s p i n Hamiltonian and numerical d i a g o n a l i z a t i o n of the same i n order t o provide e i g e n v a l u e s and e i g e n v e c t o r s . The frequency d e r i v a t i v e s are a l s o c a l c u l a t e d at t h i s p o i n t and r e t a i n e d . The second part of the program c o n s i s t s of 95 implementing the c a l c u l a t i o n of a s e t of s u i t a b l e parameter increments. T h i s i n v o l v e s simply d e f i n i n g K and c a l c u l a t i o n of [K+WK]- 1, and s u i t a b l e i n v e r s i o n r o u t i n e s were a v a i l a b l e f o r t h i s purpose. A f t e r the increments are obtained, the parameters ( p * ( j ) } are s u i t a b l y r e d e f i n e d and the program r e v e r t s t o the beginning of the f i r s t s e c t i o n to s t a r t an i t e r a t i v e c y c l e . I t was arranged t h a t t h i s c y c l e would be terminated based on any of s e v e r a l reasonable c r i t e r i a . At the stage of the t h i r d s e c t i o n of the program, the i t e r a t i v e c y c l e has been terminated, consequently the parameters have been determined. The purpose of the t h i r d s e c t i o n i s to p r o v i d e a s t a t i s t i c a l d e s c r i p t i o n of the parameters i n terms of remaining e r r o r measure. Such a d e s c r i p t i o n i s a v a i l a b l e through the r e s u l t : (79 sigma (j) 2 = m (j) (e«e)/(M-J) det[K+K ] .[27] where: sigma (j) i s a standard d e v i a t i o n i n the j ' t h parameter; (e»e) i s the sum of the squares of frequency e r r o r s remaining a f t e r s o l u t i o n of the normal equations; M-J i s the number of degrees of freedom i n the system; m (j) i s the minor of the j ' t h d i a g o n a l element of the a r r a y K +K. T h i s was e a s i l y implemented with a v a i l a b l e computing programs. The success of the method l i e s i n e f f e c t i n g the i t e r a t i v e s o l u t i o n to the normal equations. The ease of i n v e r s i o n of the matrix of normal equations i s important. One 96 problem which may a r i s e i s the d i f f i c u l t y of two or more parameters a c t i n g i n an e q u i v a l e n t manner (symmetry) or l i n e a r dependence of the parameters. In such a s i t u a t i o n , the r e s u l t should be t h a t the matrix K +K w i l l have a determinant ze r o and t h e r e f o r e may not be i n v e r t e d . Recovery from t h i s s i t u a t i o n can be had by removing the l i n e a r dependence from the f o r m u l a t i o n . R ecasting the e f f e c t s of dependent parameters upon a s m a l l e r s e t of independent ones w i l l s u i t a b l y reduce the rank of the problem. An example where t h i s type of problem i s avoided i n the i n i t i a l f o r m u l a t i o n i s the s p i n — s p i n i n t e r a c t i o n i n the t r i p l e t s t a t e . Here the e i g e n v a l u e s are l i n e a r l y dependent and the f o r m u l a t i o n i s i n terms of f i v e independent parameters i n s t e a d of s i x . In e a r l y work on the program some checks upon i n v e r s i o n accuracy were arranged and p r o v i s i o n s f o r two i n v e r s i o n c y c l e s were made, i n order t h a t the i n v e r s i o n might be a s s i s t e d i n a near i l l — c o n d i t i o n e d problem. The motive was t h a t a near symmetrical s i t u a t i o n c o u l d be handled be t h i s means. Faced wi t h data c h a r a c t e r i s t i c of a symmetrical s i t u a t i o n however, the program would not succeed. Subsequently, the a d d i t i o n a l r e f o r m u l a t i o n necessary when l i n e a r dependence i s encountered was undertaken by Dr. J . Hebden. The t r i v i a l cause of non i n v e r t i b i l i t y i s l a c k of dependence of the assigned t r a n s i t i o n s upon a parameter. Removal of such a parameter from the f o r m u l a t i o n i s necessary. 97 6.2.4 P r o g e r t i e s The purpose of t h i s s e c t i o n i s to i l l u m i n a t e by example some of the p r o p e r t i e s of the program p a r t i c u l a r l y the convergence of the method. I t i s apparent from the p r e v i o u s l y d e s c r i b e d theory t h a t t h i s method of s o l u t i o n should apply to e i t h e r a determined problem or an over determined problem, and convergence behaviour i n each case i s i l l u s t r a t e d . A c c o r d i n g l y , F i g u r e 25 d e p i c t s the v a r i a t i o n of the sum of the squares of r e s i d u a l s during the i t e r a t i o n to a s o l u t i o n . The data used i n t h i s c a l c u l a t i o n were experimental, measured from an s = 1/2 1 = 3 / 2 system. A l o g a r i t h m i c p l o t was chosen due to the s i z e of the data. I t can be seen that the method worked w e l l i n the determined problem,in two i t e r a t i o n s r e ducing the r e s i d u a l s by. s e v e r a l o rders c f magnitude. I t e r a t i o n was terminated a f t e r two c y c l e s . The convergence behaviour of the overdetermined problem i s a l s o shown, i n a neighbourhood of the extremum.,A d i f f e r e n t t e s t i s d e p i c t e d i n F i g u r e 26 . In t h i s case the resonance data were s u p p l i e d by the program F i e l d s , t h e r e f o r e the d e s i r e d parameters were al r e a d y known. T h i s type of problem s u p p l i e s a s t r i n g e n t t e s t of the method. The f i t t i n g program was given the necessary .data and the c o r r e c t g value and asked to determine a h y p e r f i n e t e n s o r . , F i g u r e 26 d e p i c t s the convergent behaviour as a p l o t of e r r o r measure a g a i n s t i t e r a t i o n c y c l e . The 9 8 25. Convergence B e h a v i o u r . (£0 Determined 0 - 6 + v -If - l Overcetern't'macl ,/4 j . -5-: -6 © • / 3 - f -•/4 •*/6 \ •7 U G r x u c w tmsnoor—*.» \ te ro j ; ion numw or 9 9 2 6 . C o n v e r g e n c e B e h a v i o u r . \ G \ \ <•> \ \ \ \ \ 0 V \ \ \ 100 r e s u l t of the convergence demonstrated i n the above f i g u r e was the d e s i r e d result,namely the h y p e r f i n e tensor used by the program F i e l d s t o generate "experimental data" was c o r r e c t l y deduced. T h i s property was e x p l o i t e d f o r t e s t i n g the f i t t i n g program throughout i t s development. The preceding examples i l l u s t r a t e the convergent behaviour of the method when a l l of the data are s u p p l i e d i n a c o n s i s t e n t manner. P a r t of t h i s data i s an assignment of the observed t r a n s i t i o n s to energy l e v e l s of the system. one c h a r a c t e r i s t i c which was o f t e n observed on f a u l t y assignments was that the convergence behaviour would be terminated; t h a t i s , the program would deduce parameters which r e s u l t i n i n c r e a s e d e r r o r measure. although t h i s behaviour i s of dubious value i t does serve to i n d i c a t e an i n c o n s i s t e n c y . Improving the r a t e of convergence of the method (hence an expected r e d u c t i o n i n computing time) was the impetus f o r an extended method. The nature of t h i s extension was i n c l u s i o n of higher order d e r i v a t i v e s i n t o the formulae, more p r e c i s e l y , the frequency changes necessary to improve the d e s c r i p t i o n were expressed i n a T a y l o r s e r i e s up to t h i r d o rder i n s t e a d of f i r s t order i n parameter increments. In the e a r l y stages of development of the program, the extended method was programmed and i n v e s t i g a t e d . The r e s u l t of t h i s was t h a t minimal i f any improvement i n terms of computational time was obtained. E v i d e n t l y the e x t r a time r e q u i r e d to 1 0 1 support the e x t r a c a l c u l a t i o n s of the extended method was s u b s t a n t i a l l y b a l a n c i n g a f a s t e r convergence. The improvement i n the r a t e of convergence was c e r t a i n l y not s t a r t l i n g . One concluded t h a t the higher order terms d i d net play a l a r g e p a r t i n d e s c r i b i n g the e r r o r s u r f a c e . For these reasons, as w e l l as r e l a t i v e s i m p l i c i t y of programming, i t was decided to implement the unextended ( l i n e a r l e a s t squares) method. In making t h i s d e c i s i o n one must agree to forego i n f o r m a t i o n c o n t a i n e d i n the second and higher order terms of the extended treatment. In p r a c t i c e , t h i s l o s s i s not of extreme concern. Although the s i g n of second order p a r t i a l d e r i v a t i v e s w i l l i l l u m i n a t e the type of extremum deduced, one may take the view that any improvement i n the e r r o r measure i s a d e s i r e a b l e s t e p . The question of l o c a l minima cannot be examined by the unextended treatment; however, r e s u l t s which are g r o s s l y i n c o n s i s t e n t with expected e r r o r measure should be e a s i l y spotted. B a s i c a l l y , the purpose of the method i s not to i n v e s t i g a t e the e r r o r s u r f a c e s but r a t h e r to achieve a minimal e r r o r measure. 102 6 t5 Some I l l u s t r a t i v e Examples The u t i l i t y of t h i s program should be i l l u s t r a t e d with some examples. These have been s e l e c t e d to i l l u s t r a t e s p e c i f i c areas. The f i r s t example w i l l i l l u s t r a t e the c o m p a t i b i l i t y of t h i s program with the program F i e l d s , and a l s o the manner i n which the o p e r a t i o n of the program can be checked. The s p i n Hamiltonian i n t h i s example i s c h a r a c t e r i s t i c of the t r i p l e t s t a t e , with an i s o t r o p i c e l e c t r o n i c Zeeman term. In t h i s example one wanted to check t h a t the l e a s t squares f i t t i n g program could s u c c e s s f u l l y t r e a t data i n the s i t u a t i o n where measurements are not known to be taken i n p r i n c i p a l planes of the zero f i e l d s p l i t t i n g t e n s o r . , The motive f o r t h i s example i s that s i n g l e c r y s t a l measurements of a n i t r e n e were being taken concomitantly. Necessary paramagnetic resonance data were a v a i l a b l e from the program F i e l d s . Zero f i e l d s p l i t t i n g parameters were assign e d values t y p i c a l of aromatic n i t r e n e s , and an X band freguency assumed. C h a r a c t e r i s t i c a n i s o t r o p y of t h i s s p e c i e s i s e v i dent i n T a b l e 2 where resonant magnetic f i e l d s t r e n g t h as a f u n c t i o n of o r i e n t a t i o n i n t h r e e orthogonal planes i s t a b u l a t e d . The resonances span a l a r g e p o r t i o n of a v a i l a b l e magnetic f i e l d s t r e n g t h . The o r i e n t a t i o n i s given i n a p o l a r angle d e s c r i p t i o n r e l a t i v e to the p r i n c i p a l axes of the zero 103 Table 2. T y p i c a l N i t r e n e Magnetic Data. plane 1 plane 2 o r i e n t a t i o n (degrees) t h e t a 84.81 84.21 84.00 84.21 84.81 85.76 87.00 88.45 90.00 91.55 93.00 94.24 phi 239.86 254.92 270.00 285.08 300.14 315.16 330. 14 345.08 0.00 14.92 29.86 44.84 t r a n s i t i o n (gauss) 6809.57 6988.38 7054.79 6988.38 6809.57 6571.86 6342. 23 6179.53 6121.16 6179.53 6342.23 6571.86 o r i e n t a t i o n (degrees) theta 168.45 153.93 139.07 124.14 109.20 94.24 79.28 64.33 49. 39 34.48 19.69 6.00 phi 23. 89 36.45 40. 25 42.28 43.68 44.84 45.96 47.20 48.80 51.35 57. 11 90.00 t r a n s i t i o n (gauss) 1703.68 1862.17 2224.50 3005.02 4705.21 6571. 86 5975.83 3851. 26 2602.30 2039.56 1777.00 1678. 56 plane 3 o r i e n t a t i o n (degrees) theta phi 79. 28 64.33 49. 39 34.38 19. 69 6.00 11. 55 26.07 40.93 55.86 70.80 85.76 134.04 132.80 131.20 128.65 122.89 90.00 336. 11 323.55 319.75 317.72 316.32 315.16 t r a n s i t i o n (gauss) 5975.83 3851.26 2602.30 2039.56 1777.00 1678.56 1703.68 1862.17 2224.50 3005.01 4705.20 6571.86 104 f i e l d s p l i t t i n g . F i g u r e 27 d e p i c t s the s i t u a t i o n : the p e r p e n d i c u l a r planes i n which the resonances were c a l c u l a t e d , and the r e l a t i o n s h i p of the a x i s system d e f i n e d by these planes and the p r i n c i p a l axes of the zero f i e l d s p l i t t i n g . The resonant c o n d i t i o n was c a l c u l a t e d at f i f t e e n degree increments i n these planes, corresponding to t h i r t y t h r e e d i s t i n c t o r i e n t a t i o n s . These data were then input to the f i t t i n g program i n two ways. In the f i r s t case, knowledge of the o r i e n t a t i o n s of the resonances r e l a t i v e to the zero f i e l d s p l i t t i n g p r i n c i p a l axes was used; t h a t i s to say the p o l a r angle d e s c r i p t i o n s u p p l i e d by the program f i e l d s was used as i n p u t data. From an a r b i t r a r y i n i t i a l guess, convergence towards the known r e s u l t was e f f e c t e d . T h i s may be f o l l o w e d i n Table 3 , where the convergence of the components i s i l l u s t r a t e d . The r e s u l t a f t e r four i t e r a t i o n s can be f a v o u r a b l y compared with the known value of the zero f i e l d s p l i t t i n g which was: D = 0.94 cm - 1 and E = 0.025 cm - 1. The f o u r f i g u r e r e s u l t i s due to roundoff i n the p r e c i s i o n of angular data. The c o n v i n c i n g agreement demonstrates that the program i s f u n c t i o n i n g c o r r e c t l y . The r e s u l t tends to d i a g o n a l form because use was made of the knowledge of r e l a t i v e o r i e n t a t i o n s of the magnetic f i e l d and the zero f i e l d s p l i t t i n g . Of course t h i s i n f o r m a t i o n i s not n e c e s s a r i l y known i n experimental measurements and may be sought. 105 27. A x i s System 106 T a b l e 3. Zero F i e l d S p l i t t i n g Convergence * D E Dxy Dxz Dyz 0 1.500000 0. 050000 0.000100 0.000200 0.000300 1 0.653717 0. 025175 -0.000018 -0.000106 -0.001943 2 0.897376 0. 024971 -0.000001 -0.000023 -0.000147 3 0.938932 0. 024988 -0.000000 -0.000000 -0.000039 4 0.939959 0. 024989 -0.000000 -0.000000 -0.000035 5 0.939959 0. 024989 -0.000000 -0.000000 -0.000035 where D and E s a t i s f y : Dxx = -D/3 + E Dyy = -D/3 - E Dzz = 2D/3 107 The second part of t h i s example corresponds to the l a t t e r s i t u a t i o n . Corresponding to lack of knowledge of magnet o r i e n t a t i o n s r e l a t i v e t o the zero f i e l d s p l i t t i n g , one s e l e c t e d the a x i s system d e f i n e d by the i n t e r s e c t i o n s of the planes which c o n t a i n the c a l c u l a t e d resonances. These axes are c a l l e d A B and C i n F i g u r e 27. Angular data r e l a t i v e to t h i s system f o r the 33 o r i e n t a t i o n s were f i t t e d . The r e s u l t i n g tensor was d i a g o n a l i z e d . Relevant data to t h i s example are l i s t e d i n Table 4 . Components are measured with r e s p e c t to the ABC system, and a p o l a r angle d e s c r i p t i o n of these d i r e c t i o n s i s given. An i n s p e c t i o n of these d i r e c t i o n s shows that they l i e e s s e n t i a l l y p a r a l l e l or a n t i p a r a l l e l to the known p r i n c i p a l d i r e c t i o n s o f . t h e zero f i e l d s p l i t t i n g used to generate t h i s example. By comparing the p o l a r angles d e s c r i p t i o n of these d i r e c t i o n s and r e f e r r i n g to f i g u r e 27, one can see t h i s more d i r e c t l y . The absence of a b s o l u t e d i r e c t i o n s d i d cause some c o n s t e r n a t i o n i n p r o v i d i n g an E u l e r angle d e s c r i p t i o n to the t r a n s f o r m a t i o n between the r e f e r e n c e (BCA) frame and the frame d e f i n e d by the e i g e n v e c t o r s . U n t i l i t was r e a l i z e d t h at an improper r o t a t i o n could e q u a l l y well a r i s e i n the d i a g o n a l i z a t i o n process, only p a r t i a l success was achieved i n f i n d i n g the E u l e r angle t r a n s f o r m a t i o n . The e s s e n t i a l f e a t u r e s of t h i s . example are t w o f o l d . F i r s t , p r i n c i p a l d i r e c t i o n s r e l a t i v e to an e x p e r i m e n t a l l y known frame can be deduced. Second, numerical p r e c i s i o n of the 108 Table 4. Zero F i e l d S p l i t t i n g Tensor components: b c a b -0.30806110 0.01972805 0.07093540 c -0.30806110 -0.07093496 a 0.61612230 e i g e n v a l u e s : # 1 = -0.33833296 # 2 = -0.28833305 # 3 = 0.62666611 e i g e n v e c t o r s : # 1 = 0.70323362 b -0.70323270 c -0.10452867 a # 2 = 0.70710630 b +0.70710726 c -0.00000026 a # 3 = 0.07391317 b -0.07391269 c +0.99452187 a p o l a r a n g l e s : (theta,phi) # 1 = (96.00,-45.00) # 2 = (90.00, 45.00) # 3 = ( 6.00,-45.00) Where: Theta i s measured from A. P h i i s measured from B i n the BC plane. 109 p r i n c i p a l values o b t a i n e d i s s u f f i c i e n t . In t h i s example the numerical values which are deduced f o r the parameters D and E are 0.939999 and 0.0249999 r e s p e c t i v e l y . These compare f a v o u r a b l y with the v a l u e s 0,94 and 0.025 which were used to generate the data. The d i f f e r e n c e i n p r e c i s i o n between t h i s c a l c u l a t i o n and the p r e v i o u s one may be r e c o n c i l e d by noting t h a t the data d e f i n i n g the d i r e c t i o n of the magnetic f i e l d i n the f i r s t c a l c u l a t i o n were s u b j e c t to round-off, while the data f o r the second c a l c u l a t i o n were not. Despite the manifold a r i t h m e t i c o p e r a t i o n s which are performed i n the f i t t i n g process, the r e s u l t remains v i a b l e . A s i m i l a r type of c a l c u l a t i o n was a l s o done on the h y p e r f i n e term, and the data f o r t h i s example were a l s o c a l c u l a t e d by F i e l d s . The s p i n system was an S = 1/2 I = 3/2 case. In t h i s example the h y p e r f i n e term i s the v a r i a b l e , and the Zeeman term was f i x e d . Four t r a n s i t i o n s a t three mutually p e r p e n d i c u l a r magnet o r i e n t a t i o n s were used as data. Convergence of the f i t t i n g i s i l l u s t r a t e d i n Table 5 where one may compare the f i t t e d , components with those used to generate the data. The components are i n good agreement, with the e x c e p t i o n t h a t two of the components have opposite s i g n s . T h i s i s not an unreasonable r e s u l t , s i n c e the data s u p p l i e d are not s e n s i t i v e to the c h o i c e between the two symmetry r e l a t e d d e s c r i p t i o n s . The other c o n s i d e r a t i o n i l l u s t r a t e d by t h i s example i s the q u a l i t y of f i t t i n g . The f i n a l R.M.S. 110 Table 5. i t e r a t i o n number Hyperfine S p l i t t i n g Convergence h y p e r f i n e tensor components: 0.0965533 -0.0001079 0.0959599 0.0618538 0.0739246 0.0606304 0.0604278 0.0604244 0.0604242 0.0604242 0.0495600 0.0677618 0.0037322 0.0783668 0.0060664 0.0664970 0.0051470 0.0662758 0.0050727 0.0662733 0.0050732 0.0662732 0.0050732 0.0662732 0.0006804 0.0038590 0.0105280 -0.0139389 -0.0022778 0.0835631 -0.0012620 -0.0045138 0.0846017 -0.0013921 -0.0034856 0.0825340 -0.0032859 •0.0049500 0.0823050 •0.0027844 -0.0047565 0.0823015 -0.0027430 •0.0047531 0.0823014 0.0027427 •0.0047531 0.0823014 r.M.S. r e s i d u a l 410. 224. 135. 2.97 . 74 .04 .016 0.0604250 0.0050663 0.0027479 0.0662752 0.0047595 0.0822996 111 d e v i a t i o n i n freguency e r r o r s corresponds to about 16 m i l l i g a u s s i n magnetic f i e l d s t r e n g t h ; the i n p u t data to the f i t t i n g program were of t h i s order of p r e c i s i o n . T h i s type of c a l c u l a t i o n a l s o i l l u s t r a t e s the power of the method and some of the p r o p e r t i e s which may be examined to ensure that the program i s f u n c t i o n i n g c o r r e c t l y . The t h i r d example of t h i s method of a n a l y s i s a p p l i e s to data which were experimental i n o r i g i n . These measurements p e r t a i n t o EPR s t u d i e s of c e r t a i n r a d i c a l s produced by X-i r r a d i a t i o n i n the some monobasic arsenate compounds ( 8 0 ) . The componds i n v e s t i g a t e d were s a l t s of ammonium, potassium, rubidium, cesium and the deuterated analogues. The s t u d i e s concluded t h a t the EPR s p e c t r a should be assigned to a f a m i l y of a r s e n i t e type r a d i c a l s , c o n t a i n i n g at l e a s t three members. The EPR s p e c t r a of these s p e c i e s d i s p l a y e d a l a r g e h y p e r f i n e i n t e r a c t i o n which was orthorhombic. Angular v a r i a t i o n s t u d i e s of X band EPR measurements on these c e n t r e s i n potassium dihydrogen arsenate (KDA) i s d e p i c t e d i n F i g u r e 28 . From p l o t s of the angular dependence of the resonant magnetic f i e l d s t r e n g t h i n t h r e e o r t h o g o n a l planes, assignments of the t r a n s i t i o n s due t o . a . s i t e were made. T y p i c a l l y f o r t y to s i x t y o b s e r v a t i o n s were assigned and f i t t e d . The q u a l i t y of f i t t i n g of the observed resonances with those p r e d i c t e d by the s p i n Hamiltonian e v a l u a t e d with f i t t e d parameters i s a l s o d e p i c t e d i n F i g u r e 28. The c a l c u l a t e d resonances are from the program 4600 — 112 KDA ^ 3600 — < 3400 — o 3000 — 2800 — 2600 — 2400 2200 — 2000 — 1800 — 0° c 45^ 9 0 0° ab ab 45^ 90" b.a 28. Angular Behaviour Of Resonant F i e l d s . KDA 113 F i e l d s . The agreement i s g u i t e a c c e p t a b l e , with most o b s e r v a t i o n s f a l l i n g c l o s e to a l i n e w i d t h . T h i s shows t h a t the s p i n Hamiltonian deduced i s an a c c u r a t e d e s c r i p t i o n of the o b s e r v a t i o n s . A f u r t h e r t e s t of the parameters deduced by t h i s method i s provided by the comparison of resonant f i e l d s to be expected along p r i n c i p l e d i r e c t i o n s of the h y p e r f i n e t e n s o r and a random spectrum of these s p e c i e s . Such a comparison i s i l l u s t r a t e d i n F i g u r e 29 . The s t i c k diagram should be compared to the KDA powder s p e c t r a l f e a t u r e s which have not been s t a r r e d . The stronger s t a r r e d a b s o r p t i o n s belong to another s p e c i e s . The t h r e e a r s e n i t e type r a d i c a l s are denoted K L and K; t h i s comparison of powder f e a t u r e s with the c a l c u l a t i o n ensures t h a t the d e s c r i p t i o n i s an a c c u r a t e one. The a v a i l a b i l i t y of t h i s method played an important p a r t i n the a n a l y s i s of the data. i n 29. Powder EPR Spectrum. KDA 115 6j_6 Tensor Q u a n t i t i e s In keeping with the o r i g i n a l i n t e n t of p r o v i d i n g a u s e f u l a n a l y t i c a l t o o l f o r a n a l y s i s of EPB data, due c o n s i d e r a t i o n of the treatment of tensor q u a n t i t i e s had to be made. The form of the i n f o r m a t i o n a v a i l a b l e a f t e r s u b j e c t i n g data to the f i t t i n g procedure i s a r e a l symmetric a r r a y of components and a s s o c i a t e d d e v i a t i o n s . The s u b s c r i p t s of the components l a b e l a s e t of orthogonal d i r e c t i o n s . The r e l a t i o n s h i p of these d i r e c t i o n s to an experimental frame of measurement i s d e f i n e d by the manner i n which the o r i e n t a t i o n dependent o b s e r v a t i o n s were s u p p l i e d to the f i t t i n g program. The d e s i r e d i n f o r m a t i o n i s the magnitude of p r i n c i p a l values and a s s o c i a t e d p r i n c i p a l , d i r e c t i o n s , r e c o v e r a b l e by d i a g o n a l i z a t i o n : T = R*»T ,»B where T was deduced, T• i s i n d i a g o n a l form, and R i s an a r r a y of p r i n c i p a l v e c t o r s . The s i x independent components of T are separated as 3 parameters f o r e i g e n v a l u e s , and 3 parameters f o r the t r a n s f o r m a t i o n R. I t i s p e r t i n e n t t h a t the program F i e l d s uses an Euler angle d e s c r i p t i o n of such t r a n s f o r m a t i o n s , and a program which deduces t h i s d e s c r i p t i o n was w r i t t e n so t h a t t h i s process would be t r a n s p a r e n t . L a t e r , t h i s was appended to the~ f i t t i n g program. 116 6.7 E r r o r Measure Fo l l o w i n g a n a l y s i s by means of the f i t t i n g procedure one has a v a i l a b l e i n f o r m a t i o n of the f o l l o w i n g nature. In a d d i t i o n to components of a symmetric tensor which have been determined so as to minimize e r r o r s , one a l s o has a set of v a r i a n c e s i n these components. These q u a n t i t i e s are not n e c e s s a r i l y e r r o r s r e s u l t i n g s o l e l y from measurements. Although p o s i t i v e numbers,they do not r e p r e s e n t signed g u a n t i t i e s ; they c h a r a c t e r i z e a d i s t r i b u t i o n of measures about a mean. The q u e s t i o n o f a s s i g n i n g v a r i a n c e s to the eigenproblem symbolized T±s means n e i t h e r a d d i t i o n nor s u b t r a c t i o n of a p a r t i c u l a r matrix of v a r i a n c e s s to T and then d i a g o n a l i s a t i o n of the r e s u l t a n t . A c o r r e c t phrasing of the problem c o n s i s t s o f a s k i n g about the t o t a l i t y of s o l u t i o n s to the t o t a l i t y of problems represented by T±s. T h i s should be kept i n mind i n the f o l l o w i n g d i s c u s s i o n . The eigenproblem of T can be solv e d with e r r o r d i c t a t e d by the procedure of the numerical method. In terms of p r e c i s i o n a v a i l a b l e on computational f a c i l i t i e s here, t h i s p r e c i s i o n i s not an i s s u e . The q u e s t i o n of.degeneracy i n T i s the f i r s t p o i n t to a r i s e . I f T has d i s t i n c t e i g e n v a l u e s , the e i g e n v e c t o r s w i l l be w e l l d e f i n e d and orthog o n a l . I f twofold degeneracy e x i s t s , then two e i g e n v e c t o r s span a plane, and any two v e c t o r s i n t h i s plane w i l l do so, the convenient 1 1 7 non-unique c h o i c e being an orthogonal one. S i n c e the v e c t o r s here are three dimensional, then the e i g e n v e c t o r belonging to the d i s t i n c t eigenvalue d e f i n e s t h i s plane. In the case of three f o l d degeneracy, any t h r e e orthogonal v e c t o r s are a convenient c h o i c e , but the c h o i c e i s e n t i r e l y a r b i t r a r y . Any of these s i t u a t i o n s may be expected to a r i s e i n EPR data. The f i r s t s t e p i n the problem i s to f i n d the eigenvalues of T, and then to decide i f the v a r i a n c e i n the components of T w i l l allow a degenerate problem to occur. Enclosure i n double v e r t i c a l bars denotes the norm of a matrix. T h i s q u a n t i t y i s the square root of the sum of the suares of a l l elements. The i n t e n t i s to s t a r t with the eigenproblem of T and then c o n s i d e r changes that may r e s u l t i n the problem T±s, One motive f o r t h i s i s that ||T|| i s expected to be l a r g e r than | | s | | ; i f not, i t i s probably d i f f i c u l t to say anything about the problem. R e c a l l i n g the nature of s, one must p i c t u r e a f a m i l y of symmetric matrices. T h i s l e a d s r a t h e r d i r e c t l y to f i n d i n g bounds upon the matrix elements of s; a theorem which i s of use i n t h i s regard has been e s t a b l i s h e d (81). From t h i s theorem one concludes t h a t the eigenvalues of T±s l i e w i t h i n an amount | | s | | of the eigenvalues of T. One may now decide i f the eigenvalues of T±s are d i s t i n c t . A r e a s o n a b l e c r i t e r i o n f o r t h i s d e c i s i o n i s t h a t i f | T ( i i ) -T ( j j) | exceeds 1.41U||s|| they are d i s t i n c t . I f they are not d i s t i n c t by t h i s c r i t e r i o n , i t should be concluded that they 118 may well be degenerate. I f t h i s i s the case, the e n t i r e p e r s p e c t i v e of the problem w i l l be a l t e r e d . At t h i s p o i n t , one has a l i m i t on p o s s i b l e s h i f t s of the e i g e n v a l u e s of T±s i n r e l a t i o n t o those of T with | | s | | . The next p a r t of the problem i s to place a value f o r the p o s s i b l e e i g e n v e c t o r s of T±s i n r e l a t i o n t o those of T. One method of d i a g o n a l i s a t i o n i s to perform a r o t a t i o n about the normal to the i j plane by an angle: 6 ( i , j ) = 0.5Tan-»{2T(i,j)/T(i,i)-T(j,j) } ...,...[28] T h e r e f o r e , i f s i s transformed i n t o a b a s i s of the e i g e n v e c t o r s of T, the magnitude of the o f f d i a g o n a l elements w i l l measure the amount by which the e i g e n v e c t o r s of T±s may d i f f e r from those of T. Again one may expect to c a l c u l a t e s ( i , j ) only f o r a p a r t i c u l a r s, which would contravene the i n t e r p r e t a t i o n of s. However, one may a s s i g n an upper bound to t h i s . Since | | s | | 2 = t r a c e { s 2 } i t i s i n v a r i a n t under orthogonal t r a n s f o r m a t i o n s . T h e r e f o r e s 2 ( i , j ) cannot exceed ||s||?/2; the f a c t o r of two comes from the f a c t that s i s symmetric. The r o t a t i o n necessary to b r i n g T±s to d i a g o n a l form i s bounded a t : 0.5Tan _ 1 { \ |s| | / 1 . 4 1 4 ( T { i , i ) - T ( j , j ) ) } where T ( i , i ) - T ( j , j) > 1.414||s|| ensures that the e i g e n v e c t o r s I and j of T are well d e f i n e d . . T h i s e x p r e s s i o n has an upper l i m i t of 17.5 degrees. E v i d e n t l y there may be a r a t h e r r e s t r i c t e d s e t of measurements to which t h i s w i l l 119 a p p l y . The question of the r e l i a b i l i t y of e i g e n v e c t o r s and e i g e n v a l u e s i s of s u f f i c i e n t i n t e r e s t t h a t a program to d e a l with t h i s problem has been produced by the Computing Centre (82). Although produced t o d e a l with rounding e r r o r i n e i g e n v a l u e and e i g e n v e c t o r c a l c u l a t i o n s , s i m i l a r c o n s i d e r a t i o n s to those r a i s e d here apply, and s i m i l a r problems i n e v i t a b l y a r i s e . They found that good upper bounds f o r problems i n v o l v i n g w ell separated eigenvalues could be c a l c u l a t e d , while e r r o r bounds f o r e i g e n v e c t o r s belonging to near i d e n t i c a l eigenvalues c o u l d be very i n f l a t e d . T h i s behaviour i s not unexpected. I t i s concluded t h a t the most p r a c t i c a l way to d e a l with the q u e s t i o n of the u n c e r t a i n t i e s i n the e i g e n v e c t o r s of i n t e r e s t here i s to implement equation [28] and r e c a l l t h a t the bound may become r a t h e r p e s s i m i s t i c . 120 BIBLIOGRAPHY # Reference Page 1. W. Lwowski, Angew. Chemie I n t e r n a t . E d i t . , v o l 6, p. 897 (1967). 2 2. A. R e i s e r , G. Bowes, R.J. Home, Trans. Far. S o c , 62, 3162, (1967) .......2 3. G. P o r t e r , Proc. Roy. Soc. Vol A303, p. 139, (1968). .........2 4. E. Wasserman, Progress i n P h y s i c a l Organic Chemistry, v o l . 8, p. 319, (1971). .......2 5. W.A. Yager, E. 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Soc, 6, 110, (1955), 117 82. J . B j e r r i n g , " E r r o r Bounds f o r Eigenvalues and E i g e n v e c t o r s " , Computing Centre, U.B.C, 1969. 1 19 125 APPENDIX Standard Deviations i n Parameters: Standard deviations in parameters are calculated according to: sigma(j) 2 = m (j) (e»e) / (M-J) det[ K+K ] [29] Where: sigma(j) i s a standard deviation i n the j»th parameter; (e»e) i s the sum of the squares of frequency errors remaining aft e r solution of the normal equations; M-J i s the number of degrees of freedom i n the system; m(j) is the minor of the j'th diagonal element of the array K+K. This may aris e i n the following manner. In the solution to the least sguare problem, the eguation P - [K+K]-»K+e ............................ [30] Relates residuals to changes i n parameters, Construct the following eguation (which could be done each time a set of measurements were made) PP+ = [ K+K ]-*K+ee + K[[ K+K]-» ]• [31] PP+ and ee+ w i l l be sguare matrices. ee+ i s a matrix whose elements are products of residuals. PP+ i s a matrix whose elements are products i n changes i n parameters. Consider repeating the measurements and constructing [31 ] a large 126 number of times. Average [31 ] over these r e p e t i t i o n s . The matrices which are f u n c t i o n s of K depend only upon the s o l u t i o n t o the problem, by hypothesis these are the same. On the other hand, the average of ee+ should behave d i f f e r e n t l y . O f f d i a g o n a l elements w i l l be averages of randomly signed products of r e s i d u a l s , which t h e r e f o r e tend to zero. Diagonal elements should approach a mean square d e v i a t i o n i n the repeated measurements. T h i s means e e + — > a 2 I , where a 2 i s a mean square d e v i a t i o n over s e t s of r e s i d u a l s . Then P+P = a 2 [ K + K ] ~ l . T h i s e s t a b l i s h e s the dependence of sigma (j) 2 on the matrix [ K+K ]. The f a c t i s t h a t i n the problem that t h i s i s a p p l i e d t o , repeated measurement i s not commonly a v a i l a b l e . Instead, the overdetermined problem i s c h a r a c t e r i s e d by a mean square d e v i a t i o n e»e/M-J 127 Programme Information Access to the programme i s a v a i l a b l e through the EPR group at the chemistry department. The method of use of the programme, more s p e c i f i c a l l y , the form of r e g u i r e d input i s f u l l y d e s c r i b e d on comment cards. These cards are to be found near the beginning of the f i r s t subprogramme on the FORTRAN source l i s t i n g of the programme. As a means of supplementing t h i s i n f o r m a t i o n by way of example, sample i n p u t and output are a v a i l a b l e from the EPR group. Since the programme s i z e i s approximately 3000 c a r d s , i n c l u s i o n h e r e i n was f e l t u n d e s i r a b l e . The e s s e n t i a l i n p u t may be o u t l i n e d as f o l l o w s . The f i r s t s e v e r a l cards supply i n f o r m a t i o n needed i n dimensioning a r r a y s which w i l l be used to h o l d input data and matrices. The form and i n i t i a l values of s p i n hamiltonian parameters are a l s o i n d i c a t e d at t h i s p o i n t . F o l l o w i n g t h i s data, the angular dependent experimental i n f o r m a t i o n i s s u p p l i e d . These are resonant magnetic f i e l d s t r e n g t h s f o r EPR, and r a d i o f r e q u e n c i e s f o r ENDOR. Next, a s p e c i f i c a t i o n of the energy l e v e l s i n v o l v e d i n the t r a n s i t i o n s i s r e g u i r e d . F i n a l l y , one i n d i c a t e s those parameters which are to be r e f i n e d i n the f i t t i n g process. 

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