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Electron nuclear double resonance studies of free radicals trapped in irradiated single crystals of sodium… Park, John Melvyn 1977

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ELECTRON NUCLEAR DOUBLE RESONANCE STUDIES OF FREE RADICALS TRAPPED IN IRRADIATED SINGLE CRYSTALS OF SODIUM FORMATE AND POTASSIUM HYDROGEN BISPHENYLACETATE by JOHN MELVYN PARK B.A., University of Keele, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department o f Chemistry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1977 (c) John Melvyn Park, 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 24r Jaunty /977 Research Supervisor Prof. CA. McDowell Abstract Electron Nuclear Double Resonance (ENDOR) measurements have been made of hyperfine couplings in x-irradiated single crystals of sodium formate and potassium hydrogen bisphenylacetate (KHBP). In sodium formate ENDOR signals were obtained from both proton and sodium ion neighbours of the CO^" centre formed on irradiation. ENDOR studies 23 of the Na hyperfine interaction together with CNDO calculations indicate that the CO^" forms a tight ion pair with the nearer Na+ cation, thus explaining the four line EPR hfs. observed. Hf. interactions have also been resolved for four pairs of nearest neighbour protons. The tensors are mainly dipolar in character, but contain some isotropic contributions which are interpreted in terms of covalent interactions. 23 Extra so-called 'forbidden' lines are observed in the Na ENDOR and a model of ENDOR enhancements involving cross relaxation with other paramagneti species i s suggested. (#2" was also observed in uv-irradiated samples: the threshold energy for radical formation was estimated to be 100+10 kcal mole A previously reported free-radical reaction in sodium formate was found to be reversed by uv irradiation, the reaction obeying second order kinetics. Proton ENDOR studies of x-irradiated KHBP confirmed the presence of the benzyl radical, for which a l l seven anisotropic proton hyperfine tensors were measured. The isotropic couplings agree with earlier EPR measurements of the benzyl radical undergoing free rotation. The dipolar couplings provide an independent estimate of the spin density distribution in the radical which is not in complete accord with earlier determinations based on the McConnell relation. The dipolar tensors imply a spin distribution close to that predicted by INDO and other calculations, which suggests that these calculations may be qualitatively correct, and the McConnell relation not s t r i c t l y applicable. Several other radicals are also present in irradiated KHBP. Two are tentatively identified as cyclohexadienyl type radicals formed by hydrogen addition at the phenyl ring ortho and para to the methylene carboxy group. - i i i -TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i i LIST OF TABLES , v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS i x CHAPTER 1 - INTRODUCTION 1 CHAPTER 2 THEORETICAL 7 2.1 The Spin Hamiltonian 7 2.2 Hamiltonian Parameters 8 2.3 The Eigenvalue Problem 12 2.3(a) EPR Hamiltonian with F i e l d Along a p r i n c i p a l d i r e c t i o n (S-I=%) 13 2.3(b) EPR Hamiltonian with F i e l d i n a General Direction 18 2.3(c) Introduction of the Nuclear Zeeman Term -'Forbidden Transitions 19 2.3(d) ENDOR Transitions 21 2.3(e) ENDOR Inten s i t i e s 25 2.4 NMR Experiments i n Paramagnetic Systems 25 2.5 The ENDOR Experiment 28 2.6 Relation of Hyperfine Coupling Tensors to Electronic Structure 32 2.6(a) Isotropic Couplings 33 2.6(b) The McConnell Relationship 34 - i v -Page 2.6(c) 8 Proton Couplings 36 2.6(d) Anisotropic Couplings 38 2.6(e) Multicentre Terms 40 2.6(f) Three Centre Terms 42 2.6(g) Applications 42 CHAPTER 3 EXPERIMENTAL 44 3.1 Experimental 44 3.2 ENDOR Spectrometer 44 3.3 Field and Frequency Calibration 48 3.4 ENDOR Experiments 49 3.5 Irradiation Units 50 3.6 Sample Preparation 51 3.7 Data Analysis 52 3.7(a) Preliminary Analysis 52 3.7(b) Determination of Hyperfine Parameters 53 3.8 CND0/IND0 Calculations 54 3.9 Crystal Alignment 55 CHAPTER 4 EPR AND ENDOR STUDIES OF C0 2" CENTRES IN UV- AND X-IRRADIATED SINGLE CRYSTALS OF SODIUM FORMATE 58 4.1 Introduction 58 4.2 EPR of uv-Irradiated Crystals 63 4.3 Proton ENDOR Studies 65 4.4 Sodium Hyperfine Interaction 74 4.5 ENDOR Intensities and Relaxation Mechanisms 82 V -Page CHAPTER 5 ENDOR STUDIES OF AN X-IRRADIATED SINGLE CRYSTAL OF POTASSIUM HYDROGEN BISPHENYLACETATE 87 5.1 Introduction 87 5.2 The Benzyl Radical 91 - 5.3 Results and Discussion: Benzyl Radical 94 5.4 Other Radicals 106 REFERENCES 115 APPENDIX 1 The E f f e c t of g-Anisotropy on ENDOR Frequencies 122 APPENDIX 2 Some Aspects of the Dipolar Hyperfine I n t e r a c t i o n 126 APPENDIX 3 Misalignment of Planes of Observation 129 APPENDIX 4 A uv-Induced Radical Reaction i n Irr a d i a t e d Sodium Formate 133 - v i -LIST OF TABLES TABLE Page I EPR Parameters for the C0 2~ Centre Produced by y and UV I r r a d i a t i o n 64 II Proton Hyperfine Tensors i n X-Irradiated Sodium Formate . . 73 III CNDO Spin Densities as Functions of Geometry 78 IV Proton Hyperfine Tensors i n the Benzyl Radical 98 V Proton I s o t r o p i c Coupling Constants (MHz) f o r the Benzyl Radical i n D i f f e r e n t Media . . . . . . 99 VI Angles 4>? Between a Vector of Proton Tensor i and a Vector of para proton tensor . . . . . . 99 . VII Proton Dipolar Coupling Tensors i n Benzyl Radical (MHz) . . 103 VIII Hyperfine Tensors f o r Radicals I and II 108 IX Hyperfine Tensors i n the Cyclohexadienyl and a-Naphthyl Radicals . » » • • • » » « . . o o . . . . . 109 - v i i -LIST OF FIGURES FIGURE Page 1. EPR and ENDOR transitions for a system with S=I=Js. 16 2. Fir s t order ENDOR transitions for a system with S=h 1=3/2. 23 3. Relaxation pathways for a system with S=I=%. 30 4. (a) Schematic representation of isotropic proton couplings induced by a-ir polarisation. (b) Dihedral angle 6 used i n calculating 3 proton couplings. 35 5. Coordinate system used i n McConnell-Strathdee calculations. 39 6. Block diagram of ENDOR spectrometer. 45 7. Typical morphology and axis systems for sodium formate. 59 8. Angular variation of proton ENDOR frequencies in (a) approximate a*3 and (b) the approximate y3 plane. 66 9. Angular variation of proton ENDOR frequencies in the approximation ay plane sodium formate. 67 10. A typical ENDOR spectrum of x-irradiated sodium formate at 77K. 68 11. Projection of part of the sodium formate lattice into the crystallographic be plane, showing the hydrogen atoms corresponding to the hyperfine tensors listed in Table II. 70 12. Schematic representation of p-orbital overlaps inducing spin density in an HC02" neighbour of the C02~ radical ion. 71 13. Sodium ENDOR spectrum in the frequency range near the free sodium NMR frequency v ^ . 75 14. Sodium ENDOR spectra obtained by irradiating EPR lines i - i v in turn, in order of decreasing f i e l d . 80 15. Appearance of a l l three sodium ENDOR lines obtained by irradiating the lowest f i e l d EPR line. 81 16. ENDOR spectrum obtained from sodium formate crystal after 11.5 h x-irradiation, showing the strong distant ENDOR proton resonance line. 84 17. Sodium ENDOR lines obtained by irradiating the second lowest f i e l d EPR line at 4.2K. 85 18. External morphology and axis system of KHBP single crystal. 88 19. Projection of part of the KHBP crystal l a t t i c e onto the ac plane. 89 20. Angular variation of proton ENDOR spectra from the benzyl radical i n x-irradiated KHBP. 93 - v i i i -FIGURE ; Page 21. Typical EPR(a) and ENDOR(b) spectra of x-irradiated KHBP obtained at 77K. 95 22. INDO overlap spin densities i n the benzyl r a d i c a l . 104 23. Proton dipolar vectors i n cyclohexadienyl. 110 24. Assignments of hydrogen addition radicals I and I I . I l l 25. Angular v a r i a t i o n of ENDOR spectra for radicals I and I I . 112 26. Relative i n t e n s i t i e s of CO2" (C) and secondary r a d i c a l (X) EPR spectra as a function of uv - i r r a d i a t i o n time. 134 27. Ratio of EPR i n t e n s i t i e s [C]/[X] plotted against uv-i r r a d i a t i o n time. Data from Fig. 26. 135 - ix -Acknowledgements I wish to thank Dr. C A . McDowell f o r supervising the work described here, and f o r suggesting the problems studied. I am g r a t e f u l to Mr. P. Markila f o r performing the c r y s t a l s t r u c t u r e analysis of sodium formate and i d e n t i f y i n g the axes of c r y s t a l s used f o r ENDOR measurements, and to Dr. J . C Speakman of Glasgow U n i v e r s i t y f o r very k i n d l y providing unpublished r e s u l t s of a redetermination of the c r y s t a l s t ructure of potassium hydrogen bisphenylacetate. Two discussions with Dr. J . T r o t t e r were instrumental i n e l u c i d a t i n g the implications of Dr. Speakman's r e s u l t s . Dr. F.G. Herring was a continual source of advice and encouragement, and Drs. J.B. Farmer and W.C Lin provided u s e f u l background information about instrumentation and the EPR spectra of i r r a d i a t e d potassium hydrogen bisphenylacetate, Messrs. T. Markus and K. Sukul provided expert maintainance of the spectrometers used. I must thank a l l my colleagues f o r t h e i r i n t e r e s t and encouragement. P a r t i c u l a r l y , I thank Dr. N.S. Dalai whose f r i e n d s h i p , s c i e n t i f i c expertise and unflagging enthusiasm were invaluable during the i n e v i t a b l e periods of f r u s t r a t i o n . I am g r a t e f u l to the National Research Council of Canada f o r a Graduate Student Bursary, and to the Department of Chemistry f o r several a s s i s t a n t s h i p s . - 1 -Chapter 1 INTRODUCTION Viewed c l a s s i c a l l y , the magnetic resonance phenomenon i s a consequence of the f a c t that a magnetic dipole w i l l precess about a f i x e d magnetic f i e l d with a c h a r a c t e r i s t i c (Larmor) frequency. I f a r o t a t i n g or o s c i l l a t o r y magnetic f i e l d i s applied i n the plane of t h i s precession, i t produces a maximum e f f e c t when i t s frequency of o s c i l l a t i o n i s equal to the Larmor frequency. This frequency e q u a l i t y i s c a l l e d the resonance condition: when i t i s s a t i s f i e d the precessing dipole w i l l strongly absorb energy from the o s c i l l a t i n g f i e l d and widen the cone of i t s precession. A quantum mechanical a n a l y s i s y i e l d s e s s e n t i a l l y the same r e s u l t . Two of the most important a p p l i c a t i o n s of t h i s e f f e c t have been i n e l e c t r o n paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR), i n which the magnetic dipoles are provided by the e l e c t r o n i c and nuclear spin r e s p e c t i v e l y . These two techniques are complementary i n the types of system to which they can be applied, and also i n the sense that EPR has the higher s e n s i t i v i t y and NMR the higher r e s o l u t i o n . - 2 -A double resonance experiment consists i n s a t i s f y i n g two such resonance conditions simultaneously, i n such a way that the energy absorption by one resonance influences the absorption by the other. Such an experiment can be invaluable i n studying (or eliminating) the i n t e r a c t i o n s between d i f f e r e n t spins i n a given system. In i t s widest sense the term electron nuclear double resonance encompasses such experiments as dynamic nuclear p o l a r i s a t i o n and the s o l i d s t ate and Overhauser e f f e c t s (1-6); the acronym ENDOR however r e f e r s to a s p e c i f i c experiment o r i g i n a t e d by Feher i n 1956 (7-9) to measure the hyper-f i n e couplings of magnetic n u c l e i i n t e r a c t i n g with paramagnetic centres. This technique proved u s e f u l from the f i r s t because i t was able to resolve hyperfine couplings which were comparable to or even hidden within the electron paramagnetic resonance linewidth. A c l a s s i c example of t h i s was the experiment by Holton, Blum and S l i c h t e r (10) on F centres i n L i F i n which s p l i t t i n g s were resolved f o r n u c l e i as much as seven s h e l l s away from the defect centre, a l l the couplings being within the EPR linewidth. The e a r l y a p p l i c a t i o n s of ENDOR were to such inorganic systems, p a r t i c u l a r l y to the study of-point defects i n a l k a l i h a l i d e c r y s t a l s . For n u c l e i such as c h l o r i n e , sodium or nitrogen, with spins greater than ENDOR can also measure the quadrupole coupling constant i f the l o c a l symmetry allows t h i s to be non-zero. Hyperfine and quadrupole parameters were measured by Cook and Whiffen (11) f o r nitrogen centres trapped i n diamond, In many cases too i t was poss i b l e to make accurate measurement of the e f f e c t i v e nuclear g-factors and to r e l a t e the observed aniso t r o p i c s to the e l e c t r o n i c structure of the paramagnetic centre. ENDOR of point defects i n a l k a l i h a l i d e c r y s t a l s has been the subject of several reviews. (12-15) - 3 -Systems of more chemical i n t e r e s t studied by ENDOR have included ligand superhyperfine i n t r a n s i t i o n metal complexes (16-18) and, hyperfine i n t e r a c t i o n s i n organic t r i p l e t state molecules (19-21), and i n free r a d i c a l s . The f i r s t two of these are again adequately covered i n the l i t e r a t u r e , so th i s d i s c u s s i o n of them w i l l be b r i e f . The analysis of ligand superhyperfine structure enables one to estimate the amount of admixture of ligand wave-functions to the wavefunction of the metal ion, and hence to estimate the covalent contributions to the metal-ligand bonding. In the higher m u l t i p l e t states one's main i n t e r e s t i s i n using the measured hyperfine parameters to describe the d e l o c a l i s e d o r b i t a l s containing the unpaired electrons. Probably the largest area of study and the one of most i n t e r e s t here has been that of organic free r a d i c a l s . For these the range of a v a i l a b l e n u c l e i i s quite l i m i t e d , and probably 90% of the work has been on proton ENDOR. Halogen and metals hyperfine couplings have been determined. But of the elements us u a l l y present i n such species, carbon, oxygen and sulphur a l l c o n s ist of at least 99% zero-spin isotopes, while ENDOR of nitrogen n u c l e i i s hindered by quadrupole r e l a x a t i o n e f f e c t s (22). Proton ENDOR has at least two advantages. F i r s t the high proportion of hydrogen atoms i n most organic molecules means that there w i l l be many possible couplings to measure, both i n t e r - and dntra-molecular. Secondly, the i n t e r p r e t a t i o n of such couplings i s s i m p l i f i e d by the fact that f o r hydrogen only the Is o r b i t a l i s occupied. The information derived from the proton hyperfine couplings can be r e l a t e d to the e l e c t r o n i c structure of the r a d i c a l , and more fundamentally can be used to i d e n t i f y the r a d i c a l i t s e l f . This use of ENDOR has proved p a r t i c u l a r l y valuable i n studies of radiation-damaged c r y s t a l s of amino acids (23-26) and other molecules of b i o l o g i c a l i n t e r e s t (27-30); i n these cases the EPR spectrum i s often too complex or poorly resolved f o r - 4 -analysis and ENDOR provides the only means of i d e n t i f y i n g the r a d i c a l . In many cases too ENDOR has shown that such unresolved spectra are due to more than one species, when t h i s was not apparent from the EPR spectrum i t s e l f (28-29). Such r e s u l t s taken i n conjunction with knowledge of the temperature of i r r a d i a t i o n and of the c r y s t a l s tructure have been used to i n f e r the mechanism of r a d i c a l formation. (27, 31) Improvements i n instrumentation have opened up new p o s s i b i l i t i e s f o r the a p p l i c a t i o n o f ENDOR and t h i s survey concludes with an account of some of these developments. The a p p l i c a t i o n of ENDOR to the study of powders or glasses rather than s i n g l e c r y s t a l s requires the use of lar g e r samples but reduces the amount of data analysis needed (32-38). Relaxation e f f e c t s become more complex and important i n t h i s case, and can be used to advantage. The method i s p a r t i c u l a r l y u s e f u l f o r the study of complex b i o l o g i c a l molecules which do not form good c r y s t a l s . In recent years ENDOR has been applied to the study of free r a d i c a l s and i o n p a i r s i n the l i q u i d phase (40-42). T h i s poses much more st r i n g e n t requirements, since i t necessitates r e l a t i v e l y high temperatures, at which r e l a x a t i o n makes low power saturation of the EPR s i g n a l impossible. As a r e s u l t radio-frequency a m p l i f i e r s producing of the order of 1 kW are needed and the consequent problems of radio-frequency interference can be severe. (40) Double ENDOR, el e c t r o n nuclear t r i p l e resonance, or TRIPLE are extensions of the ENDOR experiment to the simultaneous sa t u r a t i o n of two nuclear resonances; as a r e s u l t the r e l a t i v e signs of the two couplings can be determined. The experiment i s d i f f i c u l t and has r a r e l y been performed - 5 -on s o l i d s (43, 44), but recent r e s u l t s suggest that i t may f i n d u s e f u l a p p l i c a t i o n i n the l i q u i d phase. (45) Op t i c a l detection of ENDOR i s a technique of high s e n s i t i v i t y which can be used to measure the hyperfine i n t e r a c t i o n s i n the excited t r i p l e t states of diamagnetic molecules; i t has been p r o f i t a b l y applied to the study of b i o l o g i c a l molecules (46, 47). These and other aspects of ENDOR are well covered i n review a r t i c l e s (50, 51) The work described i n t h i s t h e s i s mainly concerns proton ENDOR studies of free r a d i c a l s i n i r r a d i a t e d s i n g l e c r y s t a l s (52, 53). The couplings measured encompass the most us e f u l range of ENDOR: i n the case of C®2~ c e n t r e described i n Chapter 4 the proton couplings are intermolecular, mainly d i p o l a r i n character and 3 MHz or less i n magnitude; f o r the r a d i c a l s trapped i n X - i r r a d i a t e d potassium-hydrogen bisphenylacetate described i n Chapter 5, the couplings are intramolecular, with s i g n i f i c a n t i s o t r o p i c parts, and i n some cases magnitudes greater than 50 MHz. The work described i n Chapter 4 al s o includes some ENDOR studies of neighbouring sodium n u c l e i , from which s t r u c t u r a l information was deduced and r e l a x a t i o n mechanisms q u a l i t a t i v e l y assessed. In Chapter 4 the emphasis i s on intermolecular i n t e r a c t i o n s and a probe of the r a d i c a l ' s environment; i n Chapter 5 i t i s on the i d e n t i f i c a t i o n of the r a d i c a l s and the determination of t h e i r geometric and e l e c t r o n i c structures. The theory of the Spin Hamiltonian and i t s a p p l i c a t i o n to ENDOR i s presented i n Chapter 2, together with a di s c u s s i o n of the ENDOR experiment i t s e l f and the i n t e r p r e t a t i o n of hyperfine tensors i n terms of e l e c t r o n i c s t r u c t u r e . - 6 -Chapter 3 describes the experimental procedures and intrumentation used to obtain the data, and the methods of data analysis and interpretation. - 7 -Chapter 2 THEORETICAL 2.1 The Spin Hamiltonian Note on Units In the following d i s c u s s i o n , unless otherwise stated, energies w i l l be assumed to be given i n frequency u n i t s (generally MHz), so that f o r example the simple expression f o r the EPR resonance condition would be written v = g8H ; angular momenta are given i n un i t s o f f c ; thus the statement '1=3/2' r e f e r s to a nucleus with angular momentum 3/2 t i . Several thorough and d e t a i l e d comprehensive accounts of the Spin Hamiltonian are a v a i l a b l e i n the l i t e r a t u r e (104-6). Rather than attempting to imitate them, t h i s d i s c u s s i o n i s i n the nature of an overview with a phenomenological bias and some emphasis on points of p a r t i c u l a r relevance to the r e s t of t h i s t h e s i s . - 8 -Historical In 1950 Pryce (54) showed that the magnetic properties of a doublet or higher state could be described by a single spin operator, and that the result was accurate to second order in energy; and the following year Abragam and Pryce (55) extended the formulation to include hyperfine inter-actions. These results are fundamental since they make the theory of magnetic resonance tractable, reducing a many-electron problem to one involving the effective Hamiltonian of a single electron. In the s p i r i t of this formulation the results of electron paramagnetic resonance experiments are often successfully interpreted in terms of a single unpaired electron occupying a unique molecular orbital. 2.2 Hamiltonian Parameters With the work of Pryce et al the spin Hamiltonian came to take on a conventional form, primarily j u s t i f i e d by the success of i t s applications. Pake and Estle (56) have discussed the type of Hamiltonian parameters which are allowed by symmetry and the dimension of the basis set of spin functions. Several of the terms they describe apply only to transition metals or to higher multiplet states of organic molecules or are negligibly small, and for the cases of interest here the spin Hamiltonian ^ i , takes the following form: i t = 3H . g.S +S.D.S^[S.A ( i ).I ( i ) - g W g NI ( i )-H+I C i )-P ( i )'I ( i )] . [2-1] » i ~ • N • ~ ~ s -Here H is the applied magnetic f i e l d vector, S the vector operator for the .total unpaired electron spin, and I ^ the corresponding operator for the ith nucleus. At X-band frequencies (-9.3 GHz), typical ranges of values of the terms are, in order: 9.3 GHz, 0-20 GHz, 1-200 MHz, 1-15 MHz, 0-100 MHz; the physical significances of the individual terms are given below. - 9 -2.2 (a) BH'g'S, the electron Zeeman term, represents the p o t e n t i a l energy of the electron magnetic moment, -Bg'S, -in the magnetic f i e l d H. For a free e l e c t r o n t h i s energy would be simply g 8H«S, where the s c a l a r g has the value 2.00232, the Lande f a c t o r f o r a free e l e c t r o n . In p r a c t i c e , the e l e c t r o n i c angular momentum i s not purely derived from the spin: due to sp i n - o r b i t coupling small amounts of o r b i t a l angular momentum are mixed i n , so that S represents an ' e f f e c t i v e ' or ' f i c t i t i o u s ' spin. This admixture of o r b i t a l angular momentum causes the e f f e c t i v e g-factor to be s h i f t e d from i t s spin-only value, and to vary with the magnetic f i e l d d i r e c t i o n , since the o r b i t a l contributions to S are themselves a n i s o t r o p i c . The f a c t that, f o r free r a d i c a l s i n the s o l i d s t a t e , g approximates the f r e e - s p i n value i s due to 'quenching' of the o r b i t a l angular momentum -- i t s precession i n the c r y s t a l e l e c t r i c f i e l d s so that magnetic i n t e r a c t i o n s are averaged out. Mathematically, t h i s appears as the f a c t that the eigenvalue of the angular momentum operator L , must be zero, because the eigenvalue of any hermitian operator i s r e a l , and the o r b i t a l must be taken as r e a l , while L z i t s e l f i s pure imaginary. For free r a d i c a l s the g - s h i f t , g o b s - g e i s generally le s s than 1% and the quantity g, r e f e r r e d to as the g-tensor, i s represented by a r e a l 3x3 to symmetric matrix (g..)- The elements of g are estimated by perturbation X J • z theory taking the s p i n - o r b i t coupling term XL'S as a f i r s t order perturbation, and for an atom the r e s u l t f o r g i s (62) <y II h ><v II h > o 1 z 1 n n 1 z 1 o r o o 1 * z z = v 2 X i — £ 2 - 2 i n E -E n o with s i m i l a r expressions f o r the other elements of g. The g-factor f o r a molecule i s taken as a sum of atomic terms (56, 62). - 10 -The absolute sign of the g-tensor cannot be determined i n a conventional EPR experiment, but requires the use of c i r c u l a r l y p o l a r i s e d microwaves. Further d e t a i l s of t h i s , together with other implications of t h i s way of w r i t i n g the electron Zeeman term, and p o s s i b l e exceptions to the formulation are given i n References (56, 59). 2.2 (b) S'D'S the e l e c t r o n i c spin-spin coupling energy, where S i s the t o t a l spin represents the magnetic i n t e r a c t i o n s between two or more unpaired electrons. The term i s zero f o r doublet sta t e s , but can be very large i n organic t r i p l e t molecules and t r a n s i t i o n metal complexes. The mechanism of the i n t e r a c t i o n can be dominated by e i t h e r s p i n - o r b i t coupling ( t r a n s i t i o n metals) or a through-space d i p o l a r i n t e r a c t i o n (organic t r i p l e t s and higher multiplets.) For a s u i t a b l e choice of axis system the d i p o l a r spin-spin i n t e r a c t i o n can be written i n the form 2 2 2 2 2 2 , 2 „ 2 J c 2 r -3x c2 t -3y c2 T -Zz i r o 7 n hg B {S x<—g > + S < + S z <—g >} [2-3] r • r r where the angular brackets denote a s p a t i a l average. This form shows the 2 2 2 2 tensor D i s t r a c e l e s s since x +y +z =r . For the s p i n - o r b i t coupling mechanism the same r e s u l t holds and D i s generally represented by a traceless symmetric tensor. 2.2 (c) S'A'I, the hyperfine energy, represents the i n t e r a c t i o n between the nuclear and electron magnetic moments. Like g and D, the tensor A i s u s u a l l y .written as 3x3 symmetric matrix. (See 59). It has two main components. (a) The d i p o l a r term S*B-I which corresponds to the c l a s s i c a l 2 - 5 i n t e r a c t i o n - y [ ( 3 r r - r U)r ]«u_ of two dipoles u. and y 2 separated by a - 11 -vector r . (U i s the u n i t dyadic) . I f both and are quantised along a common d i r e c t i o n , making an angle 6 with r , the angular v a r i a t i o n of the d i p o l a r i n t e r a c t i o n has the f a m i l i a r form B'«(3cos 20-1) [2-4] where B' i s a constant. The trace of the tensor B i s proportional to that of (uu-(l/3)U) (where u i s a u n i t vector p a r a l l e l to r) which i s given by (uu-(l/3)U):U = u-u-l/3(3) = 0 , [2 T5] so that B i s the an i s o t r o p i c part of A. (b) The i s o t r o p i c Fermi contact term 'a' or ' a ^ s 0 ' corresponds to the i n t e r a c t i o n between the nucleus and an el e c t r o n which i s i n s i d e i t , i n other words to the overlap of the e l e c t r o n i c wave function with the nucleus. The r e l a t i v e s i z e of the l a t t e r enables one to approximate the quantity a by a * •|^g3g N3 NA*6(r)Ydx = | % g N B N l * 2 ( 0 ) | , [2-6] where Y i s the e l e c t r o n i c wave function i n question. For heavy elements where ¥ i s 'dense', the f i n i t e s i z e of the nucleus may become s i g n i f i c a n t , g i v i n g r i s e to the s o - c a l l e d 'hyperfine anomaly 1 (60, 61). This i s much less than 1% of the t o t a l hyperfine i n t e r a c t i o n i n a l l cases of i n t e r e s t here, and w i l l not be considered further. The value a i s 1/3 of the trace o f A, so i t i s easy to separate the two components phenomenologically; t h e i r p h y s i c a l i n t e r p r e t a t i o n s are discussed i n a l a t e r s e c t i o n . - 12 -2.2 (d) -g^g^H-I, the nuclear Zeeman i n t e r a c t i o n i s the analogue of the electron Zeeman term; the main d i f f e r e n c e i s that nuclear magnetic moments are of the order of g^/8 e=l/1836 times the e l e c t r o n i c moment, so that the nuclear Zeeman energy i s correspondingly smaller. (At X-band frequencies, the electron and proton terms are ca 9.3 GHz and 14 MHz re s p e c t i v e l y . ) As a r e s u l t , anisotropy i n the nuclear g f a c t o r i s undetectable i n normal ele c t r o n paramagnetic resonance. 2.2 (e) I'P'I the nuclear quadrupole i n t e r a c t i o n i s formally analogous to the e l e c t r o n i c spin-spin term; i t represents the energy of a non s p h e r i c a l nucleus i n an inhomogeneous e l e c t r i c f i e l d and i s zero i f I<1. The tensor P i s proportional to the nuclear e l e c t r i c quadrupole moment Q and to the gradient tensor of the e l e c t r i c f i e l d at the nucleus. I f the e l e c t r i c f i e l d s a t i s f i e s Laplace's equation the l a t t e r tensor i s t r a c e l e s s , and t h i s i s generally assumed to hold f o r P. The a p p l i c a b i l i t y of t h i s assumption i s fit discussed i n References (64, 65). 2.3 The Eigenvalue Problem To r e l a t e the t r a n s i t i o n f i e l d s and frequencies measured i n a magnetic resonance experiment to the Hamiltonian parameters discussed above, one needs to solve the eigenvalue equation %V = EY [2-7] fo r E and Y. The most general method of s o l u t i o n , and the method adopted f o r the treatment of data described i n t h i s t h e s i s , i s by numerical d i a g o n a l i s a t i o n of the t o t a l spin Hamiltonian m a t r i x ^ . In favourable cases the equation may also be solved a n a l y t i c a l l y or by perturbation theory taken to f i r s t or second order. In recent years second order approximations to the eigenvalues of [2-7] - 13 -have been given by several authors. (66-70). The results are very complicated in the general case, but contain the interesting feature that T T tensors such as g and A occur only in symmetric terms (e.g. g *A 'A-g) so that any asymmetry in the tensors themselves would not be apparent. Since these methods are useful for preliminary analysis of data, and offer more physical insight than numerical methods, the next section provides an i l l u s t r a t i o n of their use in solving a relatively simple Hamiltonian. 2.3 (a) EPR Hamiltonian with f i e l d along a principal direction (S=I=%) We take the Hamiltonian of equation [2-1], with S=h, I=h, so that the spin-spin and quadrupole terms do not appear; g is taken to be isotropic, to and H, defining the z axis, i s parallel to a principal axis of the A-tensor; the x and y axes coincide with the other principal axes of A, so that there are no off-diagonal elements of the hyperfine tensor. One chooses as a basis in which to express_^( the set |MgMj> defined by the eigenvalues of S z and I^; for S=h and I=h as here, m5=±h, Mj=±%, so the states are conveniently denoted by |++>, |+->, |- + >» I"5**-The x and y terms i n the Hamiltonian are evaluated using the relations J + = ( . y i J y ) ; J_ = U x - i J y ) [2-8] and <MJ+l|j+|MJ> = <MJ|J_|MJ+1> = (J(J+l)-M J(M J+l)) i s [2-9] where J=S or I. The resulting Hamiltonian matrix i s , with G=g8H and N-g N3 NH, - 14 -l++> l--> l+-> l-+> <++l z k{k -A ) x y 0 0 <--1 x y -%G-52N+kA z 0 0 <+-l 0 0 h£-kN-kkz <-+l 0 0 k(A -A ) K x yJ -h£+^-kkz [2-10] and i t s eigenvalues E are the sol u t i o n s of the equation D e t ( ^ - El) = 0 [2-11] where 1 i s the 4x4 un i t matrix. The two blocks are solved separately, y i e l d i n g = kkz * ^ [ ( G + I O ^ C A ^ ) 2 ] ^ E 2 = kkz - M t G + N ) 2 ^ ^ ) 2 ] ^ [2-12] E3 = + ^ [ ( G - N ^ C A ^ ) 2 ] * 5 E 4 = -kkz - [ ( G - N ) 2 A ( A x + A y ) 2 ] J s To put these i n a form u s e f u l f o r other than numerical a n a l y s i s one has to approximate the square roots according to the conditions normally v a l i d f o r X-band EPR, v i z N « G , and ( s l i g h t l y less generally) A , A « G -x y A binomial expansion then gives E , = kk +J$(G+N) + ~ ( A - A ' ) 2 / G + d ( ( A - - A ) 4/G 3) i z lb x y x y E = +JsA -%(G+N) - J U A -A ) 2/G4(9((A -A ) 4/G 3) *• Z . J.O X V X V - 15 -h - - ^ C G - N ) + - r i ( A x + A y ) 2 / G + Z 9 ( ( A x + A y ) 4 / G 3 ) [2-loJ E 4 = - 3 f i A z - ! s ( G - N ) - ^ ( A ^ A p 2 / G + $ ( A x + A y ) 4 / G 3 ) S u b s t i t u t i n g these r e s u l t s i n t o the o r i g i n a l eigenvalue equation gives the eigenfunctions 4^ which to the same l e v e l of approximation are [2-14] A A A A 4 1 1 32 L G - N ; 1 ^ G - N } 1 Normally i n E P R , t r a n s i t i o n s are induced "by a l i n e a r l y p o l a r i s e d microwave f i e l d producing a time-dependent magnetic f i e l d Hj *cos2irvt. i s u s u a l l y perpendicular to H and can be taken as being along the x a x i s , so that i t introduces the following time-dependent term i n the Hamiltonian gBS •H 1cos2Trvt-g B MI H cos2TTVt . [2-15] When the frequency v i s equal to the energy d i f f e r e n c e (MHz) between two eigenstates of J-l< the resonance condition i s s a t i s f i e d and a t r a n s i t i o n can occur. The corresponding t r a n s i t i o n p r o b a b i l i t y between states i and j i s - 16 -9PH.S • aS.I - g^H.I F i g . 1. EPR and ENDOR t r a n s i t i o n s f o r a systein with S=I=%. - 17 -C47T2/h)H121 <y± I g 3 S x - g N 3 N I x | | 2-6 (hv-AE) . [2-16] As before, the nuclear term i s much smaller than the e l e c t r o n i c and can be neglected; a further s i m p l i f i c a t i o n r e s u l t s i f the hyperfine terms i n [2-14] are less than about 10% of the Zeeman term, i n which case the eigenfunctions are approximate eigenfunctions of S z and the strongly allowed EPR t r a n s i t i o n s correspond to the s e l e c t i o n r u l e AM s=±l. The p r i n c i p a l EPR t r a n s i t i o n s are thus t ¥ <=>v t ¥ <=>v l l Y l ^ * 4 2 2 Y 3 The corresponding energies are E^-E^ a n d E ^ E ^ ; and the resonance conditions are A 2+A 2 ^ A 2 + A 2 - V = E 2 - E 3 = - ^ + G + i(-4^ .^..- [2-17] In a field-swept experiment N and G w i l l have.different values f o r each t r a n s i t i o n , so that the corresponding second order terms w i l l make d i f f e r e n t contributions; t h i s d i f f e r e n c e however i s generally very small and w i l l be neglected. The s p l i t t i n g between the two allowed l i n e s i s then (in f i e l d units) I V V = ^ l H r H 2 l = lA zl • [ 2~1 8 ] S i m i l a r l y the centre of the two l i n e s f a l l s at A 2 + A 2 ^|G 1 +G 2| = v - 2L-] [2-19] - 18 -which i s s h i f t e d downfield from the 'g-only' value (= hv/gB) by approximately i A +A I '-pr*-' • I 2 - 2 0 ' 2.3 (b) EPR Hamiltonian with f i e l d i n a general d i r e c t i o n The main l i m i t a t i o n of t h i s d i s c u s s i o n has been the assumption that H l i e s along a p r i n c i p a l axis of A., i n which case as shown by equation 2-17 above, the nuclear Zeeman term makes an undetectable co n t r i b u t i o n to the observed spectrum. One consequence of t h i s i s that i t i s very hard to determine even the r e l a t i v e signs of the p r i n c i p a l values of the hyperfine tensor. This remains true i f we rela x the condi t i o n on the d i r e c t i o n of H, but s t i l l require S-A' I»g^8^H*I, as the following analysis i l l u s t r a t e s . We take the e l e c t r o n i c Zeeman term as the zero order Hamiltonian and evaluate the f i r s t order corrections e due to the nuclear terms. A u s e f u l device i s the ' e f f e c t i v e f i e l d approximation' i n which one regards the nuclear spin I as experiencing an e f f e c t i v e magnetic f i e l d H defined by ~ e f f ' • • - g N P N H e f £ - I = S-A-I -g NB N'H.I . [2-21] In the EPR case now considered, the hyperfine term predominates and for t h i s one makes the f i r s t order s u b s t i t u t i o n S«A - m h-A {2.22] ~ x s ~ z where h i s the un i t vector along which S i s quantised. (h=H/|H|, or a l i t t l e more generally, h = H-g /(H-g-g-H) 2 , [2-23] fo r S«A«I«g3H.) " S3 " The eigenvalues of I i n t h i s e f f e c t i v e f i e l d are - g N e N | H e f £ | m i i m s ( h . A 2 . h ) \ [2-24] and I now obeys the s e l e c t i o n rules An i j=±l. The r e s u l t i n g hyperfine pattern 2 consists of 21+1 l i n e s separated by (h«A « h ) 2 , from which one can determine 2 2 only the tensor A . The p r i n c i p a l values of A are the squares of those of A so that a l l information concerning the signs of the l a t t e r has been l o s t . 2.3 (c) Introduction of Nuclear Zeeman Term - 'Forbidden' T r a n s i t i o n s When A becomes comparable to the nuclear Zeeman energy the s i t u a t i o n becomes more complicated and one must diagonalise the f u l l nuclear Hamiltonian [2-21], to an appropriate l e v e l of approximation at l e a s t . Since the requirement that's* A.* I - g^8^H<<gBH implies a small A, a f i r s t order treatment i s again adequate, and the eigenfunctions of [2-21] are approximate eigenfunctions of S„. The nuclear Hamiltonian i s now Li mS~'~'~ " 8NBN~*~ • [ 2 " 2 5 ] (72") I f the eigenvalue of t h i s Hamiltonian i s e, i t can be shown ' that, for n e g l i g i b l e g-anistropy, the corresponding eigenfunctions have the form 2e-Azzirig+vp ^ = 2 2 2 2 ^ I m S + > [(2e-Azzm +vp) +(Axz +Ayz )m ]'1 (A ms-iA ms) i+ xz yz J i ' i- — j - | m -> [2-26] [(2e-Azzms+vp) +(Axz +Ayz )m s ] zu -with v =g.,8x,H. p 6N N This expression shows that i f the s t a t i c f i e l d d i r e c t i o n (z axis) i s along a p r i n c i p a l axis of A., or i f A i s e i t h e r much la r g e r or much smaller than vp/2 then there i s no mixing of nuclear spin functions. But in the general case, mixing of the nuclear spin states allows other t r a n s i t i o n s among the four energy l e v e l s and gives r i s e to four EPR l i n e s . The expression f o r e i s most simply evaluated using the e f f e c t i v e f i e l d approximation used above, and turns out to be % s , m i = [ • «s 2h-A 2-h-2m sg N 8 NHh.A.h +(g NB NH) 2]\ I = Psh-A'A-h- * g N 8 NH.h.A.h +Cg NB NH) Z]^m I , [2-27] with as above, H=Hh. Note that e i s now not l i n e a r i n mg. In general, one cannot expand the square root i n a r a p i d l y convergent s e r i e s , so that the t r a n s i t i o n s between l e v e l s of d i f f e r e n t mg value occur at energies which bear no simple r e l a t i o n to the magnitude of the hyperfine tensor. Furthermore e i s t y p i c a l l y of the order of |A z z/2±Vp| so that the wavefunctions can vary r a p i d l y with magnetic f i e l d strength and the o r i e n t a t i o n of the A-tensor. The r e s u l t i n g EPR spectrum consists of two p a i r s of l i n e s of strongly varying i n t e n s i t y and complex dependence on the hyperfine parameters. (72) Such spectra are very d i f f i c u l t to analyse. Methods i n v o l v i n g the use of second moments have been suggested (73) ; the extra information one 2 ..obtains i s the tensor A i t s e l f rather than A . Although the global sign of the tensor cannot be determined i n t h i s way, the r e l a t i v e signs of the p r i n c i p a l values are f i x e d by the i n t e r a c t i o n with the nuclear Zeeman term, - 21 -and make i t possible f o r instance to determine the absolute value of the contact i n t e r a c t i o n unambiguously. The success of these methods i s l i m i t e d by the r e s o l u t i o n a v a i l a b l e i n a s o l i d state EPR experiment where the linewidth, t y p i c a l l y 1-10 MHz, i s often comparable to the hyperfine couplings to be measured. 2.3 (d) ENDOR Tra n s i t i o n s At t h i s point ENDOR can provide a major s i m p l i f i c a t i o n of the anal y s i s . Postponing f o r a moment consideration o f the ENDOR experiment i t s e l f , we may view the technique as an i n v e s t i g a t i o n of the energy l e v e l s of the nuclear Hamiltonian [2-21] within a given mg m u l t i p l e t . In general a s p i n - % nucleus w i l l give r i s e to two ENDOR t r a n s i t i o n s , denoted by v + and v , corresponding to t r a n s i t i o n s within the two e l e c t r o n i c m u l t i p l e t s . (See Fig . 1). The nuclear resonance condition i s s a t i s f i e d when m [2-28] I where |mj-mj1|=1 [2-29] so that v '+ = ^h.A 2.h +g NB NHh.A.h +(g Ne NH) 2) J s [2-30] Thus the observed frequencies are now re l a t e d r e l a t i v e l y simply to the - 22 -components of the hyperfine tensor. I f l>h, there w i l l be 21 t r a n s i t i o n s corresponding to Amj=±l within each mg m u l t i p l e t ; to f i r s t order these nuclear t r a n s i t i o n s w i l l be degenerate i f the quadrupole i n t e r a c t i o n i s absent. Figure 2 i l l u s t r a t e s t h i s f o r the case of 1=3/2. If h rotates within a s i n g l e co-ordinate plane, then a plo t of 2 v against f i e l d o r i e n t a t i o n ( f or constant f i e l d strength and zero quadrupole i n t e r a c t i o n ) has the form 2 2 o v = acos 6 + bsin28 + csin^e , [2-31] a simple r e s u l t which i s very u s e f u l f o r the preliminary analysis of data; 2 the constants a, b, c are elements of the tensor (v^Ut^A) . 2 The second order corrections are of order A /4G; the ENDOR l i n e width i s ~30kHz, so t h i s f i r s t order treatment i s adequate f o r a<40 MHz, when H=3300 G, and within t h i s range of coupling constants three fur t h e r useful r e s u l t s can be derived. F i r s t l y i t follows d i r e c t l y from [2-30] that • 2 2 • N N In p r a c t i c e one cannot assign the mg values absolutely so that v + and v_ are u s u a l l y i n t e r p r e t e d as the high and low frequency t r a n s i t i o n s r e s p e c t i v e l y ; observation of both t r a n s i t i o n s thus gives a simple method of determining A to within a sign. - 23 -V, r t > i ? H>-i> H>-\> — * ! gpHm s + a m s m i 9Np N H m I F i g . 2. F i r s t order ENDOR t r a n s i t i o n s f o r a system with S ^ , 1=3/2. In the absence of a quadrupole i n t e r a c t i o n , the three t r a n s i t i o n s l a b e l l e d v > are degenerate i n f i r s t order. < - 24 -Secondly, i f A i s small compared to VpU, the square root i n the expression f o r v + can be expanded i n a binomial s e r i e s . I f a l l but the f i r s t two terms are n e g l i g i b l e the ENDOR l i n e s are symmetrically placed about Vp=|g^3^H|. The same r e s u l t i s obtained when A i s i s o t r o p i c : i f A i s replaced by a.U, the expression f o r v + immediately reduces to |vp±a/2|. In a s o l i d matrix, the p r o l i f e r a t i o n of small couplings from weakly coupled n u c l e i u s u a l l y gives the ENDOR spectrum a " c h a r a c t e r i s t i c symmetry about Vp and serves as a convenient means of f i e l d c a l i b r a t i o n . I f a i s l a r g e r than the two ENDOR l i n e s are separated by 2Vp and f a l l symmetrically about a/2. T h i r d l y , by expanding the expression f o r v + as a Taylor s e r i e s i n Avp=g^B^AH, one can show that when a<2v^, to a reasonable approximation, the separation of the ENDOR l i n e s i s unchanged by small v a r i a t i o n s i n H, so that observed ENDOR frequencies can be corrected to a constant f i e l d value by applying s h i f t s equal i n magnitude to the change i n NMR frequency. This i s useful i n p l o t t i n g the angular dependence of ENDOR l i n e s when the s t a t i c f i e l d i s varying. Within the range of g-values normally encountered i n free r a d i c a l s , the ENDOR frequencies c a l c u l a t e d above are independent of the magnitude of the e l e c t r o n i c g-tensor. However, the r e l a t i v e anisotropy of g can produce a small f i r s t order e f f e c t on v± by changing the quantisation axis of S. This point i s discussed i n Appendix 1. The above analysis has assumed throughout that m i s (approximately) a good quantum number. I f the spin Hamiltonian contains other terms which are comparable to gBH t h i s w i l l not be true. Examples are the hyperfine i n t e r a c t i o n of an arsenic nucleus (74) or the e f f e c t of a large D-tensor (75); i n these cases m„ must be replaced by the expectation value <S > which i n general w i l l d i f f e r from 0.5 and w i l l vary from state to state ; t h i s - 25 -phenomenon has been used to determine the r e l a t i v e signs of the 'ENDOR' coupling and the larger tensor, and has been extended to the case of two moderate and comparable proton couplings (76). 2.3 ( e) ENDOR i n t e n s i t i e s To evaluate the ENDOR t r a n s i t i o n moment one needs second order wave functions. These are very d i f f i c u l t to obtain a n a l y t i c a l l y f o r the general case, but have been given by Kwiram (33) and Iwasaki et a l (67), and a simple c a l c u l a t i o n f o r a s p e c i a l case was presented i n 1966 by Whiffen (71). The important consequences are that the ENDOR t r a n s i t i o n moment increases with the ENDOR frequency, and varies with the magnitude of the A-tensor and i t s o r i e n t a t i o n with respect to the s t a t i c f i e l d . As w i l l appear i n the next section, the ENDOR i n t e n s i t y also depends, but i n no simple way, on the re l a x a t i o n c h a r a c t e r i s t i c s -of the paramagnetic system. For these reasons, c a l c u l a t i o n or simulation of s o l i d state ENDOR spectra has r a r e l y been attempted. The paper by Dalton and Kwiram (33) i s a notable exception. 2.4 NMR experiments i n Paramagnetic Systems Before we turn to the ENDOR experiment i t s e l f , i t i s i n s t r u c t i v e to consider the r e s u l t s of a simple NMR experiment on a s i m i l a r system. Figure 3 shows the energy l e v e l diagram f o r a system with S=h, l=h- In the absence of induced t r a n s i t i o n s , the spin system w i l l be i n thermal e q u i l i b r i u m with the l a t t i c e , and s p i n - l a t t i c e r e l a x a t i o n w i l l maintain a population d i f f e r e n c e between each p a i r of l e v e l s ; the r a t i o of populations w i l l follow the Boltzmann d i s t r i b u t i o n , N 1/N 2 = exp(-AE/kT) , [2-331 - 26 -where AE i s the energy d i f f e r e n c e between the two l e v e l s . For X-band experiments, the exponentials may be replaced by 1-AE/kT; t h i s i s an exc e l l e n t approximation at 300K or 77K, and remains adequate f o r the present purpose at 4.2K. The separation of l e v e l s connected by EPR t r a n s i t i o n s i s ca. 9.3 GHz, while the nuclear l e v e l s 1 and 2, 3 and 4 are separated by a 14 MHz. Thus the population d i f f e r e n c e s across the nuclear l e v e l s are -3 -10 of the e l e c t r o n i c values, and t h i s i s r e f l e c t e d i n the r e l a t i v e i n t e n s i t i e s of the two types of t r a n s i t i o n . In the diamagnetic system the NMR s e n s i t i v i t y would also be l i m i t e d by the long s p i n - l a t t i c e r e l a x a t i o n times which cause s a t u r a t i o n at low power. In a paramagnetic system, t h i s l i m i t a t i o n i s greatly reduced since the f l u c t u a t i n g f i e l d s due to the r e l a x i n g electrons enhance the nuclear r e l a x a t i o n rates. As a r e s u l t of t h i s , the NMR l i n e i s considerably broadened, and except at very low temperatures the n u c l e i see only a time-averaged hyperfine i n t e r a c t i o n given by <nig>h«A«I, where <nig> i s the time average expectation value of the S z operator f o r the electron i n t e r a c t i n g with a given nucleus.' <irig> d i f f e r s from zero only because of the Boltzmann d i s t r i b u t i o n of populations i n the ^ = ^ h l e v e l s . At X-band frequencies and a temperature _3 of 77K <mg>~10 , so that the e f f e c t of a hyperfine coupling even as large as 50 MHz w i l l be only to s h i f t the NMR resonance by -50 KHz from the free proton frequency. In many cases t h i s w i l l be only a few linewidths so that the r e l a t i v e r e s o l u t i o n , s h i f t / l i n e w i d t h , i s quite low. An order-of-magnitude estimate of the r e l a t i v e EPR and NMR s e n s i t i v i t i e s i n such a paramagnetic system can be obtained as follows. The output voltage produced by a t y p i c a l spectrometer has the form AV = Qx"t;V [2-34] - 27 -where Q i s the q u a l i t y f a c t o r of the resonant c a v i t y or c i r c u i t , x, the f i l l i n g f a c t o r , V the applied voltage, which i s prop o r t i o n a l to the o s c i l l a t i n g f i e l d H^, and x" i s the imaginary part of the s u s c e p t i b i l i t y (63) A Bloch Equation treatment (58) gives u> O T 2 x o X ~ 'S o 9 o T" [2-35] ( 1 + T 2 ^ ( U - O ) o ) % T 1 T 2 Y ^ H J where T - i s the spin-spin r e l a x a t i o n time, y=gH/K and x i s given by the Curie-Law expression ^ . N g V j W i J i ^ j . . s . r l . [2-36) For s i m p l i c i t y we consider the amplitude of an absorption mode signal at the centre of the resonance, OJ=OJ , and to take account of the v a r i a t i o n of t r a n s i t i o n p r o b a b i l i t y with magnetic moment, we evaluate the maximum si g n a l as a function of H^, gi v i n g the si g n a l strength l i m i t e d by saturation. 2 2 AV i s proportional to x" Hi> an<^ t h i s function has a maximum when T 1 T 2 ' Y H l =*' Sub s t i t u t i o n of t h i s i n t o [2-35] leads to the r e l a t i o n k XoE(T2e T 1 N \ 2  X o N l T l e T2N/ i VEPR NMR where the term ai /y has been cancelled since i t i s equal to H which i s 2 5 assumed to be the same f o r both systems. From [2-36] X^g/xojj (ve/vjP =4x10 ; and T2 e/ T2N c a n ^ e r e P i a c e c ' by the r a t i o of r e c i p r o c a l linewidths, t y p i c a l l y "200, with the r e s u l t that ^ ~ - - 3 x 10 4 . [2-38] NMR This estimate neglects the di f f e r e n c e i n nuclear and e l e c t r o n i c - 28 -s p i n l a t t i c e r e l a x a t i o n times, but i n p r a c t i s e NMR experiments could be c a r r i e d out at higher f i e l d s than EPR so that the increased value o f (OQ would p a r t i a l l y compensate f o r the ease of sat u r a t i o n . As i t stands t h i s rough c a l c u l a t i o n shows why NMR i n paramagnetic systems i s a very d i f f i c u l t experiment, and to obtain d e t a i l e d measurements of hyperfine i n t e r a c t i o n s a more sop h i s t i c a t e d technique i s needed, using the s e n s i t i v i t y of the e l e c t r o n i c absorption. 2.5 The ENDOR Experiment . Now we consider what happens i f one of the EPR t r a n s i t i o n s (say 2 <=> 3) i s induced and the microwave power increased u n t i l s a t u r a t i o n occurs. The o r i g i n a l population d i f f e r e n c e between these two l e v e l s i s reduced and can be brought close to zero, at which point the EPR s i g n a l amplitude w i l l f a l l to a low value. In p r a c t i c e to achieve e f f e c t i v e microwave sat u r a t i o n u s u a l l y requires lowering the sample temperature to 77 K or 4.2K, when most r e l a x a t i o n processes w i l l be i n h i b i t e d . In t h i s case the population d i f f e r e n c e between l e v e l s 1 and 2 or 3 and 4 w i l l be increased from 6 to e (see Figure 3). A source of radio-frequency r a d i a t i o n i s now scanned u n t i l the resonance condition f o r one of the hyperfine t r a n s i t i o n s i s s a t i s f i e d ; when t h i s i s achieved a large absorption of energy occurs, tending to return the populations of the spin states to t h e i r thermal equ i l i b r i u m values, and desaturating the ele c t r o n resonance. This increase i n the o r i g i n a l EPR s i g n a l as a r e s u l t of an induced nuclear t r a n s i t i o n i s c a l l e d an ENDOR enhancement; the experimental procedure involves monitoring the i n t e n s i t y of the saturated t r a n s i t i o n while the r f o s c i l l a t o r i s scanned through the hyperfine frequencies. Experimentally i t i s often found that the strongest ENDOR signals are obtained when the EPR s i g n a l i s only s l i g h t l y saturated — within ca 3dB of the microwave power producing - 29 -the maximum signal. (40, 78) This discussion has been basically qualitative because i t i s very d i f f i c u l t to give a general description of the ENDOR mechanism which is quantitatively useful. But one can discuss some of the other factors involved i n the ENDOR mechanism as follows. The ENDOR enhancement arises from two main mechanisms, only one of which is specific to the double resonance.experiment; this f i r s t process is the increase in effective nuclear Boltzmann factor, and the removal of nuclear relaxation times as a limit on the signal strength. The second mechanism is more subtle and arises from the hyperfine interaction (50, 51) The electronic magnetisation w i l l follow the total magnetic f i e l d given by - + ~ r f s ^ n c e t n e latter varies much more slowly than the electron Larmor frequency; the resulting hyperfine f i e l d experienced by the nucleus w i l l thus have a transverse component varying at the same frequency as the applied r f . The sum of these two oscillating fields induces the nuclear transitions; this sum may be zero i f the two effects cancel, in which case ENDOR w i l l be unobservable. If the two fields add, the resultant may be much greater than alone and lead to an enhanced transition rate for the same power. This i s the physical basis of the variation of ENDOR intensity with ENDOR frequency mentioned above. In practice to take f u l l advantage of this one needs to match the impedance of the r f c o i l to that of the amplifier over the range of the frequency scan. One instrumental factor should also be mentioned: a microwave cavity equipped for ENDOR w i l l have a quality factor of the order of 2000; the corresponding value for a conventional NMR c o i l w i l l be closer to 100, so the use of an EPR signal to detect NMR in this way has a further advantage. This too is related to the higher frequency of the EPR experiment. - -l S » v P /kT 3. Relaxation pathways f o r a system with 5=1=^.1^ and T ^ are the e l e c t r o n i c and nuclear spin l a t t i c e r e l a x a t i o n times r e s p e c t i v e l y . and T represent c r o s s - r e l a x a t i o n processes. The open arrows represent allowed EPR and ENDOR t r a n s i t i o n s , and the values at the l e f t are the r e l a t i v e populations 6f the four l e v e l s when the EPR t r a n s i t i o n i s saturated. - 31 -A f u l l treatment of ENDOR mechanisms and intensities has not been developed for several reasons. In general one cannot usefully abstract a two-spin system from the ensemble of nuclei and paramagnetic centres. There are other relaxation pathways such as those in Figure 3 denoted by TlN C n u c l e a r spin-lattice relaxation), and the cross relaxation routes T and T , which may be induced by the interactions between neighbouring X XX paramagnetic centres. Note that T and T" correspond to forbidden transitions. X XX These routes should be included, together with other magnetic interactions with the la t t i c e . In fact the existence of other relaxation routes than T^ i s essential for a conventional ENDOR experiment. In the simple model discussed above, in which one EPR transition was saturated and T^ was the only significant relaxation process, inducing a single NMR transition would merely redistribute the populations of the levels and allow the EPR signal to saturate again in a time ~T^. In the absence of other relaxation processes only a transient ENDOR signal would be observed. One aspect of these other interactions i s the polarisation of distant nuclei. This has i t s origin in a solid state effect, by which the nuclei are polarised i n the EPR levels undergoing saturation. This polarisation can diffuse to 'distant' nuclei in the lattice and give rise to a 'distant ENDOR1 or 'matrix ENDOR' line. (79) Further complications arise due to contributions from the dispersion mode of the EPR signal, and from passage effects due to modulation and a fi n i t e scan rate. In particular, f i e l d and frequency modulations often produce markedly different spectra in the region of the free proton NMR frequency. Occasionally the effects of some of these mechanisms have been studied using the theory of electrical networks, with the radio-frequency t r a n s i t i o n acting as a 'short' by decreasing the e f f e c t i v e T^. The method has been applied to a study of Tm 2 + ions i n CaF2(80) } but i n general the r e s u l t i n g equations are very complicated and contain several q u a n t i t i e s which are very hard to determine. Their a p p l i c a t i o n has been l i m i t e d mainly to double resonance experiments i n the l i q u i d phase. (81) _2 In sum, ENDOR i n t e n s i t i e s are t y p i c a l l y 10 of the corresponding EPR i n t e n s i t i e s ; t h i s represents a gain of about two orders of magnitude over the NMR values for the same system. The linewidths however are comparab to the values that would be obtained by NMR and con s t i t u t e an improvement on 2 EPR r e s o l u t i o n by a fac t o r of 1-5x10 . 2.6 Relation of Hyperfine Coupling Tensors to E l e c t r o n i c Structure Since the bulk of the work described i n Chapters 4 and 5 i s devoted to the i n t e r p r e t a t i o n of measured hyperfine parameters, t h i s section surveys the means by which such parameters are r e l a t e d to the e l e c t r o n i c structure of a paramagnetic centre. In a l l cases t h i s i s v i a the spin density d i s t r i b u t i o n p(r) - the excess of a a spin over 8 spin at a, point r - which may be defined by P(r) = <Y|£s(r-r, ) 2S k> [2-39] k k z k where the sum runs over a l l electrons, p i s often used to mean the t o t a l s p i n on an atom or i n an o r b i t a l , i n which case the form of p given above must be integrated over the o r b i t a l s i n question; the term spin density although conventional, i s then not r e a l l y appropriate since what i s refe r r e d to i s a f r a c t i o n of the t o t a l spin rather than a volume density. In many cases the spin d i s t r i b u t i o n i s adequately represented by a single molecular o r b i t a l c o n s i s t i n g of a l i n e a r combination of atomic o r b i t a l s (LCAO): * = Ic . c^ . [2-40] th 2 In t h i s case the spin density i n the i o r b i t a l i s simply c^, and the sp i n 2 density on a given atom i s of the form ^c^ k where k runs over the o r b i t a l s k - 33 -centred on the atom i n question. McConnell (85), proposed an 'atomic' d e l t a function as the spin density operator, which gives a g e n e r a l i s a t i o n of t h i s r e s u l t . An advantage of t h i s simple formulation i s that i t lends i t s e l f to the d i r e c t i n t e r p r e t a t i o n of experimental r e s u l t s . Given the measured hyperfine constants a^ and the corresponding values a ^ f o r u n i t spin, one obtains simply, |c.| = ( a ^ a . 0 ) * * . [2-41] The s i n g l e LCAO d e s c r i p t i o n can give only p o s i t i v e values f o r p; a more so p h i s t i c a t e d model w i l l incorporate spin p o l a r i s a t i o n (or more generally e l e c t r o n c o r r e l a t i o n e f f e c t s ) due to the exchange i n t e r a c t i o n s between electrons and w i l l lead to negative spin d e n s i t i e s at some points. A simple case of t h i s i s provided by the McConnell r e l a t i o n which i s discussed below. 2.6 (a) I s o t r o p i c Couplings As discussed above, the Fermi contact i n t e r a c t i o n i s proportional 2 to the value of ty evaluated at the nucleus. The expression /l>6(r-r.)*dT = I c.c /<(. 6(r-r.)<J, .dx [2-42] i , j J 3 w i l l be dominated by the one centre term corresponding to o r b i t a l s centred on nucleus i , since the amplitude of an atomic o r b i t a l decreases almost exponentially with distance. In t h i s approximation only s - o r b i t a l s w i l l give r i s e to a contact i n t e r a c t i o n , since a l l higher o r b i t a l s have nodes at the nucleus: the i s o t r o p i c part of the hyperfine tensor i s thus a measure of the s - o r b i t a l spin density. For atoms other than hydrogen the main use of these spin d e n s i t i e s i s i n the c a l c u l a t i o n of h y b r i d i s a t i o n r a t i o s and the p r e d i c t i o n of geometry using p - o r b i t a l spin d e n s i t i e s and the Coulson r e l a t i o n s h i p (82). - 34 -For hydrogen atoms there i s an important s i m p l i f i c a t i o n because the gap between the n=l and n-2 e l e c t r o n i c energy l e v e l s i s ~10eV or 230kcal/mole, which means that e s s e n t i a l l y only the Is o r b i t a l i s a v a i l a b l e for bonding. Thus the s - o r b i t a l spin density of a hydrogen atom i s d i r e c t l y r e l a t e d to the amount of covalent bonding to the paramagnetic centre. This has been applied to studies of hydrogen bonding where the separation of covalent and i o n i c contributions to the t o t a l bond i s otherwise very d i f f i c u l t (83). 2.6 (b) The McConnell Relationship The above discussion i s quite adequate f o r systems of asymmetry, but for species which approximate TT symmetry other e f f e c t s must be included. On a simple model the H-atoms of an aromatic IT r a d i c a l should show no hyperfine couplings since they l i e i n the nodal plane of the spin d i s t r i b u t i o n . The observed couplings, often of the order of 30MHz, are much too large to be the r e s u l t of out-of-plane v i b r a t i o n s of the hydrogen atoms (84): the couplings i n fact result.from spin p o l a r i s a t i o n . An empirical r e l a t i o n a = Qp [2-43] had been proposed, r e l a t i n g the observed proton i s o t r o p i c coupling a to the Tr-electron density on the nearest carbon atom. McConnell and Chestnut (S5) showed that t h i s r e l a t i o n could be derived semiquantitatively from Valence Bond, Molecular O r b i t a l or Unrestricted Hartree Fock Theory. The simplest q u a l i t a t i v e d e s c r i p t i o n i s i n terms of the exchange i n t e r a c t i o n which favours the proximity of a n t i p a r a l l e l spins for electrons i n orthogonal o r b i t a l s . See F i g . 4. The p r e d i c t i o n of a l l these models i s that p o s i t i v e spin at the carbon w i l l induce negative spin at the proton, and t h i s was confirmed by F i g . 4. (a) Schematic representation of i s o t r o p i c couplings induced by a - i r p o l a r i s a t i o n . The arrows denote the r e l a t i v e amounts of a and 8 spin. (b) Dihedral angle 6 used i n c a l c u l a t i n g 8 proton couplings. - 36 -measurements of NMR contact shifts (86). Similar relations hold for N-H bonds, and have been proposed with varying degrees of success for one centre and *4N interactions, and for C-F bonds. Several attempts have been made to generalise the McConnell relationship. Colpa and Bolton (87) suggested the formula a = Q 1P+Q 2ep_Q 1p+Q 2p 2 , [2-44] where e is the charge in the C-H bond: Giacometti, Nordio and Pavan (88), suggested a = Q i P + Q ^ - P ^ where p 2 and p 2 ' are the IT spin densities on neighbouring carbon atoms. Both equations were derived for aromatic radical ions, and the values of Q 2 change sign with the sign of the ion, so the applications to neutral radicals is not obvious. Melchior (89) has given a c r i t i c a l account of these and other forms in his study of the problem. In practice the simple McConnell relationship is almost invariably used to obtain TT electron spin densities, the value of Q being chosen empirically, in the range -60 to -80 MHz. 2.6 (c) 8-Proton Couplings The ethyl radical "CH2CH,j can be regarded as a IT radical with a substituent methyl group; the McConnell relation w i l l explain the methylene (a) proton hyperfine interactions, but not the equally large methyl (8) couplings. Their origin cannot be 0-ir polarisation as described above because of the C-C bond between the hydrogens and the spin on the methylene carbon. Instead hyperconjugation or the overlap of the TT spin density with the C-H bonds is postulated (90, 92). The form of the interaction involves - 6 / -the exchange or overlap i n t e g r a l between the 2pir o r b i t a l containing the unpaired electron and the s-p hybrids comprising the C-H bonds; i t leads } to the semi-empirical r e l a t i o n s h i p . a = p ( B ( ) + B 1 c o s 2 e ) [2-45] where 6 i s the dihedral angle between the TT d i r e c t i o n and the C-H bond as shown i n Figure 4(b) . The i s o t r o p i c parts of the 6 proton couplings have been quite s u c c e s s f u l l y i n t e r p r e t e d using t h i s formula with BQ=-9 MHz and B1=122 MHz. Recently Maruani et a l (93, 94) have given more general expressions and Kwiram et a l (95) have discussed the p h y s i c a l i n t e r p r e t a t i o n of the two non-observables p o l a r i s a t i o n and overlap spin d e n s i t i e s . The cyclohexadienyl r a d i c a l * o has two B protons i n a methylene group; i n t h i s case the formula becomes more complicated because there are now two centres of spin density B to the hydrogens. Bersohn (96) had derived expressions f o r the B couplings i n semiquinone ions and Whiffen (97) showed that h i s r e s u l t s explained the unexpectedly large B couplings i n cyclohexadienyl. Bersohn's formula 2 2 2 showed that i n t h i s case p(=c ) would be replaced not by (c^ +c 2 ) but by 2 2 (c^+c 2) . For cyclohexadienyl, c i = c 2 > s o t n e r e s u l t , 4c , i s twice the expected spin density. Examination of Bersohn's formula suggests that i n the general case with c ^ c 2 the r e l a t i o n [2-45] would have the form 2 a = (c^+c 2) . (B 0+B 1cos6 1cos6 2) Now, however, BQ and B.^  themselves w i l l be functions of the geometry since - 38 -th i s determines the h y b r i d i s a t i o n at the methylene carbon. Numerical r e s u l t s by Morukuma et a l confirm t h i s (98). 2.6 (d) Anisotropic Couplings One-Centre terms For the one-centre terms the unpaired e l e c t r o n i s i n a p or higher o r b i t a l centred on the magnetic nucleus i t s e l f , and the tensor B i s given by | = y.ePN<4>|0|+> [2-46] where u g and y^ are the respective e l e c t r o n i c and nuclear moments, (P i s the 3 di p o l a r operator (3uu-U)/r . As an example we can take <)>=2p which transforms as the vector z; B takes i t s simplest form i f u i s a unit vector along z, i n ~ Sf ~ ~ which case we have " B z z = V N ( 3 - 1 ) < R " 3 > , = 2 < R " 3 > V N [2-47] B = H H X I(0-l)<r" 3> = -<r~3>y yx. = B xx e 1 e N yy and a l l other terms are zero. Thus B i s a x i a l f o r the i n t e r a c t i o n with a p o r b i t a l ; i n many cases, depending on the magnitude of other terms i n the Hamiltonian, the r e s u l t i n g angular v a r i a t i o n of the a n i s o t r o p i c coupling has the form B'(3cos 2 9 - 1) . This point i s discussed i n more d e t a i l i n Appendix 2. One centre terms often dominate the d i p o l a r tensors of heavier elements, but make no c o n t r i b u t i o n to those of hydrogen atoms due to the absence of av a i l a b l e p - o r b i t a l s . - 39 -F i g . 5. Coordinate system used i n McConnell-Strathdee calculations. The X'Z' plane contains the axis of the p - o r b i t a l , and Z1 i s the internuclear d i r e c t i o n . - 40 -2.6 (e) Multicentre Terms These are the only source of anisotropy i n proton hyperfine couplings, but are correspondingly less important f o r other elements. The two centre terms are usu a l l y s u f f i c i e n t and are considered f i r s t . where now ty and ty have a common centre, d i f f e r e n t from that of McConnell and Strathdee (99) o r i g i n a l l y t a c k l e d t h i s problem f o r the case of a magnetic nucleus i n t e r a c t i n g with spin i n a 2s or 2p S l a t e r o r b i t a l ; f o r the 2p case the i n t e r n u c l e a r vector was taken to be eit h e r i n the nodal plane or along the axis of the p o r b i t a l (TT and a cases r e s p e c t i v e l y ) . Expanding the expression f o r ty as a sum of Legendre polynomials, they were able to give expressions f o r the elements of B i n terms of the quantity a=2rz where r i s the int e r n u c l e a r separation and z i s the e f f e c t i v e nuclear charge or o r b i t a l exponent of the S l a t e r o r b i t a l . P i t z e r et a l (100) extended the r e s u l t s , i n c l u d i n g terms from S-functions o r b i t a l s were derived r e c e n t l y (-102) . B a r f i e l d (101) has given a complete For these one has to evaluate the expectation value omitted i n the o r i g i n a l c a l c u l a t i o n s . Corresponding expressions f o r 3p TT set of formulae f o r <j>,ij;=ls, 2s, 2p^, 2p^. For example i f z i s the bond d i r e c t i o n <2p y|9 z y|2s> = <2p x|e x z|2s> = - ( W 3)R" 3{15a-[4a 4+10a 3+20a 2+30a+(15/a) ] e " 2 a } , [2-48] <2s|e I2s> = R" 3[l-((4/9)a 5+(2/3)a 4+(4/3)a 3+2a 2+2a+l)e" 2 a].2 . [2-49] - 40 -2.6 (e) Multicentre Terms These are the only source of anisotropy i n proton hyperfine couplings, but are correspondingly less important f o r other elements. The two centre terms are u s u a l l y s u f f i c i e n t and are considered f i r s t . where now cb and have a common centre, d i f f e r e n t from that of (p. McConnell and Strathdee (99) o r i g i n a l l y t ackled t h i s problem f o r the case of a magnetic nucleus i n t e r a c t i n g with spin i n a 2s or 2p S l a t e r o r b i t a l ; f o r the 2p case the i n t e r n u c l e a r vector was taken to be e i t h e r i n the nodal plane or along the axis of the p o r b i t a l (TT and a- cases r e s p e c t i v e l y ) . Expanding the expression for t|> as a sum of Legendre polynomials, they were able to give expressions f o r the elements of B i n terms of the quantity a=2rz where r i s the i n t e r n u c l e a r separation and z i s the e f f e c t i v e nuclear charge or o r b i t a l exponent of the S l a t e r o r b i t a l . P i t z e r et a l (100) extended the r e s u l t s , i n c l u d i n g terms from 6-functions o r b i t a l s were derived r e c e n t l y (102). B a r f i e l d (101) has given a complete For these one has to evaluate the expectation value omitted i n the o r i g i n a l c a l c u l a t i o n s . Corresponding expressions for 3p IT set of formulae f o r cb,ij>=ls, 2s, 2p^, 2p o > For example i f z i s the bond d i r e c t i o n <2p y|8 z y|2s> = <2p x|6 x z|2s> = - ( ^ ) R - 3 { 1 5 a -[4a 4+10a 3+20a 2+30a+(15/a)]e" 2 a} .. [2-48] - 41 --3 A l l the integrals have the general form of an R variation at large distances (corresponding to a point dipole interaction) modified by more complicated variation at short distances. As long as the total dipolar interaction can be represented as a sum of two centre terms, these formulae can be used to evaluate the interaction at a given nucleus for any geometrical' arrangement of 2s and 2p orbitals. This fellows because an unpaired electron distribution given by, say, 2 2 ^ c^2px+c 2p y can be represented by the single term ( c x +cy ) 22p', where 2p' i s the 'vector sum' of the x and y terms. 2p' i t s e l f can then be resolved into a and TT components. Using this approach theoretical estimates of this dipolar coupling tensors can be made as described below; the method is that of the author's program Dipole which was used in the interpretation of the data presented in Chapters 4 and 5. Since both the main radicals studied here were of simple TT or a symmetry i t was sufficient to include only one 2s and one 2p orbital per atom. The distribution of spin density in CC^- also means that for both systems the a l l p-orbitals can be taken as p a r a l l e l . The spatial co-ordinates of the magnetic nucleus in question are read in together with those of a l l the atoms contributing spin density and the corresponding orbital exponents. The spin distribution i s represented by the coefficients c and c of the s and p orbitals on each atom, and the ' s p ^ ' total dipolar tensor is estimated as a sum of two-centre terms, as follows. The vector between the magnetic nucleus and the i t h atom is taken as the z-axis of a local co-ordinate system and the p orbital on atom i i s resolved into n and a components p - p cosG + p sine [2-50] 0" Tf - 4 2 -where 0 i s the angle between t h i s z'-axis and the axis of the p - o r b i t a l as shown i n F i g . (5 ). The s i x terras such as c c <2sI 6I2p >cos0, c <2p >Ie12p >sin© cos© S p TT p TT 0" are now evaluated using B a r f i e l d ' s equations and summed to give the t o t a l c o n t r i b u t i o n of atom i . The r e s u l t i n g tensor i s transformed i n t o the o r i g i n a l or 'lab' frame using the transformation given by Derbyshire (92) ; the contributions from other atoms are evaluated i n the same way and added up i n the lab. frame. F i n a l l y the t o t a l tensor i s diagonalised to f a c i l i t a t e comparison with experiment. 2.6 (f) Three Centre Terms Three centre terms are of the form, B = y N u e « ) , C 1 ) | e ( 2 ) | ^ 3 ) > [2-60] with a), 6 and tj> a l l on d i f f e r e n t centres. These terms are generally small since the integrand i s s i g n i f i c a n t only i f <j> and IJJ have appreciable overlap i n the neighbourhood of centre 2. The three centre i n t e g r a l s can be expressed i n terms of two centre terms by Mulliken's approximation (103) < « ( 1 ) |e<2> | ^ 3 ) > = W C D 1 ^ 3 3 , ^ ( 1 ) | Q(2) k ( i ) > + < ^ ( 3 ) ^ ( 2 ) U ( 3 ) > ] i [2-61] Higher terms are generally undetectable. 2.6 (g) Applications The t h e o r e t i c a l importance of the an i s o t r o p i c coupling tensor i s - 43 -t h r e e f o l d . F i r s t l y , the elements of B are proportional to cz (or, f o r cross terms, to the quantity c ^ c ^ ) , so that the magnitude of the coupling can provide an estimate of the spin density d i s t r i b u t i o n which i s independent of the McConnell r e l a t i o n . Secondly, i t i s a source of geometrical information. For TT r a d i c a l s two extreme cases are of most i n t e r e s t . For small r , as i n the ' c l a s s i c ' C-H fragment only one centre of spin density i s s i g n i f i c a n t , and the p r i n c i p a l values of the d i p o l a r tensor approximate (b, 0, -b) and the f i r s t two p r i n c i p a l d i r e c t i o n s r e s p e c t i v e l y define the bond and the axis of the p o r b i t a l . In general, whether or not the r a d i c a l has TT symmetry, the most p o s i t i v e p r i n c i p a l value w i l l correspond to the bond d i r e c t i o n . The other extreme i s the case of large r , when the s p a t i a l extent of the o r b i t a l s becomes i n s i g n i f i c a n t and the i n t e r a c t i o n approaches that of two point d i p o l e s ; B i s then a x i a l and the largest p r i n c i p a l value corresponds to the i n t e r -dipole d i r e c t i o n . This i s us e f u l i n i d e n t i f y i n g the protons responsible f o r intermolecular couplings; and, i f the spin density on the paramagnetic centre i s known, the magnitude of the coupling can be used to estimate r . T h i r d l y , i f the sign of the spin density i s known, as i s u s u a l l y the case, the form of the d i p o l a r tensor determines the sign of the t o t a l coupling tensor. This i s one of the few ways of determining the sign of A f o r small couplings. - 44 -Chapter 3 EXPERIMENTAL 3.1 ENDOR Spectrometer The spectrometer used for the ENDOR measurements has been described i n f u l l elsewhere (78): the main features of the instrument together with a few modifications are given below, and summarised i n Figure 6. The spectrometer i s b u i l t around an X-band EPR spectrometer and i s capable o f operating i n e i t h e r the homodyne or superheterodyne modes. Almost i n v a r i a b l y , however, experiments at the usual temperatures of 77K or 4.2K require the use of such low microwave powers that the r e s u l t i n g afc i n s t a b i l i t y and noise l e v e l s make homodyne operation impracticable. A l l the r e s u l t s reported here were obtained using the superheterodyne mode. In t h i s case s i n g l e sideband detection i s employed with an intermediate frequency of 30MHz. The k l y s t r o n frequency i s s t a b i l i s e d by a Microwave Systems Inc. Model M0S-1 frequency synchroniser. I n i t i a l l y t h i s frequency i s matched to the resonant frequency of the microwave c a v i t y , but i s held constant t h e r e a f t e r , to an accuracy of a few kHz over a period of hours. I f the c a v i t y frequency d r i f t s due to changes i n temperature as the r f heating v a r i e s during a scan, the mis-match of frequencies w i l l cause the noise l e v e l to r i s e . At 4.2K - 45 -KLYSTRON f KLYSTRON POWER SUPPLY ISOLATOR X T SYNCHRONISER 50dB DIRECTIONAL COUPLERS ATTENUATOR 20d3 DIRECT ION A U COUPLER BALANCED MIXER ^ATTENUATOR 30 MHz MULTIPLIER v AMPLIFIER ' i n ISOLATOR PRECISION ATTENUATOR ~ T T ^ FILTER CAVITY PHASE SHIFTER MATCHED LOAD MAGIC TEE SAMPLE.. CAVITY ! ISOLATOR MICROWAVE SWITCH ISOLATOR MIXER a DETECTOR TO MODULATION COILS C R O MAGNET POWER SUPPLY & F1ELDIAL MARK I I 1 ISOLATOR 30 AMP MHz LIFIER > '—. RF DETECTOR RF AMPLIFIER t AUDIO POWER LOCK-IN AMPLIFIER SYSTEM RF SIGNAL GENERATOR AUDIO AMPLIFIER SCANNING UNIT FREQUENCY COUNTER DIGITAL-ANALOGUE CONVERTER X.Y RECORDER Fig. 6. Block diagram of ENDOR spectrometer. - 46 -t h i s may need continual a t t e n t i o n , but at 77K occasional adjustment of the f i n e tuning control on the synchroniser i s s u f f i c i e n t to compensate f o r the d r i f t . The main branch of the microwave power at the c a r r i e r frequency i s led through a s e r i e s of microwave attenuators v i a a Magic T bridge to the c a v i t y . Two types of c a v i t y were used. Both were rectangular, operating i n the T E Q ^ mode and were f i t t e d with 3-turn c o i l s of copper wire to produce the r f f i e l d ; they d i f f e r e d i n the p o s i t i o n of the r f c o i l s . In e i t h e r case the c r y s t a l to be studied i s mounted against the c a v i t y wall beneath the r f c o i l . Vacuum grease i s the usual adhesive. In one designated 'bottom mounting 1 the c o i l was on the c a v i t y f l o o r , so that the r f f i e l d at the sample i s i n the same plane as the r o t a t i o n of the s t a t i c magnetic f i e l d . In the 'side mounting' c a v i t y , the r f c o i l i s on the side end w a l l , so that the f i e l d i t produced at the sample i s v e r t i c a l and thus always perpendicular to the s t a t i c f i e l d . This end p l a t e i s removable and held by brass screws; as a r e s u l t the c a v i t y Q i s somewhat reduced, but t h i s i s compensated by the greater ease and accuracy of mounting samples within the c a v i t y . Reflected microwave power from the c a v i t y i s led through the t h i r d arm of the Magic T to a balanced detector; i t i s detected at 30MHz, amplified, and passed to a PAR model 121 l o c k - i n a m p l i f i e r , f o r processing. The output from t h i s a m p l i f i e r i s used to provide the Y-drive of a Hewlett Packard Moseley 7005A X-Y recorder. The X-drive i s provided by e i t h e r the f i e l d sweep from the F i e l d i a l f o r EPR spectra, or by a s i g n a l generated by the frequency sweep f o r ENDOR. (See below) Rf power f o r ENDOR i s provided by an ENI Model 320L wideband a m p l i f i e r rated at 25W for a 50 ohm load. For low frequency ENDOR measurements a - 4 7 -IP1 Model 500 a m p l i f i e r was used i n conjunction with low-pass f i l t e r s . The impedence of the r f load i s not matched to the a m p l i f i e r over the whole frequency range and optimum sig n a l s are given i n the range 10-20 MHz; at higher frequencies the matching can be improved by s u i t a b l e choice of connecting cables. The frequency sweep i s provided by a Marconi TR1066B11 s i g n a l generator replaced by a Marconi-type 2002AS r f o s c i l l a t o r f o r frequencies below ~8 MHz, part of the output of which i s fed to a Hewlett-Packard 5326C frequency counter d r i v i n g a Hewlett-Packard HP 580A digital-analogue converter, which provides the X-drive of the X-Y recorder. In normal use the d-a converter scans modulo 10MHz i n steps of 10kHz; t h i s provides a l i m i t on the r e s o l u t i o n of the system, which i s considerably less than the ENDOR linewidth i n a l l cases encountered. Frequency modulation i s applied to the r f c a r r i e r by the l o c k - i n a m p l i f i e r , d r i v i n g a Hewlett-Packard HP 450A a m p l i f i e r , the output of which i s fed to the Marconi o s c i l l a t o r . The magnetic f i e l d i s provided by a Varian rotatable electromagnet with 9" polepieces, c o n t r o l l e d by a Mark II F i e l d i a l . Magnetic f i e l d modulation was produced by modulation c o i l s wound on the polepieces. The modulation c o i l s were driven from the reference frequency output of the PAR l o c k - i n a f t e r s u i t a b l e power a m p l i f i c a t i o n ; modulation frequencies were t y p i c a l l y "300 Hz at 77K and 50 Hz at 4.2K. The resonance mode of the c a v i t i e s used means that f o r e i t h e r cavity at a magnet o r i e n t a t i o n given by 0=0°, the s t a t i c and microwave f i e l d s are p a r a l l e l and the allowed EPR t r a n s i t i o n s become very weak. Also, when the bottom-mounting c a v i t y i s used, the s t a t i c f i e l d becomes p a r a l l e l to the r f f i e l d when 0=±9O°, so that ENDOR t r a n s i t i o n s become weak. E f f e c t i v e ranges of magnetic o r i e n t a t i o n are thus ±(20°-90°) f o r the side-mounting and ±(20°-75°) f or the bottom-- 48 -mounting c a v i t y . The l a s t half-metre of waveguide and the microwave c a v i t y are enclosed i n two concentric pyrex dewars. For low temperature operation the outer dewar contains l i q u i d nitrogen, and the inner e i t h e r l i q u i d nitrogen or l i q u i d helium. The c a v i t y i t s e l f i s cooled by conduction through a closed copper tube immersed i n the l i q u i f i e d gas, designated t o p r e v e n t the l a t t e r from entering the c a v i t y . A temperature of 4.2K has the advantage of producing a high Boltzmann f a c t o r , and a high c a v i t y Q due to increased c o n d u c t i v i t y . On the other hand the system i s then more s e n s i t i v e to changes i n temperature caused by r f heating; also, the b o i l i n g helium i s a source of noise, and has to be replenished a f t e r about 2 hours i f maximum, r f power i s used. At 77K the nitrogen i n the inner dewar i s quiescent and w i l l l a s t i n d e f i n i t e l y i f the outer dewar i s kept f u l l . The e f f e c t o f a lower Boltzmann f a c t o r i s mitigated by the greater r e l a x a t i o n rates which permit a higher fm c a r r i e r frequency to be used. In general, operation at 77K i s the more convenient a l t e r n a t i v e and most of the measurements described here were made at t h i s temperature. 3.2 EPR Measurements EPR spectra were obtained using a Varian E-3 spectrometer operating at X-band frequencies. 3.3 F i e l d and Frequency C a l i b r a t i o n For EPR microwave frequencies were measured with a Hewlett-Packard 5245L frequency counter f i t t e d with a plu g - i n u n i t HP5255A. Magnetic f i e l d strength was measured using home-made proton NMR probes containing g l y c e r o l . - 49 -The c a l i b r a t i o n points were reproducible to about 0.2kHz or less than 0.1G. For ENDOR the radio-frequency was measured using the X-Y recorder trace as described above. Frequency markers were e i t h e r added by hand a f t e r each spectrum had been run, or were included at 1MHz i n t e r v a l s by an automatic pip-marker b u i l t i n the UBC Chemistry Department e l e c t r o n i c s shop. Magnetic f i e l d s were c a l i b r a t e d using the symmetry of the proton ENDOR spectrum i n the v i c i n i t y of the free proton NMR frequency. This frequency could "generally be located to within 5kHz. Independent f i e l d c a l i b r a t i o n would provide s l i g h t l y greater accuracy, but the greater inconvenience i s r a r e l y j u s t i f i e d , p a r t i c u l a r l y as an external f i e l d probe can measure only the f i e l d outside the c a v i t y , while the ENDOR spectrum r e f l e c t s the f i e l d strength at the sample i t s e l f . In a few cases small s h i f t s i n the value of H were made to optimise the ENDOR si g n a l and t h e i r magnitudes were estimated from the c a l i b r a t i o n of the F i e l d i a l . The c a l i b r a t i o n of EPR or ENDOR t r a n s i t i o n f i e l d s or frequencies achieved by f i t t i n g the po s i t i o n s of the frequency markers to a quadratic expression, and using t h i s function to i n t e r p o l a t e the l i n e p o s i t i o n s . A least-squares f i t t i n g routine written by Dr. J.A. Hebden for a Monroe 1656 programmable desk c a l c u l a t o r was used f o r t h i s . For ENDOR data at le a s t 5 c a l i b r a t i o n points were f i t t e d f o r each spectrum and the f i t reproduced the frequencies of the c a l i b r a t i o n markers to within 5kHz; the quadratic term i n the f i t t e d expression was very small. 3.4 ENDOR Experiments To obtain ENDOR, f i r s t an EPR si g n a l was obtained; a point was selected on t h i s spectrum, and the f i e l d scan and modulation were switched o f f . - 50 -Microwave power was increased to 18-20 dB below the 300mW output o f the kl y s t r o n . This power l e v e l was a compromise between the requirements of saturation at 77K and the noise produced i n the superhet. detection. The radiofrequency a m p l i f i e r was scanned u n t i l an ENDOR s i g n a l was seen,. u s u a l l y near 14 MHz, and then modulation frequency, l o c k - i n phase, microwave power, and f i e l d strength were a l l optimised using the observed s i g n a l . Once established the optimum conditions were found to be quite reproducible from day to day. A d e t a i l e d study was not made, but i n general the strongest ENDOR signals were obtained by saturating the centre of a given EPR l i n e ; f o r CX^ -centres i n i r r a d i a t e d sodium formate, a s h i f t of 1-2 G produced as much as 50% change i n the ENDOR i n t e n s i t y . In most cases the achievement of a reasonable s i g n a l - t o - n o i s e r a t i o required an fm amplitude at le a s t comparable to the ENDOR linewidth. The fm deviations used were 30kHz f o r the strong l i n e s i n the range 12-16 MHz, inc r e a s i n g to 80kHz at higher frequencies where the i n t e n s i t i e s were smaller and the l i n e s less c l o s e l y spaced. The fm c a r r i e r frequency wqs chosen on the b a s i s of a maximum signal - t o - n o i s e r a t i o and was 1 kHz f o r sodium formate and 3kHz f o r potassium hydrogen bisphenylacetate at 77K; at 4.2K the c a r r i e r frequency was ~500Hz. 3.5 I r r a d i a t i o n Units X - i r r a d i a t i o n was c a r r i e d out using a Machlett OEG-60 X-ray tube operating at 40kV, 40mA. U l t r a v i o l e t i r r a d i a t i o n was c a r r i e d out using e i t h e r a Bausch and Lomb SP-200 or a Hanovia 679A high pressure mercury lamp. UV i r r a d i a t i o n at low temperatures was c a r r i e d out i n a pyrex dewar f i t t e d with quartz i r r a d i a t i o n windows; the sample temperature was maintained by - 51 -ei t h e r f i l l i n g the dewar with l i q u i d nitrogen or mounting the c r y s t a l on a copper rod cooled by conduction. 3.6 Sample Preparation  Sodium Formate Commercial Reagent Grade NaHCG^ (Eastman Kodak) was used and c r y s t a l s were grown by slow evaporation of saturated aqueous solutions at room temperature. The c r y s t a l s formed as colourless p l a t e s ca 0.5x0.5x0.1 cm. i n s i z e and elongated along the (101) d i r e c t i o n . (See Chapter 4) They were heated at ~120° C f o r about 20 minutes before i r r a d i a t i o n to remove traces of moisture, and stored i n a desiccator. E s s e n t i a l l y s i m i l a r spectra were obtained from samples not heated before i r r a d i a t i o n . Uv i r r a d i a t i o n produced no change i n the appearance of the c r y s t a l s f o r any of the dosages used. A f t e r X - i r r a d i a t i o n , however, the c r y s t a l s were pale yellow i n colour. This colouration increased s l i g h t l y with time or on heating, but diminished a f t e r uv i r r a d i a t i o n , and seems to be co r r e l a t e d with the presence of a second r a d i c a l . (See Appendix 4) Potassium Hydrogen Bisphenylacetate (KHBP) Phenylacetic a c i d (Eastman Kodak) was c r y s t a l l i s e d from d i s t i l l e d water by evaporation at room temperature. Following the method of Smith and Speakman (107) weighed q u a n t i t i e s of t h i s were taken with ^-molar proportions of potassium carbonate (Malinkrodt 'Analar') and heated i n ethanol u n t i l the potassium carbonate had disso l v e d . Evaporation and cooling of t h i s s o l u t i o n y i e l d e d colourless p l a t e l e t s of KHBP, melting at 143° C. ( L i t e r a t u r e value 142° C (107)). Micro-analysis gave: ca l c u l a t e d f o r K C ^ l ^ ^ C 61.9'H 4.9% found: C 61.8±0.3 H 5.1±0.3% . . - 52 -The p l a t e l e t s were dissolved i n ethanol and la r g e r c r y s t a l s were obtained by slow evaporation at room temperature. These formed as colourless p l a t e s ca 1x0.5x0.1 cm i n s i z e ; smaller c r y s t a l s (0.5x0.5x0.1 cm) were cut from these and used i n the subsequent experiments. On X - i r r a d i a t i o n c r y s t a l s of KHBP became s l i g h t brown i n colour, with a s l i g h t r e f l e c t i v e sheen. There was no apparent change i n t h e i r appearance a f t e r 1-2 months' storage at room temperature; but over 6-7 months the c r y s t a l s gradually became pale yellow i n colour. Changes i n the EPR and ENDOR spectra of X - i r r a d i a t e d c r y s t a l s of KHBP were apparent on a time scale of the order of two weeks. I n i t i a l l y 1 hour's i r r a d i a t i o n was used, but r a d i c a l decay on storage became apparent' i n the course of the ENDOR measurements and a further % hour's i r r a d i a t i o n was necessary to complete the study. 3.7 Data Analysis ENDOR data were taken i n three orthogonal planes by r o t a t i n g the magnet i n i n t e r v a l s of 2.5, 5, or 10° depending on the density of l i n e s . 3.7(a) Preliminary Analysis In general the l i n e s were very c l o s e l y spaced and often overlapped so i t was necessary to use a preliminary f i t t i n g procedure to pick out data points corresponding to a given tensor. A few points were selected v i s u a l l y and used to i d e n t i f y further p o i n t s . The preliminary f i t was to the function given i n equation [2-31] and used a least-squares f i t t i n g algorithm developed by the author and modified by Dr. J.A. Hebden who programmed i t for the Monroe 1656. I f the r e l a t i v e anisotropy of the term (%A+v U) i s small, expansion - 53 -of the square root i n [2-30] shows that the expression [2-31] reduces to simpler form 2 2 v = a'sin 9+b'sin20+c'cos 0 and t h i s form was used f o r the small couplings i n sodium formate. For the KHBP a n a l y s i s , the f u l l form of [2- ] was used, and f o r h i g h l y a n i s o t r o p i c 2 2 tensors a value of 10000 MHz was added to a l l v values before f i t t i n g , to minimise the p r e f e r e n t i a l weighting of high frequency l i n e s . This procedure 2 s h i f t s a l l v values equally but does not d i s t o r t the angular v a r i a t i o n . 3.7(b) Determination of Hyperfine Parameters The e x t r a c t i o n of Spin Hamiltonian parameters from experimental data was performed using the l e a s t squares f i t t i n g program LSF (108) written by Drs. J.R. Dickinson and J.A. Hebden, f o r an IBM 370/168 Computer. For t h i s program the values of S and I are read i n , together with the nuclear magnetic moment and the other elements of the Spin Hamiltonian [2- ] where these are non-zero. The values of these parameters can be treated e i t h e r as f i x e d q u a n t i t i e s or as i n i t i a l guesses at values to be determined. For ENDOR the e l e c t r o n i c g-tensor was put i n the f i r s t category and i n both studies was taken to be i s o t r o p i c . (See Appendix 1). The hyperfine tensor was the only quantity r e f i n e d i n the ENDOR studies,- but i n the studies of uv - i r r a d i a t e d sodium formate both A and g were r e f i n e d . The experimental data are read i n as a s e r i e s of observed t r a n s i t i o n frequencies, with the corresponding f i e l d s strengths and d i r e c t i o n s . The l a t t e r can be defined by ei t h e r two polar angles or a set of Euler angles. The Euler angle option was chosen here as i t leant i t s e l f more e a s i l y to the c o r r e c t i o n of misalignments of the data planes. (See below) For each value and o r i e n t a t i o n of H, the t o t a l spin Hamiltonian i s - 54 -diagonalised numerically; the t r a n s i t i o n frequencies are c a l c u l a t e d and compared to the experimental values, and the r e s u l t i n g set of re s i d u a l s i s used to make a f i r s t order c o r r e c t i o n to the parameters to be r e f i n e d according to the le a s t squares c r i t e r i o n . The whole process i s cycled u n t i l the errors and parameters remain constant. In general the i n i t i a l guess at the required parameters need be accurate only to within an order of magnitude and convergence i s achieved within 5 i t e r a t i o n s . It should be emphasised that the s i x independent elements of A or g, fo r instance, are r e f i n e d separately, so that no assumption i s needed about the magnitude or d i r e c t i o n of any of the tensors. The program FIELDS (109) written by Dr. J.A. Hebden was also used as a diagnostic device. This program i s an inverse of LSF i n the sense that i t uses input Hamiltonian parameters to c a l c u l a t e t r a n s i t i o n f i e l d s or frequencies. Like LSF i t i s based on a numerical d i a g o n a l i s a t i o n of the Spin Hamiltonian, with no mathematical approximations. For the c a l c u l a t i o n of ENDOR frequencies, the appropriate f i e l d strength i s read i n and the di f f e r e n c e between the s p e c i f i e d energy eigenvalues gives the required t r a n s i t i o n . For f i e l d c a l c u l a t i o n s at a given t r a n s i t i o n frequency, the input f i e l d i s taken as an i n i t i a l guess; the co r r e c t i o n to the Hamiltonian from terms i n (H,,, -HT ... ,.. i s treated as a perturbation taken to 7th order and used True I n i t i a l ) v to r e f i n e the guess, and t h i s process i s cycled to convergence. 3.8 CNDO/INDO Calc u l a t i o n s CNDO and INDO c a l c u l a t i o n s were performed on an IBM 370/168 computer using the program of Pople and Beveridge (110), obtained through the Quantum Chemistry Program Exchange. - 55 -3.9 C r y s t a l Alignment A well-formed c r y s t a l can be mounted quite accurately on the c a v i t y wall with a s u i t a b l e edge or face p a r a l l e l to the c a v i t y s i d e . A p a i r of v e r n i e r c a l i p e r s made a u s e f u l 'collimating s l i t ' f o r t h i s purpose and.the estimated uncertainty i s less than 2°. This i s o f the same order as the misalignments i n the waveguide and c a v i t y construction i t s e l f . For large couplings the ENDOR frequency may vary by more than 100kHz per angular degree, so misalignments of t h i s s i z e could cause s i g n i f i c a n t e r r o r s . However the symmetry of the c r y s t a l enables such errors to be estimated and approximate corrections made. Both the c r y s t a l s studied here had C2/c symmetry with 4 molecules per u n i t c e l l . An orthogonal axis system a, b_, c_ was used i n each case, with c*=axb. The r e s u l t i n g spectra show two s i t e s i n the ab and be* planes (becoming degenerate i n ac* and at the b-axis) , corresponding to two tensors T ^ and T t 2 ) r e l a t e d by T ( l ) = T ( 2 ) a = a, b, c* T aa aa T ( D = _ T(2) . T ( l ) = _ T(2) . T ( l ) = T ( 2 ) ab ab ' be* be* ' ac* ac* In Appendix 3 expressions are derived f o r the transformation of T (1) (21 and T by a r o t a t i o n matrix R, when the rotations are small. In t h i s case. R i s approximately diagonal and rotations about the three axes commute and can be treated independently. Such rotations r e l a t e the tensors A ^ and A^ 2^ observed i n the actual planes of observation to the true values T ^ , T ^ which would be obtained * ' as by measurements i n the true ab, be* and ac* planes. - J U -In what follows a,b and c* are taken to mean the pseudo a,b,c* di r e c t i o n s i n the actual data planes. A misalignment of ac w i l l cause a s p l i t t i n g of the l i n e s ; t h i s was not observed i n the KHBP study. A misalignment of the other two planes w i l l s h i f t the crossover points of the l i n e s from and . However, to f i r s t order i n R „ the crossover frequency i s unshifted and allows comparison to be made of a given axis i n two planes, provided that misalignments about the b axis are small. In p r i n c i p l e the equations derived i n Appendix 3 make i t p o s s i b l e to cal c u l a t e the misalignments; attempts to do t h i s however f or KHBP f a i l e d to give consistent r e s u l t s , probably due to imperfect data. In the case o f sodium formate an approximate method was s u f f i c i e n t to reduce the e f f e c t of misalignment within that of the other e r r o r s . For the larger couplings measured i n KHBP the following approach was f i n a l l y adopted. The a-axis i n the ab plane was a r b i t r a r i l y assumed to be accurate. (One needs a reference p o i n t l i k e t h i s because there i s no symmetry d i r e c t i o n i n the ac plane to define a or c; the f i n a l r e s u l t may be i n error by 1 or 2° about b) By d e f i n i t i o n , ab was now accurate to within a r o t a t i o n about a. Comparison of the crossover frequencies i n the ac plane established the p o s i t i o n of a i n the l a t t e r to within 0.3°. (The f a c t that there was no d e t e c t i b l e misalignment of ac was h e l p f u l but not e s s e n t i a l : the average of s p l i t l i n e s would serve the same purpose). c* was then defined as the d i r e c t i o n i n ac at 90° to a, so that ac was correct to within (very small) r o t a t i o n s about a and c. The corresponding frequencies were read o f f and compared to the crossover points i n be*. The 'c*' crossovers must l i e i n the ac plane, although those at b may be s h i f t e d a considerable distance from the true a x i s . Comparison of ac* and be* then established where the l a t t e r cut the former, and enabled be* to be corrected - 5 7 -f o r misalignments about b. A l l three planes were now (to within 0.3°) i n erro r be rotations about a and c* only. The r e s u l t i n g rms error f o r the methylene coupling tensor i n the benzyl r a d i c a l i n KHBP was reduced from 130 to 80kHz, by t h i s procedure. Appendix 3 shows that once misalignments about b have been eliminated the numerical average of A ^ ^ and A ^ ^ i s independent of R to f i r s t order and i s equal to | T „ | . Such averages are used i n the discussions of the r e s u l t s . - 58 -Chapter 4 EPR and ENDOR Studies of C0 2" Centres i n UV- and X-Irradiated Single Crystals of Sodium Formate 4.1 Introduction In 1961 using EPR Ovenall and Whiffen (112) i d e n t i f i e d the C0 2" r a d i c a l ion i n Y _ i r r a d i a t e d s i n g l e c r y s t a l s of sodium formate at room temperature. 23 They showed that the r a d i c a l i n t e r a c t s with a si n g l e Na nucleus, which gives r i s e to an almost i s o t r o p i c , four l i n e hyperfine pattern of s p l i t t i n g approximately 25 MHz. They measured the g-tensor and the hyperfine tensor 13 due to C (I = Js) i n natural abundance (ca 1%), and used the r e s u l t s to estimate the c o e f f i c i e n t s of the atomic o r b i t a l s comprising the Molecular O r b i t a l containing the unpaired e l e c t r o n . The e l e c t r o n i c structure of the r a d i c a l was assigned by analogy with the i s o e l e c t r o n i c species N0 o (also C„ ) f o r which the r e s u l t s of Molecular r 2 2v O r b i t a l c a l c u l a t i o n s were a v a i l a b l e , and i s [(Is)] , ( l a ^ 2 , ( l b 2 ) 2 , ( 2 a ^ ) 2 , 2 2 2 2 2 1 2 (2b 2) , ( S a p , ( l b j ) , (3b 2) , ( l a 2 ) , ( 4 a ^ ; A^ so that the r a d i c a l has 13 a symmetry. Some no n - a x i a l i t y of the C-A tensor was inter p r e t e d i n terms of p o l a r i s a t i o n of the b., l e v e l s . Fig. 7. Typical morphology and axis systems for sodium formate. a, b, c are the crystallographic axes determined by Markila; <X,8,Y are the axes originally determined by Zachariasen in Reference 122. - 60 -The odd-electron Molecular O r b i t a l i s concentrated i n the carbon s and p o r b i t a l s (c =0.39 c =0.67) so that more than 50% of the spin i s l o c a l i s e d s p 13 on the carbon atom. The r a t i o a^ s Q/B for the C hyperfine i n t e r a c t i o n applied to the Coulson r e l a t i o n s h i p (82) in d i c a t e s h y b r i d i s a t i o n of the form sp° (n~1.8) f o r the odd-electron o r b i t a l on carbon and hence implies an 0C0 angle of 130±5°, the un c e r t a i n l y a r i s i n g mainly from that i n the value of B f o r the carbon 2p o r b i t a l . Subsequently C Q ^ ~ was studied i n a range of environments, and by a v a r i e t y of techniques, by other workers; the general conclusions of Ovenall 17 and Whiffen were confirmed. Luz et a l using O-enriched sodium formate 17 measured the 0 hyperfine i n t e r a c t i o n s , and determined the spin d e n s i t i e s on the oxygens d i r e c t l y rather than r e l y i n g on values i n f e r r e d from the ca l c u l a t e d g - s h i f t s (113). Symons et a l surveyed the EPR parameters of CX^" and NO^ i n a v a r i e t y of matrices (114, 115), and discussed the e f f e c t of the environment on the h y b r i d i s a t i o n at the cent r a l atom. Hartmann and Hisatune studied the i n f r a - r e d spectra of 0.0^ i n a l k a l i h a l i d e discs and estimated the 0C0 bond angle to be 127±8° from the e f f e c t of i s o t o p i c s u b s t i t u t i o n on the v i b r a t i o n a l frequencies (116). CQ^ was also produced by the re a c t i o n between carbon dioxide and sodium metal (118). C02~ i s produced when a l k a l i or a l k a l i n e earth formates or oxalates are i r r a d i a t e d , and also by i r r a d i a t i o n of some carbonates notably c a l c i t e (126). In many of these cases the EPR spectrum of the CO^' ion shows hyperfine structure due to i n t e r a c t i o n with a metal c a t i o n , but also i n many cases (X^ i s neither the only product of i r r a d i a t i o n nor the most stable one at room temperature (115). Evidently the small s i z e of the r a d i c a l ion requires an i o n i c environment to s t a b i l i s e the species, but, as suggested by the appearance of metal hyperfine structures such i o n i c i n t e r a c t i o n s tend to have covalent character as w e l l . Thus i f other atoms are present the r a d i c a l centre tends - 61 -to s t a b i l i s e i t s e l f by forming a la r g e r species than CO^ : while CC^ i s quite stable i n c a l c i t e , other species tend to dominate i n hydrated oxalates or formates. S i m i l a r l y i r r a d i a t i o n of acetates or s a l t s of higher a l i p h a t i c acids produces mainly such species as H^CCO^ although CO^ may also be present i n small amounts p a r t i c u l a r l y at low temperatures and i n f a c t Iwasaki et al have studied CC^" r a d i c a l p a i r s i n X - i r r a d i a t e d l i t h i u m acetate (116) . In 13 t h i s study there appears to be a s i g n i f i c a n t d i f f e r e n c e i n C hyperfine parameters between single and paired CG^ ions - presumably the r e s u l t of i n t e r - r a d i c a l forces which modify the geometry of the p a i r s . Sodium formate i s one of the simplest matrices i n which to study CO^ . Unlike many other s a l t s i n which CC^ can be formed, NaHCC^ i s anhydrous and possesses a r e l a t i v e l y simple c r y s t a l s t r u c t u r e . As.a r e s u l t CC^ i s the major product of i r r a d i a t i o n and i s formed i n a p a r t i c u l a r l y simple o r i e n t a t i o n , with i t s axis c o i n c i d i n g with the b axis of the c r y s t a l , and the r a d i c a l i s stable for several months at room temperature. A f t e r the work of Ovenall and Whiffen, sodium formate i t s e l f received furt h e r attention; Whiffen and Chantry (119) obtained u l t r a v i o l e t absorption spectra of s i n g l e c r y s t a l s . B e l l i s and Clough (120) used EPR to study the thermally induced reaction of CO^ i n the, sodium formate l a t t i c e to form a new paramagnetic species. Some new r e s u l t s concerning t h i s r e a c t i o n are presented i n Appendix 4. In 1967 Cooke and Whiffen made ENDOR measurements on y - i r r a d i a t e d sodium 23 formate at 77K (121); they determined the Na hyperfine and quadrupole tensors, and showed that strong proton ENDOR enhancements could also be obtained, but did not make a d e t a i l e d study of the l a t t e r . Since the proton i s o t r o p i c hyperfine coupling can be used as a s e n s i t i v e measure of covalency i n paramagnetic species (85) an ENDOR study of the unresolved proton superhyperfine structure seemed to o f f e r a means of probing - 62 -the environment of the CC^ centre i n sodium formate. A second motive f o r t h i s study was the assignment of the sodium hyperfine s t r u c t u r e . Although, as stated above, the EPR spectra show i n t e r a c t i o n with a s i n g l e sodium nucleus, the c r y s t a l s tructure shows the.formate ion to be roughly equidistant between two Na + ions, and no wholly convincing arguments had been presented to i d e n t i f y the nucleus responsible f o r the observed i n t e r a c t i o n . The EPR r e s u l t s discussed so f a r had been i n t e r p r e t e d i n terms of the a v a i l a b l e c r y s t a l s t r u c t u r e of sodium formate published by Zachariasen i n 1941 (122); t h i s structure d i d not show the p o s i t i o n s of the hydrogen atoms, and we therefore requested a redetermination of the structure at t h i s Department. This was c a r r i e d out by Markila using X-ray d i f f r a c t i o n techniques (123) , and showed some s i g n i f i c a n t differences from Zachariasen's r e s u l t s , most notably i n the choice of axis systems. The r e l a t i o n between Zachariasen's and Markila's axis systems i s shown i n Figure 7. F o r t u i t o u s l y , the orthogonal axes a,b,c* chosen by Whiffen are within 1° of the true axes. The p r e c i s i o n of the X-ray data enabled the charges on a l l atoms to be r e f i n e d by t r e a t i n g the core and valence s h e l l s of each atom separately, with the population of the l a t t e r taken as a parameter to be r e f i n e d . The r e s u l t s , C: 0.16(3)e; 0: -0.23(l)e; H: -0.49(10)e; Na: 0.79(14)e, i n d i c a t e a considerable amount of covalency i n the formally pure-ionic Na-0 bonds, and a h i g h l y unusual Na + ...H" — C + hydrogen bond of opposite p o l a r i t y to the usual forms such as F"—H +.. .F". As determined by Markila, c r y s t a l s of sodium formate are monoclinic with a=6.2590(6), b=6.7573(16), c=6.1716(5) A; 3=116.140(6)°; Z=4; space group C2/c. The sodium formate molecule i s planar with symmetry, and the formate ions l i e i n layers p a r a l l e l to the b axis. Each sodium has s i x oxygen neighbours - 63 -at an average distance of 2.45A. Together with the Na...0 covalency mentioned above, the weak C-H...Na hydrogen bonds form continuous rows of NaHCC^ molecules along the b ax i s . The C-0 bond length i s 1.246(1)A, the 0C0 bond angle 126.3(2)°. Apart from the increased bond angle, the CC^" centre takes up e s s e n t i a l l y the same p o s i t i o n as i t s formate ion precursor; t h i s i s shown quite c l e a r l y by the absence of s i t e s p l i t t i n g s f o r any of the g, hyperfine, or quadrupole tensors. Thus any change i n the r e l a t i v e p o s i t i o n s of the c e n t r a l carbon atom and the two nearest neighbour sodium ions must occur along the b-axis. This point i s relevant to a di s c u s s i o n of the o r i g i n of the sodium hyperfine i n t e r a c t i o n . 4.2 EPR of uv-Irradiated C r y s t a l s . A f t e r a few minutes' uv i r r a d i a t i o n , c r y s t a l s of NaHCG^ showed the c h a r a c t e r i s t i c f o u r - l i n e EPR spectrum of the CO^'-.-Na* species. The spectra were i d e n t i c a l to those obtained by X - i r r a d i a t i o n , and those described by Whiffen et a l f o r y-irradiated c r y s t a l s . Prolonged i r r a d i a t i o n with a Hanovia 679A mercury lamp gave s u f f i c i e n t l y 13 13 strong s i g n a l s to show the C hyperfine l i n e s , from C (with I=h) i n nat u r a l abundance. EPR measurements i n three perpendicular planes provided data from 13 which the g - and C A-tensors were cal c u l a t e d ; t h e i r p r i n c i p a l values are shown i n Table I, where the r e s u l t s f o r y - i r r a d i a t e d samples are included f o r comparison. 13 The signal-to-noise r a t i o f o r the C s a t e l l i t e s was low, and f o r orie n t a t i o n s where the couplings were smallest, the s a t e l l i t e s overlapped the wings of the main spectrum, making the l i n e p o s i t i o n s hard to determine p r e c i s e l y . In view of t h i s and the v a r i a t i o n i n the published parameters f o r i r r a d i a t e d sodium formate (112, 121) the agreement f o r both g and A i s within IS * the experimental uncertainty and serves to confirm that the products of uv and - 64 -Y - i r r a d i a t i o n are the same. Table I: EPR parameters f o r the CO-" centre produced by Y and UV i r r a d i a t i o n Y I r r a d i a t i o n a UV I r r a d i a t i o n 1 3 . C A-tensor 546 544 436 429 MHz 422 410 g-tensor 2.0014 2.0022 2.0032 2.0032 1.9975 1.9980 a Reference 112. Interposing f i l t e r s of Pyrex, Corex, or Vycor glass between the uv source and the c r y s t a l e stablished that l i g h t of wavelength 260±20 nm i s necessary f o r the ph o t o l y s i s to occur. This value i s based on the published transmission c h a r a c t e r i s t i c s of the glasses, and a semi-quantitative estimate of the r e l a t i v e EPR i n t e n s i t i e s a f t e r equal i r r a d i a t i o n times on a given c r y s t a l . The corresponding energy i s 110+10 kcal/mole, which i s about 25 kcal/mole greater than the C-H bond strength i n sodium formate. It may be s i g n i f i c a n t that there i s a weak absorption band at 255 nm s p e c i f i c to the s i n g l e c r y s t a l spectrum of sodium formate (119). The mechanism of p h o t o l y s i s does not seem to be simple. Several attempts to produce C0 2 by uv i r r a d i a t i o n at 77K f a i l e d to reveal any paramagnetic species, although C0 2 was produced ( i n lower y i e l d ) at temperatures between 300 K and 238 K (1-2 dic h l o r o ethane/liquid nitrogen s l u s h ) . These r e s u l t s are consistent with a mechanism of r a d i c a l formation i n which absorption of energy by a sodium formate molecule produces an e l e c t r o n i c excited state of the molecule which - 65 -w i l l decay back to the ground state unless higher v i b r a t i o n a l l e v e l s are populated, permitting d i s s o c i a t i o n to occur. X - i r r a d i a t i o n of sodium formate at 77K does produce CC^ -; t h i s i s consistent with the above observations since the primary products of X i r r a d i a t i o n are high energy electrons which deposit energy r e l a t i v e l y continuously i n the l a t t i c e , and thus are capable of causing l o c a l heating e f f e c t s as well as s p e c i f i c e l e c t r o n i c t r a n s i t i o n s (124). 4.3 Proton ENDOR Studies X - i r r a d i a t e d c r y s t a l s were used f o r a l l ENDOR measurements since they had much stronger EPR sign a l s than c r y s t a l s i r r a d i a t e d by u l t r a v i o l e t l i g h t . The duration of the X - i r r a d i a t i o n was found to be important: 2-4 hours gave the strongest ENDOR s i g n a l s . This point i s discussed i n more d e t a i l l a t e r . Figure 10 shows a t y p i c a l proton ENDOR spectrum, f o r which the magnetic f i e l d i s oriented at 60° to the £ axis i n the yf3 plane. The spectrum shows the usual near-symmetry about the free proton frequency, vp, and also includes a si n g l e sodium ENDOR l i n e which f a l l s w ithin the range of the scan. The angular v a r i a t i o n of the proton spectra i n the planes of observation i s shown i n Figures 8, 9 where the s i t e s p l i t t i n g s i n the aB and yB planes are c l e a r l y apparent. Approximately 90 measurements were used to f i t each tensor, and the rms. er r o r was t y p i c a l l y 30kHz, s l i g h t l y less than the ENDOR linewidth. The main sources of erro r are small r e s i d u a l misalignments of the c r y s t a l , and the uncertainty i n the l i n e p o s i t i o n s due to the overlap of ENDOR l i n e s . The diagonalised forms of the proton hyperfine tensors obtained are given i n Table I I . Their assignments to s p e c i f i c protons were made by comparing the d i r e c t i o n of the largest a n i s o t r o p i c p r i n c i p a l value to the d i r e c t i o n s expected from the c r y s t a l s t r u c t u r e . Since most of the spin density of the - 66 -Fig. 8. Angular variation of proton ENDOR frequencies in (a) approximate a B and (b) the approximate yB plane. The ordinate is the difference between the ENDOR frequency and the free proton nmr frequency. - 67 -1 1 1 1 1— T T . 1 1 1 | i i I r - 6 0 - 3 0 t 0 3 0 6 0 9 . Angular v a r i a t i o n of proton ENDOR frequencies i n the approximate ay plane of sodium formate. The ordinate i s the d i f f e r e n c e between the observed frequency and the free proton nmr frequency. F i g . 10. A t y p i c a l spectrum of X - i r r a d i a t e d sodium formate at 77K the magnetic f i e l d i s oriented at 30° to the 3 a x i s , i n the Y8 plane. A Sodium ENDOR l i n e i s shown near 16 MHZ. CC>2 fragment is on the carbon atom, the intermolecular C...H directions serve to identify this tensor direction. As a refinement of this procedure we estimated the dipolar parts of the tensors themselves, using the McConnell-Strathdee equations (99), modified by Barfield (101) as described in Chapter 2. The results of these calculations were also used to determine the sign of the total tensor. A l l three atoms of C02~ were included, with the 2s and 2p^ orbitals (the latter being along the crystal b axis) contributing positive spin density; the radical geometry and spin densities were based on EPR data (113). The results of these calculations are also given in Table II. The quoted uncertainties in the experimental principal values are derived from either the deviation between equivalent crystal sites or the experimental rms error, whichever is the greater, and the assignments to specific protons are shown i n Figure 11. In general, the calculated dipolar tensors reproduce the observed values quite well. Some of the deviations are due to neglect of polarisation spin densities in the theoretical model. The most significant deviations occur for tensors 2 and 4, which are markedly less axial than the calculated tensors. In view of the agreement achieved for tensors 1 and 3 and the fact that even a point-dipole approximation should be quite good for the 5.4 A C H distance corresponding to proton 4, the discrepancies can hardly be due to using Slater orbitals in the calculations. Since no reasonable variation of parameters in the calculations reproduced the observed nonaxiality, and the two tensors are those with significant isotropic parts, we conclude that covalency contributes appreciably to these two hyper-fine interactions. On this basis the form of tensor 2 can be explained as follows. The 2p orbital (perpendicular to the radical plane) of one of the oxygens of the C02~ i s directed almost exactly at the carbon bonded to hydrogen 2, (<0CQ=97.5°) and similarly the 2p x orbital of the C02~ carbons is directed at 70 -lb V (+ 2.809) 6 3 (0.0) (-2.809) o2 ( 0 . 0 ) 1 (+2.809) 0 .0 ) <?1 (+2.809)64 (0.0) »c oO o H ©Na o 1 A 11. P r o j e c t i o n of part of the sodium formate l a t t i c e i n t o the c r y s t a l l o g r a p h i c be plane, showing the hydrogen atoms corresponding to the hyperfine tensors l i s t e d i n Table II.. The figu r e s i n parenthesis are the distance i n A perpendicular to the be plane from the plane containing the CO-- r a d i c a l . - 71 -- 72 -an oxygen of the formate ion, so that intermolecular overlap o f the p^ o r b i t a l s on both C and 0 i s p o s s i b l e . Luz et a l (H3) f i n d approximate spin d e n s i t i e s of r e s p e c t i v e l y +8% and -1% i n the 2p^ o r b i t a l s o f C and 0 i n the C0 2~ fragment; hence such overlap can induce spin i n the neighbouring molecule. However the carbon 2p x o r b i t a l i t s e l f cannot be the main source of spin density on that molecule: to account f o r the p o s i t i v e contact term, such spi n would have to be negative (by analogy with the ' c l a s s i c a l ' C-H fragment (111)), and t h i s would make the tensor component c l o s e s t to b more negative, i n c o n f l i c t with observation. Instead we must postulate that e l e c t r o n c o r r e l a t i o n e f f e c t s between CO^" and HCG"2~ ions induce p o s i t i v e s p i n density i n the a o r b i t a l s of t h i s HC0 2~ ion. See Figure 12. This follows i f we regard each p a i r of overlapping 2p^ o r b i t a l s as a s i n g l e o r b i t a l i n which the s p i n density due to p o l a r i s a t i o n changes sign between C0 2 and HC0 2~, as i n the C-H bond o f i r - e l e c t r o n aromatic r a d i c a l s . I f the r e l a t i v e signs of the a and TT p o l a r i s a t i o n s are the same i n the formate ion as i n C0 2 t h i s w i l l then induce p o s i t i v e spin density i n the a o r b i t a l s of the formate. The observed contact term i s a l i t t l e l a r g er than one would expect i f t h i s were the only mechanism i n e f f e c t ; d i r e c t overlap of the a o r b i t a l s of the two ions may make an a d d i t i o n a l c o n t r i b u t i o n . E s s e n t i a l l y s i m i l a r arguments should hold f o r tensor 4. The measured i s o t r o p i c part i n t h i s case i s however rather small, which makes i t more d i f f i c u l t to assess the importance o f the various mechanisms c o n t r i b u t i n g to the spin density at H4. A s l i g h t displacement of the C0 2~ fragment, as suggested by the Na hyperfine couplings, would add f u r t h e r complications but w i l l not e s s e n t i a l l y change the i n t e r p r e t a t i o n . Table I I : Proton Hyperfine Tensors In X Irradiated Sodium Formate Experimental Calculated TENSOR a1so MHz Dipolar Part Dipolar Part Principal value MHz Direction Cosines* 1 m n Principal value MHz Direction Cosines* 1 m n 1 -0.02 10.02 3.15 ± 0.04 -0.053 +0.607 -0.793 3.18 -0.019 +0.404 -0.875 -1.65 t 0.03 0.994 10.042 *0.016 -1.63 0.834 +0.491 0.254 -1.50 ± 0.01 . +0.105 0.793 +0.600 -1.55 10.557 0.724 +0.413 2 0.85 i0.02 2.95 ± 0.04 0.747 ±0.437 0.502 2.66 0.767 ±0.417 0.488 -0.945t 0.04 . -0.609 ±0.752 0.253 -1.34 -0.600 ±0.736 0.314 -1.98 ± 0.004 i0.267 0.495 ;0.827 -1.32 ±0.229 0.532 +0.816 3 0.02 10.01 3.51 ± 0.06 0.785 +0.481 -0.390 2.48 0.720 +0.621 -0.308 --1.82 ± 0.03 0.597 ±0.414 0.687 -1.26 0.476 ±0.120 0.871 -1.69 ± 0.03 i0.168 0.773 +0.612 -1.22 ±0.505 0.774 +0.303 4 t 0.127 tO.006 1.145± 0.006 0.575 ±0.779 -0.251 1.02 0.540 ±0.800 -0-.260 -0.360± 0.010 -0.656 ±0.622 0.428 -0.506 -0.558 ±0.572 0.601 -0.7871 0.005 10.489 -0.082 10.868 -0.517 10.630 0.179 10.755 The direction cosines are referred to the crystal axis system as determined in reference 5; m and n corresponds to b_ and £ of that system and 1 corresponds to the vector b_ x c_. The signs chosen consistently relate one distinguishable crystallographic s i t e to the other. - 74 -4.4 Sodium Hyperfine Interaction In an attempt to shed further l i g h t on the sodium hyperfine coupling which dominates the EPR spectrum of i r r a d i a t e d sodium formate, we i n v e s t i g a t e d the ENDOR of the sodium n u c l e i at 77K. As mentioned e a r l i e r , Cooke and Whiffen (121) made a thorough ENDOR study of t h i s coupling which appears as an almost i s o t r o p i c quartet of s p l i t t i n g ~8G i n the EPR. Our experiments corroborated t h e i r r e s u l t s , but our main i n t e r e s t was to look f o r other sodium couplings, p a r t i c u l a r l y those of the next-nearest-neighbour ions. We c a r r i e d out a survey of the ENDOR spectra down to 2.5 MHz, the free sodium NMR frequency being approximately 3.8 MHz i n our experiments. A t y p i c a l spectrum f o r t h i s frequency range i s shown i n Figure 13. In the e a r l y experiments the presence o f 3rd and 5th harmonics of the main frequency generated i n the r f a m p l i f i e r at high gain, caused spurious signals to appear at 4.8 and 2.9 MHz from the free-proton region of the ENDOR spectrum; these s i g n a l s were removed by the use of appropriate low band-pass f i l t e r s , and the r e s u l t i n g spectra resembled that of Figure 13. The spectra obtained without f i l t e r s showed no new l i n e s above 5 MHz. Angular v a r i a t i o n of the spectra was studied i n three perpendicular planes. The high density of l i n e s , coupled with the f a c t that each sodium nucleus i s expected to give s i x ENDOR l i n e s because of the quadrupole i n t e r a c t i o n , prevented us from analysing the spectra i n d e t a i l , but the r e s u l t s show that an upper l i m i t on the hyperfine coupling of the next-nearest neighbour sodium ion i s approximately 2.5 MHz. Since the signal-to-noise r a t i o f o r the nearest neighbour couplings was ~50, we can be confident that any couplings much la r g e r than 2.5 MHz would have been observed. This leads to the s u r p r i s i n g r e s u l t that the two nearest neighbour couplings d i f f e r by a f a c t o r of 10 or more, but does not t e l l us which sodium i s responsible f o r the observed hyperfine s t r u c t u r e . Nor i s the d i p o l a r part of MHz F i g . 13. Sodium ENDOR spectrum i n the frequency range near the free sodium nmr frequency v M . The break i n the spectrum corresponds to a change i n the f i l t e r s used. - 76 -the observed coupling h e l p f u l here, f o r as discussed i n Appendix 2 the observed 3.7 MHz coupling (121) i s too great by a f a c t o r of almost three to o be a through-space i n t e r a c t i o n across ~3A, and must be a t t r i b u t e d to the presence of spin density i n the sodium p or d o r b i t a l s . From the c r y s t a l structure i t seems that the sodium c l o s e s t to the two oxygens i s the nearest neighbour, since the corresponding C-Na distance i s o only 2.8A, and the observed Na...O i n t e r a c t i o n s would provide another source of s p i n density at the sodium; on the other hand, some 60% of the spin density i n C02~ i s i n the carbon 2p^ o r b i t a l d i r e c t e d towards the other Na, and although t h i s C-Na distance i s 3.9 A, the centre of gra v i t y of the sp hybrid o r b i t a l o w i l l be only -3.0 A from that sodium. Because of t h i s ambiguity, we i n t u i t i v e l y expected some r e l a t i v e s h i f t of the CO^- and sodium ions to have produced the large d i f f e r e n c e i n sodium couplings, and we t r i e d to f i n d evidence f o r such a rearrangement. The ENDOR studies show that the largest p r i n c i p a l value of the main sodium hyperfine tensor l i e s along the c r y s t a l b-axis, which i s also the axis of the formate i on so as mentioned e a r l i e r , any t r a n s l a t i o n must leave the Na-C-Na d i r e c t i o n p a r a l l e l to the b-axi s . The observed proton tensors provided no cl e a r - c u t evidence that the C02~ i t s e l f had moved; and indeed the i n t e r p r e t a t i o n given above f o r the form of two of the tensors suggests that a s i g n i f i c a n t displacement of the fragment has not occured. The ca l c u l a t e d d i p o l a r tensors are too s e n s i t i v e to small u n c e r t a i n t i e s i n geometry and spin density to provide convincing evidence, and we therefore attempted to ca l c u l a t e the r e l a t i v e spin d e n s i t i e s on the two sodium ions d i r e c t l y . We based our c a l c u l a t i o n s on the fragment - 77 -and used the computer program wr i t t e n by Pople and Beveridge (110), i n the CNDO approximation. As a check on the q u a l i t y of the method we also performed a c a l c u l a t i o n on the 'undamaged' fragment, with a l l three formate ions complete; the c a l c u l a t e d charges on the atoms were C: 0.43e; 0: -0.48e; H: - O . l l e ; Na: 0.45e, reproducing the trend of Markila's r e s u l t s . In the subsequent c a l c u l a t i o n s on the paramagnetic fragment we assumed various values f o r the two C^-Na distances, corresponding to displacements of the CC^ or sodium ions along the b-axis, and compared the c a l c u l a t e d s - o r b i t a l s p i n d e n s i t i e s on these three atoms with t h e i r experimental values. Because of the approximations inherent i n the a b s t r a c t i o n of a small part of the l a t t i c e and i n the c a l c u l a t i o n s themselves we made no attempt to optimise the geometry f o r minimum energy. With the atoms at t h e i r c r y s t a l l o g r a p h i c p o s i t i o n s the sodium spin d e n s i t i e s were Na^ 2.1% and Na 2 2.6%, and carbon s - o r b i t a l spin density was 7.3%, i n comparison to the experimental values, Na: 2.6%, C: 15%. The v a r i a t i o n of these q u a n t i t i e s with the p o s i t i o n s of the three ions does not lend i t s e l f to a graphical or algebraic presentation; however some t y p i c a l r e s u l t s are given i n Table I I I , and the main trends are as f o l l o w s . Displacement of the C0 2~ fragment towards Na.^  d i d not a l t e r the r a t i o of sodium spin d e n s i t i e s appreciably and lowered the carbon s p i n density s t i l l f u r t h e r . Displacing Na^ towards increased both sodium spin d e n s i t i e s to about 0.04 but l e f t t h e i r r a t i o close to unity; thus i t seems u n l i k e l y that the sodium ion further from the C0 2 oxygens i s responsible f o r the large hyperfine i n t e r a c t i o n . _ o Displacement of the C0 2 towards Na 2 by distances of order of 1 A a l t e r e d the sodium spin d e n s i t i e s to 0.02 and 0.004 i n favour of Na ? and brought - 78 -Table I I I : CNDO Spin d e n s i t i e s as functions of geometry Y N a x -3.932 -3.732 -3.232 -3.932 -3.932 -4.132 Y N a 2 2.825 2.825 2.825 2.625 2.825 2.625 Y c 0.0 0.0 0.0 0.0 1.0 0.1 A r l 0.0 -0.2 -0.5 0.0 1.0 0.3 A r2 0.0 0.0 0.0 -0.2 -1.0 -0.3 P N 3 l 0.021 0.034 0.038 0.025 0.004 0.019 ^Na 2 0.026 0.039 0.036 0.037 0.021 0.036 P C 0.073 0.067 0.056 0.078 0.111 0.090 The Y values are the o coordinates i n A of the respective atoms. Ar^ i s the change i n Na^..C distance from the c r y s t a l l o g r a p h i c value. A r 2 i s the corresponding value f o r Na2...C. p i s the CNDO s - o r b i t a l s p i n density. - 79 -the carbon spin density c l o s e r to the observed value; s i m i l a r l y displacement of Na 2 towards the CC^ increased the sodium spin density r a t i o and the sp i n density. A f u r t h e r p o s s i b i l i t y , displacement of Na^ away from (the r e s u l t of removing the C-H...Na hydrogen bond) gave a s i m i l a r r e s u l t . From these r e s u l t s i t seems l i k e l y that the sodium ion nearest the oxygens of the C0 2~ fragment i s the one responsible f o r the observed hyperfine couplings, and that the inequivalence of the sodium couplings r e s u l t s from s h i f t s of both Na^ and Na 2 (and p o s s i b l y a small s h i f t of the C0 2~ r a d i c a l i t s e l f ) along the b-axis i n such a way as to increase C-Na^ and decrease C-Na 2 > This conclusion i s supported by the EPR experiments of Bennet, Mile and Thomas (118), who prepared C0 2~ by depositing sodium on the surface o f s o l i d carbon dioxide. This method of preparation p r a c t i c a l l y ensures that C0 2 -Na + ion p a i r s are e a s i l y formed, and that the Na partner i s the one responsible for the dominant hyperfine structure i n the EPR spectra of C0 2 . S i m i l a r r e s u l t s were obtained i n v i b r a t i o n a l studies by Jacox and J i l l i g a n (125) . It i s i n t e r e s t i n g to note also that the Na hyperfine tensor i s not quite a x i a l about the formate C 2 a x i s . A s i m i l a r though more pronounced n o n a x i a l i t y i s shown by the quadrupole coupling as determined by Cook and Whiffen. These observations suggest that i n sodium formate the C0 2~ centre i s held r i g i d i n the l a t t i c e , a conclusion which i s r e i n f o r c e d by comparing the EPR parameters measured at room temperature and 77K (121, 112). For sodium formate there i s no s i g n i f i c a n t change i n the parameters between these two temperatures, i n contrast to observations of C0 2 i n c a l c i t e (126) where marked temperature dependences of the EPR linewidth and g-value are a t t r i b u t e d to t o r s i o n a l 13 o s c i l l a t i o n s o f the r a d i c a l ; also i n l i t h i u m acetate (116) the form of the C hyperfine tensor i s interpreted i n terms of motional averaging by nearly free - 80 -I 1 1 1 r 16 17 18 MHz Fig. 14. Sodium ENDOR spectra obtained by irradiating EPR lines i in turn, in order of decreasing f i e l d . iv - 81 -F i g . 15. Appearance of a l l three sodium ENDOR l i n e s obtained by i r r a d i a t i the lowest f i e l d EPR l i n e . The two weak l i n e s are also shown at higher gain. - 82 -oscillations about the 0...0 directions. 4.5 ENDOR Intensities and Relaxation Mechanisms During the ENDOR studies of the nearest neighbour Na nuclei some effects were observed which cannot be explained in terms of the simplest model of ENDOR (references (59, 7)). and thus have a bearing on the ENDOR mechanism i t s e l f . The f i r s t of these effects concerns the ENDOR spectra observed when each of the four EPR lines was saturated i n turn. Typical results are shown i n Figure 14, where i t can be seen that not only does the intensity of the strongest line vary, but also that other, normally 'forbidden' lines appear. ('Forbidden' in the sense that the simple model of ENDOR enhancements predicts no such lines but rather when as here a/2<vXT , saturation of the m =±3/2 lines within a Na' I given ms manifold i s expected to give one ENDOR transition each, and saturation of the m T=±% lines to give two equally intense lines separated from the f i r s t pair by the quadrupole interaction.) T h i s effect also appears without comment in the spectra published by Cook and Whiffen (121). Figure 15 shows the appearance of a l l three high frequency ENDOR lines obtained by saturating the lowest f i e l d EPR transition. The 'expected' ENDOR line is at 16.5 MHz. It should be emphasised that the results presented in Figures 14 and 15 were quite reproducible, and essentially independent of the position of the saturation point within a given EPR line. Electronic mechanisms are thus not the cause of this effect. We attempted to model the relaxation behaviour by a set of coupled equations in which the rates of change of population of the eight multiplet levels were given i n terms of the population differences and f i r s t order rate constants representing microwave transitions and T , T and T processes, - 83 -these rate constants being treated as adjustable parameters. Requirement that the system be i n a steady state produced eight l i n e a r simultaneous equations which were then solved numerically f o r the populations of the eight l e v e l s . We did not succeed i n developing a qu a n t i t a t i v e d e s c r i p t i o n of the observed ENDOR i n t e n s i t i e s , but were able to draw some q u a l i t a t i v e conclusions. The observed i n t e n s i t y r a t i o s of the strong ENDOR t r a n s i t i o n s v a r i e d too unsymmetrically with m^  to be the r e s u l t of simple T^ processes (Airig±l, or Airij=±l). The most l i k e l y r e l a x a t i o n mechanisms are through v i b r a t i o n a l modulation of the sodium quadrupole and hyperfine tensors, and cross r e l a x a t i o n . The quadrupolar and ani s o t r o p i c hyperfine i n t e r a c t i o n s are too small f o r t h e i r contributions to be dominant, but the t o t a l hyperfine i n t e r a c t i o n can be written f a i r l y accurately as a- [S I +MS I +S I )] L z z + - - + so that i t s time v a r i a t i o n can induce t r a n s i t i o n s with Airig=±l, Anij = + i ; i n contrast, the cro s s - r e l a x a t i o n mechanism can induce the forbidden t r a n s i t i o n s Am =±1, A m = ± l , and can r e s u l t from i n t e r a c t i o n with neighbouring paramagnetic centres (127, 128). As has been discussed by Kwiram et a l (127), t h i s l a t t e r mechanism can be the dominant one i n molecular c r y s t a l s when i r r a d i a t i o n r e s u l t s i n the formation of several paramagnetic species, and the e f f e c t i v e r e l a x a t i o n rate i s l i k e l y to be p r i m a r i l y due to modulation of the g-tensor of the most an i s o t r o p i c species. Some evidence f o r the importance of cross r e l a x a t i o n i s provided by our second main observation — the v a r i a t i o n of ENDOR i n t e n s i t y with i r r a d i a t i o n dose. The optimum ENDOR signal-to-noise r a t i o was obtained with i r r a d i a t i o n times of 2-4 h. Longer i r r a d i a t i o n s r e s u l t e d i n a r a p i d decrease i n the i n t e n s i t y of the main ENDOR spectrum while s h i f t i n g i n t e n s i t y to the 'distant - 84 -F i g . 16. ENDOR spectrum obtained from sodium formate c r y s t a l a f t e r 11.5 hr X - i r r a d i a t i o n , showing the strong 'distant ENDOR' proton resonance l i n e . F i g . 17. Sodium ENDOR l i n e s obtained by i r r a d i a t i n g the second lowest f i e l d EPR line at 4.2K. \ - 86 -ENDOR' l i n e (79) . F i n a l l y a f t e r 11-12 hours of i r r a d i a t i o n , the spectrum shown i n Figure 16 was obtained. Comparison with Figure 10 shows that the former spectrum i s completely dominated by the di s t a n t ENDOR l i n e . The EPR spectra of heavily i r r a d i a t e d c r y s t a l s revealed a s i g n i f i c a n t increase i n the concentration of another paramagnetic species, evidently that described by B e l l i s and Clough (120), f o r which the g-tensor has the p r i n c i p a l values 2.002, 2.006, 2.006. Further support f o r t h i s i n t e r p r e t a t i o n comes from the re s u l t s of ENDOR experiments at 4.2K, i n which the sodium ENDOR enhancements were q u a l i t a t i v e l y s i m i l a r to those observed at 77K (See figu r e 17). The molecular motions such as t o r s i o n a l o s c i l l a t i o n s which are responsible f o r s p i n - l a t t i c e r e l a x a t i o n are generally very temperature dependent, and hence such motions cannot provide the dominant r e l a x a t i o n mechanism here. On the other hand c r o s s - r e l a x a t i o n i s r e l a t i v e l y temperature independent, and has been shown to provide the primary r e l a x a t i o n pathway at very low temperatures (129). S i m i l a r l y the no n a x i a l i t y of the g-, hyperfine, and quadrupole tensors for C0^~ implies that molecular motion i s r e s t r i c t r e d i n the sodium formate l a t t i c e . Thus these r e s u l t s are consistent with the suggestion that cross r e l a x a t i o n i s a dominant r e l a x a t i o n route i n i r r a d i a t e d sodium formate, and i t s r e l a t i o n to the i r r a d i a t i o n dose may well explain the frequent f a i l u r e s to detect ENDOR sign a l s i n molecular c r y s t a l s . - 87 -Chapter 5 ENDOR Studies of an X - i r r a d i a t e d Single C r y s t a l of Potassium Hydrogen Bis Phenylacetate 5.1 Introduction This study was undertaken to c l a r i f y an incomplete e a r l i e r study i n these laboratories (130) i n which the benzyl r a d i c a l was i d e n t i f i e d i n the si n g l e c r y s t a l EPR spectra of i r r a d i a t e d potassium hydrogen b i s phenylacetate, (C 6H 5CH 2COO) 2KH, (KHBP). The s p e c t r a l r e s o l u t i o n i n that case was l i m i t e d by the inherent linewidths and by overlapping spectra from other species, and proved inadequate f o r d e t a i l e d a n a l y s i s . An ENDOR study of t h i s system thus seemed to o f f e r the best chance o f determining the sp e c t r a l parameters and hence of comparing c a l c u l a t e d s p i n d e n s i t i e s with experimental r e s u l t s . As discussed below the benzyl r a d i c a l has received much t h e o r e t i c a l a t t e n t i o n , while there have been r e l a t i v e l y few experimental r e s u l t s a v a i l a b l e f o r comparison. The c r y s t a l s tructure of KHBP has been determined by Mano j lovic" and Speakman (131) using X-ray d i f f r a c t i o n methods and a sing l e plane p r o j e c t i o n of the structure has been studied at 300 K and 120 K using neutron d i f f r a c t i o n by Bacon and Curry (132, 133). There are small di f f e r e n c e s i n the values obtained by neutron and X-ray d i f f r a c t i o n , and comparable changes i n the neutron 18. External morphology and axis system f o r a s i n g l e c r y s t a l of KHBP - 89 -Fig.. 19. P r o j e c t i o n of part of the KHBP c r y s t a l l a t t i c e onto the ac plane.. - 90 -diffraction results at the two temperatures studied. The results of Manojlovi: and Speakman showed that the crystal was monoclinic: space group I2/a=C2/c, o Z=4, B=90.6, a=28.40, b=4.49, c=11.90 A. The crystal structure is shown i n Fig. 19; the hydrogen atom lies at a centre of inversion and forms a symmetric 0...H...0 hydrogen bond connecting the two phenylacetate residues. In the course of the analysis, the direction cosines of C-H bonds inferred from ENDOR data were compared with the results of the crystal structure analysis. The agreement was generally good, except that the c components determined by the two methods differed by a sign, suggesting that some misassignment of axes had occured. Dr. J . C. Speakman (134) kindly confirmed that his published data were slightly in error, and that the correct value of 8 is 89.6°. This makes only a very small change in the numerical results but does invert the sign of third component of each vector (135) bringing the crystallographic results into agreement with the ENDOR findings. Since an additional but undetermined small change in crystal structure occurs on cooling the crystal, no attempt was made to correct Speakman's parameters and his room-temperature results were used to interpret the ENDOR data after correction of the sign of the third direction cosine. KHBP crystals had the form of plates elongated along the b axis; this feature together with the characteristic monoclinic site splittings in the be and ab planes (Ch. 3) enables the axes to be located quite easily (Fig. 18). The orthogonal axis sytem a,b,c* was chosen with axb=c*, and the magnetic f i e l d rotated in the three planes ab, ac*, be* as described in Ch. 3. Since 8 is close to 90° the orthogonal axes are of course essentially the same as the monoclinic axes a,b,c. - 91 -5.2 The Benzyl Radical The benzyl r a d i c a l i s one of the simplest odd-alternate neutral hydrocarbon r a d i c a l s , and has been the object of considerable t h e o r e t i c a l i n t e r e s t . Much of t h i s a t t e n t i o n has been devoted to c a l c u l a t i o n of the TT electron spin d e n s i t i e s since they provide a d e t a i l e d test of the wave function which can thus, i n p r i n c i p l e , be compared with experimental r e s u l t s . Carrington and Smith (136) summarised some of the early work i n t h i s d i r e c t i o n . Subsequently, Pople et a l (137) developed the INDO approximation and c a l c u l a t i o n s using t h i s method have been repeatedly applied to s p i n density c a l c u l a t i o n s on the benzyl r a d i c a l (137, 138, 139). Kruglyac and Mozdor (140) performed s e l f consistent configuration i n t e r a c t i o n c a l c u l a t i o n s to various l e v e l s of approximation and discussed the s i g n i f i c a n c e of the d i f f e r e n t methods. More recently Raimondi et a l (141) c a r r i e d out an extensive Valence Bond c a l c u l a t i o n and compared the r e s u l t s with those of other c a l c u l a t i o n s and of experiment. The experimental estimates of the s p i n d e n s i t i e s have h i t h e r t o been based on electron paramagnetic resonance studies of the benzyl r a d i c a l undergoing free r o t a t i o n e i t h e r i n s o l u t i o n (136, 142, 143) or i n an adamantane matrix (144, 145). A recent study by Jones and Wood et a l employed 1 3 C enrichment at the methylene and bridgehead carbons to i n f e r the corresponding 13 TT-spm de n s i t i e s from the corresponding i s o t r o p i c C hyperfine couplings. This method r e l i e s on a McConnell-type r e l a t i o n (85) to estimate the e f f e c t s of O-TT p o l a r i s a t i o n inducing spin i n the carbon s - o r b i t a l s . A l l previous work used the simple McConnell r e l a t i o n s h i p to obtain carbon TT spin d e n s i t i e s from the i s o t r o p i c coupling of the corresponding a proton. In a d d i t i o n to assuming the v a l i d i t y of the McConnell r e l a t i o n s h i p or 13 i t s C analogue these methods have the l i m i t a t i o n that they provide no d i r e c t s t r u c t u r a l information about the benzyl r a d i c a l . - 92 -The conformation and geometry of the benzyl r a d i c a l and i t s analogues have been the subject of several t h e o r e t i c a l studies. Shansal (146) studied the SCF energy as a function of the geometry and o r i e n t a t i o n of the methylene group and showed that the in-plane c o n f i g u r a t i o n p r e d i c t e d by simple valence theory gave the minimum energy. Lloyd and Wood (144) obtained the INDO-minimum energy geometry as a function of the p o s i t i o n s of a l l 14 atoms. Their r e s u l t s again led to a planar geometry, but the H-C-H angle was reduced to 112°, and the carbon skeleton showed considerable quinonoid character. Hitherto there was no experimental evidence a v a i l a b l e with which d i r e c t comparison of any of these r e s u l t s could be made. A major inadequacy of the r e s u l t s of c a l c u l a t i o n s of the spin density d i s t r i b u t i o n i n the benzyl r a d i c a l has been the r e l a t i v e s i z e of the ortho and para spin d e n s i t i e s . The i s o t r o p i c proton couplings are i n the r a t i o a a. para/ ortho=1.22±0.03, while the corresponding c a l c u l a t e d s p i n density r a t i o i s generally less than 1. The great majority of INDO c a l c u l a t i o n s show t h i s trend, the c a l c u l a t e d s p i n d e n s i t i e s being t y p i c a l l y Ppara~0.25, portho~0.26 f o r a r a t i o -0.96. In cases where the ca l c u l a t e d s p i n d e n s i t i e s have r e f l e c t e d the r e l a t i v e sizes of the proton couplings, the c a l c u l a t i o n have u s u a l l y been questionable on other grounds. Thus Nanda and Narasimhan (147) i n t h e i r UHF c a l c u l a t i o n s obtained ortho and para couplings of 7.07 and 7.52G r e s p e c t i v e l y , i n reasonable agreement with experiment, but at the expense of a meta coupling c a l c u l a t e d as 5.11 G compared with the experimental 1.95 G. S i m i l a r l y i n t h e i r CI c a l c u l a t i o n s Kruglyak and Mozdor (140) d i d obtain P p a r a > P o r t h o by incl u d i n g only s i x configurations, but they then comment, "This i s an agreement with the experimental s p l i t t i n g s i f the simple McConnell - 93 -CRYSTAL ORIENTATION • F i g . 20. Angular v a r i a t i o n of the observed ENDOR frequencies f o r the protons. of the benzyl r a d i c a l i n X - i r r a d i a t e d KHBP. The frequencies have been adjusted to a common free proton frequency of 14.3 MHz; the ENDOR linewidth, i n a l l cases less than 0.1 MHz, i s too small to be resolved on the scale of the Figure. - 94 -equation i s used. This agreement must be considered as to be [sic] a c c i d e n t a l f o r a more pre c i s e d e f i n i t i o n of the wavefunction by an extension of the CI basis as well as the use of open s h e l l o r b i t a l s leads to an opposite r e l a t i o n between the spin d e n s i t i e s on para- and ortho-atoms." In 1970 Kuprievich, Kruglyac and Mozdor (154) suggested that the McConnell r e l a t i o n s h i p may be un r e l i a b l e as an estimate of proton hyperfine couplings which d i f f e r by less than 1-2 G. The main exceptions to t h i s trend are the c a l c u l a t i o n s of Simonetta et a l (141) i n which increasing the number VB structures from 14 to 784 al t e r e d the r a t i o p o r t h o / p p a r a from 0.599 to 1.075. The advantage of an ENDOR study on the benzyl r a d i c a l trapped i n a c r y s t a l l i n e matrix i s that i t enables the t o t a l , a n i s o t r o p i c proton coupling tensors to be determined. The d i p o l a r parts can then be r e l a t e d to the r a d i c a l geometry, and also used to provide an estimate o f the spin density d i s t r i b u t i o n which i s independent of the v a l i d i t y of the McConnell r e l a t i o n s h i p . 5.3 Results and Discussion: Benzyl Radical The EPR spectra of X - i r r a d i a t e d c r y s t a l s of KHBP at room temperature or 77K were -120 G wide and poorly resolved (See F i g . 21). Preliminary ENDOR studies, saturating d i f f e r e n t parts of the EPR spectrum, established that, more than one r a d i c a l was present. In addi t i o n to a very large number of l i n e s near the free proton n.m.r. frequency (-14.3 MHz), ENDOR t r a n s i t i o n s were observed at frequencies up to 50 MHz. The angular dependence of the ENDOR spectra studied by changing the o r i e n t a t i o n of the magnetic f i e l d by i n t e r v a l s of 2.5°, 5°, or 10°, depending on the density of l i n e s , i n the three perpendicular planes, ab, be*, ac*. F i g . 20 shows the angular v a r i a t i o n of the ENDOR t r a n s i t i o n s due to the benzyl r a d i c a l i n the planes of observation. For t h i s - 95 -22 24 26 28 30 32 34 36 38 40 F i g . 21. T y p i c a l EPR (a) and ENDOR (b) spectra of X - i r r a d i a t e d KHBP obtained at 77K. The ENDOR spectrum was obtained by saturating the EPR spectrum at the point' shown by the arrow. The breaks i n spectrum (b) correspond to a change i n o s c i l l a t o r band at 22 MHz, and a reduction i n gain between 14 and 15.5 MHz. - 96 -figur e the t r a n s i t i o n frequencies have been corrected to a free proton nmr frequency Vp=14.30 MHz. The angular v a r i a t i o n can be described by the usual spin Hamiltonian, Jt = BH.g.s. + I ( I C i ) . A ( i ) . S - g N B N l H . I ^ ) where g i s the ele c t r o n g-tensor, S the unpaired electron s p i n operator and A.^ and 1 ^ are r e s p e c t i v e l y the hyperfine coupling tensor and nuclear spin tv* operator f o r the i proton. This Hamiltonian does not contain the nuclear-nuclear d i p o l a r term I^.d.I^ which has been.observed i n s i m i l a r systems containing methyl or methylene groups (148, 149) . The con t r i b u t i o n o f the di p o l a r i n t e r a c t i o n i n such systems i s less than 50 kHz, which i s with i n the linewidth f o r the methylene proton ENDOR s i g n a l s . The a n i s o t r o p i c g-tensor components f o r the benzyl r a d i c a l have not been reported; the r e s u l t s given i n Appendix 1 suggest that i t might be po s s i b l e to estimate the g-anisotropy from i t s e f f e c t on the ENDOR frequencies. Attempts to do t h i s by allowing the program LSF to r e f i n e the g-tensor were unsuccessful, however, the c a l c u l a t i o n s f a i l i n g to converge. Accordingly g was taken as g^soU=2.0023 U f o r the analysis of the ENDOR data; t h i s approximation i s expected to introduce a n e g l i g i b l e e r r o r . For the meta proton couplings both ENDOR t r a n s i t i o n s were used, but f o r the other couplings of the benzyl r a d i c a l only the high frequency t r a n s i t i o n was observed. Apart from the approximations mentioned above we made no assumption regarding the magnitude or o r i e n t a t i o n of any of the hyperfine tensors. The minimum number of points used to f i t a tensor was 37, the maximum 84 and the rms error varied from "TOO kHz f o r the large couplings to -20 kHz f o r the meta proton couplings. The main source or erro r i s probably r e s i d u a l misalignment of the c r y s t a l , a f t e r the corrections described i n Ch. 3 had been applied. - 97 -After the methylene proton couplings had been identified as belonging to the benzyl radical i t was possible to assign the ortho and para couplings by comparing their isotropic parts to the values obtained by EPR. However, in order to pick out the ENDOR transitions of the meta protons from the large number of lines below 25 MHz i t was necessary to predict their approximate angular variation. Accordingly the spin density obtained by EPR for the meta carbon was used in the equations of McConnell and Strathdee (99) and the approximate total hyperfine tensors reconstructed. The corresponding angular variation of the ENDOR frequencies was then calculated using the program FIELDS and used to select the appropriate set of data points. The self consistency of the results and their agreement with earlier work serve to ju s t i f y this procedure. The diagonalised forms of the hyperfine tensors so obtained are given i n Table IV; i n Table V the corresponding isotropic parts are compared with the values from EPR data. In both cases the signs of the tensors are chosen to be consistent with INDO results. Comparison of our results with those of other workers shows that there is only a small variation of the isotropic coupling constants with the environment of the radical. This allows us to neglect the effects of the crystal matrix and is consistent with the observations of. Manojlovic and Speakman (131) and of Bacon and Curry (133) that the benzene ring interacts relatively weakly with neighbouring molecules. The direction cosines in Table IV show that a l l the proton tensors have a common TT direction (perpendicular to the ring). Apart from that of meta(2) (A6=6.5°) a l l the 'T T ' vectors are within 3.6° of the mean, which i t s e l f is 7° from the normal to the benzene ring in the undamaged molecule. This result confirms the planarity of the benzyl radical and shows that no major reorientation in the crystal occurs as a result of radical formation. Similarly the 'a' direction for the para - 98 -Table IV Proton Hyperfine Tensors i n the Benzyl Radical Proton Ortho(l) rms Error,kHz Methylene(1) 74.0 Methylene(2) 90.0 50.0 P r i n c i p a l Value MHz Direction cosines * 1 m n -40 22 0. 6173 ±0. 7855 0 .0442 V -20 29 -0. 6587 ±0. 5467 0 .5175 AJL. -70. 31 -0. 4303 ±0. 2901 0 .8548 -44. 08 0. 6372 . ±0. 7707 0 .0022 -21. 23 -0. 0425 ±0. 0315 0 .9986 \ -69. 96 0. 7697 ±0. 6361 0 .0534 An -16. 21 0. 6758 ±0. 7371 0 .0020 ACT - 7. 78 -0. 5689 ±0. 5232 -0 6346 -19. 08 -0. 4688 ±0. 4277 0 7729 0rtho(2) AT, -15 .85 0 6615 ±0 .7496 0 .0197 44 .0 ACT - 7 .76 -0. 0728 ±0 .0903 -0 .9933 -18 .77 0. 7463 +0 .6556 0 .1142 Meta(l) K 2. 69 0. 6726 ±0 7397 0 .0192 23 5 A<r 4. 80 -0. 5639 ±0 .5292 -0 6340 AJL 7. 15 0. 4791 ±0 4157 0 7731 Meta(2) A* 2. 49 0. 6543 ±0 7368 0 1703 23. 1 A* 4. 86 -0. 1297 +0 1126 0 9852 AJ. 7. 00 0. 7450 +0. 6751 0 0190 Para 49. A* -17 .05 0. 6176 ±0. 7841 0. 0622 0 A<r - 9 .98 -0. 7113 ±0. 5230 0. 4696 AJ. -26 .28 -.03356 ±0. 3342 -0. 8807 * A f r corresponds to the d i r e c t i o n c l o s e s t to the CH bond; to the axis of the 2p x o r b i t a l and Aj^is perpendicular to both. t The d i r e c t i o n cosines are r e f e r r e d to the axis system a,b,c*, and the ± signs chosen c o n s i s i s t e n t l y r e l a t e one di s t i n g u i s h a b l e c r y s t a l s i t e to the other. - 99 -Table V Proton Iso t r o p i c Coupling Constants (MHz) f o r the Benzyl Radical i n D i f f e r e n t Media Medium KHBPa c Aqueous Soln. Adamantane^ Methylene -45.02 b -45.78 -43.90 Ortho -14.25 b -14.39 -14.22 Meta 4.83 b 4.90 4.76 Para -17.78 -17.19 -16.80 a This work b Average of (1) and (2) c Reference 136 d Reference 145. Signs are chosen to be consistent with INDO r e s u l t s . Table VI Angles <$>^  between o vector of proton tensor i and o vector of para proton tensor Tensor Methylene(l) Methylene(2) Ortho(l) 0rtho(2) Meta(l) Meta(2) <J>° 120.8 121.0 111.6 112.3 67.7 60.3 - J.VVJ -proton coupling i s within 5^° of the corresponding d i r e c t i o n i n the undamaged c r y s t a l . This vector i s expected to be the C 2 axis of the benzyl r a d i c a l . Table VI l i s t s the angles t[) between the ' a ' vector of the para coupling and the corresponding d i r e c t i o n s f o r the other couplings, and shows that the C „ u symmetry of the r a d i c a l i s e s s e n t i a l l y maintained i n the KHBP l a t t i c e . The s l i g h t anomaly represented by the meta(2) proton i s evidently a r e f l e c t i o n o f a small s o l i d - s t a t e i n t e r a c t i o n ; the published neutron s c a t t e r i n g f actors (133) show the two meta protons to be s l i g h t l y inequivalent. The d i f f e r e n c e i n the ENDOR parameters however i s too small to a f f e c t the i n t e r p r e t a t i o n . Since the methylene protons are much fu r t h e r from a l l other centres of spin density than from the methylene carbon and t h i s i s the largest spin density i n the r a d i c a l , these 'a' d i r e c t i o n s should l i e very close to the true C-H bond d i r e c t i o n s . C a l c u l a t i o n s using the McConnell-Strathdee model suggest that the perturbations due to the other spin d e n s i t i e s w i l l s h i f t the o d i r e c t i o n by 0.5° or l e s s . The methylene a d i r e c t i o n s thus imply an H-C-H bond angle of 118±2°. This of course i s expected from simple valence theory but i s i n contrast to the r e s u l t of the INDO c a l c u l a t i o n of Lloyd and Wood i n which the minimum energy configuration corresponded to an HCH angle of 112° (144). By symmetry the a d i r e c t i o n of the para coupling l i e s along the bond d i r e c t i o n as shown above, but f o r the ortho and meta proton couplings, the presence of s i g n i f i c a n t spin density on neighbouring carbon atoms w i l l i n general s h i f t the 'a' d i r e c t i o n of the hf tensor s l i g h t l y away from the bond, and prevents us from i n f e r i n g the CH bond d i r e c t i o n s d i r e c t l y . In order to i n t e r p r e t the a n i s o t r o p i c parts of the hyperfine tensors we made use of the McConnell-Strathdee equations mentioned above,.and attempted to reconstruct the a n i s o t r o p i c tensors using c a l c u l a t e d and experimental spin d e n s i t i e s . Using a Q value of -74 MHz and the McConnell r e l a t i o n s h i p a. =Qp^  we estimated the TT e l e c t r o n spin d e n s i t i e s from the ENDOR coupling - 101 -constants. Using these values and an assumed spin density of -0.1 at the bridgehead carbon we obtained the tensor components shown i n Table VII b where the experimental values are also shown f o r comparison. The 2p^ spin d e n s i t i e s on a l l seven carbon atoms were included i n the c a l c u l a t i o n of each tensor. It can be seen that the agreement with experiment i s quite good but that the c a l c u l a t e d value of the para coupling i s too large, and the sign of the TT component of the methylene proton tensor i s wrong. In an e f f o r t to remove the discrepancies, we considered three e f f e c t s not included i n the MeConnell-Strathdee model. F i r s t l y a p o l a r i s a t i o n , inducing p o s i t i v e s p i n density i n the carbon 2p^ o r b i t a l can be r u l e d out because i t would add an a x i a l component to the hf tensor i n a sense which would increase the discrepancy. E a r l i e r work too has shown that a p o l a r i s a t i o n can be neglected i n such c a l c u l a t i o n s . (99, 159) A second p o s s i b i l i t y i s motional averaging of the tensor components by t o r s i o n a l o s c i l l a t i o n s . This can be considered by the following model. Consider a tensor T i n a general axis system.x, y, z. I f the tensor i s rotated i n t h i s frame by an angle 8 about z, where 0(t) i s the instantaneous angular displacement, the new tensor i n the x, y, z frame i s T'(t) = R T(9).T.R(0) (5.1) where R i s the r o t a t i o n matrix S3 cos0 -sine 0 sine cose 0 0 0 1 - 102 -T'(t) i s thus r e a d i l y obtained i n terms of 0. The 'observed' tensor T w i l l be the time average of T'. To evaluate t h i s one assumes a simple harmonic o s c i l l a t i o n so that 0=0Qcoscot, and expands the trigonometric functions as power se r i e s i n 0. Thus f o r example 1 3 1 3 3 sin0 = 6 - -r6 + = 0 coscot - -r- 0^ cos cot + ... 6 o 6 o 2 2 2 cos0 = 1 - HQ + ... = 1 - ^6 cos cot + ... (5.2) The time averages of these functions are now obtained by i n t e g r a t i n g t over one period and d i v i d i n g by 2TT. The c a l c u l a t i o n i s s i m p l i f i e d by noting that sin0 i s an odd function of coswt and hence must average to zero. For 0 q<O.5 radians the cosine s e r i e s converges r a p i d l y and the f i n a l r e s u l t f o r the elements of — . . 2 T i s , to terms i n 0 , =5 ' O T = T (1-^0 2) +T„„%0 2 aa aa^ o J 88 o T = T ( l - % 0 2 ) , a,B=x,y a Z \ (5.3) T = T (1-0 ) . xy xy o T = T zz zz Note that the trace i s conserved, the xx and yy elements are mixed and the o f f diagonal elements reduced i n magnitude. The motional parameters f o r KHBP obtained by neutron d i f f r a c t i o n (133) in d i c a t e that 0 decreases from -5° to -4° between 300 K and 120 K, so at o ' 77 K we can s a f e l y take 0 Q<O.l rad, with the r e s u l t that motional averaging cannot produce an e f f e c t greater than -0.2 MHz f o r even the methylene coupling. The t h i r d c o n t r i b u t i o n we considered was the e f f e c t of off-diagonal elements of the spin density matrix. Our INDO c a l c u l a t i o n showed that several of these were s i g n i f i c a n t , (Fig.22) notably values of p=-0.23 from the ortho-- 103 -Table VII Proton Dipolar Coupling Tensors i n Benzyl Radical (MHz) a b c d Methylene D<r D x 0.86 24.26 -25.12 -0.15 26.93 -26.78 -0.09 26.48 -26.39 -3.40 26.79 -23.39 4,0 121 121 121 121 Ortho -1.79 6.47 -4.68 -2.29 7.61 -5.32 -2.27 7.26 -4.99 -1.85 6.25 -4.40 cf>° 112 109 110 109 Meta D A D«r D X -2.24 -0.01 +2.25 -2.66 0.47 2.19 -2.66 0.47 2.19 -2.28 0.14 2.14 64 66 66 79 para Pit 0.72 7.79 -8.51 -0.77 11.27 -10.50 -0.76 11.04 -10.28 -0.32 8.27 -7.95 <{,«>; 0 0 0 0 a Experimental b Calculated using spin d e n s i t i e s from McConnell r e l a t i o n s h i p . c As for b, plus terms due to off - d i a g o n a l spin d e n s i t i e s . d Calculated using the following spin d e n s i t i e s : methylene 0.64, bridgehead -0.11 ortho, para 0.16 meta -0.05. - 1 0 4 -P=-0-025 \ \ S= 0-25 \ p = -g.2 3 \' \ S= 0-034 Fig. 22. INDO overlap spin densities in the benzyl radical. The corresponding overlap integrals, S, are also shown. - 105 -methylene TT o r b i t a l s and p=+0.19 from the methylene-para overlap. Comparison with the corresponding values of the products c^c^ of the molecular o r b i t a l c o e f f i c i e n t s showed that t h e s e o f f diagonal spin d e n s i t i e s were predominantly overlap rather than p o l a r i s a t i o n terms. We estimated t h e i r c o n t r i b u t i o n to the t o t a l tensors using Mulliken's approximation (103). The overlap i n t e g r a l At B S = <2p I2p > was obtained from the INDO r e s u l t s , and the one-centre i n t e g r a l s were evaluated using the McConnell Strathdee equations. The r e s u l t of adding these terms i s shown i n Table VII where i t can be seen that a small improvement i n the c a l c u l a t e d values r e s u l t s . I t i s i n t e r e s t i n g to compare these r e s u l t s with the valence bond c a l c u l a t i o n s of Raimondi et a l (141), i n which increasing the number o f p a r t i c i p a t i n g valence bond configurations s t r i k i n g l y improved the c a l c u l a t e d spin d e n s i t i e s . Formally the e f f e c t of off-diagonal spin d e n s i t i e s i n an LCAO approach i s equivalent to the co n t r i b u t i o n of d i f f e r e n t valence bond structures as cross terms i n the evaluation of the d i p o l a r tensors, as o u t l i n e d i n Ch. 2. However a f t e r i n c l u d i n g these terms the c a l c u l a t e d para coupling i s s t i l l s i g n i f i c a n t l y l a r g e r than experimental value. No reasonable v a r i a t i o n o f parameters i n the McConnell-Strathdee model reproduced t h i s trend without r e q u i r i n g the ortho and para spin d e n s i t i e s to be approximately equal; the r e s u l t s of one such c a l c u l a t i o n are given i n Table V l l d . This observation r e c a l l s the tendency of most c a l c u l a t i o n s (136-7, 140-1, 144, 150-1) to p r e d i c t P o r t h o > P p a r a suggests that there may be some p h y s i c a l s i g n i f i c a n c e to the r e s u l t . Although the valence bond configurations included by Raimondi c l e a r l y do contribute as shown above, some doubt remains as to the magnitude of t h e i r e f f e c t . This i s p a r t i c u l a r l y true since there i s no marked trend i n the r a t i o p /p when the benzyl r a d i c a l i s studied i n environments of ortho para / - 106 -d i f f e r e n t p o l a r i t y (Table V) or undergoes F or CSL s u b s t i t u t i o n (144, 152-3). The strong v a r i a t i o n i n ° o r t n o / P p a r a w i t h the number of i o n i c configurations i n the valence bond model makes t h i s observation a l i t t l e s u r p r i s i n g . I t seems possible then that part of the discrepancy between the majority of INDO c a l c u l a t i o n s and the i s o t r o p i c proton coupling tensors determined experimentally i s due to a f a i l u r e of the simple McConnell r e l a t i o n s h i p f o r the para coupling. This suggestion has also been made by Kuprievich, Kruglyak and Mozdor (154). A d i r e c t way of deciding t h i s point would be to 13 measure the C hyperfine couplings at the ortho and para p o s i t i o n s . We know of no such studies, although Lloyd and Wood et a l (14b) have measured the 13 C couplings of the methylene and bridgehead carbons. Recent EPR studies of the s i m i l a r nitrobenzene anion r a d i c a l have 13 1 determined C and H hyperfine couplings (158); i n t h i s case however the experimental r e s u l t s seem consistent with the q u a l i t a t i v e p r e d i c t i o n s of c a l c u l a t i o n s . 5.4 Other Radicals This study of KHBP i l l u s t r a t e s the l e v e l of complexity which can r e s u l t when ENDOR studies are made on a system with a large number of protons. Since the c h a r a c t e r i s a t i o n of the benzyl r a d i c a l was the main object, ENDOR li n e s above 50 MHz were not studied. But even i n the frequency range studied, 10-50, MHz, a t o t a l of the order of 5000 data points were obtained. This volume of data presented considerable d i f f i c u l t y i n a n a l y s i s ; preliminary f i t t i n g to assign points to a given coupling was e s s e n t i a l but often d i f f i c u l t . Excluding the frequency range 14±3MHz which i s l i k e l y to contain l i n e s from intermolecular couplings there were i n d i c a t i o n s of at le a s t 40 couplings i n the frequency range studied. The majority of these must be intramolecular, - 107 -4 and imply the existence of at le a s t f i v e r a d i c a l s . However, low s i g n a l - t o -noise r a t i o s prevented the angular v a r i a t i o n of most of these couplings from being followed f a r enough to determine the tensors; r a d i c a l decay caused many l i n e s to appear i n one plane only. Thus, i n ad d i t i o n to the d i f f i c u l t y of e x t r a c t i n g coupling tensors from the data there i s the problem o f assigning them to i n d i v i d u a l r a d i c a l s . Saturation of d i f f e r e n t points on the EPR spectrum was not of great help here: the r e s u l t i n g v a r i a t i o n i n the ENDOR spectra confirmed the existence of more than one r a d i c a l but with the exception of a very few l i n e s the spectra themselves were too complex f o r c o r r e l a t i o n s to be made. For these reasons the i n t e r p r e t a t i o n of the r e s u l t s must be a l i t t l e more t e n t a t i v e than f o r the benzyl r a d i c a l . Nevertheless, an i n t e r p r e t a t i o n was p o s s i b l e , and the 8 couplings given i n Table VIII were assigned using the following arguments. Tensors 1, 2, and 3 of Table VIII show the large, almost i s o t r o p i c character of t y p i c a l 8 proton couplings i n a TT r a d i c a l . (For these large tensors there i s an ambiguity as to whether the observed ENDOR l i n e i s v< or v> i . e . |a/2 +v "| or |a/2 _ V p l > but t h i s question can u s u a l l y be resolved by varying and noting whether the ENDOR l i n e s h i f t s i n the same or opposite sense.) The s i z e of the i s o t r o p i c parts suggests that these 8 protons are i n t e r a c t i n g with two centres of spin density (cf Ch. 2); the markedly non a x i a l character of the d i p o l a r parts leads to the same conclusion and implies that the r a d i c a l s containing these 8 protons are formed by the addition of a hydrogen atom to the benzene r i n g . In such a r a d i c a l the three main centres of spin density are ortho and para to the point of H-addition. The r a d i c a l , so formed would be analogous to the cyclohexadienyl r a d i c a l described i n Ch. 2, and the r e s u l t s f o r t h i s can be used i n i n t e r p r e t i n g the - 1 08 -Table VIII Hyperfine Tensors f o r Radicals I and II Tensor i s o Dipolar Part. D i r e c t i o n Cosines MHz MHz 5 . 0 6 - 0 . 9 7 2 TO.029 0 . 2 3 2 1 1 1 9 . 3 - 0 . 0 6 0 . 2 3 2 =r0.019 0 . 9 7 3 - 5 . 0 1 - 0 . 0 2 4 ± 0 . 9 9 9 4 0 . 0 2 5 1 5 4 . 0 8 0 . 3 4 2 + 0 . 7 8 3 0 . 5 2 0 2 1 1 5 . 0 2 - 0 . 4 4 1 - 0 . 5 3 3 ± 0 . 2 9 4 0 . 7 9 3 - 3 . 6 4 0 . 7 7 4 ± 0 . 5 4 8 0 . 3 1 7 6 . 0 5 0 . 7 8 4 + 0 . 4 6 2 0 . 4 1 3 3 1 2 0 . 7 7 - 0 . 0 2 - 0 . 2 1 3 ± 0 . 4 2 6 + 0 . 8 7 9 - 6 . 0 4 - 0 . 5 8 1 + 0 . 7 7 6 0 . 2 3 5 1 8 . 3 8 0 . 6 8 8 ± 0 . 4 0 9 0 . 6 0 0 4 - 3 7 . 4 9 0 . 0 3 - 0 . 4 5 0 ± 0 . 8 7 8 - 0 . 0 9 0 - 1 8 . 4 0 0 . 5 7 1 ± 0 . 2 0 9 - 0 . 7 9 5 - 3 . 6 3 - 0 . 5 4 2 + 0 . 8 3 3 - 0 . 1 1 4 5 7 . 0 8 0 . 1 8 0 . 8 4 0 + 0 . 5 3 4 - 0 . 0 9 6 3 . 4 4 0 . 0 1 9 + 0 . 1 4 7 0 . 9 8 9 - 3 . 6 0 - 0 . 5 5 6 + 0 . 8 2 9 - 0 . 0 5 3 2 6 7 . 1 6 0 . 3 5 6 - 0 . 0 1 9 + 0 . 0 7 7 0 . 9 9 7 3 . 2 4 - 0 . 8 3 1 + 0 . 5 5 3 0 . 0 5 9 - 1 . 1 6 - 1 . 1 6 + 0 . 0 8 6 - 0 . 9 1 1 7 2 1 . 1 7 - 4 . 3 2 - 4 . 3 2 + 0 . 8 5 0 0 . 1 4 6 5 . 4 8 5 . 4 8 + 0 . 5 2 0 0 . 3 8 5 - 1 . 6 6 - 1 . 6 6 + 0 . 7 2 1 - 0 . 6 8 3 8 4 0 . 3 9 - 1 . 8 2 - 1 . 8 2 + 0 . 5 2 9 0 . 4 7 2 3 . 4 7 3 . 4 7 + 0 . 4 4 1 0 . 5 5 2 - 109 -Table IX Hyperfine Tensors i n the Cyclohexadienyl and a-Naphthyl Radicals Cyclohexadienyl Proton a(MHz) Methylene m +133.6 -25.17 7.42 -36.57 a-Naphthyl a(MHz) +101.8, 90.4 -30.15 7.7 -36.41 Dipolar part. 15.36 1.16 -16.5 0.12 3.8 -3.92 19.17 0.88 -20.08 - 110 -present data. Table IX shows the i s o t r o p i c couplings obtained by Fessenden and Schulter (155) i n t h e i r EPR study of cyclohexadienyl, together with the t o t a l proton tensors derived from the ENDOR study by Bbhme and Wolf (156) of the c l o s e l y r e l a t e d a hydronaphthyl r a d i c a l . (INDO c a l c u l a t i o n s on r e l a t e d systems give s i m i l a r r e s u l t s , but p r e d i c t almost equal spin d e n s i t i e s at the three main centres.) The main features of these r e s u l t s i n addition to the large g couplings are the c h a r a c t e r i s t i c a couplings to the three centres of p o s i t i v e spin density, and the r e l a t i v e l y large negative spin d e n s i t i e s at the two meta p o s i t i o n s . The g couplings i n KHBP can be t e n t a t i v e l y assigned by t h e i r d i p o l a r p a r t s . D-c F i g . 23. Proton, d i p o l a r hyperfine p r i n c i p a l vectors i n cyclohexadienyl. The three main centres of p o s i t i v e spin density are starred . - I l l -3 Because of the sp hybridisation of the methylene carbon, which causes the methylene hydrogens to l i e above and below the ring, the dipolar directions w i l l not l i e in the radical plane. This makes 8 couplings in an aromatic system harder to assign than a couplings; however, simple consideration of the dipolar interaction between such a methylene proton and the two B centres of spin density (Fig.23) shows that the most negative principal value, D-, of the dipolar tensor w i l l be inclined at 20-30° to the perpendicular to the ring. Similarly one of the positive principal values, D+, w i l l correspond to a direction inclined at 20-30° to the original aromatic C-H bond, but i t s projection i n the ring plane w i l l be along that bond. By projecting the principal vectors of the coupling tensors onto the radical plane i t i s possible to assign tensor 1 to hydrogen addition at C^ (Radical I) and tensors 2 and 3 to hydrogen addition at C. or C_. (Radical II) CO, i * I II Fig. 24. Assignment of hydrogen addition radicals I and II. - 1 1 2 -F i g . 25. Angular v a r i a t i o n of ENDOR spectra f o r r a d i c a l s I and I I . - 113 -These latter two possiblities cannot be distinguished by the g couplings alone as the C^-H^ and C^-H^ directions are essentially colinear. The addition at is relatively unambiguous, since the only alternative would be H addition at the bridgehead carbon which is highly unlikely because of steric hindrance. The a coupling, 4 has i t s direction parallel to C^H^ or C^H^ and i t s magnitude is similar to the para coupling of cyclohexadienyl; thus 4 is assigned to the para position of radical II, viz or C^. The ENDOR lines associated with tensors 5, 6 and 7 weakened simultaneously when the saturation point of the EPR spectrum was altered; these three tensors are therefore assigned to the same radical. Of these, tensors 5 and 6 are typical meta couplings of a cyclohexadienyl type radical, while 7 i s a B type coupling in which the proton interacts with a single centre of spin density. This latter coupling i s consistent with proton H 2 of the original methylene group in KBHP. The '6' spin density would be at the bridgehead carbon: this could arise from H-addition at either or but not at C^; furthermore H-addition at would give rise to only one 'meta' coupling, the other position of negative spin density being the bridgehead carbon. Thus the tensors 5, 6 and 7 are consistent with hydrogen addition at either or but not at Cy. The latter p o s s i b i l i t y is also opposed by chemical intuition which would predict addition to occur o- and p- to the CH2CO0 group rather than m- and p-. The f i n a l assignments of these three tensors is less certain and must be based on the directions of the 'meta' dipolar tensors. By analogy with the meta couplings of the benzyl and a naphthyl radicals, the dipolar value closest to zero corresponds to the bond direction. Projecting the corresponding principal vectors onto the ring plane shows that Cg and C, rather than C„ and C. are themore li k e l y assignments, so tensors 5, 6, 7 - 114 -are tentatively assigned to radical 11(a). The spin density at the bridgehead carbon in radical II is likely to be -0.4 from the data i n Table IX; using this value in equation 2-45 with BQ=9 B^=122 MHZ predicts a dihedral angle of 53°, in reasonable agreement with the value 64° for ^.from the room temperature crystal structure data. Uncertainties in B^ and B^ and a possible reorientation of the radical at low temperatures make this estimate reasonable. In particular B^ may be too low; a value of 150 MHz (157) has been suggested and would give better agreement with the values at C,. C.. 6' 4 The dihedral angle corresponding to proton H^ i s 139° and would lead to an isotropic coupling ~30-40 MHz. The tensor 8 with a=40.4 MHz is in reasonable agreement, but i t s anisotropic part i s anomalous; the positive principal direction is 23° from C^H^ and so the assignment of tensor 9 to proton H_ of radical II must remain conjectural. - 115 -REFERENCES 1. CD. J e f f r i e s i n Progress i n Cryogenics (Heywood § Co. Ltd., London, 1961). 2. A. Abragam i n The P r i n c i p l e s of Nuclear Magnetism (Clarendon, Oxford, 1961). 3. CD. J e f f r i e s , Phys. Rev., _117» 1 0 5 6 (I960). 4. A.W. Overhauser, Phys. Rev., 9_2, 411 (1953). 5. T.R. Carver and C..P. S l i c h t e r , Phys. Rev., 102 (1956). 6. A. Abragam, Phys. Rev., 98, 1729 (1955). 7. G. Feher, Phys. Rev., 103, 500 (1956). 8. G. Feher, Phys. Rev., 103, 834 (1956). 9. G. Feher, Phys. Rev., 114, 1219 (1959). 10. W.C Holton, H. Blum and C P . S l i c h t e r , Phys. Rev. Le t t e r s , 5_, 197 (1960). 11. R.J. Cook and D.H. Whiffen, Proc. Roy. Soc. A, 295, 99 (1966). 12. see f o r instance I.L. Bass and R.L. Mieher, Phys. Rev., 175, 421 (1968). 13. H. Se i d e l and H.C Wolf i n Physics of Color Centres, ed. W.B. Fowler (Academic Press, New York, 1968). 14. H. Se i d e l and H.C. Wolf, Phys. Stat. S o l i d i , 11, 3 (1965). 15. V.G. Grachev, M.F. Deigen, G.I. Neimark and S.I. Pekar, Phys. Stat. S o l i d i B, 43, K93 (1971) . 16. see f o r example C P . Scholes, R.A. Isaacson and G. Feher, Biochim., Biophys. Acta, 263, 448 (1972). 17. C H . R i s t , J.S. Hyde and T. Vanngard, Proc. Nat. Acad. S c i . U.S., 6^ 7, 79 (1970). 18. C H . Ri s t and J.S. Hyde, J . Chem. Phys., 49, 2449 (1968). 19. see f o r example C A . Hutchison and C A . Pearson, J . Chem. Phys., 43, 2545 (1965). 20. C A . Hutchison, J . Phys. Chem., 71_, 203 (1967). 21. C A . Hutchison, Pure Appl. Chem., 27, 327 (1971). 22. S.N. Rustgi and H.C. Box, J . Chem. Phys., 59, 4763 (1973). - 116 -23. J.N. Herak and C A . McDowell, J . Mag. Reson., 16, 434 (1974). 24. M. Iwasaki and H. Muto, J . Chem. Phys., 6_1, 5315 (1974). 25. F.Q. Ngo, E.E. Budzinski and H.C. Box, J . Chem. Phys., 60, 3373 (1974). 26. J.Y. Lee and H.C. Box, J . Chem. Phys., 59_, 2509 (1973). 27. H.C. Box, E.E. Budzinski and K.T. L i l g a , J . Chem. Phys., 57, 4295 (1972). 28. H.C. Box, H.G. Freund, K.T. L i l g a and E.E. Budzinski, J . Phys. Chem., 74, 40 (1970). 29. S.N. Rustgi and H.C. Box, J . Chem. Phys., 60, 3343 (1974). 30. J.N. Herak, D.R. Lenard and CA, McDowell, J . Mag. Reson. (to be published). 31. H.C Box, E.E. Budzinski and W.R. Potter, J . Chem. Phys., 61_, 1136 (1974). 32. A.L. Kwiram, J . Chem. Phys., 49, 2860 (1968). 33. L.R. Dalton and A.L. Kwiram, J . Chem. Phys., 5_7, 1132 (1972). 34. _ J.S. Hyde, G.H. Ri s t and L.E.G. Eriksson, J . Phys. Chem., 72_, 4269 (1968). 35. R.D. Allen d o e r f e r , Chem. Phys. L e t t e r s , T7, 172 (1972). 36. J . Helbert, L. Kevan and B.L. Bales, J . Chem. Phys., 57_, 723 (1972). 37. L.E.G. Eriksson, J.S. Hyde and A. Ehrenberg, Biochem. Biophys. Acta, 192, 211 (1969). 38. J.R. No r r i s , M.E. Druyan and J . J . Katz, J . Amer. Chem. S o c , 95, 1680 (1973). 39. H.L. Van Camp, C P . Scholes and C F . Mulks, J . Amer. Chem. S o c , 9_8, 4094 (1976). 40. D.E.B. Kennedy, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1974. 41. H. van W i l l i g e n , M. Plato, R. B i e h l , K.P. Dinse and K. MObius, Moi. Phys., 26, 793 (1973). 42. R.A. Allend o e r f e r i n Magnetic Resonance, ed. C A . McDowell (MTP Reviews of Science, Butterworths, London, 1974) . 43. R.J. Cook and D.H. Whiffen, P r o c Phys. Soc. (London), 84, 845 (1964). 44. N.S. Dal a i , S.R. Srinivasan and C A . McDowell, Chem. Phys. L e t t e r s , (3, 617 (1970). 45. K.P. Dinse, R. Bie h l and K. Mbbius, J . Chem. Phys., 61, 4335 (1974). - 117 -46. I.Y. Chan, J . Schmidt and J.H. van der Waals, Chem. Phys. Le t t e r s , 4, 269 (1969). 47. L.T. Cheng, J. van Lee and A.L. Kwiram, B u l l . Am. Phys. S o c , 15, 268 (1970). 48. M.J. Buckley, C.B. Harris and A.H. Maki, Chem. Phys. Le t t e r s , 4, 591 (1970). 49. P.D. Parry, T.R. Carver, S.O. S a r i and S.E. Schnatterly, Phys. Rev. Lett e r s , 22_, 326 (1969). 50. A. Abragam and B. Bleaney i n El e c t r o n Paramagnetic Resonance of T r a n s i t i o n  Metal Ions (Clarendon, Oxford, 1970) Ch. 1, 4. 51. A.L. Kwiram, Ann. Rev. Phys. Chem. 22_, 133 (1971). 52. N.S. D a l a i , C.A. McDowell and J.M. Park, J . Chem. Phys., 63, 1856 (1975). 53. J.M. Park and C A . McDowell, Moi. Phys. ( i n Press). 54. M.H.L. Pryce, Proc. Phys. Soc. (London), A63, 25 (1950). 55. A. Abragam and M.H.L. Pryce, Proc. Roy. Soc. (London), A205, 135 (1951). 56. C E . Pake and T.L. E s t l e i n The Physical P r i n c i p l e s of Ele c t r o n Paramagnetic  Resonance, 2nd Ed. (Benjamin, Reading, 1973) . 57. C P . S l i c h t e r i n P r i n c i p l e s of Magnetic Resonance (Harper and Row, New York, 1963) Ch. 4 and 7. 58. A. Carrington and A.D. McLachlan i n Introduction to Magnetic Resonance (Harper and Row, New York, 1967) Ch. 9. 59. A. Abragam and B. Bleaney i n Ele c t r o n Paramagnetic Resonance of T r a n s i t i o n  Metal Ions (Clarendon, Oxford, 1970) Ch. 1, 4. 60. J.M. Baker and F.I.B. Williams, Proc. Roy. S o c , A267, 283 (1962). 61. N.S. Dal a i , J.A. Hebden and C A . McDowell, J . Mag. Reson., 16_, 289 (1974). 62. H.F. Hameka i n The T r i p l e t State, ed. A.B. Zahlan et a l , (Cambridge U.P., London, 1967), A.J. Stone, Proc. Roy. Soc. A 271, 424 (1963). 63. J.B. Farmer, p r i v a t e communication. 64. C P . S l i c h t e r i n P r i n c i p l e s of Magnetic Resonance (Harper and Row, New York, , 1963) Ch. 6. 65. L.R. Walker, C K . Wertheim.and V. Jaccarino, Phys. Rev. L e t t e r s , 6, 98 (1961). ~ 66. A.L. Kwiram, J . Chem. Phys., 55, 2484 (1971). - 118 -67. M. Iwasaki, J . Mag. Reson., 16^ 417 (1974). 68. W.C. L i n , Moi. Phys., 25, 1163 (1973). 69. A. Rockenbauer and P. Simon, Molec. Phys., 28_, 1113 (1974). 70. R. Skinner and J.A. Weil, J . Mag. Reson., 21_, 271 (1976). 71. D.H. Whiffen, Moi. Phys., 10_, 595 (1965). 72. N.M. Atherton and D.H. Whiffen, Moi. Phys., 3_, 1 (1960); H.A. Farach and C P . Poole, Adv. i n Mag. Reson., 5_, 229 (1971). 73. R.E. B e l l i s and S. Clough, Moi. Phys., 4, 135 (1961). 74. J.A.R. Coope, N.S. D a l a i , C A . McDowell and R. Srinivasan, Moi. Phys., 24, 403 (1972). 75. K. Minakata and M. Iwasaki, Moi. Phys., 23, 1115 (1972). 76. V.V. Teslenko, Y.S. Gromovoi and V.G. Krivenko, Moi. Phys., 30_, 425 (1975) . 77. A. Carrington and A.D. McLachlan i n Introduction to Magnetic Resonance (Harper and Row, New York, 1967) Ch. 1. 78. N.S. D a l a i , Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1971). 79. J . Lambe, N. Laurance, E.C Mclrvine and R.W. Terhune, Phys. Rev., 122, 1161 (1961). T. Cole, C. H e l l e r and J . Lambe, J . Chem. Phys., 34, 1447 (1961). 80. J.M. Baker and W.B.J. Blake, J . Phys. C: S o l i d St. Phys., 6_, 3501 (1973). 81. J.H. Freed, Ann. Rev. Phys. Chem., 2_3, 265 (1972). 82. C A . Coulson i n Valence (O.U.P., 1961, London), Ch. VIII. 83. N.S. D a l a i , C A . McDowell and R. Srinivasan, Chem. Phys. L e t t e r s , 4_, 97 (1969). 84. G. Fraenkel and B. Venkataraman, J . Chem. Phys., 24_, 737 (1956). 85. H.M. McConnell and D.B. Chestnut, J . Chem. Phys., 2_8, 107 (1958). 86. D.R. Eaton and W.D. P h i l l i p s , Adv. i n Mag. Reson., 1_, 103 (1965). 87. J.P. Colpa and J.R. Bolton, Moi. Phys., 6_, 273 (1963). 88. G. Giacommetti, P.L. Nordio and M.W. Pavan, Theor. Chim. Acta, 1_, 404 (1963), 89. M.T. Melchior, J . Chem. Phys., 50, 511 (1969). - 119 -90. A.D. McLachlan, Moi. Phys., 3, 233 (1960). 91. C. H e l l e r and H.M. McConnell, J . Chem. Phys., 32, 1535 (1960). 92. W. Derbyshire, Moi. Phys., 5_, 225 (1962). 93. J . Maruani, A. Hernandez-Laguna and Y.G. Smeyers, J . Chem. Phys., 63, 4515 (1975). 94. J . Maruani, Moi. Phys., 30, 1685 (1975). 95. D.H. Marcellus, E.R. Davidson and A.L. Kwiram, Chem. Phys. L e t t e r s , 33, 522 (1975). 96. R. Bersohn, J . Chem. Phys., 24, 1066 (1956). 97. D.H. Whiffen, Moi. Phys., 6_, 223 (1963). 98. K. Morokuma and K. Fukui, B u l l . Chem. Soc. Japan, 36_, 534 (1963). 99. H.M. McConnell and J . Strathdee, Moi. Phys., 2_, 129 (1959). 100. R.M. P i t z e r , C.W. Kern and W.N. Lipscomb, J . Chem. Phys., 37, 267 (1962). 101. M. B a r f i e l d , J . Chem. Phys., 53, 3836 (1970). 102. F.G. Herring, C A . McDowell and J . C T a i t , J . Chem. Phys., 57_, 4564 (1972). 103. R.S. Mulliken, J . Chem. Phys., 46, 497 (1949). 104. M.H.L. Pryce, Nuovo Cimento, Suppl. 6, 817 (1957). 105. J.S. G r i f f i t h s , Theory of T r a n s i t i o n Metal Ions (Cambridge U n i v e r s i t y Press, London, 1961) . 106. N.M. Atherton, El e c t r o n Spin Resonance (Ellis-Horwood, London, 1973). 107. N. Smith and J.C. Speakman, Trans. Farad. S o c , 44, 1031 (1948). 108. J.R. Dickinson, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1974. see also J.A. Hebden, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1970 and N.S. Dal a i , J.R. Dickinson and C A . McDowell, J . Chem. Phys., 57, 4254 (1972). 109. J.A. Hebden, Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1970. Also CR. B y f l e e t , D.P. Chong, J.A. Hebden and C A . McDowell, J . Mag. Reson. 2_, 69 (1970). 110. J.A. Pople and D.L. Beveridge, Approximate Molecular O r b i t a l C a l c u l a t i o n s (McGraw H i l l , New York, 1970). - 120 -111. A. Carrington and A.D. McLachlan, Introduction to Magnetic Resonance (Harper and Row, New York, 1967) p. 106. 112. D.W. Ovenall and D.H. Whiffen, Moi. Phys., £, 135 (1961). 113. S. Sc h l i c k , B.L. S i l v e r and Z. Luz, J . Chem. Phys., 54, 867 (1971). 114. P.W. Atkins, N. Keen and M.C.R. Symons, J . Chem. S o c , 2873 (1962). 115. J.H. Sharp and M.C.R. Symons, J . Chem. Soc. (A), 3075 (1970). 116. K. Nunome, K. Toriyama and M. Iwasaki, J . Chem. Phys., 62_, 2927 (1975). 117. K.O. Hartman and J.C. Hisatune, J . Chem. Phys., 44, 1913 (1966), 118. J.E. Bennet, B. Mile and H. Thomas, Trans. Faraday S o c , 61_, 2357 (1965). 119. G.W. Chantry and D.H. Whiffen, Moi. Phys., 5_, 189 (1962). 120. R.E. B e l l i s and S. Clough, Moi. Phys., 1£, 23 (1965). 121. R.J. Cook and D.H. Whiffen, J . Phys. Chem., 71_, 93 (1967). 122. W.H. Zachariasen, J . Amer. Chem. S o c , 62_, 1011 (1940). 123. P.L. Markila, M.Sc Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1974). 124. J.H. O'Donnell and D.F. Sangster i n P r i n c i p l e s of Radiation Chemistry (American E l s e v i e r , New York, 1970). 125. M.E. Jacox and D.E. M i l l i g a n , Chem. Phys. L e t t e r s , 28, 163 (1974). 126. J.A. McMillan and S.A. Marshall, J . Chem. Phys., 48, 467 (1968). 127. L.R. Dalton, A.L. Kwiram and J.A. Cowen, Chem. Phys. L e t t e r s , 14, 77 (1972). 128. S.K. Wong and J.K.S. Wan, J . Chem. Phys., 55, 4940 (1971). 129. L.R. Dalton, A.L. Kwiram and J.A. Cowen, Chem. Phys. Le t t e r s , 17, 495 (1972). 130. W.C. L i n and C A . McDowell, unpublished r e s u l t s . 131. L. Manojlovic and J.C. Speakman, Acta Cryst., B(24), 323 (1968). 132. C E . Bacon and N.A. Curry, Acta Cryst., 10_, 524 (1957). 133. C E . Bacon and N.A. Curry, Acta Cryst., _13, 717 (1960). 134. J.C. Speakman, p r i v a t e communication. 135. J . T r o t t e r , p r i v a t e communication. - 121 -136. A. Carrington and I.CP. Smith, Molec. Phys., 9_, 137 (1965). 137. J.A. Pople, D.L. Beveridge and P.A. Dobosh, J . Amer. Chem. S o c , 90, 4201 (1968). 138. see f o r example J . Tino and V. Klimo, Chem. Phys. L e t t e r s , 2S_, 427 (1974). 139. H.G. Benson and A. Hudson, Molec. Phys., 20, 185 (1971). 140. Y.A. Kruglyak and E.V. Mozdor, Theoret. Chim. Acta, _15, 365 (1969). 141. M. Raimondi, M. Simonetta and G.F. Ta n t a r d i n i , J . Chem. Phys., 56, 5091 (1972) . 142. W.T. Dixon and R.O.C. Norman, J . Chem. S o c , 4837 (1964). 143. H. Fischer, Z. NatUrforsch., 20, 488 (1965). 144. R.V. Lloyd and D.E. Wood, J . Amer. Chem. S o c , 96, 659 (1974). 145. A.M. I h r i g , P.R. Jones, I.N. Jung, R.V. Lloyd, J.L. Marshall and D.E. Wood, J . Amer. Chem. S o c , 97, 4477 (1975).° 146. M. Shansal, Molec. Phys., 23, 441 (1972). 147. D.N. Nanda and P.T. Narasimhan, Internat. J . Quant. Chem., VIII, 451 (1974). 148. R.J. Cook and D.H. Whiffen, J . Chem. Phys., 43, 2908 (1965). 149. K. Toriyama, K. Nunome and M. Iwasaki, J . Chem. Phys., 64, 2020 (1976). 150. A. H i n c h l i f f e , Chem. Phys. L e t t e r s , L3, 594 (1972). 151. J . C Shug and D.H. P h i l l i p s , J . Chem. Phys., 59_, 1616 (1973). 152. P. Neta and R.H. Schuler, J . Phys. Chem., 77. 1 3 6 8 (1973). 153. L.D. Kispert, H. Lin and C U . Pittman, J . Amer. Chem. S o c , 95, 1657 (1973) . 154. V.A. Kuprievich, Y.A. Kruglyak and V. Mozdor, Intern. J . Quantum Chem., 4_, 73 (1970). 155. R.W. Fessenden and R.H. Schuler, J . Chem. Phys., 39, 2147 (1963). 156. U.R. BOhme and H.C. Wolf, Chem. Phys. L e t t e r s , 17, 582 (1972). 157. J.N. Herak, D. K r i l o v and C A . McDowell, J . Mag. Reson., 23, 1 (1976). 158. G.L. Swartz and W.M. Gulick, Moi. Phys., 30, 869 (1975); R.P. Mason and J.E. Harriman, J . Chem. Phys., 65, .2274 (1976). 159. P. Gloux and B. Lamotte, Moi. Phys., 25, 161 (1973). - 122 -Appendix 1 The E f f e c t of g-Anisotropy on ENDOR Frequencies In the h i g h - f i e l d approximation one replaces s by m h where h i s a unit vector p a r a l l e l to the s t a t i c f i e l d H. I f g-anisotropy i s s i g n i f i c a n t we should use m h' where s~ h- = h.g/(h.g 2.h) H • ( A - l . l ) I f the r e l a t i v e anisotropy i s small we can write h 1 = h+X (A-1.2) 2 where h.X=0 and X « 1 . The h i g h - f i e l d ENDOR Hamiltonian then becomes = ( B j h ' . A - v h).I (A-1.3) with t r a n s i t i o n frequencies given by v = [m 2 h ' .A2.h'-2m v. h' .A.h+v 2 ] * * . (A-1.4) mg 1 s ~ z - s p~ » ~ p J 2 Using the f a c t that A and A are symmetric, s u b s t i t u t i o n of (A.1-2) gives v = [m 2(h.A 2.h+2h.A 2.X+X.A 2.X) m s ~ s ~ ~ ~ ~ ~ = s -2m v (h.A.h+X.A.h)+v 2 ] % s P - ! • ~ - P J = [ v 2 + A] 5 2 , (A-1.5) 1 ISO J ' v where v. i s the frequency corresponding to an i s o t r o p i c g ( i . e . X=0) and A 3.S0 • a -i s the sum of terms containing X. 2 Since X and hence A are assumed small we can neglect terms i n X and approximate the frequency s h i f t , 6v = -v. , by - 123 -6 V " 3A = ^ V i s o + A J V A/2v • l 6 v l * '4^'^.A.h±2v pX.A.h] . (A-1.6) Note that i n the axis system defined by (h,X,hxX), X.A.h i s an off-diagonal 2 element of A m u l t i p l i e d by X; the same holds of course f o r A . Thus i f e i t h e r h or X l i e s along a p r i n c i p a l axis of A, Sv i s zero; hence there w i l l be no e f f e c t on an i s o t r o p i c A tensor. So f a r the treatment has been quite general and holds f o r any X such that 2 X <<1. To evaluate X i n terms of g we take g=gU+6, written f o r short as g+6, with g~2 and 6. . « 1 f o r a l l i . j . Then h« = .(gh+h.|)/|(gh+h.6)| ; to f i r s t order i n 6 t h i s becomes h« = h + i[h.6-(h.<5.h)h] or X = -[h.6-(h.6.h)h] = -[a-h.S.hU] . (A-1.7) . g ~ ~ ~ ~* The requirement that X.h=0 i s s a t i s f i e d since h.y=h. Also, i f h l i e s along a p r i n c i p a l axis of 6, h.5=(h.6.h)h, and X vanishes. Combining t h i s with the re s u l t i n equation (A-1.6) shows that the e f f e c t of g anisotropy i s greatest i n dir e c t i o n s away from the p r i n c i p a l vectors of both A and g and hence the e f f e c t would be most marked i f A and g have a common axis system. As an example we take g a x i a l , with p r i n c i p a l values 2.000, 2.000, 2.000+e, and l e t - 124 -h = £i+mj+nk i n the g-frame. Then h.g = 2£i+2mj+(2n+en)k = 2h+nek , so that h 1 = (2h+nek)/|2h+nek| . (A-1.8) A f t e r evaluating the denominator and using the binomial theorem to r e t a i n only terms i n e t h i s becomes h'= h+ijneCk-nh) (A-1.9) i . e . X=%ne(k-nh). Again X.h=0 since k.h=n. We can estimate the magnitude o f X i n terms of e as follows: X = (X.X)*2 = Jsne(l-2nh.k+n 2)^ = hneil-n2)^ . (A-1.10) Since |n|<l, |n| (1-n ) 2=sin6cos6=J2sin2e<0.5 X < l E ,/4 ( A - l . l l ) For a numerical estimate of 6v we consider the A tensor given by A =A =30MHz, A =50MHz, A =25MHz, A =A =0: h = -(1,1,0). yy zz xx ' xy ' yz xz ' ~ ^ ' ' J These values give h.A.h=65MHz; h.A 2.h=4325(MHz) 2. Thus using (A-1.5) with v =14.3MHz and m =-h P s v. = 47.07 MHz. I S O By (A-1.6) |6v| = —L—[\.A2.h + 28.6 X.A.h] . I f X = —(1,0,1) t h i s becomes, - 125 -|6v| = g 4 1 1 4 [ X ( 2 5 6 3 + 1072.5)] = X(39) MHz . For 6v>100 kHz=0.1 MHz we require X>o.oo25. By ( A - l . l l ) t h i s implies a g-anisotropy >0.01 which i s large f o r an organic free r a d i c a l but not t o t a l l y unreasonable. Since the angular v a r i a t i o n of v w i l l be s l i g h t l y d i f f e r e n t s from that of v. , i t may be pos s i b l e i n favourable cases to estimate the i s o ' 3 * r e l a t i v e g-anisotropy from ENDOR data i n any event, i f the g-tensor i s known to be h i g h l y a n i s o t r o p i c , allowance must be made f o r t h i s when large a n i s o t r o p i c hyperfine tensors are to be determined. - 126 -Appendix 2 Some Aspects of the Dipolar Hyperfine Interaction The d i p o l a r hyperfine i n t e r a c t i o n has the form 3uu-U - < u e . ( - ^ ) . y N > [A-2.1] r where the angular brackets denote a s p a c i a l average over the wavefunction of the unpaired el e c t r o n . I f we take y^=+g^3^I and y/e=Pg£S where p i s the spin density, both y e and y^ are independent of the s p a c i a l v a r i a b l e s and can be taken outside the brackets. Comparison with the standard form S.B.I then shows that 3uu-U B = pgegNeN<-^-^ r y e y M 3uu-U r Thus f o r a given geometry and spin density, the hyperfine i n t e r a c t i o n s of d i f f e r e n t n u c l e i w i l l s c a le as g ^ y . y i . "N N For 2 3Na(I= 3/2, y N=2.2161g N) and 1HCI=Js, y N=2.7917^) , the g N values are i n the r a t i o 0.265:1, so f o r a given spin d i s t r i b u t i o n the d i p o l a r tensor 23 f o r a Na nucleus w i l l be only 26.5% of that f o r a proton at the same p o s i t i o n . o For an interatomic distance r =2.5A and p=l, the largest p r i n c i p a l value of a proton coupling tensor i s ~5 MHz, the exact value depending on the s p a t i a l d i s t r i b u t i o n of the l o c a l spin density; thus f o r un i t spin density the corresponding maximum value f o r a sodium hyperfine tensor would be -1.3 MHz. For p~0.6 as i n the Na +...C0 2~ ion p a i r i n sodium formate the maximum through-space d i p o l a r coupling to the sodium.atom would be ~1 MHz. - 127 -For a system with cylindrical symmetry about an axis defined by u (A-2.2) can be written § =.gBgNeN(3uu-U)<l3> . r One-centre interactions between a nucleus and spin in one of i t s p orbitals form the simplest example. The dipolar energy is then proportional to g given by 8 = y g.(3uu-U).y N . If one makes the f i r s t order substitution Ug=yeh(=pg6msH/H) this becomes 3 = u:.(3(h.u)u-h) ,y M . «-e ~N Two extreme cases of most interest: (a) The nuclear spin is quantised along H : unless B is very small (so that the nuclear Zeeman energy dominates), this requires a large isotropic hyperfine term. Then u^ry^h and 8 = y ey N(3(h.u) 2-l) = y ey N(3cos 26-l) where h.u=cos8 . (b) The nucleus experiences a f i e l d dominated by the dipolar interaction i t s e l f , so that is quantised along the vector 3(h.u)u-h. In this case 8 = u ey N|(3(h.u)u-h)| = y ey N[9(h.u) 2-6(h.u) 2 +l]^ 2 k = V eV N[3cos Q+iy . In general i f there i s an isotropic hyperfine interaction, a, and a significant nuclear Zeeman term as well as the dipolar f i e l d , y w w i l l be - 128 -quantised along the resultant of a l l three e f f e c t i v e f i e l d s : W e f f aC3(h.u)u-h)p eii N<r- > + (a +v 2 P 2SI = b cose u + ch where b=3p eP N<r~ 3>/(2SI) , c = ^ a + v p - ^ p ^ r " ^ / ( 2 S I ) The corresponding u n i t vector i s ^, b u cos6 + ch b u cos6 + ch (c 2+b 2cos 26+2bccos 2e) 2 [c 2+ (b 2+2bc)cos 2e] Then g va r i e s as 8 = p y N[3cos6u-h].h' [(2b+ c)cos 9-c] y e p N 2 2 5 e * [c +b(b+2c)cos Z6] 2 which reduces to case (a) when o>b and to case (b) when c=-b/3. - 129 -Appendix 3 Misalignment of Planes of Observation We assume that data have been taken i n three planes close to the orthogonal c r y s t a l l o g r a p h i c planes ab,bc*,ac*. 'Close' i n t h i s context means <5°. b i s the unique axis o f the c r y s t a l , and the symmetry requires that the elements of two symmetry r e l a t e d tensors and obey the r e l a t i o n s h i p s T<P = T.<?> a l l i i x . 1 1 T ( i ) = XC2) T ( i ) = T C i ) 23 23 12 " 1 2 CD _ . T(2) T 3 " T 3 (The subscripts 1, 2, 3 are associated with vectors a,b,c* r e s p e c t i v e l y . ) This implies that there w i l l be ' s i t e s p l i t t i n g s ' i n the ab and be* planes, with the s i t e s becoming degenerate at the 'crossover' points at the a,b,c* axes, and everywhere i n the ac plane. I f the planes of observation are s l i g h t l y misaligned, these crossover points w i l l be s h i f t e d . Those corresponding to a and c* w i l l - n e c e s s a r i l y - remain i n the ac plane, but w i l l be s h i f t e d by r o t a t i o n s about b. The cross-over point corresponding to b w i l l be s h i f t e d i n an a r b i t r a r y d i r e c t i o n due to r o t a t i o n s about the other axes. A misalignment of ac* w i l l r e s u l t i n the l i n e being s p l i t . The misalignment can be represented by a r o t a t i o n matrix R, which i n general w i l l be d i f f e r e n t f o r each plane. The data observed i n a given plane of observation correspond to an 'apparent* tensor A which i s r e l a t e d to the true tensor T v i a the misalignment; thus - I T A = R .T.R = R .T.R f o r e i t h e r s i t e . st » a ~ s *» " Under the conditions applying here v i z small r o t a t i o n , R w i l l approximate the i d e n t i t y matrix. Thus cross-products of off-diagonal elements of R - 130 -can be neglected and diagonal elements can be replaced by unity. This i s equivalent to commutation of the rotations about the three axes, so that these rotations can be treated as independent. With these approximations the elements of A can be evaluated: A l l = T l l + 2 T12 R21 + 2 T13 R31 A22 = T22 + 2 T12 R12 + 2 T23 R32 A33 = T33 + 2 T23 R23 + 2 T13 R13 A12 = T12 + T11 R12 + T22 R21 + T13 R32 + T32 R31 A13 = T13 + T11 R13 + T33 R31 + T12 R23 + T23 R21 A23 = T23 + T22 R23 + T33 R32 + T12 R13 + T13 R12 If rotations about b are eliminated (so that R_,=R, „=0 and R. . =-R'. . 31 13 I J j i for i ^ j ) these equations reduce to A l l = T l l " 2 T12 R12 A22 = T22 + 2 T12 R12 + 2 T23 R32 A = T - 2T R 33 *33 23 32 A12 = T12 + ( T i r T 2 2 ) R 1 2 + T13 R32 A13 = T13 " T12 R32" T23 R12 - 131 -A23 = T23 + ( T33" T22 ) R32 + T13 R12 Note that the trace i s conserved. Also, by v i r t u e of the symmetry r e l a t i o n s f o r T^ P and l[2? the numerical mean value of A ^ and k[2? i s T. . . Since f l ) (2) (1J (2) A v ' and A v v are thus symmetrically displaced from T and T the l i n e s corresponding to the two s i t e s w i l l s t i l l have the correct crossover frequency. A more d e t a i l e d proof of t h i s i s given below. E i t h e r set of equations can be used to obtain the R^ .. by f i t t i n g the experimental data. One method which i s d i r e c t l y a p p l i c a b l e uses the di s p l a c e -ment of the crossover point from an 'axis' i n the misaligned plane of observation, At axis 1, i n the plane of observation, the f i r s t order ENDOR frequency f o r s i t e (1) w i l l be given by 2 _ f w A 2 i U ) . v A ( l ) . „ 2 'CD ~ C^ C A hi + v p A n P. 2 2 2 3 with C A ) 1 1 = A-Q + + A 1 3 . The ENDOR frequency f o r s i t e (2) w i l l be the same expression with A^ P C2) ' 1 J replaced by A> / . I f i s the true ENDOR frequency at axis 1 given by p 11 p then v T 2 = C%CT 2 ) . n + v T n + v p 2 , V ( i ) " V ' T = * " A i r T l l > < A l l + T l l > + ^ A 1 2 - T 1 2 ^ A 1 2 + T 1 2 ) + ^ A 1 3 - T 1 3 ^ A 1 3 + T 1 3 ) ] + v p ( A n - T n ) where ( i ) r e f e r s to s i t e (1) or (2) and the same superscript i s understood for A „ and T „ . To a good approximation the l e f t hand side i s 2 v T . A v ^ where A v ^ i s the s h i f t from the 'true' frequency, while the products on the r i g h t hand side are e s s e n t i a l l y of the form 2T..AA. . where AA..=A.,-T.. 6 J I J l j i j i h l j - 132 -As noted above, i f R 1 3 = 0 l A i j J I a n c* l A i j ^ I a r e s y m m e t r i c a l l y placed about |T..I, so that AA^P=-AA^ and hence Av,' . =-Av,„. . Thus for small misalignments the ENDOR lines from sites (1) and (2) in the region of the crossover are symmetric about the true lines; and hence the crossover frequency i s unchanged. The same holds for other axes by permutation of subscripts. - 133 -Appendix 4 A. uv-Induced Radical Reaction in Irradiated Sodium Formate Bellis and Clough identified as (H.C(O).o.C02)"2 the free radical produced when X or y irradiated sodium formate is heated. Denoting this species by X, the reaction . co2" i X is essentially complete after ~30 minutes at 120°C or after -1 year's aging at room temperature, as is easily shown by the change in appearance of the EPR spectrum. In the course of our studies on NaHC02 we found that this reaction could be reversed by ultra violet light. The optimum wavelength lies in the range 270-300 nm. The kinetics of this reverse reaction were studied by uv-irradiating crystals containing radical X in the EPR cavity. At suitable orientations the spectra of X and CG^- were sufficiently separated for the relative peak heights and linewidths to be determined quite easily, thus enabling the relative concentrations of the two radicals to be estimated. The results of one such experiment are shown in Fig. 26. For more than 65% conversion the total concentration of radicals remained constant: the small fluctuations in the value of [C02~]+[X] are attributed to changes in cavity temperature. A plot of [C0 2~]/[X] vs time is essentially linear (Fig. 27), implying second order kinetics. For a second order reaction, the slope of this plot is k 2C where k 2 i s the rate constant and C the i n i t i a l (=total) radical concentration. Using this fact i t was possible to estimate the dependence of k 2 upon light intensity and temperature. By interposing different neutral density f i l t e r s and measuring the - 134 -F i g 26. Relative i n t e n s i t i e s of ZQ{ CC) and secondary r • spectra as a function of u v - i r r a d i a t i o n time. - 136 -corresponding values of k^C from the r e s u l t i n g l i n e a r p l o t s i t was found that the r e a c t i o n i s e s s e n t i a l l y f i r s t order with respect to l i g h t i n t e n s i t y , the experimental value of the exponent being 1.16±0.1. S i m i l a r l y by carrying out the i r r a d i a t i o n s at room temperature and 77K the a c t i v a t i o n energy E i n the simple Arrhenius equation k„=A exp(-E /RT) was found to be E =0.5±0.2 kcal/mole. a The uv-induced r e a c t i o n was e s s e n t i a l l y r e v e r s i b l e : heating the sample at 110°C f o r -30 minutes restored the presence of X, and t h i s cycle could be repeated several times, although with a small loss of i n t e n s i t y on heating. These r e s u l t s are d i f f i c u l t to i n t e r p r e t , p r i m a r i l y because of the second order k i n e t i c s , which imply d i f f u s i o n through the c r y s t a l l a t t i c e . The following model covers several features of the r e s u l t s . (i) Light i s absorbed, e i t h e r by the sodium formate l a t t i c e , or by species X i t s e l f , to form an a c t i v a t e d species X* hv + X ^> X* ^H" = k I [X] =k1 [X] . * X decays with a high p r o b a b i l i t y to X: k , X* —I X . ( i i ) A small f r a c t i o n of the X* formed d i f f u s e s through the l a t t i c e u n t i l two X* species react to form CC^ , k X*+X -=^ - 2C0 2" . • The rate equations are ^ = - ^ [ X ] + k ^ [ X * ] , - 137 -k 2 [ x * ] 2 + k x[X] - k_ x[X*] dt d[C0 2"] " k 2 [ X * ] 2 dt * * I f k2[X*]<k^<<k ^ , [x"] w i l l be present i n small proportions, almost i n equilibrium with X; the o v e r a l l r e a c t i o n w i l l then be driven by step ( i i ) giving r i s e to the observed second order k i n e t i c s with an e f f e c t i v e rate constant k 2k^/k ^. Numerical s o l u t i o n of the rate equations confirms t h i s r e s u l t . This model i s not e n t i r e l y s a t i s f a c t o r y i n that i t o f f e r s no clue as to the nature of X* or i t s mode of d i f f u s i o n . The r e l a t i v e l y high rate of reaction suggests that X* may be a species which ' d i f f u s e s ' by a s e r i e s of hea d - t o - t a i l reactions with neighbouring formate ions: a l t e r n a t i v e l y X might be an electron or an e x c i t a t i o n rather than a chemical s p e i c i e s . This p o s s i b i l i t y i s favoured by the nature of step ( i i ) , the formation of C0 2 . The absence of any spin-spin s p l i t t i n g s c h a r a c t e r i s t i c of r a d i c a l p a i r s shows that the formation of C02~ from X* does not produce two C0 2 ions i n close proximity, so some e a s i l y d e l o c a l i s e d form of energy i s probably instrumental i n the r e a c t i o n . X* + HC02" HC02 * + X 

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