"Science, Faculty of"@en . "Chemistry, Department of"@en . "DSpace"@en . "UBCV"@en . "Gardner, Christopher Leonard"@en . "2011-10-13T16:00:35Z"@en . "1964"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "Hydrazoic acid has been photolysed in a krypton matrix at 4\u00C2\u00B0K and the products studied by electron spin resonance spectrometry. This study showed that NH\u00E2\u0082\u0082 radicals have been produced as a secondary product in the reaction. In addition a broad, intense resonance at g = 2 and a weak, half field resonance has been tentatively assigned to the imine (NH) radical. This suggestion is shown to be consistent with theoretical considerations.\r\nDiazomethane has also been photolysed in krypton and carbon monoxide matrices at 4\u00C2\u00B0K, and the products studied by e.s.r., in an attempt to detect the methylene (CH\u00E2\u0082\u0082) radical. The results of this study were complicated and a complete analysis was not possible. It is suggested that some of the features may be explained in terms of an overlap of the spectre from CH\u00E2\u0082\u0083 and CH radicals. There are difficulties in such an explanation however.\r\nA study has been made on the line shapes of poly-crystalline samples of aromatic triplet state molecules.\r\nIt is shown how the experimentally observed spectra of the photoexcited triplet states of axially symmetric molecules such as triphenylene and non-axially symmetric molecules such as naphthalene and phenanthrene can be explained in terms of a line shape calculated from a first order perturbation treatment. This model gives a good explanation of the observed line shapes, however better agreement with the observed field positions is obtained if a second order correction is included. Line shape calculations have also been made for molecules, such as the substituted imines, where spin-spin interaction is large. It is shown that the calculation is in agreement with the observed spectra of phenylimine and benzenesulfonylimine. In addition, it is shown how the experimentally determined value of the spin-spin interaction constant, D, can be related to the spin density on the nitrogen of the substituted imines. The spin densities calculated in this way are in good agreement with spin densities calculated on the basis of the H\u00C3\u00BCckel theory."@en . "https://circle.library.ubc.ca/rest/handle/2429/37927?expand=metadata"@en . "ELECTRON SPIN RESONANCE STUDY OF SOME TRIPLET STATE MOLECULES CHRISTOPHER LEONARD GARDNER B.So., University of B r i t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Chemistry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLTJMBIA June, 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r ref e r e n c e and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that, copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of (L^EMKTRY The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGJKAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY CHRISTOPHER LEONARD GARDNER B.Sc, The University of B r i t i s h Columbia, 1961 THURSDAY, JULY 2ND, 1964, AT 2:30 P.M. IN ROOM 261, CHEMISTRY BUILDING of COMMITTEE IN CHARGE Chairman: I. McT. Cowan N. B a r t l e t t M. Bloom J.A.R. Coope J.B. Farmer R.F. Snider R. Stewart External Examiner: CA. Hutchison University of Chicago ELECTRON SPIN RESONANCE STUDY OF SOME TRIPLET STATE MOLECULES ABSTRACT Hydrazoic acid has been photolysed i n a krypton matrix at 4 K and the products studied by electron spin resonance spectrometry. This study showed that NH2 r a d i c a l s have been produced as a secondary pro-duct i n the r e a c t i o n . In addition a broad, intense resonance at g = 2 and a weak, half f i e l d resonance has been t e n t a t i v e l y assigned to the imine (NH) r a d i c a l . This suggestion i s shown to be consistent with t h e o r e t i c a l considerations. Diazomethane has also been photolysed i n krypton and carbon monoxide matrices at 4\u00C2\u00B0K, and the products studied by e.s.r., i n an attempt to detect the methyl-ene ( C H 2 ) r a d i c a l . The r e s u l t s of t h i s study were complicated and a complete analysis was not possible. It i s suggested that some of the features may be explained i n terms of an overlap of the spectre from C H 3 and CH r a d i c a l s . There are d i f f i c u l t i e s i n such an explanation however. A study has been made on the l i n e shapes of poly-c r y s t a l l i n e samples of aromatic t r i p l e t state mole-cules. It i s shown how the experimentally observed spectra of the photoexcited t r i p l e t states of a x i a l l y symmetric molecules such as triphenylene and non-a x i a l l y symmetric molecules such as naphthalene and phenanthrene can be explained i n terms of a l i n e shape calculated from a f i r s t order perturbation treatment. This model gives a good explanation of the observed l i n e shapes, however better agreement with the observed f i e l d positions i s obtained i f a second order c o r r e c t i o n i s included. Line shape c a l c u l a t i o n s have also been made for molecules, such as the substituted imines, where spin-spin i n t e r -a ction i s large. It i s shown that the c a l c u l a t i o n i s i n agreement with the observed spectra of pheny-limine and benzenesulfonylimine. In addition, i t i s shown how the experimentally determined value of the spin-spin i n t e r a c t i o n constant, D, can be re l a t e d to the spin density on the nitrogen of the substituted imines. The spin densities calculated i n t h i s way are i n good agreement with spin densities calculated on the basis of the Huckel theory. GRADUATE STUDIES F i e l d of Study: Chemistry Topics i n Physical Chemistry Quantum Chemistry Advanced Theoretical Chemistry Topics i n Chemical Physics S t a t i s t i c a l Mechanics Modern Physics J.A.R, Coope R.F. Snider A. Bree R.M. Hochstrasser R.F. Snider W.C. L i n CA. McDowell B.A. Dunell R.F. Snider M. Bloom Related Studies: Topics i n Inorganic Chemistry Topics i n Organic Chemistry Crystal Structures El e c t r o n i c s N. B a r t l e t t W.R. Cullen D.E. McGreer J.P. Kutney R.E.I. Pincock J. Trotter R.E. Burgess PUBLICATIONS J.B. Farmer, C L . Gardner and C A . McDowell, Energy Transfer Between T r i p l e t States Detected by Electron Spin Resonance Spectroscopy, J . Chem. Phys. 34, 1058 (1961) J.B. Farmer, C.L. Gardner and CA. McDowell, Electron Spin Resonance Line Shape of T r i p l e t Triphenylene i n Rigid Solution, Submitted for p u b l i c a t i o n i n Molecular Physics i I wish to express my gratitude to Professor G. A. McDowell and Dr. J. B. Farmer for t h e i r guidance and encour-agement throughout the course of t h i s work. I have benefited greatly from the many discussions on the t h e o r e t i c a l aspects of t h i s work that I have had with Dr. J. A. R. Coope, for these I thank him. Thanks are also due to Mr. J. Sallos who b u i l t the superheterodyne and 100 Kc. spectrometers and kept them i n good running order, to Mr. R. Muehlchen who was always a great help with the low temperature apparatus, and to Mr. M. Symonds who l i q u i f i e d the helium. I g r a t e f u l l y acknowledge re&e'ipt of an NRC \"bursary and two N.R.G. studentships. i i Abstract Hydrazoic acid has been photolysed i n a krypton matrix at 4\u00C2\u00B0K and the products studied by electron spin resonance spectrometry. This study showed that NHg radicals have been produced as a secondary product i n the reaction. In addition a broad, intense resonance at g \u00C2\u00BB 2 and a weak, h a l f f i e l d resonance has been t e n t a t i v e l y assigned to the imine (NH) r a d i c a l . This suggestion i s shown to be consistent with the-o r e t i c a l considerations. Diazomethane has also been photolysed i n krypton and carbon monoxide matrices at 4\u00C2\u00B0K , and the products studied by e.s.r., in-.--.an attempt to detect the methylene (CHg) radical-.: The r e s u l t s of t h i s study were complicated and a complete analysis was not possible. It i s suggested that some of the features may be explained i n terms of an overlap of the spectra from CH\u00E2\u0080\u009E and CH r a d i c a l s . There are d i f f i c u l t i e s i n such an c explanation however. A study has been made on the li n e shapes of p o l y c r y s t a l l i n e samples of aromatic t r i p l e t state molecules. It i s shown how the experimentally observed spectra of the photoexcited t r i p l e t states of a x i a l l y symmetric molecules such as triphenylene and non-axially symmetric molecules such as naphthalene and phen-anthrene can be explained i n terms of a line shape calculated from a f i r s t order perturbation treatment. This model gives a good explanation of the observed li n e shapes, however better i i i agreement with the observed.field positions i s obtained i f a second order correction i s included. Line shape calculations have also been made for molecules, such as the substituted imines, where spin-spin interaction i s large. It i s shown that the c a l c u l a t i o n i s i n agreement with the observed spectra of phenylimine and benzenesulfonylimine. In addition, i t i s shown how the experimentally determined value of the spin-spin i n t e r -action constant, D, can be related to the spin density on the nitrogen of the substituted imines. The spin densities c a l -culated i n t h i s way are i n good agreement with spin densities calculated on the basis of the Huckel theory. i v CONTENTS Page. Acknowledgement 3 (i) Abstract ( i i ) Chapter I. General Introduction 1 a) The T r i p l e t State i n Chemistry 1 b) Zero F i e l d S p l i t t i n g of the T r i p l e t States 3 c) H i s t o r i c a l Discussion of the e.s.r. of T r i p l e t State Molecules 6 Chapter I I . Description of the Apparatus 12 a) 100 Ke , X-Band e.s.r. Spectrometer . IS b) Superheterodyne Spectrometer and Associated Cryogenic Equipment 13 (i) Superheterodyne Spectrometer 13 ( i i ) L i q u i d Helium Dewar 15 Chapter I I I . Electron Spin Resonance Study of the Photolysis Products of Hydrazoio Acid Trapped at 4\u00C2\u00B0K. 17 a) Introduction 17 b) Experimental Procedure E l c) Experimental Results 22 d) Discussion of Results 23 Chapter IV. Electron Spin Resonance Study of the Photolysis Products of Diazomethane Trapped at 40K. 37 a) Introduction 37 b) Experimental Procedure 39 V Page,. c) Experimental Results 41 d) Discussion of the Results 45 e) Conclusions 48 Chapter V. Electron Spin Resonance Line Shapes i n P o l y c r y s t a l l i n e Samples of Aromatic T r i p l e t States 50 a) Introduction 50 b) Experimental Procedure 53 o) General Discussion of e.s.r. Line Shapes i n Po l y c r y s t a l l i n e Samples 54 d) Calculation of the e.s.r. Line Shapes for a Po l y c r y s t a l l i n e Sample of T r i p l e t State Molecules with A x i a l Symmetry. D small compared to g^ SH. 58 e) Calculation of the e.s.r. Line Shape for a ::\u00E2\u0080\u00A2 P o l y c r y s t a l l i n e Sample of T r i p l e t State Molecules with les s than A x i a l Symmetry. D small compared to gjSH 62 f) Calculation of the e.s.r. Line Shape for a Po l y c r y s t a l l i n e Sample of Tr i p l e t State Molecules with A x i a l Symmetry. D large compared with the Zeeman S p l i t t i n g 71 Appendix 1 82 Appendix 2 85 Appendix 3 87 Bibliography 90 v i LIST OF FIGURES to follow Page. 1. Block Diagram of 100 Kc Spectrometer 13 2. Block Diagram of Superheterodyne Spectrometer 15 3. Liquid Helium Dewar 15 4. Apparatus Used i n HNg Preparation 22 5. E.S.R. Spectra of the Photolysis Products of HN2 23 6. E.S.R. Spectrum of the Photolysis Products of HN3 Half F i e l d Line 23 7. Energy Level Diagram for R i g i d NH 25 8. Plot of Cos 26 (H) as a Function of gfH for NH 26 9. Energy Level Diagram for NH i n the K- 0 and K\u00C2\u00BBl Rotational States 29 10. Apparatus for CHgNg Preparation 40 11. E.S.R. Spectra of the Photolysis Products of CHgNg i n a Krypton Matrix 41 12. E.S.R. Spectrum of the Photolysis Products of CHgNg i n a CO Matrix 44 13. Experimental and Calculated E.S.R. Spectrum of Tr i p l e t Triphenylene 61 14. Experimental and Calculated E.S.R. Spectrum of T r i p l e t Naphthalene 68 15. Plot of Cos 2 \u00C2\u00A9 (H) as a Function of gjSH for Phenyl-imine 71 16. Predicted Spectrum of the z-Transition of Phenyl-imine 72 17. Experimental and Calculated E.S.R. Spectrum for Phenylimine 76 I, GENERAL INTRODUCTION a) The T r i p l e t State i n Chemistry ( l ) (2) For a long time i t has been recognised that when some compounds are i r r a d i a t e d two types of photolumineseence can be observed; a short l i v e d fluorescence and a longer l i v e d phos-phorescence, each having i t s own ch a r a c t e r i s t i c spectrum. In 1935 Jablonski (3) postulated the existence of a metastable state l y i n g below the lowest excited singlet state, and sug-gested that phosphorescence was associated with the t r a n s i t i o n from t h i s metastable state to the singlet ground state. It was not u n t i l 1944 however that Lewis and Kasha (4) showed that t h i s phosphorescence accompanied the quasi-forbidden t r a n s i -t i o n from the lowest t r i p l e t l e v e l to the singlet ground stat6. The excited t r i p l e t state i s characterised by a r e l a t i v e l y long l i f e t i m e because transitions between states of different m u l t i p l i c i t y are forbidden. The t r i p l e t state i s believed to play an important role i n the chemistry of reactions because of i t s long l i f e t i m e and i t s position as the lowest excited state. While i t i s w e l l known that the ground state of p r a c t i c a l l y a l l molecules i s a singlet state, there are a number of ex-ceptions to t h i s . These exceptions, such as Og (5), NH (6), the dinegstive ions of some aromatic molecules (7), substituted imines (8) and methylenes (9) which have t r i p l e t ground states, are a l l characterised by an o r b i t a l degeneracy or near-degen-eracy. s The difference i n energy \"between the t r i p l e t , and singlet states with the same configuration i s a result of the exchange int e r a c t i o n between the electrons. In the approximation that the singlet and t r i p l e t states are described by the two elec-tron coordinate wave functions, \u00E2\u0080\u00A2 n ( %M %w - %(\u00E2\u0080\u00A2)) and (1-1) the energy of the singlet and t r i p l e t states i s given by (10), TP TP 0. \u00E2\u0080\u00A2\u00C2\u00BB J s 0 1+S 2 (1-8) where Q \u00C2\u00BB Coulomb i n t e g r a l ' /Mi\u00C2\u00AB%(:0 VHilfl\u00C2\u00A5\u00C2\u00BBMelt,c-?_ J = Exchange i n t e g r a l * /SiCO %W V \u00C2\u00AB&ti) % ( 0 dt, dt_ r , , , (l-\u00C2\u00A7) S = Overlap i n t e g r a l * J HVi) WO V i a the interelectron i n t e r a c t i o n . For electrons i n the same molecule H^ A A N A H^ B w i l l be ortho-gonal and hence S vanishes. There w i l l now be a separation of 2J between the singlet and t r i p l e t states. Which of these states has lower energy depends.on the sign of J. The experi-mental r e s u l t s are i n agreement with J being p o s i t i v e . On thi s basis we can now understand why the f i r s t excited state of most molecules i s a t r i p l e t state. The ground state of most molecules i s a singlet state as the-Paula P r i n c i p l e forbids a 3 t r i p l e t state when there i s no o r b i t a l degeneracy. In those molecules with degenerate ground states the Pauli P r i n c i p l e i s already s a t i s f i e d and a t r i p l e t ground state i s allowed. For electrons on di f f e r e n t atoms or molecules the states need not be orthogonal and S does not vanish. In th i s case the s p l i t t i n g i s no longer symmetrical. Which state l i e s the lowest i s s t i l l determined by the sigh of J which i s , i n general, negative. This leads to a singlet ground state, b) Zero F i e l d S p l i t t i n g of the T r i p l e t State In the previous section we have seen, i n a q u a l i t a t i v e fashion at le a s t , how exchange interaction s p l i t s the singlet and t r i p l e t l e v e l s i n a molecule. In the following discussion we w i l l be considering almost exclusively the magnetic proper-t i e s of molecules i n the t r i p l e t state. Experimentally (11) i t i s found that the three spin com-ponents of the t r i p l e t state are usually non-degenerate even i n the abscence of an external magnetic f i e l d . There i s in fact a s p l i t t i n g which i s termed the zero f i e l d s p l i t t i n g . Two interactions, spin-spin and spin-orbit, can lead to such a s p l i t t i n g . For o r b i t a l l y non-degenerate molecules composed of \" f i r s t row\" elements i t i s well established (12) that the zero f i e l d s p l i t t i n g may be accounted for almost e n t i r e l y on the basis of spin-spin i n t e r a c t i o n . The reason for t h i s i s that the f i r s t order spin-orbit term vanishes for o r b i t a l l y non-degenerate molecules and the second order term i s small compared to the f i r s t order spin-spin i n t e r a c t i o n . For mol-ecules containing heavy atoms this i s no longer true as the 4 second order spin-orbit contribution becomes important (13). We w i l l now consider the derivation of the spin Hamil-tonian of the spin-spin i n t e r a c t i o n . The spin-spin i n t e r a c t i o n arises from what may be considered as the c l a s s i c a l i n t e r -action of the magnetic dipoles of the electrons. The Hamil-tonian representing (14) this i n t e r a c t i o n i s given by where the summation i s ca r r i e d out over a l l electrons. In the approximation that the t r i p l e t state i s described by a two electron t r i p l e t state function, (1-4) takes the more simple form. It i s worthwhile noting that i t i s not necessary (15) to i n -clude a contact term i n (1-5). The reason for th i s i s that, i n order that the Pauli P r i n c i p l e be s a t i s f i e d , the s p a t i a l portion of the t r i p l e t state wave function must be antisym-metric. This antisymmetry precludes the p o s s i b i l i t y of having r i 2 = 0-(1-5) may be rewritten i f we proceed as follows. In terms of i t s components (1-5) becomes (1-4) (1-5) 12 - BXuZ,xLSiG)Sxl2)+Sx(0Sz(v)] -3yazlXSt(i)SyU) + Si,l.)Sx(x)]] (l-b) 5 The las t three terms i n (1-6) can always he eliminated by a transformation into the p r i n c i p a l axes coordinate system. For molecules with s u f f i c i e n t l y high symmetry, such as naphthalene and triphenylene, the symmetry axes and p r i n c i p a l axes coin-cide. A suitable reorganisation of (1-6) gives us (1-7) 31 = D' (Sz (1) Sz (2) -1/3 S(l).S(2))-*-E\u00C2\u00BB (Sx(l)Sx(2)-Sy(l)Sy(2)) where D\u00C2\u00BB- 3/2g 2/S 2(rf 2-3zf 2) and E' = 3/2g 2jg 3(yf 2-xf g) r12 r12 We are dealing with a system however where and are coupled to give a resultant spin S with S - l . I f we substitute S = S(l)+3(2) and S^ \u00C2\u00BBSv(l)4-S\u00E2\u0080\u009E( 2), (^ \u00E2\u0080\u00A2= x.y.z) into (1-7) and average, over the s p a t i a l coordinates, the following spin Hamil-tonian i s obtained. ^Hspin-D(S^ - 1/3 S 2)+E(S^ - s|) (1-8) where TJ = fg 2 p 2 /^SE-^SEX and E = fg 2|5 2 /^12-^2^ r12 / \ r12 A much more elegant derivation of (1-8) i s possible (Appendix 1) i f tensor methods are used. The parameters D and E are termed the zero f i e l d s p l i t t i n g or spin-spin i n t e r a c t i o n para-meters. It was shown by Stevens (16) that, i n the absence of an external magnetic f i e l d , the most general form of the Hamil-6 tonian for an S = 1 system i s given \"by (1-8) neglecting hyper-fine i n t e r a c t i o n . I f an external magnetic f i e l d i s present a Zeeman term must he added to (1-8) and our Hamiltonian becomes TR= g/sH.s + Dfs 2, - 1/3S 2 K E(S^ - Sy) ( 1_. 9 ) where the g-tensor has been assumed i s o t r o p i c . This Hamiltonian has been found adequate to explain (11) the fine structure of the observed e.s.r. spectrum of molecular t r i p l e t states. c) H i s t o r i c a l Discussion of the e.s.r. of T r i p l e t State Molecules Although electron spin resonance had been observed i n t r a n s i t i o n metal ions with t r i p l e t ground states i n the early stages of paramagnetic resonance studies and the theory given in d e t a i l by Stevens (16), the early attempts (17) of detection of excited t r i p l e t state molecules f a i l e d . The reason for t h i s f a i l u r e probably l i e s p a r t l y i n the lack of s e n s i t i v i t y of the instruments used and pa r t l y because a p o l y c r y s t a l l i n e sample was used. The f i r s t successful detection of an excited t r i p l e t state by e.s.r. was r e a l i s e d by Hutchison and Mavvgum (18) i n 1958. In these experiments they were able to obtain an oriented as-sembly of naphthalene molecules by \"dissolving\" the naphthalene i n a single c r y s t a l of durene. By i r r a d i a t i n g the c r y s t a l at 77\u00C2\u00B0K with u l t r a v i o l e t l i g h t , they were able to detect the elec-tron spin resonance of the photoexcited t r i p l e t state of naph-thalene. They found that the angular dependence of the e.s.r. 7 spectrum could be explained i n terms of the Hamiltonian (1-9) with , . D= \u00C2\u00B1 0.1008* 0.0007 cm\"1 ; E = \u00C2\u00A5 0.0138* 0. 0002cm'1 and g = 2.0030* 0.0004 In t h i s work the A m =l t r a n s i t i o n s * ( i . e . the t r a n s i t i o n between neighbouring spin levels) were observed. Because of the magnitude of the spin-spin s p l i t t i n g parameters D and E, the resonance f i e l d varied over a range of about 2000 gauss as the orientation of the c r y s t a l with respect to the magnetic f i e l d was changed. T This detection of naphthalene i n i t s t r i p l e t state was the f i r s t investigation of an excited molecular state using e.s.r. It provided a means for studying these excited species not previously a v a i l a b l e . Much work has been carried out dn such photoexcited t r i p l e t states recently. Other metastable 2 \"*\"3 states such as the E state of Cr i n AlgOg (19) and the excited t r i p l e t state of M centres i n KC1 (20) have also been detected. Soon after the work of Hutchison and Msp^um appeared, two papers by van der Waals and de Groot (21,22) were published *In the t r i p l e t states of aromatic moleotiles the spin-spin int e r a c t i o n i s smaller than the Zeeman in t e r a c t i o n and at high f i e l d s the eigenstates of the Hamiltonian are approximated by the eigenstates of Sz quantised i n the f i e l d d i r e c t i o n . Because of t h i s , i t has meaning to l a b e l eigenstates i n terms of the iMs) states they approximate, and one talks of A m \u00C2\u00BBl and 2 t r a n s i t i o n s . The s i t u a t i o n i s not well defined for D the same order of magnitude or larger than the Zeeman term however. In t h i s case the A m \u00C2\u00BB l or 2 terminology does not r e t a i n much significance. 8 describing the detection of the e.a.r. A m = 2 t r a n s i t i o n i n a system of randomly oriented aromatic t r i p l e t states. They showed that, unlike the A m \u00C2\u00BB1 t r a n s i t i o n s , the A m = 2 tran-s i t i o n was comparatively is o t r o p i c making detection possible i n a random sample. For molecules with a x i a l symmetry, they were able to derive an exact expression for the l i n e shape of th i s t r a n s i t i o n and show how the zero f i e l d s p l i t t i n g parameter could be obtained from the experimental e.s.r. spectrum. This technique has the advantage that i t i s experimentally simple and applicable to almost any system. In the discussion that follows we w i l l be mainly concerned with molecules having a x i a l symmetry. It seems worthwhile to i l l u s t r a t e some of the ideas i n terms of such a model. T r i p l e t molecules having a x i a l symmetryi.imay be described by the Hamiltonian in the molecular coordinate system. The term i n E vanishes because the x and y axes are equivalent. If eigenstates of are chosen as basis states, the Hamiltonian may be written i n the matrix form, ^ a p i n = g^H.S + l X S 2 . - l / 3 S 2 ) (1-10) S_, quantised with respect to the molecular coordinate system (1-11) iflZ gpHSin\u00C2\u00A9 1 1/3 D-gj5H Cos\u00C2\u00A9 0 where 9 i s the angle between the z-axis and H. 9 An expression for the energies of the spin l e v e l s as a function of f i e l d i s e a s i l y obtained for H oriented along one of the molecular axes These are found to he i) W \u00C2\u00B1 1 = 1/3 D \u00C2\u00B1 gpH WQ= -2/3 D (1-12) i i ) 9 \u00C2\u00BB 9 0 \u00C2\u00B0 An exact expression i s not e a s i l y obtained for an ar b i -trary orientation of the molecule i n the magnetic f i e l d as the secular equation does not factor. In this case the problem i s most e a s i l y treated using perturbation theory. For aromatio hydrocarbons the spin-spin i n t e r a c t i o n i s usually small com-pared to the Zeeman s p l i t t i n g and one can treat the spin-spin i n t e r a c t i o n as a perturbation on the Zeeman l e v e l s . In this way i t becomes clear why the A m \u00C2\u00AB1 t r a n s i t i o n i s anisotropic but the A m = 2 t r a n s i t i o n r e l a t i v e l y i s o t r o p i c . To the l i m i t of f i r s t order perturbation theory (see Chapter 5) the energies of the spin levels are given by i s strongly orientation dependent. On the other hand, Am-2 E * l =*g/5H +D/6(3Cos 20 -1) E 0 = - D/3 (3Cos 2 9 -1) (1-13) A m = l transitions occur when the resonance condition 10 tr a n s i t i o n s w i l l occur when the resonance condition , r\u00C2\u00BBv0 = 2gj3H i s s a t i s f i e d and hence, to f i r s t order, i s independent ,of orientation. Higher order terms do however introduce some ani-sotropy. Reoently Yager, Wasserman and Cramer (23) have reported the detection of Am -1 tr a n s i t i o n s i n a randomly oriented sample of photoexcited naphthalene. At f i r s t sight, t h i s seems surprising as the anisotropy i s expected to spread the resonance over several thousand gauss. It i s now clear (24) that ,the li n e s occur when the s t a t i c magnetic f i e l d (H) i s oriented \u00E2\u0080\u00A2 along one of the molecular axes. For these orientations there i s a large change i n the number of absorbing molecules for a small change i n H. It i s these large changes that are i n fact recorded experimentally. In Chapter 5 we show how both the lin e shapes and positions of these transitions can be explained i n terms of a t h e o r e t i c a l l i n e shape based on a f i r s t order perturbation model. In addition to the photoexcited t r i p l e t states,we have mentioned previously, there are two other broad c l a s s i f i c a t i o n s of t r i p l e t states which have been studied by e.s.r. over the l a s t few years. These are, f i r s t l y , the thermally accessible t r i p l e t states which result from the exchange coupling of two doublet states, giving r i s e to a ground singlet state and a low l y i n g t r i p l e t state. Secondly, there i s the group of mole-cules which have a t r i p l e t ground state. 11 An early example of a thermally accessible t r i p l e t state was investigated by Bleaney and Bowers i n 1951 (E5). They studied the electron spin resonance of copper acetate. In t h i s compound two cupric ions are held i n close proximity by a cage structure of the acetate groups. Exchange forces are strong enough to couple the two spins to give a singlet ground state and a low l y i n g t r i p l e t state. A similar phenomenon has been observed by Chesnut and P h i l i p s (26) who studied single c r y s t a l s of charge transfer compounds of tetraeyanoajuinodimethane (TCNQ). Their results show the presence of a thermally accessible t r i -plet state. Hirota and Weissman (27) have reoently shown that some negative ion r a d i c a l s form dimers with a low l y i n g t r i p l e t , stat e. Ground state t r i p l e t molecules have also been investigated by electron spin resonance. Molecular oxygen, which has a ground state, has been investigated both i n the gas phase (28) and i n a elathrate (29). Some organic methylenes,(9) and imines (8) which have t r i p l e t ground states have recently been investigated. This work: reports an attempted investigation of the two ground state t r i p l e t molecules CHg and NH. Appendix 2 consists of a reasonably complete l i s t of ref-erences describing e.s.r. studies of t r i p l e t states. This l i s t has been mainly concerned with the work done on organic t r i -plet states and leaves out a large volume of work on the e.s.r. of t r a n s i t i o n metal ions. References to these works are a v a i l -able, for example, i n G.E.Pake, \"Paramagnetic Resonance\" (30). 12 I I . DESCRIPTION OF THE APPARATUS a) The 100 Kc, X-Band e.s.r. Spectrometer The 100 Kc, X-band e.s.r. spectrometer used for the i n -vestigations at l i q u i d nitrogen temperature i s , i n almost a l l d e t a i l , i d e n t i c a l with the Varian #V-4500 e.s.r. spectrometer. The microwave power, from a Varian #V153C klystron, d e l i v -ering approximately 300 mw over i t s tuning range of 8.6 to lOGc/sec., i s fed d i r e c t l y into a wave guide run and, via a magic tee and accessory equipment, to a Varian #V-4531 multi-purpose cavity operating i n the TE012 mode. Accessory equip-ment includes a Polytechnic Research and Development Co. Model 1203 f e r r i t e i s o l a t o r and a Model 159A variable attenuator, a Hewlett-Packard Model x 810 A s l i d e screw tuner and DeMornay/ Bonardi terminating load and d i r e c t i o n a l coupler. The detection system employed i s as follows. The magnetic f i e l d i s modulated at 100 Kc/sec by a pair of sweep c o i l s sur-rounding the sample. The microwave frequency i s held at the resonant frequency of the cavity by means of an automatic f r e -quency c o n t r o l . When the main magnetic f i e l d i s swept through resonance, the bridge unbalance, due to the absorption of microwave power i n the cavity, i s detected by a s i l i c o n diode detector as an a.c. sig n a l . This signal i s controlled i n amplitude and phase by the slope of the resonance l i n e . The a.c. s i g n a l i s fed v i a the 100 Kc reciever to a phase sensitive detector thence to a demodulator to eliminate the unwanted 100 13 Ko signal and to a recorder (Leeds and Northrup Speedomax H), where i t i s displayed as the f i r s t derivative of the absorption curve vs magnetic f i e l d . The 100 Ko transmitter and receiver were b u i l t i n t h i s laboratory from Varlan c i r c u i t s but, as opposed to the ITarian u n i t , on separate copper lined chassis. The 100 Kc c r y s t a l was enclosed i n a heat shield for greater thermal s t a b i l i t y . Fig. 1 shows a block; diagram of the spectrometer. The magnet f i e l d was supplied by a Varian #V-4012A 12 inch magnet, having a 2*5 inch pole gap. The f i e l d was monitored by a proton resonance magnetometer. The magnetometer consists of a probe c o i l , containing g l y c e r o l , which i s inserted i n the magnet gap beside the cavity, and i s connected to a marginal o s c i l l a t o r , which i s frequency modulated at 20 cycles/sec. The proton resonance i s displayed on an oscilloscope and a General Radio Co. Model 1001-A signal generator tuned to zero beat. The frequency of t h i s generator i s measured with a Hewlett-Packard Model 524B counter. To measure the microwave frequency, a Hewlett-Packard Model 504A transfer o s c i l l a t o r was tuned to zero beat with two of the microwave harmonics. The frequency of t h i s o s c i l -l a t o r was then measur'ed with the Hewlett-Packard Model 524B counter. b) Superheterodyne Spectrometer and Associated Cryogenic Equipment i ) Superheterodyne Spectrometer AF Uni C t Re<(< Powi Supp ctor ir Lo ad Shifter Klyttron isolator \u00E2\u0080\u0094 Attenuator NUgAc Tec NMf\ Probe Modulation Coils Magnet Coils Cry t t& l Oetector 10 0 Kc Magnet-ometer CRO S i .na I Generator Frequency Counter I0O Kg. 06ciUator t o o Kc Modulator Phase Shifter Phase Detector Recorder t\u00E2\u0080\u0094 integrator Fig. I . 100 Kc. E.SK S p e c t r o m e t e r . 14 A l l e.s.r. measurements at l i q u i d helium temperature were made using a superheterodyne spectrometer designed and b u i l t i n t h i s laboratory. The superheterodyne spectrometer has the advantage that, while i t s t i l l uses a 400 cycle f i e l d modu-l a t i o n , the s e n s i t i v i t y of the spectrometer i s comparable with the conventional 100 Kc e.s.r. spectrometers used i n many laboratories. The microwave system, with accessories, i s e s s e n t i a l l y the same as that described for the 100 Kc spectrometer with the exception that a low frequency modulated microwave cavity i s used which, i s , i n t h i s case, made of s o l i d brass. The detec-t i o n and A.F.C. systems d i f f e r markedly however. The r e f l e c t i o n signal from the cavity i s mixed with a signal coming from a second Varian #V153C klystron, d i f f e r i n g i n frequency by 30Me/sec from the f i r s t , i n an L.E.L. Model #XBH-2 mixer-preamplifier. The 30Mc beat signal i s then fed into an L.E.L. Model #30B50 narrow band, I.F. amplifier and from there into a conventional 400 cycle, phase sensitive detection system. With such a system there i s some problem with the auto-matic frequency control of the f i r s t klystron. It i s desirable to take t h i s s i g n a l after the I.F. amplification and 30Mc de-modulation. Because of the narrow band pass of the I.F. am-p l i f i e r however, there i s no A.F.C. s t a b i l i s a t i o n u n t i l the two klystrons are closely tuned to the 30Mo frequency d i f f e r -ence. This problem was overcome i n the following way. I n i t i a l l y the f i r s t klystron i s s t a b i l i s e d from a signal taken after the 15 preamplification. The second klystron i s then tuned so that a signal i s \"being passed \"by the I.F. amplifier. At t h i s stage, the f i r s t klystron i s s t a b i l i s e d from a signal taken after the I.F. a m p l i f i c a t i o n and the f i n a l adjustments are made. Fig. S shows a block diagram of the spectrometer, i i ) Liquid Helium Dewar The l i q u i d helium dewar used i n t h i s work was constructed in t h i s department, following c l o s e l y the design of Duerig and Mador (31). F i g . 3 shows a schematic diagram of the dewar. Of a l l metal construction, the dewar consists of an inner helium container which i s surrounded by a vacuum envelope and a l i q u i d nitrogen jacket to minimise heat loss. The vacuum -7 -8 envelope i s pumped to 10 - 10 mm Hg using an o i l d i f f u s i o n pump and a rotary o i l pump. The pressure i s measured with an N.R.C. type 507 i o n i z a t i o n gauge. The microwave cavity i s attached d i r e c t l y to the bottom of the dewar, as i s a 0.15 i n . sapphire rod which s i t s i n the centre of the cavity and on which gaseous samples are condensed. The dewar i s f i t t e d with a s i l i c a r a d i a t i o n window and the cavity i s slotted at one end to allow i r r a d i a t i o n of the sample i n s i t u . A hole i n the bot-tom of the cavity permits the i n s e r t i o n of a 1/16 i n . outside diameter, 0.005 i n . w a l l thickness, stainless s t e e l c a p i l l a r y through which the gaseous samples enter. Helium was l i q u i f i e d for these experiments i n a C o l l i n s cryostat. The helium gas was recol l e c t e d during the experiment and an elaborate set up for cleaning and recompressing the gas A. o _: ti O o ^ O Q. < o 3.3 JO V Q. V O u o 5 s*-< t < u. n \u00E2\u0080\u00A2 a <-*-> a) O o \u00E2\u0080\u00A2\u00C2\u00A3 0 > 0 k k o \u00C2\u00BB' 5** S3\" I 4\u00C2\u00AB \u00C2\u00A3 \u00E2\u0080\u0094 i * V L L L o J\" d O ^ \u00C2\u00B0 o * /I o O I-\u00E2\u0080\u00A2a u-. \u00E2\u0080\u00A2a a. r i I. \u00E2\u0080\u00A2\u00E2\u0080\u00A2> * 3 5 H a *<5 a 3 o i 0-\u00C2\u00ABf Z 4-> , _< C w 3 CO OJ \u00C2\u00A3 o c +-> Q> -C C Q> CL_ OJ CO ttJ <\u00C2\u00BB 'Hi .2 \u00C2\u00A3 LIQUID HELIUM TARGET C A P I L L A R Y F i g . 3 . L lc ju id H e l i u m D e w a . r 16 enables i t s reuse. The procedure used i n transferring the l i q u i d helium to the dewar was as follows. Several hours prior to the transfer, the l i q u i d nitrogen dewar was f i l l e d i n order that the helium dewar would be cooled by radiative transfer. The l i q u i d helium was transferred from the storage vessel to the dewar by applying an excess pressure of about \ pound to the top of the storage vessel thus forcing the l i q u i d helium through a 3/16 inch outside diameter, 0.01 inch w a l l thickness, stainless s t e e l transfer tube enclosed i n a vacuum jacket. The transfer could be followed by monitoring the change i n the resonant frequency of the cavity as the dewar cooled and then by the quantity of the gas which was collected. 17 I I I . ELECTRON SPIN RESONANCE STUDY OF THE PHOTOLYSIS PRODUCTS OF HYDRAZOIC ACID (HNg) TRAPPED AT 4\u00C2\u00B0K a) Introduction The o p t i c a l spectrum of the imine r a d i c a l (NH) i n the gas phase has \"been known (32) for a long time. The most intense hand i s a hand at 3360A* which i s ascribed to a t r a n s i t i o n from the Y\" ground state to an excited T T s t a t e . Funke (33,34) ob-served t h i s spectrum during the thermal decomposition of am-monia and made extensive measurements on the system. Ramsay(35) has made measurements on the spectra of NH and NHg which were obtained by f l a s h photolysis of hydrazine. Thrush (36) studied the NH system obtained by f l a s h photolysis of hydrazoic acid. He also i d e n t i f i e d NH 2 and gave tentative evidence for the production of N\u00E2\u0080\u009E. On the basis of these studies he suggested 6 the following k i n e t i c scheme for the gas phase photolysis of HN\u00E2\u0080\u009E 6 HNg+ h-9 -NH+-N2 NH-*-HN3 \u00C2\u00BB\u00C2\u00BBNH2-*-N3 (-3-1) NH2+-M3 \u00E2\u0080\u0094\u00C2\u00AB-NHg+N 3 Recently measurements have, been made by Dixon (6) on the 2 TT\u00C2\u00AB 3 Y\" t r a n s i t i o n of NH. This species was obtained by the fl a s h photolysis of HNCO i n the gas phase. On the basis of the above gas phase o p t i c a l r e s u l t s . i t i s d e f i n i t e l y established that the ground state of NH i s a 3\u00C2\u00A3~ state and NH i s expected to. be paramagnetic. We f e l t that, 18 because df t h i s , i t would he worthwhile to oarry out an elec-tron spin resonance study of t h i s molecule. The hydrazoic acid system was chosen \"because the decomposition i s exothermic and no recombination of the fragments i s expected i n the matrix. It was hoped that some indi c a t i o n of the nature of the trapped NH r a d i c a l would be obtained i n t h i s way. From the r e s u l t s of Dixon (6) the following important constants are obtained for the L\" ground state. i ) Rotational constant BJJ-16.3454 cm'1* 0.0015 cm\"1. The energy of the r o t a t i o n a l levels i s given approximately by the expression Ek=B\u00C2\u00A3K(K>1), where K i s the r o t a t i o n a l quantum number i i ) Spin-spin s p l i t t i n g constant D = 2X\" = 1.856 *\u00E2\u0080\u00A2 0.014 cm\"\"1\" . Coope (37) has e&lculated a value of D=1.63 cm\"3' using atomic SCF functions, i n reasonable agreement with the exper-imental value. In recent years much work has also been carried out on the detection of the NH r a d i c a l trapped i n a matrix at low temperatures using infrared and u l t r a v i o l e t techniques. Dows, Pimentel and Whittle (38) deposited the products from a discharge i n hydrazoic acid on a window at 77\u00C2\u00B0K. The infrared speetrum was determined as the sample was allowed to warm. In addition to bands that could d e f i n i t e l y be assigned to ammonia, hydrazoic acid and ammonium azide, they observed bands that they attributed to N 2H 2 and (NH) where x probably equalled four. They found that these bands increased on warming 19 suggesting the presence of a precursor that could not he de-tected. They suggested that t h i s precursor was, i n fact, NH. In another study, Becker, Pimentel and Van Thiel (39) photo-lyzed HN that had been trapped i n an inert matrix at 20\u00C2\u00B0K. After photolysis they found features which dissappeared on warmup. They suggested these were due to the presence of NH. These studies cannot he considered as providing conclusive evidence for. the production of NH. Conclusive evidence has been provided by the u l t r a v i o l e t studies of Robinson and McCarty (40) however. In these exper-iments they trapped, at 4\u00C2\u00B0K, the products of a discharge i n a mixture of ammonia oi hydrazine with an inert gas. Under these conditions they f i n d that the spectra of both NH and NHg can be observed. In a more recent investigation, McCarty and Ro-binson (41) have analyzed the r o t a t i o n a l structure of the NH r a d i c a l i n rare gas matrices. They fi n d that i t i s possible to c o r r e l l a t e the structure of the NH trapped i n the s o l i d with that i n the gas phase and, on t h i s basis, conclude that NH i s undergoing e s s e n t i a l l y free r o t a t i o n i n the matrix. Keyser and Robinson (42) further investigated the photo-decomposition products of hydrazoic acid i n a rare gas matrix at 4\u00C2\u00B0K, the i d e n t i c a l system that we study here. From the i r electronic spectra they are able to show that NH i s produced i n considerable amounts, with NHg being produced as a secon-dary produet. They estimate the r a t i o of NH to NHg to be^200:1. They also give tentative evidence for the production of Ng. EO, In the l i g h t of the above evidence, i t appears that the photolysis of HNg i n an inert matrix may be described by the following reaction scheme. HN\u00C2\u00BB + hV \u00E2\u0080\u0094\u00E2\u0080\u0094*fNH +N? NH + HNg -NHg + N 3 possibly accompanied by HNg+- hV - \u00C2\u00BBH +N 3 Schnepp and Dressier (43) followed the photolysis of am-monia i n a matrix at 4\u00C2\u00B0K. They were able to detect the elec-tronic spectrum of NH, as w e l l as NHg, when the wavelength ,of o i r r a d i a t i o n was shorter than 1550A. It i s i n t e r e s t i n g to note that Foner et a l (44) observed a broad resonance, as w e l l as a resonance due to NHg, when they carr i e d out an e.s.r. study on the discharge products of ammonia which were trapped i n an inert matrix. A possible ex-planation i s that t h i s broad resonance was due to NH although no explanation was given by the authors. It has been found (40) however that NH i s produced under such conditions. This i s also i n agreement with gas phase e.s.r. measurements which have been made by Farmer and Ferraro (45) on the discharge products of ammonia.>In t h i s case, when i d e a l conditions are chosen, a spectrum believed to be NH has been detected. Recently Smolinsky, Wasserman and Yager (8) have reported the e.s.r. detection of a series of substituted imines formed i n an organic glass at 77\u00C2\u00B0K by the photolysis of the corres-21 pending azide. These imines are found to have t r i p l e t ground states. In Chapter I we have seen that ,the fine structure of the , e.s.r. t r i p l e t state spectrum may, for o r b i t a l l y non-degenerate molecules, he explained i n terms of the spin^spin i n t e r a c t i o n of the two electrons. As NE, which has a ^T\" ground state, i s o r b i t a l l y non-degenerate the Hamiltonian ~3{ = gSH.S-*-D(S2 - 1/3 \u00C2\u00A72) (3-3) should be adequate for the description of the magnetic pro-perties. The term i a E has been dropped for symmetry reasons. S t r i c t l y , terms of the form A(S.K) and A 1 ! 2 should.be added to (3-3). These are neglected however as the r o t a t i o n a l mag-netic moment i s small compared to the spin magnetic moment. b) Experimental Procedure Hydrazoic acid.(HNg) was prepared i n small quantities before each experiment to minimise the danger of handling t h i s material. Because of the explosive and toxic nature of hydra-zoic acid, i t was necessary to prepare the compound i n a fume hood with a protective wire screen. Adequate protective c l o t h -ing was also worn by the operator. Hydrazoic acid has the following physical properties (46). Colorless l i q u i d m.pt -80\u00C2\u00B0C. b.pt 37\u00C2\u00B0C The method used i n the preparation of HN3 was that given by Dows and Pimentel (47). This method has the advantage over 22 methods which employ concentrated H 2 S O 4 or H^PO^ i n that no v o l a t i l e impurity (such as SQg). contaminates the HNg as i t i s produced.. An excess of stearic acid was placed i n a 100 ml f l a s k and about 0.3 g sodium azide added. This was then heated i n 0 vacuo to about 90 C where the stearic acid i s molten. The hydrazoic acid was removed as i t was formed by condensing the material i n a l i q u i d nitrogen trap. A trap-to-trap d i s t i l l a t i o n was used for p u r i f i c a t i o n . Krypton was thenpassed through the trap containing the HNg and the mixture collected i n a storage bulb by condensation into a sidearm. A matrix r a t i o of 1: 150 - 250 HNg/Kr was used i n these experiments. To prevent de-composition, the sample was kept refrigerated u n t i l shortly before use. F i g . 4 shows a schematic diagram of the apparatus used i n t h i s preparation. The gas .mixture was deposited on a sapphire needle at 4\u00C2\u00B0K. Radiation of the sample was oarried out using a General E l e c t r i c #A-H6 u l t r a v i o l e t lamp. A water f i l t e r was used between the quartz lens system to reduce the heating effect of the lamp. A l l e.s.r. measurements were made using the super-heterodyne spectrometer described i n Chapter 2. c) Experimental Results Figure 5 shows e.s.r. spectra i n the free-electron region, obtained i n three separate experiments. In each instance there i s a broad resonance (^40 gauss) on which i s superimposed a number of narrow l i n e s . The two components of the main spectrum 23 occur i n varying proportions, so two d i s t i n c t paramagnetic species must \"be responsible. The narrow line s were less obvious when care was taken to avoid warming of the sample (Fig.5a). In experiments where i t was necessary to r e f i l l the helium reservoir (Fig. 5b and 5e), warming may have occurred. Hence the narrow l i n e s are probably caused byaa secondary reaction. It i s possible i n fact to i d e n t i f y t h i s spectrum as being due to NH,,. The e.s.r. spectrum of NH g has been w e l l established by the work of Foner et a l (44). These authors give Ag= 24.12 gauss and A^ = 10.40 gauss. This agrees w e l l with the values A H = 24.8 ^0.5 gauss and AJJ =\u00E2\u0080\u00A2 10.7^ *= 0.5 gauss taken from Fig.5c. The broad underlying resonance we ascribe to the NH r a d i c a l . In addition a weak resonance due to H atoms was detected. A caref u l search of the range 0-12,000 gauss yielded a single, weak resonance i n the h a l f f i e l d region (Fig. 6) apart from the features mentioned above. This i s also ascribed to the NH. d) Discussion of Results In a discussion of the experimental results i t i s neces-sary to consider how the products are trapped i n the matrix. Previous studies of the e.s.r. of t r i p l e t molecules have been mainly concerned with \"large\", r i g i d l y trapped molecules. The NH r a d i c a l on the other hand, appears to be su b s t a n t i a l l y free to rotate i n i t s matrix environment, according to the spectroscopic evidence c i t e d above (40,41). The e.s.s. spectrum predicted for these two conditions are quite d i f f e r e n t as w i l l now be discussed. E.5.R. Spectrzt of t h e Photolysis Products o* H N 3 . I Fig. 6. E.S.R. Spectra of the Photolysis Products of H N 3 . (c\) H a l f F i e l d L i n e , (b) Hydrogen Atom S p e c t r a . 24 i) The R i g i d System In a r i g i d system the axis of any i n d i v i d u a l molecule i s fixed and may be defined by the polar angles (9 , $ ). On the other hand, the r o t a t i o n a l angular momentum of the mole-cule i s indeterminate as a consequence of the uncertainty p r i n -c i p l e . The energies of the spin levels may be obtained d i r e c t l y from (3-3) as a function of the angle\u00C2\u00A9 . In a p o l y c r y s t a l l i n e sample, the e.s.r. line shape i s a composite involving an average over a l l orientations. This type of treatment has been carried out by de Groot and van der Waals (22), K o t t i s and Lefebv.re (24) and Farmer, Gardner and McDowell (48) with, however, p a r t i c u l a r reference to the case, relevant to aromatic hydrocarbons, i n which the z e r o - f i e l d s p l i t t i n g i s smaller than the Zeeman energy. In that ease i t i s natural to describe the spin states as derived from the | M ^ states quantised with respect to the f i e l d d i r e c t i o n and speak of highly anisotropic Am-=1 t r a n s i t i o n s (hV 0 = g^H \u00C2\u00B1 D/2 (3Cos 2\u00C2\u00A9 -1) to f i r s t order) and weak, much less anisotropic Am=2 transitions (hV0 = 2g\u00C2\u00A3H to f i r s t order). In the NH r a d i c a l , by contrast, (gp)'1!)*- 20KG. , and the zero f i e l d s p l i t t i n g i s larger than the Zeeman energy for a l l f i e l d s available. The language of |ms^ states and A m tra n s i t i o n s here loses s i g n i f i c a n c e . Instead the eigenstates of the system approximate to the zero f i e l d states. We have seen previously that the e.s.r. r e s u l t s for NH should be explained adequately i n terms of the Hamiltonian (3-3). I f we take as basis states, eigenstates of S z quantised with 25 respect to the molecular z-axis (2-3) may he written i n the following matrix form. (3-4) 1/3 D + gjftH CosG Ijtfi&ft Sin9 0 g\u00C2\u00A3H Sine -8/3 3) 1^2 g)3H Sin\u00C2\u00A9 0 1/J2 gpK Sine 1/3 D-g(8HCosO H has been assumed (as can be done without loss of generality) to l i e i n the xz;-molecular plane and to make an angle 9 with respect to the z-axia. An exact solution for the eigenvalues i s e a s i l y obtained for 9 = 0\u00C2\u00B0 and 90\u00C2\u00B0. The eigenvalues are found to be a) 9 * 0\u00C2\u00B0: E =1/3 D\u00C2\u00B1 gfiH, -2/3 D A- (3-5) b) 8 = 900; E- -D/g\u00C2\u00B1 7D a i-4(gpHt)r , l/3 D These two cases are i l l u s t r a t e d i n Figure 7. Lines are drawn corresponding to tra n s i t i o n s which can occur for a microwave frequency of 9000 Mc/see. For 9 - 6 \u00C2\u00B0 i t i s seen that three tra n s i t i o n s are possible (although not a l l are allowed), whereas only one t r a n s i t i o n i s possible for 0 = 90\u00C2\u00B0. For other orientations the secular equation does not faetorise and an expression for the eigenvalues i s not e a s i l y obtained. The position and number of resonances possible for a given orien-t a t i o n i s best seen using a graphical method similar to that used by Ko t t i s and Lefebure (24). DeGroot and van der Waala (22) have shown that resonance w i l l occur when the condition (3-6) r *Q(|Q- 2^0(9wf*{3a*-DM(g\u00C2\u00BBttfl. M&^U&HF ' 4 * 26 i s s a t i s f i e d . The r e s u l t s are shown for NH i n Fig.8. Tallies of D = 1.86 em\"1 and \u00C2\u00A3 = 0.310 cm\"1 have been used. From Figure 8 i t i s seen that; i ) One t r a n s i t i o n i s pdisfcbbei!. for a l l orientations. This is the one corresponding to the t r a n s i t i o n between the levels which have energy l/3 D i n zero f i e l d . This resonance w i l l extend from H = 1500 - 8500 gauss. i i ) For orientations near 8 = 0\u00C2\u00B0 (from 0\u00C2\u00B0 - ~6\u00C2\u00B0) three tran-s i t i o n s are possible, the one given above plus two between the le v e l s which have energies l/3 D-gj9H and - 2/3 D for 0 = 0 \u00C2\u00B0 (see F i g . 7). This resonance w i l l occur over the f i e l d region H= 16,700 - 23100;gauss. The question now arises as to whioh, i f any, of these resonances one can hope to observe. Experimentally one can not hope to see the second s i t u a t i o n as i t l i e s w e ll beyond the available magnetic f i e l d s i n our laboratory. It should be noted that t h i s resonance i s expected to be weak as the number of molecules mating an angle near 0 = 0 \u00C2\u00B0 i s small. On the other hand one might expeot to obtain a detectable signal from the f i r s t case. In Chapter 5 i t i s shown that a s i n g u l a r i t y occurs at the high f i e l d l i m i t of the composite spectrum. This s i n g u l a r i t y occurs at the f i e l d value of (gy8H)2= Do\" -r h 2 . For D = 1.86 cm\"1 and b~ 0.31 cm\"1 4t 1*111 occur at 8770 gauss. As we have mentioned previously, exper-iments have been recently performed (8,49) on a series of sub-s t i t u t e d imines r i g i d l y trapped i n a matrix at 77\u00C2\u00B0K. It i s Cos*9 = euMD^H)1 * { 3 i a - D 2 - 3 ( 9 A H ) 2 } D S ^ / S H ) * i = 0 31 Ocm\"' 2-0 F i g . 8 . Graphical Representat ion of Cosi0(H) as a, F u n c t i o n of Magnet ic F i e f d S t r e n o t h . 27 demonstrated i n Chapter 5 that the observed e.s.r. r e s u l t s for these imines may be explained by the occurrence of the singu-l a r i t y i n the composite spectrum described above. The fact that no such resonance was observed i n the present system could thus be taken as meaning that NH i s not being r i g i d l y trapped i n the matrix, i i ) A Freely Rotating NH Radical The s i t u a t i o n now i s the opposite of that discussed i n i ) . The molecule i s i n an eigenstate of r o t a t i o n a l angular momentum. The d i r e c t i o n of the molecular axis, on the other hand, i s defined only i n terms of a p r o b a b i l i t y d i s t r i b u t i o n given by the square of the r o t a t i o n a l wave function. Before we consider the effect of this \"randomness\" of orientation, i t i s desirable to give some consideration to the coupling of the angular momenta i n the NH r a d i c a l . The ground state of NH i s a zXm\" state. As such, the r a d i c a l should be approximated by a Hunds case (b) coupling scheme. In such a scheme both K, the r o t a t i o n a l quantum number, and S, the spin quantum number, are good quantum numbers. K and S couple further to give a resultant angular momentum J. The eigenstates of the coupled system, j-KSJMj^, may be written as a linear combination of products of spin and r o t a t i o n a l states i n the usual way. For a given K, J may assume the values J = )E-S| , \K-S+l\ \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 K+S . In our case, where S=1, J may assume values J=K + 1, K and K - l . These J levels w i l l be referred to as the 28 spin multiplets corresponding to the r o t a t i o n a l quantum number K. When the spin-spin i n t e r a c t i o n i s considered, these spin multiplets are no longer degenerate but are, i n general, s p l i t . We w i l l now investigate t h i s s p l i t t i n g of spin multiplets. The K = 0 r o t a t i o n a l state i s represented by the zeroth order spherical harmonic YQ\u00C2\u00B0^I which i s a constant. This means that a l l molecular orientations are equally probable. Expressed in the laboratory coordinate system, the spin-spin Hamiltonian (3-3) may be written as _ ^ 2 (3-74 The Hamiltonian (3-7) has been derived under the condition that we are dealing with a spin 1 system. I f we assume that the system i s i n an eigenstate of r o t a t i o n a l angular momentum a further s i m p l i f i c a t i o n of (3-7) can be made. In p a r t i c u l a r , an average of the space portions of (3-7) should be made over the r o t a t i o n a l wave function. For the K=0 r o t a t i o n a l state i t i s found that this averaging leads to a vanishing of a l l the terms i n (3-7). It i s thus found that, i n the K= 0 l e v e l , there i s no spin-spin s p l i t t i n g . Using similar arguements one can derive an expression for the s p l i t t i n g of the spin mul-t i p l e t s for any K. It i s found that for a l l K+0 the spin multiplets are s p l i t . These arguments were used by Kramers (50) to derive his o r i g i n a l expression for the s p l i t t i n g of the spin multiplets i n 0 g cine to spin-spin i n t e r a c t i o n . Kramers' re s u l t s may be stated as follows. 29 1. For K=0* there i s no s p l i t t i n g 2. For K+O the s p l i t t i n g between the three J levels (J-K - r-1, K and K-l) i s given by the following expressions. (3-8) E(K-*l)~D/3 - (K> 1)D ; E(K) = D/3; E(K-l)=D/3 - KD 2K-*3 2K-1 Appendix 3 gives a derivation of these r e s u l t s using tensor methods. One may now ask, what happens when a magnetic f i e l d i s applied? The answer to th i s i s that the 2J>1 degeneracy, (cor-responding to the values mj=-J, J) w i l l be removed. The energy of th i s s p l i t t i n g i s given. by Wmj - gjfJHmj providing the separation between levels of differ e n t J i s large compared to the Zeeman s p l i t t i n g . ( i . e . we are not i n the Paschen-Back region). The value of g j , the appropriate, g-factor, i s obtained from the following expression g.T\u00C2\u00BB gvU-l} + 2 ~ 2 (3-9) K J U + I ) J U + I ) J T J ^ T ) The term in(g.J) can be neglected as g K i s small. Substituting (\u00C2\u00A7.J>- J2-*-S2-K2 y i e l d s g . T . i 3 + g k 2 \u00E2\u0080\u00A2 2 7^ ' In the special case that K = S - 1 i t i s found that gj - 1. Fig.9 shows the s i t u a t i o n for K\u00C2\u00BB0 and K -1 for the cases where no magnetic f i e l d i s applied and where one i s applied. *For K=0 there i s of course no degeneracy i n J as J\u00C2\u00ABS. The only degeneracy that remains i s a three f o l d spin degeneracy which i s not l i f t e d . 1-64 . , , r - , > o &000 (tooo 6000 8000 lopoo H( gauss) .9 . Energy LtvtL Diagram for the K=0 and K=/ Rotat ional States of N H . 30 The above discussion has been based on the assumption that the NH r a d i c a l can be described by a Hands ease (b) coup-l i n g scheme. This assumption implies that the r o t a t i o n a l quantum numbers K are good quantum numbers. For NH t h i s turns out to be a good a p p r o x i m a t i o n f o r Og (28) however i t i s a poor one. This i s the reason, for example, why the experimental r e s u l t s (50), for low K, do not agree with Kramers formulae. It may be shown (51) that the spin-spin interaction \"mixes i n \" states of K=*2 with the state K. The approximation we have used i s thus only correct to f i r s t order. For NH the separation between the r o t a t i o n a l levels (33.4 em\"'1' between K = (J) and K = l ) i s much greater than, the spin-spin i n t e r a c t i o n (D = 1.86 cm' 1). The second order term w i l l be very small and may .be neglected for our purpose. In Og, however, these terms are, for small K, of the same, order of magnitude and the molecule does not conform to a Hunds case (b) coupling s i t u -ation. The averaging out of the spin-spin i n t e r a c t i o n i n the K\u00C2\u00BB 0 r o t a t i o n a l state i s somewhat similar to the averaging out of the anisotropic part of the hyperfine i n t e r a c t i o n i n s o l -ution e.s.r. work. It should be pointed out that the s i t u a t i o n i s not e n t i r e l y analogous however. In the s i t u a t i o n that the molecule i s defined by a de f i n i t e r o t a t i o n a l quantum number, the averaging occurs because.of the,required indeterminacy of orientation. The only state where the spin-spin interaction i s eliminated i s the K= 0 state where a l l orientations are equally probable. In a l i q u i d , however, /bhe averaging occurs by a 31 rapid, physical tumbling of the molecule. Such a tumbling i n the case of NH would not lead to a narrow resonance at g = 2, as the tumbling provides a mechanism (52,53) for s p i n - l a t t i c e . r e laxation. The l i f e t i m e i s expected to be so short i n fact that, because of uncertainty broadening, the l i n e width w i l l be of the order of several thousand gauss. In the l i g h t of the discussion given i n ( i ) and ( i i ) above, we consider an interpretation of the experimental r e s u l t s . It i s clear that the observed spectrum cannot be explained i n terms of a system of r i g i d NH r a d i c a l s . We have seen however that no zero f i e l d s p l i t t i n g i s expected i f the NH r a d i c a l i s i n the K=0 r o t a t i o n a l state. In t h i s case one should observe the normal Zeeman t r a n s i t i o n at g = 2. Presumably, one might also expect to see hyperfine in t e r a c t i o n from the hydrogen and nitrogen nuclei i n the molecule. For the K> 1 r o t a t i o n a l state transitions are expected to occur at twice the g= 2 magnetic f i e l d . The prominent feature i n our experimental r e s u l t s i s the broad, underlying resonance at g = 2. I t i s t e n t a t i v e l y sug-gested, that t h i s i s due to the trapping of NH r a d i c a l s i n the matrix and that these r a d i c a l s are undergoing e s s e n t i a l l y free rot a t i o n . As a l l experiments were done at 4\u00C2\u00B0K, only the K=0 r o t a t i o n a l state i s expected to be appreciably populated. Less than one molecule i n 10,000 w i l l be i n the K = 1 state i f thermal equilibrium i s assumed. 32 The broadening may be accounted for by a alight h&n.der-ance of r o t a t i o n due to a deviation of the l a t t i c e sttrroundinga from octahedral symmetry. Coope (54) has shown that such a hindrance introduces a small spin-spin i n t e r a c t i o n term. The magnetic properties are now described by an e f f e c t i v e Hamil-tonian, P y. - gpH.S + D 1 ( S * - 1/3 S 2) (3-10) i f i t is assumed that the l a t t i c e s t i l l has a x i a l symmetry. The parameter D' i s defined by (3-11 D'\u00C2\u00BB ( Torient( \u00C2\u00A9 , $ ) \ 2 ( Torient ( 6 , (j) )) D where ^ orient ( \u00C2\u00A9 , ) i s , i n Coope's terminology, the orienta-t i o n a l wave function and i s an eigenfunction of the Hamiltonian li =BK2 + V( 9 , ji), where V(8,4>) i s the l a t t i c e p o t e n t i a l . The problem now reduces i t s e l f to a s i t u a t i o n similar to that of the aromatic t r i p l e t state molecules. Theoretically, a l i n e shape of the form shown i n F i g . 13b i s predicted, ne-glec t i n g hyperfine i n t e r a c t i o n . The inclusion.of hyperfine interaction w i l l r e s u l t i n an overlap of six such spectra. Some broadening w i l l probably occur as the sample w i l l not be magnetically d i l u t e . This i s probably the reason that no struc-ture was resolved. Experimentally we observe a l i n e which has approximately a 40 gauss lin e width and a 140 gauss t o t a l width. In terms of the model proposed above, th i s i s consistent with D1 being of the order of 0.005 cm\"1. This theory also provides an explan-ation for the occurrence of the h a l f f i e l d l i n e . It i s well 33 known (Si) that there i s some in t e n s i t y of the flm =2 tran-s i t i o n . An estimate of the in t e n s i t y of the Am = 2 t r a n s i t i o n as compared to the i n t e n s i t y of the Am =1 t r a n s i t i o n can he made i n the following way. For DT small, the eigenstates of the system w i l l approx-imate the states l lV , lo)and 1-1^ quantised with respect to the f i e l d d i r e c t i o n . We take these as the zeroth order states. In terms of the laboratory coordinates, (3-10) i s given by (3-13) 31'-- |(5Cosxe->Xsi -it) * I Coses^ee'^(StST *STSI) +SJ Siv*e e**^S* The eigenstates, corrected to f i r s t order, are then given lay, j J This y i e l d s , (3-14) These states are orthogonal to f i r s t order. For D'/gp3 small the normalisation constant w i l l be olose to unity and i s neglected. In our case, the r . f . f i e l d i s perpendicular to the s t a t i c magnetic f i e l d . The t r a n s i t i o n p r o b a b i l i t y for the A m r S t r a n s i t i o n i s proportional to, l/vulc |u> \\ z _ D'2 0OS 2 8 Sin 9 (3-15) 34 We substitute the value g|3H = o/z obtained from the f i r s t order energies. We thus get, \(Vi l s x l ^ - i M 2 = 2D1 2 Coa 29Sin 2 e l \ X | \" ^2 (3-16) The Am = l t r a n s i t i o n i s allowed i n zeroth order, the tran-s i t i o n p r o b a b i l i t y being given by, 2 | ( l ( S x | 0>| 2 = 1.0 (3-17) An estimate of the r e l a t i v e i n t e n s i t i e s of the Am-= 1 and Am =2 transitions can now be obtained by an integration over a l l angles. T I( A m \u00C2\u00BB 2) Jx I Cos 20 Sin 6 46 = 4/15 D' 2/^ 2 1 * (3-18) I ( A m \u00C2\u00BB l K [ s i n e d G -1.0 ft The r a t i o i s then given by, I( Am = 2) _ 4 2ll (3-19) I( Am = 1) TH ^2 For D'= 0.005 cm-1 and 0=0.30 cm - 1, one obtains an Am =2 i n t e n s i t y of 0.075$, Experimentally the observed i n t e n s i t y r a t i o i s estimated to be i n the range 0.3 - ifo. The objection may be raised that the s i g n a l at g =2 that has been assigned to NH i s i n fact due to some other doublet species formed i n the reaction. It seems possible to reject most other suggestions however. For example, i) Thrush (36) has given tentative evidence for the production of Ng i n the gas phase photolysis of HNg. The ground state of Ng i s a 2 J f g state and, unless the o r b i t a l momentum i s com-35 p l e t e l y quenched, the resonance w i l l not appear at g=2. In any case the concentration of N g should not exceed th& com-bined concentrations of NHg and H. In these experiments the species responsible for the broad resonance was i n at least a four f o l d excess. i i ) Recently i t has been reported (26,27) that the coupling of two doublet states can lead to a\"dimer\" which has a singlet ground state and a low l y i n g t r i p l e t state. One might consider such a coupling to occur for two NHg r a d i c a l s which are near-est neighbours. This explanation seems inconsistent for sev-e r a l reasons. F i r s t l y at 4\u00C2\u00B0K, i t i s expected that only the singlet state w i l l be populated and the dimer should be dia-magnetic. Secondly, the broad species i s i n excess of the NHg. S t a t i s t i c a l l y t h i s i s improbable, as the pr o b a b i l i t y of having two NHg r a d i c a l s as nearest neighbours i s small. i i i ) One might also suggest the p o s s i b i l i t y of multiple trapping s i t e s . This seems u n l i k e l y i n view of the previous investigations of NHg (44) where no such phenomenon was observed. In conclusion, the res u l t s may be summarised as follows. In the photolysis products of hydrazoic acid trapped i n an krypton matrix, we have detected, using e.s.r., a s i g n a l whieh could d e f i n i t e l y be assigned to the NHg r a d i c a l . In addition, we have t e n t a t i v e l y assigned a broad but intense signal at g=\u00C2\u00BB2 and a weak, h a l f f i e l d signal to NH ra d i c a l s trapped i n the matrix. This assignment has been shown to be consistent 36 with what i s expected t h e o r e t i c a l l y for NH undergoing essen-t i a l l y free rotation, however a positive i d e n t i f i c a t i o n was not possible. 37 IV. ELECTRON SPIN RESONANCE STUDY OF THE PHOTOLYSIS OF DIAZOMETHANE TRAPPED AT -4\u00C2\u00B0 K. ay Introduction In recent years many studies, both t h e o r e t i c a l and expe-rimental, have been made on the methylene r a d i c a l (CHg). The main interest i n t h i s problem has been the nature of the eon-fig u r a t i o n of the two non-bonding electrons which can lead to a singlet or t r i p l e t state as the ground state. Theoretically there has been much controversy on t h i s point. Foster and Boys (55) predicted that the ground state 3 r> should be a bent t r i p l e t state, with a bond angle of 129\u00C2\u00B0. Walsh (56) l a t e r decided that the ground state should be a \u00E2\u0080\u00A2^ A^ state. By drawing e o r r e l l a t i o n diagrams he was able to show that, i f the molecule was to assume a lin e a r configuration, the ground state should be a state. Using a modified v a l -ence bond approach Jordan and Longuet-Higgins (57) concluded that the ground state should be a linear ^Zg . The sama con-clusion was also reached by Pedley (58) from heat of formation considerations. The experimental evidence which has been reported tends to support the state as the ground state. Woodworth et a l (59) photolysed diazomethane i n the presence of ols-2-butene and found ois-2-dlmethylcyclopropane as the; product indicating stereospecific addition. They pointed out that t h i s i s consis-tent with the addition of singlet methylene to the double 38 bond because of spin conservation considerations.-Anet et a l (60) and Prey (61) repeated this experiment i n the presence of a high pressure of inert gas. They found under these con-di t i o n s the addition was not stereospecifio and, i n addition, 3-methyl-l-butene was formed. They suggest that t h i s scheme i s consistent with the addition of t r i p l e t methylene to the o l e f i n . The methylene i s expected to be formed i n i t i a l l y i n the singlet state and then degraded to the t r i p l e t ground state. The f i r s t direct evidence supporting the 3\u00C2\u00A3g ground state was obtained by Herzberg et a l (62,63) i n the gas phase f l a s h photolysis of diazomethane. Their experiments revealed fea-tures which indicate the ground state i s a 3 j - state. Bands were also found due to a bent ^A^ state. This could be shown to be higher than the 3\u00C2\u00A3~ state as a lower pressure of inert gas was required to produce these features. More recently Wasserman et a l (9,64) have reported the detection of several substituted methylenes which have t r i p l e t ground states. Several attempts have been made to trap and detect C H 2 i n an inert matrix at low temperatures using infrared and u l t r a -v i o l e t methods.,Infrared studies have been carried out by Pimentel et a l (65,66). They have suggested that the disappear-ance of certain bands on warmup which correlate with bands they ascribe to ethylene indicates the presence of methylene. The more recent investigations of Goldfarb and Pimentel (67), 39 and Robinaon and McCarty (68), using infrared and u l t r a v i o l e t spectroscopy, are complicated and often c o n f l i c t i n g . Robinson and McCarty found bands near 3S00.R whioh they suggest might be due to a t r a n s i t i o n from the 2 I g state of methylene. Arguments by Herzberg (8) show t h i s to be u n l i k e l y however. An e.s.r. investigation of the photolysis products of diazomethane i n a krypton matrix at 4\u00C2\u00B0K was car r i e d out by Gerry (69) i n t h i s laboratory. In these studies no signal was found that could d e f i n i t e l y be ascribed to methylene. The investigation did show however thait a large concentration of methyl radioals was produced. The following discussion also reports an attempt to detect the methylene r a d i c a l produced by the photolysis of diazo-methane i n an inert matrix. It was f e l t that this investigation might be more successful than the one reported by Gerry for two reasons. F i r s t l y , a large improvement had been made i n the instrumentation with a change from a 400 cycle spectrometer to a superheterodyne spectrometer. Secondly, work that has been recently reported by Wasserman et a l (8,9) indicated that, i f the CHg were r i g i d l y f i x e d , a resonance should be expected at a much higher f i e l d than had previously been investigated. b) Experimental Procedure Diazomethane was prepared before eaoh run i n order to minimize the decomposition of the product. Precaution was taken i n the preparation because of the toxic and explosive nature of diazomethane. One small explosion occurred during the course of t h i s work. 40 The method used for the preparation of diazomethane was that described by Gerry (69). This method consists of adding about 1 gram dry Eastman Kodak N-methyl-N-nitroso-p-toluene-sulphonamide (Diazald) to ^ 1 5 ml of a saturated solution of KOH i n ethylene g l y c o l . I t was necessary to prepare the sat-urated solution of KOH i n ethylene g l y c o l a short time before using as, otherwise, polymerisation would take place. The solution was degassed to remove any trapped a i r . This process took approximately one hour. Dry diazald was then added to the ethylene glycol-KOH solution i n vacuo - Figure 10 shows the design of the reaction vessel and vacuum system used i n t h i s preparation. The diazomethane produced i n the reaction vessel was removed by constant pumping and the product c o l -lected i n a l i q u i d nitrogen trap. At t h i s stage i t was always found that the diazomethane was contaminated by an ethylene impurity which appeared as a white s o l i d condensed below the yellow diazomethane. The ethylene was removed by allowing the i n i t i a l portion from the trap to d i s t i l l o f f before any diazo-methane was collected. The diazomethane was d i s t i l l e d into a second trap and from there i t was transferred to the sample bulb. The diazomethane was di l u t e d with enough krypton to give a matrix r a t i o of 1:150-250 diazomethane to krypton. The sample was then stored at l i q u i d nitrogen temperature u n t i l shortly before using to prevent decomposition. I t was found b e n e f i c i a l to keep the room as dark as possible during the preparation to prevent photolysis of the product. A mass spec-\u00C2\u00A3 3. 41 trometric analysis of the product by Dr. D.C.Frost confirmed that the bulk of the sample was diazomethane. A l l e.s.r. measurements were made using the superhetero-dyne spectrometer and associated cryogenic equipment which has been described i n Chapter I I . A General E l e c t r i c #A-H6 mercury lamp was used i n the photolysis. The photolysis was carried out under two d i f f e r e n t conditions. I n i t i a l l y the photolysis was carried out using a Corning #4308 f i l t e r which passes wavelengths greater than 3400&. The f i l t e r oould then be removed and the photolysis carried out using an u n f i l t e r e d lamp. In. an attempt to remove some of the infrared r a d i a t i o n , a water f i l t e r was placed between the quartz focussing system. o) Experimental Results i ) Photolysis of Diazomethane in.a Krypton Matrix a) In the g=2 region, after a short exposure with an A-H6 lamp f i l t e r e d so as to pass.only wavelengths > 3400&, a f i v e l i n e spectrum was obtained as i s shown i n Figure 11a. I t i s seem to consist of a four equally spaced l i n e s i n the r a t i o of approximately 1:6:6:1 with a centre l i n e not quite sym-me t r i c a l l y placed between the two inner lines and of somewhat larger l i n e width. A l i n e was also observed w e l l below (85 gauss) g =2. b) When the f i l t e r i s removed there i s a r e l a t i v e decrease i n the i n t e n s i t y of thetwo inner peaks and an increase i n inten-s i t y of the outer peaks, the r a t i o of the four equally spaced li n e s eventually becoming 1:3:3:1 with the centre psak becoming neglig i b l e (Figures l l b - d ) . 33\u00C2\u00AB10q. I 332b 9. SV1I< I 6*0-3lbctn' ( M 2 . 3 . 2. 3. 4. E.S.R. Spectrai of-the Photolysis Products of C H a N 2 in & K r M a t r i x and E can be derived from the ex-perimental data. Recently Yager, Wasserman and Cramer (23) as w e l l as others (89,90) have reported the detection of Am=l t r a n s i t -ions i n a randomly oriented sample of aromatic t r i p l e t states. I t i s now clear (24) that the observed t r a n s i t i o n s arise from the large change i n the number of molecules that can absorb, for a small change i n H, when the s t a t i c magnetic f i e l d i s oriented along, one of the molecular axes. This provides an excellent method for determining the spin-spin i n t e r a c t i o n parameters D and E as the f i e l d p o s i t i o n of the peaks i s sen-s i t i v e to changes i n 3) and E. Also, the i n t e n s i t y of these tra n s i t i o n s i s high compared to the Am =2 t r a n s i t i o n s . In the following discussion we show how the observed line shape and position of these A m = 1 t r a n s i t i o n s can be explained i n terms of a l i n e shape calculated from a f i r s t order pertur-bation model. In 1962 the e . 3 . r . detection of a number of substituted imines and methylenes were reported (8,9,49,64). The eharac-52 t e r i s t l c feature of the spectra of the substituted imines i s that they consist, i n general, of a single high f i e l d reson-r ance l i n e . No detailed explanation of these measurements has been given and no e.s.r. spectra have been published. In the present work we report the detection of two of these imines, phenylimine and benzene.sulfonylimine. The experimentally ob-served spectra are consistent with a l i n e shape calculated from a perturbation treatment where the zero, f i e l d levels are taken as zeroth order l e v e l s and the Zeeman in t e r a c t i o n i s treated as a perturbation on these l e v e l s . The model used i s based on the assumption that the molecules have a x i a l symmetry. It i s shown that t h i s assumption may be j u s t i f i e d by consi-dering the orders of magnitude of the terms contributing to the spin-spin i n t e r a c t i o n . There i s no contribution to E from the dominant one centre term on the nitrogen. This discussion also shows that a linear r e l a t i o n between the spin-spin i n t e r -action parameter D and, the TT -electron spin density on the v nitrogen i s expected. It should be pointed out that the methods described i n t h i s section are only applicable under the l i m i t a t i o n s that the derivations imply; namely that gj8H>>D for the \" a m = l li n e shapes\" and D*> g^H for the case discussed with reference to the substituted imines. For intermediate, eases where the Zeeman in t e r a c t i o n and spin-spin interaction are of the same order of magnitude, such as the substituted methylenesthese methods f a i l and one must use a more exact method such as that outlined by K o t t i s and Lefebv.re (24). 53 b) Experimental Procedure i ) Preparation of the Samples l j Triphenylene.: Triphenylene purchased from K. and K. Laboratories, Inc. was dissolved i n E.P.A. (a mixture of ethylether, isopentane and ethylalcohol i n the r a t i o of 8:3:5 by volume. This mixture forms a transparent glass on cooling to 77\u00C2\u00B0K ) the concentration being about 10~2M. The solution was sealed i n a t h i n walled, 4 mm s i l i c a tube after a thorough degassing. 2. Phenylazi.de: Phenylazide was prepared following the procedure given i n Organic Syntheses (91) with the following modifications. East-man Kodak phenylhydrazine hydrochloride was used as the s t a r t i n g material and a corresponding reduction i n the amount of cone. HC1 used i n the preparation was made. The quantities of s t a r t i n g materials was reduced to one quarter of those given i n the above reference. About 4 ml of crude phenylazide was obtained which was vacuum d i s t i l l e d . The 2|- ml portion b o i l i n g between 52-53\u00C2\u00B0C at 6 mm Hg pressure was used for the experiments to be, described. Samples of phenylazide (^ 10\" 1 M) were dissolved i n a mixture of 1 part eyclohexane to 3 parts decalin by volume. This matrix was used rather than E.P.A. because of i t s greater r i g i d i t y at 77\u00C2\u00B0K. The solution was sealed i n a thi n walled 4 mm s i l i c a tube after a thorough degassing. 54 3. Benzenesulfonylazi.de The prooedure need was that given by Dermer and Edmison (92). 32.8 gm of product were obtained after f i n a l p u r i f i c a t i o n . The samples, used were prepared hy dissolving the benzene-sulfonylazlde i n a mixture of 1 part oyelohexane to 3 parts decalin to give a solution 10\"^M. The solution was sealed i n a thin walled, 4 mm s i l i c a tube after a thorough degassing. A l l samples were i r r a d i a t e d at 77\u00C2\u00B0K i n a Varian #V-4531 multipurpose e.s.r. cavity. Light from a General E l e c t r i c #A-H6 mercury lamp was focussed through a s i l i c a o p t i c a l system. Spectra were obtained with the 100 Kc spectrometer that has been described previously. The magnetic f i e l d was calibrated using an n.m.r. magnetometer. o) General Discussion of e.s.r. Line Shapes i n P o l y c r y s t a l l i n e Samples In c a l c u l a t i n g the e.s.r. l i n e shape that arises from a p o l y c r y s t a l l i n e sample, i t i s necessary to consider which orientations of the molecule contribute to the li n e for a given value of the s t a t i c magnetic f i e l d . The t o t a l l i n e shape i s obtained by averaging over a l l orientations with an equal p r o b a b i l i t y d i s t r i b u t i o n assumed. In obtaining an expression for the line shape the f o l -lowing factors must be considered. i ) Line Shape Due to a Single Orientation It i s w e l l known that various interactions, such as spin-spin and s p i n - l a t t i c e interactions, give r i s e to a resonance l i n e which has a non-zero l i n e width because of the f i n i t e 55 l i f e t i m e of the spin states. It i s necessary to specify a suitable l i n e shape funotion, fdff-H), to describe the shape of the l i n e . In the following discussion, as has been assumed in much of the published work, the assumption i s made that the line shape may be approximated by a Dirac delta function, i(H'-H). This means the li n e i s assumed to have zero li n e width. Provided the line width of the composite spectrum turns out to be large compared with the line width for a single orientation t h i s approximation i s a good one ( 9 3 ) . The as-sumption greatly s i m p l i f i e s the line shape ca l c u l a t i o n s . Some broadening of the spectral features i s obtained because of th* f i n i t e line width. i i ) T r a n sition P r o b a b i l i t y The i n t e n s i t y of the l i n e for a given t r a n s i t i o n i s de-pendent on the p r o b a b i l i t y of the t r a n s i t i o n between the two levels occurring. As the eigenstates of the Hamiltonian are, i n general, dependent on the orientation of the molecule i n the magnetic f i e l d , i t i s to be expected that the t r a n s i t i o n p r o b a b i l i t y w i l l also be a function of orientation. In many provious studies, the assumption has been made that the tran-s i t i o n p r o b a b i l i t y i s independent of orientation. For tha Am-1 t r a n s i t i o n s we also make t h i s assumption. Some jus-t i f i c a t i o n for t h i s assumption can be given however. The i n t e r a c t i o n of the molecule with the microwave f i e l d H-j_ i s given by, JA m.w. = gpS.Hi (5-1) 56 In moat e.s.r. spectrometers, i s oriented so i t i s perpen-dicu l a r to the s t a t i c magnetic f i e l d H. For convenience we aaaume that the x-axis coincides with H]_. In the l i m i t of f i r s t order perturbation theory we take as eigenstates, states of Sz quantised with respect to H. The t r a n s i t i o n p r o h a h i l i t y for the t r a n s i t i o n m \u00E2\u0080\u0094*.mg 1 i s then given by an expression of the form, (5-2) lim3 = |(ms-r l | s x|m g)j 2 = S(S + l)-m a (ms -r l ) whieh i s independent of the orientation of the molecule. Further, for the t r a n s i t i o n s 10) \u00C2\u00AB-|l) and | - l ) \u00E2\u0080\u0094 * | o ) , W_^ = WQ. The assumption that the t r a n s i t i o n p r o b a b i l i t y i s the same for both A m =1 transitions and i s independent of orien-t a t i o n s i s correct to f i r s t order. On the other hand, i n the f i r s t order theory, the Am = 2 t r a n s i t i o n i s forbidden. This t r a n s i t i o n only becomes allowed (See Chapter I I I ) when higher order terms are considered. Hence, the t r a n s i t i o n p r o b a b i l i t y i s strongly dependent on the orientation of the molecule i n the s t a t i c magnetic f i e l d . As shown by de Groot and van der Waals (22), these factors must be included i n a l i n e shape ca l c u l a t i o n for the Am =2 t r a n s i t i o n . i i i ) Averaging over a l l Orientations For any orientation, specified by the Euler angles (94) (j) , 0 and V , the energy i s , i n general, a function of 0 and *P but independent of\u00C2\u00A9 . When considering the contributions from 57 a l l orientations, i t i s necessary to average over a l l values of 0 and V . The number of molecules having angles between 0 \u00E2\u0080\u0094 * Q * d 9 and dT i s given by dN(0J4O=SinededH' (5-3) We are now i n a pos i t i o n to obtain a general expression for the e.s.r. li n e shape of a p o l y c r y s t a l l i n e sample. For convenience we w i l l only consider the c a l c u l a t i o n for a single t r a n s i t i o n . The t o t a l l i n e shape must of course be obtained by a summation over a l l t r a n s i t i o n s . In p a r t i c u l a r , we w i l l be concerned with the two A m = l tran s i t i o n s i n photoexcited aro-matic hydrocarbons. The l i n e shape function may be written i n the general form I f we make the assumptions described above ( i . e . W constant and f(H'-H) = (^(H'-H), the following expression re s u l t s Im\u00C2\u00A3(\u00C2\u00BB')^jHl dfH'-H) S i r r e d * (5-5) We consider the evaluation of thi s i n t e g r a l i n the following special cases: (a) The molecule i s a x i a l l y symmetric. In t h i s case the resonance f i e l d i s dependent only on 0 . The angle V may be integrated out d i r e c t l y . We then f i n d (5-6) 58 From the properties of the d - f u n c t i o n t h i s reduces to give (5-7) dH (b) The molecule does not have a x i a l symmetry. In t h i s case the resonance f i e l d i s a function of both GandH^. The line shape function may then be expressed as In the discussion that i s to follow, we consider f i r s t the c a l c u l a t i o n of the e.s.r. li n e shape from a p o l y c r y s t a l l i n e sample of t r i p l e t molecules that have a x i a l symmetry. The more general case for molecules with less than a x i a l symmetry i s then discussed. Throughout t h i s discussion i t i s assumed that the spin-spin interaction i s small and may be considered as a pertur-bation on the Zeeman lev e l s . This s i t u a t i o n i s encountered i n the photoexcited aromatic t r i p l e t states which have reoeived considerable attention recently. The calculated l i n e shapes are compared with som6 experimentally observed spectra. d) Calculation of the e.s.r. Line Shape for a P o l y c r y s t a l l i n e Sample of T r i p l e t State Molecules with A x i a l Symmetry;B small compared to gjiH. For molecules, such as triphenylene, which have a x i a l symmetry, the fine structure of the e.s.r. spectrum can be (5-8) 59 adequately explained in I',terms of the spin Hamiltonian J i = g\u00C2\u00A3 H.S DtSg - l/3\u00C2\u00A72) (5-9) For aromatic hydrocarbons the magnitude of the spin-spin in t e r a c t i o n i s less than the Zeeman interaction. In such a case i t i s convenient to consider the Zeeman levels as zeroth order lev e l s and treat the spin-spin i n t e r a c t i o n as a perturbation on these l e v e l s . This may be done using standard perturbation techniques (95). We rewrite (5-9) i n terms of the laboratory (primed) coordinate system, where the z' di r e c t i o n i s taken as the d i r e c t i o n of H, by making the following substitution for Szp S z = CosD Si* + Sin9 e\" 1^ S^ + Sin O e 1^ SJ (5-10) 2 2 I f we chose as basis states, states of S z quantised with respect to the laboratory axes, we may represent 3(' = D(S2-l/3\u00C2\u00A72) by the following mattix. (5-11) l/6(3Cos 2 e -1) ly^Cose Sine e\" 1^ i S i n 2 e e - 2 i ^ 1/05COS0 Sine e 1^ -1/3(3CQ>s2e -1) l^CosOSinee' 3^ \u00C2\u00A3Sin 29 e 2 i ^ l^Cos\u00C2\u00A9 Sin\u00C2\u00A9 e 1* l/6(3Cos 2 9-l) \u00C2\u00AB D ? From (5-11) we obtain the following expressions for the energies correct to f i r s t order. B * 1 * \u00C2\u00B1gsH + D/6(3Cos2\u00C2\u00A9 -1) (5-13) E 0 = -D/3(3Cos 2e _D From (5-12) one obtains the following resonance conditions for the two Am = l t r a n s i t i o n s . 60 hv\u00E2\u0080\u009E= g/5H \u00C2\u00B1D/S(3Cos 6 -1) (5-13) In an e.s.r. experiment i t i s -usual to use a fixed microwave frequency Vo and vary H u n t i l the resonance condition i s s a t i s f i e d . In terms of the resonance f i e l d s we may rewrite (5-13) as . 0 J I - ^ H Q - H ! =D'/2(3Cos^0 -1) form=0\u00E2\u0080\u0094*m = 1 (5-14a) and ft 0 (5-14b) J l 2 =H 0-H S=-D'/2(30os 2 e-1) for m = -l\u00E2\u0080\u0094*m= 0 In these expressions we have substituted H 0 = hVo/gp and Dl^gjo We now consider the li n e shape c a l c u l a t i o n for the tran-s i t i o n m=0\u00E2\u0080\u0094\u00C2\u00BB\u00C2\u00BBm = 1. We have seen previously that the composite line shape i s given by the expression dCos9 1(A)* cU. (5-15) By rearranging (5-14a) and d i f f e r e n t i a t i n g i t i a e a s i l y found that 1(A) dCose 2L. (zJh,,\ where -D* <>Pl1< Df E (5-16) A similar expresaion i a found for the lin e due to t r a n s i t i o n (5-14b) _ J -(5-17) This type of cal c u l a t i o n i s w e l l known i n connection with n.m.r. problems (96) and has also been applied by Burns (97) to the e.s.r. powder spectrum of the C r + 3 ( s - 3 / S ) ion i n an a x i a l c r y s t a l l i n e f i e l d , a syatem which i a adequately de-aoribed by our Hamiltonian (5-9). 61 Figure 13a shows the f i r s t derivative e.s.r. spectrum from the photoexcited t r i p l e t state of triphenylene. The line s numbered 1 to 4 originate from A m = l tra n s i t i o n s of the t r i p l e t state. The low f i e l d l i n e at 1416 gauss originates from the Am =2 t r a n s i t i o n of the t r i p l e t state. The strong resonance i n the centre of the spectrum i s caused by free r a d i c a l s generated i n the EPA matrix. The weak l i n e at 3137 gauss on the low f i e l d side of the free r a d i c a l resonance is\u00C2\u00AB believed to be a double quantum t r a n s i t i o n i n the t r i p l e t state. This has recently been reported by de Groot and van der Waals i n the spectrum of deuteronaphthalene (89) and deuterophenanthrene (90). A l l l i n e s assigned to the t r i p l e t state dissappear when i r r a d i a t i o n i s stopped but the free r a d i c a l resonance remains. Figure 13b shows the t h e o r e t i c a l l i n e shapes given by equations (5-16) and (5-17). The innermostj/feingularities, cor-responding to 6=90\u00C2\u00B0, are associated with the zero slope points of lines 2 and 3, while the outermost ones, correspond-ing to 9 = 0\u00C2\u00B0, are associated with the inner edges of l i n e s 1 and 4. In the p l o t t i n g of the l i n e shape, D was chosen such that l i n e s 1 and 4 and t h e i r t h e o r e t i c a l counterparts were coincident. With t h i s condition a value of D= 0.132 cm-\"1 i s obtained. This result agrees w e l l with that of D= 0.134 em\"1 obtained by de Groot and van der Waals (22) from the measure-ment of the low f i e l d A m = 2 t r a n s i t i o n . Although the model used describes adequately the observed e.s.r. l i n e shape there i s some discrepancy i n the predicted gauss 1849 2430 (I) (2) 3877 (3) 4681 (4) (b) 1849 2557 gauss 3973 468! Fig.13. E x p e r i m e n t a l and C a l c u l a t e d E.S.R. Spectra of Tr iplet Triphenylene. 62 po s i t i o n of the l i n e s . I t i s noticeable that the i n t e r v a l between line s 1 and 2 of the experimental spectrum i s not equal to that between line s 3 and 4, as i s predicted by the f i r s t order model, but i s appreciably less. Closer agreement i s achieved by including the second order corrections to the energy l e v e l s . These are given by E^ I l ^ l V ^ - E ? ) (6-18) t+w> Irom (5-11) i t i s found that E ( 2 ^ l = \u00C2\u00B1 D 2Cos 2 9 S l n 2 9 \u00C2\u00B1 P 2Sin 4e ; E* 2=0 2 gpH 8g|3H 0 (5-19) \u00E2\u0080\u00A2 5 S i n g u l a r i t i e s again occur for 9 ^ 0 and 90\u00C2\u00B0. The second order correction affects the s i n g u l a r i t i e s i n the th e o r e t i c a l spectrum i n the following way. At \u00C2\u00A9 - 0\u00C2\u00B0 the correction vanishes and no s h i f t should be observed. At 0 - 90\u00C2\u00B0 the correction adds a term D2/8g|SH to both t r a n s i t i o n energies. Experimentally this should be observed as a s h i f t of l i n e s 2 and 3 toward lower f i e l d . The calculated s h i f t i s 100 gauss for li n e 2 and 68 gauss for l i n e 3. This i s i n reasonable agreement with the observed s h i f t s of 127 gauss and 96 gauss. It i s probable that, higher order terms contribute to the s h i f t . The t h i r d order correction does not contribute however. e) Calculation of the e.s.r. Line Shape of ,a P o l y c r y s t a l l i n e Sample of T r i p l e t State Molecules with Less Than A x i a l Symmetry. D small compared to gftH We now want to extend the ca l c u l a t i o n of the previous section to molecules with lower symmetry. In such a case the 63 e.s.r. fine structure i s described by the Hamiltonian ^ = gpH.S + D(S 2 - 1/3 S 2) + E(S 2 - S 2) - (5-20) We may rewrite (5-20) i n the form 3t = gp.S+D(S 2 - 1/3 ,S 2)+E/2(S 2-rSf) (5-21) where S \u00C2\u00B1 \u00E2\u0080\u00A2= S x\u00C2\u00B1iSy. As before we w i l l treat the spin-spin interaction as a pertur-bation on the Zeeman le v e l s . We wish therefore to express (5-21) i n terms of the laboratory coordinates. To do t h i s wa make use of the ro t a t i o n operator R which defines the r e l a t i v e orientations of the molecular (unprimed) and laboratory (primed) coordinate systems. In p a r t i c u l a r we obtain the re-l a t i o n , r 1 s y The rotation matrix R i s given by R = Cos^ S i n ^ 0 - S i n t Cos y 0 0 0 1 CosG 0 -Sin\u00C2\u00A9 0 1 0 SinG 0 Cos 9 (5-22) (5-23) Cos<\u00C2\u00A3 Sin^> 0\" -Sin)\u00E2\u0082\u00AC. Z 1 \u00E2\u0080\u009E \u00E2\u0082\u00AC Sir \9 e 1 >4> (5-24) 64 where we now have the r e l a t i o n , 'a, S_ R1 V (5-24) d i f f e r s from that given by Rose (98) because of the basis chosen. In an |ms^ basis with respect to the laboratory system, the following matrix for ZHi\u00C2\u00ABD(S 2 - 1/3 S 2) i s obtained, l/6(3Cos 29-1) l ^ C o s S Sin\u00C2\u00A9e _ i* \u00C2\u00A3Sin29 e\" 2 i* ~K.| = D 1/gCos\u00C2\u00A9 SinQ e i (^ -l/3(3Gos 29-1) l/02Cos 6 SinQ e-\u00C2\u00A3Sin2Q e 2 i ^ l/fgCos 8 Sin9 e 1* l/6(3Cos 29 -1) (5-25) The diagonal elements of the matrix 3 t ? * S(S 2 + S 2) are 2 -\"K^ps Sin 2Q Cos2T ; y, 0 0 - - S i n 2 6 Cos2H> (5-26) In the special case that \u00C2\u00A9 = 9 0 \u00C2\u00B0 the matrix of \"3tg i s given by 3 l z = E 1 -4 2. -Cos i f ->Tx*iWfe (5-27) From expressions (5-25) and (5-26) the following expressions for the f i r s t order energies are obtained. E \u00C2\u00B11= gpH+D/6 (3Cos 29 -1)+ B/E S i n 2 9 Cos2V E Q= -D/3 (3C6s 20-1) -ESin 29 Cos2H> (5-28) 65 In the spe c i a l ease that 9 = 0 \u00C2\u00B0 and 90\u00C2\u00B0 the second order corrections are found to he l ( 2 ^ l ( 8 = 90\u00C2\u00B0)= ^ D 2 * E 2 ( 2 S i n 2 2 ^ + Coa^H* ) ' IH ( gflS 8gflH ) f?\ ~ (5-29a) E i 2 , ( 9 = 90\u00C2\u00B0)= 0 and . . / . E^ 2)\u00C2\u00B1l (9= 0\u00C2\u00B0)= * E ; E l 2 M 9 ~ 0 \u00C2\u00B0 ) = 0 (5-29h) 8g\u00C2\u00A3H Using the expressions (5-28), we may calculate the e.s.r, l i n e shape. We have seen previously that the general li n e shape function i s given hy I M *J^9(|\u00C2\u00A3)c>e (5-30) From the f i r s t order energies one obtains the resonance con-d i t i o n P 9 h^=gp\u00C2\u00B1(D/2 (3Cos \u00C2\u00A9 -D+SE/2 Sin^ O Cos2f ) (5-31) This yields the following resonance f i e l d s J \ , =H rH 0= D'/2 (3Cos2\u00C2\u00A9-1) -h3E'/2 S i n 2 9 Cos 2 f (5-32a) J,^H 2-H 0= -D'/2 (3Cos 20-1) - 3E'/2 Sin 29Cos ZV (5-32b) where H Q - W\>Q/gp ; DT= D/gp ; and E'-=E/gp We now consider the c a l c u l a t i o n of the li n e shape from the t r a n s i t i o n (5-32a). The procedure used follows cl o s e l y that given by Bloembergen and Rowlands (84). We may rewrite (5-32a) i n the form 66 J v ^ D 1 Co3 39- (DT + 3E')Sin 28 Cos 2 V- (D*-3E f)Sin 28 S i n 2 V (5-33) 2 2 The parameter E 1 may always he chosen of opposite sign to D' hy a suitable choice of molecular axes. The assumption we make that D' > 0 >Ef does not therefore result i n loss of generality, The corresponding l i n e (5-32h) with E' > 0 >D f i s the mirror image, about h f * 0, of the one discussed above. From (5-33) i t i s found that )\u00E2\u0080\u00A2= (J^-D'Cos^e + (D1 2 (D'-t- 3E' )Sin 28 )\"*\" (5-34) 2 (^ vV A-3), 326 t-3E')Sin 2e)\"^(-J >,^ D'Cos8^-In order to obtain an expression for the li n e shape an integration over 0 i s necessary. Only ce r t a i n values of B are allowed however i n certain f i e l d regions because of the following r e s t r i c t i o n s i ) 0 < Cos 2* 1 . i i ) I t i s necessary that | 1 be r e a l . This further r e s t r i c t s the value Cos2\u00C2\u00A9 can have. Taking these points into consideration i t i s found that: i ) In the f i e l d region -\u00C2\u00A3(Df-*- 3E')< J\u00C2\u00BB( < DT we have 2X+ D!-3S' >Cos28 > 2Jt,-H),+ 3B' S ( ] > \u00C2\u00BB - K \u00C2\u00BB ) 3(D\u00C2\u00BB+ E') i i ) In the f i e l d region -^(D f-3E 1 )< -fc, < -i(D'-\u00E2\u0080\u00A2-3E\u00C2\u00BB) we have 2Jt,-vD,-3El > Cos28> 0 2(D'-E') 67 The actual l i n e shape i s given hy the expression i s given by (5-34). This i n t e g r a l i n an e l l i p t i c i n t e g r a l (99) and by a suitable substitution can be brought into the form of a complete e l l i p t i c i n t e g r a l . This has been discussed by Franklin (99) and the re s u l t s are given by Bloembergen and Rowlands (84). For the case we have just discussed where Df> -i-(DT+3E' )> -ftD'-SE') the line shape i s given by a) I L W \u00C2\u00AB cU (5-35) (l -i n the f i e l d region -fc(Df + 3E\u00C2\u00BB H-^D 1 where (5-36) o ) I U , ) \u00C2\u00AB 0 everywhere else. The i n t e g r a l appearing i n (a) and (b) above i s a oomplete e l l i p t i c i n t e g r a l . Its value for varying K i s given i n mathe-matical tables. This i n t e g r a l has a logarithmic s i n g u l a r i t y at 68 K = l; from the above expressions for K t h i s i s found to ocour at the f i e l d value 4^ = + 3B 1). The l i n e shape calculated from the t r a n s i t i o n (5-32b) i s the mirror image, about A^\u00E2\u0080\u00940, of the one that has been calculated. Figure 14b shows the calculated l i n e shape when the values, corresponding to naphthalene, of D = 0.1008 cm\"1 and E = -0.0138 cm\"1 are chosen. Experimentally, the f i r s t derivative of the e.s.r. absorption l i n e i s recorded. One expects there-fore to see sharp lines corresponding to the s i n g u l a r i t i e s i n the- t h e o r e t i c a l spectrum. One should thus observe three pairs of l i n e s with separation of 2D1, DT-*-3E? and D'-3E' centred about Jl=0. The calculated r e s u l t s are compared with the experimental e.s.r. spectrum (Figure 14a) which has been recorded by de Groot and van der Waals (89). It i s clear that the tkeerejfc.4aai.lspectr.um i s i n good agreement with the experi-mental r e s u l t s . The calculated line shape provides a good explanation of the experimentally observed l i n e shape. In p a r t i c u l a r , i t now becomes clear why the inner two lines are e s s e n t i a l l y symmetric i n molecules with less than a x i a l sym-metry while very assymmetric line s are obtained i n molecules, such as triphenylene which have t r i g o n a l symmetry. It i s noticeable that while the shape of the e.s.r. spectrum i s i n good agreement with the calculated spectrum based on a f i r s t order approximation, there i s some discrepancy i n the observed and calculated li n e positions. We have at-tempted to account for t h i s by the addition of the second \ Fig./*. Experimental and Calculated E.S.R. Spectra of Trip le t N a p h t h a l e h e . 69 order energy corrections, (5-29a) and (5-29b). These correc-tions add a term to a l l the t r a n s i t i o n energies and hence s h i f t a l l the resonanoe f i e l d s to lower f i e l d . A comparison of the experimental line positions with l i n e positions c a l -culated using f i r s t order and second order perturbation theory i s given i n Table 1. The calculated results for phenanthrene are also included and compared with the experimental r e s u l t s of de Groot and van der Waals (89). The agreement between the observed and calculated line positions i s seen to be somewhat improved by addition of the second order correction. 70 Table 1: Comparison of the Observed, and Calculated A m = 1 Line Positions i n Naphthalene and Phenanthrene. a) Am =1 Transitions i n Naphthalene l a t Order D =0.1008 onT 1 E -0.0138 om\"1 Line Positions 2nd order 6 = ^ = 0.3089 cnT 1 4373 gauss 4053 3620 2980 2547 2227 4368 gauss 4011 3564 2935 2473 2214 Experimental* 4380 gauss 4064 3540 2950 2480 2228 *M.S. de Groot and J.H. van der Waals, Mol.Phys.,6,545(1963) b) Am-=1 Transitions i n Phenanthrene D\u00C2\u00BB0.1044 om\"1 6*Kv 0 \u00C2\u00AB 0.3089 em\"1 E = 0.0196 om\"1 Line Positions 1st Order 2nd- Order 4597 gauss 4417 3481 3121 2185 2005 4555 gauss 4409 3420 3058 2169 1926 Experimental\" 4609 gauss 4398 3400 3012 2133 1975 *M.S. de Groot and J.H. van der Waals, Physical,29,1128(1963) 71 f) E.S.R. Line Shapes in a Polycrystalline Sample of Triplet State Molecules with Axial Symmetry; D large compared, with the Zeeman splitting In the previous two sections we have discussed line shape calculations for th\u00C2\u00AE = l transitions\" in a polycrystalline sample. We now wish to consider the situation where the spin-spin interaction is large compared to the Zeeman interaction. We have seen previously that the magnetic properties may he adequately described by the spin Hamiltonian (5-9). If we take as basis states, eigenstates of S z quantised with respect to the molecular axis, then (5-9) may be expressed in the matrix form (1-11). For H parallel to one of the molecular axes the eigenvalues are given by expressions (.1-13). An expression for the eigenvalues for other orientations is not easily found. The position and number of resonances possible for a given orientation is best seen by using a graphical method similar to that used by Kottis and Lefebvre (34). Le Groot and van der Waals (33) have shown that resonance w i l l occur when the condition (5-37) Gos20 - 3Pg-*-9L(gi3H)2* { 3 62-P2-3 (gflH) Z\ 7-302+4L2*13(gpHT2~' 2 7 I K g j B H ) a is satisfied. Figure 15 shows a plot of Cos 20 vs g^H with D=1.00 cm\"1 and 6 = 0.3030cm\"1. The curve shows that: i) Only one transition is possible for a l l orientations. This corresponds to the transition between the levels which are labelled |1^ and | - l ) in zero f i e l d . Cosae(H) = aD3^D(3|3H)l=t{3 and 1-1^ i n zero f i e l d . Now consider which of the above resonances are l i k e l y to be experimentally detectable. For D=1.00 cur 1- and 6 = 0.303om\"\"l i t i s found (Figure 15) that only about five percent of the molecules w i l l be able to contribute to the second resonance discussed. Because of the small percentage of molecules and the large spread i n f i e l d , i t seems u n l i k e l y that t h i s reson-ance w i l l be detectable. At any rate, i t w i l l have considerably less i n t e n s i t y than the resonance from the f i r s t case discussed. Q u a l i t a t i v e l y one can see that t h i s resonance i s expe.cted to be of the form shown i n (Figure 16). I f any signal i s detected i t should appear as an \"absorption\" i n the derivative curve corresponding to the low f i e l d l i m i t . On the other hand, i t i s much more probable that an absorption should be detectable experimentally in, the f i r s t s i t u a t i o n . In t h i s case i t can be shown that a s i n g u l a r i t y occurs i n the composite spectrum at the high f i e l d l i m i t . An exact expression for t h i s l i n e i s not e a s i l y obtained. An approximate expression may be obtained i n the following way however. For D>> g^H, we take as zeroth order states, the states | l > , 1 0 > and \ - l ^ quantised with respect to the molecular z-axis. We are interested i n the t r a n s i t i o n between the levels , . l a b e l l e d 11> and l-l)> i n zero f i e l d and have energy D/3. I f we substitute t h i s approximate eigenvalue into the diagonal element (-2/3 D - E) of the secular determinant we obtain E (e=o\u00C2\u00B0) cm\"' -02. A -OH --Ob -0-8 1 7000 8000 DM.OOcm1 6-0-30cm q o o o 10,000 H(gauss) l i ,000 I30OO i*,ooo 1(H) 1000 &000 qooo which, from (5-41) i s seen to be equivalent to 8 \"=90\u00C2\u00B0. The approximate expression above w i l l only predict the f i e l d p o s i t i o n of the s i n g u l a r i t y accurately provided D \u00C2\u00BB o . In applications we make lat e r i n t h i s section t h i s condition 75 does not hold rigorously. In any determination of D, we make - D D 1\" T i l i s correction w i l l not affect the general shape of the l i n e hut i s s i g n i f i c a n t i n predicting the p o s i t i o n of the s i n g u l a r i t y . As has been mentioned previously, Wasserman et a l (8,49) have recently made e.s.r. measurements on a series of sub-s t i t u t e d imines.which have t r i p l e t ground states. The charac-t e r i s t i c feature of these i s that they a l l consist of a single, high f i e l d l i n e which sometimes may be observed to be s p l i t into two components* The i n i t i a l assignment (8) of these authors was incorrect. However i n a l a t e r publication (49) they attribute t h i s l i n e to either an x y - t r a n s i t i o n ( 0= 90\u00C2\u00B0) or a z - t r a n s i t i o n ( \u00C2\u00A9-=0\u00C2\u00B0). From the s p l i t t i n g observed i n certain cases they conclude that the x y - t r a n s i t i o n i s probably the most reasonable assignment. I t was f e l t that a more d e f i n i t i v e conclusion could probably be reached by a l i n e shape study of these species. Because of the high f i e l d p o s i t i o n , i t seems certain that the spin-spin i n t e r a c t i o n parameter D must be large. Further since, i n general, only the single resonance i s observed, the E term must be very small. I t seems reasonable therefore to attempt an explanation of the r e s u l t s using the l i n e shape discussed above. Although a large number of these imines have been detected by Wasserman et a l , none of t h e i r spectra have been published. For t h i s reason the spectra of the imines formed by the photol-y s i s of phenylazide and benzenesulphonylazide were repeated. 76 The spectrum of phenylimine i s shown i n Figure 17a. The fi e l d , p osition of the zero slope point i s found, to he 6715 gauss whioh agrees w e l l with the value of 6701 gauss given by Wasserman et a l (49). Figure 17b shows the calculated, spectrum based, on the expression (5-42). I t i s clear that the experimental r e s u l t s are i n agreement with such a model. According to t h i s model, the s i n g u l a r i t y occurs at the f i e l d p o s i tion where (gyjH)2= o D1. It should be pointed out that the exact expres-sion i s given by (gpK) - 6 (D+&). The approximation i s only good when Q i s small compared with D. In our case t h i s cor-re c t i o n i s s i g n i f i c a n t and must be made. This correction should not however change the features of the calculated curve appreciably. For phenylimine one obtains a value of D'-D+i =* 1.30 em-1 or D=1.00 cm\"1. I t i s i n t e r e s t i n g to compare this with the value of D =1.86 cm\"1 obtained from the elec-tronic spectrum of NH (6). I f i t i s assumed that one of the unpaired electrons i s l o c a l i s e d i n a nitrogen 2p o r b i t a l and the other i s i n a IT -molecular o r b i t a l , i t can be shown that the parameter D should be approximately l i n e a r l y proportional to the \"TT-electron spin density on the nitrogen. We demonstrate : t h i s with p a r t i c u l a r reference to phenylimine. Similar arguments can of course be used for the other substituted imines. We l a b e l the centres i n the molecule as shown below. I D-l-OOcm-' I 1,77b 0-0-3 03cm1 II (b) I WIS \u00C2\u00ABH> The E-j^ term vanishes from symmetry. In this expression i s the nitrogen Sp o r b i t a l . Evaluation of the above expression y (101,10a) using Slater o r b i t a l s y i e l d s , 1 1 T ^ j l S - S C \" a g-f In t h i s expression a Q= Bohr radius and Z = the ef f e c t i v e nuclear charge of the Slater o r b i t a l s . For an estimate of the contribution from the two centre terms a point charge approximation i s adequate. One here assumes the electrons to be l o c a l i s e d on the atomic centres and the values of D and E are obtained by replacing the expec-ta t i o n value, /3ZJ^|\u00C2\u00A3r_\u00C2\u00A32\ and/ x f . 2 - Y ? g \ 1 by the sharp *IS 79 values 3ZT\u00E2\u0080\u009E - r f p and & 1 2 \" Y I 2 g i v e n \u00C2\u00B0y i * * 1 6 nuclear R 1 2 5 positions. Making t h i s approximation one obtains the following From simple Huckel theory, one i d e n t i f i e s the c o e f f i c i e n t s 2 +h Cr as being equal to the \"IT -electron spin density on the r * n centre. Further, to f i r s t order i n the change i n Coulomb i n t e g r a l , the spin densities i n the imines are expected (103) to be the same as the corresponding a l k y l r a d i c a l analogue. One may treat phenylimine therefore as having the same IT -electron spin density as the benzyl r a d i c a l . This r a d i c a l i s odd alternant, as are most of the carbon anologues of the imines that were investigated by Wasserman et a l ( 4 9 ) . Huckel theory predicts (104) that there w i l l be zero spin density on the \"unstarred\" centres i n such molecules. For phenylimine these occur at the 2 and 4 positions and hence there w i l l be no contribution to D and E from the terms D^g, F^g, D-^ and ET_4. This i s important as i t i s seen that the magnitude of 1; 3Z|g - rf_g i s s t i l l large for the nearest -neighbour,two \u00C2\u00AB\u00C2\u00BB, centre t.\u00C2\u00BB. One thus sees that, to the approximation used, the measured value of D should be very nearly proportional to the TT -electron spin density on the nitrogen. The spin density measured i n t h i s values -1 80 way might he expected to he a few percent low aa a correction should he made for the two centre contributions. Observe that the predicted value for S (0.00\"6 om e l for phenylimine) i s small, i n agreement with the experimental r e s u l t s . We take the experimentally observed value (6) of D=l*86cm~ for NH aa correaponding to \u00E2\u0080\u00A2=\u00E2\u0080\u00A2 1. The spin density on the nitrogen of the substituted imines i s taken as, (5-51) Using t h i s r e l a t i o n one obtains an experimental spin density of 0.54 for phenylimine. This may be compared with the value of 0.571 obtained (49) from Huckel theory. The e.s.r. spectrum of benzenesulphonylimine was similar i n form to that of phenylimine. The f i e l d value corresponding to the centre of the spectrum i s 7791 gauss at a microwave frequency of 9093 Mc/seo. This gives a value of D=.1.45 cm\"1 and jjg = 0.78. Using expression (5-51) we have also calculated values of D and j ~ from the r e s u l t s for the imines reported by Wasserman et a l . The r e s u l t s , shown i n Table 2, are compared with nitrogen spin densities calculated using (49) Huckel theory. 81 Ta\"ble 2: Comparison of the Experimental Values of D and fa to the Calculated Value of ft . Imine D(exp) fa(exp) j^(Hucke NH 1.86 1.0 Benzene sulfonyl 1.45 0.78 3-nitrophenyl 1.05 0.56 0.571 3-methoxyphenyl 1.00 0.54 0.571 Phenyl 1.00 0.54 0.571 4-nitrophenyl 0.98 0.53 0.540 4-methoxyphenyl 0.96 0.52 0.513 4-biphenylyl 0.92 0.49 0.516 2-naphthyl 0.89 0.48 0.529 1-naphthyl 0.78 0.42 0.450 2-anthryl 0.76 0.41 0.471 1-anthryl 0.65 0.35 0.381 82 APPENDIX 1 An AlternatiyeDerivation of the Spin Hamiltonian This derivation follows c l o s e l y a derivation shown to the author hy Dr. J.A.R. Coope. If the Hamiltonian (1-5) i s expanded i n terms of i t s components and a d i f f e r e n t c o l l e c t i o n of terms made i t pos-si b l e to write (1-5) i n the form ( A l - l ) fix I where S\u00C2\u00B1 \u00E2\u0080\u00A2= Sx * i S y ZH may thus he written i n the form ZH - - f f j j r ^ S ^ M A ^ ( O ^ ) (Al-2) l*\*-X where s(2)(i > 2) and are the spherical components of the second order tensor pperators S_f 2^(l,2) and These operators are defined hy (2^ i) SV ; (1,2)* i ( S i S 2 + S 2 S i ) - l / 3 ( S i . S 2 ) 1, with components , s S < 2 ) ( i , 2 ) - l/{5(3S z(l)S z (2)-Si .S2) S ( 2 ] 1 , i ( S i (1)S 2(2)+- S* (2)S Z(1)) (Al-3) S ( 2 ) * 2 = i(s\u00C2\u00B1 ( 1 ) S \u00C2\u00B1 (2)) fa) \ and i i ) A (e,^J\u00C2\u00BB(3XX-l)/^ where A ^ \u00C2\u00A312 r i 2 with components n. 83 The algebra of such tensor operators was f i r s t derived by Racah (105,106) and i s also given i n texts (98,107). The usefulness of w r i t i n g (Al-1) i n t h i s tensor form l i e s i n the fact that, by u t i l i s i n g the Wigner-Eokart theorem (107), one can almost immediately derive the spin Hamiltonian of the system. The Wigner-Eckart theorem implies that the matrix elements of two n**1 order spherical tensors are proportional to each other. In our case we are interested i n obtaining matrix elements of the form (\u00C2\u00ABSm\u00C2\u00AB|3ll\u00C2\u00ABSK> ' - 3> z(^m slEt-rS (^.,M^l&,^U^<> U l - 5 ) where <* labels the space variables. As S ( 2 t l , 2 ) and are defined i n di f f e r e n t spaces the matrix elements may be s p l i t into the product of two matrix elements ( ^ s U ^ A ^ l e , ^ (Al-6) A separation of this form i s not possible for a system of n-p a r t i c l e s . It can be made however for the two electron case we discuss here. The Wigner-Eckart theorem states that we may write the (2) matrix element involving S m ( l , 2 ) i n terms of the matrix element of the t o t a l spin tensor operator S m , where the components S ^ are of the same form as (Al - 3 ) , (Sm s|s (2)(l, 2) | Sm|) - K<^Sms|s(g) | Sm^ (Al-7) Since (Al-7) must hold for a l l ma,ms and m, K may be evaluated 84 from the special case that (SS [ s < 2 ) (1,2) | s s ) = K ( s s | s ( 2 ) | s s ^ o r / . to\ i \ (Al-8) K \u00E2\u0080\u00A2=. (SS |S^ n(l \u00C2\u00BB 2)lSS )) ( s s | s ( \u00C2\u00A7 ^ s s ) We are p a r t i c u l a r l y interested i n the case with Si=S 2=|- and S = l . Remembering that the coupled state 111) may be written as | l l ) ' J-ei) I i i ) \u00C2\u00BB K may be evaluated as follows. K , ( H U & | g s z C i ) s z ( 2 ) - g i . S 2 1 H)l hi) 3S\u00C2\u00A7 - S 2 | l l ) 2 Substituting into (Al - 7 ) we f i n d that i X 4.Sl\u00C2\u00AB,l3lWSM{> =-3>'(SWl4H-,r < \" I A -\" ' 6 ' * ) I \" > SS |smi> ( A l - 9 ) The spin Hamiltonian i s thus found to be ^ s p i n v - g 2 ? 2 ! ! ^ 1 ^ <*1A^Q>*)H) S ( 2 ) (Al-10) m X In the coordinate system which diagonalises (\u00C2\u00AB<| A-m^ej<^)W) (which i s the molecular coordinate system for molecules with s u f f i c i e n t l y high symmetry) it i s found that \"^spin takes the form Xspin'DJS 2 - 1/3 S 2 ) + E ( S 2 - S 2) ( A l - l l ) where D - f g 2 ? 2 / r g 2 -^3zf 2 j^and E = |-g2jS2 / y j 2 - \ ^ r i 2 \ r i 2 ' whioh are the re s u l t s we obtained previously. See also McLachlan (108) for a similar but somewhat more complicated derivation. 85 APPENDIX 2 Bibliography of e.s.r. T r i p l e t State Results a) Single Crystal Measurements: 1. C.A.Hutchison and B.W.Marvgum,^.Chem.Phys,29,952(1958) 2. C.A.Hutchison and B.W.Majrvgum, J . Chem.Phys,32,1261(1960) 3. C.A.Hutchison and B.W.Mangum,J.Chem.Phys,34,908(1961) 4. S.Foner,H.Meyer and W.H.Kleiner,J.Phys.Chem.Solids,18, 273(1961) 5. A.Schmillen and G-.von Foerster,Z.Naturf., A16,320(1961) 6. R.W.Brandon,R.E.Gerkin and C.A.Hutchison,J.Chem.Phys., 37,447(1962) 7. R.W.Brandon,G.L.Closs and C.A.Hutchison,J.Chem.Phys., 37,1878(1962) 8. A.W.Hornig and J.S.Hyde,Mol.Phys.,6,33(1963) 9. J.Vincent and A.H.Maki.J.Chem.PhysT,39,3088(1963) 10. C.A.Hutchison,Record of Chemical Progress,24,105(1963) 11. J.Vincent and A.H.Maki,Bull.Am.Phys.Soc.,8,620(1963) b) Am =2 Transitions i n Photoexcited Aromatics: 1. J.H.van der Waals and M.S.de Groot,Mol.Phys.,2,333(1959) 2. M.S.de Groot and J.H.van der V/aals,Mol.Phys. ,F,190(1960) 3. B.Smaller,J.Chem.Phys.,37,1578(1962) 4. L.H.Piette,J.H.Sharp,T.Kuwana and J.N.Pitts,J.Chem.Phys., 36,3049(1962) 5. G.von Foerster.Z.Naturf.,A18,620(1963) 6. P.Kottis and R.Lefebvre,JTcEem.Phys.,39,393(1963) 7. M.Ptak and P.Douzou,Compt.Rend. ,257,43*811963) o) Am-1 Transitions i n Photoexcited Aromatics: 1. W.A.Yager,E.Wasserman and R.M.R.Cramer,J.Chem.Phys.,37, 1148,(1962T\" 2. M.S.de Groot and J.H.van der Waals,Mol.Phys.,6,545(1963) 3. M.S.de Groot and J.H.van der Waals,Physica,29,1128(1963) 4. J.B.Farmer,C.L.Gardner and C.A.McDowell,Mol.Phys., d) Ground State T r i p l e t States 1. R.W.Murray,A.M.Trozzolo and E.Wasserman,J.Am.Chem.Soc., 84,3213(1962) 2. A.M.Trozzolo.E.Wasserman and R.W.Murray,J.Am.Chem.Soc., 84,4990 (19 6 2) 3. G.Smolinsky.E.Wasserman and W.A.Yager,J.Am.Chem.Soc., 84,3220(1962) 4. G.Smolinsky,L.C.Snyder and E.Wasserman,Rev.Mod.Phys., 35,576(1963) 86 5. R.E.Jease.P.Bloen,J.D.W.van Voorst and G. J.Hofrtink.Mol. Phys.,6,633(1963) 6. R.Braslow,H.W.Chang and W.A.Yager,J.Am.Chem.Soo.,85,2033(1963) e) Thermally Accessible T r i p l e t States: 1. D.B.Chesnut and W.D.Philips.J.Chem.Phys.,35,1002(1961) 2. D.B.Chesnut and P.Arthur,J.Chem.Phys.,36,2969(1962) 3. N.Hirota and S.I.Weissman.Mol.Phys.,5,537(1962) 4. H.M.McConnell.D.Pooley and A.Bradbury,Proc.Nat.Acad.Scie(U.S.) 48,1480(1962) 5. M.T.Jones and D.B.Chesnut, J.Chem.Phys. ,38,lIB\"l, (1963) 6. D.D.Thomas,H.Keller and H.M.McConnell,J.Chem.Phys.,39,2321(1963) 7. E.A.Chandross and R.Kreilick,J.Am.Chem.Soo.,86,117(1964) 8. E.A.Chandross,J.Am.Chem.Soc.,86,1263(1964) f) Energy Transfer Between T r i p l e t States: 1. J.B.Farmer,C.L.Gardner and C.A.McDowell,J.Chem.Phys.,34,1058 T1961) 2. R.W.Brandon,R.E.Gerkin and C.A.Hutchison,J.Chem.Phys., 37,447(1962) 87 APPENDIX 3 Derivation of the Spin Hamiltonian for a Rotating T r i p l e t State Molecule 3 r -For a molecule i n a y_ state the Hamiltonian may he written as K =Ho (3Cos 2e -1) 4 1 = T Cos 8 Sin\u00C2\u00A9 e (A3-4) A l - | S i n 2 G e \u00C2\u00B1 2 ^ The scalar product i s given by (A3-5) I f the molecule conforms to a Funds case (b) coupling scheme, both K, the r o t a t i o n a l quantum number, and S are good quantum numbers. The Hamiltonian TM., may be replaoed, i n the 88 following way, \"by an e f f e c t i v e spin Hamiltonian, which operates i n the manifold with K and S fix e d . We chose as basis states, the states |KSmKms^ =- i K m j r ^ Sms^ (A3-6) Any other state can, of course, be written as a lin e a r com-bination of t h i s basis set and any matrix element of \"31, , with K and S fixed, as a lin e a r combination of matrix elements of the form (A3-7) We may separate (A3-7) into a product of matrix elements as follows. (A3-8) I* The Wigner-Sofcart theorem (107) states however that the matrix element involving the second order tensor A ^ i s proportional to the matrix elements of the tensor operator K^ 2^KK - l / 3 2 l , that i s Substituting (A3-9) into (A3-8) we f i n d that the e f f e c t i v e spin Hamiltonian may be written as Ztt spin = < ) C < U ( , > U < > D l t - r S ^ K ^ (A3-10) The prop o r t i o n a l i t y c o e f f i c i e n t (KK) A(3)I KK )involves an integration over three spherical harmonics. This i s e a s i l y evaluated by wri t i n g the r o t a t i o n a l wave function |KK^ as the sectoral harmonic Y ( \u00C2\u00A7 k i n K 6 e i K ^ 89 On evaluation one finds ( K K ( A ( 2 ) 1 K K > -2 (A3-11) ) [ = (2K-l)(2K* 3) HOC [ K L \u00C2\u00A3 ' \ KK.y The i d e n t i t y 2Jr\. \u00E2\u0080\u00A2 v - = ' T a \"3\" may he proven (109). From t h i s i d e n t i t y and by substituting (A3-11) into (A3-10) i t i s found that the spin Hamiltonian may be written as 2D J (S.K) 2 -V-i(S.K) - g 2 K 2 l C-l) 12K+3) 1 ~ 3 1 ^Kspin _ -2T2K> (A3-12) 2 2 2 By substituting S.K*|-(J -S -K ), the following energy expres-sions are obtained for the three J values (J-=K-tl, K and K-l) corresponding to S-1 and a given K. E,<-+1 =D/3 - (K+l)D ; % * D / 3 and E K-l\u00C2\u00ABD/3 - KD *- (2K+3) {W-D which are Kramers r e s u l t s (50). The author i s indebted to Dr. J.A.R. Coope for showing him t h i s derivation. 90 BIBLIOGRAPHY 1. Raid,C.:Quarterly Reviews,12,205(1958) 2. Porter,G.:Proc.Chem.Soe.,291,0ct.l959 3. Jablonski.A.:Z.Physik.,94,38(1935) 4. Lewis,G.N. and Kasha,M.: J.Am.Chem.Soc ,66,2100(1944) 5. Van Vleck,J.H.:Phys.Rev.,31,587(1928) 6.. Dixon,R\N. :Can. J.Phys. ,37,1171(1959) 7. Hojjtink,G. J. :Mol.Phys.,2,85(1959) 8. Smolinsky,G.,Wasserman,E. and Yager,W.A.:J.Am.Chem.Soc. 84,3220(1962) 9. 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