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OPTICAL AND ELECTRON SPIN RESONANCE SPECTROSCOPY OF MATRIXISOLATED SILICON AND MANGANESE SPECIES By ROBERT FRANCIS FERRANTE A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1977 To my parents Digitized by the Internet Archive in 2010 with funding from University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation http://www.archive.org/details/opticalelectrons00ferr ACKNOWLEDGEMENTS The author extends his deep appreciation to Professor William Weltner, Jr. whose encouragement, professional guidance, patience, and support made this work possible. Thanks are also due to all the members of Professor Weltner's research group, particularly Dr. W. R. M. Graham and Dr. R. R. Lembke, for their collaboration, assistance, and advice during the research. The author greatly appreciates the expert craftman ship displayed in fabrication of experimental apparatus by A. P. Grant, C. D. Eastman, and D. J. Burch of the Machine Shop and R. Strasburger and R. Strohschein of the Glass Shop, as well as the maintenance of electronic equipment by R. J. Dugan, J. W. Miller, and W. Y. Axson. The author would also like to acknowledge the support of the Air Force Office of Scientific Research (AFOSR) and the National Science Foundation (NSF) during this work. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES viii ABSTRACT xi CHAPTERS I INTRODUCTION 1 The MatrixIsolation Technique 1 References Chapter I 9 II EXPERIMENTAL 11 Introduction 11 Experimental 11 Apparatus 11 General Technique 26 References Chapter II 29 III ESR THEORY 30 Introduction 30 Atoms and the Resonance Condition 30 The Hyperfine Splitting Effect 33 2E Molecules 43 The Spin Hamiltonian 43 The g Tensor 47 The A Tensor 52 Randomly Oriented Molecules 53 Molecular Parameters and the Observed Spectrum 61 3E Molecules 7 The Spin Hamiltonian 76 4Z Molecules 94 The Spin Hamiltonian 95 E Molecules 105 The Spin Hamiltonian 106 References Chapter III 113 IV SILICON SPECIES 117 Introduction 117 Experimental 118 ESR Spectra 119 SiN2 119 SiCO 125 Si2 133 Optical Spectra 133 Si and Si2 133 SiN2 137 SiCO 144 Si(CO)2 150 Discussion 152 References Chapter IV 171 V MANGANESE SPECIES 175 Introduction 175 Experimental 177 ESR Spectra 179 Mn Atoms 179 Mn+ 179 MnO 184 MnO2 190 MnO3 196 Mn04 205 Discussion 207 Mn Atoms and Mn 207 MnO 212 MnO2 215 MnO3 217 MnO4 220 References Chapter V 232 BIOGRAPHICAL SKETCH 237 CHAPTERS Page LIST OF TABLES TABLE PAGE I ESR data of SiN2 and SiCO in their 3 ground states invarious matrices at 40K 131 II Si2 absorption bands in argon matrices at 40K 135 14 III Ultraviolet absorption spectrum of Si N2 in an argon matrix at 40K 140 IV Vibrational frequencies and calculated force constants (mdyn/A) for SiNN and SiCO molecules in their ground 3E states 143 12 V Absorption spectrum of Si CO in an argon matrix at 4K 147 VI Comparisgn of stretching force constants (mdyn/A) for relevant molecules XYZ 154 VII Total density matrix elements for SiCO, SiN2, and the free ligands CO and N2 157 VIII Spin densities in SiCO and SiNN 164 IX Comparison of vibrational frequencies and electronic transitions of CXY and SiXY molecules 167 X Field positions (in gauss) of observed fine and hyperfine structure lines of Mn+:Ar at 40K. A = 275 G; v = 9390 MHz 185 XI Magnetic parameters, observed and calculated line positions for the I+1/2)+I/2) perpendicular transition of MnO ( E) in Ar 191 XII Magnetic parameters, observed and calculated line positions for the I+1/2) >I1/2) perpendicular transition of MnO2 (4E) in Ar 195 XIII Spin Hamiltonian matrix for the states M, m) for Mn03 (2A) including interaction with the 55Mn (I = 5/2) nucleus 199 TABLE PAGE 2 XIV Magnetic parameters of MnO (2A ) in Ne; observed transitions in Ne an Ar 203 XV Magnetic parameters, observed and calculated line positions for MnO4 (2T1) in Ne 208 XVI Summary of magnetic parameters and derived quantities for manganese and some manganese oxides 210 vii LIST OF FIGURES FIGURE PAGE 1 Basic design features of the liquid helium dewar used for ESR studies 13 2 Variabletemperature modification of liquid helium dewar used for ESR studies 14 3 Basic design features of variabletemper ature liquid helium dewar used for optical studies 17 4 Basic design features of cryotip assembly used for optical studies 20 5 Zeeman energy levels of an electron interacting with a spin 1/2 nucleus 36 7 6 Zeeman energy levels of a SS ion with I = 5/2 37 7 Absorption and first derivative lineshapes of randomly oriented molecules with axial symmetry 58 8 Energies of the triplet state in a magnetic field for a molecule with axial symmetry; field parallel to molecular axis 82 9 Energies of the triplet state in a magnetic field for a molecule with axial symmetry; field perpendicular to molecular axis 84 3 10 Resonant fields of a E molecule as a func tion of the zero feild splitting 87 11 Theoretical absorption and first derivative curves for a randomly oriented triplet state molecule with axial symmetry 90 12 Theoretical absorption and first derivative curves for a randomly oriented triplet state molecule with orthorombic symmetry 92 4 13 Energy levels for a Z molecule in a magne tic field; field parallel to molecular axis 102 viii 14 Energy levels for a 4 molecule in a mag netic field; field perpendicular to molecular axis 103 15 Resonant fields of a 4E molecule as a function of the zero field splitting 104 16 Energy levels for a 6E molecule in a magnetic field for 0 = 00, 300, 600, and 90 109 17 Resonant fields of a 6 molecule as a function of the zero field splitting 111 18 ESR spectra of SiN2 molecules in argon matrices at 40K 120 19 ESR spectra of SiN2 molecules in various matrices at 40K 124 20 ESR spectra of SiCO molecules in argon matrices at 40K 126 21 Effect of temperature upon the ESR spectrum of Sil3CO in an argon matrix 128 22 ESR spectra of SiCO molecules in Ar and CO matrices 130 23 Ultraviolet absorption spectrum of SiN2 molecules in an agron matrix at 40K 138 24 Infrared bands of SiN in argon and nitrogen matrices a? 4K 142 25 Absorption spectrum of SiCO molecules in an argon matrix at 40K 145 26 Infrared spectra at 40K of an argon matrix containing vaporized silicon atoms and 13CO/Ar = 12CO/Ar = 1/375 148 27 Infrared spectra at 40K of an argon matrix containing vaporized silicon atoms and Cl60/Ar = C180/Ar = 1/375 149 28 ESR spectrum of Mn in argon at 40K 180 29 Zeeman levels and observed transitions for Mn+ in argon 185 FIGURE PAGE FIGURE PAGE 30 ESR spectrum of MnO in argon at 40K 187 31 ESR spectrum of MnO2 in argon at 4K 193 32 ESR spectrum of MnO3 in neon at 40K 197 33 ESR spectrum of MnO3 in argon at 40K 202 34 M = 3/2 component of MnO in argon at various temperatures, and in neon 204 35 ESR spectrum of MnO4 in neon at 4K 206 36 Molecular orbital correlation diagram for MnO3 218 37 Molecular orbital correlation diagram for MnO4 223 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTICAL AND ELECTRON SPIN RESONANCE SPECTROSCOPY OF MATRIXISOLATED SILICON AND MANGANESE SPECIES By Robert Francis Ferrante December, 1977 Chairman: William Weltner, Jr. Major Department: Chemistry The 3 molecules carbonyl silene, SiCO, and diazasilene, SiNN have been prepared by vaporization and reaction of silicon atoms with N2 or CO and trapped in various matrices at 40K. The electron spin resonance (ESR) spectra indicate that some or all sites in some matrices induced slight bending in the molecules, and that the species undergo torsional motion in the solids. Isotopic substitution of 13C, 15N, 180, and 29Si was employed to obtain hyperfine coupling data in the ESR and shifts in the optical spectra. 1 In solid neon, assuming gl =gi=ge, D = 2.28 and 2.33 cm1 for SiN2 and SiCO, respectively. Hyperfine splitting in 14 15 29 argon yield A = 17 ( N), 21 (15N) and 95 (29Si) MHz for SiN2 and 84 (29Si) and <14 (13C) MHz for SiCO. These confirm calculated results, in the complete neglect of differential overlap (CNDO) approximation, that the electron spins in both molecules are largely in the pn orbitals of Si. Optical transitions (with vibrational progressions) were 0 1 observed beginning at 3680A (SiN stretch, 470 cm ) and 0 1 o 3108A (NN stretch, 1670 cm ) for SiN and 4156A (SiC stretch, 750 cm1; C0 stretch, 1857 cm1) for SiCO in Ar. Infrared (IR) spectra in Ar indicate vNN = 1733 and NN 1 n 1 V = 485 cm for SiN and v = 1899 cm for SiCO. SiN 2 C0 Calculated stretching force constants are k = 2.0, SiN kN = 11.8, kCO = 15.6, and ksiC = 5.3 mdyn/A, the 1 latter assuming vSiC = 800 cm The CNDO calculations suggest r bonding of Si to the ligand, which is stronger in SiCO than SiN2, and some ligand + dSi backdonation, also stronger in SiCO. An attempt was made to correlate these vibrational and electronic data with those for CCO and CNN. Annealing an argon matrix containing SiCO to 350K led to the observation in the IR of 1 Si(CO)2, a n! 1 silicon counterpart of carbon suboxide, with v = 1899.6 cm1 A corresponding treatment of a SiN2 matrix did not produce N2SiN2, nor was N2SiCO observed when both ligands were present. The molecules MnO, MnO2, MnO3, and MnO4 have also been prepared, by the vaporization and reaction of manganese atoms with 02, N20, or 03, and isolated in various inert gas matrices at 40K. ESR has been used to determine magnetic parameters which are interpreted in terms of molecular geometry and electronic structure. MnO is confirmed to have a 2 22 6 + ground state with gl = 1.990(7), assuming gl = ge, and a zero field splitting in accord with the gas phase value IDI = 1.32 cm Hyperfine splitting due to the Mn (I = 5/2) nucleus are JAI I = 176(8) and IA1! = 440(11) MHz. MnO2 is S 4 2 a linear E molecule with probable configuration U6 , xii D = 1.13 cm1 (assuming g11 = g = 2.0023), IAI I = 353(11), IAi = 731(11) MHz. Mn03 exhibits very large hyperfine splitting IAII = 1772(3) and JAI = 1532(3) MHz indicative 2 of an sdz2 hybrid 2A ground state of D3h symmetry. The spectrum of MnO4 is consistent with a C 3 molecule 2 distorted from a T1 electronic state in tetrahedral symmetry by a static JahnTeller effect. The g and A tensors are slightly anisotropic: gll = 2.0108(8), gl = 2.0097(8); JAIII = 252(3), JAII = 196(3) MHz. The electron hole is almost entirely in an oxygen rbonded orbital with one oxygen atom displaced along its MnO bond axis. Warming to 350K did not induce thermal reorientation. xiii CHAPTER I INTRODUCTION The MatrixIsolation Technique Molecular spectroscopy, the primary tool for the investi gation of intimate details of molecular geometry and electronic structure, has been routinely applied to a large assortment of stable chemical species in the solid, liquid, and gaseous phases. The advent of highspeed electronic instrumentation has extended the range of spectroscopic techniques to allow the study of unstable or shortlived molecules and fragments. However, some molecules are still very difficult or even impossible to observe because of their short lifetimes, reactivity and/or method of preparation. Among these are molecules that exist only under high temperature conditions, such as stellar atmospheres or in arcs, and fragments whose reactivity precludes production of sufficient quantities for normal analyses. Even when such species are observed, anal ysis of their spectra is often greatly complicated by their production in a multitude of electronic, vibrational, and rotational states, as can occur in laboratory methods of generation of such radicals via arcs, flash photolysis, etc. Utilization of the matrixisolation technique can overcome many of these difficulties. The technique of matrixisolation spectroscopy was pro posed independently by Norman and Porter (1) and by Whittle, Dows,, and Pimentel (2) in 1954. Basically, the high temperature species, reactive molecules, or radical frag ments are prepared and trapped as isolated entities in inert, transparent solids, or matrices, at cryogenic temperatures. They do not undergo translational motion, but are immobilized, thus preventing further reaction and preserving the specimen for conventional spectroscopic analysis. The common technique of treating solid samples by diluting with KBr and forming a compressed disk of finely dispersed solid in a KBr matrix can be considered a crude form of matrixisolation. It has been demonstrated that the information gained by this tech nique is gaslike within a few percent. The matrix material can be any gas which will not react with the trapped species and which can be readily and rigidly solidified. Many different substances have been used for this purpose, including CH4, CO, N2, CS2, SF6, 02, as well as aliphatic and aromatic organic, However, the solid rare gases Ne, Ar, Kr, and Xe are usually employed because they are relatively inert chemically, transparent to radiation over a wide wavelength region, and offer a wide range of melting points and atomic sizes. The choice of matrix gas is determined to a large extent by the effect the solid matrix will have upon the trapped molecule. Neon, because it is the least polarizable, is expected to perturb the molecule least and generally makes the best matrix. Unfortunately, its trapping efficiency is not as great as the other rare gas solids, and it is difficult or impossible to achieve isolation of some species in solid Ne. Argon is better in this respect, and is generally used as the matrix medium. The heavier rare gases are found to perturb the trapped molecules to a greater extent than Ne or Ar, and are therefore, less desirable as matrix materials. Temperatures sufficiently low to condense the matrix gas can be attained with either physical refrigerants (nitro gen, hydrogen, or helium in the liquid state) or mechanical closedcycle refrigerators utilizing JouleThompson expansion of high pressure hydrogen or helium gas. Readily available and inexpensive, liquid N2 (boiling point 77.40K) is useful for some matrix materials, but is limited to relatively stable guest species. Liquid H2 (boiling point 20.4K) is often used, but it entails a fire hazard in addition to the normal dangers of handling cryogenic fluids. Thus liquid He is the most suitable of the physical refrigerants and the only one useful for condensation of solid Ne, which melts at 240K and permits solid state diffusion at about half that temperature. Temperatures of liquids H2 and He can be reduced to 150K and 1.20K, respectively, by pumping on the liquid. Closedcycle refrigerators which can attain temper atures near the boiling point of He are commercially avail able. Their advantages include convenience, elimination of the need to replenish cryogenic fluids while experiments are in progress, and low cost of operation after the initial investment. The substrate on which the matrix is condensed is usually chosen to be transparent in the spectroscopic region of interest. Some suitable materials are CsI, KBr, NaCI, and Au (for reflectance systems) in the infrared region, quartz, sapphire, and CaF2 for the visible and ultraviolet regions, LiF for the vacuum ultraviolet and sapphire or other non conducting material for microwave spectroscopy, including electron spin resonance. The polished crystal plates are mounted on a cold block which makes good thermal contact with the liquid refrigerant reservoir or the expansion cham ber of the JouleThompson refrigerator. In variabletempera ture dewars, the cold block is isolated from the reservoir, but cooled by introducing a controlled leak of refrigerant into the mounting block. Several methods can be employed for production of the guest species which are trapped on such substrates. One common procedure is vaporization of a nonvolatile material from a hightemperature Knudsen cell in a vacuum furnace. These cells can be constructed of carbon, or the refractory metals, Mo, Ta, or W. To prevent degradation of the cell material from contact with the hot vapor, the crucible can be lined with C, A1203, or BN. The cells are heated by resistance or induction methods, and temperature up to 29000K can be achieved in this manner. The vapor effusing through a small orifice is collimated into a crude molecular beam and deposited simultaneously with the matrix gas in such proportions that M/R (the ratio of the number of moles of matrix material to the number of moles of the trapped species) is 500:1 or greater. In many cases the compositions of vapors so obtained have been characterized mass spectrometri cally, greatly facilitating the analysis of the resulting spectra. The matrix gas can also be doped with a reactant gas by standard manometric techniques to produce other unstable reaction products. Another high temperature source is thermolysis of a gaseous compound by passing it through a hot W or Ir tube, the resulting products being cocondensed with the matrix gas as above. An additional common method for generating unstable species is to subject a volatile parent compound to highenergy radiation, as produced by microwave or electric discharges, ultraviolet lamps, laser sources, or gamma rays, or by electron or ion bombardment. This exposure can be performed during or after sample deposi tion, and the resulting changes observed spectroscopically. Combination of the hightemperature and photolytic procedures can produce other unstable species for study. A large array of spectroscopic techniques can be applied to matrix samples, including infrared (IR) and Raman, visible (VIS) and ultraviolet (UV) absorption and emission, electron spin resonance (ESR) and Mossbauer spectroscopy. Matrix isolation spectroscopy has several advantages over gas phase work. First of all, there is the ability to observe normally unstable or highly reactive species at leisure, using conventional or only slightly modified spectrom eters. Another prime advantage, particularly important for hightemperature species, is that the molecules are always trapped in their ground electronic and vibrational states. With most work done below 250K, there are no "hot" bands, as l the thermal energy is only 17 cm Because states other than the ground state are thermally inaccessible, sensitivity is increased over that of hightemperature gasphase work, and analysis is aided since the originating level for spectro scopic transitions is always the ground state. With long deposition times (up to 48 hrs with automatic flow control), sufficient concentrations of molecules can be accumulated in order to observe species of low abundance or spectral features with low absorption coefficients. In addition, controlled diffusion experiments can be conducted in order to follow the formation of new species, polymers, or clusters. Finally, it is possible to observe preferential orienta tion of some species in matrices; the equivalent effect is observed in single crystals, but for most species considered in matrix work, these would be impossible to prepare. Of course, there are some disadvantages to the technique, primarily the frequency shift from gasphase values, caused by perturbing effects of the matrix. Frequencies in neon matrices are generally higher than those in argon, and the data in these solids often bracket the gasphase value. Neon does give the closest agreement, with vibrational frequencies i shifted 10 cm or less. Electronic transitions show a similar trend, the Ne values differing from gasphase by up to 200 cm. For trapped molecules, all transitions up to 200 cm .For trapped molecules, all transitions exhibit shifts of the same order of magnitude, and usually the same direction; trapped atoms show no such regularity, and often include absorptions that have no apparent corre spondence to gasphase transitions. Magnetic parameters are also influenced by matrix effects, usually exhibiting, trends related to the atomic number of the matrix gas. Theoretically, hostguest interactions causing these perturbations are not well characterized, although some effort has been extended in this direction. Samples of such attempts can be found in (312). Another common observation in matrix work is that the shapes and widths of bands vary widely, depending upon the extent of interaction between the absorbing species and the matrix. Usually Ne shows the narrowest lines, up to I 10 cm full width at half maximum (FWHM) in the IR, up to 0 20 A FWHM for electronic absorptions. The broadening effect usually increases with the atomic number of the matrix gas, but lines in Ne are occasionally rather broad, also. Line shapes are also somewhat matrixdependent. However, for a given matrix, the perturbations are useful in identification of progressions of vibrational lines in a particular excited electronic state, since such bands always show the same shape. The departure of the lineshapes from the usual Lorentzian form are generally attributed to occupation of multiple sites of similar energy in the rare gas lattice, and/or the simultaneous excitation of lattice modes (phonons). Site splitting of a few wavenumbers, Angstroms, or gauss are common, but can often be partially or completely eliminated by the irreversible process of annealing the matrix. This is done by allowing the matrix to warm up, and then rapidly quenching it to the original low temperature. A final disadvantage is the loss of information about rotational levels of trapped species. Some molecules, such as H20, HC1, NH2, and NH3 do rotate in matrices, but rota tional sturcture is usually lost in broad vibrational envelopes. This introduction is not designed to discuss in depth the various aspects of the matrix isolation technique, but to illustrate the method and possibilities of its applica tion, as well as enumerate some advantages and disadvantages. More extensive details can be found in several recent reviews (6, 7, 11, 1322) and references contained therein. References Chapter I 1. I. Norman and G. Porter, Nature, 174, 508 (1954). 2. E. Whittle, D.A. Dows, and G.C. Pimentel, J. Chem. Phys., 22, 1943 (1954). 3. M. McCarty, Jr. and G.W. Robinson, Mol. Phys., 2, 415 (1959). 4. M.J. Linevsky, J. Chem. Phys., 34, 587 (1961). 5. G.C. Pimentel and S.W. Charles, Pure Appl. Chem., 7, 111 (1963). 6. B. Meyer, "Low Temperature Spectroscopy," Elsevier, New York, 1971. 7. A.J. Barnes, "Vibrational Spectroscopy of Trapped Species" (H.E. Hallam, ed.), Wiley, New York, 1973, p. 133. 8. R.E. Miller and J.C. Decius, J. Chem. Phys., 59, 4871 (1973). 9. A. Nitzan, S. Mukamel, and J. Jortner, J. Chem. Phys., 60, 3929 (1974). 10. G.R. Smith and W. Weltner, Jr., J. Chem. Phys., 62, 4592 (1975). 11. S. Cradock and A.J. Hinchcliffe, "Matrix Isolation, A Technique for the Study of Reactive Inorganic Species," Cambridge University Press, Cambridge, 1975. 12. B. Dellinger and M. Kasha, Chem. Phys. Lett., 38, 9 (1976). 13. A.M. Bass and H.P. Broida, "Formation and Trapping of Free Radicals," Academic, New York, 1960. 14. W. Weltner, Jr., Science, 155, 155 (1967). 15. J.W. Hastie, R.H. Hauge, and J.L. Margrave, "Spectro scopy in Inorganic Chemistry," Vol. 1 (C.N.R. Rao and J.R. Ferraro, eds.), Academic, New York, 1970, p. 57. 16. W. Weltner, Jr., "Advances in High Temperature Chemistry," Vol. 2 (L. Eyring, ed.), Academic, New York, 1970, p. 85. 17. D. Milligan and M.E. Jacox, "MTP International Review of Science, Physical Chemistry, Series I," Vol. 3 (D.A. Ramsay, ed.), Butterworth, London, 1972, p. 1. 18. A.J. Barnes, Rev. Anal. Chem., 1, 193 (1972). 19. A.J. Downes and S.C. Peake, Mol. Spectrosc., 1, 523 (1973). 20. L. Andrews, Vib. Spectra Struct., 4, 1 (1975). 21. B.M. Chadwick, Mol. Spectrosc., 3, 281 (1975). 22. G.C. Pimentel, New Synth. Methods, 3, 21 (1975). CHAPTER II EXPERIMENTAL Introduction The general experimental procedure including cryogenic, hightemperature, spectroscopic, and photolytic apparatus is discussed in this section. Specific details peculiar to individual molecules investigated will be presented with the discussion of those species. Experimental Apparatus In this research, three separate cryogenic systems were employed, two for optical and IR studies and a third for ESR experiments. An ESR and an optical dewar utilizing liquid He as physical refrigerant were adapted from the design of Jen, Foner, Cochran, and Bowers (1), and modified for variabletemperature operation as described by Weltner and McLeod (2). Both systems are comprised of an outer liquid N2 dewar, which serves as a heat shield, surrounding the inner liquid He dewar. The sample substrate is attached to a copper block suspended from the bottom of the inner dewar by a small tube which permits passage of a controlled leak of refrigerant into the block. The inner dewar is positioned such that the trapping surface is directly in the path of the sample inlets. Pertinent details of the ESR dewar, described by Easley and Weltner (3), and Graham, et al.(4), are indicated in Figures 1 and 2. The stainless steel inner dewar, capacity approximately 2.1 liters, is jacketed at the lower end by a copper shroud which is part of the outer (liquid N2) reser voir surrounding the upper portion. This jacket extends down to encase the microwave cavity and maintain it near 770K. A slot approximately 3.5 cm long and 0.6 cm wide allows the matrix gas and furnace vapor to reach the sample substrate. At 900 to either side of this slot, two rectangular openings of approximately 2.4 cm x 0.6 cm are provided. These points correspond to the location of interchangeable windows, sealed by Viton "O" rings to the outer vacuum chamber, which permit visual examination of the matrix and serve as ports for photolysis or spectroscopic observations. The trapping surface is single crystal sapphire, obtained from Insaco, Inc. It is a flat rod, 3.3 cm long, 3.1 mm wide, and 1.0 mm thick, securely enbedded by Wood's metal solder into the copper block, as indicated by Figure 2. A Chromel vs Au0.02 at. % Fe thermocouple is also attached to the copper block so that the temperature can be monitored. Although the temperature of the sample substrate itself is never determined, single crystal sapphire has high thermal conductivity at 40K so that it rapidly equilibrates to the temperature of the mounting. Also shown in Figure 2 is the construction of the variabletemperature modification. The copper lower can is Zcr acz L J < LL J  a. aa < > 0 0 C/) < 0 0 L) I >Y ra OC e o_ ( c V ....   r C 0 Oa) 0 oro 0 ." \ )) , I 0  a)a *0 \ LI 0 ' J  U) Ua >I SL4 connected to the main liquid He reservoir by a thin stainless steel tube; a vent for the lower can is also provided to exhaust the gaseous He as it evaporates. Outside the vacuum vessel, the He outlet is equipped with a valve to control the flowrate of liquid He into the lower can. The main liquid He reservoir is pressurized to about 2.5 psi to supply an uninterrupted flow through the mounting block. To vary the temperature, the outlet valve is closed, and as the He evaporates, it forces the liquid refrigerant out of the lower chamber, allowing it to warm. The change in tempera ture is monitored with either a Leeds and Northrup model 8687 potentiometer or a Newport model 2600 digital thermometer. After sample deposition is completed, the entire inner dewar assembly is lowered approximately 3.8 cm with respect to the fixed outer can and vacuum chamber, utilizing a vacuumtight bellows arrangement mounted at the top and not indicated in the Figures. In this manner, the rod is positioned in the center of the copper Xband (=9.3GHz) microwave cavity; this location corresponds to the maximum intensity of the circulating magnetic field of the microwave radiation injected into the cavity. The front end of the cavity is slotted and aligned with another interchangeable window in the outer vacuum chamber located just below the sample inlets. In this way, the sample can be photolyzed and ESR spectra recorded simultaneously. The back end of the cavity is fitted with a standard copper waveguide coupling mounted just outside a mica window, which serves to maintain high vacuum conditions in the cavity. With the sapphire rod in position, the dewar is separated from the furnace assembly by means of a gate valve and disconnected from it. The entire dewar assembly is then rolled on fixed tracks to the proper position between the poles of the ESR magnet. When so aligned, the alternating magnetic field of the microwave radiation is oriented perpendicular to the static field of the external magnet. However, the inner (liquid He) dewar can be rotated 3600 on bearings to permit detection of resonance signals with the flat surface of the rod oriented at any angle with respect to the external field. Following the same basic design, the dewar used for optical studies is diagrammed in Figure 3. The stainless steeliriner (liquid He) dewar, of capacity 8.6 1, is surrounded by a'liquid N2 container and copper sheath. Four circular open ings in the sheath, each of 3.5 cm diameter, are located at the level of the sample substrate mounted on the lower cham ber of the inner dewar. At one of these openings, the fur nace and matrix gas inlets are attached on the outer vacuum chamber. This opening can be sealed with a gate valve when deposition of vapor from the furnace is not desired. At 90 to either side of the sample inlet, the openings form part of the optical path of the spectrometric instruments. Interchangeable windows (approximately 4 cm diameter) are mounted with Viton "O" rings to the outer vacuum chamber at these points. The window materials are chosen to match the spectral region of interest; CaF2 is used for the visible VALVES Figure 3. Basic design features of variable temperature liquid helium dewar used for optical studies. 0 and UV regions (20007000 A), and CsI is used for visible 0 and IR studies (3500 A50p). All optical crystals were ob tained from Harshaw Chemical Co. The fourth port is located at 1500 to the sample inlets, and interchangeable windows can also be mounted on the outer chamber at this point. These are usually either quartz or LiF, and serve to admit photolyzing radiation to the sample in the ultraviolet or vacuum ultraviolet regions, respectively. Such sample photolysis cannot be accomplished simultaneously with deposi tion. As in the case of the ESR dewar, the entire inner assembly can be rotated on bearings through 3600, to align the sample window with any of the above. The sample substrate is a polished optical crystal, 2.2 x 1.1 cm, chosen to match those on the outer vacuum casing and the spectral region of interest. It is mounted in the lower chamber of the variabletemperature inner dewar with all four sides in contact with the cold copper block. At all points where the window is in contact with the copper heat sink, a thin gasket of indium metal is inserted. This material has good thermal conductivity and is sufficiently plastic that it conforms to all contours of both surfaces when the window mounting frame is firmly screwed into the copper block from above the substrate. A chromel vs Au0.02 at % Fe thermocouple is mounted to the copper immediately adjacent to the window. Temperatures at this point are measured with either the potentiometer or a Cryogenic Tech nology Inc. digital thermometer /controller. Variable temperature operation is achieved in the same manner as described above for the ESR dewar. The third piece of cryogenic apparatus employed in this research utilizes an Air Products model DE202 Displex cryotip. This is a twostage, closedcycle He refrigerator which makes use of the JouleThompson effect as compressed gas at 300 psi is expanded, with a pressure drop of over 200 psi. The vacuum housing for the cryotip is very similar in design to the liquid He optical dewar, except that the fourth window discussed above is located at 1800 to the sample inlets; thus photolysis can be conducted simultaneously with sample deposition. Internally, there are a few modifications. The cryotip unit (shown in Figure 4) is constructed of stainless steel, with the exception of the final expansion chamber. There is no liquid N2 outer dewar, but its function as a heat shield is taken over by a nickelplated copper shroud attached to the first expansion stage, maintained at 40600K. This extends down to and surrounds the sample substrate holder, with two openings cut at 1800 apart. The entire unit is rotatable through 1800, to align the sample window with any two opposite ports in the external vacuum housing. The second expansion stage is terminated with a copper cold tip. The copper sample window holder is firmly screwed into this tip, and good contact is assured with an indium gasket. The circular sample windows (2.6 cm diameter) are of the same materials as employed in the other optical dewar, ELECTRICAL He GAS <C THERMOCOUPLE and HEATING WIRES EXPANDER Ist STAGE 2nd STAGE COPPER COLD TIP TARGET WINDOW RADIATION SHIELD MATRIX GAS INLET GATE VALVES Figure 4. ROTATABLE JOINT FURNACE ASSEMBLY VACUUM PUMPS Basic design of cryotip assembly used for optical studies. and are secured with indium gaskets and a copper retaining ring. The chromel vs Au 0.02 at.% Fe thermocouple is mounted on the sample holder. The temperature is varied between 100K and ambient by an electrical heating wire wrapped around the second stage expander cold tip. The temperature is measured and automatically maintained at any preset value with the CTI thermometer/controller. The vacuum chambers of all three of the above systems are pumped by 2 inch oil diffusion pumps, with liquid N2 cold traps, backed by mechanical forepumps. Dewars employing liquid He refrigerant attain an ultimate vacuum of approx 8 imately 5 x 10 torr. All pressures are measured with BayertAlpert type ion gauge tubes and Veeco RG31X control circuits. Vaporization of nonvolatile materials was accomplished in vacuum furnaces of identical design attached to each of the cryogenic systems. These are watercooled brass cylinders 20.3 cm long and 15.2 cm in diameter. Furnaces associated with the liquid He dewars are pumped by an oil diffusion pump of minimum two inch diameter intake, backed with a mechanical forepump and equipped with liquid N2 cold traps, 6 attaining pressures below 1 x 10 torr. The furnace mounted on the Displex cryotip utilized the same pumping system as the cryotip assembly. A schematic of the furnace apparatus is illustrated in Figure 1. Interchangeable, demountable flanges of various design are inserted into the furnace body. For resistance heating of samples, these flanges are equipped with water cooled copper electrodes. Tantalum cell holders are securely bolted with Ta screws to the ends of the electrodes, and inserted in the holders are Ta cylinders 2.5 cm long, 6.4 mm O.D., and of varying wall thicknesses, filled with the solid to be vaporized. The open ends of the cell were sealed with tightfitting Ta plugs.. The effusion orifice of 1.6 mm diameter is directed towards the target window. These cells could be aligned at any angle to the horizontal; the vertical position is necessary for samples which are melted to produce sufficient vapor pressure for deposition. Not shown in the figure is a watercooled copper heat shield placed 1 2 cm in front of the cell and fitted with a central 2 cm hole to allow passage of some of the sample vapor to the target. The cell was heated by passing up to 500 amps at up to 6 volts through the electrodes. Cell temperatures were measured with a Leeds and Northrup vanishing filament optical pyrometer through a flangemounted Pyrex window which was shuttered to prevent deposition of a film on the window. Induction heating of samples was performed in the same furnace body, but equipped with a flange holding 10 turns of 6.5 mm hollow copper coil, the axis of the helix aligned towards the target window. A Ta Knudsen cell, 2.2 cm long, 1.4 cm O.D., 3 mm wall, is supported, coaxially with the coil, on three W rods (1.5 mm diameter and 8 cm long) attached to a screw mechanism mounted on the furnace flange in place of the electrodes. In this way the cell position could be adjusted in the coils to provide maximum coupling. Tempera ture measurements were achieved as above, except that the cell was provided with a blackbody hole, which eliminates the necessity of emissivity corrections. The watercooled RF coils were attached to a Lepel 5 kW high frequency induc tion heater. With both methods of heating, the distance from cell orifice to the trapping surface was approximately 12 cm. Matrix gases or gas mixtures were usually admitted to the dewars through the copper inlet shown in the figures. This was connected to a copper manifold equipped with fittings to connect to Pyrex sample bulbs. The manifolds were pumped by a 2 inch oil diffusion pump, equipped with liquid N2 trap, and backed by a mechanical forepump, which 5 gave pressures less than 1 x 10 torr. Flow rates were adjusted with an Edwards needle valve, and pressure changes monitored with a Heise Bourdon tube manometer. Gas mixtures were produced in a similar vacuum system by standard manometric techniques. The rare gases, neon and argon, were Airco ultrapure grade, and used without further purification except for passage through a liquid N2 cold trap immediately prior to deposition. Electron spin resonance measurements were made with an Xband Varian V4500 spectrometer system employing super heterodyne detection. A12 inch electromagnet useful from 0 13 kG provided the static magnetic field, which was modulated at low (200 Hz) frequency. The output of the instrument was recorded on a Moseley model 2D2 XY recorder. When signals were weak, a Nicolet model 1072 signal average, equipped with SW71A sweep and SD72A analogtodigital converter plugin units was used to improve the signalto noise ratio. The magnetic field was measured with either an Alpha Scientific model AL67 or a Walker Magnemetrics model G502 NMR gaussmeter in conjunction with a Beckman 6121 counter. The microwave cavity frequency was determined with a HewlettPackard highQ wavemeter. 0 Absorption spectra were recorded from 7000 2000 A using a Jarrell Ash 0.5 meter Ebert mount scanning mono chromator. Gratings ruled with 1200 lines/mm and blazed at 0 5000 and 3000 A gave a reciprocal linear dispersion of 0 16 A/mm in first order. Detectors used were the RCA 7200, 0 for the range 3700 2000 A, and either the RCA 1P21 or 0 931A, for the range 7000 3500 A, each operated at 1000 VDC. The photomultiplier output was processed by a Jarrell Ash 82110 electronic recording system and displayed on a Bristol model 570 strip chart recorder. Continuum light sources were a General Electric tungsten ribbonfilament lamp for the visible and a Sylvania DE 350 deuterium lamp for the UV. Radiation from these sources was passed through the matrix and focused onto the spectrometer slit with quartz optics. The spectra were calibrated with emission lines from a Pen Ray low pressure Hg arc lamp. A PerkinElmer 621 spectro photometer (purged with dry N2 gas) was used in the IR region from 4000 300 cm with an accuracy of 0.5 cm1 Photolyzing radiation in two spectral regions was avail able from either a high pressure Hg arc lamp or a flowing H2He electrodeless discharge lamp. The Hg lamp consists of a watercooled General Electric type AH6 Hg capillary lamp operated at 1000 W, the output of which was focused onto the sample with quartz optics. The radiation from this lamp consists of the characteristic Hg lines and a strong base continuum. When this lamp was in use, the dewars were equipped with quartz optical windows for transmitting the radiation to the sample. The flowing H2He electrodeless discharge lamp is constructed after the design of David and Braun (5). It consists of a quartz tube, 15 cm long, fused 4 cm from the end with a larger diameter quartz tube to form an annular space 6 cm in length. The annulus has provision for inlet of the 10% H2 in He gas mixture (Air Products), and the central tube is connected to a mechanical pump, which can 2 evacuate the entire system to about 3 x 102 torr. This effectively seals the quartz body against an LiF optical window by means of a brass fitting equipped with "0" rings. The LiF window is mounted in the dewar photolysis ports. The gas flow is adjusted to give a pressure of about 1 torr with a gas regulator. An 85W Raytheon PGM 10 microwave genera tor, operating at 2450 MHz, was used with a tunable cavity to excite a discharge in the flowing gas. The emission was 0 characterized by the intense Lyman a line at 1216 A. Color centers which developed in the LiF due to the high energy radiation could be removed by annealing the windows at AIr Or1 4:M Ci.i^ I,^,,, General Technique Preparation of the matrix samples was achieved in the following manner. The dewars, furnace assemblies, and gas manifolds were readied at least one day before an experiment was run, and allowed to pump out overnight. If a good vacuum was maintained, the liquid N2 cold traps associated with the diffusion pumps were filled; this brought the furnace and gas manifold assemblies near their ultimate vacuums. The sample cells were then slowly heated and the samples allowed to outgas at low temperatures (about 2000C below deposition temperatures) while the dewars were prepared. While preparation of the Displex cryotip involved only checking the He and cooling water pressures, and switching on the device, preparing the liquid He dewars was somewhat more involved. First the outer, and then the inner dewar was filled with liquid N2. The inner dewar was constantly purged with dry N2 gas when not in use to prevent formation of ice in the narrow channels of the variabletemperature chamber. Filling this container with liquid N2 served to precool it, and minimize the quantity of liquid He wasted for this purpose. After the lower chamber, on which the sample substrate was mounted, had reached liquid N2 tempera tures, that refrigerant was pumped out by pressurizing the chamber with N2 and He gas. When the liquid N2 had been re moved and recovered, the dewar was flushed with He gas and allowed to warm up 10200K. This assured complete removal of the N2, which could solidify as liquid He was added. It was found to be very important that a positive pressure of dry gas was applied to both dewar openings, especially when it was cold. When the dewar had warmed slightly, liquid He was transferred by a vacuuminsulated tube into the dewar, which was sealed with a pressure cap when transfer was completed. These preliminary activities took approximately one hour to perform. A charge of liquid He lasted approxi mately 4 8 hours, depending on the rate of flow through the lower chamber, which was set with a needle valve. The Displex cryotip also took about one hour to reach operating temperature, but it could maintain that temperature indefi nitely. With the deposition surface at a sufficiently low temperature, the gas manifold was sealed from its pumps and filled with the matrix gas or gas mixture. To prevent formation of a solid residue from vaporization on the surface, the gaseous sample alone was deposited on each side of the substrate for approximately five minutes. The rate of gas deposition through the entire run was controlled with an Edwards needle valve to obtain a flow of about 0.3 mmole/ min. During this time the nonvolatile sample was heated to its deposition temperature; formation of the metal film on heat shield and furnace viewing port indicated that sufficient material was being vaporized. With the initial deposit on the sample substrate, the gate valve separating furnace and dewar was opened and furnace vapors were co condensed with the matrix gas. During sample deposition, typical pressures observed in the furnace and dewar were 4 5 1 x 10 and 2 x 10 torr, respectively. Deposition times varied from 1/2 to 2 hrs, depending on the species being formed; the sample substrate was rotated 1800 periodically, to form an even coating on the surface. When deposition was completed, the dewar and furnace were isolated, the gas flow stopped and hightemperature cell allowed to cool. The matrix samples thus prepared were then observed spectroscopically. 29 References Chapter II 1. C.K. Jen, S.N. Foner, E.L. Cochran, and V.A. Bowers, Phys. Rev., 112, 1169 (1958). 2. W. Weltner, Jr. and D. McLeod, Jr., J. Chem. Phys., 45, 3096 (1966). 3. W.C. Easley and W. Weltner, Jr., J. Chem. Phys., 52, 197 (1970). 4. W.R.M. Graham, K.I. Dismuke, and W. Weltner, Jr., Astrophys. J., 204, 301 (1976). 5. D. David and W. Braun, Appl. Opt., 7, 2071 (1968). CHAPTER III ESR THEORY Introduction The interactions of paramagnetic atoms and molecules with magnetic fields, which gives rise to the electron spin resonance phenomenon, is discussed in this chapter. Details of the theory applicable to atoms (or ions) and doublet, triplet, quartet, and sextet state molecules encountered in this research will be presented in separate sections. More extensive treatments of the basic theory presented here can be found in a number of excellent references (113), Atoms and the Resonance Condition The paramagnetic substances with which we are concerned are those which possess permanent magnetic moments of atomic or nuclear magnitude. In the absence of an external field such dipoles are randomly oriented, but application of a field results in a redistribution over the various orienta tions in such a way that the substance acquires a net magnet ic moment. Such permanent magnetic dipoles occur only when the atom or nucleus possesses a resultant angular momentum, and the two are related by p = yG [i] where p is the magnetic dipole moment vector, G is the angular momentum (an integral or halfintegral multiple of h/2n = "i where h is Planck's constant), and y is the magnetogyric ratio. The motion of these vectors in a magnetic field H' consists of uniform precession about H at the Larmor precession frequency w =yH. [2] The component of G or p along H remains fixed in magnitude, so that the energy of the dipole in the field (the Zeeman energy) W = VIH [3] is a constant of the motion. The magnetogyric ratio which relates the magnetic moment to the angular momentum according to Eq. [1] is given by Y = g(e/2mc) [4] where e and m are the electronic charge and mass, respectively, and c is the speed of light. The factor g = gL is unity for orbital angular momentum and g = gS = 2.0023 for spin angu lar momentum. Including this factor with Eq. [1] and defining the Bohr magneton as 8 = ei/2mc, we have (along the field direction) UL= gLmL [5a] [5b] S = gsmS Because the angle of the vector p with respect to the applied field H is space quantized, only 2G + 1 orientations are allowed. These allowed projections along the magnetic field are given by mG where mG is the magnetic quantum number taking the values mG = G, Gl, ..., G. [6] This accounts for the appearance in Eqs. [5] of the factors mL for orbital angular momentum and mS for spin angular momentum. 2 If only spin angular momentum arises, as in a S1/2 atom, the 2S + 1 energy levels separate in a magnetic field, each of energy E = mSH (7) EmS geBmSH, [7] with equal spacing ge H. However, the angular momentum does not generally enter as pure spin, so that the g factor is an experimental quantity and mS an "effective" spin quantum number, since some orbital angular momentum is usually mixed into the wavefunction. In orbitally degenerate states described by the strong (RussellSaunders) coupling scheme, J = L + S, L + S 1, ...,IL Sland E = gi m H where S+ S(S + 1) + J(J + 1) L(L + 1) J 2J J + 1) is the Lande splitting factor. This reduces to the free electron value for L = 0. Taking the simplest case of a free spin, mj = ms = 1/2, and there are only two levels. Transitions between these levels can be induced by application of magnetic dipole radiation obtained from a second magnetic field, at right angles to the fixed field, having the correct frequency to cause the spin to flip. Thus the resonance condition is hv = geSHo [10] where H is the static external field and v is the fre 0 quency of the oscillating magnetic field associated with the microwave radiation; this frequency is about 9.3 GHz 2 2 for the Xband. Thus for a S or P atom, the ESR spectrum will consist of one line corresponding to the particular g value of the atom. The Hyperfine Splitting Effect If only one line were observed in the general case, the ESR technique could offer only a limited amount of informa tion, the g value. However, there are other interactions to consider which increase the number of spectral lines and the information that can be obtained. One of the most impor tantisthe nuclear hyperfine interaction. Usually at least one isotope of an element contains a nucleus having a nonzero magnetic moment. The magnetic moment of the odd electron can interact with this nuclear moment and split the single ESR line into hyperfine struc ture. This effect can be pictured as follows. The magnetic field "felt" by the electron is the sum of the applied external field and any local fields. One such local field will be that caused by the moment of the magnetic nucleus; this is, in turn, governed by the nuclear spin state. It is then clear that, in the case of nuclear spin I = 1/2, for example, the local field in which the electron finds itself will be one of two contributed by the nucleus, since there are 21 + 1 nuclear levels. Hence, there will be two values of the external field which satisfy the resonance condition, that is, r (H' ) = (H' AM) [11] where A/2 is the value of the local magnetic field, A being the hyperfine coupling constant, and H' is the resonant field for A = 0. One example of this phenomenon is 1 2 the H atom. This is a pure spin system, S1/2, with I = 1/2; its ESR spectrum consists of two lines separated by A = 508G, centered around g = ge = 2.0023, as shown in Figure 5. A more detailed look at the paramagnetic species with nonzero nuclear spin in a mangetic field indicates that there are several interactions at work. One is the inter action of the external field with the electrons, which has already been considered. An analogous term results from the precession of the nuclear magnetic moment in the exter nal field. The nuclear magnetic moment pI is related to the nuclear g factor gI by the relation g = [12] N where 8N = eli/2M is the nuclear magneton and M is the proton mass. The third term describes the interaction between the electrons and nuclei. Thus the Hamiltonian can be written H = gHJ + hAI.J g B HI [13] where the underscore indicates that the quantities are operators. Except in very strong fields, the interaction of the nuclear moment with the external magnetic field (the nuclear Zeeman term), which is represented by the last term in Eq. [13], is small, and will be neglected. Also omitted from this Hamiltonian are even smaller effects, such as the nuclear electric quadrupole interaction. Reference to Figures 5 and 6 will indicate the behavior of the levels as a function of external field strength. The two limiting cases of very weak and very strong field are of particular interest. The Zeeman effect in weak fields is characterized by an external field splitting which is small compared to the natural hyperfine splitting; that is, hAI.J > gBHJ in Eq. [13]. In this case, the orbital electrons and the nuclear magnet N I 0) o ro O N I II 4 0)0 0 , 0 0 z Z tl _n) I i N2 " z a Wj ,. 4J S'\\/4 0 u w z U LJ \0 Z l< E 0) (1) Cl *il (H b I) 0 7 L C\OJ C0J c\j J C\J r+ I C\J + .0 L) Q C) 0 0 C\ O 0 0 O O I remain strongly coupled. A total angular momentum F = I + J exists, which orients itself in the external field. F takes the values I + J, I + J 1, ..., II JI. The component of F along the field direction, mF, has 2F + 1 allowed values, the integers between F and +F. In the case presented in Figure 6, with both pN and A positive, these components are arranged, in order of decreasing energy, mF = F, F 1, ..., F. Each individual hyperfine level splits up into 2F + 1 equidistant levels in the weak field, giving (2J + 1)* (21 + 1) Zeeman levels altogether. Note that in both Figures 5 and 6, the levels are not all degenerate even at zero field. This effect, produced by the hAIJ term, is called the zero field splitting. In the PaschenBack or strong field region, the splitting by the external field is large compared to the natural hyper fine splitting. The strong interaction with the external field decouples I and J, which now process independently around H. F is no longer a good quantum number, but there exist m and mi, the components of J and I along the field direction. In this case, each Zeeman level of the multiple characterized by a fixed m is split into as many Zeeman hyperfine lines as there are possible values of mi, that is, (21 + 1). Since there are still (2J + 1) levels for a given J, there are, exactly as in the weak field, (2J + 1)* (21 + 1) total energy states. In contrast to the weak field situation, the levels here form a completely symmetric pattern around the energy center of gravity of the hyperfine multiple. This pattern manifests itself in Figure 6. Also recorded in that Figure are the m values of the different hyperfine groups. Values of m are, in order of decreasing energy, 5/2, 3/2, 1/2, 1/2, 3/2, 5/2 for mj = 0, 1, 2, 3; this order is inverted for the remaining mj groups. The situation in intermediate fields is somewhat more complicated. The transition between the two limiting cases takes place in such a way that the magnetic quantum number m is preserved. In weak field, m = mF; in strong field, m = mI + m In this region, the Zeeman splitting is of the order of the zero field hyperfine splitting. For a 2S1/2 state, as in Figure 5, the general solution for the energy levels over all fields is given by the BreitRabi equation (14). In terms of the quantum numbers F and m = mF, it is W(F, AW I Hm AW 4m X2 21/2 14 W(Fm) = 2(2 + 1) I + ( + 21 + 1[14a hA where AW = (21 = 1) [14b] andX ( J/J + II /I) H0 and X = [14c] The plus sign in Eq. [14a] applies for F = I + 1/2 and m = +(I + 1/2), ..., (I 1/2) and the minus sign for F = I 1/2 when m = (I 1/2), ..., (I 1/2). The zero field hyperfine splitting is AW. The limiting cases of weak 2 2 and strong fields correspond to X <<1 and X >>1, respectively. For the general case of intermediate fields, the energy values of the Zeeman levels can be derived from the following key equation given by Goudsmit and Bacher (15): Xm + 1 1 (I + m + 1)(J mj + 1)] m + 1, m 1 I I J [A Xm 1 m (I m + 1)(J + m + 1)] = 0, M 1, m + 1 2 I J I J where A is the hyperfine coupling constant, gj and g'I = I are the electronic and reduced nuclear gfactors, respectively, and the XXA are coefficients in the expansion of the wave function; the other symbols have their usual meaning. Here, AWH is the energy of the level with respect to the center of gravity of the hyperfine multiple. This relation yields one system of homogeneous equations in XX for each value of m = mI + m The resulting secular equations are solved for the energies of the Zeeman levels at any field. Such a calculation was performed to produce Figure 6. With a multitude of levels available, it is necessary to explain the observed ESR spectra in terms of the selection rules. Since transition between Zeeman levels involve changes in magnetic moments, we must consider magnetic dipole transi tions and the selection rules pertaining to them. In the pure spin system with I = 0, the single line observed corresponds to the m = 1/2  m = 1/2 transition. In s s general, the criterion is that Amj = 1, corresponding to a change in spin angular momentum of T. Since a photon has an intrinsic angular momentum equal to I, only one spin (nuclear or electronic) can flip on absorption of the photon, in order to conserve angular momentum. With the fields and frequencies ordinarily used in ESR work, the transition usually observed is limited to the selection rules Amj = 1, Am = 0; the opposite is true in NMR work. It is, however, possible to observe the Amj = 0, Am = 1 NMR transition with ESR apparatus, if the zero field splitting (propor tional to the hyperfine coupling constant A) is large enough relative to the microwave frequency. If this does not occur, only the ESR lines will be observed, resulting in a multiple of 21 + 1 hyperfine lines for each fine structure (Amj = 1) transition. Thus the H atom spectrum (Figure 5) will consist of two lines, while that of Mn ( S3, I = 5/2) will contain 36 individual lines, if all are resolved. These interactions of the electron with a nucleus are related to fundamental atomic parameters, which can be deduced from the observed spectrum. They are most simply categorized as isotropic and anisotropic interactions. The anisotropic interaction has its roots in the classi cal dipolar coupling between two magnetic moments. This interaction is given by 42 e'N *3( e'r) ( N' r) E 35[16] 3 5 r r where r is the raduis vector from the moment e to N and e N r is the distance between them. The quantum mechanical version is obtained by substitution of the operators, gBS and gNN I, for the moments e and N', respectively, yielding I(LS) 3( r) (Sr) Hdip = rg N 3 5 [17] r r For a hydrogenic atom with nonzero orbital angular momen tum (that is, p, d, ... electrons), this yields L(L + 1) 1 aJ e N J(J + I) ; [18' a more exact relativistic treatment also adds a multiplica tive factor [F(F + 1) I(I + 1) J( J + 1)]. For s elec trons, a similar dipolar term yields 2 a = gg (3cos 8 1) [19 a = geBgiSN 3 91 r where 0 is the angle between the magnetic field direction and a line joining the two dipoles. However, the electron is not localized and the angular term must be averaged over the electron probability distribution function. For an s orbital, all angles are equally probable due to the spherical symmetry, and the average of cos2 over all 0 causes the function to vanish. Thus the classical dipolar term cannot be responsible for the hyperfine structure of the 2S1/2 hydrogen atom. The actual interaction in the selectron case is described by the Fermi contact term (16). This isotropic interaction represents the energy of the nuclear moment in the magnetic field produced at the nucleus by electric cur rents associated with the spinning electron. Since only s orbitals have finite electron density at the nucleus, this interaction only occurs with s electrons. This yields the isotropic hyperfine coupling constant a = 7T 9g Io l2 [20) as 3 e~ (0) [20] where the last term represents the electron density at the nucleus. This term has no classical analog. The a value s is proportional to the magnetic field at the nucleus, which can be on the order of 105 G. Thus unpaired s electrons can give very large hyperfine splitting. 2 E Molecules The Spin Hamiltonian The discussion to this point has involved only those terms in the Hamiltonian of the free atom or ion which are 44 directly affected by the magnetic field. However, it will be useful to begin the discussion of paramagnetic molecules by consideration of the full Hamiltonian, which can be written, in general, H = + H LS+ S + SH + IH [21] E LS SI SH IH Each of the terms can be described as follows. The term H expresses the total kinetic energy of the electrons, :E the coulombic attraction between the electrons and nuclei, and the repulsions between the electrons: 2 2 P ZAe E e H =  + [22] E i 2m A,i r. i>j r [22 1 i] where pi is the momentum of the ith electron, r. is the i 1 distance from electron i to nucleus A of atomic number ZA, and r.i is the distance between electrons i and j. The BornOppenheimer approximation (17) has already been invoked to separate out the nuclear motions and nuclearnuclear repulsions. This term yields the unperturbed electronic levels before spin and orbital angular momentum are con sidered. Eigenvalues of this term are on the order of 105 cm1 The energy due to the spinorbit coupling interaction is usually expressed in the form H = XLS [23] LS where L and S are the orbital and spin angular momentum operators and A is the molecular spinorbit coupling constant. The magnitude of this interaction is of the order 2 3 1 of 10 10 cm . The hyperfine interaction arising from the electronic angular momentum and magnetic moment interacting with any nuclear magnetic moment present in the molecule may be expressed as SI I= gI Nge + 3 3 5 r r S I 8T6 (r)S* + [24] r 3 This term corresponds to the sum of the isotropic and anisotropic hyperfine interactions discussed for atoms above. The Dirac 6function indicates the isotropic part which has a nonzero value only at the nucleus. The 2 1 hyperfine interaction has a magnitude of about 10 cm The electronic Zeeman term, SH' is primarily responsible for paramagnetism. It accounts for the interaction of the spin and orbital angular moment of the electrons with the external magnetic field, according to H = B(L + g S)*H [25] These energies are on the order of 1 cm These energies are on the order of cm1 These energies are on the order of 1 cm The interaction of the nuclear moments with the external field, that is, the nuclear Zeeman effect, is given by H =? Y Y H), [26] 3 1 and is usually too small (10 cm ) to be significant. It is evident that this total Hamiltonian (which still excludes higherorder terms and field effects which could be observed in crystals) is very difficult to use in calculations. However, experimental spin resonance data obtained from the study of the lowestlying levels can be described by use of a spin Hamiltonian in a fairly simple way which does not require detailed knowledge of all the interactions. These levels are generally separated i by a few cm (by the magnetic field), and all other electronic states lie considerably higher. The behavior of this group of levels in the spin system can be described by such a spin Hamiltonian, and the splitting, which may usually be calculated by first and second order perturba tion theory, are precisely the same as if one ignored the orbital angular momentum and replaced its effect by an anisotropic coupling between the spin and the external magnetic field. This was first shown by Abragam and Pryce (18). Just as the g factor becomes anisotropic and not necessarily equal to g = 2.0023, the S cannot represent a true spin but is actually an "effective" spin. This is defined, by convention, to be a value such that the observed number of levels equals (2S + 1), just as in a real spin multiple. Thus we can relate all the magnetic properties of a system to this effective spin by the spin Hamiltonian, since it combines all of the terms in the general Hamiltonian which are sensitive to spin. Nuclear spins can be treated in the same manner, and the spin Hamiltonian corresponding to Eq. [21] can be written (neglecting the nuclear Zeeman term) H pin= 8HS + IAS [27] spin 0 where the double underscore indicates a tensor quantity. The g Tensor As alluded to previously, the anisotropy of the gtensor arises from the orbital angular momentum of the electron through spinorbit coupling. Even in the case of E states, which have zero orbital angular momentum, the interaction of a presumably pure spin ground state with certain excited states can admix a small amount of orbital angular momentum into the ground state, and change the values of the components of q. This interaction is usually inversely proportional to the energy separation between the states. The spinorbit interaction can be described as (5) H = AL*S = A[L S + L S + LS ] LS  xx yy zz [28] This term is added to the Zeeman term in the Hamiltonian, thus H = H. (L + g S) + AL*S .[29] For an orbitally nondegenerate ground state represented by IG, MS>, the first order energy is given by the diagonal matrix element (1) = <,MSIg HzSz G,Ms> + ISIHz + ASz Ms where the first term is the spinonly electronic Zeeman effect. Because the ground state is orbitally nondegenerate, <ILzI > = 0, and the second term vanishes. The second order correction to each element in the Hamiltonian matrix is given by H Z !M M 1 [31] s S n (0) (0) n G where the prime designates summationover all states except the ground state. Since g98H.S will vanish. The operator matrix can then be expanded to HM M [ S1 S (0) (0) n W W [32] n G and the quantity Axz Ayz Azz _' (0) (0) n G n G Axx Axy Axz factored out where the ijth element of the tensor is given by = n (0) (0) w  n G = A [33] second rank [34] where i and j are any of the cartesian coordinates. This simplification yields HMs = M (MS2i H.A.H + 2AH*A.S + A 2 SAMs' > . sg [35] The first operator represents a constant contribution to the paramagnetism and need not be considered further. The other terms represent operators which act only on spin variables. When combined with the Zeeman term of Eq. [30], the result is the spin Hamiltonian 2 H = B. (g 1 + 2AA)*S + X2S*A*S spin e = BHg'S + S*D.S , [36] where S= g 1 + 2AA [37] and D = X2A [38] with 1 being the unit tensor. The SD*S term is operative only in systems with S 1 1 and will be considered later. The other term in Eq. [36] is the spin Hamiltonian in the absence of hyperfine inter action. It is evident that the anistropy of the g tensor arises from the spinorbit interaction due to the orbital angular momentum of the electron. This may be expanded to show the gtensor as BSgqH = B[S S S I g g g g H [39] xyz xxxyxz x [39 gyx yy yz y zxgzygzz z where S S and S are the components of the effective spin along the axes. Strictly, g is a 3x3 matrix and is referred to as a symmetrical tensor of the second order (the symmetry implies that the unpaired electrons are in a field of central symmetry). The double subscripts on the gtensor elements may be interpreted as follows: gxy is the contri bution to g along the xaxis when the magnetic field is applied along the yaxis. These axes are not necessarily the principle directions of the gtensor, but a suitable rotation of the axes will diagonalize it; then the diagonal components are the principle directions of the gtensor with respect to the molecule. It is noteworthy that, if the molecule has axes of symmetry, they must coincide with the principle axes of g; if it has symmetry planes, they are perpendicular to the principle g axes. Three cases of interest, with regard to molecular symmetry can be outlined. If the system is truely a spin only system, g will be isotropic and the diagonal elements equal to g If it is isotropic but contaminated with orbital momentum, the principle components will be equal but unequal to g e In the former case spin xyz ex H0 g H [40 = ge[Hx S + H S + H S ]. S xx yy zz For a system containing an nfold axis of symmetry (n > 3), two axes are equivalent. The unique axis is usually designated z and the value of g for H II z is called g ". For H I z, the value is gi. Thus Hspin = (gHS + g HYS + g HS ). spin Ixx I yy zz [41] Finally, for systems where there are no equivalent axes (orthorhombic symmetry), gxx yy g 9zz and H spin (g H S + g H S + g H S ). [42] spin xx xx yy yy zz zz The A Tensor It has already been noted that the hyperfine inter action is composed of isotropic and anisotropic parts. While the isotropic coupling is that which is observed in liquids, the anisotropy due to dipoledipole interac tions can be observed in fixed systems, such as molecules in a rigid matrix. If Eq. [24] is expanded (and the L*I term is dropped since this is a Z state), the interaction can be seen to assume the form of a tensor 2 2 H r / rAr2 3x \ /3xy\ 3xz \ dip xz \Sf Kr5 [x dip = ( NSxS Sz r x r 5 r 3 r2 3y2 /3yz [43] r r r [43] 2/3xy\ 3y r23z2 3" \ (r z z r r r = hS*T*I When the isotropic part is added, the spin Hamiltonian with hyperfine becomes H spin= BSgH + hS.A*I spin [44] where A = A 1 + T [45] with A0 being the isotropic hyperfine coupling constant and 1 the unit tensor. Thus the element A.. = A. + T.. [461 13 iso 13 and, as with the gtensor, certain cases can be selected due to the molecular symmetry. An isotropic system has A = xx A = A = A. Systems with axial symmetry are yy zz iso characterized by A = A = A = A. + T and Az xx yy iso xx zz All = Ais + Tz iso zz Randomly Oriented Molecules Having discussed some basic theory of ESR in mole cUles, one must consider how these effects manifest them selves in the spectrum. Anisotropy will appear in the spectrum of a rigidlyheld molecule, but there is a difference between the effects observed in crystals and in matrices. In a single crystal, with a paramagnetic ion or defect site, for example, the sample can be aligned to the external field and spectra recorded at various angles of the molecular axis to the field. In the matrix, the samples ordinarily have a random orientation with respect to the field, and the observed absorption will have contri butions from molecules at all angles. This was first considered by Bleaney (19, 20), and later by others (2126). Solving the spin Hamiltonian,Eq. [42],in the ortho rombic case and assuming the g tensor to be diagonal, the energy of the levels will be given by 2 2 2 2 .2 2 2 21/2 E = HH(gl sin Ocos2 + g2 sin Osinc + g cos2)/2 = BgHSHH [47] where SH is the component of the spin vector S along H, gH is the g value in the direction of H, 6 is the angle between the molecular z axis and the field direction, and { is the angle from the x axis to the projection of the field vector in the xy plane. For axial symmetry g 2 2 2 2 1/2 (g1 sin + g11 cos ) and the energy of the levels is given by 2 2 2 2 E = BSHH(gl sin 6 + g 1 cos 6). [48] Thus the splitting between energy levels, and therefore transitions between them, are angularly dependent. Consider first the case of axial symmetry. As a measure of orientation, it is convenient to use the solid angle subtended by a bounded area A on the surface of a sphere of radius r. The solid angle is the ratio of the surface area A to the total area of the sphere, that is, 2 S= A/4wr If all orientations of the molecular axis are equally probable, the number of axes in a unit solid angle is equal for all regions of the sphere. If the sphere is in a magnetic field, the orientation of the axes will be measured by their angle 6 relative to the field. Taking a circular element of area for which the field axis is the z direction, the area of the element is 27(r sin6)rd6, and the solid angle dQ it subtends is 2 2nr sind6 dQ 2  = 1/2 sin6d6, [49] 47r2 and if there are N0 molecules, the fraction in an angular increment d8 is N0 dN =  sinEd6. [50] Assuming the transition probability is independent of orientation, which is approximately the case, the absorption intensity as a function of angle is proportional to the number of molecules lying between 6 and 8 + dO. Since g is a function of 6 for a fixed frequency v, the resonant magnetic field is hv 2 2 2. 21/2 H = (g2 cos 6 + g sin 2 [51] and from this s2 (g0H0/H) g sin 6 = 0 2 [52] g 2 gi 11 where go = (gl + 2gi)/3 and HO = hv/g0B. Therefore 2 2 g0 H0O sinede = H 3 H 2 ( 2 0 [o 2 2H 1/2 S) H. [53] The intensity of absorption in a range of magnetic field dH is proportional to dN IdN Ie HdH ILdH [54] where the two factors on the right are obtained from Eqs. [50] and [53], respectively. From the above equations H = hv/gll = g0Ho/gl at 6 = 00 H = hv/gB = g0H0/g9 at 0 = 900. [55a] [55b] At these two extremes, the absorption intensity varies from IdN N 091.1 2g H0(g11 2g12 dN = m dH at 0 = 0 [56] at 6 = 90. [57] Tf plotted against magnetic field, this absorption (after considering natural linewidth) takes the appearance of Figure 7a, for gll>gj. Since the first derivative is usually observed in ESR work, the spectrum would appear as in Figure 7b. If the g tensor is not too anisotropic, gll and gi can be readily determined as indicated. In general, the perpendicular component can easily be distinguished in such a "powder pattern;" the weaker parallel peak is often more difficult to detect. If there is also a hyperfine interaction in the randomly oriented molecules, the pattern of Figure 7b will be split into (21 + 1) patterns, if one nucleus of spin I is involved. Such a spectrum for I = 1/2 is shown in Figure 7c, where, because gl1 % g, and Ai < Al the lines for m =+1/2 and 1/2 point in opposite directions. If instead, gl were shifted upfield, relative to gll and A1 % All the spec trum would appear as two patterns similar to Figure 7b, and separated by the hyperfine splitting. The spin Hamiltonian for an axially symmetric molecule is similar to that for an atom, but now incorporating parallel and perpendicular components of g and A: H sp = B[g H S + g (HS + H S )] spin II zz xx yy [58] + Ai S I + A (SxI + S I ) II zz xx yy 0 a 0 .0 *H H rQ 0 0 wc) >1> > Or  S*H  (0 M O   0 4J t i rnv M .SH  0 0 4) 4 U U I Q >U 0a) C4 tP 0 U C rOO > W r1 0) >i O O (* 4' >i U> () r4 (d rl (t (L) H 0 4 4 Q) 0 O0) SH H t r rX C4  07 This omits the small nuclear Zeeman term and assumes the symmetry axis is the z axis. Equation [51] can be rewritten to include the nuclear hyperfine effect (8) to first order as hv K H 8 g mI [59] where 2 2 2 2 2 g = g cos 6 + g sin 2 [60] and 2 2 2 2 2 2 2 2 Kg = All gl cos + Ai g1 sin [61] The intensity of absorption IdN/dHI can again be derived and is found to be 22 2 dN 0 2cos6 (g0 _g )goH 'dH' 2 2 9 2g [62] + m g11 Al 2 g2 A12 K(g g ) 1 +I  n9*I U^L + ml g 2K g Here sinOd6 cannot be solved explicitly so that dN/dH cannot be written as a function of only magnetic parameters. Equations [59] and [62] must be solved for a series of values 8 to obtain resonant fields and intensities as a function of orientation. However, H = (g0H0/gl ) (mIAll /Bgll ) at 0 = 00 [63a] H = (g0H0/g ) = (miA /Bg) at 6 = 900 [63b] and again IdN/dHI  Thus a superposition of the typical powder pattern results, with the relative phase of the lines determined by the magnitudes of the magnetic parameters, as discussed above. In Figure 7c, the lines do not overlap and analysis is simple, but this is not always the case. In general, the best approach is to solve the given equations by computer for a trial set of g and A values, and match the calculated spectrum to the observed. A similar treatment can be applied to molecules of orthorombic symmetry and instead of the two turning points at g11 and qg, there will be three corresponding to gl' 92, and g3. Such a spectrum is considered, including hyperfine interaction with a spin 1/2 nucleus, by Atkins and Symons (11) and Wertz and Bolton (5). As mentioned in Chapter I, molecules trapped in rare gas matrices do not always assume random orientations. During condensation of a beam of reactive molecules in solid neon or argon matrices, some preferential orientations of the molecules relative to the flat sapphire rod may occur, and in some cases the alignment can be extreme. This non random orientation can easily be detected by turning the matrix in the magnetic field; a change in the ESR spectrum indicates some degree of preferential orientation. The degree of orientation appears to depend upon the size and shape of the molecule, the properties of the matrix, and other factors which are not completely understood (27). Two examples are the molecules Cu(N03)2 (28) and BO (29). The latter case shows very strong orientation such that with the magnetic field perpendicular to the rod surface, the parallel lines were strong and the perpendicular, weak. With the rod parallel to the field, the perpendicu lar lines became very strong and the parallel components disappeared entirely. This indicated that the BO molecules were trapped with their molecular axes normal to the plane of the condensing surface. This orientational behavior is analagous to that usually observed in singlecrystal work. Molecular Parameters and the Observed Spectrum Having discussed the nature of the interactions appearing in the spin Hamiltonian and the form of the observed spectrum, it is time to consider the relationships between the spectral features and the paramagnetic species themselves. This will begin with the exact solution to the spin Hamiltonian in axial symmetry, and presentation of the secondorder solutions which are usually adequate, and conclude with the molecular information revealed through g and A components. A thorough discussion of the spin Hamiltonian Hspin g 11H S + gi (H S + HIS ) + A I S + A (I S + IS ) [64] jII zz xx yy has been given by several authors (8, 30, 31). Considering the Zeeman term first, a transformation of axes is per formed to generate a new coordinate system x', y', and z', with z' parallel to H. If the direction of H is taken as the polar axis and 6 is the angle between z and H, then y can be arbitrarily chosen to be perpendicular to H and hence y = y'. Thus only x and z need to be transformed. With H = Hsin9, H = HcosO, and H = 0 x z Y H = 8[gl cos8Sz + glsinS x]H. [65] Choosing the direction cosines z = g cos9/g and x = gisin6/g, with g defined by Eq. [60], then H = gg[ S + S ]H [66] and the Zeeman term becomes and the Zeeman term becomes H = gS''H z [67] where S = S + S z zz xx S = . S + 2 S X XZ ZX S = S y y For the hyperfine terms hf = A S I + A (SII + S I ), hf  zz xx yy and I is rotated by I = nI nI Z zz xx I = nI + .nI X XZ ZX I = I ' y y where the n. are the direction cosines for the nuclear 1 coordinate system relative to the electronic coordinates. By inverting Eq. [68] to obtain the S. analogs of Eq. [70], 1 and substituting into the Hamiltonian of Eq. [69], the Hamiltonian of Eq.[64] is transformed into [68a] [68b] [68c] [69] [70a] [70b] [70c] S= ghS + KI 'S + KI I 'S ' z z z K x x 2 2 + A A sinecoseI S + A I'S [71] K z x iy Y with the definitions = A1 g1 cose/Kg, = A g sin9/Kg, 2 2 2 2 2 2 2 2 and K g = All g1 cos + A g1 sin 8. Dropping the primes and using the ladder operators S = S + iS and S = S iS x y x y, this can be rewritten in the final form 2 2 +  A A2 g S+S S Hspin signSz + KSz +kO  S = gHSz + KS Iz [ + I 2 cosesin 2 z + I AI + ) (s+I+ + SI ) 4K 4 + ( A+ A (S+I + SI) [72] 4K 4 This Hamiltonian matrix operates on the spin kets IMs,MI), and can be solved for the energies at any angle. An example of the use of this exact solution will be presented later. The exact solution is difficult to solve at all angles except e = 0, but elimination of some of the offdiagonal elements (those not immediately adjacent to the diagonal) results in some simplification and is usually adequate. The solution is then correct to second order, and can be used when gH>>A11 and A as is often the case. The general secondorder solution is given by Rollmann and Chan (32) and by Bleaney (20) as 2 2 2 A All + K 2 AE(M,m) = g H + Km + [ 1 [I(I + 1) m ] 8G K A 2 + 4( A)(2M l)m [73] where K is A and A1 at 6 = 0 and 900, respectively, and G = gH/2. Also, M is the electron spin quantum number of the lower level in the transition, and m is the nuclear spin quantum number. Note that the first two terms on the right result from the diagonal matrix elements and yield equi distant hyperfine lines; this is the first order solution given in Eq. [59]. The last two terms cause increasing spacing of the hyperfine lines at higher field which is referred to as a "secondorder effect." This solution can routinely be applied because the hyperfine energy is usually not comparable to the Zeeman energy. The g tensor has appeared in the derivation of the spin Hamiltonian, and it is seen, from the above equations, how the values of the principle axis components can be determined from the ESR spectrum. Now we shall consider its relationship to a molecular wavefunction in the linear combination of atomic orbitals (LCAOMO) approximation. The usual form applied, Eq. [37], is a result of the secondorder perturbation treatment, yielding the first order corrections to the g components <0 ILn) g g 6. 2E [74] ij e 13 E n n where the primed summation is over all excited states n which can couple to the ground state 0 and E is the energy of that state above the ground state.. The Kronecker delta is 6... 13 Because the correction is caused by the spinorbit inter action, only certain states can couple with the ground state. Specifically, these are the states such that (33) Fn e L 00 + A lg [75] 1 that is, the direct product of the irreducible representations of the ground and excited state with the representation of the angular momentum operator (which transforms as rotations) must include the totally symmetric representation of the symmetry group. A specific example, which will be encountered later, is that a Z state can only mix with a H state. Usually there will only be one such state of energy low enough to make the term significant. The spinorbit coupling constant C can be assumed positive or negative, depending on whether the excited state involves excitation of an electron or a "hole," respectively. Both this constant and the orbital angular momentum operator L can be written as sums of atomic values: 67 ZL. = k [76a] k1 k k i = k [76b] where k indicates a particular atom in the molecule. Actually, ,k decreases rapidly for large rk (ar3) so that k is essentially zero except near atom k, where it may be assumed to have a fixed atomic value Xk. Thus Eq. [74], for the perpendicular component of a E state diatomic molecule, becomes g9 = Z nikk Ix ( lkI) > [77] n k,k' Here, E and H are the LCAO wavefunctions @ = Eaix(i) [78a] 1 j = Zb X(j) [78b] j where x(i) and X(j) are A.O.'s in the ground and excited states, respectively. Then the second matrix element in Eq. [77] will reduce, for a diatomic, to sums of terms involving integrals of the type (X(i)kl kxIx(J)k) (atom k only) [79a] x x(i)kIk'. Ix(j)k ) (both atoms) [79b] (X(i)k l,' X()k> (both atoms) [79c] k' (X(i)k' X(J)k,) (both atoms) [79d] The first matrix element in Eq.[77] can be simplified if Ak is assumed constant near k and zero elsewhere. Then it becomes similar to the integrals of Eq. [79a]. The integrals in Eqs. [79c, d] require that the origin of the operator kx be moved from atom k to atom k'. This introduces a linear momentum term according to x = k +i RP [80] k ky where R is the interatomic distance. Fortunately, the elements involving P are usually zero or small, so that the term can be neglected and the integrals in Eqs. [79] are then all of similar form. Eq.[79a, b] involve the application of the angular momentum operator to the atomic functions involved. The nonzero elements have been tabulated (34) and are given below: (PxLy lP> ) (d IL Id 2 2) (dxz Lxdyz) = (dz2 LxIdyz) = i [81c (dxyl dyz) = (dyz Lz dzx) = i [81d] (d xL xd y) = (dx2 2 L Id = (dx 2 21L d z)= i [81e] (ilL qi') = (i' Lq i> .[81f] Overlap integrals which appear because the A.O's are centered on different atoms have also been tabulated (35, 36). Detailed discussions of this approach can be found in Stone (37) and Atkins and Jamieson (38). Secondorder corrections to the gfactor have been determined by Tippins (39), utilizing thirdorder perturba tion theory. This degree of the theory must be used to calculate corrections to gI and the result, analogous to Eq. [74], is ( <(nl L 10 2 gil = g 1/2  EE [82] n 0  Thus it can be seen that gll is always very close to or less than g since the correction factor is small and squared. On the other hand, gl can be greater or less than g and the difference can be quite significant. Thus, if wavefunctions are available, or can be constructed, the values of Ag = g ge can be calculated and compared to those determined experimentally. Examples of this approach can be found in references (3, 5, 11, 40), and in the discussion of the MnO molecule to follow. Alternately, one can approximate the energy separation of the lowest interacting level from the ground state. The hyperfine coupling constant has been shown to consist of isotropic and anisotropic parts, and its tensor nature has been discussed. The spin Hamiltonian, with hyperfine interaction, can be written as H s = HII.g.S + aLI + bI*S + cI S [83] spin zz where a = gegNn 3~ [84a] Sc 2 s 1 84b SN g N \ 2r c = 3gg KN N 3 [84c] 2r where the angular brackets are, as usual, quantum mechanical averages. This definition has been given by Frosch and Foley (41). Neglecting the small L.I term, we can compare Eq.[83] with Eq.[58] and identify the observed splitting as A = b + c [85a] Ai =b [85b] The isotropic part can be written in terms of the parallel and perpendicular components as A + 2A A. II c 8rr 2 Aiso 3 =b + 3 8 NBN [86] The anisotropic or dipolar component is given by A A 2 Adp = c 3cos 01 [87 dip 3 3 gePN N\ 2r3 Thus the observed spectrum is related to the fundamental quantities I(0) 2 and (3cos281/2r3) for interaction with that nucleus. If the L*I term is included, the values become All = b + c Agl a [88a] AI = b + Ag a [88b] where the Agi = gi ge have already been discussed. These small corrections can be approximated from the observed Agi value and the value (1/r3) for the particular nucleus. The dipolar coupling constant can be considered further. The dipolar part of the Hamiltonian (Eq.[171) can be written as g N (3 cos 1) SI [89a] 3 r and the energies of the levels IMs,MI) are given by 3cos 81 E = geN MI r3 1). [89b] For an electron in an orbital centered on the nucleus in question, the anisotropic hyperfine coupling follows Eq. [89], but has an additional term to represent the average direction of the electron spin vector in the orbital. The hyperfine splitting is the separation between adjacent levels IMSMI) and IMs,MI1) and equals A 3cos20 (3cos2l) [901 dip =e 9NcoN 3 3s) [90] where a is the angle between r and the principal axis of the orbital, and 0 is the angle between the latter and the direction of the nuclear magnetic moment vector. The value of (3cos2a1) can be evaluated for the atomic orbital functions; the values are 4/5 for any p orbital and 4/7, 2/7, and 4/7 for the dz2, dxz,yz, and dx2y2 xy orbitals, respectively. These quantities are used to give the principle value of Adip for an orbital, that is A Atomic values of the quantities A. and Ap have been evaluated for many iso dip atoms, and a useful table can be found in Ayscough (1) or Goodman and Raynor (42). Equation [90] assumes that the nuclear moment vector PI is aligned with the external field, that is, the applied field is much stronger than the field at the nucleus due to the electron. This strongfield approximation is actually valid only when (a) the applied field is large, (b) the anisotropic coupling is small (and hence the field at the nucleus small), and (c) the isotropic coupling is large (since the field caused by the electron reinforces the applied field). It is found that the field at the nucleus due to the electron is much larger than the field at the electron due to the nucleus. If the strong field approxi mation is not valid, the dipolar interaction will vary as 2 1/2 (3cos6+1)/2. However, the numerical values of the aniso tropic coupling at the turning points will be the same, and only the signs will differ. In a crystal, the difference will be discernable, but for the powder patterns obtained in matrices, the strongfield analysis is sufficient. A more thorough discussion can be found in references (5, 42). The calculated atomic A. and A dip values are often iso dip used with experimentally determined molecular values to derive coefficients or spin densities on a particular atom in a molecule. Using an LCAOMO wavefunction described as = ZaiXi, the values of As. and Adip at a particular nucleus x can be written dip e= g NN (iI (3cos2 1/2rx)1> [91a] x 87r 9 Nlg2 [91b] iso 3g e N Nl (0)x2 [91b Since these integrals are expected to be small except near atom x Ax = ai(P d )2A (atom) [92a] dip i x x dip A = a.(s )2 A (atom) [92b] 1so I x iso where the ai are the coefficients. Although this implies that atomic properties remain unchanged in the molecule, which is unlikely, it is quite useful in comparing trends to model wavefunctions. Examples of its utility will be given later. A more quantitative approach to calculating A tensor components, based on the intermediate neglect of differential overlap (INDO) molecular orbital approximation, has been developed (4346). 3 Molecules A theorem due to Kramers states that, for all systems with an odd number of electrons, at least a twofold degeneracy will exist which can only be removed by applica tion of a magnetic field. This would apply to cases with S = 1/2, 3/2, 5/2, ...; the first has been considered in detail. In the triplet case, S = 1, and two noninteracting electrons can be described by four configurations: a(l)a(2), c(1)8(2), 8(1)a(2), and 8(1)B(2). In a molecule of finite size, interactions will occur and the configurations can be combined into states which are symmetric or anti symmetric to electron interchange. These states are a(1)a(2) [93a] (1/T ) [a(l)B(2) + 8(l)a(2)] (1/1) [a(l)8(2) B(1)a(2)] [93b] 8(1) P(2) [93c] The multiplicity of the symmetric states (on the left) is (2S + 1) = 3; this is a triplet state. Because of the Pauli principle, this state may exist only if the two electrons occupy different spatial orbitals. For systems of two or more unpaired electrons, the degeneracy of these spin states may be lifted even in the absence of a magnetic field; this is termed the zerofield splitting (ZFS). If the number of unpaired electrons is even (S 1 i, 2, ...), the degeneracy may be completely lifted in zero field. Additional terms in the Zeeman Hamiltonian H = BH.g.S are required to account for this. The Spin Hamiltonian It was shown in the derivation of the spin Hamiltonian (Eq.[36]) that the anisotropic part of the spinorbit coupling produces a term S*D*S which is operative only in systems with S > 1. However, at small distances, two unpaired electrons will experience a strong dipoledipole interaction, such as has been considered in the anisotropic hyperfine interaction: Sg2 S *S 3(S *r)(S *r) HSS 3 2 5 [* 94] r r In this equation, F is the vector connecting the electrons, and the Si are the spin operators of the individual electrons i. If the scalar products are expanded, and the Hamiltonian expressed in terms of a total spin operator S = S + S2, 1 2 2 2 2 2 and considering that r =x + y + z the matrix form of 77 the Hamiltonian can be written 2 2 2  S=/2) [ s s s r 3x 3xy 3xy H =(1/2)g 8 S [ S ] 55 S SS xyz 5 5 5 x r r r 2 2 3xy r 3y 3yz S 5 5 5 y r r r 2 2 3xz 3yz r 3z2 5 5 5 z r r r . =S DS [95] This term, representing the dipolar interaction of the electron spins, should be compared to the last term of Eq. [36]; the latter evolved from Eq. [31] by treating the spinorbit coupling interaction as a perturbation on the Zeeman energy and assuming that the space and spin parts of the electronic wavefunctions were separable. It can be seen that the two terms are identical in form, except for a numerical constant. These spinorbit and spinspin contributions to D cannot be distinguished experimentally. Whatever the origin of the interaction, the D tensor can be diagonalized and the fine structure term becomes 2 2 2 H=D S + D S D S [96] xxx yyy z zz where D + D + D = 0, that is, D is a traceless tensor. xx yy zz This can be written in terms of the total spin as 2 2 2 SD.S = D[S 2 (1/3)S(S + 1)] + E(S S 2 z _x y + C/3)(D + D + D ) S(S + 1), [97] xx yy zz where D = D (D + D )/2 and E = (D D )/2. zz xx yy xx yy The last term is a constant, proportional to the trace of D which is zero for pure spinspin interaction, and does not appear in the spin Hamiltonian. The terms involving D and E account for the removal of the degeneracy of the three triplet wavefunctions in the absence of an external magnetic field. While the effects of spinspin and spinorbit inter actions are not separable, some qualitative statements can be made about them. It is found, in general, that organic triplets show little influence from spinorbit coupling. The g values of such molecules are usually very close to ge, and the zero field splitting can be ascribed almost completely to spinspin dipolar interactions. Then one would expect the value of D to be approximately inversely proportional to the molecular volume; this trend can be seen in the methylene derivatives C6 H C C6H5 and 1 1 NC C CN, where D is 0.4 cm (47) and 1.0 cm (48), respectively. Introduction of a heavy atom increases the spinorbit coupling (since XaZ4 ), thus D for CN2 is 1.16 cm i (49) but for SiN2, discussed below, it is 2.28 cm1. If the molecule involves transition metals, which usually have large spinorbit coupling constants, the zero field splitting is due mainly to the spinorbit interaction. The solution to the general Hamiltonian for triplet states of randomly oriented molecules is given by Wasserman, Snyder, and Yager (50). Here, the special case of linear molecules will be briefly discussed. Neglecting hyperfine and other spinorbit interactions, the spin Hamiltonian for a linear triplet will be spin = g9) Hz + g(HS + HS ) + D(Sz2 2/3) [98] where z is the molecular axis. If y is chosen arbitrarily to be perpendicular to the magnetic field, then H = 0 and 2 Hspin = g BH zS + g0(HS ) + D(S 2 2/3). [99] Choosing as a basis the orthonormal spin wavefunctions S+ 1) =I'aal) [100a] 1 0 (') =(l//l2B2 +1"2) [100b] I 1) = 1 ,2) [100c] and considering the effect of the spin operators on the func tions as S a =(1/2) S x =(1/2),a S a =(1/2)iB S 8 =(1/2)ia x x y y 2 2 S a =(1/2)a S z =(1/2) S a =(1/4)a S 2B =(1/4) z z Z Z [101a] [101b] then the Hamiltonian matrix will be I + 1) D/3 + G Gx /2 x S0 ) G Gx/2 2/3 D Gx/ 2 I 1) [102] D/3 Gz z where G z = g 8Hz and Gx = gjBHx. The eigenvalues for Hi z (Hz = H, H = 0) are W+1 = D/3 + gl BH W0 =(2/3)D W 1 = D/3 g 11 H. 1+ 1) I 0o I 1) [103a] [103b] [103c] At zero field, the + 1) and 1) states are degenerate, and the appropriate wavefunctions are T =(1/2 )(I + 1> I 1)) [104a] T =(1/V2)(I + 1) + 1>) [104b] Tz = 0) [104c] In these wavefunctions the spins are quantized along the X y, and z axes, respectively: S IT ) = S T y = S IT ) = 0. S' x x yy z z The eigenvalues for Hilz are plotted against H in Figure 8, and all yield straight lines. For H z, H = H and H = 0, and the roots of the secular Sx z determinant are W1 = D/3 [105a] W2 = (1/2) D/3 + (D2 + 4g 2H2 ) /2] [105b] W3 = (1/2)D/3 (D2 +4g 2H2) 1/2 [105c] with the eigenvectors x = 1/i) + 1) ) [106a] I HI O o 0  ,I r4 0II I~ LLJ J 4O "0 4O rU 0 0  I4 0 C~~j ,I OaO 4 (  roD Q )4 4) G)  4" AJI I II  o 4 Cr  'N I X~ N I co f 1 py = cosa jt + 1) + ) + sinal0) [106b] 1 F 1 z = sinea I + 1) + I ) +cosaI0), [106c] where tan 2a = 2gl3H/D. As H approaches zero, where a = 0, these functions reduce to the functions T, Ty, and T x z given in Eq.[104]. The eigenvalues for H z are plotted versus H in Figure 9, and only at high fields, where ar/4, do the lines become straight. In the intermediate region, y and iz are mixed and lead to a curvature of the energy with H. It is evident that the energy levels, and hence the fields at which transitions between them occur, are very dependent upon the orientation of the axis of the molecule with respect to the fixed magnetic field. This will, in general, cause the spectra of randomly oriented molecules to be broad and difficult to observe. The relative transition probabilities are given by 2 2 2 2 hij k = g cos i yl( IS j)I [107] where i, j, and k are any of the molecular axes (z is axial) and y is the angle between the oscillating magnetic field of the microwave radiation (perpendicular to the fixed field) and k. Then transitions are allowed between levels characterized by the following wavefunctions: 84 I Ll 0 0 t Vb N 0 1 I 3 WU K N 3^ 0 I: NQU 4 S r( N o N + f + + I I I >~KN lI (13 X 4, 0 4 0 o' '0 E r41 *H H 0 4J; r4 U H (0 34 04 m )rl * r X 4 0 >1 H 4 *oh4 a) ^ cl nj i0 r (1) T T z (2) Ty T z y z 2 (3) y < X xy2 (4) z  X xYI' where the symbols on the right are the usual designations given to the observed ESR lines. These transitions are indicated in Figures 8 and 9, and all correspond to AM = 1 transitions. Also indicated in Figure 9 is a dashed line representing the forbidden AM = 2 transition. This transi tion (0y c 'z) is allowed for Hosc H, but is has a finite transition probability (9) when H is not parallel to any of the x, y, or z axes, even if Hos IH, as in the apparatus employed here. The AM = 2 notation is acutally a misnomer, since the spin functions I + 1) and 1), corresponding to the infinite field Ms values, mix significantly at finite fields. Thus M is not a good quantum number and the transition could actually be described as AM = 0, since the eigenfunctions each contain contributions from spin functions of the same M. Thus if D is not too large, the transition will be observable. i. Employing the exact solution to the spin Hamiltonian matrix for triplet molecules, which may be bent (E ; 0), the resonant fields of the transitions are (50) 86 H = T [(hVD) E ] 11 zl g 8 1 2 2 1/2 H [(hv+D) E21/ z2 g [I 1 H  xl gjy 1 Hy2 gi y2 = 9 H =  AM=2 [(hvD+E) (hv+2E) 1/2 [(hv+DE) (hv2E)] 1/2 [(hvDE) (hv2E) I1/2 [(hv+D+E)(hv+2E)] 1/2 2 2 2 1 (hv) D +3E2 1/2 gB 4 3 In Figure 10, the resonant fields of these transitions for linear molecules (E = 0) are plotted as a function of D, for a fixed microwave frequency of 9.1 GHz, where the energy equals 0.3 cm1. The z and xy lines are so marked. As can be seen from the Eqs.[108], the effect of a nonzero E term in the spin Hamiltonian is to split each xy line into [108a] [108b] [108c] [180d] [108e] [108f] [108g] 0.9 0.8 0.7 0.6 0.2 0. Figure 10. I 2 3 4 5 6 7 8 Hr (Kilogauss) 3 Resonant fields of a 3 molecule as a function of the zero field splitting. 